ADVMCED
CLASSICS
David Pines, Series mimr Anderson, RW., Bmk Nod Bethe H. and Jackiw, R., fn te Qwncum Mechanics, Third Ediidm kynman, R., Pbmn*Hdrm Inmw~om Feynman, R., Q w n t ~ mEkctrody~mks Feyaman, R., Sttt~isdcalM e c b ~ c s Feynman, R, The They of^ F z ~ & r n e d P~ocesses Negele, J. W. and Orland, H., Quantum ManyMParticle Systems NoziPres, P,, Theoq of Interating Femi Systems Parisi, G., SmdsLical Fie@new Pines, D,, The ManyOBodyI-"roblem a i g g , C., Gauge TkMies of Ehtl Strong, and Elecltrompetie herwtions Setnwinger, J ., P a ~ i c k s Sources, , anti Fie& , Volume X Schwinger, J.,Particles, So~rces,arzd fib,Voium I1 Schwinger, ., Partktes, Sources, a d Fie&, Volume IXI
m&,
OHN W, NEGELE Massachusetts Institute of Technology
HENRIORLAND Service de Physique Theorique, CEA Saclay, France
i
-
/' A
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Editor's Foreword
Perseus Books's FTontiers in Physics series has, since 1961, made it ~ossiblefor leading physicis~to communicate in coherent fashion their views of recent developments in the most exciting and active fields of physics--without having to devote the time and energy required to prepare a formal review or monograph. Indeed, throughout its nearly forty-year existence, the series has emphasized informaliq in both style and content, as well as pedagogical clarity, Over time, it was expected that these informal accounts A d be replaced by more formal counterparts-textboks or monogmphs-as the cutting-edge topics they treated gradually became integrated into the body of physics knowledge and reader interest dwindled. However, this has mt proven to be the case for a number of the volumes in the series; M a y works have remained in print c m an on-demmd basis, while others have such intrinsic value that the physics community has urged us to extend their life span. T h e Advamelrl Book Classics series has been desimed to meet this demand. X t will keep in print those volumes in F~ontiersin Physics or its sisrer series, Lecture nts in Physics, that continue to pmvide a u n ~ w aceount of a mpic of lasting interea. And through a s i d l e priming, Ehese cl;zssic=swill be made available at a compatatively modest cost to the reder, The manybody problem, the study of quancum many-pa~ticlesystems, is an essential part d the education of graduate students in both chemistq and pkysics. In the presm infomal text/monograph, John Negele and Henri Orland provide advanced undergduates and beginning graduate students with a sdfcontained introduction to the key physical ideas and maefiematical techniques cufrmtly employed in Get& as diverse as nuclear physics, at-omic physics, condensed matter pt-tysics, and theoretical chemistv. Written in lucid fashion, and containing humewark problems at the end of each chapter, Qwntum Many-
ParEick Sysenzf can seme as both a text for a one-emester inrrductory course or as a reference book for the novice and experienced researcher alike. I am very pleased that the Adwarned Bmk Ckrssics series wiH now make it readily accessiOIe to new geneliatiom of readers.
David Pines Aspen, Cdorado A u ~ s 1998 t
PREFACE
The problem of understandkgthe properties of quantum systems possessing large or Infinite numbers of degrws of free-dom pemdes all of theoretical physics. Hence, the thear&icail methods and the physical insight which have bwn devdoped over the years for quantum many-particle systems campdse an esential part of the educatjon of students in diwiptlnes as diverse as solid state physics, field thwry, atornlc physics, mndensed matter physics, quantum chemistry and nuclear phydcs. During the past decade, we have taught one and two-semester courses on the quantum tkmry of many-particle systems to graduate students in them disciptfnes at the Massachusats institute of Technology, and this book Is;an autgrowth of these lectum. Compard to the texts that appeared in the early 1970%.we have presented standard topiia from a dimrent pasrspective and Included a number of new developments. Because of the physical appeal and utllity of the kynnnan path integral, we have used functional integrals as the foundation of our presttntatiion. Eunctbnal integral teichniques proliide an economical formalism for d&ving familiar results, such as perturbation wpansions, and yfeld valuable new apvaximations and insight into such proWems as quantum callective motion, tunneilng daay, and phase transitions, Because of the power and physicat lnsights provMdrsd by these twhniques and their prevalence in the literature, we believe it is essential to teach them to students at this level. Order parameters and brobn symmetry play crucial roles in charactedzing and understandjng the phases in whlch matter exists and the transitions betwen these phases* These concepts, which are familiar frorn the Landau theory of phase transitions, arise quite naturalry frorn our generat development In krms of functional htegvals, and are discuss& in detail in this text. Another new taptc is the use of stochastic methods for many-body probterns. Techniques have misted for a long ltime to use Markov random walks and Monte Carlo M e s 6f phydcal "rnterat evaluation of integrals to calculate quantum mt3chanieai ob to any desired degree of accuracy" In the past, such mhnlques have rewived less attention than analytic methods involving summations of diagrams having undeternind convergence propeeies or other ultimately uncontroltd approdmations. We believe that stochastic methods are Intellectually interesl-ing in their awn right and that they provide a powerFul tool to obtain definitive answers to certain classes of otherwliw insolvable problems;. Hence, we have included a pedagogical introduction to stschastic methods, showing hew to calculate obsewables of interest, stressing the physical connection with path integrals, and demonstrathng how to tailor the method to the physics of the problem under consideration. The scope of this book is Intend& to be sufficleMly broad to serve as a tat far ;a one- or two-semester graduate course. Thus, in addltion to these new topics, we have also inctuded the basic body of methodology found In otder texts, such as perturbation theory, Green's function techniques, and the Landau theory of k r m i liquids.
X:
PREFACE
Our pedagogical objective Is to convey the essentfat Ideas and to prepare the student to read and understand the relevant research lterature. VVt? have attempt& to p r e ~ n the t formalism tersely, without undue emphasis on technical details and t o show how it applies to a broad variety of interesting physlcat systems. Homework problems are provided at the end of each chapter, and are crucial to a thorough mastevy of the subject. Instructive alternative treatments of formal developments in the text are often presented as problems, as well as detaited caIcutations which are too lengthy for the text. One model system, particles in one dimension interacting via a &-function two-body potential, Is us& extensively to illustrate methods presented in the text, For this system, both exact sotutlons and a multitude of common appraxlmations can be worktzcf out in detail analytically. Finally, the organization of the book is as follows: We assume only an understancftng of elementary quantum mechanics and statistical mechanics, so we begin in Chapter f, with a thorough, self-contained treatment of second quantfzatian and coherent states. Chapter 2 presents the general formalism of path integrals, perturbation theary and its resummations, and non-perturbative approximations in the formally simple case of the grand canonical ensemble at finite temperature, Specialization to zero temperature and the canonical ensemble is discussed in Chapter 3. Chapter Iaddresses the role of order parameters and braken symmetry in many-body theory and shows how mean field theory embodies the essential physical content of the Landau theory of order parameters and phase transitions, The next chapter develops the general properties of Green" functions, and thdr application in describing fundamental excitation! and physical observables, The phenomenotogical description and rnlcroscopic foundation of the Landau theory of Fermi liquids a-re preanted in Chapter 6, Chapter 7 describes a number of fu&her developments of functional integral techniques, including alternative functional integral representations, the treatment of quantum mean field theory and tunnefing decay, and the study of high orders of perturbation theory. The final chapter presents stochastic methads. As in all such eflFafls, we arc?indebted to many people far thdr Fnvaluahte assistance in writing this book. This book was originally stimulatd by Davld Pines and benefited from the editorial guidance of Rkk Mixter and Allan Wytde. Although it is impossible to fist at1 of the teachers, coHeagues, and students whose insights have contributed to t h h work, we would particulady like to acknowledge the contributions of Re Balian, J, B, Blaizot, E, Brain, C. De Domfnic'is, C. ttzykson, S. E. Koonin, S, tdbler, S. tevit. G. RIpk, and R, ScttaefFer. Portions of eartier drafts of the manuscript w r e typeset in %X by Meredith typesetting, as well Pollard, Karf Kawafski. and Dany Bunet* The majority of the as the final editing, improving the layout and appearance of formulas. and preparation of tables was performed by Roger L. Gilson, to whom the authors are particularly indebtad for his outstanding work. We also wish to acknowledge the excellent technical art work by Don Souza and the work of the Addison-Wesley production department in providing the final copy.
CONTENTS
Chapter
........ 1 Quantum Mechanics of a Single Part'rcle . . . . . . . . . . 1 Systems of Identical Parltdes . . . . . . . . . . . . . . . 4 Many-Body Operators . . . . . . . . . . . . . . . . . . 9 Creation and Annihilation Operators . . . . . . . . . . . . 11 Coherent States . . . . . . . . . . . . . . . . . . . . 20 . . . . . . . . . . . . . . . . 20 Boson Coherent States Grassmann Algebra . . . . . . . . . . . . . . . . . . 25 krmton Coherent States . . . . . . . . . . . . . . . . 29 Gausslan Integrals . . . . . . . . . . . . . . . . . . 33 Problems for Chapter 1 . . . . . . . . . . . . . . . . . 31 General fc;rrmatlmmaf Ffnlfe Temperature . . . . . . . . 41 introduction . . . . . . . . . . . . . . . . . . . . . . 47 Quantum Statistical Mechanics . . . . . . . . . . . . . 157 Physical Response Functions and Green" Fmct'ions . . . . 49 Appradmation Strategies . . . . . . . . . . . . . . . . 53 FunctEonal Integral brmulation . . . . . . . . . . . . . . 57 kynman Path Integral . . . . . . . . . . . . . . . . . 57 !maginay-Time Path Integral and the Partition Function . . 63 Coherent State Functional Integral . . . . . . . . . . . . 66 The Partition Function far Many-Padlrcte Systems . . . . . 68 Perturbation Theory . . . . . . . . . . . . . . . . . . . 74 Wick% T h r e m . . . . . . . . . . . . . . . . . . . 75 Labeted bynman Diagrams . . . . . . . . . . . . . . . 78 Unlabeled Gynman Diagrams . . . . . . . . . . . . . . 82 W ugenhoftz Diagrams . . . . . . . . . . . . . . . . . 88 Frequency and Momentum Representation . . . . . . . . 92 The Linked Cluster Thearem . . . . . . . . . . . . . . 86 Calculation of Obsewables and Gran" fundions . . . . . 3.1 Im&uciblle Diagrams and Integral Equations . . . . . . . . . 105 Genwating Function for Connected Green" Functions . . . . 105 The EfFwtive Potential . . . . . . . . . . . . . . . . . 108 The Self-Energ and E)yson%Equation . . . . . . . . . . 112 Higher-Order Vcr&w Functions . . . . . . . . . . . . . 115 Stationaq-Phase Approximation and Loop Expansion . . . . . 220 One-Dimensional Inhgrsrl . . . . . . . . . . . . . . . 12% kynman Path tntegraf . . . . . . . . . . . . . . . . . 123 Many-Particle Partition Fundion . . . . . . . . . . . . 124 Pmblems for Chapter 2 . . . . . . . . . . . . . . . . . 131
li Secand Quamttratlon and Coherent States 1.1 1.2 1.3 1.4 1.5
X
CONTENTS
. . . . . . . . 138 hynman Diagrams . . . . . . . . . . . . . . . . . . . 138 Obsembles . . . . . . . . . . . . . . . . . . . . . 138 Zero-Temprtrature krmlon Propagatars . . . . . . . . . . 142 Fermion Diagram Rules . . . . . . . . . . . . . . . . 146 Bornas . . . . . . . . . . . . . . . . . . . . . . . 253 mme-Order& Diagrams . . . . . . . . . . . . . . . . . 157 The ZerolTemgerature Llrnit . . . . . . . . . . . . . . . $64 Problems for Chapter 3 . . . . . . . . . . . . . . . . . 167 Order Fsrametarr and Broken Symmetry . . . . . . . . 1.16 introduction . . . . . . . . . . . . . . . . . . . . . . 176 Phases of Two Familiar Systems . . . . . . . . . . . . 176 Phenomtlmolo_gicaltandau Theory . . . . . . . . . . . . l79 Broken Symmetq . . . . . . . . . . . . . . . . . . . 184 General Formulation wit h Order Paramaers . . . . . . . . . 185 Infinite Range tsing Modd . . . . . . . . . . . . . . . 185 Generalizations . . . . . . . . . . . . . . . . . . . . . 190 Physical Exampies . . . . . . . . . . . . . . . . . . 192 Mean Field Theory . . . . . . . . . . . . . . . . . . . 195 Legendre Trandorm . . . . . . . . . . . . . . . . . . 195 Femomagnetlc Transition for Classleaf Spins . . . . . . . . 187 Appticaleion to Genwal Systems . . . . . . . . . . . . . 204 FtuctuatFons . . . . . . . . . . . . . . . . . . . . . . 207 Landau Ginzburg Theory and Dimensional Analysis . . . . . 207 Qnetoop Gorrwtions . . . . . . . . . . . . . . . . . 2 l 1 Continuous Symmetry . . . . . . . . . . . . . . . . . 214 One-Loop carredions far the s - g model . . . . . . . . 2 i 7 Lower Critical D~~~?Rs!oII. . . . . . . . . . . . . . . . 220 . . . . . . . . . . . . 222 The Andersan-Higgs Medranism Problems for Chapter 4 . . . . . . . . . . . . . . . . . 226 Perturbatlcrn Ttreew at Zero Tamperaturn
Chapter
5 5.1
. . . . . . . . . . . . . . . . . . . . . . 235 . . . . . . . . . . . 235
Introduction Definitions . . . . . . . . . . . Evafuat'ion of Observabtes . . . . Analytic Properties . . . . . . . . Zero Temperature Green's Cnctions Finite Temperature Green's Functions Physical Content of the Self Energy . Quasiparticle Pole . . . . . . . . Effltt~tiveMasses . . . . . . . . Optical Potential . . . . . . . . Linear Response . . . . . . . . . The Response Function . . . . . Random Phase Approximation . .
. . . . . . . . . . . 23'1 . . . . . . . . . . . 240
. . . . . . . . . . . 240
. . . . . . . . . . 241 . . . . . . . . . . . 249 . . . . . . . . . . . 251 . . . . . . . . . . . 255 . . . . . . . . . . . 259 . . . . . . . . . . . 262 . . . . . . . . . . . 262 . . . . . . . . . . . 265
CONTENTS
5.5
Chaptu
6
6.1 6.2 6.3
Zero Sound . . . . . . . . . . . . . . Matrk xFornn of RPA ........ Sum Rules and Examples Magneric Susceptibility of a Fermi Gas . . . . Static Susceptibility at Zero Temperature Static Susceptibility at Finite Temperature . Dynamic Susceptibility at Zero Temperature Dynamic Suxeptibility at Finite Temperature Problems for Chapter 5
. . . . . . . 471 . . . . . . . . . . . . . . . . . . 276 . . . . . . . 277 . . . . . . . 281 . . . . . . . . . 281 . . . . . . . 282 . . . . . . . 284 . . . . . . . 284 . . . . . . . . . . . . . . . . . 285 Tha Lsndau Theory of Fermi Liquids . . . . . . . . . . 296 Quariparticles and thdr Interactions . . . . . . . . . . . . 296 Observable Properties of a Normal Fermi Liquid . . . . . . . 299 Equilibrium Properties . . . . . . . . . . . . . . . . . 299 Nonequilibrium Properties and Collective Modes . . . . . . 305 Miuoscopic Foundation . . . . . . . . . . . . . . . . . 313 Calcutation of the Quasiparticfe Interact'ian Problems for Chapter 6 . . . . . . . . .
Chaptar
. . . . . . . 332 Representations of the Evolution Operator . . . . . . . . . 332 The Auxiliary Field . . . . . . . . . . . . . . . . . . 332 Overcomptete Sets of States . . . . . . . . . . . . . . 336 Ground State Properties for Finite Systems . . . . . . . . . 340 The Resolvent Operrtor . . . . . . . . . . . . . . . . 340 Static Hartre Approximation . . . . . . . . . . . . . . 342 RPA Cow~tions . . . . . . . . . . . . . . . . . . . 344 ...... The Loop Expansion .. . . . . . . . . . 348 Transition Amplitudes . . . . . . . . . . . . . . . . . . 350 S-Matdx Elements . . . . . . . . . . . . . . . . . . 352 . . . . . . . . . . . . 353 Cotlectjvte!Excitations a ~ T~nniltnng d Example of One Degrecl3 of Freedom . . . . . . . . . . . 354 Eigenstates of Large Amplitude Collective Motion . . . . . 360 Barrier Penetratkon and Spontaneous Fission . . . . . . . 364 Conceptual Questions . . . . . . . . . . . . . . . . . 370 Large Orders of Perturbation Theory . . . . . . . . . . . . 371 Study of a Simple Integral . . . . . . . . . . . . . . . 372
7 Further Development of Fvnctlonal lntegrslr T.1 7.2
7.3
7.4
7.5
Bore! Summation . . . . The Anharmonic Oscillator Problems for Chapter 7 . .
Chapter
. . . . . . . . 313 . . . . . . . . 324
B 8.1 8.2
. . . . . . . . . . . . . . . 373 . . . . . . . . . . . . . . . 376
. . . . . . . . . . . . . . . 382 Stochartic Methods . . . . . . . . . . . . . . . . . . 400 Monte Carlo Evaluathon of integrals . . . . . . . . . . . . 400 Central Limit Theorem . . . . . . . . . . . . . . . . . 80% Importance Sampling . . . . . . . . . . . . . . . . . 483 Sampling Techniques . . . . . . . . . . . . . . . . . . 405 SamNing Simple Functions . . . . . . . . . . . . . . . 4W
AV
CONTENTS
Markay Processes . . . . . . . . . . ... Neumann-Ulam Matrix lnvctrston Microcanonical Methods 8.3 Evaluation of One-Pavlicfe Path Integral . Observables . . . . . . . . . . . . Sampling the Action fnitiai Value Random WaIk . . . . . . Ttlnnding ........ 8.4 Many Particle Systems Path fntegral in Caordf n a b Representation Eunctlanat fntegrals Ovw Ffdcis . . . . 8.5 Spin Systems and Lattice Fermiens Cht;ckerboard Dwsmpodtbon . . . . . Special Methods for Spins . . . . . . Problems for Chapter 8 . . . . . . . .
. . . . . . . . . 408 . . . . . . . . . 412 . . . . . . . . . . . . . . . . . 413 . . . . . . . . . 416 . . . . . . . . . 416 . . . . . . . . . . . . . . . . . . 417 . . . . . . . . . 429 . . . . . . . . . . . . . . . . . . . . . . 423 . . . . . . . . . 426 . . . . . . . . 426 . . . . . . . . . 431 . . . . . . . . . . . . 434 . . . . . . . . . 435 . . . . . . . . . 438 . . . . . . . . . 440 References . . . . . . . . . . . . . . . . . . . . . . . 447 lndex . . . . . . . . . . . . . . . . . . . . . . . . . 455
CHAPTER 1 SECOND QUANTIZATION AND COHERENT STATES The quantum mechanics of a single particle is usually formulated in terms of the position operator 2 and the momentum operator fi, All other operators of physical interest may be expresd in terms of these operators, and a natural representation for quantum mechanics, the coordinate representation, is defined f n terms of eigenfunctions of the position operstor, In this chapter, an analogous formalism is developed for systems composrtsl of many idenlcical particles. For these systems it is useful to define operators which create or annihilate a partkclie in specified states. Operators of physical Interest may be expressed in terms of these creatlon and annihilation operators, in which case they are said to be expressed in "second quantizedfYorm, The eigenstates of the annihilation operators are coherent states. A natural representation for the quantum meichanics of many-particle systems, the holomorphic representation. is defined Sn terms of these coherent states, As a prelude to the formaflsm for many-particle systems, it is useful to begin by reviewing some elementary aspects of quantum mechanics,
1-11QUANTUM MECHANICS OF A SINGLE PARTICLE
I#),
Quantum mechanks descdbes the state af a particle by a state vector which belongs to a H i f b e ~space U. This WlfbePt space U is the vector space of complex, square: integrable functions, defined In configuration space, Using Dirac notation, the scalar product of vectors in U is:
Then by definition, a vector finite:
14) belongs to the Xilbert space K if the norm of 14) is dsr \#(?)l2 4 +m
Of particular importance are the vectors 1 3 and position operator and momentum operator $
-
(1.2)
)g), dgenvectors of the quantum
8
$13= ?IF)
(1.31
Atthough these vectors do not belong to )C, because their- norm i s not finite, they span the whole Hilbert space U. This is reflected by the following closure relations:
2
SECONR QUANTIZATION AND COHERENT STATES
where 1 denotes the unit operator in U, A state vetor f'i") rapresents a state In which the passlxticlcr:i s locatli?& at point P, and a stab vector l#%) represents a particle with a momentum ji. The overtap of these vectors 1s given by: (F\F') = 6(81(~- F] (1.7)
and
The wave functSon of a pa&icle in a state 14) is giwn in coardinate rewsentotlon by:
and represents the probability amplitude for findkng the particle at point i. In coordinate representation, the operators ;C" and fk act as follows:
and
Thus, in coordinate repremntation we may write:
and
)id $=--mm idP fi
'
b r a patticle of mass m in a local pdentiaf V[?),the Hamiltonian is
2.1 QUANTUM MECHANICS OF A SINGLE PARTICLE
S
In this ease, the timer-independent Schrwinger equation
has the familiar Corm in coordinate representation
F;or particles having some internal degrees of freedom, such as spin 8 or $=spin i', the state veGtor l#)has several components, eaek corresponding to a d'rfferent set of values c;,and P, of the internal dwrea of freedom, Analag-ous to the dPlfinition af the states li") in Eq. (1,3), we may define ai state [Far),which is a state of the particle localized at point 3, with a projectfon of the spin n and isosph r7.These states span the Wltbefl space of the partitle:
and their overlap is given by:
Whenever na amb.lguit;'res will arise, we; will use the notation:
and
and the canvention:
b(z - S') = 6,,t 6,,r
- .
6f5)(1 P )
With this notation, equations (1.18) and (1.19) simplify to:
and
4
SECOND QUANTIZATIOM A N 0 COHERENT STATES
1.2 SYSTEMS OF IDEHTCCAL PARTCCLES The Hilbert space of states for a system of N identical partides is the space of complex, square integrable functions. defind in the configuration spa= of the N padcites. The wave function +N(i?l,h, ...,%), which represe-nts the probability amplitude for finding particles at the N positions Pl, F'%, FM, must satisfy the condition:
...,
As we have defined it, the Wllbert, spaw the single-particle Hilberl space X :
h is dmply the Nth tensor product of
tf (la))is an orthonormat basis of U, the canonicat orthanormal bads af UpC is corrstructect from the tensor poducts:
Note for future reference that the states defined in this way utilize a c u w d bracket in the ket symbol. These basis states have the wave ftlnctions:
The overtap of two vectars of the basis is given by:
and the completeness of the basis is obtained from the tensor product of the completeness relation far the basis { (a)):
where I now represents tke unit operator in h.Physjcally, it is clear that the space XN is generated by linear combinations of products of singfe-particle wave functions. Thus far. in defining the Hilbert space RN. we have not taken into account the symmetry property of the wave function. In contrast to the multitude of pure and mixed symmetry states one could define mathematically, only totally symmetric and antisymmetric states are otrsemed in nature, Particles occurring Sn symmetric or antisymmetric states are called Bosons and Fermions respectivefy, The wave functlon of 1V Bosons is totally symmetric and thus satisfies
1.2 SYSTEMS O F IDENTICAL, PARTICLES
..
5
.
where (PI.,P2,. ,PN) represents any permutation. P, of the set (1,2,. .,H). The: wave function of N Fernrfons Ss antisymmetric under the exchange of any pair of particles and therefore satisfies:
denotes the sign. or parity, of the permutation P , and is defined as the Here. parity of the number of transposfttons of two etements which brings the permutation (PI.,P2,. ,PN)to its original form (1,2,. ,N). Although the symmetry requirements for Bosons and Fermions are uttimatrzrly found& on experlnnant, It may be proven within the wntext of quantum field theory that gkve general assumptions of locality, causality and torentz Invariance, particles with integer spin (O,1,2, ...) are Bosons and paflicles with half-integer spin .) are Fermfons. Familiar samples of Bosons indude photons, pions, mesons, gluons, and the, 'H@atom. Examples of krmfons include protans, neutrons, elect~ons,muons, neutrinos, quarks, and the " H e atom. Composite particles composed of any number of Bosons and an even or odd number of Fctrmions behave as Bosons or FermS~ns respectively at enwgies sufl"lcientty low compared to their BjncfEng energy. %r convenience, we shalf adopt the following unifid notation for Bosons or Fermions: (1.32) ( F P Z S ~ P* ~ ,S ~ P N=)S'+ ( i l , ? ~ , if^)
..
..
(4,g,. .
-
0
where P is the padty of the permutatjon, and g is 4-1 or -X for ffosans or Eermions resptllctivety. These symmetry requirements imply corresponding restrictions of the Nilbert of belongs to space XN of N-particle systems. A wave function $ ( ? l , . , the Nllibert space of N Bosons, BM, or the Mifbert space of Itl Fermions, FN. if it is symmetric or antisymmetrk respectively, under a permutatisn of the paeicles. We will define the symmetrlzation operator PB and the antisymmetdzation operby their adion on a wave fundion $(Fl,. ,PN): ator Pp in
..
..
For sample, for two Bosons:
and for two krmlans:
The manifestly hermitian operator For any wave function of UN:
P{ ; may be shown to be a projector as follows.
d
SECONO QUANTIZATIQN AND COWEREHI" STATES
when P P denotes the group compwition of P and P. Since = the summation over P and P" can be replaced by a summation over Q 5= P P and B:
Since this equality holds for any waw function 4, the equality holds for the o p eratw itself and the symmetrization and antisymmetdzation operators are projectors. These opuators project )(N onto the Hilbert space of Bosons BR and the Hilbert space of kmSons TH:
Using these project8cs, a sl(stem of b w n s or kmions, with one particte in state and one particle Tn state QN is representd as follows:
arl ,one particle In state aa,
.
...,
Note that the% symmarized or antisymmetdze;d states util:irite a curly bracht fn the ket symbol. The PaufS excludon principle, that two k m l o n s cannot occupy the same state, two states are identical, is automatlcalty satisad for antisymmetdc states. SCIPPOM?
and no aceeptabfe many-firmion state exists In this ease. Ram Equations (1.373, it follows that If lax aH)ts a basis of the Hilbert space #N. then P{:)la2.. .aru) is a basis of BM or 7 ~The . closure relation (1.30) in Xnr bwomes a closure relation in or
...
rH:
.
Further, i f the bast a) is orlhogonaf In U. then the basis lal.. e t H ) is orthogonal In or h. aN) is orthoganal in KM, ;and the basis
...
1.2 SYSTEMS OF tDEFJTiCAL PARTICLES
7
The scalar product of two such vectors constructed from the same basis [a)is:
Becaure of the orthogonality of the basis [a). the only non-vanishing terms in the right hand side of (2,41] are the v u t a t i o n s P such that:
...
...
If a;, ,a;, is a permutation of a&, ,aH. the overlap may be evaluated straightfomardly. For Fermions. since there is at most one particle per state la). no two identical states can be present in the set {aI ,W ) and therefore, there exists only one permutation P which transforms a% ,aH into a:, ,ah. The sum in (1.41) thus reduces t o one term, and i f the states lai) are normalized, we obtain
...
{(I:.
..a;IIa1...LIN)=
...
...
( Fermions )
.
For Bosons, many particles may be in the same state, and therefore, any permutation whlch does interchange particles in the same state contributes to the sum (1.4%). The overlap (1.41) is thus equal to the total number of permutaticms which transform (al a M )into (a:,, ,a",). If the set of states Car.. ,m) represents a system Bosuns in state ag, , np Bosons in state c+, where with nl Bosuns in state al, the states @ ,or,l,are distinct, the overlap Is given by:
...
..
...
...
.
The results (1.431) and (1,103b) may be combined efficiently by specifying a state wlth particles in states a%,. , a in~terms of occupation numbers n, for each stata of the basis !a). h r Bosoms, the oecuparion numbers are a pdon" nat restrict&, whereas For krmions they can take only the value O or 1, In both cases, the sum of the occupation numbers, which cuunts the total number of occupied: states, must be equal t o the number of particles M:
..
For aampltir, the threBoson state wlth partides in states al,a~, comesponds t o nl = 2 , =~1, and = O for i 2 3. The thr(tte-Fermion state with parEicles in states ~cl,aa,ascorresponds to n g == I,n%== l,tibs =z 1, and I= O for i 2 4, Using the convention that O! = 1, the formulas (1.43) are equivalent to the single expression
8
SECOND QUANTiZATION AND COHERENT STATES
Finally, an orthonorrnaf basis far the Wllbert space BAT or Fhl is obtained by utilizing aw1: (1.45) to normalize the states iai
...
Mote that in contrast to the states defined in (1.Z'I) and (1.371, the normalized symmtltric or antisymmetric states Befind in (2.46) utilize an angular bracket in the ket symbol. Because orthonormality was us& 'in the ealeulat'ron of the normalization hctor, it will be understaad henceforth that whenewr the symbsl rag...a ~is)utilized, the basis (lad))is orthonsrmal, h )constructed fvom an orthonormal The overlap betwwn a tensor product basis and the synrmetdzed or antisymmetrized stdh fa1. a ~ is)
fai ...
!a)
..
where S (Mii)denotes a permanent far lsosons
Pcr(Mii)
E
C M Z , P X M ~. ,MN,PN P~ P
and a determinant for Ferrrnions
det(Mij)
C ( - 1 ) P ~ t , p ~ ~ a , MN,PN ~2 P
(I.488)
In coordfnatr? representatian, wa thus obtain a basis of permanent wave functions for Bosons
and a basis of Slater ddwminants far GrmTons
Similarly, the overtap of two normalizd Soson or Fermion states is (@l**.P~lal ...aAT) =
i
Finally using (1.40) and the normalization in (1.46), the closure relation in
(1.50) or
11.3 MANY-BODY OPERATORS We now consider rnatrtx elements of many-body operators in the canonlcat basis of and From the%, the representation of operators in the spaces fattaws straightforwardly using the symmetrization and antisymmetrizatlon operators P,;,. Eq. (1.33). l e t Q be an arbitrary operator In BM or TN. fndependent of whether the particles are Bosuns or f'ermions, their indfstlnguishability implles that O is invariant under any permutation of the partictes. Thus, for any states, and any permutation P:
h.Eq. (1.27),
We begin by considering the care of onebody operators. An operator 0 is a one-body operator if the action of 0 on a state al.,.aM)of N partic[es is the sum of the action of 0 on each pa~fticle:
where the operator operates only on the S*" particle. For example. the kinetic energy operator in the (p') basis, acts as:
and a local one-body potential in the (s) basis acts as:
The matrix element of a one-body operator ?l between two states (al...aN)and
l& ...h) is given by
and thus for two non-orthogonal states:
SECOND QUANTI;ZATION AND COHERENT STATES
10
The one-body operator LI is entirely determined by its matrix elements (alol~) in the single-particle Wilbea space U, Similady. an operator P is a WO-body operator i f the action of P on a state al...aH)Of N pavtfctes is the sum of the action of V on all distinct pairs of particles:
C
Pial . . . a n r )
(1.56)
fijl~l...a~)
l
where Vdioperates only on part,icles i and j. The restriction i j ytdds a summation over distinct pairs. The symmetry requirement (1.52) on implies that = P*. and the expression [l,%) can be rwritten as:
P
Pii
The matrix elements of P are given by:
and for non-o~hogonatstates,
(P
Thus. a two-body operator V i s entirely determined by its matrix elements (aa' IF@) in the Wilbevt space H2 of hnro-particte syst~ms, Further, a two-body interaction P is said to be local. or velocity independent. when it is diagonal in configuration space (and spin and i m p i n 7f appropriate): fn this c a ~ Equation , (1.57) takes the simple form:
In general, we define an n-body operator al...&#) in the following way:
.& as an operator which acts on a state
That is, the action of .& on a state of N particles is the sum of the action of l? on all distinct subsets of n-particles. Anatogows to the previous casrts, the matrlx elements of A satisfy:
and an n-body
operator is entirr?fy determined by its matrix elements of n-particle systems.
...a,~hl@l. ..a,) in the Hi!bert space X,
(a1
1.0 CREATION AND ANNIHILATION OPERATORS
I1
l,4 C R E N I Q M AND ANNIHILATION OPERATORS Creation and annihitation operators provlde a conwnient representation of the many-pafticle states and many-body operators introduced in the preceding sections. These operators generate the entire Wilbert space by their action on a single reference state and pravlde a basis fau the algebra of operators of the Nilbert space. For each singleparticle state [X) of the single-particle space U, we define a Boson or Fermion creation operator a! by its action on any symmetrized or antisymmetrized Xg . X N ) of or as folfows:
..
For the present treatment, it is convenient to use an orthonormal basis ([Xi)), which the definition of a: may also be written
for
where n l is the occupation number of the state [A) in [X1.. . X N ) . Note that whemas we have previously denoted other operators by a circumflex. 6, since the symbols a and at wilt be reserved exclusively for creation and annihilathon operators, no circumflm will be used. Physically the operator a: adds a particle in state IX) to the state on which it operates, and symmetfizes or antisyrnmetrltes the new state. tn the case of Fermions, since there can be at most one Fermion in a given state, Eq. (1-63b) takes the shpter farm:
a: (AI
...AN) =
[AAl
a
...AM)
if the state fX) is not present in
lX1 ,..AN) .
5f the state \X) is present in [X1 ...AM)
(1.64)
In the foliowing, to avoid a distfnctiofi between Bosons and Fermfans, we shalt use Eq. (i.631" tn addition to the many-papticle states we have umd thus far, it is u d u t to define the vacuum state, denotrta 10). which represents a state with zero particles. A natural extension of (1.63) is the definition that a! acting on the vacuum 10).creates a particle in state IX) 4:
10)= 1x1
.
f 1.65)
The vacuum state 10) i s a physical state with no particles, and must be distinguished from the zero of the Nlltzert space, As a physical example, consider an atom in an excited state whkch can emit a photon. The fnitiat state of the system Ts a dfret product of the excited atomic state and the photon vacuum, After d ~ a y the , state is a product of the final atomic state and a one-photan state. The creation operators a1 do not operate within one space 8, or Fa,but rather operate fram any space 8,. or F, to or &+g. Hence, it is useful to define the Fock space as the direct sum of the Boson or Fermion spaces
12
SECOND QUANTIZATIQN AND COHERENT STATES
whwe by definition: and 81=Z;i=U
.
A generral state 14) of the Fock space is a linear combination of states with any number of particles. b r instance, the state:
belongs to the Fack space and represents a system of particles in whlch there is probability $ for having no particles, probability for having one particle in state /X). and probabiliw for having two particles in state Suitable bases for the hck space tctiti thrsr unnormartizrsd or narstates of the proper symmety. (IO), iXl), fX1X2)t bIX2.. Am), .) or IA1X2), lXIAa.. As), ,) and in subrequent derivations we will choooe the most convenient form. The closure retation "r the b e k space may be written
1.
i
.
. ...
I Xl... X>,(X
l...
..
*
(%*@g)
Any basis vector /XI ...A N ) or 1441 ...AN) may be generated by repeated action of: the creation operators on the vacuum 10):
and
Thus, the creation operators generate the entire Fock space by repeated action on the vacuum. The abrence of factors of N or nk in Eq. (1.70a) is the reason for the definition of the states ) and their use throughout this section. The symmetry or ntisymmetry properties of the many-pafllcite slates impose commutation or antlcommutation relations between the creation operators. For arty state lAl.. .XN) and any single particle states IX) and
1.4 CREATION AND AfaNlXILATlON OPERATORS
2.3
where we have used the permutation praperties of the states and, as usual, is 4-1 for Bosons and -1 for Fermions. Since Equation (1.71) holds for any state. the at's satisfy the operator equation
Thus for Bosons, the creation operators commute:
whereas they anticommute for Ferrmlons:
It is convenient t o combine Eqs. (1.72) in a unified form with the following notation
The operators a! are not self-adjoint. and therefore we define the annihilation opwators ar as the adjoints of the creation operators a:. The commutation relations of annihiretfon operators folto~immediately from the adjoint of Eq. (2.73)
and their action t o the left on bras Follows from the adjoints of Eqs. f 1.64) and (2.65). The action of ax t o the right is obtained by evaluating Its matrix element betwilten two arbitrary states. From the adjoint of Eq. (1,53a), we obtaln
The r.h,s. of 11.75) is non-rert, onty if m -f- 1 =. n, so one eflect of ax acthng to the right is t o decrease the number of paflfdes in the state on which it acts by one. In particular, when acting on the vacuum it yO&s O for any state [X):
Similarly
(al.:
=0
.
From Eq. (1.76) it i s evident that the vacuum is the kernel of the annihilation operators, tn order t o calcutate the action of ak on a many-particle state, consider an orthonormat single-particle basis (Ia))such that \A) i s one of the basis states, Using the cl~surerefation (%.SS),we obtain:
24
SECONE) QUAPSTIaTBQM AND COHERENT STATES
-
.p,)
..
,ap) equal ta a permutation are mn-vanishing and the scalar product is given By (1.41) with the
Onty the terms with P = (n 1) and the set (X, ar,.
of result
where
a indicates that the state pc has been removed from the many-particle state
lh .* pm)For a normatizad Sate,
fa1.. '8,)with mupation number nx for the state
X we may write
We observe from Eq. (1.78) that the eRect of ar acting on any state is to annihilate one partide in the state X from this state. h r the c a e of Bossns, the general result Its mare convenientty w p r e s d In oc-cupation number representation. We denote by Ing,, nb,, na, .) a rymmetrized state with np, particles in l&). np, particles in I&) and so on. Since r p ~non-vanishing equal terms adse in Eq. (1.78),,the Boson result may be written
..
For hrmions, It i s convenient to wrlte Eq. (1.78) in the form (-l)'-' l@z 0
...)A ...a,)
if !A) is occupied If IX) is unoccupied
.
(~..rsa)
As expwteds phydcally, if there are no particles presRt tn state IX) to be annihifated, ax yields zwo. To close the algebra of the creation and annihilation operators, W need to w l u a t e the commutators of creation and annihilation apefators, As in the previous ca!culation, we consider the care in which (X) and I@) are two states belonging to the orthonormal basis { la)). Then, according to Eqs. (1.63) and (1.78)
nratlal.. .a,) = olx~pz~. . a,) n
and
Hence
..
for any state lax quations:
.
+
(1.82) arailag.. .a,) = (6rp gotaA)lal.. a,) ,a,), M) the creation and annihilation operators satisfy the operator [aA,a;':]-c
= aka:
- $a:ar
.
=
(1.8Sa)
Thus, Besons satisfy the cornmulathon relations
and krmions satisfy the anticommutation relations
Creation and annihifation operators in another basis may be obtain4 straightfarwardly from the operators { a i , o a ) . Consider a transformation which transforms the orthonormaf basis (la))Into another basis (I&)) as fatlows
By this equation and the definition of the creation operators a; and a t .
...a,) = C(&~G) ;)lp&% ...h,) = E(a/&)a!I&, ...6,) .
obl&l.. .em)= lee, 0
(1.85)
a
Since this result holds for a set of bads states satisfy the operator equation a; =
.&,),
the creation operators
C(al&)a$
and ann5 hilzltion operators stitidy the adjdnt equation
A convenient mnemonic for this result is the fact that the action af the operator Eq. (11.86a) on the vacuum xeproduces the transformation for singte particle states, Eq. (X ,841, Commutation and anticommutation relations for ai and a$ are computed straightfollwardty from (1.86). For example
36
SEGQMR QUAMTIZATIQN AND COHERENT STATES
and simitarfy
[@L a!]-$ = [a&a-]-f B B 1
t
=0
.
In the event that the basis ('G)) is oflhonormal. Eqs. (1.87) produce the same com,, and the transformation (1.84) is mutation relations for a: and a& as for a& and a canonical. Of particular importance is the {Is)) basis defined in Eq. (1.20). where) .1 represents 17~7)In this ea=+ the creation and annihilation operators are traditionally denoted by +'(S) and $(S) and are called geld operators. Their commutation relations follow from equation f1.87a),
The expansion of these operators on a h s l s {la))forlws from (1.86)
where #a(%) is the coardinab represt!ntat"ron wave functlon of the state [a) tn additfon to generating aft the many-particle statm in the %ck space by repeat& action on the vacuum, a; fundamental propew of creation and annihilation operatars is that they provide a basis for all operators in the %ck space. That Is, any operator can be expresd as a linear comibinatlan of the set of al! products of the operators
(.!,@.l*
A convenient twhnlque for mpre~ntlngan opemtor in terms of creation and annihilation operators i s to first use a basis in which it Es diagonal and then transfsrm to a generat basis, To reprewnt the operatar In the diagonal basis, we define the number operator 6, : A, = aia, (1.901
.
Using Eqr. (1.63) and (1.78). we obsewe that the action of A, on a state (4)is to count the numkr of partides in state lar) in the state 14):
.
aM). The operwhere C:, , 6 yields the number of particles in state lo) in ator 10 which counts the total number of particles in a state is given by:
1.4 CREATION A N 0 ANPSIXfLATlON OPERATORS
We now anrider a one-body operator. basis la)
17
0,which is diagonal in the orthonormal
Using (1.55). and (1.41) we obtain
Since this equality holds for any states, we obtain the operator equation
Thus, in order to obtain the action of 0, we must sum over all states la) multiplying U , by the number of particles in state la). We may now transform from the diaganal representation ts a general basis, udng Eqs, (1'86). Then
where
Eor aampte, using field aperators in the (P) representation, the kinetic enera operator 9,and a local ontbody operator l?may be written
18
SECOND QUANTIZATION AND COHERENT STATES
and in the
(9)
representation, the Mn&e energy is:
P
Similarly. a two-body operator may be expressed in terms of creation and annihilation operators. Using the basis V,@ in which P is diagonal.
We calculate a general matrix element as befwe:
zi+j
The fador $ ,,,V , is a sum over all distinct pairs of particles prerent in the which counts the number state [al..a x ) .so we need to conrtwct an operator of pairs of particles in the states far) md If' la) and 18) are digerent, the number of pairs is man# whereas if lar) = [p),the mmber of pairs is n. (n, - 1). Hence, the operator which counts pairs inay be written
.
la>.
or in twms of creation and annhhtlatlon operators,
Using this operatar, Eq. (1.102) may be rewritten
and hence.
P satisfies the operator equation (l.lost)
Anafogaus to the case of a sntl-body operator, the action of a two-body operator is r of single-particle states la) and l@) and multiplying obta'lned by summing ~ v e pairs
1.4 CREATION AND ANNIHILATION OPERATORS
10
the matrix element [aatV a@) by the number of pairs of such particles present in the physical state. Transforming from the diagonal representation to an arbitrary basis. the general expression far a two-Body potential is
Mote the order of annihilation operators in this expression which i s requird to remove the factor from Eq. (1.104). and thus yield a simple result for both Fermians and Bosons. Indeed, one of the great virtues of representing aperators in terms of creation and annihilation operators is the simplicity and convenience of Eq. (1.107). which concirety handles all the bookkeeping for Fermi and Bose statistics. Occasionally, it is canwniant to use the f i l l w i n g variant of Eq, f i ,107a) In which the matrix element i s manifestfy symmetrized or antkymmetdzed:
[l.107b) As an example, if (1,107a) yields:
P is an interaction which is diagonal in the ($1representation. Eq.
In terms of its antisymmetrized matrix elements,
Eq. (d.tO'lb) may be umd with:
The precding dedvations for one- and two-body operators may be straightforwardly generalized to n-body operators, with the resuit:
The only technical complication "r the general derivation is to carefully count n-tuples of particles in states lal). /a,)when not alt ( a i ) are distinct (See Problem l,i). Flnafty, far future rderencc?, it is useful to define normat ordering for a manyc partide operator. Ely definition, an aperator Ss in normal order if all the creation operators are to the left of all the annihilation operators, Far example, the first line of Eq. (1.104) is not in normal order whereas the last line is in normal order. Clearly, any apressian which is not In normat order may be brought into normat order by a wquence of appticatlons of the commutation or anticommutation relations for creation and annihitatian operatars,
..
20
SECOND QUANTIZATI0H A N 0 COHERENT STATES
2 3 COHERENT STATES Thus far we have used permanents or Slater determinants as a natural basls for the Fock space B or 7, Another ctxtremely useful basis of the Fock space is the basis of coherent states which is analogous to the basis of position dgenstatcs in quantum mechanics. Although this is not an orthonormal basis, it spas! the whole Fock space. Just as the position states I'it) are defined as eigenstates of 7,the coherent states are defind as eigenstates of the annihilation operators. To see why annihilation operators are selected rather than creation operators. it is usdui t o consider the dstcnca! sf dgenstates of the: crestton and annlhitation operators. If we denote by 14) a general vector of the Fock space. we can expand 14) as: 00
Thus,14) necessarily has a component with a mtnimum number of paeicles and 'rf we apply any creation operator t o 14). we see that the minimum number of particles in 14) is increased by onee Thus, the resulting state cannot be; a multiple of the orig;inal state and therdore a creation operator cannot have an dgenstate. QR the other hand, i f we apply an annihilation operator to (4). this decreases the maximum number of partides in 14) by one, since 14) may contain components with all particle numbers, nothing a priopi forbids 14) t o have dgenstatw, Assuming we have succedd in constructing an eigenstate 16) of the annihilation operators, then (a,~rz) aa l&> = 60116) When more than one annihilation operator acts upon such a coherent state, a sbnificant digerence arises between Bosons and Eermians. The commutation or anticommutation relations (1.W) imply corresponding retations for the eigenvafues g
For krmions, the dgenvalues anticommute, and in order ta accommodate this unusual fgature we sub~quenttywilt need ta introduce antic~mmuthgvariables catted Grassmann numbers. Far Bosons, however, the dgenvaluet-s commute and we wilt be able to procm4 straightforwardly using ordinary nurnbws, Hence, we shall begin by considering Bomns.
BOSOM COWERENT STATES For Bowns, the elgenvalues 4, of the annihilatfon operators may be real or complex numbers. 1is convenient to expand a Bossn coherent state in occupation number representation
..
..
where as usual n in ,, . n,, .) denotes a normalized symmetrized state with n, particles in state /al). n, particles in state /a2), and {lai))is an orthonormal basis.
...
1.5 COHERENT STATES
21
The dgenvafue condition. Eq. (1.112) for an annihilation operator aai acting on
19) implies the following conditions on the coemeients for all (m,) 64.
-
Aml m@$
(1.21s)
...(IPI I)...
Refating each coeficient by induction to the coeRcient for the vacuum state which we arwtradly set equal to 1, we obtain
Substituting (1,916) in (1.114) and using the fact that
Note by taking the adjoint of (1,152), that the adjoint of a cohennt state is a left eigenstale sf the creatfon operatQrs:
The action of a creation operator a& on a coherent state is given by a!
I+)= .:C.
0 . 4 10)
d = -14) 34,
(l.lz~a)
with the adjolnt relatiorr
The avedap of two eoherlent states is given by:
x (nu,.
..m, ...,:nI ...m:,)
.
(1.121)
22
SECOND QUANTBZATlON AND COHERENT STATES
...na, ...lnhl ...nh, ...)
Since the basis (a)is orthononnal, the scalar product (nag is equal t o 6nalnb, 6napnbp which leads to:
...
...
A cruciaf property of the coherent states is their overcompleteness in the Fock space. that is, the fact that any vector of the Fock space can be expanded in terms of coherent states. This is expressed by the closure relation
where 1is the unit operator i n the Fock space, the measure is given by:
and the integration extends over aft values of Re& and h#,. As explained in Problem 1.2, one may verify Eq. (1.123) straightforwardly by integrating the left hand side t o obtain the familiar completeness refation Eq. (1.51). A more economical proof is provided by Schur's lemma, which in the present context states that i f an operator commutes with all creation and annihilation operators, it is proportional t o the unit operator in the Fock space. Using equations (1.120).
so that evaluating the commutator of Eq. (1.123) and integrating by parts,
By taking the adjoint of Eq. (1.126). we obsem that the left hand side of Eq. (1.123) commutes with all the creation operators as well as the annihilation operators so it must be proportionaf to the unit operator. The proportionality factor is calculated by taking the expectation value of the left hand side of (1.123) in the vacuum:
This proves Eq. (1.123).
This completeness relation provides a useful expression for the trace of an operator. Let A be any operator and let (In > 1 denote a complete set of states. Then
In quantum mechanics. the completeness d the position eigenstates allows us to represent a state I+) = dz*(z)lz) where +(z) = (%l$) is the coordinate representation of the state l$) Analogously, Equation (1.124) implies that any state I$) of the foek space can be repre-entedi as:
I
where by definition:
*fQIU)
= (413r)
I$).
is the coherent state representation of the state and 6 denotes the set (4;). The coherent state representation for Bosans is often referred to as the holomorphic representation, which arises from the fact that JI is an analytic function of the variables 4:. Physically. +(#*l is simply the wavefunction of the state I$) in the coherent state representation; that is, the probability amplitude to find the system i n the c o b e r a state 14). Just as it is US&^ to know how the operators % and j3 act in coordinate repreentation, it is useful to exhibit how the operators o i and a, act in the coherent state representaaon, Using Eqs. (1.119)and (1.120) we find:
and
(4loLlf) = 4: f(4*1 Thus we can write symbolicatly
d aa = -
84:
and a! = 4;
*
24
SECONL) QUAMTlaTIQN AN0 COHERENT STATES
wtticlil is consistent with the Boson ~ommutationrules:
Aslda from factors of i , the behavior of a an$ d in the coherent state representation is thus arralogous to that of % and p" in coordf nate repremntation. The result, Eq, (1.130), ylefds a simpfe mpression for the Schradinger equation in the coherent state representation. If H(&!, a ), is the Hamiltonian in normal form. then projmtfon of the SchrSdingw squation
on the left by
(41 yieids
b r a standard Hamittonhn with one- and Wo-body operators, It reads:
In the space of holomorphic functions using (1,123):
$(p),the
unit operatw is okained by
which implies:
plane I ~for the familiar representation Note that this Is just a general form In the C O M P of a &function 6(2 S') = $c'~('-~'). Another useful property of coherent states is the simple form of matrix elements of normal-ordered operators between coherent states. If we denote by ~ ( o L , a , ) an operator in normal form. the action of the .a to the right and a: to the left on cohennt states imm-edfately yfdds
-
1.5 COHERENT STATES
25
where A($;, $L) is the normal form of the operator where the creation operators a$ have been replaced by yb: and the annihilation operators a. have been replaced by +b. For wample, a twolbody potentiat is written:
From their definition, it is clear that coherent states do not have a 6xed number of particles. Rather. the occupation number m, for each state a is Poisson distributed with mean value [q%,I2
(l.139) Thus the distribuition of particle numbers has the average value
and variance
= @l*W)
(46146)
=
C,:,a= p
(l*l4M)
Q
goer t o In the thermodynamic limit, where R -+ +m.the relative width 8 = zero. and the coherent states become sharply peaked around X,reflecting the fact that the product of PoEsson distl.ibutions approaches a normal distribution.
CRASSMANN ALGEBRA In order t o construct coherent states for fermions which are dgenstates of annihilation operators, we have seen in Eq. (1,113) that it &If be messary t o use anticommuting numbws, Atgebras of anticanmutfng numbers are call& Grassmann algebras, and in this section we briefly summaize their essential propeflies. Fer our present purposes, it i s suf'Ficitsnt to view Grassmann algebra and the definitions of integratlon and digerentiatian as a clever mathematical construct which takes care af all the minus signs associatd with antisymmetry without attempting to attach any physicai significance! ta it, A complete treatment of Grassmann algebra is given in the treatise by Berezin (19&5), A Grassmann algebra fs defined by a set of generators, which we denote by (&),ar = 1,. ,m. These generatws anticommute:
..
so that, in part;icutar:
e:=o.
26
SECONE) QUANTIZAT$OM AND COHERENT STATES
The basis of the Grassmann algebra is made of at! distinct products of the generators. Thus, a number En the Grassmann atgebra is a linear combination with complex weficients of the numbers ( 2 , &, ,&, E,, , &, fa, 5,, ) where by convention the Indlca are order4 a1 as .< .a,. The dimension of a Grassmann afgebra with n generators ts 2% sin- dis.einct bads elements are produced by the two poss5bifities of 'rncludlng a generator O or 1 t h e s for each of tz generatars. Hence, a matnix representation of Grassmann numbers requires matrices of dimension at least X P, In an algebra with an even number n = 2p of generators, one can define a a n Jugation operatian (callled involution in some t e ) in the following way, VVe salect a set of p genexatsrs and to each generator f a , we assadate a generator which we denate The Coltowing properties define canjugation in a Grassmann atgebra:
...
..
c:.
...
em,
tf X Is a comlpIex number, (Xtml* = X*e: and for any product of generators:
To simpltffy notation, we now consider a Grassmann algebra with two generators. VVei can denote the generatars by E and and the algebra is generated by the Four nlfmbws (1, t, Bsauw of praperty t1.145b). any analytic function f defined on this algebra is a flnear function: = fo+fif (1.145)
e*,
c,e*a.
f(e)
and this Is the form we will obtain for the coherent state representation of a wave function. Sfmffarly, the eoherent state representation of an operator in the Grassmann algebra will be a funetfon of axtd and must have the form
c*
As for ordinaq complex functSons, a ddvattve can be detind for Grassmann variaMe functions. It is defined to be Edentkaal to the conptex derivative, except that in order for the derivative operator to act an 6, the mriable ( has to be anticommuted through until it is adjacent to For instance:
&.
With them definitions:
d -A[
c*
I
6) =: a1
.
t*
( l .148a)
(l.248~)
3
Note from Eq. (1.148~)that the operators $& and anticommute. In defining a definite integral. there is no analog ofthe familiar sum motivating the Rienann integral for ordinary variables. Hence, we define 1ntt;gration over Grassnann variabtss as a linear rnapplng which has the fundamentaf prope&y of wdinary inbgrals over functions vanishing at infinity that the Fntegraf af an exact diff'erentlalform is zero. This requirement implies that the integral of is zero, since lis the derivative of (. The only non-vanishing integral is that of C, since is not a derivative, Hence, the definlte integral is defined as follaws:
and as in the case af a derivative, in order to apply f1.149b). one must first antimmmute the vartable 6 as rctquired to bring It next to de. A shmpte mnemonic for this defiinition is the fact that Grassmann integration is identical to Grassmann diferenttation. Since have been 4lefine-d arbitrarily to be conjugate variabies but half of the generators are otherwfsar: equivalent to the generators &, lt is natumt to def'ine fntqratfan for conjugate vadabtss in the same way:
(z
Mate, however, that in contrast to a Riemann Elntttgrat in which d z is an infinitesimal real variable, d r is not a Grassmann number and it makes na sense to apply Eq, (1.144) to the quantity &(E.) to try to relate Eq. (1.149) to Eg. (1.150). The fotlawlng; examples Illustrate the application of these integration rules. Udng f1.145), we obtaln:
(1
C
dtL f (C) = ff and using (2.146) we get:
28
SECONID QUANTImTIQN AND GOHEREFST STATES
and
The motivation for this deFinitEan of integration fs that with them canve~tions, many results look similar to those of complex fnttqratlon. for instanea. consider the definition of a Grassmann S-functian by:
To verify that this definition has the desired behador, wre use Eq. (1.145) to obtain
for any funcDion f (0, Motivated by Eq. (1,135) for Boson coherent states, 'lt will be u d u f to define a scalar product of Grassmann functions by:
where f (0 is defind by (1.545) and
With detinltion (1,155).W sea that:
and it can be shawn that Grassmann functions have the structure sf a Hifbert space. The results we have presented fw the case of twu generators 5 and generalize straightfaforwardly to Zp generators h i$;. (i as shown in ProMem 1.3.
...ep, ..
1.5 COHERENT STATES
29
FERMION COHERENT STATES ff we try to construct coherent states for Fermions, we immediatety encounter the difficulty that if we expand them according to [f .114), the coefficients must be Grassmann numbers. Therefore, in order to construct coherent states, we must enlarge the Fermion Fock space. We first define a Grsssmsnn algebra g by associating a generato! E, with each annihilation operator a.. and a generator with each creation operator.:a We then construct the generaftzed Fock space as the set of linear combinations of states of the Fock space 7 with coefficients in the Grassmann algebra 9. Any vector I$)in the genefatired Fock space can be expanded as:
where the X , are Grassrnann numbers and the It$,) vectors of the h c k space. In order to treat expressions containing combinations of Grassrnann variables and creation and annihilation operators. it is necessary to augment the definition of the Grassmann variables to specify the commutation relations between c's and a's and the adjoints of mixed expressions. To obtain results analogous to those obtained previously for Bosons, it is natural and convenient to require that
and (&)t
= $tf*
where denotes any Grassmann variable in {E,, €2) and ii is any operator in {aL, a,). We now define a Fermion coherent state It)analogous to Boson coherent states by
so that the second line in Eq. Note that the combination €,a: commutes with (1.1591 reproduces each non-vanishing term of the expansion of the exponential in the first line. Although the coherent state belongs to the generalized Fock space and not to 7. as we shall see, the crucial point is that any physical Fermion state of 7 can be expanded in terms of these coherent states. We now vedfy that coherent states as defined in (1.159) are eigenstates of the annlhilation operators. For a single state a. the anticommutation relations of a,,a&, and yield the relation.
30
SECOND QUANTIZATIQ-NAND
COHERENT STATES
Using Eqs. (1.154) and (1,160) and the fact that a, and cE, both wmmute with the combination &a; for p $ a. we obtain the desired eigenvalue conditions.
Similarly, the tldjolnt of the coherent state Is
and is a left-eigenfunction of a!
The action of aL on a coherent state is analogour to the Boron result. Eq. (1.120). and dlgers only in sign:
(1.1Ma) and dwrilarly one may ved@
d
(4th = +-(iFI aP
*
The ovdag of two coherent states is easily cateuiated:
(I*f 646)
1.5 COHERENT STATES
31.
As in the Bason case, the closure relation may be written
where the L denotes the unit operator In the physical Rrmion k c k space. 7. This closure relation may be proved using Schur's lemma as we provd Eg. (1.123) once integration by parts has been derived for Grasrmann wriables (See Problem 1A). Here. we pmsent an afternative prod. M define the operator A t o be the left hand side af
To prove (i.i6'1), it is rufftcient to prove that for any vectors of the basis of the Sock space: (1.168) (ax a & % f B*pm) b . = (a1 a,[#l... L>
...
...
.
Using the eigenvalue propetty of the coherent states (i.26iJwt? obtain
and the analagous adjornt equations. Thus
Now consider the integrals which may arise in Eq. (5,370jfor a particular state 7:
Thus, the intewal in Eq.(l.l?O) is non-vanishing only if each state 7 i s either occupied in both ( a x ,..a,( and l&,. . . h or ) unoccupied in both s t a b , which requires that n=n and {al. .a%) fs some permutation I ) of (P1 PG). In this case, the integral is easily evaluated by writing E,, G ,E;, ..egm= ( - ~ ) ~ f ~.taIE:, . .[G. and
.
...
.
...
..
..
noting that an even number of antlcommutatians is requlred to bring the intwral over each state into the form of Eq. (1.1?1), so that the value is just The left hand side of Eq. (1.168)thus yields the result previously derived in Eq. (1.50)for the right hand side, so the equality is established for any vectors in the Sack space. As in the ease af Bosons, this completeness relation provides a usdul expressiion for the trace of an operator. Because the matrfx elements (hl[) and (i$f+~) between
32
SECOND QUANTI-TIQN
AND COHERENT STATES
in the Fock space and coherent states cantaln Grassmann numbers, it
states
gallaws from the anticommutation rdatlons that
-
(dtilO(fl+,.) (-.E\J"li)(Jtiltl*
(l.172)
Mm-. 'ff we deOine a eom@eteset of states (In)) i~ the h c k space, the trace of an operator A may be written ' R A=x ( n l ~ l n ) 11.
d@d(,ca
':'"x(nlf) (EIA~~) t3
d ~ d & c - CQ
(-tlC ~ In)(n/lt) 1)
~ d @ d e , e - ~ ~ ~ : ~ " .( - ~ ~ ~ ) ~ ) (1,1731 LX
The overcompleteness of the firmtan coherent states alfaws us to define a Grassmann coherent state repfisentation analogous to the coherent state representaticm far Bosons in Eq. (1,129a)
where!
(et$> = tbClt"*) WSEhin thls representation, it follows from Eqs. fl,llil and ) (1,1164) that the creatfon and annf hilation operators satisQ:
Thus, as in the Boson case, the operators a. and a i are represented by the operators respectively. The anticommutation relation (1.83) is repre~entedby ae: and
fz
As in the Boson case, the matrix element of a normal-ordered operator ~ ( a i a , ) betwen tws cohwent states is very simple:
1.5 COHERENT STATES
33
However, in contrast to the Boson case, the expectation vahe of the number operator is not a real number:
and it is meaningless to speak about the average number of particles in a Fermion coherent state. Finally, we conclude this section by contrasting the physical significance of Boson and krmlon coherent states. Boson coherent states are the physical states which emerge naturally when taking the classical limit of quantum mechanics or of a quantum field theory (See Problem 1.5). In the classical limit. when the field operators are assumed to eommute, the definition of a dasoicat field #(S) at each point of space is identical to saying that the system is in the coherent state = eldai9(')*t(')[0). for example, a classical electromagnetic fieM can be v b w d as a coherent state of photons, In contrast, b m i o n coherent states are not contained In the hrmion k c k space, they are not physically ob~wable,and there are no classlcat fields of krmions, Nevertheless, hrmfon coherent states are very u d u l in formally unifying many-brmfon and many-Boson problems, and we shall use this propefiy extensively In the fatlowtng e hapters, tn the subsequent development, this physical diference wilt give rise to some significant differences in the treatment of flosons and Fermions. For example, application of the stationa~3(-phase approdmation 160 an apresslon formula&$ in terms of Boson coherent states yketds a u d u l expansion around a physical ctasslcat field configuration. For krmians, no correspondfng physical solutian exists, and the Fernrfon degrees of freedom wlll have to be integrated out ttxpticitty,
I+)
GAUSSiAN INTEGRALS In the ensuing format developmmt, we will frquentty evaluate: matrix etements of the evolution operator in coherent states, teadfng to Integrals of exponential functions which are pofynomfats in compla variables or Grassmann variables, In the case of" quadratic forms, these are straightfaward generaliizatSons of the famflf ar Gaussian integral, and we presnt several useful integrals in this w t i o n for future rderence, For brevity, we derive the identitiw for the special cass: of symmetric and Hermitfan matrices, and refer the readcit. to the standard rderences for the general case, VIze begin by proving the foilwing identity for multi-dimensionaal Integrals over real variables: J ~ A ~ J ~ dzg* * d z a C - f a * ~ s j+~ sf r ~= $ pet (1.179) where A is a real synrmeMc positive definite matrix and summatian over repeated Latin indices is undwstod throughout this s ~ t i o n ,This "rentfty is established straightforwardly by changing variables to reduce it to diagonal form and ushng the famEIiar Gaussian integral
34
SECOND QUANTlZAT1BPJ AND COHERENT STATES
-
Performing the transformations g* = z i AG'J, and zk = orthop;anal transformation which diaganalftzes A, we obtain
o,'~~,where Q
is the
which proves Eq. (l.iit9). Note that the positivity of A is essential for convergence of the Gaussian intagral. A similar identity far integrals aver pafrs of conjugate ceompfex variables is
(l.182) which is valid far any matrix H with a positive Hermitian part. For the special ease of o Hermitian matrix. It i s proved in the same way as Eq. i1.179) by defining the and its complex conjugate, transforming Hto diagonal transformation yi = z~--HG'J~ form and evaluating the diagonal integral
Finally, we wish to establish the analogous identity for Grassmann variables
where, for simplicity, 1Y Is again assumed to be Hermitlan but not necessarily positive definite and {qi, q:, y, 1 are Grasrmann variables. Note that Eq. (1.184) differs from Eq. (1.182) by the appearance of the determinant in the numerator instead of the denominator, Ta prove this identity, we need to deriw two additional resules for Grassmanfi variabiltjs: the transformation law for art Integral under a change of variables and the formula for a Gaussian integral. A Gaussian integral involving a singfe pair oP conjiugate Grassmann variables is easily evaluaterf as follows,
5.5 GOHERENT STATES
35
Note that for a single variable, the Grassmann Gaussian integral yields a in contrast to $ in Eq. (1.183) for the ordinary Gaussian integral. Hence. if we can bdng the multivariable Grassmann integral (1.184) into diagonal form, we expect to obtain the product of eigenvalues, and thus the determinant of X, in the numerator instead of in the denominator as for cornpiex variables. In order to transform Eq. (1.184) into diagonal form, we need to derive the law for linear transformations of Grassmann variables
which differs from the transformation law for complex variables by the appearance of the Inverse of the JacoMan instead of the Jzrcolbian, The derivation i s facilitated by retabetling the variables as Fottows:
and writing
$' = Mr)fb' * "*
The only non-vanishing contributions to Eq. fi.186) come from the term in the polynomial containing each 5 as a factor, which we write as 6 . Thus, we must evaluate J in the equation
pnfz)=",
The left hand side yields p(-%)" and the right hand side is evaluated by noting that the only non-vanishing contributions arise from the (2n)! distinct permutations P of' the variables {q) generat& by the product. Thus
so that
SECOND QUANTliEATlON AN0 COHERENT STATES
36
wttict, proves Eg. (1.186) for a general linear transformattan. finally, Eq. [1.184) may now be proved by defiining the transformations pi .= = t$ - RC'*$;, diagonalizing X with a unitary transformation LT. -a* pi, * notlng that all the Jacobians are unity, and defining = ~,;l'~$ and = U&. using the Gaussian Integral Eg. (1.185). Thus
~~;'$~,~z
C
=
fi
=detR
(1.190)
m=%
which provies Eq, (1.184). Note, as In the prevlous case, that the derfvatlon may be! generailzed to a non-Hermktian matrix W. It may appear curious that the Gausslan integral for Grassmann variables requires no restrictions 615 the matrix B whereas for ardfnary variaMes H must be positive deflnlte for the integral to canverge, krmally, the dfstfnctrion arises because;the ctxpansion of the exponential cm'." terminates at first order. yielding a finite integral inerpective of: the sign of a. ThSs formal property, however, reflects a fundamentat diference betwwn firmions and Bosclns, and It Is the Pautf principle restriction that occupation numbers be either O or 1 which guarantas finite resut& for krm9ons Smespecaue of the eigtmvalues of H. This point Is illustrated by the simple case of the parttaon function of naninteracting partides, which will be shwn in the next chapter to yield Gaussian integrals of the form of Eq, ( l , i 8 b j , For non-intwaeritng particles in the Grand Canonical ensemble, the parlition function may be written a nu
where nu deno&s the occupation number of the state er, For krmions, the series for each a terminates after two terns
M no
restriction is imposed on(c,
- p)
However. for Barons.
1for all a which requires so that the paltition function is finite only if c-@('.-') that (ea p) be positive for all a. Thus the operator (B- pN) must be positive definite for Bosons, but no such requirement arist?sfor hrmionr;. The satlent results from this irttroductov chapter are collwted for subsequent rdwenc-le in Table 1.1,
-
PROBLEMS FOR CHAPTER 2
37
Table 3.1 SUMMARY OF PRINCIPAL RESULTS O F CHAPTER 2. Formulas are wrlftsn In a unlflsd form for Fermlons and Bosons using the conventions that: for EXosone f and g denote complex variables and g = 3-1 whereas for F-srmlons f and rj denora Grassmann vartableti and $ -- -1.
PROBLEMS FOR CHAPTER 1
The auX?jecdof quantum many-body fheaq is too vwt ts treat, all the formalism and iHu~trativeexmplers adeq~srtelyh the text, Thas $he prablema me intend4 ta be an integrd p& of the earn@,aind in eomtrwL to rnost f a t s , enGkely new and' separate topics will &sn be introduced, Redera are stsrongly encouragttd to read throngh all &heproblemer and wlve those which appeasl appropriate, As a guide,
38
SECOND QLTANTtZATfON AND COHERENT STATES
introductory comments will be made concerning the problems in each chapter, and particularly crucial problems will be denoted by an *. Chapter I was necessarily completely formal, and Problems 1-4 are intended to develop expertise with the formalism and fill in details omitted in the text. Problem 5 is intended to develop a physical appreciation for Boson coherent states. Problems 6-8 d e d with the physics of the Hartree Fock approximation. It is a%. sumed that students have been exposed to the conventional treatment of the Hartree Fock and Fenni gas approximations in elementary quantum mechanics courses, so these topics are not repeated in the text, However, this mean-field physics is fundamental to all the subsequent physical applications, so it is optimal to have it clearly in mind before proceeding to the general formalism. Fox those unfamiliar with these appruximations, sufficient details a m provided to work out everything using the treatment of pennanents, determinants, and matrix elements of one- and two-body operators provided in this chapter. Problem 9 is particularly important because it introduces the problem of particles interacting in one spatial dimension with 6-function forces. Most of the methods developed in later chapters may be worked out analytically for this system, so it is essential to explore the elementary properties developed here as a foundation for subsequent applications. PROBLEM 1.2 Show that the operator which counts the number of n-tuples in the states lal),laa), lam)may be written
...
from which Eq. (1.110) immediately follows. One straightforward method is to repeat the argument leading to Eq. (1,103) for three particles and then generabe to obtain
from which (1) may be proved by induction. Note that the complications associated with non-distinct states only arise for Bosons. PROBLEM 1.2 Derive the completeness relation Eq. (1.123) by integration. First consider one single-particle state a and let In) denote the state with n particles in one obtains a. Show that by writing # in polar form q5 =
Now generalilie to a set of single-particle states { ! a ) )noting that the closure relation Eq. (1.89) may be written
where { n a ) denotes a complete set of occupation numbers.
PROBLEMS FOR CHAPTER 1
39
PROBLEM 1.3 Generarae the properties of Grassman v&ables demonstrated to the case of 2p generators in Section (1.5) for the pair of generators f, {&.. (p(*, .tg). In particular, det h e t b general f o m of a fa&bn f (Eh) and an operator A({:, fa), show that &lfTr and anticommute, determine if an analogous pmperty holds for integration, find and verify an expremion for the and generalise Eq. (I.157) for (f lg) pdimensional &-function sP -
.
c
..
k, (z $0,
.
PROBLEM 1.4 Prove the claaure relatian Q, (1.116) for Femiona using Schw'g lemma. As in the Bomn case, [a,, #)(#I]= (fa - &)/#)(+l# m one must show
for m y A* First, prwe R. (1) as it stands, establishing that the left-hand side of Eq. (1.116) must be proportional to ndty, and evaluate the constant of proportionality by calcnl&ing the mat& ekemen$ h the sera-pmticle &age, Then note that R. (1) is a speeid case of integration by p e a for Grwmann v&acblw, Show tbab the generd rub for inkegation by p&s k
C
g
where acts to the left and the vKiable f a must be anticommuted to the right to %c! adjacent to the duivative. Note in particular that the sign in Eq. (2) lis revemd &om the nsual mlation far complex vdabkm and thaf expressions like$ d(:dea [A((:, (a)]B(c: :,, do not reproduce the right-hand side.
n,
&
c&)
PROBLEM 1.5 Coherent States and ths3 Clasrrfcal LImlt of the Harmonic OSConsider a arbple h
cillrator
a) Show Ghat a minimum ancertainty wave pwbet wives the thedependent Schiiidinger quistbn, giving a probhiiity demity
40
SECOND QUANTIZATION AND GOWERENT STATES
b) Now cans2er the Hebnberg reprwntatian coherent state
Eomi the eigenvdae eonditiam I
show that; the eoordinsta sgace wave functkn #(g)
-- (q(4)~stisfiea
sa that
Thns, neglecting the inconsequential phase factor e m i t i , the SchrMjdinger wave function is obtained from the Heisenbug wave function by the substitution 4 -,#e-*Ut. Hence, show th& the cohwent rstaLe wave function has the properdies
where ( ) denotes the coherent state mpectation value, Thw, the cahaent state 14) is the m h h u m uaedainty wave paekef of part mean vdaw of the c ~ o r d h a t ea d mamentnm s a t s y the elwicaf eepib. (a). nic nnotian with ampfitude propo&ionacf Gions of motion and exwafe shpge?h c i a t d wifh the wc33tator paund state is simply to 141. The proba;bility density tr-latctd, by this c b i c d mo and the ~lxgc?etationvalue of the emergly di@em fmm the classical m a l t $ ((p)% +wg(z)" only by the rero point enugy #b. ation and the distiribution of ohlcgtator gnanta P(n) =E C nm StkEag'e to- show $hat probable =%ator state in the coherenf rrfate has with E, = hw (n $) = hw(l#la = (X). her applkatiofls af cohemnt et;atrxl may be found in the faEowing dwo refere n c ~ .The preceding treatmemt of a shgle w a a t o r made k g e n e r a k d qraatised el=tr~magneticfield by R. Glauba (1963). The S-nst* for m U a & o rcaapld to m m b i t ; r q the-dependent field is solved by IP, a d M.M.NieLo (1965).
+
+ 4)
PROBLEMS FOR CHAPTER 1
41
PROBLEM 1C6 . The Martrara-.F.ockAppraxlmratfon a) Show that the expat&ion value of a Nmiltonian with a one-body kindie emergy operator Eq. (1.98) a d Local spin independent two-body potential Eg. (1.156) in a Slatw deteminm&i%
By the Ritr v ~ i a t i o n prhiple, d &hebeat dttteminaatd ;tpproimation to the exact k g (H) with rwpecf t o the singlep&lcle wave wave function iLs o b t h d by mh function 4,. Hence, introduce Lagranga mul<iplierswith 7:P = 78. to mahtain or~ ~ ~ ( a with l f l )respect ] to each wave function. thonomaXi8y and vary [ ( X ) Cab The v&ation is asimpEfid by natmg that vasia&ionwithi respwk to the red a d h a g h w p&s of 4, is quivdent do hdapenaent vdaction with respwt to 4; and (QI,. -ally, p d o m a unitw trsndonnatba on the wave functiom 4, (which cannot altm the detemlnant) ta diagonslir;e:~~8and obt& the Hastree-RcIr equ*
-
wltera the local Emtree potentid is
the non-laeaL exchange potential is
m d the d i a g o n h d Lawange malfipliem aow denoted c,. Recall that z includes bath 8pwe and spin v&abla, and e t e the spakid W-reti-Faek quatiom expEcitly for the c= of spin S, warning the n m b m of pmticles N L an inkmultiple of the spin degeneracy (25 1). b) Rtrpeat the dkvatioa for the ewe of Bomns *&h all pssticlm in s aingle &ate 4. Cornpm the equalioms with the F e r ~ a sresu1.t , in the special cam 2S I = N. E x p l G this c o m p a r b ~ physieirclly, c) Show that fhe ts&d e n s r ~of the Haftrecb-Foek solutioa for be h d d e n
+
+
$2
SECOND QUAPJTImTION AND COHERENT STATES
Abm prove Koopmm' thmrem wMeh aka- th& if oae eva1u;tl;ers the E&rFa& enanrgim of tbe: EP and (AT l)-g&icle syskm. wing Lhe H&reeFock wave functiotrer for the N-peicfe s-ysdem,
-
where e, is the eigenydue of the Neh single particle state. Hence, to the elrtent to siOLgbpMicle wave fmctiiaas only change to order 6 when the p e i e l e numb= cfrmges by l, -e, spwifiafis the removal energSP for the last pdicfe to order 'fa. d) NW, define s ane-body potcmtial hpmch that diagonahing & = I"+UHp reprodncw the H&m+Foek eqaations. h the submquemt study of pefLwbation theory, it will be a d u i to r e g d a"+ UHF as the nnp&urtted Hamiltoaian md treat V(%+ -- zj)- UHF W the pertwbation. PROBLEM 1.7
Stabtffty and EqutlfbrCum CondltDonr for Unlforrn Mattet
adpamie limit of a vstem of N p&iclw a&rero tempsrakwe c o n h d in a box of vollunne Ir in a stacte with enaerw E having mngorn dendty. Denote the density p = $ and the enagy per particle c(p) = a) Show that the p r e ~ n r eis given by P = b) Show th& the eqraubdnm density, oftctn caUd the saturation density, for seXfcboud urtconflnd balk matter occm at, the density whi& mhirxliaies elp). c) Show that the chemical patential p = E ( N + 1) E ( N ) is given by p = E + Thus, far ac seKboumd satnrating v s k m sLt eqdibrium deasity p ==: e, df E x a i n e the stab%$y wigh mspect to long wavelength demity sluetuations acer foams. Gowider dividhg the volume 'V in haff, a ~ cdculake d the total aetrgy if one half has darritiy p li and the &her ha& has demity p - 6. Show that this enerp exceeds that of the unifom configuration, and thus the aystem is stabk, if &(pc!p)) > O which is eqnivslent to > 0. When we subaeqpently examine stabhty with respect to finite wavelengt$ fluctuations using the random phase appro~magio~, this reslxf$ will be rwavered h the bng wavefengdh l h i t .
p2g.
-
R.
F
+
F
PROBLEM 1.8 Unlfarm k r d Gas A p&iculmly s b p j e srdatianw solution to the II&reeFock qaatianas for a trwlatioadly invariat system is a SXattrr inan6 of plme wavm. fn a fixlite box with pdodic b a n n d q conrlititiona, the n o m h d wave fumtiom have the f o m
gni
where kBi = and X. is a spinor representing spin, ismpin, color, or any other ~ron-spatidgnmtum number, E the degenerxy wmciatecf with the non-spatial qnantnm numbers is denoted n., the lowest M = momentum states are occupied. In the continoum limit in D dimensions, F(k,) -r (&ID J" ddDkF(k)and one simply integrates over all g lem khan the Femi momentum k f , which is specified
5
xz1
PROBLEMS FOR CHAPTER i
43
Mass Density fg/cc) Fig, 1.1
Equatton of srats of neutron star matter
a) Consides a non-interacting gas, for which (+lTi$) is the exact eaerw. Find the enwgy per pwiele ars a functioa of density, for non-relativhtic and dtr* relativistic .pin $ fernions in three dimensions. b) As a concrete physical =ample, calculate the pressnn (dynes/cmy as a function of mass density (gm/cm5) for a non-relativistic neutron BM. Taking into =count the fact that w a k interwfions allow a p t o n ~d electron to f o m s neutron ,1 f E=, f m, -t- m, -- EFl, 4m, [and a neutdno whiGh =apes) and thus calculiitrct the prwure vemucs mass density for a neuf;raZ mixture of non-relativistic protons and xleatrons and ultra-relativikstic electrons, Amma the validity of &his p < 10"E and compare your resnlts for the ation for 10"E neulron gm md the neutron-praton-e1ectron g= with the complete micrascopic calculation including nuclem forcm shown in. the graph in Fig. 1.1. c) Consider a D-dimensional sygkem, of Fernions with spin degeneracy rt, = 25+ 1 and m- m interaefing with the webody potential u(;ii --5?i) = -a&[?:-Fi)# where a > 0. Calculate the Femi gas energy per particle is ~ ( n )in the tfremodynamic limit. Deternine whether or not the system is stable in 1,2,and 3 spatial dimensions. d) Now,add a repulerive threebody potential v(iji, Q,iiik) = -%)&(c -?k) with 19 > O and calculate ~ ( nin ) t h r dimensions, ~ Wow doer, this campme with the results in (c) when n, = 2? Explaia pbysieally. &pIah why the eMe n, == 4 h relevmt to nuclear matter, that is, a system having equal densifim of neutrons and protons lintertteting wiGh naclew fore= bat na Coulomb forcehl. Find values far a and guch that the minimum h e ( p ) o c c m at the experiment4 values po == 0.18frn"~ a d efpa) = -16ZIIPeV. Skefeh c(pf for
+
9
SECOND QUANTtmTIOPJ AND COHERENT STATES
44
t h w valaers. '17~hgthe raults af Prablem 2.1, show the density regio-rrs for which the system has pwitive premare asld for which it ihl stable a i g ~ s denaity Buctuations. Calculate the compreaaion modulus K = k$ momentum kF comerspondixrg to eqailibdann density*and campme with the experh e n t d vdme K, = tZOOMcV. Row wonM one measwe a in a bu&et of liqGd $He? E m waald a m m e a w e it in a finite nnefeas? PROBLEM lt.9" A model gmblenr whichwa as an a m p b for many topks is, mmy-body theoq h a a n e d h e m i ~ n vstem d of pal~ticfe~ htaacting via &-function forces. In unit. such that = 1, the Hamiltonian is
&
g
a) Show that both =and ig( may be removed from the problem by appmpriata ~ a h ofg the bngth and energy. M a t does this h p l y coneemkg a, pedarbation expansion in g far lstaitea wieh a finite number of pmticla? Far unzorm translaLion-
ally invariant systems, show that f is the only dimensionlesa parameter. Note that high density comapondls to the weak caupfinrg Emit. b) Solve for t b exact5 boand &atee for an att~activeinteractio~aar foflows. Note that the totagly aymmet~efunction of the form free t36hrMkgw eqartCion when the 2'8 coincide sda&ewaw functions and energia for any N b w m n, is the Begenerwy ZS $- 11 for wme dditiond quantum number we will cdl. spin:
spin wave function (spin singlet for N == whwe X denotes ac t o t a y an$. Note the behavior of the si%eand 25 1) a d G b a norm emerw as N --* m. Do you expect there to be any Fermion bound staees for A" > 2S l? Thk Harniltonim w m first solvd exlsct;Xtj by HA,Bethe, (2931)and the wavefunctian is of$en refern& to ars the Betbe ansatz. c) For sn attractive intteractian and N Bomns or N ra, Femions, solve for the ap ate bound state in the Hbree-Pock appmximat ion (gee ProbXem l .B). Note that &heIfaree-Fock equakion
+ +
<
PROBLEMS FOR CHAPTER 2
45.
is t h weB-known cubic SehGdkg;l?;("equatian. with l o c h e d solutions of the form
. Eence, show that:
awe- with the exaet E to l e d k g order in N. For thorn who Note that enjoy chdlengkg htegr&, it is intmmthg ta eailculate the exact anebody density (Ibtributian from the e x x t wave function (2) and &m that it apeee wit& the Haft;r*Fwk d e ~ i t y
to leding arder h N, (The h t e v d is done by P. Calogero and A. Degwpe* (197Sj.) Thew two results whieh awee to leadring order in N are special cma of the genenral resalt provd latm that the mean field ation. gives the leding term a that there k no expwian in a expansion. Since the =&g argument parameter in the Hamiltonian, is the only possible expansion parameter for the b o n d stiree problem. ets &am the W of a vmiiftiond wave d) Note: that, the energy in part (6) centclr of madss faaction wki& is not translationally hv&ant, m that same s p ~ o w nnokion is included, A bettcer approximation k to miaimifae the intrinsic Ham3tonisn
ft
where, in onits such that
& = 1,
tern v a b h e s for the pemaneats or d e t e m h m t s we are Expl& why the udag and ahow that the I-lleree-Fock enerw abbained by nzinhking the inb&ntsic = - (NS p ) thueby eliminating one half of the 0 ( H 2 ) Hamiltonian is hcrepmq* ef %m~latioadlyinvdant solutions. Using the rwalt for a Fermi gas wave-. function obtaind in Problem 8 part (c), show that for kmioncl with spin degene r x y n, = 2 5 1the enwgy per p-icle i~ f&
-
+
5
E
Note that for an attractive g, has a minimum at kmim = with #lmin = Campme thk H m r w Pock e n a w per p&icle with that of' a condew4
.
49
SECOND QUANTtZATtON AND COHERENT STATES
d d a mate (and fhurs non-hterwting) gad3 of H&lpee Pock bound states, each containing 2 5 1 particles (see psrt (d) above). S h m that the ratio is
+
ao that in this appro~malionit?is enag&k%ilyfavorable for the t~nZorm,phase to break up hko cluster^. Using the csiteson in Problem 1.7, show that the unBorm aolntion is stable against long wavelength Buctuations for all kF > f$which inciuda the equilib~unrdensi'ty, (We will see later &ha& is urilstabh with respect to luetaafiona of waveleagth eompzurable to the sise of the Bound state.) The following references treat the 6-funcGion probbm in peater detail* The exact bound sfatm me diacamd by J, NeGuirrt (1965) and the complete S-matrix L calculated by C.N, Yang (M?). Time dependent mean field solutians and perturbative comections to the RastreeFock approximation me given by B. Yoon and J.W. Negele (1917) and &he model. has been sofved w i g fnnct.i;on$t integral teckniqnes by C,R, Nohl (1976). Maay of the salient raults for the mode8 and the h e t that id is the nan-relativbtic limit af the sigrna model are digel~38din Section Y1.A of the review 'by 3.W. Negele (1882).
CHAPTER 2 GENERAL FORMALISM AT FINITE TEMPERATURE
The ultimate objective of the quantum theoq of many-particle systems is to understand experimental!y observable prope&ies of a divgrse range of physical systems, Techniques will eventuafly be present& which are suitable for small finite systems as wet! as extensive macroscopic systems, for obsenrables in isolated systems at zero temperature as well as finite-temperature systems. Once the appropriate theoretical groundwork has Been established, specific experimental observabfes in a variety af systems will be treated fn detail. The "rtent of the present chapter, hswever, Es to present the essence of the generaf theow in the particularly simple and convenient case of systems at finite temperature whlclil may be treated On the grand canonical ensemble, To place the subsequent formalism in proper persppeetlve, quantum statistical mwhanics is reviewed, the obsemables and response functions accessible experimentally are summarized, and the general strategy of systematic apprexlmations is discussed. Eeynman path integrals are presented and funetionat integral techniques are developed for the many-body problem. Perlurbation theory is derived and detailed rules are presented for Feynman diagrams. Finally, generating functions are derived for Green" Functi~nsand irreducible diagrams, and the statjionarjr-phase approximation is applied t o the functional integral,
QUANTUM STATISTICAL MECHANICS In quantum statistical mechanics, the themodynamic properLies of a system in equilibrium are specified by the assumption of the equal occupation of accessible states. The physical problem is Idealized by considering an appropriate enwmblle of identical systems, and the probability of observing a particular state is given by the ratio of the number of qstems in the ensemble in that state to the total number of systems in the ensemble, Three ensembles are commonly used. Tha microcanonleat ensemble describes an isolatd system, which has fixed energy E and fixed particle number, The probability of obsenring a state of energy E' $ E is zero and that of observing a state of energy E' =. E Is constant, q u a l to the Inverse of the total number of states of energy E, The canonical ensemble dexrbes a system of fixed particle number in equll'tbrium with a thermal reservoir with which it may exchange enera at a spmlfied temperature. By enumerating the number of states in which the system and the reemair combined have a f i x d total energy, it follows that the probaMlity oaf obsewfng this ;t;ystem alone with energy E is proportional to e- ft where k is Boltzmann's constant. The average energy of the system is fixed and is, controlled by the temperature T. The grand canonical ensemble describes a system in equitibviurn with a particle resemoir, with which it may %change particles, as well as a thermal resewoir, Enumeration of the states in which the system and resenroir have fixed total energy and pal-tkcle number leads to the result
48
G E N E M L FORMALISM A T FINITE TEMPERATURE
g the system alone with energy E and paelcle number that the probability of o In addition to the werage enerm of the system, the N is proportional to o" average particle number fixed and is controfled by the chernicat potential p. The thermodynamic limit for a physical system of N pavticles containd in a vofume V is defined by hMng the iimlt as N and V &to ts inZinEty such that the ratio remains constant. An essential result of thermodynamics is that the three p= ensembim are quivalent for describing the macroscopic prapeflies of mo& system in the thermodynamic @nit. One Is then f m to select the ensemwe on the basis of formal convenience, and as in the a s e of ctassical statistical mechanics, the grand cansnieal ensemMe oflms the greatest formal simgficitjf, The principal case In whSch the thrw ensmbles yield nonquivatent results arises bte has divergent ffuctuations in one enmmble and is constrain& not .t;o fluctuate in another en~ntble.This situation occurs in physical systems composd of a mixture of phases such as a Ifquid-gas transition or Erase candensation, ff one looks a t a subsystem, the fact that dro@etsof one pham can move in the other phase fs responlble for the dlvergenm of the ffuctuations of the padcfe number in the grand canonicat enwmbiie, whereas these fluctuations are zero by defin3tion in the canonical ensemMe. Given the fact that the probability of obsenring a state of enerm E and particle number N is proportional to e-@(S-@M),where = the thvmal average of an operator k may be expressed as
.
h,
where {I+,)] denotes an orthonormal basis of the Fock space and 10 is the particle n u m k operator, Eq, fi.92), It la convenient to define the partition functlcn, Z,
and the grand canonical poleential, f;L
By mplicit digere~tiationd the Funetion flfp,V, T ) and use of Eqs, (2.1- 2.3) the farnitfar thermodynamic relations nay be establish&:
where P is the pressure, Duheln relation
Ocl is the tnternal enwaI and S is the entropy. The GibbsU=TS--PTr+pN
(2.51
2.5 INTRODUCTION
49
foIlows from the fact that the internal energy U(N,V,S) is an extensive function of extendwe variables and may be comblned wlth Eq. (2.k)to yield
Since the thermodynamk properlies of a system at equiijbrium are spcscifred by 0 and derivatives thereof, one of the immediate tasks in this chapter wilt be to develop methods to calculate the grand potential, Q,The techniques developed for R wilt atso appty directly to thermal averages of operators, Eq. (2.1). allowing the calculation of any equllibrium properties of interest.
PHYSICAL RESPQMSE FtEPJCTtORIS AND GREEN" FUNCTIONS Before embarking on the development of the generat formalism, it is valuable to specify conveniently calculable quantities which characterize the range of ttxperimentally accass",ble obsemables to be addres~din physical systems, In addition to measuring the equiiibrium properlies discussed abave, expen'nrentaflsts learn about physlcal syz;. tems by measuving thdr response to a diversity of external probes. The results of such measurements are conveniently expressed in twms of response functions or Green's functions. The relation of s p ~ i f i cexperimental obsewables to these quantities wfll be derivd in detail in later chapters, and in the present siection we simply present general ~ from experiments. arguments to show how they a u i naturally Consider a system which at initiaf time t# is in an eigenstate Iqa(gi)) of the hamiltonian 8. Subsequently. let it be subject to a time-dependent extunal field
where the field couples to the system through an operator denoted 61 and the states and operators are in the Schralnger representation. A convenkent repremntation of the evolution operator C;07 a system wlth a timedependent hamiltonian is given by a time-ordered wponentiaf , The time-ordered product of a set of' time-dependent creation and annihilation operators, denoted {&), 1s defined
where according to our standard convention 6 is --l or 11 for Ferrmions or Bosons respectively and P is the permutation of (1~2,. n) whkh orders the times chranotoglcaily with the latest time?to the left:
..
and which orders creation operators to the left of annihilation operators (normal order] at equal times, The time-order& exponential is dean& by
50
GENERAL FORMALISM A T FIPSITE TEMPERATURE
*hem e =. and t , == E, -tne. As shown 'rn Problem 2.3, it may be expanded in a Taylor series as fsflows:
Uslng this time-ordered exponential, the ewfution operator may be wrltten
since it satlsfiw the equation of motion
and the boundary condition a(tg,ti)
I
.
The response of a wave function orka;inal!y in dgenstate l$,) at time t g to an Z~finitesintalperturbatran by an external Fteld acting bewcrran time $6 and t is thus given by the funcaonaf derivative
where the operator bfXl(E)in the Heisenberg representation is related to the operator in the Schriidinger representation by
and the state I$(") in the Heisenberg representation is related to the state the Schviiidtinger representat'ron by
l+($))
in
Finally, cansider the expectation value of an operator 9 2 evaluated at time $3 in the state i$(tz)).The response of this expectation value to an infinitesimal pe&urbation in the external field is given by
-09
(2.15) where a B function has been insert& to extend the upper !hit of the time integraf and the lower limit t:g has been extendedi to --W to include all possible variations of U($%). This equation may be weighted by the Boltzmann factor Z-I e - ( B a - p N a ) and 1. Hence, the response of a measurement of (&(tl)) summed over a complete set to a peeurbation mupled t o is specifid by the response function
where the brackets denote the thermal average, Eq. [2.1). The physical response of a system to an external potential is thus characterized by conetation functions of the form ( b ~ " ) ( t a ) b ~ * ) ( t lThis ) ) . result that a transpofi coefacient characterizing the dissipation in a system Is spedfied by a matrix element of the thermodynamic or ground state fluctuations of an operator is o&en called the fluctuation-dissipation thwrem, For example consider measuring the magnetkation of a spin system in the presence of a time and spafially wrying magnetic field. Since the magnetic field couples to the spin through 8(z) . B(z), the operators d1 and 6 2 are spin operators d and the response function is given in terms of the spinspin correlation function (ii"(z2, tl)dfs2s L ~ ) ) Fouder transforming to momentum and frquency then directly speciffes the dynamic magnetic susctlptibiliv ~ ( k , u )Simitady. . an etmtromagnetfc field couples to a system of charged partfeles though the vator potential A, and in a gauge in which O = o,E = Thus, the response elwtrical conductivity, is of the current to a varhtion In the electric field, nthat is the n
9.
-&a.
given by the cunent-current correlation function ( I ( z l t, l ) 3(zz,tz)).This argument is evidently quite general, and a large class of experiments i n which a system is subjeded to a weak controllable aternal probe is characterized by simple correlation functions of operators at dlflerent times, In addition to these experiments involving an external potential, there Is an Imporlant class of inclusfve scattering tsxperiments used to study many-body systems w hi& may be characterizd simitady by correlation functions, Consider an aternat particle which interacts with the constifuents of a system through a weak potential v. Let a parlicle with initial momentum k scatter from the system In inftiial state and final partlcle transferdng energy w and momentum if teadlng to a final state
S2
GENERAL FORMALISM A T FINITE TEMPERATURE
momentum S -
a. The matrix element for this transition is
where ir(g) = G(--9) is the buder transform of the potential v(lsl) and the Fourier transform of the density operator is defined by
IF the interaction is weak, the inclusive cross section may be calculatd in Born a p proximation by summing over the unobsewed states ftbg) of the iFystem having mergy
Efl=Ea+u
VVritiag the C-function in terms of a time integral and u s i ~ g completeness, this may be recast in the follawing form
Summing over a complete set of states $@,) with thermal weighting factors yields the desired result that the inclusive crass m t t o n is specifid by the comefaf;tt~lt function (fitH, (z, t ) ~ f(g,~0 )) )* The inelusive cross section or structure function ~ ( qw,) has been measured for a variety of systems of physlcal interest. Striking examples which wHI be dis~ussctdi'in Chapter 5 are liquid MeS and Hea, for which the combination Bf neutmn scattering and X-ray scatterjag provide high-precision results over a wide dynamical range. fram Eq. (2.20). the integral &@(g, W ) yidds the buder transform of the two-body cornlation function (@(S)b(j)) which has now been measurd to impressively high accuracy for liquid He. Equation (2.20) apalies quafly well to a finite qstem at zero temperature in its ground state, and mtenslve high-prttcision data are now available from Inclusive electron scattering from finite nuclei. From these considerations, we are led ta the concfusion that a wide range of observables of direct experimental interest may be express-cscl in terms af the thermat average of products of operator at different times, This motivates the definition of Green's functions, whfch an one hand are the thermal averages of timeordered prsducts of operators that are the most convenie~tto calculate irt pert:urbation tfimry and on
the other hand can be related by suitable analytic continuation to the quantities arising from experimental obsemables. The n-body real-time G r ~ n "function Os defined by
where atB)($)and a?lt(t) denote annihilation and creation operators in the Heisenberg representation. Eq. (2.14). T denotes a time-ordered product. and the brackets denote a thermal average. Aside from @-functionsrestricting relative times arjdng from the time-ordered product, all t he response functbns discussed above may be express& in terms of two-pa&icle Green" functions with suitable choice of the time arguments. We will demonstrate in Chapter 5 that when g is Faurier transformed from time to fritquency o,the e @ ~oft the 8 function is merely to disHact! poles infinitesimally above or below the real axis and that physical obsentables may be obtahnetd staightforwarrlty from Green's functions. When one writes a reat-time Green's function explicitly In terms of the thermal weighting factw c - @ ( " - @ ~ ) and Heirenberg operators ~ ' ~ ' o , e - ' ~ ',the dmuttaneous appearance of real and irnalfZnay times multiplying H is formally inconvenient, tt Is therefore advantageous to define thermal, or imaginary-time, Green" ffunctfons as follows:
) abH)(r)are the imaginary-time Heisenberg representation of a: whve o h ~ ) + ( rand and a,
and T now orders operators in imaginary time. Note that ak(r) and a,(r) are not Harmitian adjoints, In Chapter 5 we wflC also show how real-time Grmn" functions can be obtain& easily from imall;inal.)r-time Grwn" functions by analytic continuaition. The essential conclusion of this swtion is that the phydcal obsewabtes of interest may be obtained by analytic continuation of Imaginary-tfme Green" functions. VVdl wilt subwuently show in the remainder of this chapter how thermal averages of timeordered products of Heisenberg operators in imag-inary time emerge naturally from path integrals and how La evaluate them in perturbation theory,
APPROXIMATION STRATEGIES We will prmntly devdop a number of techniques which formally appear to provide a systematic sRquene of successive approximations for many-particle systems. To maintain a reallstk perspective, hawever, we should consider a t the outset what level
54
GENERAL FORMALISM A T FIN1TE TEMPERATURE
of mathematical rigor ane can actually expwt and how fn practice one may select and assess an appwimation scheme. The most familiar systematic approximation scheme from elementary quantum mechanfcs is perturbation theory, lt Is based an the bellef that the behavior of a physkcal system Ts continuous "t same "smaltl" parameter describing the digerence beween a sotvable problem and the actual system. It is crucial to note, hwever, that in general perlurbatian theary yields an asymptotic rather than convergent series. Under appropriate circomstances, it may yield a useful physical approximation, but It cannot be applld systematicalfy to abritraw precision* The sauent features of asymptotic cirxpansions for our prewnt purpostr-s may be itlustrat& by the simple integral
conesponding to the classical partition function quadratic plus qua&'rc potential
Z = l d z ~ - ~ (for @ )a particle in a
We will sub*quently st3e that this intarat is highly sugp;estiw of the path integral desuibing the quantum mechanics of a particle In the s m e potential, E'lepending on the signs of the quadratic and quafiic terms, the potential has three regimes of qualitatively digereat behavior sketched in Fig, 2.1 Physically, the clasdcal or quantum behavior in this potential changes completely when 9 changes sign, since a pafliticle is localized in (a) and not in (h). Thus. we expect that the theory is nonanafytfc at g = Q and that an =pansion in powers of g has zero radius of convergence. This physical behavior is reflected in the integral Z(g) which diverges far g a O and 75 thus non-analytic at g = Q. The perturbation series for Z(g) is easily obtained by expanding c - f s 4 , with the result
where the asymptotic behavlor in the tast tine is obtained using Stirling's formula n! = atik"+*e-rr. Since Z , grows like n!, the sedes diverges as expected from the non-analyticfty at g = 0, Later on when we think In terms of dhagrams, we will understand the growth of Z,,in terms of the proliferation of the number of diagrams with increasing order and the factor (h-l)!!'In the present example which m w h e f m s the factor will be recognized as the number of diagrams generated in performing the Gaussian Integral of z4"using Wick's theorem. The crucial point for our present disicussion is the fact that, under appropriate circumstances, a finite number of terms of an asymptot"te series may give an excellent approximation. The residual error after n terms Is bounded by the (n 1)""tem
+
so the approximation continues to improve as long as gmZmdecreases. Figure 2.2 shows R, and p[Z,I as a function of n for several values of the coupling constant g. 7's assess the precision, one should compare the m"rn5murn error with the total shift due to g, /Z(g)- Z(Ofl = &. As g decreases, the minimum error decreases and occurs at larger values of n so that at g == .01, for example, one attains the extmmely htgh precision of &&l&= 1.1 x 10-'@ after 25 terms. From the asympt&ic expression for gnZ,. Eq. (2.266). we note that the minimum occurs for n $, so
gm~,I
-
.
- t This exponential dependence on that the minimum error is the inverse coupling constant is ergacteristic of perturbation theory and indicates the exceedingly high precision obtainable from a divergent theoty for weak enough coupling. Quantum efectrodynamics is a good example of a theory in which such weak coupling is realEzt3d. As in our physical argument far the quarlic potential, it is clear physically would change that electrodynamics is non-analytic at a = O since negative a = the signs of all electromagnetic forces. Nevertheless. is so small that perturbation theory is adequate for all csneeivabte purposes. The real problem comes at large coupling constant where the series begins to diverge in very low order. At g == 0.1,for aamptie, one must stop at the third term
$-
56
GENERAL FORMALISM AT' FINITE TEMPERATURE
Awrn~totlcaxganslon of Z(g), For fire values of the coupllng Zf, ts ~ t ~ t t eatd each Integer n and the residual error & ts plottod at each half Integer n
Fig. 2,2
+ 4.
after oniy attainfng an accuracy of 25%. If one wishes t o do better, one must either =tract the non-anatytk part of Zfg) using the Barel summation dis-cussed in Section 7.5 or find an alternative =pansion. For other regimes af the potentfitis sketched In Fig. 2.1, other expansions are more appropriate than (2.26). For large positlvt! g. it Ss m m reasonable t o make a strang coupling expansion by starting with the solution ts the qurartlc patentEa1 and treating the quadratic potential as the perturbation, For case (c), the natural approach is the stationaw phase approximation which will be diseusmd at length in Swtion 2.5, In this
= *eq-fs' case, the partition function is written and the expantsat Is expanded around its twa maxlma y m i n h m mf the potential.
=
.*e
9
=tar% earresponding t o the
2.2 FUNCTfQNAC INTEGRAL FORMULATION
57
The general features illustrated by the quadratic ptus quartic potential are peainent to the richer and more complicated case of the many-body problem. We wifl find that di@erent approximations wlll address diffemnt aspmts of the physics, Perturbation theory "i the two-body interaction, while at best an asymptotic =pansion, will sewe to or"gnize aur thinking and etucMate much of the g e n ~ astructure l of the problem, Various resummations wilt focus on digerent parts of tke physics such as short-range ar long-range corretations. The stationary pham appraximation will address stlll other aspects of the problem such as large amplitude coliective motion and tunnelling, In all of these expandons, there wilt be some formal apansbn parameter which prov'ides no real mathematical control on the problem. The most we can ask in pradice, as in the =ample in Fig. 2.2, is that successive terms in the series of approximations decrease.
2.2 FUMCTXQNAL thfTEGRAt FORMULATION Functional htegrals provide a powerful tool for the study of many-partidesystems, The partition function is repre~ntedby an integral over field conflguratlans whlch provides both a physically intuitive descdption of the system and a useful staut;ingpafnt for approdmations. Approximations w hleh arise naturatly from functional integrals;include perturbation expansions, loop expansions araund stationary solutions, approximations in terms of solftons or "Istantone, and stochsstic approximations, Before proceedkg to the general ease of many-gartkcle systems, it Ss instructive to illustrate the essential idea with hynmtln path Integrals, The essence of the path integraf was Introduced in a germinal paper by P,A,M, Blrac (1933) and developed exlensively by R P . kynman (1948,1949,1950). All these historic papers are published in the reprint volume edited by J, Schwinger0953).
THE FEVNMAN PATH IhlTEGRAt For physical claritb we will first Introduce the Feynman path integral in real time. Subsequently, in order to reprewnt the partition func~on,we shalt pedorm an anatytlc continuation to f maginary time where this path integral Is closely refattad to the Wfener integral and Is mathematlcalty well-defined, Consider a matrix element of the evolution aperator For a parEicIe governed by the 6) Hamiltonian B[#,
Whweas the matrix elements of the evalution operator cannot be evaluated exactly for finftc! time internals, far infin"resima1 time intervals they may be calculated to any desircsd degree of accuracy, Thus, the bask idea of the kynman path integraf is to break a Rnite time internal Znto infinitesimal steps, evaluate the evolution operator for each step, and chaln the matrix elements together to obtain the result for the finite internal. Let the time fntemal It E i be divlded into M equal steps sf size c
-
58
GENERAL, FQRMALISM A T FINITE TEMPERATURE
with intermdiate tf mes denoted
With this notation,
to E ti and
t~
It/
and it will be conve~Sentto use the same convention for initial and finat coardinates zo
= rt7;
and
By insefi'lng the closure relation Eq, (1.23) evolution operator may be written:
ZM
M
~f
g
- 1 times,
(2.29d)
the matrix element of the
The key step is to find an appropt.iate approximation for the matrix element of the infinitesimal evofutfon operator, which may be wr"lten
For our purposes In obtaining a, practical functional Integral, we: desire an approximation to the infinitesimal evolution operator which not only reproduces the exact evolution of a wave functlsn in the limit s -4 0, but also yields acceptable resuits when acting on position and momentum eigenstates !B) and Ip) as in Eq. f2.31). We obtain such an agprodmatbon, which generalizes diirectty to the subquent treatment of coherent state functional inkgrals, by considering a form of normal-orderd apanentiat Far operators tlxpressd in terms of and 9,we wilt define an operator to be in normal form when all the $3 appear to the Idt of all the $3,and the result of reordering an operator O ($,g) into normal form will be denoted :Q(p^,&): , For example, the Hamittonian for a single particle In a potential
.
is fn normat form and
2.2 FUNCTIONAL tNTEGRAL FORMULATION
59
The Hamiltonian for a particle in a magnetic field described by a vector potential A(%) may be rewritten in normal form as faflows
For any H[$,&) in normal form.
and For the spwlaf case Eq, (2.321,the leading correction is
Thus, if the infinitesimal evolution operator is approximated by the normat-order& evolution operator, the error is of order 6% times an apwator which may be expreswd in terms of multiple commutators of the operators comprisSrrg the Mamiltonan. When acting upon a normalizable. differentiable wave function $(z), the error term is rZ times a finite number, so that in the limit e: --, 0, we are assured that yhetds the correct evolution af the wave function, Fuehermore, in contrast to other approximations which are valid tie first order in E , the normal-ordered evolution operatar may be used in the integral in Eq. (2.31). :e"fwl@5'f:
(zf-4:cif H(PvB):
=
d5pm
[p,) (p, :e-'tx(@*') : lz,-,)
For the cast! of a particle in a potential, Eq, (2.321, this integral over p is a Gaussian integral yielding
At
this
point,
approximating the infinitesimal evolution operator
by
ar some other apression valid :e : instead of l - it to order e: may well appear artificial. The essential issue is to obtain an agproximatisn
60
GENERAL FORMALISM A T FINITE TEMPERATURE
Fig, 2.3
A typlcrab traJwtory contrlbuttng to a path Integral.
yielding convergent momentum integrals both in the p r e ~ n case t of reaf time and in subsequent wpressisns continued to fma#nary time, Whereas individual terms in the Taylor series wpanston yield powem ot p which diwrge, since 3 is bounded from below. both e - * k and ewi*& are bounded. Mathematically p r e c i ~derivations of sucwsdw levels of generality are provided by Slmon (1979). Trotter (1959), and Kato (1978). Althaugh it is crucial to use a form in which 9 has been exponentiated, there still remains =me arbitrarinesr~in the approximarisn Eiq. (2.37). As shown in Problem 2.4, one may nplace V ( S , - ~ )by V(%,) or i(V(z,-l) V ( z.)) and in applications such as in Chapter 8 where e remains fin'tte, thls fuedarn may be clxpioited to improve the approxina~on, The problems arhiag when the WamIItsnian contains terms in which $ and % are combined are ctxR"lt& 'in Problem 2.5 for a particle in a magnetic field. The ultimate justification for any exprmsion such as Eq. (2.38) used to appdmate
+
Is.-g) is that it reprodurns the correct evolution of the wave function, and this may be verified straightforwardly (see Problems 2.4 and 2.5). Using Eg. (2.38) and the notation of Eq. (2.29). the matrix element of the evolution operator Eq, (2.30) may be written
.
U(j~ft~,siiti)~ M+oc
(2.39) The set of poTnts (S,, 21,. ,zM) ddnes a t r 4 - w as sk&ched in Fig. 2.3. For notational convenience, fn the limit M -+ cm we will often denote this trajectory by z(t) with starting point S(@) =: zi and endpoint %(gt) =" z t , but it Is crucial to note that tbis natation does not imply continuity or diferentiability. Rather, the trajectow should always be thought of as a set of M points z(tk)Index& by the discrete times
.
2 2 FUNCTIONAL lPaTECRAL FORMULATION
v
61
g.
In the same spirit, it is convenient to represent by the symbol Again. no differentiability is implied and the precise definition of is given by the finitedigerenee t?xpression, With thls notation, the Riemann sums in the exponent may be indicated symboiicalfy the
%
and
The feynman path integrat, which is deZind as the limit of fq. (2.39) as M is denotd
-
U f t f t l , sit&)
D [ ~ ( t ) f6'e
.-,act.
P [~(t)]e'"'""" (z*al)
where
fz,az) repreents a sum over all trajectories staelng at paslition si at time ti and ending at position at time t f , the action Slz(t)j is
and the Lagrangian L[z(t))Is
The matrix element of the evolution operator between states ri) and Ist) is thus the sum over afl trajectories beginning at sg at time ti and ending at 31;/ at time t j of the exponential times the action along the trajectory. Several remarks concerning the Feynman path integral. Eg. (2,411 are prmane at t h h point. Because the path integral is an =act reprexntation of the evolution operator, it may be used as the starting point for the formulation of quantum mechanics (see Feynman and Mfbbs (1965)),"fhe superposition principle, which may be written at any time t in the form
62
GENERAL FOWhrlALlSNI A T FINITE TEMPERATURE
is apreswd in terms of path 'fntegrars as
and quantum mc?chanical intederence arises directly from the sums over trajectories. A natural approximation to the path integraf, Eq, (2.41) In the limit as h -+ O is the stationary-phase approximation. As will be shown in detail In S ~ t i o n(2.51, the dominant contributfon to the transition amplitude in this lm"r tames from trajectories surrounding the classical trajectory joining S$ to sg Finally, since the measure in Eq. [2.$2) is still ill-de#ned when E. goes to zero, it is usctful to note that the functional integral may be normallzlld by salutions of an analiytIcalty solvable reference problem, Folr wamplie, one may require that when the patential V is set to zero, the transition ampfltude is
.
The functional integrai in Eq, (2.41) is cat!& the Lagrangian form and requires that the HamittonOan have quadratic momentum dependence as in Eq, (2.32). The Wamiltonian form af the functional integral is obtained by substituting Eq. (2.37) in Eq, (2.30) without pedorming the p integration, in which case the matdx etement of the evolution operator becomes
The trajectories z(t) obey the same boundary conditions as in the Lagrangian fDrm and the trajectories p ( t ) have no boundary conditians. The Wamiitonian f o m of this functionat integral is cr prz"ors"more general than the Lagrangfan form, but requires care In the ordering of the non-commuting operators % and $3 when U contains mixed terms In 2 and $9 (see Problem 2.5). At this point, it is u s h l to note that path integrals automatically represent timebl)and &@, h) k operators acting at times tl and t g order4 products, Let OlfP, with $1 t2 and let t , denote the discrete time in Eq. (2.30) closest to 5% and ,b
>
2 2 FUtdCTlOMAL INTEGRAL FORMULATION
63
denote the dfxrete time closest to $3. Then
Thus, although there is no expticlt indication in the notation that oprators have been time-ordered, in order for the operators Ol(B, tl), Q%(&,t z ) , and &[g) t o be replaced by the c-numkrs &(s(gl)), &(z(ta)), and Hip@),z(l)], it is implicit in the construcLion of the functional integral that each operatw had to act on the complete set of states introduced at the corresponding discrete time. Operators depending upon the momentum are treat4 in the same way by letting them act an the complete set of momentum states introduced at the corresponding time. The fact that functional integrals necessarily yield time-ordered products is the reason for the asse~ionin Section 2.1 that tfme-ordered products are the quantities which arise naturally in the formalism and that physical response functions should ultimately be evaluated in terms of them, From the definition of the time-ordered exponential, Eq, (2.9), and the path Integral, it is evuent that both quantities deal with the non-eommutatfv~@of operatars in quantum mechanics in the same way. tn Both cases the: time internal is divided Into suRicienHy small subinlewalr that the commutator terms V] become negligible. There is no reason why the continuous parameter must necessarily be the phydcal time, and we will see that it is also uM?fUI to use tsmprature [or imagenaw time) as the formal par?meter for developing path Integrals.
[g,
IMAGINARY-TIME PATH INTEGRAL AN I) THE PARTITION FUNCTION The partition function for a single particle may Be written
64
GENERAL FORMALISM A T FINITE TEMPERATURE
and may be thought of as a sum over diagonal matrix elements of the imaginary time evolution operator U(zlrt, 3~i7;f=rr (%S11 evaluataid for the intewai 9 --q = FA, Note that for the one-particle problem, we work in the canonical ensemble and there i s no chemical potential, Given the observation that the essence af a path integral or time-arderd expnentiaf i s the subdivision of the I Into suaidentfy smalt intervals that commutators oT the quantum operators appearing in H may be nelylected, aII the steps In the derivation of the real-time path integral may be repeat4 for the ease of Imalginav time. For the Hamiltonian, Eq, (2.321,we obtain
where c = &(r$ -rc). Thus, the imaginary-time path integral is a sum over trajectories starting at (aq, Q) and ending a t (z/,q )of the exponentfat of a modified action in which a change fn sign of the Wnetic term yields the Wamiltonlan instead of the Lagrangian, An alternative derivation, which shows explicitly how the Lagrang"tan in the: realtime case 7s transformed into the Hamllton'ian in the imaginary t h e case, is to prtvform an analytic csntlnuatlon of Eq. (2.41) to Imaginary t h e . This continuation, known as a Wick ratation betcaum It may be view& ;as a rotation of the integration contour in the ecmplrtx d-plane, is eflectd by the varhable transformation
Thus*
2.2 FtlPSCTIONAL INTEGRAL FORMULATION
65
and the action, which is called the Euclidean action, bttcomes
The kinetic energy thus changes signs bt3c;ause each time derivative acquires a factor d i. The same dgn rewrsal arises in the dassical equations of motion in imaginary time, and the interpretation of a particle moving in an inverted potential mifl subsequently provide a picturesque way to visualize the stationary solutions to path f ntagrals in classicatly forbidden regimes* For the imaginary-time path integral. the measure appearing in Eq. (2.52) is equivalent to the WSenar measure defined in the study of continuous stochastic processes (Wiener. (1924.1932)) and the functional integral can be given a ilgorous mathematical definition. This path integral wSIl provide the Foundation af the stochastic method presented in Chapter 8, and the nature of the trajectories which contribute to it wilt be studid more thoroughly in that context. Using Eqs. (2.50) [2.52), the paeition function may be expressed
-
The pafiit'lon function Ss thus a sum over at periodic trajectories of period &A and the shorthand notation fn the last line emphasizes the fact that the i n t ~ r aover i S, at the endpoint of the interval Is equiwfent to the integral over each of the internal zk*s within the intewal, k r notational clar"ty, except when we are sprtcifically conceraed with the classical limit in which A -* 0,Et will be convenient to use units in which
P i = l. FSnaIly, the Fqnman path integral in real ar imaginary time may be stra@htforwardIy wtended to many-particle systems, For warnpie, using the symmetrized or antisymmetrized states defined in Eg, (1.38) where we used = Ifrf for B~sonsor krmhns, the partitbon function for an N-particle system may be written
As in the case: of a simple variable, the time intewal may be divided into infinitedmal steps. Hwever, new we have the additional choice of inserling at each step the
66
GENERAL FORMALISM A T FINITE TEMPERATURE
closure relation Eq. (1.30) with produd states or Eq. (1.40) using symmetrized or antfsymmetdzed states. Efthw choice yields =act evolution and the symmetry or antisymmetry of the final states in the tram sufices to impose the proper statlsticr. The use clrf product states yields the slmCtfest farmer result completttliy analogous to Eq. (2.52). and fsr the case of a Wamlttonian of the form
the partition function may be written
In the case of stochastic evolution of path integrals for hmions, the alternative choice of using antisymmetl.itf?:dstates at intermdiate steps wSII prave advantagwus In certain appflcations, as dkussect in Chapter 8.
COHERENT STATE FUktCTCOlsJAL INTEGRAL Eor a general many-particle Namiltontan rrxpresmd in smond quantixd form, a hnctional integral representation for the many-body evaiution operator may be obtained using the coherent states 14). Eq. (1.118) and (1.160) instead of the position and momentum dgenstates used for the hynman path Integral. Rwatl that the relations for Fermions and Efosons have the Idenucal form tabufated in Table 2.f at the end of Chapter 1 where the integration variables are understood to be complex variables for Bosons and Grassman vairlabtles for Fermi~ns.W wftf evafuate the matrix element of the evolution operator between an initial coherent state having components g,&, and a final state (4 ts 4:,, As before, the integral [ti,tf] is broken into M times s&ps , a closure relatian in the notation of Table 1.1
.
Is 1nwrte-d at the kth time step, and we use the natation at the end points:
For second quantized operators. the appropriate form of normal ordering is that defined in Section (1.4) with all creation operatws to the left of annihilation operators. and we will assume that X(oi,aa) is written in normal form. As in the path integral ease.
2.2 FUNCTIONAL lNTEGRAL FORMULATION
69
where the term of order e2 ES e2 times an operator which is finite when acting on a normalized. digerential wave function +(#:). Thus. using Eqs. (1.137) or (1.177) to evaluate coherent state matrix efements of the normal ordered exponentials, the matrix element of the evolution operator may be written
Note that in the case cif Fermions, since there is no metric in the Grassman algebra, all are finite. For Bosons, the argument is analogous t o the integrals indicatd in Eq, (2.a) that for the path integral case. In real time c - p H is oscillatory and the factor c @ ~ p ~ b * b arising from the measure produces convergence. In imaginary time we again rely upon the physical fact that the Hamiltsnian i s bounded from below, which implies that Es also bounded from below, Hence H(#:,~F#ask-I) is bounded and the Gaussian factor from the measure again ensures convergence, As in the case of the path integral, it is convenient introduce a trajectory O,(t) to represent the set (#a,r+o,a.. and to introduce the notation
and
H(#:.k;
do.k-l)
= H(#:(t), b.($))
in which cafe the exponent in Eq. (2.60) may be rewritten symbolically
(2-sia)
68
CENEML FORMALISM A T FINfTE TEMPERATURE
where the Schriidinger Lagrangian operator is ih& - H. As in the Feynman path integral, the trajrxt-ary and derivative notation fs purely syrrrbotic, and for any case in which ambiguity may arise, the correct physical quantity is catcuiated by perForrning the intqral: over the discrete action In Eq. (2.61~)and then talirng the lirnlt M -+ W. With this notat-Son,
where
Note in the discrete eixpresslon, that the boundary condlttans specified +a,o and that there were no variables #:,, or #,,M. and that all the internal conjugate variables and +a,r for k = 1, M - 1 are integrated. In the trajectory notation *and #:(t) and +&(g) are associated with variable displaced by one time step. are specified by the boundary condttlons 4a,k-1.respectively, so that # i ( t f ) and but (Ba(tI)and #z(tt) wrrespond to internal variables af intaration not subject to boundaq conditions. The boundary term appearing in the uponent. Eq. (2.61~)represents a term left aver at the end of the path from our grouping of twms in defining the derivative Eq, (2.61a). Mad we chosen the altmadve convention $(-#:,L+l = (-$#:(t)) +(g) .the remaining boundary term would have been 4;Qbo. Both results correspond to the same fundamental discrete exprestiion and are thus equivalent. If a symmetric formal axpression is desired, one may use the average of the two, One significant dieerence between the coherent state functionaf integral, Eq. (2,621 and the Feynman path integral is the dependence upon h. In the Feynman case. appears as a constant multiplyin$ the entire expanent, so that the stationary-phasts apransion immediately yields the classical Ilrnit, In the present case, the adion contains h within the Lagrangian as well as a multipilicatlve factor, s;o that the stationary-phase method yields a result quite distinct from the classicat limit.
d;,,,
+
THE PARTITCQM FUNCTION FOR MANY-PARTICLE SYSTEMS As in Q. (2.57). the partition function for a many-pa@iiclesystem may be expressed as the trace of an imaginary-time evolution operator. Using Eqs. (1.128) or (1.173) for the trace with Boson or Fermion coherent states and units such that h = 1. the partition function may be written
1.2 FUNCT10NAL lklTEGRAL FORMULATION
€59
When the antinuation a f Eq. f2.50) t o imaginary time is substituted in this =pression, the trace: imposes the periodic or antipertodic boundary conditions
The equivalence oF the interior and extedor coherent state Intevals is emphasized and the resulting partittan function is by relabelling 4, r
Using the trajectory notation, this may be rewrRteo
with the usual understanding that the derivatives and integrals are defined in terms of the discnte expressloo, Eq. (2,621. Note that the int6igration is over complex variables satisfying pedodic boundary conditions for &owns and Grassman variables satisfying antiperiodic boundag conditians for brmions. In a formal sense, the problem has now been rduced to quadratarts and we only need to develop techniques to evaluate the intttgral in Eq, (2.62). Our aterall approach will be to g r o q the one-body part 6f W (#*, +) tagaher with the other quadratic terms in the exponent and to develop a perturbation series in which the exponential of the many-body part of I;l(P,#) is expanded in a Taylor wries. This will give rise to a series of integrals of the products of a Gaussian t h e s polynomiafs which may be ewaluatctd straightforwardly using the tshniques developed in the next swtion. The thermal Green" function, defined In Eq[2,22), has a simple farm expressed in terms of a coherent state path int;egrai. It is usdul to give the creation and annihltat'ron operators a formal 7 label.{ai (r),a, ( r ) ) ,denoting the time slice r upon which they are defined, This purely formal 7 label Ss introduced to allow the time-ordering operator to approprktety Interlace operators with no explicit .r-dependence, and when the evoluthn operator is repre~ntedby a functional integral, the operators {aL(r),o,(r)] on the time slice P wfll be replaced by the coherent state variables ($z(r),$,(r)). To fslcilitate nanipulatian of the time-ordered product, ft is convenient to write the thermal Greenes
76
GENERAL FORMALISM A T FINITE TEMPERATURE
function, Eq, (2.22). as follws:
where the permutation P arranges the times In chronotogfcal order and Bai Is an annihiladon operator for zi n and a creation operator for i n. Using the definition of the Weisenbetg operator, Eg(2.23). and lthe fact that a functional intyrai corresponds to a time-order4 product,Eq[2.49]. the Green" ffunction may be written as follows:
<
As preparation for the generat case of a many-body Hamittonian, it Is u d u f to evaluate the pavtition function for a system of non-interact'ing particles described by a one-body Hamlltonian, Far convenience, we choose a basis in which HQ is biapnat:
The discmte expression for the partition function, Eq. (2.65), may be written
2.2 FUNCTtQNAt INTEGRAL FQRMOLATION
71
where, with the convention that the time index 'increases with increasing row and column index
and where
The determinant of
may be evaluated by expanding by minors along the first row
Substitution in Eq. (2.59) yields the familiar result for non-interacting paeicles
Note that $3 arising from the definition of the functional integral corresponds t o a specific discrete approximation to the continuum operator $ c, - p with periodic or antiperiodic boundary conditions. As seen in Problem 2.6,other diwete approximations to c, p give inequivalent results emphasidng the fact that Eq. (2.65) rather than the continuum shorthand. Eq. (2.66) is the defining expression.
+
&+ -
cler.
Finally, we evaluate the single-pat.rlctt? Green" function for non-interacting part;go. Let r, comespond to the time g$ and r, correspond to the time r$ for
integws q and r. Using Eq(2.67) far the non-intencling Waniltonian,&,we
obtain:
The inverre of S in Eq. (2.70a). with a defined by Eg. (2.70b) is
Hence, for q 2 r
- ,-(a.-@)(%-")
W ",.
(1 + p,) (2.75a)
wkwe n, is the famitlar Born or Fermloa occupatian prcrbability
2.2 FUNCTIONAL INTEGRAL FORMULATION
'1'3
Similarly, for q 5 r
The two results specify the singte-panlicle Green's function when 7, 2 7, and when T~ 7, respeetlvely, so there only remains the case in which creation and annihilation operators act at equal physical. times, as occurs for ctxampfe whenever a swondquantize-d operator is eivatuated at a specific time, Using the fact that the time-ordered product is defined to be equal to a normal-ordered product at equal t h e , the equal-time propagator may be obtained two eguivalent ways. If the operator oLoa is surrounded by evolution operators, subdivision of the intewal in the usual way yields
>
e-eX
l h ) ( 6 rlaLa@e-"
[dk-z)(+k-~f
0
+
(I
#*a,k4#&--Xe-cH(+;f+r-g) ($.??a) Thus, like the operatars In H,the creation operator Ss evaluated one time step later than the annihilation operator, and the appropriate expression far the Green" function corresponding t a equal times Is Eq, (2.76) rather than Eq. (2.75). Alternatively, the time-ordered product may be written T[a8(?)a& ( r ) ] = $a! (r)aB( T ) = aa(r)aL(7) li,@ in which case the evalution operator Ss expressed
and 4, and 4; are evaluated at equal timer. Thur.(Ta.(r)a~(r)) = S$ - l = cn, as before, Combining these results, the single-particle Green" function may be: wr'rtten
= S a a ' f i ( ~ - g-Q)
f 2,788) where the S~fi~itesimat sewes as a reminder that the second term contributes at equal times, (A convenient mnemonic for the q is the fact that the time r' assodated with the creation operator is always shifted one time step tater). Although we have derived Q carefully as the inverse of the discrete rtxpression appearing in the exponent of the partition function, it may be obtained directly as the inverse af (BT r , - p) by solving the difirential equation
+
74
GENERAL
FORMALISM A T FINITE TEMPERATURE
subject to the b o u n d a ~condition
The only ambiguity In the continuum derivation is the result at equal times. Whereas the dfserete =pression definlng the Functional Integral produces the physical result at cr-quat time, ather discrete approximations to the continuum apression may be i n c o r r ~ t (see Problem 2.6).
2.3 PERTURBATION THEORY
In this wction, we consider the catw of a Hamittonian which has been decompos-t;d into the sum of a onebody operatar Ha and the residual Wamiltonian V, which in general may eontaln a ane-body interaction as well as many-body Enterztctions, and devertop a systematic perturbation expansion in pawers of V. The basis wilt be chosen to diagonalize H. = C e,a:a. and we will write the normal-ordered many-body part
v ( ( I L o ~...
..
Q
.) The starting point is to express the Grand partition function as in terms of thermal averages defincFlc) with respect to HQ. To establish notation which will be used subsquently, we wilt define equivalent expressions "I terms of operators and functional integrals, As in Eq, (2.69), creation and annihilation operators will be given a formal r tabet denoting the time sllce upon which they are defined, These opemtorr {a&( r ) ,a, ( r ) are replaced by the coherent state variables ($: ( r ),(l. ( r )1 on the conesponding time dice in the functional integral and should not be eanfusd with the Heisenberg operators {ahHbt(r), a&X))defined in Eq. (2.23). The operator form of the parlition function is written
where the thermal average of an operator F is written
tt is crudat to note that all the operators In F are subjwt to the time-orb~ngoperator included in the definition of (F)o. Equivalently, in terms of functional integrals. the partition function may be written
2.3 PERTURBATION THEORY
75
where the thermal average is defined
Mote that the time-ordering which was explicit for operators is impticit because the functianat integral always represents the-order& products. The partitfan function of the non-interacting swem, Zcr,appearing In Eqs. (2.58) and (2*81) nay be written
Bttcause of the equivalena of Eqs. (2.80) and (2.81). we will henceforth pass freely of operators and of complex ar Grassman variabfes. between thermal averages ( Note that anaiagous expresdon will alsa be introduced later for thermal averages ( ) defined with respect to the fuif Wamittonian by replacing by M in Eqs, (2.80b) and (2.81b) , The perturbation expansion is obtained by expanding Eq. (2.8ia) in a power series
We wfll praceed by deriving a. form of Wick" theorem t a evaluate the thermal averages of the products of +* and Jt which arise in Eq, (2.83) and then develop a systematic set OF rules for constructing Feynman diagrams. In the form we will use it, Wickt$ tthmrem corresponds ta the following identity for the intwrat of a product of a poiynomlaf with a Eausshan
JP(+*+)@
#;Mij@j
C
= P s ~ M ~ *: lKi;t,irl ,~~
(2.84) whcre, as usual (+*,g) denote conptex or Grassman wviables. P(+*$) is the appropriate measure, and we have simpllfid the notation by letting j denote the state and
Tb
GENERAL FORMALISM AT FfFbtTE TEMPERATURE
time labels, We wltl first prove this identity and then rejate it to evaluation of Green's functions, thermal averages, and the traditional statement of Wick's thwrem. The identity Eq, (2.84) is a ~neralizationof the result Eq, (2.93) and may bederivd Fn the same way usO~gthe generating function
Differentlatlon of the first line of Eq. (2.85) with respect t o the sources J and yields
J"
(2.80 Note that in deriving this result, we used the fact that all the terms in the exganent are even in the J B sand $'S and thus commute with $, 4*, J and J* and the fact that an odd number of "rterchanges is requird for each digerentiatisn with respeet t o J . Differentiation of the second line of Eq. (2.85) yields
Equating the two expressions Eqs. (2.86) and (2.87) proves Eq. (2.84). We now apply this Identity t o the case of physical "Interest by defining MiI' t o be . p). Eq. (2.69). replacing +j by $a,k,where the discrete matrix representing (at H a denotes the basis states in the diagonal re-presntation and k dencltes the time point on a mesh of M points with Ar = As shown in the preceding section. M ;' is then the single particle Green" function, Eq. (2.18):
+ -
6.
The identity (2.84) then states that the n-particle Green's function for a non-interacting system is the sum of all permutations ol the products of one-particle Green" ffunctlons
2,s PERTURBATION THEORY
77
We may establish contact with the traditional statement of Wick's thwrem by defining wntractians af time-dependent operatom, Let 5, ( 7 ) denote any creation operator o i ( r ) w annihihtion operator .,(F) and let $.(r) denote the conerponding complex or Grassman wriable q5; [ r ) or ( r ) . A contraction is then defined as (g*m)
where the thermal avwage is de8ned by Eq, (2.80) and the explicit T-voduct may be omitterl because: the operators: are nsessartlily tine-ordered by the definition of the themal average. An ttqulvalent definition is given by
= ($a( ~ ) 6 (r'))o ~#
(2.91)
( ~ ) $ a( # rf)
where the thermal average is given by Eq, (2.81). From Eq, (2.88) we obtain n
and
Be?eause the ttxwtation value of two creation or annihilation operators is zero in any state of definite partlcle number, or by expIIcit integration of the corresponding Gaussian integral, the fotrowing contracttons vanish: n a! (r)aL,(r') = $: (r)$:. (r') = O and
+
Given t h e definitions, n&e that with M n (4 & - p) the left hand side of the and each factor in the right hand identity Eg. (2.84) is (+i,+i2 side cornsponds to a contraction (li+; = (ar H0 - Cl)ij'. Thus:, in this care, the thermat average Es $WR by the sum wer all complete sets of cantractSans, where a complete contmction Is a configuration in which each 1s contracted with a $* and the overall sign is specifid by gP where P Es the permwtion such that hp,is contracted with If one consider4 the upectation value of a pmduct of an unequal number of $'S and +**S, it would still be equal to the sum of all contractions since the complete wpatation value would vanish and at !east one contraction in each complete set of contractions would also vanish. Thus, the general statement of Wick" theorem, using the notation of Eqs, (2.90)and (2.91) to denote creation or annihilation operators, is the following:
..
.
+
+
=
C all complete contractions
.
"P
GENERAL FORMALISM A T FtNtTE TEMPERATURE
-
+$&a%a;&a%*;gaz(~l 4)$a,(~2
- r;) (2.9.1)
Tke utility of Wick's thwem Ss now wident, since the pawer series for in Eq. (2.83) may be reprerented as the sum of all contractions in which ringlcpaa4icle propaptors g,(r 7" connect the p d u c t s of the potential V in aft possiMe ways. Although we have us& a dlagonail basis for notational convenience, Wick's thwrem hdds in a general bads [a)which is not an eig[enbasls of Ea. in which case the only dfgwence Is that the singre-pafilcle ppagatms are no longer diagonal. An alternative derivation of Wick" tthwrem as an operator identity is presntd in Problem 2.8.
-
LABELED EEYNMAN DIAGRAMS Using Wick's theorem. the pnturbation ucpansion f w
& is obtained from the
mdes Eq. (2.83) by enumerating art the csrnprete sets of contractions contributing to the thermal avelrage sf products of the potential V. R is convenfent to reprwnt this expandon in terms of diagrams, and in this w t i a n we! wlll develop simple rules far constructing diagrams which vwide an enomical representation of the enormous number of contractions, Eor slmptkcity, we will conddw the special caw in which V is a two-bdy interaction
and submuentry address the generalizatlan of the rules to m-body interactions. The term of order n in V in Eq. (2.83) for in this c a x is
-&
A faithful represc~lntationof all the camplete sets of contractions contributing to is given by label& diagrams defined to reproduce each of the contributions to Eq. (2.96). Each contraction will join some $;,(ri) to rome (l,,(rj) yielding a propagalor 8e,qjh,(ri 9 ) which will be represented by a dirrcted line odginating at (b& (ri) and terminating at +,,(ri). Each interaction will yield a vertex having two incoming lines, conesponding to ( T ~ ) (Ti) $ ~ ~and two outgoing liner, conespanding to +:j (q)$zd(ri). The n interactions in Eg. (2.96) will thus be mpnmnted by n vertices with two outgdng Ihns sai,B and two incoming lines 75,6i acting at time 78
(&).
-
2.3 PERTURBATION THEORY
79
corresponding to the factor (2.97)
Each outgolng !he will be connected to an ingolng fine with a directed line cormsponding to the propagator
(2.98)
The set of alf possible ways of connecting interactions with propagators corresponds precisely to the set of att the contractions arising from Wick's theorem. so summation over a complete set of distinct diagrams will faithfully reproduce each of the desired contractions. At order n = 1, there are two diagrams corresponding to the two CORtractions contributing to Eq. (2.961.
Mote that because the propagator is diagonal in the singte-particle Indices, each directed line in a diagram is labeled by a single index which i s summed over all states. Recalling the form of the propagator a t equal times, we obtain (2.1000)
FM Fermions, this is just the finitetemperature generalization of the expectation value of the energy in a Slater determinant composed of eigenfunctions of HQ. Expanding the partition function to first order in the grand canonical potential yieids
For Fermions in the low temperature limit, the occupation numbers simply sum over the: occupied states reproducing the interaction energy derived in Problem 1.6.Although in
80
GENERAL FORMALISM A T FINITE TEMPERATURE
generat the singloparticle eigenfunctions ia) may not coincide with the seif-consistent Hafirm-Fock wave functions (we Problem 1-61, in the special caw of uniform matter. . will yield a plane wave basis in which case Eq. (2.100) any Wanslationally-inmriant H corresponds to the Wartre%ck energy, The diagram mles presented th& far conectly afcount for all the contractions. plopagators. and matrix elements, and we have only to augment them by the rules for the m a l l dgn and factor and a cardui general definition of summation over all distinct diagrams. In higher orden, one could imagine defwming a diagram in such a way that two
may be drawn the fotlawing ways
Although these drawings look diaerent, they are nol dtstinct Biagcams; rather they are merely ddormations of the sane diagram. The es%ntiat feature of the prepagator llne labeled S is that it beit;ins at the right side of the intwaetion at r g , and ends at the left side of the interaction at ra and whether the dlagran is drawn so that the amow is pointing u p a r d or downward 5s immaterial. For subsequent developments, it wilt be useful to Introduce a pwise general deflnitlan of dfstinct diagrams, imagine that the interactions and propagators czamprising a diagram are arbStrarily flexible and may be IIN& a@the plane upon which they are drawn and twhtd and defomd In all possible ways. At! tabels and arrows associated with interactians and propagators, however, must be retain& as the diagram is deformed, Two diagrams are dofind to be distinct if they cannot be made to coincide with respst to topological structure, dirmtfon of arrsws, and labels by some defomation. By this definition, the twa diagrams drawn above are clearly not distinct, baause dehmtng the left diagram by raising the interaction 72 above rl will make it coincide precisely with the right diagram. The summation over all distinct labeled diagrams degnd in this way then c o r r ~ t l ycounts each contraction once and only once. Since the state label on each propagator is summed over af! single-particle states, It is superfluous to "lnclude theset labels an each diagram. The only essential role these labels have played thus far is to distinguish the variables $: and tSt, aswciated wFth the left-hand side of the interaction (aPlvl76) from the variables +$ and da associated with the right. Thus, we may make the replacement
and a labefed diagram is now specified by the times G ,the arrows on prapagators, and the left (L) and right (R) deslgnaticn OR interactions.
2,s OERTURBATIOEI THEORY
82
Ili faetw asmiated with each diagram. Each Flnatly, we must eonsfdw the ipll.Etd by a sign term of neh order in Ep. (2.96) contains the ovcrall factor Boson9j. Each asbodated with the specS8c contractions (whlcC is always con&actian will m u i t in o diagram fn which the singt+pa@ide papagatws fwm -me number of do& loops, n,. W =ample the f i r e d e r dire& diagram, hbxEq. (2.%) has n, = 2, the: first-wdw c?xc)rangsdiagram, Eq, [Z.Wb) has n, = f and the con&actlm in Eq. (2.101) has n, = 5, Each closed Imp m s p b n d s ta a cycle of Intwactions , eonnrzlcts to one side of anat the l&W rtgkt side of some me dde of yet anolther v&= and on unul it finally original veea. Far example, the contraction in loop c a m p r i d of the 7,6, p prepagatarrs may be Eq. (2.1Oi) generating tlcG rewritten sehematically:
=
pondkng ta the I& and right whwe we have swifically 'rndicatd the wdables sides of each v&ex and supprred all othw lab& since we are only conccerned with the signs. Since at each nerto?xgL is feparatd from Icd,; by two variables, +lfa$R+L may be rewdtten +i+b'pbg+R and the d a d loop has the f ~ m (2, IQ39
Tbe pairs included in the cycle corresponding b the e l o d Imp under eonddwation d without a R ~ t i n gthe dgn until the cycle of conbaetions has the fdtming farm: (2.1We) with this c l o d loop is now obvious. Bysq. (2.92) the interior ch yidd +g whereas the ougw contraclzion Ji" 9 producw the factw gg. Any c l ~ loop d may bo o t m s d In a lmilar Cwm, in nMch the pairs indveci In the cyck of a n k m t b n r are separatd By pairs of variables not invotvd in the c;on&actton, The sign is not dtereb by cammuting all the palus; included in the cycle to the [eftaf all the other pairs. Thus the contribution of a lelngle cycle has been the contractions, which may s u h u e n t l y b analyxd by cycles. Hence, each clold loop contributes a factw g 5n already included in the diagram rules and a diagram with nb c l o d labops thus squirm a factw gm&. tn summary, we have mow derived the rules for condmct5ng labdd diagrams which v w M e a faithfal ntpresentatian of the campaete m t of contractions eclintributing to the nthorder perrurbation expan.ion of The ruler statcd M o w an dirtinpished fmm rub~puentmriants by the rubreript k indicating that the diagrams m labeied.
v$
9.
fr. Draw all distinct labded disgams composed of n verttcm ),-c-,( co~nard by diraaed liner b . Two diagrams am dirtinct if they unnot be deformed ro ss to c d n d o campldy, including all timas faMs Q, !&-right lab& L-R and each disttnct diagram, ewluote tk s o-n m p a g a m . contribut;ron as fdtws.
82
GENERAL FQRMALlSM A T FINITE TEMPERATURE
23; Assign a single-particle index to each direct& line and incrude the corresponding factor
where the infinitedmaf q Is lncliudttd in the 6 funcrfons to indicate that the mmnd term is to be used at equal times. Sr, For each vertex, "tndude the factor
4+g, Sum over all singtemparticle indices and integrate all times over the intenrag [6,8], fig, Muftigly the result by the factor gn" where n~ is the number of etosd loops of single-paalcle propagators in the diagram.
The number of tabeted diagrams prallferates dramatically with order. At order n in the interaction there are 2rr variabfes +* ta contract with 2n vaPiabXes yielding (2n)I diagrams, At third order there are thus 720 diagrams and at fourth order 40320 diagrams so that canbinations quickly bwome unmanageable, Since many distinct diagrams have the same numericat contrkbutbon, tt is wo@hwhFfeInvesting; Jtame addim tlonaf effort in developing a systemat9e method to account effidrsntfy for atf dlagrams which have the same contribution. For a general intewction u which has no special properties other than the symm* try (2,504)
+,
there are t;wo types of transformatlorrs which leave the: value of a diagram Invariant: permutation of the time labels and -hang@ of the extremities of each v@&-, Since all the time labels 7371 rmare integratect over the same interval. [o, p]. any permutatisn of the time labels leaves the value of a diagram invadant, For n interactions, there are! n! permutations which yield the same value. Any vefiex in a diagram is connected to the rest of the diagram by faur propagators, and we may schematically represent the dependence of a diagram I? upon one particular v e ~ e xas follows
...
2.3 PERTURBATION THEORY
83
The graph rhbtalned by exchanging the extremities af the vefiex is
By the symmetry of the matrix element, Eq. (2,1011) the value of the diagram '?l equals that of .'l For a diagram with n interactions there are exchanges which yield the same value, Note that tsxtremity =change symmetry cowesponds to interchanging the spatial integration variables in a potential matrix element such as Eg. (1.109). ro that all the transformations Ieaving a Iabeled kynman diagram Invariant correspond to permutations of spatial or time variables of integration. The most general transformatton which leaves the value of a labded dlagram invariant is a combinatisn of a permutatlon of the time labels and an exchange of vertm ~tremities, Some such transformations acting on a given diagmm generate distinct diagrams while others simply generate deformations of the adginat diagram whkh must not be counted as dfstlnct diagrams. Thus, the combinatorial task facing us is to calculate the number of distinct diagrams which have the same value sa that we can multiply this number by the value of any one such diagram. To this end, we note that the set of transformations af an nth ordet diagram generated by any permutatlon of the time labels and any exchange of extremities of ve~lcesis a group, which we will denote by G, and has grin! elements, Condder the action af this group of transformations, G , on a labeid diagram l?, Some set of transformations GP will transform L" fnta a deformation of itself and the rest of the trangormations wOIl yield diagrams which are distfnct from F. The set Gr defined in this way is a subgroup of G. We define the symmetrjl factor 5 of the dhgram as the number of dc3fornations of ]I" generat& by the action of 6, Ely the definition of Gr,S is the number of demerits of Gr and since Gr is a subgroup of G, S is a divisor of 2nn!. Now condder a graph I"' distinct from E" which 5s obtained from I' by a transformation in G. Since I" ccorresponds to some permutation of the 7 and ILR labels on I', it transforms into a dekmatlon of itsdf under Gip and S of the remahing graphs are therefore deformations of.''l Continuing in this way by selecting a graph distinct from all the previous graphs and identifying its S deformations, the sRnr! diagrams generated by G acting on l? can be grouped into sets of S diagrams, such that the diagrams in each set are deformations of each other and diagrams in dierent sets are distinct. We therefore conclude that the group G of tranzformations which leave the value of a labeled diagram invariant generates exactly distinct diagrams with the same value. AI1 dfstinct diagrams will thus be e o r r ~ t l ycounted i f we take the value of one diagram and multiply by Finally, a single diagram .from the set of 2"lal diagrams generated by the group of transformations G is specified by an untabefed dlagram. An unlabelied diagram
v
v
.
GENERAL FORMALISM A T FINITE TEMPERATURE
84
is obtain& from a label4 diagram by removing the tlme labds and the L-R labels on the vefws. Thus It is compoM of eompfetegy untabeld vertices conn-4 by dlrwtts$ Unes. As befsre, two diagrams are distinct 5f they cannot be ddwmed so as to coincide. Wwever, since there are now no time labels or E-R labels, the condirion that two diagrams coindde is less stdngent: they now only need to have the same topolo@cat aructure and the same direction of awws on vopagatws. With the tlme and L-R labels removed, all the permutation and aremity mchanp transformations fn G transform an unlabded diagram into a de;formation of itself. Thus the contribution of all contractions. that is of all distinct Babeled djagrams, is obtained by calculating all distinct unlabeled diagrams and multiplying by Note that the factor 2"n! cancels the factor i"q. (2.96). The rules for ~alculatingthe n" order contribution to the perturbation expsnsion of using unlabelled Feynman diagrams are summarized bdow. These rules are designated by the subsdpt P for hynmran and we will subsquendy assume Eeynman diagrams to be unlabled unless ttxplfcitly stat-rid to the contrary.
h
v.
&
) (
1~ Draw all distinct unlabeled diagrams cornpored of n vertices - - - connected by directed llnes 4 . Two diagrams are distinct if they cannot be deformed so as to coincide campletefy including the direction of arrsws on propagaters. For each distfnct unlabeld dlagram, evaluate the csntribution as Pdlws* 2pll Calculate the symmetry factor S for the diagram. This may be awmpl"ihd by addfng times and L-R labels to the djalfram t s m a k it a labefd diagram. Then S Ss equal to the numb@of transformations wmposd of time permutations and vertex extremity exchanges which transform the lahltld diagram into a defsrmation of "i!M?lf. Asdgn a time label q to each vertex, and a single-particle index to each directed 13me, b r each directed line Include the faiel;or
I p b r each veeex, Include the factor
5p Sum over all Sngle-paNcle indices and integrate all timer over the interval [0,fl] (33
Multiply the result by the factor and S Is the symmetry factor.
gm" where n~ is the numbw of clased loops
At thls point, some detaftd aamples may be useful "t cctarifying the definitions and rules. First we consider the definition of distinct diagmms. Consider& as tabeld diagrams, the fotlowlng sets of diagrams are distinct
2.3 PERTURBATION THEORY
85
In the first three cases, any deformation which superimposes the directed propagators interchanges a set of L-R labels and in the tast casc;?it interchanges the rr - 72 labels. However, when these diagrams are considered as unfabelcsd diagrams, in each case the left diagram is a deformation of the Fight diagram and thus not distinct. The dmptest examples of symmary factors for unlabetd Feynman diagrams are the first-order diagrams which we have already consided in Eq, (2.99)" Since there is onfy one time, the only symmetry operations are ve&ex =changes. There are two transformations, unity and the exchange of the vertex extremitia, which transform the labeled direct and exchange r2f agrams
into thernst;ilves, Hence the qmmetry factor is 2, Aecding to rule 6 p , the overalf fador for the direct term is and for the exchange term is which agree with Eqs, (2.99). In second order, both time permutations and extremky exchanges are possible. For example, the diagram
-i
-B.
is transformed Into a deformation of Itself by permuting rl and Q, by simultaneausty interchanging the .extremities of both vereices, and by a combination of both transhrmations, Thus, counting unity, we obtain S = 4. In contrast, the diagram
has Iower symmetry, Simply interchanging rl and ra yietds a distinct diagram, sinercl the 7%vertex fs then connected to the central cfosd loop by the right end and the verta is connlxted by the left end, Hence, the onfy non-trivial transformation which yields a deformation of the origiginal diagram Is the simultaneous Interchange of the timas and exchange of the extremtlties of both vertices, yielding S = 2, The diagram
86
GENERAL FORMALISM A T FINiTE TEMPERATURE
has even tower symmetry: alf ctsmbhations of time permutations and vertw extremity exchange yield distinct diagrams and thus S = I, Additbanal features arise in higher-order examples, Consider the diagram
for which verta atremfty exchanges comb;ined with any permutation always yield d i s t i m diagrams, Any cyclic permutatian of (Q, Q, rs) yietds a deformation. However, although any cydic pemutatlan of (rStQ, Q) yields a diagram oF the same topological structure, the direction of the arrws on the large dssed losp Is rever=d and the dfagram is distinct. Hence, we obtain S .=; 3, For the third-order direct ring diagram
deformations are obtained for each permutaaon of (rz,72, r9) when it is combined with the appropriate combination af =changes of extremities. For trxampte, one may verifjr that the permutation (Q, r l , r3f must be combined with the simaltaneows exchange of the extremities of atl three vevtices and that (79173,75) reqtlSms the exchangf! of the atremlties of the vertices fabefed by rz and 73. The symmetry factor is therefwe 31 == 6. Higher-order dIrst ring diagrams containhng n interactions, such as,
may be analyzed similarly, In this case all cycfie permutations of [%W.. .7%) and eycI5e permutations of [G.. .73r271) mmbind with appropriate vertex extremity exchanps yEetd deformations and S is therefore 2%. h r future reference, it is useful to note that from the symmetry factors we have cferivd, the fadors from rule far the first-order exchange. graph and all dlrwt ring . If we regard the product of an interaction and twa propagators as a matrix fn the time and shngle-paeicle labels
2.3 PERTURBATION THEORY"
(Ef 1
Fig. 2.4
(F)
f ?
(Gf
gl
(HI
87
$
Second order unlabeled Fsynman dla(lramr wlth symmetry factor
-$.
then the sum of the first-order diagrams and all direct ring diagrams Ss
By now it may have bsome evident that the enumeration of dlstinct unlabeled dlagrams and the calculation of symmetry factors are sustzeptible to human fallibility. Wena it is umful to derive a sum rule which pr~vtdesa condswncy cheek on alf the diagrams and symmetry factors contnibuting ta a @ven order of pert-urbatkn thww. We have already shwn that the number of distinct Iabeled diagrams corresponding . Therefore, the totat number of distinct label4 to a @wen untabeled diagram is diagrams. which is 2nl. must be obtained by summing over all distinct unlabeled diagrams. Hence
v
where the sum Is omr all dtstfnet unlabelrzd diagrams. This sum rule is trriviatly satisfied in first order where the dirwt and exchange diagrams each have S = 2. The case of second order perturbation theory is more interesting. Whereas there are 24 labeled diagrams in second order, there are only 8 unlabded diagrams which are shown fn Fig. 2.4 together with The symmetry facton indeed satisfy the condition C = 311, and it is a useful exercise to vwity each symmetv factor and to check that there are no other distinct unlabeted second order diagrams. Thus far, we have restricted our attention to the cam in which V is a twebady interaction, The prweding diagram analysis may be generallrecl stra'ightforwardly to
9.
B
88
GENERAL. FQRNIAtlSM A T FINITE TEMPERATURE
treat a reddual Wamittonian V ..:H Hacontaining arbitrary many-body operators, In addltIon ta two-body vert'lces, a labeled diagram may have one-body vertices reprew n t d by .
and mbody vertices represented by a, a, a,,
- - a,
The number4 points at which 'Incoming tines m& outgoing lines will be d e n o d as lis given by all distinct unlabeled diagrams joints. The order contribution to c o m p o ~ dof ta vertices, where now these n vertices lnvoiue atl possible combinations of the one and many-body vertices included In the thmuy. The symmetry facwr S is now equal ta the number of transfarmations c s m p o d of time permutations among identical verticm and permutations of the number& joints of each many-bedy ve&a which yield a deformation. As an example, for a residual Wamlltonlan, V ,containing a one-body potential U and a thre-body potential W, one possible graph is
The symmetry factor is 2 since the onty transformation which produes a deformation of the original diagram fs the exchange of joints f, and 3,
WUGENNOLTZ DIAGRAMS The untabeled hynman diagrams derked abwe treat direct and =change matrix elements separately, For many purposes It 3s simpfer and more convenient to combine them in a single symmetrizd ar antisymrnetrtzd matrix element. The resulting cfiaEframs, called Hugenhottz diagrams, are readily derived starting from the symmetrizd or antisymmetrized version af the residual WamBtonian, V". For simplicity, we again consider the case af a two-body potential.
Since we no longer wish to distinguish direct and exchange diagrams, this verLex wilt now bs represented graphically by a dot with two incoming and We outgoing fines
2.3 PERTURBATION THEORY
89
Accarding to W"rck%theorem, we again construct diagrams by drawing R vertices and mnnecting at1 incoming lines with outgoing lines. Note, however, that by the symmetry W antisymnzew of the vertex, two contradions corresponding to the =change of the incoming or outgoing lines asmclatd with a given vert:ex are equal:
Thus, in contrast to the previous F=eynmandiagrams, we wilt no longer necsd to distinguish the! two incomfng W outgoing lines of a vertex. A diagram which d-6 not distinguish the two Incoming tines or two outgoing lines at each v@= repesents many sets of contractions, To count the number of contractions assodat& with a given Wugenholtz diagram tt fs @%fur ta define an equivalent pair of lines as two directed pripagaton which begin at the same vertu. end at the same ve&a [passibly the dginal vertex), and point in the same dirwtion, First, consider the two outgaing popagators from a given verta v<, and assume that they are not an equivztlent pair. Then they terminate at digerent verlices, say , In addition other vopagatws ttsrnrfnating at uj and tlk cannot belong to and ~ k and equivalent pairs. Now asrociate the factor \/Z with each end of every line which doer not belong to an equivalent pair. Then the produd of the a s accounts for the twa choices for contracting non-equivalent outgoing lines at any vertex, such ss vj and the two choices for contracting non-equivalent incanfng Iines at any vertex, such as v$ and %. I t is mnvenient to use the four f a c t m of fi associated with the non-equivillent pair of lines originating from a vertu: to cancd the factor f assodated with that vertu from Eq. (Z.ili9j. Smondly, cmsider the earn in which the two outgdng propagators from a given vertex v6 are an equivalent pair, This pakr of lines must terminate at the same veaex and therdore has only 2 equal contractions. When combined with the factor associatad with the weex at which the pair odg'lnated, we see that each equivalent pair produces an overall factor Hence all the antractions arising from Wick's theorem are conectly counted if for each Hugenholh diagram one eliminates the factor arising from the n interactions and includes a factor of for each equinlent pair of propagators. Since we do not disttnguish pairs of Iines entering or teaving vertices, label4 Wugenhdta diagrams onty poswss time fabds and awws denoting the dirtletion of propagators. The symmetv factor S is then the number of permutitions of the time labds whkch yield a dhrmation of the odginal labded Xugenholtz diagram, and the numb@ of distinct tabled Hugenhdtz diagrams cmespnding to a given unlabdd diagram is Finally.-the sign of a Hugenhdtz diagram is most conveniently determined by wlecting any order of assigning the propagatm to the I-& and gght sides of a conventional v e e x in a corresponding Feynnan diagram, and applying the previous sign rute ( - L ) ~ $ ~ "to the, resulting diagram, By the equivarlenee ntth respetrt to in-terchangt? at a ve&ex, atthough dlgwent assignments may v d u c e dWwent Feynman diagrams, they must yield the same contribution. Far example
i
i.
4
9.
%
90
GENERAL, FQRMACtSM A T FINITE, TEMPERATURE
These arguments give rise to the Following rules, designat4 by the subscript H, for calculating the nth order contribution to the pelZurbation wcpansion of using W ugenhdtt diagrams:
5
X
I& Draw all distinct unlabeled diagrams composed of n vertices connected by directed lines 1 .Two diagrams are distinct if they cannot be deformed so as to coincide completely, fncluditlg the d i r ~ t l o nof amows on propagators, Fbr each distinct untabeled Hugenholtz diagram, evaluate the contributian as foffows. fta Calculate the symmetry factor S for the diagram, This may be done by adding tlme labels to each veeex in the diagram. Then S is qual to the number of time permutatlans whkh transform the labeled diagram into a deformation of itself. S a Asdgn a time labd 74 to each vertex and a s'ingleparticle indm to each directed line. b r each sfir~tedtlne include the factor
For each v e f i ~ include , the factor
5~ Sum over a l single-particle indices and Integrate ail times wer the interval 10,fl. The symmetry factor S 1s dean& above Multiply the result by the factor of lines, n,, is the number of pairs of in [2&). The number of eqdw tines beginning at the same vertm, terminating at the same vert.ex, and oriented in the same direction, The number of c l o d loops nk Ss obtained by repfacing each Hugenholtz ver912?x
'X:
Y
B
by a conventianal ueltex
and counting
@ Y
8
the number of closed loops as for a kynman diagram, Note that the order of the tabels on the conventional v e w must agree with the matrix elements in [G). It is an instructive exercise to generalize thee Hugenfroftz diagram rules to the case bt n-body interactions [see Problem 2.9). At this point, i t ts umful to rsalcutate in terms of Hugenholtz dtagrams some of the contributions we previously evraluatd using bynman dfagrams, The onfy diagram for n = 1Is
(2,123) There is one equivalent palr of lines and bercause na time permutatians are possible, S =2 l, Thus the contdbution is equal to that Bn Eq. (2.91)).
2.3 PERTURBATION THEORY
91
In swond order, whereas there are 24 labreled dlagrams and eight unlabeld Feynman diagrams, there are three Hugenholtz diagrams. The diagram
corresponds to the two unlabeled Feynman diagrams G and H of Fig. (2.4). It has two quivalent pairs and the symmetqy factor is 2 since interchange of the upper and lower Interactions is a dehrmatfon of the origfnal diagram, Note that the two contributions from (arF)blrG) (yG vJaj9)and (apJv)67)(761ulSu)are equal, reproducing the direct graph (X) of Fig. (2.4) with the mrrect weight f and that the contributions from (aplv176)(7S[vlflu) and (a@lv(67)(761vlua)are equal. reproducing the exchange graph (G) of Fig. (2.4) with weight f . Similarly, the graph
has no equivalent pairs and symmetry factor 2 so that when atl the combinations of direet and =hang@ matrix elements are enumerated, the thme Feynman graphs (D), (E), and (F) of Fig, (2.4) are reprodused with the proper weight. The counting for higher-order ring diagrams proweds as for the ease of kynman diagrams, The genedc nthorder ring diagram
has no equivalent palrs and a symmetry factor S = 2%to account for the cyclic .rm)and (ra ,, r2rI) which field defarmations of the orSginaf permutations of for Hugenholtz ring diagrams diagram. Thus, for n = 1 and n > 2, the factors are identical to those we dedved before for Feynman direct ring diagrams. However, for n = 2, because of the presence af two pairs of equivaient Itnes, the comct factor of is one half of the general ring factor Hence, with antisymmetrized matrix elements, the formula - # n [ l n ( l - cvgg)) overcounts the second-order diagram, and a correction tern must therefore be subtracted explidtly, As in the case of unlabeleci kynman diagrams, it is valuable to derive a sum rule which provides a consistency check on the enumeration of Wugenholtr diagrams and the calculation of symmetxy factors. We have already seen that the number of distinct For each distinct labeled diagrams corresponding to a given unlabeled diagram i s Iabeled diagram, there correspond contractions. where a factor of 4 arises for each vertex from w hlch a non-equivalent pair of lines originates and a factor of 2 occurs for each vefiex from which an equivalent pair originates. Therefare, the total number of
.
..
&
&.
&
9.
92
GENERAL FORMALISM A T FlNITE TEMPERATURE
Fig. 2.5
Second and thlrd order Hugenholtr dlaprams wlth actors $$ x
contradions, which is (2n)!,must be obtained by summing unfabded diagrams. Thus, we obtain
6
X
&.
over all distinct
where the sum is over ail distinct unfabeifed Wugenhoftr diagrams. In Fig, 2.5 at! the second-order and third-order Hugenholts diagrams are shown, together with the factor x Again. it is an instructive exercise to verify the diagrams, symmetly factors and sum rule, In comparison with the 720 original contract;'ions, the 8 third-ordw W u e nhalts diagrams. represent a significant simpliftcatSon.
h.
FREQUENCY A M 0 MOMENTUM REPRESENTATION Esr systems which are homogeneous in space such as a liquid or gas, it is fiequentty advantageous to perform the perturbation expansion in momentum representation. Similarly* for pratrlems which are homogeneous in time, it is often useful to buder transform fvam time to frequency.
2.3 PERTURBATION THEORY
93
The following conventions will be used in defining Fourier series and Fourier transforms. In one dimension for a periodic funetion F ( z ) in a box of length L. we write the kuder series as L
(2.127a)
and the correspnding Fourier transfwm f w an infinite sysEem as
For a function of time g(r) which is periodic or antiperiodic in the interval (0, p). we write the hurler series as
where For periodic functions W,&
=
2~73 Ff
and for andpedadic functions
Consider the evaluation of a dbagram for a spatially homogenesus system, In w511 be transtationatfy invariant so that the eigenstates will be plane this case waves, and it i s most convenient to formulate the problem n i_ a finite box with periodic boundary conditions. Denoting the discrete momentum km 3 (kgm,kgw,kEm) and volume V = L,L,&&, the matrix element of the potential evaluated with normalized dgenstates is
e[f, - E,,)
.
(2.130)
GENERAL FORMALISM A T FINITE TEMPERATURE
94
Thus, each interaction consewes momentum, with the Kronctcker 6 rqulring that the sum of the momenta of propagators entefing the vertex equal the sum of the rnomenta leaving the vertex, and carries a factor of f. Every dtagram can be decomposed into connect4 parts, that is parts in which an uninterruptd chain of propagators connects any vertex to any other vertex, For example in Figg,23, diagram A has three connected parts, diagrams B and C have two cionneaed parts, and diagrams D through H have one connect4 paut. The diagram rules for a connwted diagram with m interactions give rise to the following contributions, There wlll be 2m propagators gb,(q - q f , each summed independently over a complete set of momentum states k,. The m interactions will give rlse ta the product of m momemtum-cansewing Kronecker S's and a factor of V"? Since the diagram 5s linked and momentum Ss consewed at each ve@c?x,one of the m Kronecker 6's is redundant. Thus, there are (m - 1) constraints on the 2m momenta for the propagator, leading to a sum over (m+ 1) independent momenta, In the contfnuum limit, each sum over an independent mometum is reptaced by an integral F(k,) -, dSkF(L). Combining the factor Vm+' from these independent momentum integrals with the fac'tsr V""" fmm the m matrix elements thus yfetds an overafI factor of the volume It, for each connected diagram. A diagram having m, connwted parts yields a contributkon proportbonal to Urn@and it is evident that only completefy connected diagrams, having n, = 1, produce extenstve contributions. The diagram rules presented previously for Eeynman and Wugenholtz diagrams are thus modified as follows in mometum representation:
Era
&l
Ilr;.,~ Assign momentum labels to each dlrwttzrd llne as fslfows, Within each c o n n ~ t e d part of the diagram containing m interactions, select m 1propagators to labd independentfy and use conservation of momentum at each vertt31C to label the remaining propagators in terms of combinations of the independent momenta. Assign time labels as before and for each directed fine incf ude the factor
+
The single-particle energy eh may include the contribution of a general nan-tacal oncbody potential in addition to the kinetic energy 4 p For each vertrilx include the factor
g.
Esr each vertex include the factor
5 ~ , nFor each independent momentum. perform the integral / times over the interval [O,B].
2.3 PERTURBATION THEORY
95
the previously defined factor by Vn@where ncis the number of connected parts in the diagram,
~ , F , H Multiply
Fourier transformation from T to w for a problem which is homogeneous in time proceeds analogously, except that periodic or antiperiodic functions of 7 are integrated The propagator g ( q l 72) defined in Eq. (2.98) is periodic over the finite interval [O, or antiperiodic with period @ in each of the times and therefore also in the relative time. Although the relative time ranges from -@ to /J, it is sufficient to calculate the Fourier coefkients by integrating over the range [O,@l. Using Eq. (2.746) for napthe for Bosons and wn = for krmions, definitions Eqs. (2.129) with w,, = and noting that eiawn = g, we obtain
a].
@$?=
9
(2.131~) and hence
j-
1
le-iwnr
Par(?) == Wn
iwn 2
(2,131b)
( € a- P )
One additional consideration Is required to treat the special case of a propagator which begins and ends at the same vertex, In order to invoke the proper 6-function in ga(7) corresponding to the occupation probability nk, the argument is always shifted q to q a ( 7 - V ) . (Seepforexample. Eq.(2.78)). This same result may by an ;n~nites~ma~ be obtained by multiplying ga(wn) by the factor eiwaq. in which case
Q3 WtS
The time integration associated with the vertex
%Wn,
)--.-(
%Wn,
QzWn,
yields the integral
96
GENEML FORMALISM A T FINITE TEMPERATURE
Thus, at eveq vc?&ex, we obtain a factor 6 and a Kroneckr 6 requiring that the sum of the frequencies associatd with the propagators entedng the veeierx equal the sum of the frequencies of propagators leaving the veeex. As was the case wlth momentum con9t3wation. a connected diagram with m "rnlctractbns wilt have m 1Independent frequencies. Since in a general nth Order diagram each of the 2% propagators has factor of and each of the n time integrals yields a factor B, the weall contribution is proportional to The diagram rules prewntd previously Car fc3gnman and Mugenholtx dlagrams are then modifid as follows In the frequency representatfon.
+
if
3p,a Assign frquency labels to each. directed line as Collo-~s,Wthln each connmtd part of' a dlagram containing m Interactions, w l ~ m t 1 propagators to labet indeplendently and us@fmquency consmation at each ve&a to fa bet the rematnlng propagators in Ertrms of comMnations of f ndependent morneota. Assign singleparticte indleas as before, For each d l r w t d ttne include the factor
+
For propaa;rttors beginning end ending at the same veFtt?x, include an addttional factor edCJa@. 5p,g Sum over all single-particte tabefs and sum over all independent frquencies W,. 6p,a Multiply the previously defined factor by where m is the number of interactions,
h.
The generalkation of thew rules using momentum and frequency representation to %-body interactions is straightfarward, It Is often convenient to use momentum and frquency repremntatfon simuitaneously in which cam one associates with each propgetw a four-momentum (U,,k) which is consewd at each wertgx. In the zero temperature fimft which wifl be discusd in detail in Chapter 3, 18 4 W, the Fouder stlriw in w, bewmes a burier Integral, and the diagram rules can be ttlcpvressed f n the covatiant form famll'tar in relativistic field theory.
THE LtltiKEf) CLUSTER THEOREM We have seen that the expansion for Z contains all powers of the volume V , with an Indivlduaf diagram with n, conneatd parts being pcoportlonal to I r m a , fn contrast, the ground potential = -AlnZ = -PV is an utenslve quantity so it must be @. posstltzle to regroup the expansron tg obtain an wtensive expansion for Q, Whereas it is plausible that It may be express& in terms of connected diagrams, the eombtnatodal factors are not a prricrk obiifous, The linked cluster thwrem states that In. Z 5s in Fact given by the sum of aff connect& diagrams, We wltl derive this theorem udng the replica technique, both for the sake of brevity and to introduce this usduf method. A standard derivation is given fn Problem 2.10. The basic idea of the replica met-had is to evaluate P for intepr n by replicating the system nc times and expand the resuft as Cotlaws.
Thus, if we evaluate Zrrfor Integer n Tn peeurbation theory, InZ will be given by the coefFicients of the terms proportional to n, A more general statement of the method is to calculate ZRfor Integer n,continue the function to n = O (which is valid by Carison's thwrem) and evaluate an appropriate expression involving the continued function to calculate the obsemabte of interest, tn the present case, we calculate
We will first evaluate Zrr for integer ta by perturbation theafy, and according to equation (2.1333, Log Z will be given by the coemcient of the graphs proportional to n. Since
+a (P)= ;#art02
(2.136) VVe may write P as a functional integral over n sets of fields ($zc(T), $ z ( r ) ) where the index ~r runs from 1 to n.
(6)"
(6)"
The Feynman ruler for are the same as those for except that each propagator now carries an fndex B , all propagators entering or leaving a giwn ver;tex have the same index B , and all c's are summed from 1 to n. It 'is evvident that each connected part of a diagram must carry a single index g,which when summed from 1 to n, yields a factor n. Thus, a graph with n, connected parts is propartional to nme and the graphs proporlional to ra are those with only one connctcted part, that is the connected graphs, As a cansequence, we crbtain the linked duster theorem:
f l - no=
--Bl
(all connected graphs)
where R. is the grand potential of the unperturbed system
CALCULATION OF OBSERVABLES AND GREENS FUNCTIONS The 1Snked wpansion for the expwtation value of any n-body operator R is analo p u s t o that for ZZ, One way to prsced, as autlined in Problem 2 . 1 l . i ~to calculate O(A) for the Hamiltonian = .l? X& and evaluate ( R ) = &R(x). An alternative
+
98
GENERAL FORMALISM A T FINITE TEMPERATURE
method, which has the advantage of showing how symmetry factors are simplifictd, is to use the replica technlque again. Using the definftion for the unpeeurbed thermal average, Eq. (2,81b), the expectation value of R may be written
Let us again introduce n fields and define
(+2(7), $:(T))
where the Index c7 runs from I to n
Note that the operator R is calculated with the field +p,$: associated with o = 1 and is evaluated at 7 =; 0. By separating the a z: 1 component from the (m - 1) other components we ~bsewet hat
and thus the desired expectation vafue is obtained for ra =. 0:
The perturbarion expansian for R, Is obtained by expanding Eq. (2,139a) in powers sf V
and diagram rules for untabeled diagrams, tabeled hynman diagrams, Hugenhoftz dlagrams, or the momentum frequency transform af any of these diagrams are develop& as a straightfo~wardgeneralization of the rules In the preceding sections, We wilt describe here the caM of a two-body operator R and unialbeled Feynman diagrams. The expansion of Eq. (2.141) consists oP ail distinct diagrams containing one vertex (oralRIq6)at time r = O and any number of vertices (aPlv176) at other times which are integrated from O to p, As In the case of the linked ciuster proof, all propagators now carry an F~dexa, all propagators entering or leaving an interaction
2.3 PERTURBATlON THEORY
99
veflex have the same tnda, and ail Indlces are summed from 1 to n, In addition, the index asaciated with R Is one so that all the propagators entwing and leaving f2 are restricted to have cr = 1. Now, consider a diagram composed of a set of fntteraction verlices u connwted to R and One or more additional unconnected parts, Since R is constrained tcll carry a = 1 and afl propagatws and u vertices consem c7, cr must be 1 everywhere in the portfan of the diagram linked to R, fn the additional unconnectd parts, hwever, there is at feast one free summation over o leading to an owralt factor of at least one power ~f n. Hence afl diagrams with diwonnected parts vanish when n is set equal to zero and ( R ) is given by the sum of afi connected dfagrams linked to 8, Having play& their role !n the proof that only connected diagrams contribute to ( R ) , the o i n d i c ~which are constrainecl to be l in all components linked to the vertex R are supeduous and may be omitted from the finat diagram rules. The symmetry factors for diagrams contributing to ( R ) are much simpler than for cowesponding contt-ibutions to the grand potential. Shce the symmetry of a diagram is redttcd by singling out a particular vertex to be fabeled R, It Is clear that S, which specifies the number of combinatians of -tlime perl-urbatlons and vertm extremlly achangs which trajndorm a label& diagram Into a deformatlart of itself, wilt be rduced, In fact, as we shall now demonstrate, the symmetry is reduced so much that the only possible values of S are 1 or 2. Tcl see how the symmetry factors are constrafneti, consider the typicaf labeled and the diagram in which nine @-verticesat times 71 .g are denoted by
..
R-vertex at time 0 Es denoted by
One may see that there are no permutations of time labets which prod&@ a deformation of this diagram by the following agument, f ach diagram may be consMered as a series of cfosed Fermion loops connected by one or more vert;iiclrts, and i n this ctxampfe there are eight such loops, Consider any loop which has one time fixed. Since the propagators compridng the loop are directed, there are no permutations of the vertex time labels within the loop which yietcf a ddormation. Therefore, atlt the time labets in the loop are fixed, tf another loop Is conneted t o the first loop by at least one interaction, then at ireast one of its time labels is fixed and there is no freedom t o permute time labels within that toop. By Induction, one observes that no permutations of time fabefs within closd loops are gossfble. In this example, the time O fixes the labels T ~ ~ ?a, T G ~which in turn fix the labels .rs and re in the next loop, The only remaining possiblilky, then, is to permute times beween loops, However, since In the case of the diagram (2.1142aj the sub-diagrams cannected to the left and right sides of the R-vertex are tapoIogicafly inequivalent, none af these permutations produce a deformixtbn. The following diagram is an example in which exchanging times between digerent
100
GENERAL FORMALiShrl A T FINITE TEMPERATURE
loops will! produce a deformation.
Note that the subdiagram conniected to the left side of the R-vet-tex is topologkdy equivalent to that connected to the right dde. Interchanging 71 with and TS with re and simuftanwusfy interchanging the extremities of att the vefiiees thus yields a deformation of the orkginal diagrams as shown a t the right, This simultaneous extremity exchange and exchange of all! the times in the topologically equivalent subdiagmms connected to the R-vertex is in fact the on!y operation which can produce a deformation of the diagram, We have seen that no other operations which change time labels produce delormations. and by anaiyzlng the L-W labels in each closed propagator loop of a general diagram, one can also see that pure extremity exchange without time permuations can never produce a deformation. Thus. the symmetry factor for diagram (2,1425) 3s S .= 2. Another example is the exchange diagram shown in betow whlch has symmetry factor S = 2 because of the time and extremity permutatfuns shown at the right
3
(2.14%)
The general rule for the symmdry factor rmulting from this analysis Is simple. For a trwo-body operator R, S = 2 if the exchange of the mtrernities of ;R cornbind with some time permutation and exchange of interaction extremities yields a deformation; a,
Qz
B=
Ctm
13,
82
B3
.. Sm
othennrise S = 1. For an m-body operator R,
the argument is easily
generalized. The symmetry factor S Is equal to the number of permutations of tbe m joints of R, which, when cumMned with time and interaction extremity exchanges, yields a deformation. The rules for cafcutating the ptborder contribution to the perturbation expansion of the expectatlon value of a two-body operator ( R ) using unlabeled Feynman diagrams may b.e strmmarfr& as follows:
1. Draw all distinct unlabeled connected diagrams composed of one R-ve&w connected by directed lines Two diagrams 4 so as to coincide completely including the dirwtion of amows on propagators. For each distinct unlabeted diagram, ewluate the contribution as follows. Calculate ths symmetry factor S For the diagram. If the exchange of the atremltles of R combined with same time permutation and exchange of interaction atrefnities yields a deformation of the original dhgrarm, S = 2. Otherwise, S = 1, Assign a time label 76 to each of the p v-vertices, associate the time 7 = O with the R-vertex, and assign a single-particle index ta each dlrwted Ifne, For each
1.
2.3 PERTURBATION T H E O R Y
"I (E) 1
Fig. 2.6
(F)
5i
(G) t
Unlabeled Feynman diagrams far
101
[Hf C
(R)wlth Rctor
direct& line indude the factor
4, For each u-veeex include the factor
and for the R-vertex include the factor
5. Sum over all single particte indiees and integratt? the p times over the intervat
al.
[Q,
6. Multiply the resutt by the factor and S is the symmetry factor. The eight unlabeled diagrams contributing to (R) in orders p = O and 1 are shown in Fig. 2.6. together with the factors The contribution at order Q is gjven by diagrams A and B
g.
and a typical contributian a t order f is that of diagram E
102
GENERAL FORMALISM AT FINITE TEMPERATURE
The derivation of the diagrammatic expansion far the ima@nary-timeGreen's function
is completely analogous to that for the expectation value ( R ) wit h R(+;(O)+, (Q)) replaced by $ ,, (81) $a,(fln)+:k (bti(8;). Let us denote each of the n external points $a6(l13i) by a solid do-t at time (h)with a propagator for state entering it.{?. and each of the n external points $:;(B,!) as a solid dot at time
...
(a;) ...
I
with a for state a: leaving it.a%;. These external points are to be treated analogous to the R-ve&ex in the previous derivation. tntroduction of m replicas tabeled by indices with the constraint that (bai (pi) and $ :: LB,!) be as~ciatedwith oi, yields. in the Ilmk m -* 0, the sum of all linked diagrams in which the interaction vertices, v , are linked to the 2% external points. An example of a diagram contributing to the thre-particle Green*s function 42f) is the following,
It conslsts of three direct4 tines begfnnlng at the external pdnts pi,ph and fl& and terminating at the aternat points &,hand as well as an arbitrary number, In this case six, of closed loops. In contrast to a diagram for a threebody operator. where the ppagators entering and leaving each of the three joints may be exchana;ed, each of the external gcints in (2.146) is fixed. Hence, its symmetry factor is even lower than that of the corresponding contlributlon to a threbody operator, and as we shalt now demonstrate, S = 1, The analysis is similar to that far the expectation value ( R ) . In the present case, because there 1s no fredom to exchange propagators at the external points, there are no permutaions involving the times along the three propagators starting and terminating at these external points whkh can produce a deformation. Since these times are now uniquely fixed, propagator loops connected to these lines by interaction verlices in turn have at! their times fixed and by t nduction all times are fixed. For =ample, in the dial fix times ra and which in turn fix gram (2.146). propagators beginning at 8: and 78 and r~which then fix ?go. Similarly, the fact that propagators cannot be exchangd
2.3 PERTURBATION TMEQRV
303
at external points prevents any exchange of vertex extremities From producing a deformation. Since no time permutations, extremity uchanges. or combinations thereof can yield a deformation of the original diagram. S = 1. The difference between the case of the expectation value ( R ) and a Green's function is illustrated by the diagram below which shows the contribution to analogous to diagram (2.142b).
GF)
The simultaneous interchange of 71 with Q and Q with r 4 and the exchange of a l interaction extremities which produced a deformation in diagram (2.142b) yields the disti& label& diagram in the right of (2.1471Whereas the qmmetry factor i s simpler for Green" functions than for any other quantity we have evaluated, the sign of the contractions contributhng to a diagram is slightly more complicated. FOrst, consider the sign associated with the contractions of (+, (p%). . +,m(fln)$;L . P,(: (&))Q without any interaction vertices. The contraction (+a,(B1)+a. (82) \be. (B;) +,: (@;)+a: (@I)l in which $;,(pm) is contracted with $,,(@m) for all m has sign +l,and for any permutation P such that JI., (pi) is contracted with $,,,(@p,) the overall sign is (g)P. Adding any number ofoii~teractionvertices to produce a diagram with nk closed loops modiRes the overali sign by the factor (g) by the same argument as presented previously for diagrams for the grand potential. An equivalent dgn ruie is to close up a Green" ffunctlons diagram by deforming it so that each eaernal point @L coincides with FRL,The diagram then looks like $Zi do,) where the factor gm arises from the expedalion value of :$(mg
.
(g)
.
...
.
+
...
placing the $*+ in normal order. The sign is then where iiL is the number of closed loops tn the deformed diagram, Appfication of these rules to the diagram (2,146). yields -1-1. Using the first method, we note that (123) is an ewn permutation of (512) and the slx closed loops yleld an additional factor =. l. According to the second rule, joining the external == 55+7 == f polnts /!lk with 8,, producits a mventh closed loop, so that in summary, the rules for calcutating the rth order ~ n t d b u t i o nto the perturbation expansion of g(")(al~g,. .a,, @n\ai@i,. a&@;) using unlabeled Feynman diagrams are as fottows:
cm*"&
..
.
1. Draw all distinct unlabeted connected diagrams composed of n atemal points $, (pi) / ,n external points 1 (;: (p,!)/ .and r interaction vertices connected by directed lines ) Two diagrams are distinct if, holding the external pdnts fixed, the vertices and internal propagators cannot be deformed so as to coincide complete1y t ncluding the direction of awows otl propagators. The contdbution For each distinct unlabeled diagram is evaluatd as follows:
.
304%
G E M E M L FORMALISM A T FINtTE TEMPERATURE
Fig. 2.7 Dlaemms contrtbutina with overall sign factors.
to one- and two-partlcfe Green" ffunctlctns
B,
and time pi. Assign 2. Each external point Li conesponds to a specified state an in-ternaltime label 7; t o each of the r interaction verlices and for any propagator which is not cannwted to an external point assign an internal dngfe-particle index, Far each directed line include the factor
Propagators where r and r-enote either 'Internal times or external times connected t o an interaction verta wifl only have one single-panticle index at, in which case the factor bar, is supefiuous3. Far each interaction vefiex include the factor
4, Sum over all internat single-particle indices and integrate the r internal times ~i sver the inkwal [O,@I. where nL is the number of closed 5. Multiply the result by the factor propagator loops and cP is the sign of the permutation P such that each propagator line originating a t the external point $:',(@L) terminates a t the external point + a P m ( B P ~ ) ' Examples of graphs contributing to the one- and two-pa&icle Green's functions, togetheu with the overall sign factor are given in Fig. (2.7).
2.4 IRREDUCIBLE DIAGRAMS AND tNTEGRAt EQUATIONS
105
2.4 IRREDUCIBLE DIAGRAMS AND INTEGRAL EQUATIONS With the foundations of dhagrammatlc pertlrrbatlon theory p m e n t d fn the preceding swtion, 01Is now passibfe to dedve aact integral equations refattng ctonnectd Green" ffunctions and Iveducible ve&ex functions, SInce these equatims include contributions af all orders of perturbation theory, they are us&l in defining cansistertt approX.rmatians fnuolvtng infinite rwurnmatlons of diagrams and in treating the renormatbtion of dlvergent field thwrles, They atso lead naturally to the ePfective potential, X', which will be usefut in understanding spontanwus symmetry breaking in Chapiter 3, and to the sdf energy, C, which governs the propagation of a dngle particle in a many-body medhm.
GENERATING FUNCTION FOR CONNECTED GREEN'S FUNCTIONS As seen in the precding secti~~s, E t is often uwfuf to define a generating function by addfng to the phydcat Namlttanian under ctonsideration addktlonaf terms in which fieid operators are coupfed to wtemal sourws. The simplest passibility is to couple the field operatws to an atexnaf source J by addlng the term
where the sources & ( r ) are complex M Grassman variaMes far Bosons and Fermions respatively, 0 t h more ~ general sourcw, such as the bilinear form
are useful For spaclfie applications. Similar to the werating funetlon, Eq. (2.85) used ta derive. Wick's theorem. the generating function for imagina~y-timeGreen" Function is defined as the partition function for the full Hamilt~nianpius the source term Eq. 62.1483. Using the fundianai integral representation for the partition function, Eq, (2.641, the generating funaion may be written
106
GENERAL FORMALISM A T FfHtTlE TEMPERATURE
where, analagous to the thermal average with respect to H. in Eq. (2.81), we define the thermal amrage with respect to the full Wamitonian H as
(2,160b) Differentiation with respect to the sources J:(r) and
Ja(7) yields
and
so that the n-particle imaglnarqt-time Green" function may be written
Although the dhagrams for Green" fundions dedvd in Sectfon 2.3 have all the interaction vertices linked to the external legs, the diagrams are not aft con~wted. are disconnected: that is. For example. the first four diagrams in Fig. (2.7) for they may be separated into distinct subdiagrams in which the wternal points and vertices of one subdiagram are not connected by any interactions or propagators to any other subdiagrams. Since the s u n of all disonnc?ct& dt agrans simply cornsponds to combinations of produds of fewer-particle Green" functions, it is usdui to deal with (a,,ra; a . , r, lai, a',,r',) is defined connected Green's functions, w hwe as the sum of all connected diagrams linked to the external points (cut, rl; a,, r,) and (a:,4;., ark, TA).The last diagram in Fig. (2.7) is an example of a contribution to gi2! The vneratfng function for connected Grec?n%Functions, which we shall denok W (J;(r), J, (r)).may be obtained from the generating function g(J: (r),J, (7))by using the replka technique once again, The functional [ ~ ( J : ( T )J ,,( r ) f ] P may be written as a functlo~alintegral over p distinct fields @+:($ $): and the resulting Grwn's functian diagrams w i l have the property that aft connected diagrams will be proportionai to p and afl disconnected diagrams will contain at least two factors ofp. The terms proportional to p are singted out by
$p)
.
...
r:;. ..
...
2.4 IRREDUCIBLE DIAGRAMS A N 0 INTEGRAL EQUATIONS
107
so that
w ( J : f r ) tJe17)) =
B(Jo*[rlrJ o ( ~ 1 )
and
Note that by Eqs. (2.150) and (2.153). W ( J * ,J ) = -@(n(J*,J ) - O(0,o)) so that physically, W represents the dl&rence betwtsen the grand canonical potential In the presencte and absence of sources. The structure af the terms produced by this generaang functlon is iflus&ated by the examples of' one- and hrvo-particle connected Grmn% ffunctions. +;(r)J, ( r ) ] by .F*+ $'J and Abbreviating C, dr[J:(r)+, ( r ) {J,",( ~ 1 1$g, , (Q), Jai (41,+a: (4))by {J;, $;, Jlt, $16 1 in obvious notation, we obtain:
+
+
Sueessive detrivatkvei act an bath the numerator and denominator of clack term, yiefding the familiar structure of ai curnufant expansion in which all possible linked and unilnked cambinations of n$% a d m v % are generated. Note that because of the preen= of Grassman variables, the order of the factors is important in calculating dm*wtives. Except in the pwsence of spontanmus symmetry breaking, as adses for example in the case of Base candensation treated in Setlon 2.5, all expwtatbon values involving unequal numbers of cmation and annihilation opemtors vanish, greatly simpliQlng the genera! result. F=or the one-body connect& Green's Functlon fn the absence of symmetv breaking we thus obtain:
The two-body connected Green" Function ts evaluated similarly:
108
GENERAL FORMALISM A T FtNITE TEMPERATURE
The only non-vanishing terms gentsratd by the two remaking derivatives, assuming no symmetw trtrealring, are those tn which derivatives acting on nunwator terms add two +'S to (+*$*) and one g(r to each of the faclors (p), yielding
Note that the factor
in the last term arises fmm the fact that mutted through an odd number of Grassman variables to act on campact graphical representation of Eq, (2.158a) is
& had to be com(+G-(J'*+@'J)). A
where / denotes an internal propagator connected to an external point $.(r) and the abbreviation e ~ e hfar , exchange indicates the sum of all posslble permutations, P, of the external points, in this case 2, with associated factor gP, ttigher order connecled Green" fanctIons are evaluated similarty, and tt Is straighflorwarb to verify that the thm-body wnnected Grmn 'S function satisffes the following equation, w hieh Es also evident from the diagram rules for Glen's functions;
Since the G r e n k ffunc.t'ons only invotve etqurtf numbers of qb and +*, a generathg function with a bitinear m u m , Eq. (2,1491, may also be used, and it Is an instructive exercisie to repr~duwthe prseding results with a bilinear wares? [see Problem 2.12).
THE EFFECTIVE POTENTIAL In the presence of soureer. the operators (a! ( F ) , a, ( r )1 acquire non-zero upectation ~ l u e s .Let us define the avwage field
2.4 IRREDUCIBLE DIAGRAMS AND INTEGRAL EQUATIONS
209
and its compla conjugate field
denotes a thermal averagt: with respect to H Here, in an obvious notation ( plus the source term of the form introduced in Eqs. (2.80) - (2.82) for Ha and in Eq. (2,1505) for W and the 7 arguments of $, and J, have been suppressed for compactness, Instead of dmltng with the generating function W as a function of the sources J* ,J ,it is useful to perform a Legendre transformation to obtain a function of the fields $*, 4. The motivation for performing this transfermation Is evident from considering the familiar example of a spin system in a magndc field, which will be treated in detail in Chapter 4. Denoting the Hamittonian of the spin wadable as X ( & ) , the free energy as a function of an external magnetic field .@ is given by
from whlch it follows that the magnetization is given by
A state function which depends upon the magnetization Instead of the externat magnetic field is obtained by inverting the datlon (2.16ib) to obtain H ( M ) and defining the Legendre transform
It follows that
G satisfies the rwiprocity relation
Whereas both P(W) and G ( M ) contafn the same physical information, in the case of the first-order ferromagnetic phase transition, G ( N ) has bater analytic properties and wilt thus be preferable ta approximate. Below the critical temperature, since M is non-vanishing and has the same sign as H , equation (2.161b) implies that P ( H ) has non-zero posttive slope for positive H and a non-zero negative slope for ~egativeH $so that it has a cusp at the ori@n. In contrast, G ( M ) is a smooth daubte well, anti the discontinuity associated with the First-order phase transition arises from moving from the solution to Eq. (2.163) in one welt t o the solution in the second welt. In the case of the generating function W[Jz(r), J,(t)]. the equations (2.160)for 4. (J,', J,) and 4; ($2, J,) are inverted to obtain the sources as functions of the fields J: (4:, 4,) and J , (#:, da) and the effective potential (or effective action) is defined as the Legendre transform
210
GENERAL FORMALISM A T FINITE TEMPERATURE
As in the example C ( M ) ,the effective potential satisfies the reciprocity relation
-6,,
aJ (4- 3J; (4 --4 (#) 346: (+l
6 (s - r') J, (7') - $6;(7')---$-34, 1 .f
and the companion equation
When the sources are set equal to zero. Eqs. (2.165) show that the effective potential is-stationary. That is, if we denote the fields in the absence of sources by 4z(r) and #,(r), then
As we have already mentioned, ih the case of & w e condenslrtion Egs. (2.166')have non-zero solutions {J:(r),J,(T)) and these solutions will be studied in Section 2.5. In the absence of symmetry breaking, the fields {dz(r),$,(r)) are zero and all Green*s fundions which do not have equal numbers of creation and annihilation operators vanish. The eewttve potentfat is a f~eneratingfunction far vertw functkons, These vertw functions are generat& by differentiating the eflective potential I"[#: (7) ( r ff In the same way as connected Green" ffunctions are generated from GV[J;(r)J,( r ) ] :
Note that evaluation at J:: = J, = O is equivalent tu evaluation at the stationary of Eq. (2.166). solutions The vertex functions k*,r defind n d j in this way have several lmporEant properties, One feature 3s that they are one-particle irreducible and thus cannot be disconnected by removing a single inter~alpropagator, Another significant property is the faer that connected Green" functions may be construct& from vertex functions using only tree diagrams, that is, dlagrarns containing no crosed propagator loops. This propePty is atremely useful 'rn renormatlzation of field thearies, since ail the divergences arise from loop integrals which are isolatd in the v e ~ functions a ,'l and in the ddnition of ccnsistent truncated expansions, These prope~lesare most easily seen by deriving a hierarchy of integral equations satisfied by the vertex functions and Green's functions.
{e,4.)
2.4 IRREDUCIBLE DIAGRAMS AND INTEGRAL EQUATIONS
118
TW E SELF-ENERGY AN0 IDYSONeSEQUATION
rpo
Before proceeding to the general care, it is useful to study the vertwc function An ecanomical method to d&ve the desiredl integral equations for vertex functlans is to take successive derivatives of the generating function W [J*,J ] with respect to h:(8) and +z.(r') using the chain rule and Eqr. (2.165) to write functional d e r i ~ t i v e rof a functionaf E"
6.F' SJa,(l.a) +------ss,, (Q)&+at)l.(
and similarly
The lowest order quation, a general matrix form of Dyson's quation, Ts obtained by &W differttntiat'ing each of the quantities +aa (e) =" and 4,: (vs) = -S
with respect to
+,
-
(rl)
and
. ) l (
Calculating
in detail, we obtaln
which may be rewritten in the more compact notation
l
where 1 denotes the variables (al, r l ) and d2 implies a sum over a2 and an integral aver 72. The rema'inlng three derivatives yield the equations
112
GENERAL FORMALISM A T FINITE TEMPERATORE
and
The equations (2.170) may be expnessed in the matrix form
which shows that the matrix of secand derivatives of I' is the inverse of the matrix of P* PM and flM is second derivatives of W. Thus, the matrix composed of the inverse of the matrix of connected GTWR'sFunctions
rs*o,flsb"4e
In order to understand the properties and physical sknlficance of XIYdr, we new consider a system with no fymmetry breaking, in whiich case Grwnk fundons with unequal numbers of and Jl's vanish and Eqs, (2.170- 2.272) slmpfify ta
and
r,.,(i,
2) = [g$')]-'(1,2)
*
For a non-interacting system, rewdttng Eq. (2.78) for the single-particle Green's functions with explicit {a,r ) dependence yields
where U;, Ss the single-particle Hamittonfan, Thus, for a non-interacting system
2.4 IRREBUCIBLE DIAGRAMS A M 0 INTEGRAL EQUATIONS
113
The one-particle vertex function is a special casts, and rather than deal with it directly, it is conventknal t a define the self energy Z: as the digerencte between the vertex function for the interacting and non-interacting systems
2) as the matrix $ with l, the arguments and Simplifying the notation by writing $!')(l, lab& suppressed, equat"ron (2.577) may be exwessed
and an the right by Q yidds the Dyson which when multiplied on the left by $la equation
or, exhlbiting the explielt ( a ,r ) dependence,
The graphical expansion of the selFenergy E:ls evident from expressing the Dyson equation and Its series apansion in diagrams. With the following graphical notation for the ant?-paPt.lclc?Gren'S ffunctions non-"iteracting Green's functions, and self energy,
equation (2,179a) yields
The foifowing two definitions are: requSrctcS t o specify the diagrams contributing t o the self energ;y E. A diagram is n-particle imducible if it cannot be separated into two or more discann~teddiagrams by cutting n internal prapagators, An amputat& diagram attached t o the points (alrl),(az%), (a,r,) has no free propagator G ( O ) attach4 t o these paints: hence each point must connect dtrectfy t o an interaction
...
154
GENERAL FORMALISM A T FINITE TEMPEMTURE
vertm, Wttk them definitions, it is evident from (2.179~1that -E(aarz,alq) is the sum of aft one-particle irreducibte amputated diagrams connecting the pdnts (CBrr rg) and (a2,ra). 'To see this, imagine enumerating aft the diagrams for the one-par9icle Grwn's function according to the rules given in Section 2.3. Amputate the aternat propagators, and let the set of one-particle trretlucible dlapams define -C. Flnally, consider the set of graphs generated by substituting; this set of diagrams into the last line of Eq. (2.179c). By construction, each original Green's function diagram which could have been disczonnected by cutting n digerent pvopagators wifl be generat4 once and only ance by the nth term in this expansion (2.179~) The ruies for calculating the self enerw C(a@,a"?) thus follow directly from thoses enumerated in Sectbn 2,4 for the oneparticle Green" functions, The rules for the ~ t order h contribution using uniabeied Feynman diagrams are summadzcld as failws:
1. Draw all dktinct, untabeted, one-parzicle Srrducible, amputated diagrams camposed of n interaction veelces with the label (at p ) asstgned to one outgain& arraw of an interaction the labd (a', assign4 to one ingaing awow of an interslctjan ye&=, and all other arrows of the interaction vertices connected by dirsted lines Two diagrams are distinct if, holding the points (&,p) and (af,@)fixed, the linm and propagators cannot be deformed to coindde completely including the dtretion of amows on propagators. The cantribution for each distinct unlabeld diagram is watuatd as folilows: 2. Assign an internal time label r+to each interaction verlcex which is not ass'rgned to one of the aternat time values @ or For every directed line, assign a singlee-partlcfe'Index 7 and include the faetor
where T,# denote either internal or atemal times. 3. For each interaction veeex include the factor Q
k
Note that if the aternal points {a,@)and {a',@')) are associated with the same interaction ve&ex, since the interactian is instantaneous the factor &(p p@)must also be includ&, 4. Sum over all internal single-partfcle indkes and integrate all internal times Q over the Intern1 fa, p]. 5. Multiply the result by the factor where nb is the number of closed propagator loops and the extra minus sign accounts for the fact that (2.179~) spalfies -C,
-
Ta illustrate the% rutes, the first and second order contributions to the self-energy are the following: a, B
-- - -Or = 56(8 - P') C ( ~ ? I (0) ~I~'T)~T Q
2.1 1RREE)UC;lBLE DIAGRAMS AND INTEGRAL EQUATIONS
115
Note that amputated diagrams, such as (2.180) are drawn with incoming and outgoing lines which do not begin or end with solid dots, indicadng that the external labels are assigned to the vertie;x and no propagator is present. In contrast, diagrams haAng propagators in the external legs such as the Green" function diagrams in Fig. 2.7 are drawn with solld dots for the external points and Et Zs understood that a propagator is t o be incfuded between each externat point and the interaction t o which 11; is connated. The physical interpretation of 53 as a self enerw or egective one-body potential is evident from using Eq, [2,1Sf)t o rewrlte (2,178) in the farm
The full Green's functions specifying the propagation of a particle in the many-body medium is obtained from the propagator in a non-interacting system by adding t o (aljfib!aa)all the irreducible graphs describing the particle's inleraction with the rest of the system, The contdbutions (2.180a and b) are instantaneous like f i and specify the shift in energy due to the Wartre Fack mean firstd. These and the higher order contributions will be df;e-cussedin detail i n Chapter 5 ,
HIGHER-ORDER VERTEX FUNCTfQNS The essential features of 6, which dlflers from the one-particle v e ~ e xfunction only by the tilvial term are that it is one-particle inedudble and that the full one-particle Green" ffuncion rs obtained from It by Eq. (2,179) which involves no loop hntegrats. We will now demonstste that these two features generalize t o n-particle vertex functions. We begin by introducing an econsmicah graphical representation for differentiations
r$.?
ilci
GENERAL FQRMAtlhSM A T FINITE TEMPERATURE
of the form which led
(2,1701, Let
and
With this notation, Eq, (2.170a) may be written
+
5
= 8 (.1,31
.
(2,183)
To emphasize the essential structure, we shall condense the notation still further by omitting signs. disregarding the direction of the arrows, letting represent either . Then a functional and letting represent either or applied t o increases the number of legs by one: derivative
6
6
Using the chain rule
&W
&r
-&= g&.Eq. (2.168). the functional derivative 6 applied t o
adds a leg containing
=
G:
c[#] = G[@] for successive values of
With this compact notation, evaluation of n yields the desired hierarchy of equations. %r of (2.183)
0;n
= 1, we recover the abbreviated form
= 6 Evaluating successive dedwtives by letting (2.185) yields
(z.rsk)
6act on F's using (2.184) or on G's using
+-
(2,18Cibf
where the integers denote the number of disrinct ways of arranging the external la bds. Multiplying each of the extwnal legs associated with in Eq. (2.186c) by G1 = (I'l)-', we obtain
6
In the case of no symmetry breaking, this simplifies to
238
GENERAL FORMALISM A T FINITE TEMPERATURE
Note that in spite of our condensed notation, the signs and factors of g are obvious: In the absencr? of symmetry breaMng, only terms with equal numbers of derivatives with respect t a #*,QT W J * , J are non vanishing by Eq. (2.168). so that lngaing or outgoing legs of the farm and attached to 9);' in Eq. (2.186d) have factors (-1) and (-9) respectively. Since has two incoming lines and hvo outgoing lines and. by E 9 (2.354). 2 factors of 9. the terms containing and T10*20 in Eqs. (2.186d) and (2.187) have coemcient +I leading to the overall minus sign in (2.188a). By the same arguments as applied to Eq, (2.179~)far the self enerw, quation (2.188) shows that b4*,%+ Is the sum of all one-partlcte Imdueibie amputated connected diagrams with four external legs, The diiagram rules for I'ae*2+ are fdentktzat to those given for the self-energy, except for the obvious modifications that there are two external labels (al, p&),(aa, asslgned to outgoing arrows of interaction vertices, two external labels (a;, G),(ah,B;) assigned to incoming amows of interaction vertices, and the factor for a diagram with n interactions and n~ closed propagator loops is ( - l ) m - l ~ m ~ g P where cP is the sign of the permutation such that each prop agator originating at the vertex with aternat label (a", p&) germinates at the vertex with external label (ap,, @pm). Examples of cantributians to the two-particle verta function are the following
$A2)
f ust as Z: specifies the self energy or eRt3ctive one-body potential for a particle propagating in a many-paaicie system, r a s e z d conesponds to the eRectiw two-body interadon between two particles propagating in a many-pa@Sclerrtdlum. In addition to showing that is camposed of one-particle irreducible amputated connected diagrams. Eq. (2.188a) also demonstrates that the No-particle Green" function is obtained from a tree diagram conposd of Green's functions and vevtex functions of the same and lower order, These two properties are completely general, as is evident from the structure of the hierarchy Eg. (2.186). As a final example, the corresponding result for the three-pavticle Green's function and vertex function from Eq. (2.184e) in the absence of symmetry breaking is the following:
rag*24
2.4 IRREDUCIBLE DIAGRAMS A M 0 INTEGRAL, EQUATIONS
129
again demonstrating the one-particle irreducibility and tree-diagram structure. Analogous eguatbons in which vertt3x functions ere a-par-tfcle irrducibte amputated c~nneeteddiagrams may be derived straightforwardly by the same approach we have used starling with sourcm coupled to all contEnations of n creation and annfhiltion operatars, For exampte udng the billnear source, (2.149). the principal steps in deriving the two-particle irreducible theory are the fattowing. The genwating function for connected Green's functions is
from w hlch we define the expectation values
and legendre transform
with the reciprocity retations
Using the schematic notation of Eq. (2.184
- 2.188) in which -&represents --d..&em@' -.d.-64:;
are suppressed. a functional derivative legs by two
applied to
. & represents &.&,or & and indices and directions of propagators
adds a leg containing
Evaluation of successive deriativer Eq* f2.18fjj
immsse the number of
I
t
and applied t o &;;W
&r
=
S:
g[fj= 5[%] yields a hierarchy of the form of
120
GENERAL FORMACtSM A T FINITE TEMPEWTURE
The first quation shows that the matrix of faur-point verrex functions Is the inverse of the: matrix of two-particle Grwn" functions, In the absence of symmetw-bresrMng, all 9's and E"% must have equal numbers of tncoming and outg;oing Lines and the pairs of lines in these equations represent all comb-inations of aligned or anti-aligned propagators consistent with this restriction. By the same arguments used for Eqs. f2.186), Eqs. (2.1%) show that the P's are composed of Wo-particle irredudbfe amputated connectetd d'iagrams and that the %-particle Grenk functions is obtained from tree graphs involving fewer paeicle Green" functions and 2%-point and fewer potnt ve&ex fttnctions. It is an instrudwe exercise to derive these equations in detail (see Problem 2,121.
2.5 STATIONARY-PHASE APPRQXIMATEON AND LOOP EXPANSION Whereas peflurbation theory is valuabte for the formal devefopments of Se~tion
2.3 and is d i r a l y applicable to a I'lmited class of physical problems with weak interactions characterized by a smatf expansion parametw, many problems of physical Interest f nv~Ivestrong; many-body interactions for which a perturbation expansion f n the ifiteraction strength Is inappropriate. A natural approach for such prr~blernsis to reorganize the p&urbation tlxpansion Into a series tn powers of a new smatt parameter. We have already swn wamptes of such infinite resummations of perturbtion theoq in Section 2.4, where solution of Dyson" quations with same finite set of self energy diagrams or, in general, calcula~onof Grc?2?nPs functions with any fEnite set of diagrams for n-paeicte inducible vertex functions I' resums an infinite number of terms of the oviginat perturbation series, Other physically motivatd resummations will be present4 subsequently f ~ sperzifre r systems, tn view of the general lack of mathematlcal control on the convergence properties of the adginat series and the obvious ambiguity associatd with regrouping divergent series, any such resummations must be understood ultimately on physical grounds, In this section, we cons9dw a specific systematic regrouping of terms obtained by ap#ying the stationary-phase approximation to the functional integral far the partition funclion, This regrouplng will be seen to be ordered in the n u n k of loops occuring in the kynman diagrams. For certain systems, this approximation generates an asymp totic expansion in a small parameter, such as ti or the number of degrees of freedom associated with an internal symmetq, We first review the stationary-phase approximation in the case of a onedimensional integral, and then generalize to the cast?s of the kynman path integral and the partitisn functlan for many-parlicie systems, As usual, t k primary emphasis will be on the essrtnce of the method and Its physical interpretation, rather than mathematical rigar.
2.5 STATIONAWV-PHASE APPiROXIMAftON AND LOOP EXPANSION
121
The stationary-phaw approximation, also d w r d to as the saddte point approximation or m e t h d of stmplerst desent, is a method for developing an asymptotic expansion in powers of $ for an integral of the form
where L Is a real panmeter and in general f ( l )is an anatytic funetion in the cornpiex I-plane, Fw simplicity, we will first consider the special care of a real function f(t) with an absolute minimum at t == to. As L increags, the integral becomes sharply peahd around the point to, and the dominant contribution t o the S~tegralaries from the vicinity of to. Expanding f(t) around to, recognizing the fact that ft(to) = O and f M ( t o )> O since &J is the minimum, the integrat may be rewdtten:
ft
fc)
and and the change of variaMes where derivatives evaluated at to are denoted has b e n introduct3d t o rescale the Gausslan t o unit widtk, As r = (t- h] 8 W, the Mrms witk n S go to zero and we may expand I(L) in p w e r s of $. For the proMems of physical interest, we will usually be interested in wcpanding the logarithm of I(L), and i n the wesent example, it Is straightforward to expand the iexponential i n Eq. (2.149). perform the Gausdan integrals, and exponentiate the result to the desired power of (see ProMem 2.13). A more wonomical derivation, however, may be obtained by utMliz'lng our knowtdge of Wick" theorem and the link& clustar expansio~, Just as we defind contractions in connection with Eq, (2.881, we may define the contraction of r as
-
>
-+
i
The coefiFicients of rnfQT
tt
> 3 in Eq, (2,198) a n rc?golrdd as vertices with n lines
Diagrams representing all pssiMe contractions contributing to Eq. (2,198) are obtained by drawing any n u m k of vertices V",,V,, . Vm, and eonnecthng them with propagators equal t o 1, in addition to the vertices, V,, a: diagram of order N" has the . By the link& cluster thwrenr,
..
I(L) = C-efifi,(sum
of aII linked diagrams)
3.22
CENEML FORMALlSM A T FINITE TEMPERATURE
The following diagrams contribs~teto lowest order in
$
Counting the 9 x 2 ways of contracting the three lines of each vwtex in diagram (a) and the 9 x 3 ways of picking one line from each vertex to contract in diagram [h). the contribution of these two diagrams is
Similarly, counting the three distinct contactions,
diagram (c) contributes
so that the asymptotic expansion of I(e) is thus
The contribution of order ft involves diagrams containing Vs,VsVs, (V4)%, V4(V8)2, and (Vs)' and is treated in Problem 2.13. The previous dixussion generalizes straighfomardly to a complex funetion f ( t ) which is analytic in some region of the complex l plant?. Conslder the case In which f ( t ) has a single stationary point to such that f' (to) = 0. Since by the Cauchy integral formula Re f (h)Is equal to the average of Re f (g) on a circle centwed on h, must be a saddle pdnt for the function Re f ($1. For an arbitrary eontour passing through h,h f f t ) will vary along the csntour giving rise to arM@arily rapid oscillations in the integrand as i! -4 m. However, by selecting a contour such that h f ( t ) has the constant value h f(b)in the vidnity of tal the integral assumes the furm
in the region of to. Furthermwe, writing fM(to) = to w a n d order around $0
-9
in polar f o m and wcpanding
+
it is evident that the two dirwtions 4 = and 4 = -$ 3 which keep Im f constant correspond to the dirmtfons of maximum positive and negative curnature for Re f ( l ) . Hence, the countour c is deformed from the real axis such that in the vicinity oT to it cdncides with the cum having h f ( t ) = h f (to)which passes through the in the direction of steepest dexent. and the saddle point of the function eeReIft) resutting real integral, Eq, (2,201f)is evaluated as in the previous case, In the case of multiple stationary points, the anaiysis is compl5csrted in two respwts, The first, essentially technical, complicatbn, Is the ntxessity of globally analyzing the surface of e-'RCg(L) in order to connect the positive and negative t axes at infinity with a contour traversing a squencr?.of intermcldiaite saddle points. The more
2.5 STATIONARY-PHASE APPROXIMATION
123
A N D LOOP EXPANSION
substantive problem is the fact that Integration of an infinite expansion amund each of two aparate stationary points has the potential For double counting wntrlibutions to the integral, For well-separatc3d stationary potats, it is often assumed (without justification) that tow-order contributions from each stationary point may simply be added. When tw stationary points come sufficiently cfos together, the cornb'ined contribution of both stationary points Is treated by an appropriate form of uniform approximation (Berry (1966). Miller (1970). Connor and Marcus. (1971)). FEVNMAN PATH INTEGRAL The Feynman path integral far the wotution operator of a pauticle in a poterrEIal
V(%)is the limit of a product of integrals owr the variables z h at each time slice k, Eg. (2.39). so that the stationav-phase approximation may be apptjed straIrghtCorwardly to each integral in the product, Far conven'ience, we will use here the continuum notation of Eq, (2.41f
t $ <m(%t)'-v(.t'))l U(zljtf; S&&]
E)[z(tfje
CC
."m"
D [z(t)le f 'IZ(')I (2,zos)
with the understanding that dt and ( are defined by the discrete expressions (2.40). Due to the multiplicative fact6 in the exponent, it 5s evident t h i t the stationary-phase approximation will generaLe the semi-classical expansion of the evolution operator in powers of 8. Since S(z(t)]is the classical action, stationarity of S[z(t)j yields the EulerLagrange equation of motion for the classical trajeetq ~ f t )
with the boundary conditions
Expanding the action around the classical trajectory z, ( t ) and introducing the change of variables
1
rl(t) = -(ztt)
6
- g@($))
(2.208)
we obtain
(2,209)
where q(t) is requird to vanish at the end points because z(t) and z,(t) satisfy the same bounday conditions. lintroducing an infiniteslmat real term,
124
cr.
GENERAL FORMALISM A T FINITE; TEMPERATURE
to render the integral well-d fined and noting that the quadratic form -e - i m g iV@E(ze(t)) q(t) actually represents a discrete sum qkAkcqt.
-
the Gaussian integral may be performed using; Eq, (1.179) with the result
where the cfetermtnant may either be calcufated directly from the discrete expression for the quadratic form and the measure Eq, (2.42) or from the appropviatsly normalized product of efgenvatues En of the equation
It Ss an instructive exercise to explicitly evaluate the determinant for the case of the harmonic oscillator to obta'rn the exact propagator (see Problem 2.14). Physically. the leading term in Eq. (2.210) of order is given by the classical trajectory, wPth the next term of order unity comsponding to the sum of ail possible quadratic fluctuations around the classical trajectory, Higher order terns in jZ may be obtained by summing linked diagrams in which the veeices ~ ( " ) ( z , )are connected by propagators [m& i :VV"(z.)]-Ias in the pnvious seetion. It is interesting to note at this point that application of the stationary-phase approximation to the imaginary-the path integral. Eq. 12.54) corresponds to an anatopus epans'lon around the stationary solution
Were the minus sign associated with the transformation (2.53) has been group4 with the potential to indicate that the stationary trajwtory corresponds to the classical solution In the invaed potential. Thus, in tunneiing and barder penetration probrerns for w hlch the ctassicalty forMdden region does not suppor2: a ppropriate classical solutions, these stationary solutions In the inverted potential witl serve as the starESng point for a quantum mechanical apansion.
MANY-PARTICLE PART#TI_ORSFUNCTION The coherent state functional integral for the many-pairticfa evolution operator, Eq. (2,621, may be writEen In coordinate representation as
2.5 STATIONARY-PWAf E APPROXlMATiOM AND LOOP EXPANSION
125
l
Whereas an explicit factor multiplies the integral dt in the exponent, in contrast to the Eeynman path "rtegral in the previous sstion, Ir atso appears in the integrand of the exponent, Thus application of the stationary-pha~apgraximation to Eq. (2.213) doat not stdctly yield a semi-classicat expanslon in powers of h. A similar situation arises "t field theory where, for example, the action assoclat& with a scalar field has the Form I (2.2~4) -a,4ar4 ![11)2+" A#' 2 2 h!
-
and fr appears expficitty f n the mass term, In either case, to interpret the stationaryphase approximation as an expansion in &, one must imagine two separate h's, with the h appearing within the inlegrand as a fix& consbnt and only the multiptfcatSve Factar as the expansion parameter. In the absence of a strict expansion in an wptidt small parameter, our present to that of peeurbatreatment of the statieionary-pham approximation wilt be analo-f~ous b ton ttrwry, We will intraduce ai parameter t multiplying the action, which for the sake of the derivation wili be assumed to be large, just as the format: parameter X which is often introducd In the potential In perturbation theory is assum& to be small. In this way, it is straightfonnrard to develop the stationary-phase expansion and di~smonstrate its correspondence to perturbation theory, Frt~nrsubsequent application to specific expansion. the physical condiexamples and the treatment of the closely related tions wilt bwome apparent under which a suitable expansion parameter arises and the method be-comes preferable to perturbation theory, Wit h the introduction of the paranaer 8 and suppressing factors of Pc, the partition function Eq. (2.66)may be wittten
&
d
-t$
D(+* (S, r)+(z, ~ ) b
~ ( l=)
J ..cf.,.,(&- g-&)+(..r)
P
-4 1 $~ ~ @ 4 * ( s * r ~ 4 * ~ ~ * ~ 1 @ ~ ~ - ~ ~ 4 ( ~ * r ) 4 ~ ~ ~ ~ ) X @ a * (2.215~) Variatbn of the exgwlnent yields the following ttquatians for t k stationary solutions (B,* and 1(6:,
with the boundary conditions
One trivial solution about which to expand for elther krmions or Basons is: the solution 4,(s, 7 ) = (bF(z,r) =10. In thls case, z(e)is evaluated in the usual way using
126
CENEWL FORMALISM A T FINITE TEMPERATURE.
i
g
perturbation theory with propagators (& - p)-' and interaction Lv(z - g). The &-dependence of each linked diagram for the grand potenthat is the0 where nv denotes the number of vertices and nl indicates the number of internal propagator lines. Since each weeex is connected to four fines and each line is connected to two vePtices, m1 = 2nv. Finally for a translationally invtlidant system, let n~ denote the number of momentum toops, defined as the number of independent momentum integrals perfarm4 in the diagram. Mote that the number af momentum toops diflfers from n ~ the , numbw of closed propagator toops defind eariier to determine the sign of a Feynman diagram. Rwafting from Se~tion2.3 that there are - jt momentum const?nring&-functions, the number of Independent momenta 1s nM ==: nr -r;sv -t- 1, The &dependence of a general diagram may therefore be express& as L"nr or t-nftd+l. The former result may atso be obtain4 trivially by rescaling the fields 4 in Eq. (2.215) by a factor and observing that only the ratio occun in the rescaled action. Thus ordinary perturbation theory may be regarded as an expansion in and is equivalent to a loop expandon in the number of independent momentum integrals. Note, however, that when expanding around &I, == 4," = 0,the leading contribution to R it of order la t. In addition to the trivial solution 4, = 4," =: 0,Eqs. (2.215) admit non-trivial static and timedependent scsfutions, ln the case of krmtons, the stationary solutions in terms of Grassman variabfes ile outside the space of physical obsemables and the functional integrals over the sh"lt& variables must be performed to obtain a physlcal result, The most e%cEent way to proceed for krmions is thus to introduce an auxiliary field to enable the tntegrat'rons over Grassman variables to be done exadly, and then apply the stationary-phase appradmation to the integral over the auxiliary field. We defer this treatment of Fermions until Chapter 7, and address the simpler case of Bomns here. k r BOSORS, appllcati~n of the stationary-phase approximation to D(#*&)c~("&) may be regarded as approximating a double real integral $dudvcF("*")where u and v are real variables representing the real and imaginary parts of 4. Shce the general method of steepest descent requfres consideration of at! complex stationary points of u and V, in principle we should cansfder solutions for which QS; is not nwessarily the complex conjugate of 4,. Again, the most generat case wit! be drr;fttmd and we consider here the special. castt that (a,* is the complex conjugate of 4., Because of the opposite signs of the time derivatives in f 2.215b) and (2,215~3,such solut!ons must newssadly be time-independent and satisfy the static Warlree quatlan:
h
5
The general solutbons to Eq. (2.216)for translationally invariant systems are plane waves- We wilt assume for the present treatment that ii(O),which is the zeromomentum Fourier transform or volume integral of V ( $ - y), is positiw. In this case the solution which mfnlmizes the action is the aer+momentum Bose condensate
2.5 STATiONARY-PHASE APPROXlMATtON AND LOOP EXPANSION
127
with actOon
where is the volume of the system, and we have chosen the arbitrary phase to make real. The significance of the phase of 9, will be addressed in Chapter 4 in the general discussion of order parameters, With the change of variables
the action becomes
W here
=I 2
and
l' 1 dr
dzdyj+*(x, r)+(xt
[:$i:j)]
B
It Is convenient to simplify the notation by suppressing (z,r) arguments, using the matrix D. Eq. (2.219~). and abbreviating the vertices in S(') and S(') as g3 and 94. so that the partition function is written as
Although the quadratic form appearing in this functional integral differs slightly from that in Eq. (2.84) used in establishing Wick's theorem due to the presence of the $P$" and +$ terms, it is shown in Problem 2.15 that a straightforward generalization of the usuat linked dlagram expansion is obtained. Writing the inverse of the matrix D in the block form (2,221) the contractions of the fieJds
/
(v,9) are given by
t(.**r)o(,f.) +(l)d'(2) ** (1)Y(2)
{
$(3)+(2) (b* (1)*(2)
1
128
GENERAL FORMALISM A T FINITE TEMPERATURE
Note that the pos'itivity of D rquires that $C cowespond to a minimum of the action. The propagators cowesponding to these contractions are written as foltows:
As usual, arrows entering or leaving a dot indicate a $Jor +* respectively and the new feature arising from the shifted fields $ isi the introduction of lines with two arrows pointing in opposite directions. The vertices comespondlng ta the cubic and quartic terms in Eq. (2.219) are denoted
By calculating the linked diagrams composed of these vert:Ices connected by the propagators (2.2231, the partition function may be written
where the diagrams denote the sum of afl contractions of the indlcatesi topolagy, for =ample OC) = + (2.2256)
a+ o---o 0
and sim"tarty for the remaining diagrams, Note that in contrast to the expansion of Z(e) about 4 =. 0,which corresponded in Eq. to ordinary perturbation theory in powers of v W the expansion about (2.2P5a) yields contributions to hl of order l and order unity. Such terms cannot be obtained by expansion "i powers of L, sol the stationary-phase expansion corresponds to an infinite resummatian of perturbation theory. This resummation may be understood physically in terns of Base condensation, The zero-momentum mode is macroxopicsliy occupied, with amplitude $, =
#,
fi
controlled by the chemical polentlal, The shift in variables (2.218) then gives rise to verlices in which one or more of the extremities has a zero-momentum condensate factor +e which we will denote by the symbol rather than a field vadabte $J or $* hndicated in the usual way by or -. \Ne will wrkla the quadratic maaix D from Eq. (2.219b) and its inverse C from Eq. (2.221) in the following schematic form : D=G,-'+v=G-~ (2.22~rrf
-
2.5 STATIONARY-PHASE APPROXIMATION AND L 8 0 P EXPANSION
129
where
and wllf denote Go and G by broken and solid propagators:
The Dyson equation G -- C& matic Form:
- &VG
following h m (2,226a) then has the diagram-
Iteration yields the series:
and
where Internal fines are summed over Go propagators with arrows in each direction. The propagator G thus sums self-energy insertions anatogsus to the exchange term of Eg. (2,180b)where the factor JIz plays the rote of the occupation number R, for the zero momentum state, Note that in dedving Eq. (2,L99), direct terms analogous to the Hartree self-energy of the form ir(0)+2 exactly cancelled the chemical potenrcial, so in fact both direct rand exchange terms have been resummed to ail orders, The term of order L(@) in Eq. (2.225a). i$In(det D),also has a simple diagrammatic interpretation, We first write det
D = dct G ( ',
+ V ) = det G,'
det (l
+ GoV) ,
(2.228)
130
GEIVEWL FORMALISM A T FINITE TENIPEMTURE
is the parlition function & = e-flno
Since from Eq. (2.68). det
non-interacting system and because the block form for G,'. two such matrices w'ith equal determinants, it foDlows that
for the
Eq. (2.226b) contains
Using the relation Bet &f
ttrl"lld
we obtain the BesSred result
The factor (GOV)"c~rre~ponds to a series of propagators and interaaions of the form in Eq. (2.2271, which are then connected into a dosed loop by the trace and weightd by the symmetry factor accounting for the number of rotations and reflections of the resulting diagram. Thus, the term of order @ sums the one-loop diagrams
&
whwe again the line without an airrowr denotes a sum over Go propagators In each eif rectlan The first fen orders of the expansion of the grand potential in Eq. (2.225af thus are clearly ordered by the number of momentum loops.nM. with L dependence L-("M-". The leading term LS, is Just the classicat action Eq. (2.217) with no loops, The second term. of order L(@), sums the one-loop contributions shown in Eq. (2.231b) and the sums the two loop diagrams shown in Eq. (2.225)" third term, of order tml, The fact that the stationary-phase expansion in powers of systematically orders diagrams by the number of loops is easily seen as fottows. We will explicitly draw the condensate factor on the ve~icesin (2,224) as an external line x, that the
.
i
connect to three internal lines and one external fine and the ve&ex
connects to four internal lines. Mow, consider a genera1diagram obtained by contracting nv ve~lceswith propagators, Of these nv vertices, assume n v s are of the form of
PROBLEMS FOR CHAPTER 2
(2.232a) containing an external condensate line and the remaining nv
131
- nvs are of
the form (2.232b) with no external lines. Since the propagators are independent of L.
the overall L-dependence of the diagram from (2.232) will be L - " Y + ~ " Y ~ Jf one3 counts the lines coming out d all wrlices, the internal tines connecting to two vertices will be counted twice3 and the external condensate lines will be counted once, giving the topological constraint (2.233a) 2nr n v ~ 4;nv
+
-
where nd denotes the numbw of internal lines. As In the analysis with no condensate, the number of independent momentum loops, n ~aRer , acceunting for the tzv - 1 momentum conservation constraints is
sr using f2.233a) nM = % v
- -21 - a ~ 1~ .f
162-2336)
Thus. for an arbitrary diagram. the overall L-dependence may be written L-("W-') establishing the loop expansion. An analogous analysis shows that the series for the eFectiwe action Is also a loop expansion, In contrast to the expansion around 21 = 0, for which the loop expansion caiincMed with at pe&urbation expansion, in the presence af a condensate diagrams with tap^ loops contain from nw - l to 2(nM f j veetces of the form of Eq. (2.232) as well as the infinite resummation of interactions Eq. (2.227) contained in the propagators.
-
PROBLEMS FOR CHAPTER 2
The first two pfoblems review elementq mpctets of qumtum afatistical mechanics used ia this chapter. Problem 3 indicates the derivation of the theordered exponerrlfial used in the text, for thwe uxtfmstur with it, The two most cmcid pmblem for undterstwding path integrd are Problems 4 and 5, whi& we ernphia~ s i e d by an *, The next two probbm treat d k r e t e and continuum de~vationsof the propagatom, For thoae who wish to bsodea their undemtandhg of FundmenLa1 derivatiom in the t a t and establieth contact wikh traditional ddvations in the original Ekrature, alternative de~vadionaof Wick" tthmrem, tfin bked-clagter theorem, the evaluation of expectation valirea, and the stationw phase expansian we present4 in Problems 8, 10,11, md 13, Generahakio~sof topics introducd hthe text we treated in Problems Q, 12, and 15 a d a specific a m p l e of the waluation of the detemhant wiaing fram qua&aitic fiuctualtiane is given in Problem 14. PROBLEM 2.1 Shaw that RI defined in. qamdurn sbaitisticaf medmics by M. (2.40) irr e
132
GENERAL FORMALISM A T FINITE TEMPERATURE
PROBLEM 2.2 Showthattheprobabilityforasystemincont~twiththerma1 and p&icIe r e m w o k to be in a state 4 t h energy E* a d contaking N, p&ieles is proportional to Let the combination of the reservoir and system have total energy E: an<X particle number N. Use the definition af the entropy to show that the density of states for the reservoir to have energy E - E, and particle . is proportional to ciS(B-B'r"-Na) from which the final result is number N - N obtaimed by lexpandiag in E, m d Na. PROBLEM 2.3 Show that the two (2.9) and (2.10), me equivalent. Con
f the theordered exponential, &s,
intewd which h= been subdivided of time sligw M and show that the terms of order c" o b c ~ ( t d ) ] agree with the terns of order s" obtained from
El
PROBLEM 2.4" The validity of an ap ation to the matrh element of"an infinia:tseh~haJ, evol'tldion opc?s.zcdormay be checked by rrfxowirmg that the wave fixnetion evolves propaly to leading order in e, that is
Beg& with %hes h p l e case b which the Hamiltonian is the kinetic energy operatom. (with fi = m =. 1for convemience) and show
ATovv, iaGXude a potential
V ( $ ) and show
Dexnanstrate that the same anewer k obtained when V ( % )is raplcrced by V(z) or by any h e a r combinittion qV(g) -t- (1 - q)Vf,), Show that when the qmmetrie combination $V(@) + $V (S) is used, the evolution i.actually correct through order = ea. The same result may be obtained more economically by noting e-iet e - i a C - i c f + 0 f e a I ,d evaluating S ~ ( l l c - ~p)(plr-'"e-~~W ~ ~ ~ ly)*
PROBLEMS FOR CHAPTER 2
133
The fomnlation of path izltew* far a p e i c l e in a magneeic PROBLEM 2,5* gwhe~ the 1 M m g t o ~ w cant field d i b i t s the generaf problems which. a d p are cambbd. Rmafl that; the Lavamgiaa for a p&icle i)il a magneeic field is
whve h = m = 1 and the faetor 5 haa been absorbed in the vector potential A. b g b with the n a m d e r d m d f o m of the quantum IZamilto&an, E&, [2,34), md as in h o b h m 2.4, show that the evolution opt?ra;lor
yields the proper quatian of mation to l e d k g arder ia s, Now, shaw that the foaowing altmative expressions aho give the comecd evolution:
Note that this last reralt taakis Eke a pozth intepal cont&ning the Lagrangim wigh a definite d h r e t e ;itpprodma&i~ian:
By compdng with the prcrviau~resultIt,it is clem &fiat the dterrrative djsere~e appro~rnakion
yields a different answer, which is only repaired by the inelusion of the 3~Ah (y) tern. Contrast this dependence on the point at which A(2) ia evaluated in the tern $A($) wit& the hdepezrdenct? af the; point at which V(%) is evdaated demomtrated in Problem 2.4 and thereby expldn how the orde&ng of *=atom in ordinary quantum mechmics irs refl;ectd in path btep&,
134
GENEWL FORMALISM A T FINtTE TEMPERATURE
PROBLEM 2.6 Contrwt the d h m t e m&* degakg q r m i o n of the fuinctiond int interwtkg parficlm with the dtemative expression (& - p r.)
+
S, R, (2.69f, the p&ition fmction for nonapprogmation to the contbuam
Show that w h e a S yi~ldsthe cornat p a i t i a n function, h a b o n eige~vdues with porritive real p e s (so that Gamian gnhwals convage) amd yields the w m ~ t propagator for 44' at equal times, S yields an in t peition hetion, h a eigenvduw with negative real p&s, and edutrtiaa of &ha- a d - t h e pmpagabr using h,(2.76b)giva the wong m u l t This exmias e m p h ~ h e the s fxt, thaG one mwt take the h i t as M -+ oo of the d k m b rwullf o b t h 4 from the definhg a p r w i o n for the functiio~aili n k p d . PROBLEM 2.7 Obtain the propae;ator for the non-interwting ~ystemEq,(2.77) by solving the diffweatial Q. (2,78ai) with p d a d i c or a~tjlpe~odic b o u n d w conditio~a,Qne strsigh&fomwdway to procede is to h d the p e ~ a d i cor aatipexiodic eigenfunetions ( f a ) $ , ( r ) = X, $ , ( r ) and Mite G(r, r') = C, (r)$G(7') , thmby obtaining the sum in Eq. (3.131).
&+
PROBLEM 2.8 The Operator Form of Wck% Thwtm. A1thoug.b we will o d y \Nick%theo~emt;o a p x w t h e m 4 aver&%- or weand state e x w t a t i o ~vdmes as soms w u products of propagatom, the origin4 theorem by Wick (2950) is aa opaator identity; Let, T and N denote the t i m e d e h g and normal-od opwa60~~1 rmptively. ReeaU that both aperabm bclude a fBGta (; if aa &it nambv of inturhmges take place m d that Tnomal-arders operaton at equal times. Define the contraction of two opun$ors Al .nd Aa .s the diauenee W
With these defini$ionca, Wick" tthwrem statea that the time-orderd product of n operatom is equal ta the sum of the normal-ordered products of all passible coatrwtions.
PROBLEMS FOR CHAPTER 2
135
The proof is by indnctian, and the theorem holds for n = 2 by Eq. (1). Since N and T are distribmtive, it is sufficient to consider the &'s to be creation or annihilation operators. Consider n l operatom and denote the operator with the ewliest time as 8. Assuming the thearem holds for n operators,
+
contractlons of A's
+
show that Eq. (S) yields Widr'a theomm for n l particles as follms: If B is an ann.ih.ilation operator, expXah how the righbhand side yields the n o m a l product of all contractions except pairings with and show that all pairings with B vanish. If .& ier a m a t i o n operator, commute it to the left of all the creation operatars in each n o m a l product, and show that this comnzutation generates the normal product of 4 coatraetions. c1 Finally, let a,(#) = O for all annihilation operatom. Show that AiRj ia the
all compfetclte
contractlons W here
General&@the Hugenholtr; diagram. rubs to the cme h which ... up to n-body interxtionrs, The principal issue k what to do about m-tuplers of equivalent lines, Mate that in general am m-duple of equivdent lines couM s t d at an nl-body vertex and terminate at an nz-body vertex and that multipk m-trrpllm could originate or terninate at the same ve&ex. PROBLEM 2 3
tf contain8 on(e-bodyl twolbody,
PROBLEM 2.10 The Lfnke&Clurter T)Feor+m, Show that the exponential of the sum of a31 link& graph yield8 the sum of all d i a g r m s with the zrpprap~atefactore ushg unlabele4 Feyaman d i a m a w , First, let, a, denote the contribution of some b k d ctusfer and consider s graph composd of n of them clustern. b p f a i n why the symmekq fwtor is nfs: where 8, is the s y m m e t ~factor in a,. Hence, s h m $ear. that the contribution of any number (including 0) of clusters a. ia Generalise t o show that ex. "'yields the sum of all diagrams with the correct symmetry facton.
PROBLEM 2,11 Expactatlorr VaCucs of Operators, Consider the Hamiltonian #(X) = W XR and define the grand potential e-p04*) = n[e-@(Y+*R-@N Show that & a ( ~ ) f=~. R=[ ~~- @ ( ~ - P ~ ) R=] (R). Hence explain why (R) is given by the sum of actl Einked diag-rms cont&ing any number of intaration t e r n s from W and a s H g h operator R. Show that the rules and syrnrnetfy fwtom are those p r w a t e d in S e t i o n 2.3,
+
236
GENERAL FORMALISM AT FINITE TEMPERATURE
PROBLEM 2.12 h Section 2,4, the genwatirtg function, eflFeetive pateatid, vertax fanetion@,a d hierachy of intepd quations were d h v d in d e t d for tke linear source, Eq.(2.148). Repeat the steps for the bilinear source, Eq.(2.91) and dhve tbe eqnstiona, indicated sdematicafly h Eqs.12.191- 2.1%). PROBLEM 2.13 The Statfonary Phase Apprax'imetlan. Sdarstkg &am &.
- C", f
12,IW),wpand G to order &, pe*om the raulting Gangaizcn , and urponentiate the reaalt; to obtain Eq. (2.203) complete with Q(&) As in, &,(2.202), o b t h the a m e rwult by evduating the diagrams:
PROBLEM 2.14 Propagator for the Harmonic Oscillator.. Since %hezlietion in the Feynmaa path integrd For the h. nkc aw3laGoP h s qa&atic fa= in the coordinates, the s t s t i o n ~ - p h ~resalt, e EQ.(2,211) isl exact. Solve far the propagator for a particle in one spal-id &measion moving in the patantid Viz) =L $2 by explicitly evaluating a dkrete path integral and show:
It; is convermient to we the one-dimmacional. analog of Eq.(Z.59) with V ( Z ~ -re-~ ) placed by the symmetrised form !j(V ( z r ) V (sh-%)).Note that the determinant g h the ftxponent( of the pldhewiond tfi&iagoaaI.mat* occ
+
c m be evd~atedby expmding in
om to obtain the **tern
or
which may be aolved by diagonaliring a 2 x 2 rnagk.
rwursion mlation
PROBLEMS FOR CHAPTER 2
137
PROBLEM 2.15 Genc?ra&geWick%theorem aa presented in Swtion 2.3 to the ease of the functional integrtrl Eq. (2.220) encountered in Boae condensation. Firnrt igicroduce soarcm and show
where L7 and G are definttd in Eqs, (2,219b) and (2.2213, Noke tkat althongk these are half if many fields +4 iua the dimension of D,the infegral aver reail amd h % g i n q p&s yields zr real, quiadragie form of the proper dhension. By diEerentiadion with (12) = G9e3(21) and G++f12) ==; respect ta {Jr J*) and uSing the properties G++* Gge(21) prow (2.223). Finany, show that the integral of any even number of $'g and $*'S weighted with c-t(3**)D(is given by the sum of all contractions.
t*
CHAPTER 3 PERTURBATION THEORY AT ZERO TEMPERATURE In this chapter, the general Formarism introduced in Chapter 2 for the grand canonical ensemble at finite temperature is specialized t o systems of fixed particle number at zero temperature, The coherent state functiona[f integral formafism may be applied directly i n this special case by rqlacing the trace occurring in the partition function by the expectation value in a non-interacting ground state, The methodology of enumerating all contractions with Feynman dhagrams is unchanged, and the diagram rules are modified only by simple changes in the propagators and associated Factors, VVe begin by expressing phydcal observables in terms of matrk elements in the non-interacthng ground state which are strictly analogous to corresponding finite temperature traces in Chapter 2 and which may then be evaluated in perturbation theory using Wick's theorem. As a preliminary step, propagators for the zero-temperature expansion are first evaluated in the zero-particle vacuum, and the result i s then applied t o a many-Fermion system by transforming t a particle and hole operators defined relative t o the non-interacting ground stale, Given the perturbation theory for Fermions, 8osons may be treated directly by introducing a formal spin variable with spin degttneracy equal to the particle number, so that restricting the non-interacting ground state t o be an antisymmetric singiet state i n this variable renders the overall wave functbon symmetric in the physical variables, In subsequent sections, time-ordered Goldstone diagrams will be introduced and the zero-temperature limit will be discussed.
3.1 FEVNMAN DIAGRAMS The finite temperature diagram expansion was obtained by expressing the abservables of interest in terms of thermal averages of operators. Eq. (2.80b).
where all operators including those in have a formal label r denoting the and .l@ time slice upon which they are defined. Expressing the thermal average by the equiva) a series of lent functional integral. Eq. (2.81 b), and expanding F(+: (q) $a ( T ~ ) in then yields a series of Gaussian integrals which produce all polynomials In ($:, the contractions represented by Feynmart diagrams.
...
The first task i s t o derive expressbns for the ground state energy, expectation values of operators and Green's functions analogous to Eqs. (2.81a). (2,138) and (2,145) where the thermal average has been replaced by an appropriate expectation value, Let us begin with the ground state energy and denate the eigenvalues and
3.1 FEYNMAN DIAGRAMS
539
t plane
I"&, 3.1 normatized N-particle Hamittontan l ? =.
f ontours In the complex t-plane,
nfunctlons of the one-body Hamiftonian
HQ and
complete
and The basic idea is seen by considering the folfow'rng quantity
where the right-hand side i s obtained by inserting a complete set of states I$,}(+,f in the numerator, As long as \aa)is not arthogonal to i'lXTo} and fgo)is non-degenerate, in the limit Im To --+ --so the contributions of all excited states are exponentialfy damped and the desired ground state energy may be obtained. To recast Eq. (3.3) into a form analogous to Eq. (3.1). and t o prepare for the subsequent treatment of Green's functions, it is useful to select a contaur from -+To to 5% in the complex E-plane. Let the distance from -$Tot o $ !Eo along this contour be specified by a real variable g. The time-ordered product for a set of operators having times on the contour is then defined by ordering the operator in terns of increasing and a path integral is defined by breaking i into arbitrarily small intervals and introducing a complete set of states at each slice in the usual way. There is considerable freedom in choosing the contour, since by analytkity the result will be independent of the shape of the contour and the only requirement on To is that its imaginary part must ultimately ga to --W, One passible contaur is G,shown in Fig. 3.1, corresponding to the imaginaq time used in Chapter 2, For our present purposes,
140
PERTURBATION THEORY A T ZERO TEMPERATURE
however, it is preferable to use the contour C2, which is rotated by an arbitrariiy smafl angle 6 relative to the real axis and may be expressed in terms of a real variable f as
In addition t o reproducing the familbr results from conventional derivations in real time, this choice also enables us to treat real-time Green" functions without the need for explicit analytic continuation. Throughout this chapter the time variable t w"t\ be understood t o be on the contour C2and the -ie factor wiil only be included explicitly when necessary. With this preparation, we now define the following ground state average specified by .Howhich is the zero-temperature anaiog of Eq. (3.1)
(3.5) As before, it is cruciaf to note that T time orders along C2 all the operators in E" as well as interleaving them with the operators in B0(aL(t),a,(t)). Note that H. enters the definition implicitly through the fact that .HO defines IQa), as well as explicitfy. Using this definition, Eq. (3.3) may be rewritten "t the following farm
2
= Tlim ~ - - * o o-To ( 2 l n / ( @ o / @ ~ ) l - i ( E ~ - ~ ~ ) ~ ~ ) (3-6)
=Eo-& which is analogous to the finite temperature result following from
Eq. (2.80a) that
.Fad
and application of fi - 0, = $ ln(edt ')0. AS before. expansion of e-' d' Wick's theorem yields the sum of all diagrams which, when exponentiated, yields the oo to exist, it is crucial that each sum of a!! linked diagrams. In order for the limit linked diagram be proportional t o To for large To. This Todependence follows from the d t l .. dt, ( V ( t l ) V ( t , ) ) H , fact that the time integrals in the nthorder term -J
-
.
.. .
may be expressed as (n 1) integrals over relative time, each of which converges because of the choice of contour and positive definite excitation energies in propagators, and one integral over absolute time which yields a factor G. Physicalty, ground state Feynman diagrams are always of finite time extent because they represent time histories of virtual excitations which by the energy-time uncertainty principle, can onty be sustained for limited times. In the same way. expectation values of operator and Green's functions may be expressed in forms strictly analogous t o the finite temperature results in Chapter 2.
3.1 FEVPtMAN DIAGRAMS
141
Let R(O)denote an operator & acting at time t = 0. Then. the ground state expectation value, of R. may be expressed In the same form as Eq. (2,138) as follows
Similarly, by repeating the steps in Eq. (2.673, the: rz-particle Green" function defined In Eq. (2.21) may be expressed in the same form as Eq, (2,145). As in Eq. 12.67). let g,, denote an annihitation operatar for i n and a creation operator for d > n, let ,P denote the permutation which arranger the times t p i in chronological ) the Heisenberg operators defined in Eq. (2.14a). order and let ( a ( H ) t , a ( H )denote Then
<
. . .gap,, (tp2n)e-i'tP'n-
X e - i ( t ~ ~ - t ~a,,, ~ f H( tep 2 )
= lim
(-i)"
gP
1 ( 4 Q / @ 0 ) / 2e-iEoTo
~~'o~'~)('~/@o) II, m
f g m )(gm
l*,)
142
PERTURBATION THEORY AT ZERO TEMPERATURE
The left-hand sides of Eqs, (3.7) and (3.8) are of the form of Eqs, (2.238) and (2,145), respectively, and by the standard replica argument are given by the sum of all linked diagrams containing the operators R or a,,(tl). .ai,m(tz,). Thus, all zero-temperature observabies of interest may be obtained by evaluating expressions of
.
ep
dt ~ ( a ~ ( t ) a ( t ) )
the form ( e-'"S-= using Wick3 theorem and the appropriate O(d) zero-temperature propagators derived in the next section, It is useful to note in passing that there exists an alternative method to obtain the same diagram expansions h real time by considering the evolution operator for a Hamiltonian H,(t) = .Ho+ c - c i t I ~ in which the interaction V is adiabatically turned oR a t large positive and negative times. As outlined in Problem (3.11, a theorem by GellMann and l o w (1951)shows how to obtain an dgenstate of the interacting system by applying the evolution operator for H E ( $ )to an eigenstate of the non-interacting system and taking the limit as e --, 0,from which observables may be evaluated straightforwardly.
ZERO-TEMPERATURE FERMION PROPAGATORS The generating function for a non-Interacting N-Fermion system with one- body Wamiftonian .Ho at zero-temperature i s defined
where we have used a single-partkele basis which diagonalizes H@with ehenvajues e,, J,*(tf and J,(L)denote Grassmann sources and @a) denotes the H-Fermion ground state of Ho. tn the usual way, the average defined in Eq, (3.5) for any product of creation and annihilation operators may be obtained from appropriate derivatives of M H o with respect to the sources evaluated at r* = r = 0, As a first step toward the evaluatbn of ,MXg, it is useful to calculate the related quantity
has been replaced by the zero particle state 10)and the where the N-particle state denominator is temporarily ignored. It i s also convenient to pick the zero of the energy scale such that all ea > 0. Since the Fermian zero-particte state 10) is the Grassmann $) with $ = 0,the discrete functional integral for M o ( J * , J ) may be written
3.1 FEVNNLAN DIAGRAMS
143
where, using the notation of Eq. (2.70)
and
The essential diRerence between the matrix $(E,) in the present case and Eg. (2.70) for the partition function is the boundary condition, Expanding by minors, DekS = 1 and the inverse i s
Thus. Scl(e,),.,
vanishes for 4
< r and for
q
> r, we obtain
Hence, the non-interacting single-particle Green"$ function, or contraction, is given by
l44
PERTURBATION THEORY A T ZERO TEMPERATURE
Note that since normal-ordered operators have creation operators represented by a coherent state one time step later than the annihilation operator, as in Section 2.2, an infinitesimal q" has been introduced in Eq, (3.15) to emphasize that the result vanishes at equal time. The fact that the propagator (3.15) is non-vanishkg only for positive times (t t') whereas the thermodynamic propagator (2.78) has components for both positive and negative times arises divectly from the digerence betMleen the boundary conditions = $($) = O in the present case and the antiperiodic boundary conditions $(p) = -+(Q) in the finite temperature case. One also observes in Eq. (3.15) an example of the general feature that the imaginary part in Eq. (3.2b) defining the contour 0%makes the propagator converge at large values of Jt- t'l. Physically. Eq. (3.15) for the non-interacting Green's function is obvious. For t > t'. a: creates a one-particle statq in eigenstate or of time t' which propagates with the evolution operator until it is annihilated at time tt' = t. For t t', a,(t) acts on 10)giving zero, The generating function M H o defined relative to the N-Fermion non-interacting ground state fao)can be reduced t s the simple form of M. by introducing a partieiehole transformation. Let the basis states of HQ be labefed in order of increashg dgenvalues E , . Then, the ground state Slater determinant is obtained by occupying the lowest N states
-
+*(y)
<
N
and occupation numbers may be defined
In the familiar case of a translationally invariant system, the states a are plane waves, the Fermi gas ground state includes all momenta up to F e Fermi momentum kF, and the occupation numbers are specified by nk = @(hr;. - fkt). It is crucial to the present treatment that be non-degenerate. This requires that there be a finite energy gap E N + L - E N between the last occupied state and the first unoccupied state, since otherwise if E N + ~= c ~ the . state a k + , ai10) would be h).(Degenerate cases, such as arise In partially-filled shells in atomic and nuclear physics, require an appropriate form of degenerate perturbation theory such as the Bloch-Horowitz (1958)expansion.) Here, we will assume the existence of a gap. and shift the energies such that:
IQ0)
nf:~:
This shift is useful in defining propagators which converge as It - t" -+moo, Finally, we define particle-hole operators , h,) by the canonical transformation
{bi
3.1 FEVRIMAN DIAGRAMS
145
For an unoccupied state m 3 N,b: cmates a particle whereas for an occupied state a 5 N. bL destroys a particle creating a hole so corresponds to the set of particle-hole creation operators relative t o IQo). For both m > N and a 2 N. Eqs. (3.16) and (3.19) imply b,fgio) ==Q . (3"m)
{bi)
is the zeroso that ( b , ) Is the set of annihilation operators retative to I@@) and particle, zero-hole vacuum. Thus, when caherent states are constructed using the operators the state I$) with = O will be may be written In terms of parllcle-hole operators,
{hi),
+,
and the generating function is
where the sum ~ f E, = has~been cancelled out of the numerator and denominator. Because IiQo) i s the zero-pa@icle, zero-hole vacuum for ail the operators ( b , ) , Eq, (3.20). the evaluation of Mx,(J*, J ) by a discrete functional integral i s identicai to that for M. Eq, (3.11). Note that since Det S(")= 1, the denominator is unity. As before, the numerator factodzes into products of integrals for each a,so that the Green's function is diagonal, Thus, the generating function may fie written
(3.23)
where the natation S(-c,) arises because the hole state energies enter with a minus sign in Eq. (3.22). For particle states, cu > N,the Green" function is cafcutated as before in E q . (3.15)
PERTURBATION THEORY A T ZERO TEMPERATURE
1414
Recall that the -r)hrises because at equal physical times the creation operator (rj is evaluated one time slice later than the annihilation operator (g) so that the @ function yiefds zero. For hole states, a 5 H,we obtain the new result
The case of equal times for hate states ts difirent than for particle states, Since at equal times, the time-ordered product of attk is defined t o be the normal-ordered product, for a N , it follows that the time-ordered product of b i b , at equal timer is -babL. Hence, when a path integral is evaluated, c-cH b,bi eWLX is broken up as eeeH so that b, and 6: must be evaluated on the ba D(+*+)C-$*@ same time dice and the 19 function in Eq. (3.24b) is satisfied at equal times. Combining the tw results for particles and holes, Eq, 63-24], the complete Green's function for the non-interacting system may be writ;ten
<
I$)(+/bi
W here
iGo, (g
- g')
= e-"a('-")[B(t
- tf - 11')(1- n,) - @(t' - t +
]
.
(3.2Sb)
Note that this result is precisely of the form of the thermal propagator. Eq. (2.78). with the occupation numbers n, replacing the thermal weights ( e @ ( t a - ~ ) l)-', real time t replacing -ir, and no chemical potential, Physkcally, the first term represents the propagation of particles in the system. When a is an occupied state and t > t f . a! acting on creates an additional particle which propagates with energy E. until it i s tater annihilated at time 7. Similarly, when a i s an occupied state and L < t t , a , acting on destroys a particle in state rr leaving a hole which propagates with energy e, until is is fater filled at time t' by the re-creation of a parllcle in state a, so that the second term represents hole propagation. Fkdlly, note that the sign of E , from (3.18) ensures that along the contour Ca. Eq. (3.4). both the particle and hole terms go to zero as It t'[+ --too.
-
/ao)
-
FERMION DIAGRAM RULES Feynman diagrams for expectation values of operators and Green's functions are obtained for many-Fermion systems i n the same as i n Sectbn 2.3. Multiple d'rEerentiation of ,MHo (J*, J) with respect to J* and J yields Wick's theorem as i n Eq. (2.84) with the contractions defined by iGo, Eq. (3.25). Thus, all the resuits i n Chapter 2.3
$:G,
may be applied at zero temperature with the replacements 7 -+it.!J d t -+ dt. (-l)n -+ (-2")" and propagators g, ( r r'). Eq. (2.98). replaced by iGou(t - t'). Eq.
-
(3,25b), The result. Eq. (2.137a). that 0
- R.
is given by
-$
times the sum of all linked
diagrams where each unlabelted Feyxrman diagram contains the factor the zero-temperature counterpart for the Fermhon ground state energy:
E.
- W.
= lim to-c@
-1G
all linked diagrams
(3.26)
where each unlabelted Eeynman diagram with n interactions, nr, closed foops and (-X)"& a ~ d propagators iGo,. For symmetry factor S contains the faetor example, the Wartree-Foek diagrams yietd
in agreement with Eq. (2,iQOb) and Problem (1.6). Slmiilarly, a direct second-order contuibutbon to .Eo- WOis
Note that, because the t integral is evaluated along the contour C2,the contribution at the upper limit is zero and that the sum over yiefds two equal contributions when either of the pairs (a@), is occupied an8 the other i s unotcupictd. It will
148
PERTURBATION THEORY A T ZERO TEMPERATURE
be useful henceforth ta distingukh sums over states by the convention that upper ease Roman labels denote oc-cupied states, lower case Roman labets denote unoccupkd states, and Greek labels run over ail states: : occupied states,
: unoccupied states,
: all states
.
(3.29)
<
Thus, the restrictions a, b > N and A, B N indicated tsxpticitly i n Eq. (3.28) will be imptied by this notation. Diagrams for expectation values of operators and Green" functions are obtained in the same way, A diagram for (R) with p interactions, nL Fermion loops and symmetry factor S has the factor (--X)"& A contributian to an n-parlicle Green's function dGm(qtz,. antm a%, . crag;) with r interactions and tab Fermion loops has the factor (-i)' (-X)P (-1)"~where is the sign of the permutation P such that each propagator line originating a t (aA,t",) terminates at (ap,, tp,). The single-particle progagatom are in all cases iGo(t), The frequency representation of a zero-temperature propagator Is given by a Fourier integral rather than the Fourier series obtained at finite temperature, Eq. (2.1;31b) because the time integral extends m r the infinite domain (-cm,m) instead of the Rnite interval (0, @). The frequency transform of the non-interacting zero-temperature Green" function is defined by the relations
..
..
.
To Faurier transform propagators, we need the Identity
with inverse dt
e-i'a'@(ft) = f
ti w-e,fiq
(3.31b) *
Equation (3.31a) is established by contour integration as follows. Consider the case +iq for which there is a single pole below the real axis. For t 0,e-'@" vanisher in + oo so that the contour may be closed in the upper the upper half w plane as half plane without ttncl~singany poles and thus yields 0,For t > 0,the contour must be closed in the l w e r half plane encircling the pole at w .=: s(, riq and yielding -2wi times the residue e'e-"d. Hence, the intwral is given by Eq, (3.30) far all t . The proof for -iq is analogous. Using Eqr. (3.31) and (3.25b). the propagator is
-
occupied states (hole states)
W pfane
unoccupied states (particle states1
Fig. 3.2
The poles of Go,(w)
In
the complex w plane.
where we recall that e, is positive for unoccup"te8 states and negative for occupied states. An alternative derivation in which the integral for eo,(w) manifestly converges is obtained by using the contour C2specified by t = ;(l - i i ) .Eq. (3.41, along which the propagator Go, ( t ) converges as the real variable f -. Am. Then. ~ f ' ) ( w ) should be defined as the Fourier transform with respect to the real variable ?i from which we obtain
Noting that e a t is a positive infinitesimal for unoccupied states in the first term and a negative infinitesimal for occupied states in the second term. Eq. (3.32b) reproduces the previous result (3.32a); The location of the poles of Go,(@)In the complex u pfane is shown in fig. 3.2 and is cttaracteristic of the general structure of one-particle Green" functions established subsequently in Chapter 5. The essential Features are that the poles for unoccupied, or particle, states are displaced below the real axis, occupied, or hote, states are a b m the real axis, and because of the restriction to a non-degenerate ground state, there is a gap between the iowest particle state and highest hole state. Our convention of shifting the zero of the energy scale such that e ~ < , O and ea > O corresponds to measuring energles and w relative to a Fermi energy or chtrmicai potential toealrzd in the gap, As at finite temperature, an additbnal factor Is required to treat the spechat case of a propagator which begins and ends at the same physical time, VVe have seen
PERTURBATION THEORY A T ZERO TEMPERATURE
150
previously that the equal time propagator is --a, which is assured in Eq. (3.25bf by writing i ~ ( O ) ( t ) = e-'"gj@(t qy(1 - na) 8 ( - t r)')n, As before. this result may be enforced by multiplying iG(w) by the factor eCWQsince
-
h
=
-
+
-
-
1.
-
e - i ~ ( t - q t ) i ~ O a ( W ) C - i ~ a ( t - ~ ' ) / ~ q')(l (t n,) @(-t
+ qr)n, ] . (3.33)
%K
From Fig. (3.2). it is also clear that since the factor c'@"' requires that the contour be closed in the upper half plane, the integral over w must pick up only the occupied state contributions, VVc! now consider the factors arising in the zero-temperature frequency representation for the ground state energy. Recall that a linked diagram at finite temperature with n interactions had n l independent frequency sums C,* and an overall facof Hence, when the Fourier series are replaced by Fourier integrals, tor for Sdw F(wn). each independent frequency sum is replaced by an F(@,) 4 integral and the diagram has an overall factor p. This is completely analogous t o the overall volume factor Y obtained for each linked diagram in momentum representation. SimiIarfy, at zero temperature, each linked diagram acquires an overalt factor To which cancels 1in Eq. (3.26) yielding a constant in the limit To4 m. ?-a The contvibutron t o Ea-Wo of a linked diagram with n interactions and nL closed Fermion loops in calculated as fo:oflows, Interaction vertices are included as at finite temperature. The propagator i & t l ( w ) . E q . (3.32)' is included for each Fermion line with an additional factor c'w'' for propagators beginning and ending a t the same interaction. After utOIizing the (n l) frequency conservatban conditions at interaction vertices. freare performed for the (a+ 1) remaining independent frequencies. quency integrals
+
&
I
&
-
E
From Eq, (3.26). the result should be multiplied by the factor i s the symmetry factor. A slightly more convenient rule is t o use propagators Eo,(w). integrals dw and t o include the factor (i)%" for 2 n propagators and (&)"+l for n+ X independent momenta in a new overall factor S * As an example, let us evaluate once again the second-order direct rcont~bution considered in Eq. 13.28)
We already know from the time representation that (a@) must be particles and (76) holes or vice versa. This is reflected in the fact that the oniy non-vanishing contributions in Eq. (3.34a) arise from terms in which the poles associated with n, and ngi
are on one side of the real axis and the poles associated with n, and n s are on the other side as may be verEed by consMering the other alternatives: ff the n, and n, l poles are in the same half plane. the integral dw ill ,,+Ws-Y1-Lskiv may Be etas& in the other half ptane yidding zero. The same argument applies to "8 and W , so that the n, and % pales must be in the opposite plane h r n the m, pole. Finally, the ra, and n s poles must be in the same half plane, since otherwise 1 l ub-u,-e, iq yields zero. The contribution in which (up) are dW7 ,-re particles and (76) are holes is equal ta that in which particles and holes are interchanged. so we may calculate twice the contribution of the former case. Using the convention that Cab) denote particles and (AB) holes, we obtain the following result
l
Closing the M , and wb contours in the lower half plane and the o~ contour half plane yields
- - -
Sn the upper
f ince E , 4b e~ e g IS po~ltivedefinite, the iq is irrelevant and Eqs. (3.34b.c) reproduce the result (328). The general structure of calculations in the frequency representation is of this farm, with one contour integral being required for each independent frequency. For many caiculations, the time representation or the time ordered diagrams introduced in a tater section are therefore more convenient. The rutes for the contribution to the ground state expectation value of operator R from a Ihnked diagram with p interactkons and ral; cfosed Fermion loops are obtained anatagousfy, Since the operator R is evaluated at time t = 0,there is no overall factor of To. The finite temperature rutes are appiieb with the mod"lfrcatIon that each independent frequency corresponds to the integral $ $.each Fermion propagator corresponds t o iGo,(w) with an additional factor c-'@"' if it begins and ends a t the same vertex. (--1)"k" For example. the lowest-order contribution for and the overall factor is
vP
152
PERTURBATION THEORY A T ZERO TEMPERATURE
a No-body operator R is
( R )Irjl =
B
where Eq. (3,331 was used t o evaluate the frequency integrals. The frequency representatian of the many-particle Green's function
...
...aLwb) = d t l .,.dt, c&: ., .dt; X e2i C ; , I ( ~ l t'j - ~ f t(altl, e ...a,t, ( &it:, ...a;%) I
iQcml(az~l, an@n &:W;,
j ' i ~ ( " )
(3.36)
is evafuated lay the same modification of the finite temperature rules, For a diagram with r interactions, nr. closed Fermion loops and permutation P such that fhe Fermion lines originating at ( a i , w i ) terminate at (a>i,W;&), the contribution to imG, contains independent frequency integrals Fermion propagators i&E(w) and overall factor (-i)r(-l)P(-l)mfi. For exampie. the Hartree contribution to the one-particle Green's function yietds
5 g.
= GO,(@)co,
(W)
C ( ~ D I. V I ~ ~ ) B
(3.31)
To relate the self-energy to irreducible contributions to the one-body Green*$ functian, we need to recognkze a sign diflerence arising between the finite and zerotemperature cases as a result of our other sign canventions, Physically, the self-energy C represents the energy associated with propagation in the medium and should be defined with a sign such that the combination $- X enters into the total Green's functions. Since at finite temperature Go"l = 8, + H o , the self-energy was defined In Eqs. (2.178) and (2.181) so that G-'(r) = ~,'(r)+ C = a, H. C from which it followred that G(r) = Gofr) Go(rfZ;G(r). (Note that since we are only interested in an weratl sign, the matrix indices on C and C are suppresseti and the [abets ( r ) and ( t )are only used to distinguish the finite and zero temperature cases, respectively.) At zero temperature. ~ , a f ( t= ) i& E, with t, entering with the opposite sign so that the appropriate definition of the self energy is
+ +
-
-
'
G- ( t ) = G ,' ( t )
- C(t)
3.1 FEYNMAN DIAGRAMS
l53
and the Dyson equation is
The self energy. C. is thus defined in terms of one-particle irreducible amputated Green's function diagrams as before, but without the overall minus sign. The result for the Hartree insertion, Eq. (3.37) is consistent with the general argument. since after amputating Ga and G ~;he. contribution to the self-energy is the correct Hartree expression CD(BDlvlaD). For future reference, it is also useful to calculate the second-order direct contribution t o the self-energy. The second-order contribution t o the Green's function is
= B and (76) = Separating the two non-vanishing contributions (ay) = (AB), (AB). p = b and evaluating the W integrals as in Eq. (3.34) yields the result (atBfvlab)(ablv(aB) l g ) = x bw + c B - c a - c b + i q 8
B
-5
(a'b[v(AB) (AB[vlab) -w-cb+ch +eB + i q
b
(3.396) The correspondence between finite and zero temperature rules is summarized in Table 3.1. The finite temperature rules enumerated in Chapter 2 showed in detail how to evaluate observables using the diagram elements and factors displayed in the left column. When these elements and factors are replaced by the entries on the right. the corresponding zero temperature observa bles are obtained. BOSONS The ground state of a non-interacting N-Boson system is a Base condensate
where a8 is a creation operator for the lowest single-particle state of the non-interacting Hamiltonian Ho. By Eqs. (1.63b) and (1.79a).
l54
PERTURBATION THEORY AT ZERO TEMPERATURE
eh"."
for equal times
eiWq
far equat times
Table 3.1
Summary of Correspondence Between Finite and Zero Temperature Diagram Rules
so that a0 and a! yield factors depending on the occupation of the ground state.
Thus. the expectation value of an arbitrary product of operators in the Bose ground state cannot be separated into "tndependent pvoducts of contractions and there is no zero-temperature Wick's thgorem as in the Fermion case. Twhnically, the point a t which our prwbous derivations for Fermions cannot be generalized to Sosons i s the constirudion of a canonical transfermation analogous to Eq, (3.19)such that all the new annihilation operators annihilate the ground state,
3.1 FEYMMAN DIAGRAMS
155
There are two options for treating Bosons. The first method treats the Bose condensate as a classical e-number field (Bogoliubov. 1947: Hugenholtr and Pines. 1959). For large N . there is negligible diqerence between fi and 0 ( ~ - ~ / SO % in ) the thermodynamic limit one may define new operators 2
=D-
+
which eliminate the problem of factors depending on the occupation of the condensate. Since (3.43) and
bol@~(N - m)) =
b i l @ (~N - m ) ) =
I@B (M- - 1)) I@B(N
1))
bo and bi may be replaced by the c-number f i . All the remaining operators af i > O annihilate on the state that Wick" theorem may be obtained as before, The resulting diagram expansion is quite similar to that obtaheed In Chapter 2 for a Bose condensate and contains four Ends of vertices with all passible combinations of a's and h's, For example, atataca yields a conventional vertex connecting to four whereas atbtbb has only one propagator and three "condensate" propagators)---
< -- - <
lines) . The general expansion is obtained by enumerating the diagrams connecting all four kinds of vertices and a simple special case is treated in Problem
(3.3). A second alternative which is simpler and more general is ta make the Bason problem into an equivalent Fermion problem by introducing a formal spin variable with spin degeneracy equal to the particle number (Gentile, 1940, 1942: Brandsw, 1971). This method has the advantage that it is valid for any pau-licfe number, does not involve the bookkeeping associated with four diEerent interaction vertkes. and involves only trivial modifications of the Fermion rules derived in the previous stzcthn, To avoid confusion with the physical spin and t o make the language more cotorful, Iet us call the format spin variable cotor. Each physical single-particle state now has N degenerate substates with different color projections, We now treat this augmented problem, with both physlcal and cotor variables, as a many-fermion problem and use zero-temperature perturbation theory to find the totally antisymmetric ground state. The non-interacting ground state, Go, is a determinant with all particles in the lowest physical dngle-particle state and having each color projecthn occupied once, Thus, i s the product of an antisymmetrk color-singlet wave function times a wave function which Is totally symmetric in the physical variables, The Wamiltonian in the eniargtld
156
PERTURBATION THEORY A T ZERO TEMPERATURE:
Fermhn space is the product of the physical Hamiltonian and the unit operator in Since the unit aperator l,r, cannot change the color color space, H = fi'(~)l,,~,,. wave function. the interacting ground state I@) = limr, T,,-, eSgHTlea)must also factorize into the product of an antisymmetdc color singlet times a totally symmetric wave function in the physical space corresponding t o the Boson ground state. The only assumption, that c - ' ~ ~ [ @yields ~ ) the ground state instead of an excited eigenstate. is a reflection of the ever present assumption that we have selected a non-interacting ground state which is not orthogonal to the true ground state, The Boson diagram rules are extremely simple. Each propagator is summed over physical single-particle labels and over cotor projections. Since H contains the diagonal unit operator in cofor space, there can be no cotor flips and the color projections must be the same for all Ferrnion propagators comprising a closed toop. When propagators are summed over aft physical quantum numbers and cofor prdectisns, each closed propagator thus acquires a factor of N and there Is no other remnant of the artificial color vrtdabtes, Since the non-interacting Boson ground state has only a single physical orbital occupied, the sum ovtfr M occupied orbitals {A) for Fermions is replaced by a single occupled orbital which we shall denote by (0). The sum aver unoccupied orbitals is denoted as a sum over an infinite set of labels (a) as for Fermions, Thus, there are only two changes relative to Fermlons: each closed toop contributes a factor N and the sum over occupied states (A) Is replaced by a single state (0). Several comments are useful concrtrning the N-dependence and signs. Direct and exchange graphs now d i h r by a factor of N. For example, the Mavtree-Fock diagrams, Eq. 13.27) now yield
For an infinite system in the thermodynamic lim"rt, the overall N dependence must be determined by considering the volume dependence as well as the formal Ilf dependence. Each interaction vertex contributes a factor as before. so the Hartree = Np. Counting independent momenta is now different than energy goes as for krmions, however, since occupied orrespond t o momentum k -- 0.
$
For example, the second order diagram corresponding t o a single sum
E,
--+
will have momenta 0,0, g. and
-S
dk so that the two interactions, one
mamegtum sum, and two closed loops y first appear unintuftive, they play Whereas the minus signs in the Bosan arising from successive depletion a very simple role In generating the factors of the condensate, Consider, for example, evaluating the matrix eiement arising from four potential interactions
by summing contractions. By successive application of Eq. (3.41). the result must be M = N(N- 1)(N- 2 ) ( N - 3). Since the non-vanishing contractions of the operators
3.2 TIME-ORBERED DIAGRAMS
a; i f O are unique, we may represent
M
557
'by the graph
and enumerate all the ways of joining the ends of the upper vertices and lower vertices with downgoing condensate propagators. The 4! contractions generate the following topafogies the indicated number sf tines:
Mote that in (3.46~1the dashed interaction lines are omitted and all distinct permutations of the labels are counted, fncluding a factor of (-1) and N for each closed loop, as required by the Boson diagram rules, we obtain M = " N -6NS llN2 BN = N ( N - I)(N 2 ) [ M 3) as required, Thus, the Fermion minus signs combined with the factor of N from the eolor sums do just what is required t o generate the correct Bose factors, A detailed application of these rules to an Interacting Bose gas is given in Problem 3.3,
-
+
-
-
It is often useful to explicitl~separate the particle components c-ir*(t-t')8(t - t') and hole components --c-'*('-' )8(tt-t) of the zero temperature propagator Go,(t-t') which are distinguish& by the relative time order of t and t< This ismay be done by rewriting each term in the ttxpansion of the exponential of the interactlsn In the farm
Thus, we enumerate time-ordered diagrams representing all contractions with a specific relative time order and omit the factor of *. Adopting the convention that time increases in the upward direction, an upgoing line will always denote a particle state a tf
denote a hole state
A
= c-'~*("-').
/" = c-'ea(t'-t)
and a downgoing line will
't
It will be convenient to use the Hugenholtz
-
convention, in which all matrix elements are antisymrnetrized and all I, IE exchanges are accounted for by where p, is the number of equivalent pairs of lines. Two diagrams are dktinct if they cannot be deformed t o ccrincjde completely in tapotogy
558
PERTURBATION THEORY A T ZERO TEMPERATURE
and direction of arrows while maintaining the same relative time ordering, Thus, In contrast t o the Feynman diagrams (2,10Ib),the following diagrams are distinct
whereas in both cases the following dhagrams are equivalent
With these conventions, there are no symmetiry Factors, and physically the diagrams simply enumerate each time history once and onty once. The integrals over relative time may be perf'ormed by breaking each propagator into a product af factors involving all Intermediate times, For example, if t , is later than,L with the same number af intervening times, particle and hole propagators arc! written as follows:
Note that the hole energies e~ entel into Eq, (3.49) with tbe opposite sign from the particle energies because the relative time entering the propagator is the time at which 'the arrow termhates minus the time at which it begins, The segment of the total propagator for an arbitrary diagram between times t , and then contributes the Factor
where
Physically, Sm is just the excitation energy for a state with p particles and g holes For ground relative to the non-interacting ground state between times t, and $,+l. state observables p = q so that the intermediate state is a p-particle, p-hole excitation and with our sign convention that E , > O and aA .I:Q, gmis positive. Consider now the Integral over all the ordered times i n an arbitrary linked diagram with 7a interactions
3.2 TiME-ORDERED DIAGRAMS
limit
159
Performing the integral over each C m , l .f m 5 n -- X, along the contour Cz in the To--, oo yields
where > O since S,, and i j are positive, Hence, the time integrals of the propagator arising in the ground state energy ykeld
In addition to antisy mmetriited Hugenholtz matrix elements for each interaction vertex (a@[vlyii)=z (aPloi(l 7 6 ) -- ( 67) and Eq. (3,51c), the expansion for the ground state energy E. WOcontains the usual factor (-i)R-'(-l)"g from Table 4.1, the factor from the Hugenholtz rules which together with the explicit enumeration of all time orders completely replaces the symmetry hctor S, and a factor (-l)ns to account far the minus sign associated with each of the hole line propagators. Each energy denominator is positive definite, so that the itj is superfluous. Thus, the final formula for the ground state energy is
&
-
-(Ca
- C, ra) i s computed between each successive Interaction including all the particle lines and hole lines present in that' time interval and tab and n, denote the number of hale lines and equivalent pairs, respectively. An antisyrnmetrized matrix element is included for each interaction, and tar, is obtained by countkg the number of cjased loops in the corresponding diagram with direct matrix elements at each vertex, Wistoricaffy, the expansion for the ground state energy In terms of linked timeordered diagrams. Eq. (3.52). was first obtained by Goldstone (1957) using the Dyson expansion, and is known as the Goldstone expansion. Time-ordered diagrams are t i s instructive to compare this therdore also frequentfy catied Goldstone diagrams. X result with Bdtlouh-Wigner perturbation theory, which has the wrong N dependence in higher orders (see Prolslem 3.6). As shown i~Problem 3.7, Rayleigh-Schradinger perturbation theory may be obtained from Briltouin-Wigner perturbation theory by = )m and systematically expanding energy demoninators has the proper N-dependence in each order. As observed by Brueckner (1959),the terms in Rayleigh-Schrijdin(yeuperturbation theovy which would correspond tcs unlinked diagrams cancel in tow orders of perturbation theory (sec? Problem 3.8) and it is the generality of this cancellation which is established by Goldstone's theorem or Eq. (3,521,
A global energy denominator
xm(w,-x,
PERTURBATION THEORY A T ZERO TEMPERATURE
It60
As an example of the evaluation of a Goldstone diagram for the ground state energy we wilt evaluate the second-order contribution which includes both direct and exchange terms
There are two equivalent pairs, two hole fines, and the associated direct diagram has two closed loops, Hence,
The direct term agrees with the previous, more lengthy calculations in Eq. (3.28) and
(3.343, The rules far the expansion of the ground state expectation value of an operator are analogous and are left as an exercise, One new feature arises in the ewtuation of Green" functions using time-ordered graphs. In addition to r interactions, a oneparticle Gmen" function contains a creation operator at some time ti and an annihilation operator at some time tb. For the case tr > ti the generat form of the relative times and a specific example are shown below,
Including att factors as in the case of E. -- W Oand Faurler transforming from retatlve times Z;f - ti to frequency w in the usual way, the one-particle Green's function may be written
iGi(w) --
{-qr
T T * &
an,
into a product of factors for each time interval between tl and t j and <: t i , w combines with Sm at that It enters precisely as a hole line, Similarly, if intermediate times as a particle line, Thus, performing the time integrals as in Eq. (3.51b) and retaining the irj factors. the general result for the one-particle Green's function for either time order ifs
Breaking e-'@('f-'i)
ti as in
Eq. (3.491, we see that w may be grouped with S, for all t ; 5 m 5
3.2 TINE-ORDERED DIAGRAMS
1161
W here W ,
=
C, r, - CA( A - sgn(tl - ti)@ Eafa- C A ~ A
intewals between tf and ti intervals outside tf and ti
.
(3.576)
A simple mnemonic to correctly include w in the appropriate demoninator is t o add a fictitious propagator, denoted by the dotted line it^ (3.553, running from the annihilation operator at t g to the creation operator at t i e Depending upon the direction of the arrow, it is treated like a particle or hole with single-particle energy o in computing enertgy denominators. (It must, however, be ignored when calculating nh, n h and m,). To illustrate these rules and tlo show how several time-ordered graphs reproduce a single Feym a n graph, we will calculate the second-order one-particle irreducible diagrams contributing to the one-particle Grettn's hnction, Consider first the following time order which includes a two-particle one-hole: intermediate state:
-
__- (-~)l+l -
2&
(-ed#
+
(d9 W
i it))(-C.
1 4ab)(ab l v l dB)
- rb + LB + W + i q ) (-td
+
W
i iq)
where the sign has been determined by eoosidering the direet term which has one closed loop. Separating the direct and exchange terms, the self energy i s
C(d'wldw)= ,,I
(d% 1 C &B
t
v(ab)(ail1 v l d B ) - (d'B / v f crb)(crb 1 v 1 B d ) w -- E~ 3- i q
+
-
(3.583) the direct term of which agrees with (3.395). In this case, the time-ordered rules again provides the result more economkczllly than the sequence of frequency Integrals required for Feynman diagrams. In contrast, there are five difFerent time orders involving an intermediate state with two holes and one particle (in addition t o d and b) represented by the following diagrams:
As shown in Problem 3.10, each term is evaluated as in Eq. f3.12a) and the sum of the five resulting terms yields the self-energy,
162
PERTURSATION THEORY AT Z E R O TEMPERATURE
The overall minus sign relative to Eq. (3.58b) results from the additional hole line, The direct term agah agrees with Eq, (3.39b) and this case illustrates how a general class of topologicalfy 'identical time-ordered dkgrams combine to give a simple result, Whereas most of the diagrams appearing in the Goldstone expansion have the direct physkaal interpretation of enumerating a time histow of the N-particle system, several special cases arise which merit comment. One case is the presence of diagrams which appear to violate the Pauli principle, Although there is a temptation to eliminate such graphs on physical grounds, our dedvation shows that they must be included, Their role i s easily understood by considering the following pair of linked and unlinked graphs which have a propagator for state a appearhng twice at the same time:
Because the right-hand graph has an additknal loop relative; to the left-hand graph and all matvix elements and denominators are identical, the two contributions t o cancel identically and there is no mistake. Since the exponential of (@trole-"aT@l@,) the sum of all linked dlagrams necessau'rly generates all possible unlinked diagrams irrespective of labels, including the right-hand graph in 13.60).it is essential to retain the left-hand graph to preserve the physical cancellation. Similarly, there exist canceling pairs of graphs such as the following
far whkh the linked partner has no physical meaning. Qnce a particle has been scattered out of an occupied state A into an unoccupied state a, it Is impassible to haw a subsequent interaction involving state A corresponding to the left-hand diagram. However, this lrnphysicail contribution exactly cancels the unlhked Pauli violating diagram on the right, and by the preceding argument, must be retained. Note that no corresponding cancellation arises if A in the left-hand graph is replaced by a particle line, since closed toops beginning and ending at the same vertex as in the rlght-hand graph are necessarily hate lines. These special cases only arise far graphs involving for large systems. For very duplicate labels which occur with relative probability small systems, where a large fraction of the graphs are in some sense unphysical, there is no particular advantage t o usbng the linked cluster expansion, and it may be preferable to use Buillouin Wigner perturbation theory or same other formatism which is not deliberately rearranged according to paeicfe number. Goldstone diagrams provide a convenient framework for physicalty motivated re summations of diagrams. The first resummation we consider here corresponds to the independent particle or mean-field approximation, The use of the Hartree-Fock basis to include the average effect of at! the other particles has already been discussed in Problem 1.6 and in Chapter 2 in connection with the use of the Hartree-Eock self-eneru to define single particle propagators, In terms of diagrams, it is convenient to decompose H into H. = T ( I H F and V = CijV ( Z { zj)- UHF and to enumerate all diagrams Then containing both the two-body potential v and the one-body potential -&F.
+
-
3.2 TIME-ORDERED DtAGRAlVIS
163
far every diagram one can draw with the potential -UHF, there is a corresponding diagram with a Marlree-Fock self-energy
may be defined as in Problem 1.6 t o identically cancel all Wartree-Fock selfand bp. energy insertions
Thus, use of the Martree Fock potential pePfarms an infinite summation of all possSble insertions of the WarEree-Fack self-energy in propagators
and includes the physical effect of the mean fidd on the single particle wave functions and propagatars. Just as the independent-particle approx!mat'ron describes the propagation of a single p a ~ i c l eby replacing its interaction with at! the other particles by the mean field, the next resummathon corresponds to the independent pair apgroximatbn which treats the two-body interaction between a pair of parllicles in the medium white replacing the "Iteraction of the pair with aI1 other particles by the mean field. In terms of diagrams, one sums all orders of ladder diagrams in which two interacting paPticles rescatter any number of times in intermediate particle states (using Hartree-Fock propagators from the previous resummation). Thus, Gofdstona diagrams are reexpressed in terms of the Brueckner reaction matrix
which satisfies the integral equation
The hierarchy of resummations may be continued with the solution of the three-body Bethe-Faddeev equation which, using single-parlicle Wartree-Fock propagators and twobody reaction matrix elements, sums all succc?ssiverescatterings af three particles in intermediate partick states. Since time-ordered diagrams clearly separate pafiictes and holes, this sequence of resummations gives rise ta an expansion ordered in the number of hole lines. The hole-lhe expansion is use-ful in treating dense quantum liquids, such as llquld Helium and nuclear matter.
264
PERTURBATIQN THEORY A T ZERO TEMPERATURE
3.3 THE ZERO-TEMPERATURE LIMIT The objective of this final section is to compare the zero-temperature perturbation theory derived in this chapter with the zero-temperature Ihmit of the finite temperature theory of Chapter 2. Followhg the arguments of Kohn and tuttinger (1960).the essential point is evident from considering the ground state energy for Fermions in the lowest order of perturbation theory in which the two theories can diffw. A general format proof may be found "I the work of Luttinger and Ward (1960). At Finite temperature, the energy is given by E = n+pN+r"S, where the chemical potential p must be adjusted such that N = yieids the desired particle number. Thus, we must compawe m il T,fCI - -t- ( p - po)l\Tj with the zero-temperature - WO,Since finite temperature propagators with n, = result zero-temperature propagators with n, ..:f. or O and each zero-temperature graph for - 6 corresponds to the zero-temperature limit of some finite-temperature graph for Ct no,the two tsxpansions agree in most respects, There are just two diRerences between the zero and finite temperature theories which, for simplicity, we will discuss here for the case of translationally Invariant systerns:
no
-%
-
1) The fSnite temperature expansion contains ( p - p o ) N where the chemical potential f,,= IV, need not agree for the non-interacting system, defined such that with the chemical potential of the interacting system defined by =,I N.
2) The finite temperature expansion admits anamalous graphs such as
in
which the same state appears as both an upgoing and downgoing fine, Mote that since we have assumed translational invariance throughout this section we will label states by single-particfe momenta. At finke temperature, nk(l - na) does not vanish at the Fermi surface whereas such anomalous graphs are never present i n the zero temperature theory. f a understand the role of these two effects, it 3s useful t o note first that; they both vanish for finite systems, The chemical potential enters Q l]-' and as dto only through the occupation numbers nk = eB(ek-') - no)contains p only through T W , nk -+ @ ( p- E ~ ) . Hence
&?;(a
-+
- E&).
+
-
For discrete energies c* in a finite periodic box, the &-function Bit is never satisfied so that = and hence p = po. Similarly, consider the lowest order (second order in the potential) anomalous diagram -+ & ( p
F
3,s THE ZERO-TEMPERATURE LIMIT
3.65
where Uh denotes the Hartree Fock potential
The h t u r e that. an anomalous diagram yields B(1 - nk)m is general since oppositely directed propagators of the same k yield the factors (1 - n k ) ( - n k ) with no exponential so that integration over relative time yields an additional fador p. In the zero-temperature limit, this factor yields
-
Hence, like the term IV(p m), anomalous diagrams produce S-functions which are never satisftd for discrete r k . Therefore, For finite systems the T --, O limit agrees with the zero-temperature expandon. For infinite systems, we wish to compare finite and zero-temperature theories in the thermodynamic limit, so we consider the double limit lhT4 (Linrv+,), , In this case both the anomalous diagrams and N I P - h ) are nan-vanishing, but as suggested by the fact that bath terms yield the sane 6-function, it is possible that they may cancel- The criterion for cancellation is easy to see in lowest order. &et us separate the Grand potentbl into the following terms:
where hfs denotes all the diagrams which have zero-temperature counterparts and CZA denotes all the remaining anomalous diagrams. Note that thb ground state energy calculated in zero-temperature theory may be written
-+
This follows because W O =.I hZo N p ~ E * = E * @ and each of the diagrams in flz goer; Over to its zero temperature counterpart evaluated with propagators evaluated at 1" = 8 and PO. Thus, we may write the difwrence between the zero-temperature limit of Gn'tte-temperature theory and E. as
where p and ~;losatisfy
PERTURBATION TWEQRV AT ZERO TEMPERATURE
'366
Since ftz is first-order in the potential and QA is second order, it is straightfomard t o use (3.69b) t a solve for p m to first arder and to use this result to obtain A t o second order. The details are exptained in Problem 3.2 and denothng the order in the potential t o whkh quantities are evaluatttcf by superscripts, the result is
-
where the average of any function of k at the Fermi surface is defined by
and U denotes the Haeree-Fock potential, Eq, (3.65b). The physical essence of the result, Eq. (3.10) is the relation between the symmetry of the non-interacting zero-temperature ground state and the exact ground state, When A = 0,both theories yield the physical ground state. When A < 0,the 2' -+Q limit of the Plnlte-temperature theory yields the ground state whereas the zero temperature theory produces an excited ctigenstate with some incorrect symmetry it has inherited from the non-interacting ground state, Since A measures the fluctuations In II;, at the unpe@urbed Fermi surface, it will vanish for a spherical Fermi surface and a HarltreeFock potential Uj, which depends only on I k 1. This will occur, for example, when a is the kinetic energy or any single-particle energy depending on 1 k 1 and when o(q 3)is a central potential, The cancellation has also been demonstrated far spin 112 particles interacting with tenmr forces. The feature which can destroy the canceltation and produce ~egativeA is a digerence h the symmetry of the non-interacting problem, as reflected In the Fermi surface, and the interacting problem, as reflected in U,A simple example is a system of spin112 particles in an external magnetic field. Taking .& as the kinetic energy, the Fermi surface is two equal Fermi spheres for spin up and sph down. Repeating the derivation of Eq, (3.70)for a one-body perturbation -@Bog simply replaces U ( k ) by the matrix element of the one-body operator, -pBm where m is the spin project'ran. In this case, the vafiance of pEZm evaluated over the tvvo Fermi spheres with m = =1=Is 1 non-vanishing, This corresponds to the fact that the perturbation cannot flip spins so there is no way t o get from the non-interacting state with tvvo equal Fermi spheres to the true interacting graund state with unequal Fermi spheres, We wit! return t o this =ample again in Chapter 5. In conclusion, it is evident that the decompositian H. = T U, V = er U plays a difFerent role in the finite and zero temperature theories, In both cases, V must be small in some sense far the perturbation expansban to be useful, In addition, a t zero-temperature & defines I@@)from which we determine = limilm -+ -m e - i T @ ( H ~ + V ) Thus 19)may- be trapped in the space of some and sinnply produce the lowest elgenstate in that space, In symmetry selected by
-
+
-
IQ)
PROBLEMS FOR CHAPTER 3
167
contrast, the finite temperature theory evaluates a trace which necessadly has access to all possibfa symmetries and will not become trapped. The ultimate concfusion,then, is that there is nothing Fundamentally wrong with the zero-temperature theory. Rather, ta do sensible physics, one must pick an intelligent choice for & such that f@o)has the dght symmetries and corresponds to the correct physical phase of the system, and such that V = u - U Is sufficiendy snafl to abtain reasonable clonvergence, PROBLEMS FOR CHAPTER 3
Several problems we designed to establish connections betwen the formalkm we have presented and traditional treELLmtfnt~,both to deepen understanding of the physics and to facilitate reding dhe original literature. Problem 1 shows haw to obtain the linked-cluster expansion uskg the Dyson expansion and the GelXMann Law theorem. Problems 8-8 dernonstrake how BrilEouirr-Wignctr perturbakiorr theow, which h a inappropriade dependence on the pa~ticlenumber N, can be reamanged to obtain Raybigh-SchrSinger pertarbation theofy which, stfter $he cancellation of unlinked terms, hils the correct N-dependence, A IiGtle known but interesting property of B r i l l o u i e perturbation theam is derived h Problem 8. Problems 2 and 10 work out details af results cited in the text wsociated with the cancellation of anomalous diagrams and combining top~logicdlyequivalent t i m e ordered diagrams. Zero temperature theov ist applied to the dilute b s e gaha in Problem 3 and to Fermiions in one dinrensian interwting via 6-fiznction forcm in Problems 4 and 5.
PROBLEM 3.1 Zerotemperatureperturbationfheorymayabobeobtai~edby the following traclitbnaf derivation. Let $ and H --- HO .Hldenote the wave funclion and Hamilbanian in the Schijdinger representatlion, define the evolution opemtor such that +(t) E U ( t ,to)+(to) and allow HI to be time dependeat. The comesponding quantities in the ixlter~tianrepresentstiorr, denoted in tkia problem by a ckcumflex, are defined as follows:
+
a)
born the time-dependent Sehrijdinger equation, show that
with the boundary condition fi(0) = 1 which may be integrated to obtain
dt'
By iterating, obtain the Dyson expansion
B, (t')O(tf - to) .
168
PERTURBATtQN THEQRV A T ZERO TEMPERATURE
of Ha with energy WO b) The essential idea is to start with an eigenstate agd tarn an the inLeraetion V very slow'ty so that \#a) adiabatically evolves into an eigen~tateof the fully interacting Hamiltonian. If is convenienL to pwameterke this switching-on by definingl!& = Xc-€IgIP, and to consider the limit c -t 0. The come~pondhgevolution operatos is &fiaed by &(L, to). The GeIl-Mann Low theorem (L961)then crtates %bat
+
is arr etigenetate of HQ AV, assuming that the right-hand side exisls in all orders of pe&mbafion theoq. The Emit does not ex&& sepwatelty for the numerator, so the denominator i s ctssenkial to remave an o6lhewke infinite phwe ~sociatedwith nnhked diapams. To prove the theorem, &QW
Note fhat $be heeraction representation operator8 me evaluaked at t = 0 and so me equd 60 the comesponding SchrGdinger opcsrador. Thus, ehaw
+
where E(E)= W O
(c) Obtain the familiar diagram expansion by substituting the Dyson expansion for U.(O, - W ) in the numerator and denominator of
Use the operator form of Widr's theorem in Problem 2.8 and pedarm the time integrals over relative times to show that the numerator yields the sum of all diagrams times the product of dl posgib1e linked diagrams. Similarly, show linked to al(0) that the denominator just cancels oft the product of all possible linked diagrams a d &Er& the final expasion may be d t t e n :
PROBLEMS FOR CHAPTER 3
169
Note that in co~trale31to the ddvations in, the t;ex$, this de~vationis3 not r~ericted to the g r o u ~ dstate, but i m principle may be applied to m y eigenstate. PROBLEM 3.2 CancetlatZon of Anonalwr Diagrsms. a) D h v e the h t h e of Eq. (3.10a) as follows. Note that nois bdependent is 6wt order in the irzterwtion, and fllR is erecand order. Show of the potentiaf, by q a n a i n g the left hand side of Eq. (3,69bf wound p0 that to h& d e r in the potential,
Sinzifmly, axgand Eq. f3.119a) wound h,
a d show that t b b t line of Q. (3.70a) L obtained when &l second order terns we retaked.
Using Eq. (2.72) show b) Now evaluate all the quantities appearing in il(').
Using Eg. (2.100b) which evaluates the Hartree-Fock contribution a!'),show
where U is defined in Eq. (3.6513). Finally, using Eq. (3.6Sa) for fif' show of'
-fi Er
31"-0
6(eL-P)@. Substitntion of these quantities in the result of part (a) yields the and result, Eq, (3.70). P RQBLEM 3.3* Boos Gas. Considcsl:the binding energy per p d i c l e of a mute Base gm with a repulsive interaction. Daute means that the scatterkg length a h m u d lem than the average particle spacing (n)-'l3. In this h i t , S-wave scattering dominates and the refevanL law momenlum matrk elements of the gotentiaf w e independat of the momentum. To Gmt or&erin %a5,the answer is
It is instructive to solve the problem two different ways: Make w e of the faet that the number of p&icbs excited out of the condensate is macb smaller t h m the number left in, 80 that hgeraetiona between. excited
170
PERTURBATION THEORY A T ZERO TEMPERATURE
pasticles may be neglected compwed to interactions between condensed and excited particles and between. condensed pwticles. Hence, o b t a k the effective Bamiitonian:
No may be removed by writing
The resulting Hea may be diagonali~edby the kansformrstian:
and its adjoint;, where A, is specified by the mquirement that He@be diagonal in a,. FinaUy, write out the s e c a d ordw per&urb;ationexpresdon for the scatterbe length in terms of V and efiminatre V from the ground state enww. The reaulding intepah expressed in, terns of a converge and yield $fie desired reault, The excitation spectrum i s given by the coefficient of aria, in the diagonalked HeE.Note that there ia no gap and thag $he enerw is l k e w in p as p -+ 0. Coxngme this slope with the hsouxtd velocity calculated h m the appropriate derivaf ive of the toed enerw m a functkn of density.
Is) Mow, spproach the probbm diagramakicrtlly, using number-conseming % e r et emperature Bason. diagrams, 1) Firs&,note t h s ~the scatterkg length pertains to fhe T-matrix whieh is the ladder sum of a21 possible successive i&eractions between two pwicfes. Shce S-wave scattering dominates in the lorw demity i h i k , define a pa3euao-potential, V = U6'(2% - zz) which reproduces the scattering length, and use V to represent all lddelrs of the true intmrtction, Calculate U ,
2) f i r a diagram with m inferaetions, Gnd the topologies which h%ve the leading N dependence, where N is the number of Bossns. Show that any Hwfree insertion )- - -0 increases the order of V and N by one. Hence, remove all such insertions by appropriate definition of Ha. Qf the remaining diapamns, show &ha6 the ring: diagrams dominate, Qpical ring diag-rams containing only fommd-going chains and containing a single bxkwwd-going segment are shown below:
PROBLEMS FOR CHAPTER 3
371
3) The easiest set to sum is all ring diagrams with forward-going chains of the fa- shown at the left. Note that whereas ewh ring diagram individudly divwgces, the sum is well-bdaved. Show that the result; is
This result was obtained by Brueckner and Sawada (1957)and the (nas)'/2 t e r n agrees with the result in part 1 to within 4%.
4) Sum all ring diagrams with one bukward-going segment and determine the remaining fraction& emor. PROBILEM 3.4 One Dimenstanat Interacting hrmi Gas. Consider a translationally invariant system of Fernions in one dimension interacting with the attractive &-functionpotential of Problem 1.9 and having spin degenerwy 2S 1. Recall that gip is the expansion parameter for a uniform density.
+
a) Evaluate the second-order energy and compare with the lowest order result calculated previously. Sketch the energy ks a function of density for both cases and dedermhe how the mbimurn. changes in secand order.
b) Calculate the sum of all ladder graphs with intermediate states above the F e m i sea:
Them, evaluate
-1
C
( M NI G(EM+ E N ) I M N ) - ( M NI G ( E u $. E&) I N M )
M.X
to obtain the contribution to the pound state energy. You may be lefk with one integral which cannot be done analytically, in which case expand in the low density limit, and compare with the first and sewnd order resulls. Eow does this result affect the comparison with the energy of Iocali~ledrsolutions in Probhm 1.91
PROBLEM 3.5* li/N Expansion of Ground Stare Expgctat'ian Values for the One-Dimensional S-hnctian Problem. X& waw shown in Problem 3.9 that the Hartree-Foek approximatian yields the comect N3 term in the energy for a bound state af M Fermiona with spin degeneracy 2 5 i1= N .
-*#Pu"
a) Uou have abeady found the lowest eigenfunction and eige~value of l. tr (z)+ UHF(z)#(z)= e +(z) for the HF potential
PERTURBATION THEQRV AT ZERO TEMPERATURE
172
of all higher eigenfixnctions and eigenvdues. (Hht: Look in Mome and F e s h b d (19fi3for piay wonnd with e entiah and hmerboEe tangents.) Note that atl eigenPunctions are funcfions of
Naw obtain tha complete set
consider enurmerdhg aIf tht?-ordered graphs for the energy in thk piZ (IHP(zg))). Prove that the N-dependence of an w b i t r q diagram is Nem1+%where C i$ the number a f clased loops and I is t;he number of intawtioas. Hence, emumerate all the graph of order N3,Doea this explah why the Bfi"energy w w eomect to order Ns?
b) Now
HF basis. (i.e., X. =
xi(W +
c) D d n e the class af graphs of d e f Na, One of them should akeaxly be included in the HF enerm. Write the e x p r m i o ~for th this class. (Note evaluation of thk expremio~t,yields almost all of the O (NZ) error.) d j Now consider tl_ie expectation vdue of the onebody density opera$or,
h(%)= f;". G(s: - Q), D d v e a formula for the N-dependttnce of an m b i t r w density d i a p m an4 evsluate the leadhg order ap
ation. For what other opr?r;ztora
is i%possiMe to derive generd wpressioas for the N-dependence of individud ilia,-
pams?
PROBLEM 3.6 and
Brillourn-Wfgnar Perturbation Theory.
+
f & HI)f$a) =. Ea with the normdiaatian condition sad define the prejector Q I=. 1 - 1+o)(401
Let .Ha
==
(&ol+o)= (+ol&)
WOl&~) ==
1
a) Show that
a d thus
of volume V in the limit cm at fixed p, and let Ha be the khetic energy operz~torand & a twois the usual Eartree-Fock body potential v(ri - rj). Show that whereas (+olH2 is energy which is proportional to N,the next order correction C,+@ independent of N. In a n e ~ k the g N-dependence, it la useful. to nrse momentum conservation to enumerate all states 1)4, to which H& can connect It#@) and to e x m h e the fwtors of It a d 1V wkhg from m&* elements, nromexrt;um sams, and energy denominators. Consider whether this problem persists in higher order and explain the implications for convergence of the perturbation series for large N.
b) Now, consider N Fermiona in a periodic box
N
--+
PROBLEMS FOR CHAPTER 3
173
PROBLEM 3.7 Raylelgh-SchredEnger Perturbation The~ry, The form of pertubation theory used k mwt quantum ntechaaia tatbooks is RayfeighSdr6dinger perturbation theoq. Let &l#,) = a d (EO XHl)i&)=. &l+a), where we now aocia8e a f o m d expicn~ictnpasmeter X a63 follows; E0 = Xnr, and = Xmln) We use the same narmaliradion as before (Qlol$o) = I t which implies (Oln) = 6ma, denote Q = 1 - 140)(#ol, and note that eo =: and IQ) = l&o). a) Sy equakixlg terns of: each power of h in the ScSriidinger equat;ion, o h & the recnmion relations
+
WItite ouii the explicit expressions for In) and
6,
though
€3,
b) This rault may also be obtained from Brillouin-Wigner pepfurbakian theory by expanding Lhe energ-y.denaminador
and mbstitutiag the series f;or tution through, €8.
(m- E@)ewh time it wkes, C
c) Gompme &heN dependence of ~2 far N Fermions in a box iateracting via a two-body potential with the rmult obtafned h Problem 3.8. Note that it wils the by which solved this problem. replacement of
PROBLEM 3,8 Csnltlettatton of Unllnkad Terms. Although, W shown ia the preceding problem, 62 hw the comecd H-dependence, Rayleigh, SchrGdhger p e ~ w bation theor;). aka contai~lst e r m with higher powem of N. For the t h e o do ~ make sense, o m must prove th& the@@ term8 cancel, 2
140) eonkributing to es has, contria) Show that the term (+olHl butions proportional to N2i t i ~well arr cong~butionsproporl;ional to N. Associate dig agrams with tltrege eantributions and relate the cannectedness to the N-dependence. 2
Show that the term -(dtolW1 140)(40IjFfr f40)in E 3 e x ~ t l y cancels the N 2 contribution in part (a). With sufficient effort, similar cancellation of unlinked diagrams can be vc?rified in the nex6 few ordem of perturbation tfiatoq, and it WM on this basis that the linked cluster theorem was firs&conjmfured.
b)
P RQ BLEW 3.8 Variational Properly of Britloutn-Wfgner Perturbation Theory Consider the fallowing trial wwe fanetion:
174
PERTURBATfQN THEORY A T ZERO TEMPERATURE
where x is to be m g w d d ai)
21t3
a parameter.
Show that
m
l+,). Noh that A&(&) comesponds to the where b % ( z ) Ei (4olH1 nthterm in the Brillouin-Wigner expansion for the energy in Problem 3.6. b) Show (XN( Z ) ~ X N( a ) ) 2 1 and use the variational principle to obtain a bound on State explicitly any corrrdii.tions x must sadisfy for t h i ~bound to be valid" C) Thus, show & 5 W. + AEm(8)for z = + AE,(a). Verify $h& this a safisfim the mnditiona estabhhe-d in fb). Nence, show that all odd
~2~
order8 of B ~ l ~ u ~ - V V i g pe&arbation, ner theoq yield upper bounds on the graund state enerfsfr.
PROBLEM 3.10 Summation of Tapotogteally Equivatent Time-Ordered
Qlal-
grams.
a) Consider the energy denominators -S:si contributing to each of the Green's fanction d i a p a s in Eq. (3.59a). Show that the sam of fhe first &WO diaa;rms yields
Shilariy$combhe the thifd and fourth diagrams in
E'hally, akdd the a t h diagram to
Eq. (3.59a) to obtajn
and d d the resulk to 53+4to obtain
Hence, by including the proper signs and factors, obtain the self-energy Eq. (3.5913).
PROBLEMS FOR CHAPTER 3
175
b) h a m the slgebra ia (4,it is evident that individual timeodered diagrams havhg glob& enagy denamhatom depending on the enthe state of the 8yrst;ern may be combhed rrrud that the find resalt is a product of components each of whi& has eaergy deaaminatom depending upon pmp~ilgatamwithia th& compoaemd, h t b cage of the Green's function, we obtained Gdt(w) E@), where Q and @&-ere independent of propagatom in C and enerw denominatom in C were i;mdependent of d md d'. Beghe, Randow, and Fetschek (t(F83) showed that Bhis g e n e r a k d time orde6ag is quite genrtrd. As a s h p l e example, comkme the enerm
mu)
fo obt&n a produet in which the energy denominator of each uni6 only h v o ~ v eits~ own propaga$om. Thh result may be generalirmed $0 the sum abdispamrs of w b i t r q comple~tyin which the position of the top interactioxr in the right bwd iaserttion b held fixed a d fhe lower inkcll.a;ctiom asume all thrsarderhgs consistent wikh the topology,
CHAPTER 4 ORDER PARAMETERS AND BROKEN SYMMETRY One of the goats of many-body theory is to understand the phases in which matter exists and the transitbons between these phases, This chapter addresses the fundamental role of order parameters and symmetw breaking in characterizing phases and understanding phase transitions.
To prepare for the subsequent quantitative treatment of order parameters, mean field theoy, and fiuctuations, it is useful to review briefly several elementary aspects of phase transitions. PHASES OF TWO FAMILIAR SYSTEMS
A ferromagnetic material, such as iron, and a typkaaf substance having solid, fiquid, and gas phases Illustrate the essential features of how the phase of a thermadyniamic system changes as a function of the temperature and an external thermodynamic field. In addition to the temperature and external field, it wZII be useful to characterize the state of the system by a thermodynamic variable conjugate to the external field and to study the surface of the equation of state in the space of three variables: the temperature. the external field, and the canjugate variable, The surface of the equation of state and phase diagrams for a ferromagnet are shown in Fig. 4.1 The magnetic field is the external field which together with the temperature determine the state of the system, and the phase dhgram in the H - I' plane is shown in the upper left of Fig, 4.Z. Betow the critical temperature G,the system is spontaneously magnetized and the state of the system is characterized b~ the magnetization l@. the thermal average of the microscopic spins. If we choose X to be along the B axis, these will be two ferrornagnetic shzltes at low temperature, the f state for positive H and the 4 state for negative: H. These two states are separated by a phase boundary, indicated by the soilid line, This p h a bounday ~ terminates at a critical point C at temperature T,,and above the criticat point, the system has a single paramagnetic phase. The system may either be brought from the ferromagnetk 4 phase to the ferromagnetic f phase discontinuously, by selecting a path which crosses the phase boundary, or cantinuously by choosing a path which travels around the critical point into the paramagnetic phase, The magnetization M is the thermodynamic variable conjugate to .H, and a more complete characterization of the phases of the ferromagnet is provided by the surFace of the equation of state In the space of M , H, and T sketched in the upper right of Fig. 4.1, The vertical shaded surl'itce denotes the region of phase coexistence separating the 7 and ferromagnetic phases, Isotherms below the critical temperature: T<,at and above the critical temperature TB,are shown on the the critical temperature T",, surface of the equation of state, The top view of this sufiace along the negative M axis corresponds to the phase diagram in the H - T" plane we have already discussed.
4.1 INTRODUCTION
Fig. 4.1,
g77
Phase ctlagrams; for a fsrromagnc41, The suflace of the equatlon
In thca space of magnetlratlon M ,external magnstlc: field H,and temperature T Is shown tn the upper right, Pr@=tlons of thlo surface are shown In the H T,.kf I",and hf g ptanas. tsothermr are shown at the critical temperature F,, at rc below G, and at T3 above of state
-
-
-
Viewed from the right, along the negative T axis, this surface corresponds to the set of isotherms plotted in the M - fir plane in the lower right, For TB above T,,the eumes are continuaus and represent the dngle paramagnetk phase. For !Kc Below TG, the cuwed segments af the isotherm represent the fernornagmetic f and 4 phases and the vertlcai segment f oining points A and B reprgsents the region of coexistence of the two phases at zero field. Mote that the magnetization is non-analytic at A and B and that these pcllnts correspond to a single point an the phase boundary in the .R - a" pltane. Along the criticat isotherm, the coexistence region is reduced t o a single point, the critical point, and the two phases coincide. The final view of the ferrornagnet phase suvface, the front view af the H ads, yields the boundary of the coexistence region In the M - T plane shown at the lower left. We observe that at zero H Ficttd, M is non-zero below I",, approaches zero a t Tc, and is zero above TGand we will subsequentty use this behavior to identify M as the order parameter, The analogous phase su6ace and projections for a tlypiical* substance having solid, v o t e that water is atypical in the sense that ice is less dense than water at the freezing point and is not considered in the present diseuss'ton.
278
ORDER PARAMETERS AND BROKEN SYMMETRY
Fluid
Fig, 4.2 Phase dlaprams far a substance havlng solld, liquid, and g a s phases. The surPacrt o l the equation of state Is sketchad In the space of the denslty p, the pressure p,and the temperature T. To S ~ O Wthe reqlon of the crltlcal polnt In sufficient detatl, all three axes havs suppressed zeros. Prolsctlons and isotherms are shown a s In FIG. 4.1,
liquid and gas phases are shown in Fig, 4.2. The pressure is the external thermodynamic field, and the P - T phase diagram is shown at the upper left. A t low temperature, depending upon the pressure, the materlaf can exist as a gas, liquid, or solid. At high temperature, the liquid and gas phases become ind'rstinguishabte and the material has a unique fluid phase, The liquid-gas phase bounday termhnats at the critical point marked 6 , having critical temperature T, and pressure 8. In the present discussion, we wilt be particufarty interested In the behavior in the vicinity of the critkcal point, and its similarity to the prreceding example of the ferromagnet, The tiquld, gas, and fluid phases are analogous to the ferrornagnetic ferromagnetic $ and the paramagnetic phases respectively, and as in the prevbous case, it is possible to go continuously from the gas to liquid phases by passing around the critical point through the fluid phase. Arthough the thermodynamic variable conjugate to the field P is often chosen to be the volume, the analogy to the magnetic: case is most: evident if we use the density p as the conjugate variable (p = $ where v is the specific volume). The surface of the equation of state in the space of p, P , and I" is shown in the upper right and isotherms in the p - P plane are plotted in the tower right. Again, the vertical segment
A -- B of the isotherm Tx below the cdtical temperature represents the region of phase coexistence, this time between gas and liquid. The projection onto the p I" plane in the lower left yields a phase boundary near the critical point c analogous t o that already observed for the magnetization as a function of temperature, and will subsequently lead us to identify Pfiquid Pgaa as the order parameter. Thus, the two surfaces of the equations of state in Figs. 4.1 and 4.2 show the same essential behavior in the vicinity of the critical point, The magnetic case Is somewhat simpler t o visualize because the coexistence region lies precisely In the M 2' plane whereas the tlquid gas coexistence region is a cuwed surface, In both cases, below the critical temperature a thermodynamic variable, M or p, varies discontinuously across the coexistence surface as a function of the codugate field, H or P and the discontinuity approaches zero at the critical point. Since non-analytic behavior in thermodynamic wrlables such as ME, IT) is characteristic of phase transitions, It is usltfuf to note the possbble reasons for which the c-* could besum of analytic functions comprising the partition function Z = come non-analytic, The energies Em could be non-analytic functions of the parameters of the system as in the case of the quaelc oscillator in Section 2.t. The finite sum over n could diverge as in the case of the hydrogen atom, Alternatively, non-analytidty could arise from taking the zero temperature limit or from taking the thermodynamic Iimit in which the number of degrees of freedom goes t o infinity, Except in special cases, we shall see that the orkin of non-anafyticity its in fact the thermodynamic limit.
-
-
-
C,
PNEMQMENOLBGICAL LANaAU TWEQRV Considerable insight into the structure of the general theory developed subsequently is provided by a purely phenomenological theory by Landau (Landau and Lifshitz, 1958). In this theory, an order parameter for a phase transition is defined as a quantity which vanishes in one phase, called the disordered phase, and is non-zero in the other phase, called the ordered phase, An order parameter may not exist for some transitians, such as the Kosterlitz-Thouless transition in a two dhensional classical spin system, and when it does exist, it is not unique (since any power also satis5es the definition). For our present purpose, it wilt always be possible to choose the order parameter t o be the thermodynamic average of an obsembte, We will also use the Landau definition that a transition is first order if the order parameter Is discontinuous at the transition point and scteond order if it is continuous," In the case of a ferromagnet. we observe from Fig. 4.1 that the magnetization-per spin is an order parameter, We wilt use the convention that the totali magnetization M is defined as the thermodynamic average of the spin and that 8%denotes the magnetization per spin
The magnetization vanishes In the high-temperature paramagnetic phase, and the phase transition is second order since m is continuous. The physical order in this case is the
*
Mote that for n > 2, this d i f i r s from the Ehrenfest deFinition of an n ' h r d e r transition as one in which is the lowest discontinuous derivative.
$843
ORDER PARAMETERS AND BROKEN SYMMETRY
Fig. 4.3
The Landau functlon l ( m , HIa") for a farrctmagnat.
alignment of the microscopic spins, For the case of the liquid-gas phase trans-ition, it is evident that pli,,g,ct - Peru, is a possible order parameter and that the transitFan fs also second order. The phenomenologirca! thwrg is based on the Landau function Lfm,H,T], a function of the order parameter, which we will denote as m in this discussion, the canjugate field, which we! will denote as H , and the temperature .'2 The Landau function has the property that at any fixed H and T,the state of the system i s s p e l f i d by the absolute minimum of 4 with respect to m. The only physicaf constraint on the farm of l is that it be consistent with the known symenetries of the system, One of the great accomplishments of this theory is its description of the non-analytic behavior at phastt transitions in terms of the discantlnuous jumps fn the position of the absolute minimum of a function which is Itself vaying continuously with T and H. Although we will eventually see a clear relation between the Landau f'unction and an energy functional appearing in a functional Onlttgrat for the exact partition functjon, for the present let us simply see: how the observed ferromagnetic phase transition i s descrilbd by the functian g(m, H,T ) sketched In Fig.
4.3
re,
First, consider the case a" > shown in the top row of Fig. 4.3. As an external field ir applied tending to align 6% with 8. the minimum of & is continuously
4.2 INTRODUCTION
183
d'tsptaccld kom negative to positive m consistent with the continuous behavior in the paramagnetic region of Fig. 4.1, Below I',, hwever, the absolute minimum jumps discontinuously From -m to when N passes through zero, reproducing the discontinuous behavior in the ferromagnetic region, Thus, the doubte minimum provides a very simple descriflon of a first-order phase transition. Futthermore, if we fook ahead an4 anticipate the fact that the minimum of the Landau function will correspond to the stationary phase appmxlmation to the exact theopy, which must be corrected by fluctuations around the minimum, we sm that since there is atways a barrier between the two minima far 3" .c= Te, there ES no reason for the fluctuations t o change appreciably as H passes through zero. at zero Reld as the tempemture decreases Now consider the dependence of through T, as shown by the middle d a m n of FEg, 4.3. As T approaches l",, the minimum bsomes flatter and flatter, and one might expect the fluctuations t o grow without bound, infinitesimally belaw T,, the dngle minimum b'rfurcates into two infinitesimally generating , the strucseparatd minima, which continue to separate with decreasing !l' ture obsenrd In the m- 2" phase diagram of Fig. 4.1. This generic behavior of a single mhinrutn Battening out and finatfy formlng two degenerate mfnfma thus describes a second-order phase transition. tn discussing classicaf or quantum spin systems, there Is a conventional nomenclature designating the number af spin components, or tltquivalertt!y, the dimensionaliq of the space in which the spins can move. [sing spins have a single component which fs restrict& to be parallel or antiparatid to a fixed axls and is spwih'd by the values =fiL Spins f n the z y model have two components in a plane and Hdsenberg spins have three components in three dimendons, A generic name for a spin model with n components it the O(n) model. tn the case of !sing spins, which can onfy have the values f 3, the magnetization would only paint in one of Wo dktinct diretions, and the Landau function would Iie In a plane as denotd by the sotid cuwes in Fig. 4.3. f i r S- y spins, the Landau function is a sudace f n three dimensions, and this three-dlmensionaf character is indicated by the dotted lines in the bottom representing the local minimum for each angle. Similarly, the Xeisenberg model is described by a four-dimensional suIf51ce which we wilt not attempt to sketch, Because the order parameter m changes corrt!nuously at a seeond order phase transition, the essential physics may often be seen by expanding C in the immediate nelghborhood of the transition as a power series In m. To the w e n t to which only a few turns are relevant* and the form of these terms is highly constrained by symmetry. one may identify essentially universal behador shared by classes of diverse physical systems. Such uniwsatity is already evident in the basic similarity of the phase suvfaces In the vicinlty of the criticat points in Figs, 4.1 and 4.2. For the case of a magnetic system. let us expand L(m, H,T ) through fourth order (the sixth order care is considwd in Problem 4.1).
-
* Thue is no guarantee that &he series may be truncated, and we shall subsequently see by dimenshnal analysis alone that all orders are relevant in two dimensions.
182
ORDER PARAMETERS AND BROKEN SYMMETRY
As dtscussed in connection with the m"rddle column of Fig. 4.3, the conditions for a second order transition are
a3E -i3L!- - iS21 ---a "m---
dm
am
am
9
a4.c > O dm
(second order)
.
(4,Sa)
The second derivative must vanish because the curve changes from concave upward to concave downward and the third derivative must vanish t o insure that the critical pohnt is ii fact a minimum. Similarly. from the bottom row, the conditions for a first order transition are =l
l(m,) =. l(m, 1
0
(firs$ order)
.
The form of l may be further restricted by using the symmetry of the problem and expanding each of the coeRicients a,(N,T ) in the vicinity of the critical point in terms of the reduced parameters lE T and h zz W - HC-- H a, ( H , T)= 6, i-cnh
+ dmt .
f 4.3tt)
Since, by inversion symmetry, l ( m , H , T ) L(-m, - H , 571, the only invariants are even powers of m, even powers of h or combinations of odd powers of m and odd powers of H so that c, =. O for even n and kr, .= d, = O for odd n. Furthermore, since a2 must change sign at the critical point, b2 == 6 and dz r O and since a4 > Q in the vicinity of the critical point, b4 r O and dz is "trelevant. Hence, the most general form for the Landau Function near the criticat point is . ; L
In the present case, this expandon succinctly summarizes the essential features of Fig. 4.3, tn the case of crystals, with point group and translational symmetries and more eompIicated order parameters, the Landau rule that the symmetry group of the tower symmetry state must be a subgroup of the higher symmetry state for a second order transition provides a powerful tool for identifying the invariants which may appear in and thus characterize the possible phase transitions. To avoid misunderstanding, it is important to distinguish the Landau function from another function which also has the property that its minimum specifies the state of the system. Recalf from the discussion in connection with Eqs, (2.262 2.163) that
-
if the free energy is defined
c-@~(")
=
re-@('-'*^^
then
and the Gibbs free energy is the Legendre transform with the property
Fig. 4.4 T h e Helmholtz Free energy F ( H ) , the Clbbs free energy and the tandau function t ( M ) for the fBrromattl)~tbelow G,
I'(M).
E
At zero field, then. the state of the system is specified by the condition = O so that when M is an order parameiter, I"(M) appears to satisfy the definition of the Landau function, In generat, at finite field, one is tempted to consider the fcsftowlng function as the Landau function
= O and its minimum specifies the state. (Note that the parameter is distinct from the function W ( M ) , ) Above T,. where: the isotherms in Fig. 4.1 are smooth curves, becomes qualitatively like the top row of Fig. 4.3 aed there is no problem with this assoclatian. Below the critical point, however, there is an essential digerence between the behavior of f and E which i s shown in Fig. 4.4 for = I'). Note that by Eq. (4.5~) = O a t the points the zero field case (where M = &Mo corresponding to values of M in the ferromagnetic phase and the cuwe bends conthnously upwards outside these points, However, because the isotherm M < &, = F(R = Of is a constant has H = O for all values of -& between these points. This non-anatytic behavior of I' is thus qualitatively different from the continuous double well of the phenomenologicat Landau function. Although, by definition, it describes the phase transition exactly, it does not embody the essential feature of the tandau function af describing a phase transition in terms of competing minima of a continuously varying smooth function. One useful resuit of the correspondence between and L: outside the region of phase wexistence is a clarification of the eartier discussion of fluctuations, The magnetic susceptibility directly measures the spin fluctuations%since
NT
E
3
184
ORDER PARAMETERS AND BROKEN SYMMETRY
By Egs. (4.5) the susceptibility X is also directly related to the second derivative of
and hence the spin fluctuations diverge as the curvature of critical point.
l
approaches zero at the
BROKEN SYMMETRY The Landau function sketched in Fig. 4 3 clearly Q'isglays the phenomenon of symmetry breaking whkh we wSII obsenre in mean field theory. Atthaugh the Landau function for H = Q, 2" fl", displays all the symmetries of the magnetic Mamiltonian, its minima break the symmetry, For the case of [sing spins, the system is invariant under global spin reflections Si --+ --gi and the tandau function is a symmetric doubife welt possessing the same symmetry. However, each of the two mlnima of the double well break the symmetry and transform into each other under spin reflection. Far s- y and MeEsenberg spins, the Landau function is a surface with a continuous set of degenerate minima reflecting the symmetry with respect to conthuous global spin rotations. The con6gurations corresponding to each of these minima break the spin rotation symmetry and transform into one another under spin rotation, This structure Ss quite general, F"or a Hamiltonian with a symmetry group 9, above T, the disordered phase will be invariant under $whereas l below T,, the degenerate disordered phases will not be invariant and will transform among each othw under the action of 9. SOnce the thermodynamic prope&ies of a majcroscopic magnetic system betow T", are non-analytic at H = 0,some care is required In discussing the zero field properties. On the one hand, phydcalty we know that the magnetic field of a ferrornagnet can point in some spclcific direction. OR the other hand, the strict definition of a thermodynamic atrerage as the sum over all states with the appropriate Boltzmann weights would ykld zera net magnetization since all degenerate orientations of the spins are weighted equalfy, To reconcile these two facts, one must examine carefully whether all the states of a magnetic system in zero field aetwfiy are accessed with eqltaf probability. To avoid ambiguity, we shalt define the order parameter at zero field as the limit of the order parameter In the presence of a weak external conjugate field as the field goes to zero* t"or a magnetic system, the zero field magnetization is defined
We first consider the case of Ising spins, For an arbitrarily weak positive H,the symmetry of the Landau function is broken and the minimum at positive M, denoted by A in the lower right of Fig. 4.3, is slightly lower than B. Using the f a d that H couples to the spins through the interaction - H E i Si. the Boltzmann weight then yields the ratio of probatitlities for obsevving the system in state B and state A as
4.2
GENERAL FORMULATION WITH OROER PARAMETERS
185
where, by Eq. (4.1). m is the magnetization per spin, In the thermodynamic limit, M -+ oo so that PB -+ O for any H" and as H -+ 0,the system is in state A with M = MO.The zero field state thus depends upon the hisbry by which it is prepare$, and If instead we had taken a negative fietd, it would have approached state B with M =5 -Mo, The crudat role of the thermodynamic limit in this argument is clear. If N remained finite as H -+ 0, then -+ 1 and the two minima would be equally populatwt. In fact, for a large but finite system, all the thermodynamic functions wauld be analytic, At! the sharp edges of the phase surface in Fig, 4.1 would be smooth4 and the isotherms below T", would be very steep but not infinite at H = 0,yfelding a unique solution M = 0. For infinite N.however, the behavior is non-analytic at H = Q and one must define the state of the system by a specific path on the sudace of the equatran of state as H -+ 0, Heuristlcatty, if we interpret the Landau function as the free energy for the moment, because there is in some sense an infinite barder between A and B, the system is not ergodic at H .= Q and may be trapped in ctither state A or B depending on the histouy, The s&uation can be stlgtrtly more campiicated in the case of continuous symmetry, For z - y spins, the Landau function Is a surface wRh a minimmu corresponding to the circle denoted by dashed tines in Fig. 4-3. As lil' -+ 0,the minimum becomes flat, and all the degenerate configurations can be accessed without having ta cross an infinite barder, Similarly, for Heiisenberg spins in zero fietd, there is a sphere of degenerate configurations comespanding to all the orientations of fixed %( in three dimensions. We wilt see that thermal fluctuations can excite collective Goldstone modes in whkh the system samples degenerate configurations at the bottom of the valley, Dependkg on the dfrnensionatity of the space and tensorial character of the order parameter, these co!fective modes can either destroy the order by equally mixing alf the degenerate phases, or have no effect on the order because their effect is suppressed by phase space factors,
4.2 GENERAL FORMULATION WITH ORDER PARAMETERS We now seek to formutate a general theory which in leading approximation enbodies the physics of the Landau function of the order parameter. Since the order parameter fs singled out as the coltctctive variabte representing the essential degrees of freedom governing phase transition, it Is natural to east the partition function in the form of a functional integrai over the order parameter, and in the stationary phase appraximation, the theory has precisely the tandau form. The corrections to the SPA systematicatly correct the leading Landau contribution. To introduce the basic ideas, we first present a simple illustrative model and then dkcuss the generat case,
INFINITE
RANGE ISING MODEL
In order to relate the preceding discussion to the general theory involving a functional integral over an order parameter, it is instructive to consider an admittedly artificial model, which has the important property that the stationary phase approximation is exact. In this model, each of the lsing spins (& = &l) interacts with every other spin Si with an exchange energy -$. The scaling of the exchange energy as $ in this model is required in order to have a thermodynamic limit. In the presence
186
ORDER PARAMETERS AND BROKEN SYMMETRY
of a magnetic field H , the partition function
Z is
means a summation over all the 2N spin configurations in which each = *l. ~i~m?\;ie We now introduce an extremely useful technique which we will use frequently and replace the inconvenient spin sum C{&) eC by the integral over an auxiliary field of the easily evaluated spin sum C{si)eSi . Using the Gaussian identity.
and noting that
we obtain
The magnetization m and susceptibility X are obtained by taking derivatives of (4.13) with respect t o H :
As mentioned i n the introduction. we see that as long as
P
and N are finite, -r +oo or
Z(P, H, N) is an analytic function. Singular behavior occurs only i f N
p --+
+a.
L(# H ) , which we shall later identify with the Landau function, as the exponent in the expression for 2: It is useful t o define a function
so that:
4.2 GENERAL FORMULATION WITH ORDER PARAMETERS
187
In the thermodynamic ! h i t , Rf -+ W, the stationary phase method applied t o Eq. (4.25b) becomes exact. The stationarity condition:
Z',
yields
p = dh@[Hf PJ) Let us denote by p8 the soiutlons for Eq. (4.16) which are separated focal minima of the function l ( p , I Z ) . Then we know that in the thermdynamic limit:
and
U"), they correspond to the different stable or for If there are severat minima metastable phases in which the system can exist, The probability t o find the system In the phase pg Es given by the Boltzmann factor:
If one minimum, po, is an absolute min"rmu mf E ( p , E ) , then in the limit N -+ os P(h)-4 1 and P(ps) 0 for at1 S if 0, Thus, in the thermodynamic limit, by Eq. (4,17b)p = p0 and we observe that the stationary value af this auxflliary field corresponds t o the mean magnetization. Let us study consider the case H = O for which
Eq, (4,116)
in mare detail. First
and the stationary condition is p=tA@Jp
.
The soiutions of Eq, (4.19bf are the intersection of the straight line p and the hyperbolic tangent graphed in Ftg. 4.5a. There are two quafitatively dihrent types of sdutians, depending on the slope of the hyperbolic tangent curve at the origin p. If BJ I, that i s T > J , there is only one solution y = 0,carrespondhg to zero magnetization, and this solution is identified with the paramagnetic phase, The shape of t ( p , 0 ) is shown in Fig. 4.5b for T r Te and has a single minimum at p == 0.with free energy P I N l f p , 0)=L -T fn 2. The susceptibil"ty is obtakned by dilferentiating Eq, (4,16b) with respect t o H , H = 0. I=;
288
ORDER PARAMETERS AND BROKEN SYMMETRY
Fig. 4.5 Sketches of the Landau functlon t ( p , H ) and graphical solutfon far the s;tatlanary gotnts for the Itnftntte-range Islng modal. and in the paramagnetfc case (m = 0)
,fj....
=.
T T-J
*
The susccrgtibillty diverges at 2", = J , which we thus identify as a criticat point. ICPJ > I- that is 2' < II',, Eq. (4.19b)has three solutions, which are the extrema of l[&,0)graphed SR Fig. 4.5c, Since the stationary solutions at the minima determine the physical solution, we ignore the soluthon p = 0 corresponding to a maximum. The two physicaf solutions are the mlnimurn A, corresponding to the positive magnetization f phase and the minimum B corresponding to the negative magnetization 4 phase, Both and 4 phases have the same Free energy, and thus both phases can be present in the system. Close to 2",, the magnetization ma to of one phase can be catcurated by expanding Eq. (4.19b) to third order in p around p = (1. 1 Pa PJllo - $(@J)%; (4.21e) which yields:
4.2 GENERAL FBRMULATIQhl WITH O R D E R PARAMETERS
189
The mode! prwides a concrete =ample of the need to define the zero field magnetization carefully as the limit in which an external field is turned off. Simply arguing in terms of equal Bsltzmann weights, one would naively calculate the zwo field magnetization ts be
However, as we shall now see, introduction of an infinitesimal field H , which we
wilt assume t o be positive, lifts the degeneracy between the two phases and leads to a unique absolute minimum of the Landau function. The stationarity condition 1.6 th p(H + pJ), Eq, (4.16b). is solved graphically for a small magnetk field in Fig., 4.54. Away from the criticat pdnt, uslng the definition of the zero field susceptibility, the shift 6y away from the zero field solution induced by a small field E is
-
So, as shown in Fig, 4.5, for T > T,, there Is only one solution, point Q, with magnethation m = x o H , which goes to zero as H goes to zero. The minimum of the tandau functional is displaced in the direction of H , tf T I", as shown in Fig, 4.5 there are three solutions: points A, B and Q'* The solution 9'. corresponding to p ~ "t= is again a maximum of the Landau function and thus we must on!y consider the two phases A, corresponding t o f, and B, corresponding t a 1. The magnetization of these phases is given by:
Thdr corresponding Landau function and Boltzrnan factors are gikn by
Thus, we see that the introduction of a smafl magnetic field Is sufFicient to lift the degeneracy between the 1 and 4 phases, and that:
So, the probability far the system to be in a phase opposite to the magnetic fieid is zero in the thermodynamic limit (as long as H does not go to zero as $ or faster) and the magnetization is
190
ORDER PARAMETERS AND BROKEN S Y M M E T R Y
which becomes equal to ~s;owhen H goes to zero. Thus, an infinitesimal field selects the phase with magnetization parallel to the field and the system acquires a finite magnetization. The non-anaifyt'rcity of the free energy at H = O comes from the fact that at H = 0, the magnetization of the system fflps. The free energy thus has the form sketched i n Fig, 4.4 and the magnetization as a function of H behaves as the M H phase diagram In Fig. 4.1. Because the stationary-phase approximation is exact for this model, we see clearly how the system locks onto one or the other of the two degenerate free field solutions as is turned off, depending on the way the system is prepared. tn practice, we will see that the process of turning off an external field means that whenever we encounter discretely separated degenerate minima in the Landau function, we calculate observables by taking into account only one of the mhima,
-
GENERALIZATIONS The features we have just seen for the infinite-range lsing model are very suggestive of the structure which will emerge in the next section when we address more general systems. in preparation, it is useful to dkcuss more generally some of the salient concepts, Order Parameters. In many eases of physical interest, we will be able to express the partition function as an integral Z = Jdpc-flL('vX) such that at the stationary point, corresponds to the thermal average of a microscopic observable and satisges the Landau definition of an order parameter. in the infinite-range lsing model, ye = m = (j$ S<)and we found that m was zero above T,= T , and non-zero below T,. The essential issue is to identify the physical order parameter. Once it is known, it is straightforward t o formulate a suitable integrat for Z. Thus far, we have restricted our attention to global order parameters which specit"y the value of some physical obsewable averaged over the entire system, It will also prove useful subsequently to consider a tocal order parameter specifying the average value of the obsewable at a specific position, Thus, for the magnetic example, the tocal order parameter is the local magne6ration nq = (Si) and the global order parameter representing the integral over all space. In is the total magnetization m = Ci Fourier space, m is just the iC, =. O mode, VJe shalt we that it is e a q t o generalize the Gaussiian transformation for a single global order parameter p to a general Gaussian integral over a local order parameter pi defined on alt the lattice sites. Landau Energy. The exponent t ( p , H ) appearing in the integral for the partition function naturally satisfies the definition of the landau function. The absolute minimum of L(p,H ) dominates Z in the stationary phase approximation. so the minimum of t specifies the state of the system. Phase Boundary and Critical Polnt. The order parameter varies discont"lnuously at a first-order phase transition and continuously at a second-order phase transition, In the infinite-range lsing model, m varied discontinuously as H crossed zero below Te and the transithon was thus first order. This discetnthuity in H ctearty arose from the N -+ oa limit since the ratio of the thermodynam'ic weights of two nearly degenerate wells in Eq. (4.10). = e-lN@"H. would go to one as H -,O at finite N. The magnetization approached zero continuously at zero field at the second order
xi
4.2
GENERAL FORMULATION WITH ORDER PARAMETERS
191
critical point Tc. For second-order transitions. the behavior near the critical point is characterized by the critical rtxponent B, defined by
?.
The definitions of For the infinite range lsing model, by Eq. (4.21b) we observe ,B = critical exponents introduced in this and subsequent sections are summarized in Table 4 3 at the end of the chapter for reference. Conjugate Field. A field which couples linearly t o the microscopic variable whose expectation value is the order parameter is called the field conjugate to the order Si parameter. In the lsing example. H is the conjugate field because of the X coupling in Eq. (4.10). The conjugate field, or symmetry breaking field, breaks the degeneracy of the degenerate double welt and drives the discontinuous transgtion below Tc from finite positive m at positive H' to finite negative m at negative H. A second order transition occurs for some specific value He of the conjugate field a t the critical temperature- The behavior of m at Te for H near He is described by the critical exponent 6 defined m -. IH- ~ ~ 1 % (4.30a)
xi
.
Iil-*N,
For our lsing example, Ha yields
=. 0,
and expanding Eq. (4.16bf to third order In m a t T,
3 --E
J
(4.308)
so that 6 = 3. Susceptibility, For the fsing example we have seen In Eq, (4.6) that the susceptibility for the order parameter, defined as the linear response of the order parameter t o an infinitesimal conjugate field, is also the fluctuation of the order parameter, This relation holds in general for all cases In which the conjugate field couples linearly to the order parameter, Thus, in general, the susceptibility diverges at a second order phase transithn, and the critical exponent 7 chauacterhzirrg this divergence is defined by
For the infinite range emmple, according t o Eq., (4.20) 9 = 1, In the case of a first order transklion, there is no divergence in the susceptibilky, Universal Behavlor, In the present formulation, the crkical behavior of a system is described by an integral over an order parameter of an action which is a functional of the order parameter. To the extent to which the functionals of the order parameter are equivalent for d'rEerent systems, the criticat behavior of these systems will coincide. The discussion of the power series expansion of the Landau function is highly suggestive that only a finite number of terms may be relevant t o the cdtiical behavior, and the form of these terms may in fact be highly constrained by~symmetrycondderations. Thus, It is not surpriskng that a large number of physkally distinct systems may belong to the same universality chss in the sense that they have identical critical exponents. For example, the ferromagnet lrPi"eOs, the antiferrornagnet FeF2, the fluid X e , the alloy
592
ORDER PARAMETERS AND BROKEPI SYMMETRY
@-brass, the molecular crystal N f i G L , and the three-dimensional Ishg model are all characterized by ai spin-like order parameter and belong to the same universality class, Symmetry Breaking. Very often, a phase transition occurs through the mechanism of a symmetry break"tng, In the magnetic examp'te, the Hamittonian H =I: SiSi is invanant with respect t o reversal of all the spins Si -. -Si. Normally. we would expect that (Si) -. (-Si) = 0. The appearance of a finite magnetization i s the manifestation of a broken symmetry, in which (S;) (-S;), More generally, let us consider a Wam'rlt;onian %(gi) invariant under a group of transformations on its degrees of freedarn (g&
& zij
Then, all states (B:) obtained from ( q i ) try the action of the group G have the same energy, and thus the same Boitrman factor, The conjugate field must have the property that when added linearly to the Hamittonian, it lifts fat least partially) the degeneracy of X, For example, in the magnetic case, fir S; breaks the invariance with respect to spin reversal, Hence, the symmetry group Q@ of the perturbed Hamirtonian
must be a subgroup of the group 9 ( g t strictly inclded in tonian U t then satisfies V g E $5 #U"(gqgf= #"(g)
9). The perturbed Hamil(4.34aj
and
VS E g, 11 4
Q',
~'(sQ~I Z M ' l ~ i j
(4.34b)
We will see in Section 4 that the nature of the residual subgroup Qtccontrots the stabilb a d excitation spectrum of the ordered low-temperature phase, In general, the knawfedge of the order parameter is guided by general symmetry considerattons, and by some physical intuition of the nature of the phase transition and of the conjugate field, In some cases, however, the order parameter can be a non-measurabte quant"ly, not uaiquely desned, or its physical nature can be very complex [as in the case of spin gf asses).
PHYSICAL EXAMPLES Table 4.1 summarizes the order parameter governing the phase transitions for a var"t@ of physical systems, as well as their coupling to conjugate fields. The antiferromagnet is similar t o the ferromagnet we have discussed at length with the excegtian that the interaction J SiSj favors antialignment of adjacent spins. For bipartite lattices, such that every sprn on one sublattice A is surrounded by nearest neighbors on the other sublattice 13 and vice @erg@,m and cFJ for the ferromagnet are simply replaced by staggered fields rns and Hs defined with opposite signs on the two sublattices. The ferroetectric transition in polar crystals is analogous to the ferromagnet with the spins S; and magnetic field W replaced by the polarization 1Z; and electric field E. Similarly, the binavy alloy transition is analogous t o the liquid-gas transition we have described.
4.2 GENERAL FORMULATlaN WITH ORDER PARAMETERS
593
Table 4.1 order parameters, the caupllng to conlugate fields, and the number o-l components I\f for a varfety of physical phase transltlons, The new form of tnnsition included in Tabtrt: 4.1 is that assaclated with Bose condensation, which undediies the supsrconducting and superfluid transitions. For on"entation, recall the slmpk case of Bose condensation of a non-interacting gas. The density of partictctcs In a box of volume 3f 'is given by the sum over the discrete momentum states frr the box 1 I (4.55) P @#(&-P) - 3 *
.=-C v
The density increases monotonicalfy as ifr approaches zero, at which point the p = O mode becomes macroscopicalty occupied and must be separated from the rest of the sum which may be reptaeed By an integral
.
when f ( 3 f 2 ) = 2.612,. is the Riemann pfunctlon. Bose condensation, that Is. macroscopic occupatkn of the p == O mode, thus sets En at the critical temperature
betow which the condensate density may be written
194
ORDER PARAMETERS AND BROKEN SYMMETRY
Fig. 4-6 Bosa condensate denslu as a function of temperature (a) and sketch ol the phase surface o f the complex order parameter &1 (b). The condensate density thus has the behavior sketched in Fig. 4.6a characteristic of an order parameter or a function of: an order parameter, To identify the local order parameter and understand the similarity to the magnetic example, it is helpful to recall the stationary-phase treatment of Bose condensation in Section 2.5, We wrote the path integrali in the space of coherent states which have the property = + ( z f , The actlon in the 14) = dr'(r)'f (~~10) resulting path integral. Eq, f2.215a), depended dds only through bitinear combinations cji'(x, t ) &1(z"t ) , so the action was invariant with respect to the global gauge transformation O(z,t ) -+ cia$(z, 6 ) . In the stationary-phase approximation. the acdon was dominated by a constant solution, with #(l;) = O [appropriate for the high temperature phase) or $(z) = 4, where 4, is a complex constant with magnitude and arbitrary phase refiecling the gauge symmetry, Thus, in this approximation, the condensed phase is characterized by a state in which ( 4 ( z ) )= 9, which is analogous to the magnetic case characterized by the state with (S(zi)) = 6%.We therefore identify ( $ ( S ) ) as the local order parameter and $ dz($(z)) as the global order parameter. Note that because the solutbn is spatially uniform, only the zero momentum mode is non-vankhing. Also, by the properties of coherent states that the condensate density is n o = jCbcj2 ancf. as expecte parameter. Because the order parameter 4, is complex with an arbitrary phase, the phase diagram has the structure sketched in Fig. 4,6b, The two degrees of fredorn in the complex order parameter 4 re analogous t o two components of 6% in the z: -- y model for which the magnitude Ss fixed but the orientation Is arbitrary. As in the spin ease, there is a Goldstone mode associated with the degenerate phase angle. The analogy w i t h p i n may be extended further by considering the addition of external sources S d z l $ t ( z ) . J ( z ) ~ * ( z ) G ( z ) ]as in Section 2.4 analogous to the coupling to an external magnetic field dzi f i ( z i ) * $(xi). For finite sources J*,J . the Hamiltonian is no longer number-conserving, so for the exact eigenstate we know ($(z)),*, J f 0. It is only in the care of identically vanishing sources that number conservation implies ( $ ( s ) ) ~ . , ~ = = ~0,just as for the zero field ferromagnet, equal Boltzmann weighting of all states would imply ( $ ( g i ) ) = 0. Thus. in the zero geld limit, the Bose condensate exhibits broken symmetry, and depending upon the path
6
+
I
4.3 MEAN FIELD THEORY
195
along which the sources are turned ofF, the condensate has a definite phase. IF one wants t o construct a solution of fixed paelcfe number, one must project by taking the appropriate linear combination of the degenerate solutions. For example, if we write 4, =t pei4, a state of definite particle number may be projected fram a coherent, state as follows
Once the order parameter for Bose condensation Is recognized, the order parameter for superconducting or superfluid transitions follow naturally. For superconducting metafs, pairs of electrons at the Fermi surface, form bound states (Cooper pairs) which behave eRectively as Barons and thus undergo Bose conde~satian.~ Since the bound (z)q4( S ' ) , there states are spin singlets, and the annihilation operator for a pair is In liquid SHe. the short is just one complex order parameter A = ($t(z)Gl(zr)). range reputsion is so strong that pairs $8 relative S-waves are much less bound than in relative pwaves, and by antisymmetry the p-wave pair rave Function must be a spin triplet. Coupling spin 1 and angular momentum 1 invaives 9 diRerent comptex amplitudes, and the resulting 18 real components of the order parameter give rise to an exceedingly rich physical phase structure.
The mean Fieid theovy, first introduced by P, Weiss in the study of the ferromagnetic transition, provides a simple yet powerful theory of phase transitions. We will see that its validky depends cruc'ially on the spatial dimension d. For d sufficiently large, greater than an upper critical dimension d,, the mean field theory is very good at all temperatures, It yields the exact critical exponents, which means that It correctly takes into account the divergences which occur a t the critical paint, and It provides a starting point for systematic corrections, Below d, but above a lower critical dimension del the mean fietd still works well except close to the critical point, where it yields the incorrect critical behavior, Below the lower critical dimension, de, the mean field theory is invalid and its predictions are quariitatively wrong.
LEGENDRE TRANSFORM To fo_rrmulate the theory in terms of the order parameter G5i rather than the external Field H ; , It is natural to use the Gibbs free energy, r(%i),which is the tegendre transform of the free energy F ( & ~ ) .Hence. let us review several basic ideas from Section 2.4, where the source Ja corresponded t o the external Field Hi, the field 4, corresponded t o the order parameter p,and the generating function for connected
ORDER PARAMETERS AND BROKEN SYMMETRY
396
Green's functions -W(J,) conespondd to the free energy F(Ni). The order parameter is given by
and the Gibbs free energy is defined
where &friij) is obtained by inverting Eq. (4.401. Below the transition temperature, we know that the zero fieid order parameter depends on the path by which approaches zero, so the equation of state
B
also depends upon the path. tn contrast, the reciprocity relation
rr-
i -
TF
-
.-
5
-z
er
&a
m
I;, V)
0
5
m
*J
=r
8
N
Q)
W
E J
Q,
0
VI
Q,
leads to the unique zero-field equation of state
which admits a set of broken symmetry solutions which correspond to ail the different orientations assumed by the magnetic fidd when it Is taken to zero. Since $'(Bi) is the generating Function for connected Green's functions, the Green's function specifying the spin-spin correlation function and response of k to is calculated in the usual way:
9
tkPL
$2
rrre*
*-
5
m
71 e:
W
r;S
a-
& 5 0 ,SCt;;
=z 2%
g *c 3
, ,ZJ
2:
i
Q)
tt: U
cp
I;, V)
I
(B
U
Q,
E
8
C.
D
A
V)
Q) C1
QI
$3
$5=
X
0,
8
g .2?
J-)
;ITT
For a translationally invariant system, G depends only on the relative distance (?$-Fj) and the Fourier transform i s defined
4.3 MEAN FIELD THEORY
597
P .
:
W
U
.--25
S!
...a
U
I-
U ils V1
5
9
0
111
m .".
I-
-E
5 bO
" .a .
5
._I
X
r3-
rZ,
us
V) =Z
0
a
QI
V)
8
r:
Q)
1
P
0
8
k:
CO
and in the long wavelength limit yields the static susceptibility, Eq. (4.6)
By
the same argument used in Eq, (2.169), the Green's function may be expressed in terms of the Gibbs free energy using the relation
so that
Flnalfy, as discussed in Sections 2.4 and 2.5, we recall that the perturbative expansion of the Legendre transform is identical to the loop expansion for the free energy. Hence, by using the stationary phase approximation t o obtain the loop expansion of the free energy around the mean field, we will obtain a systematic perturbative solution t o the equation of state. Eq. (4.44).
FERROMACNETIC TRANSITION FOR CLASSICAL SPINS M
"a
1-
c9
-.E
o- E
=z
-m z
'W 3 bB
2
.. "13
5
= 0 ;5
* n Ea, m &E
S
ce=
f z 2 8
2 *EG
z=
$35
Q1
6:
*2
a.
f:
U"
U
,
ti
It
0)
CO
0 5
3=
Q
3%
V"
U
8
S
E;,
01
*-, CO
-
S3 .%?
S=
,,
3g
E
m "3
2 &
(Q
g
To illustrate the mean field theory, we shall study the classicat Q(nf spin model on a D-dimensional cubic lattice with lattice spacing a. The Hamiltonian is
with the nearest-neighbor interaction Jjj
-I
J if E and j are nearest neighbors Q
otherwise
and the constraint 2 '$~x(si",~=1
where Si denotes the n components {S); of a unit spin on the iahsite of a cubic lattice, and denotes the n components of a spatially varying magnetic field on the bth site. Recall that the cases n = 1, 2, 3 correspond to the Ising, X - y, and Heisenberg models, respectively. Note that in the absence of a magnetic field, the scatar products Si gj are invariant if all the spins are rotated by the same angle so
Z q . 3 S .% W E
3$ -" $&,g. !'Y3
-
Q ET =0 I
S$$,,
GS
Q g
0
C"r
2 % j2 ~$'=
" g
g320 % ( , P& c*8gapg n, n, =is "D 0' =Ic""n"q -* :3. =J
z3lg
g Pqz 0
O R D E R FrARAMETERS AND BROKEN S Y M M E T R Y
198
,Cb, W-r
9
V)
-Em
5:
s.3
g.,%
a n %
32.
B1
Vt
@
-'
zas"
I3
( D %
3-
$
0
-v 3 3
g,,
X = %
aff
ZC;
S?
==: D;)
" g
-2
g,
S ' *
g9
3
;g
-E -*
S* 2 S g 8. -*
3
7
q we 2s
m*
S + &=
a,
Q v ,
the Hamiltonian has O(n) symmetry. in the presence of a constant magnetic field Bi = the symmetry group is reduced to O(rz - l),the global rotations of spins around the direction of B. To evaluate the partition function, as in the case of the infinite-range Ising model, we introduce an integral over an auxiliary field using the Gaussian identity Eq. f1.179)
W here
-N/2
(F)
(D
F;:
V".
2%. d
3
I??.
S 3
'5"
F;:
&U 3
4
e
7 -3
m.
11.
E: B1
%.
b" R, ID
i?
%'
6
I '
5" =r
P1
The constant C = (det J.~)-'/' and the factor Y nwmalizing the spin sum correspond to additive constants in the free energy which wili not affect our results. In the case of one-component k i n g spins, the spin sum on a single site appearing in (4-41) is simple
W
Cr )--r
g
le"l
Q)
CI
0
09
P3 P -
I
N
"11 CO
V1
E;
%
Y
S
-t W
ior
?a
8-1
B
f
m
I--r
W
""L
D,
@"'-
r
Ir=
3
For higher n, it may be evaluated in polar coordinates yiefding
which may be expressed in terms of Bessel functions, Alternatively, the spin sum may be expanded in powers of 4 as in Problem 4.3. For simplicity, we shall first consider the single component lsing model, for which
The mean-field approximation is obtained by applying the stationary-phase approximation
4 3 MEAN FIELD THEORY
where the stationary solutions
199
satisfy the mean-field equations
Since F(Hi) = S ( ~ ; ( H ~X )i ), in this approximation and zation is given by
Using Eqs. (4.54b) and (4.55) to determine
g = 0, the magneti-
H;(mi)
and using the relations ch(th-'z) = (1 - za)-'i2 and th-'(g) = $In Legendre transform is
It is straightforward to verify that the equation of state given by H; = am, is "tentical to Eq. (4.55). Specializing to the case of unworm magnetbation t?l; =. m, noting that the number of nearest neighbor bonds on a cubic lattice in L) dimensions with N sites is D N ,and expanding t o fourth order in m, we obtain
200
O R B E R PARAMETERS AMD BROKEN S Y M M E T R Y
We observe that r(m) has the form of the zero field Landau fundion sketched in Fig. 4.3 and given in f q . (4.4) and thus, the system has a second-order phase transition at
rr(,=2JD For a uniform field. the equation of state Hi =
,
(4.59)
Eyields
which implies the rero-field magnetization below r",
cormsponding t o the critical exponent
= ?. At G. Eq. (4.59) yields
so that 6 = 3. The spin-spin correlation function and susceptibility near the critical point are obtained using Eqs, (4.48b)and (456):
Since Jjj is a function of the distance between sites and vanishes except for nearest nefghbors, its Fourier transform is
= 2$aD
cos g,a
It is customary t o simplify the formulas by using a = 1 and measuring all lengths in units of the lattice spacing. The Fourier transform of Eq. (4.62) yields the low-g behavior at: the Green's function near the critical point
4.3 MEAN FIELD THEORY
203
F"or !l r ' I',,the magnetization 5s zero, so
and, by Eq. (4,471 X+
+
=
1. m
--
Note that we wilt use and subscripts to distinguish quantities above and below the cdticat point. The Green's fundion (4.66a) implies an exponential decay in coordinate space
from whkh we define the: correfatbon length
As a p e e t d physically a t a swond order transition, the cowetation laingltr diverges, and thfs divergence Is measurd in genera[ by the criticat exponent Y
In the mean field approximation, we thus find v = and 7 = I from Egs. (4.624 and b) ,respectively, Beiw the critical point, the magnetization is given by Eqs, (4.60) so that we obtain
1
PG(?) @=a ,(Tc T ) + J
~ S
2" < T,
(4.66a)
4
with the same critical exponents v = and 7 = 1. At T",, the dway of the correlation funafon Is no longer exponential. Rather, it is
where the r dependence is easily recognized by changing variaMes & S 3 If[. The general bahaviar of the correlation function near a critl~afpoint 35 parametedzed by the critical exponent q
202
ORDER PARAMETERS AND BROKEN SYMMETRY
so that in the mean field approximation, q = 0. The final thermodynamic quantity we consider Is the specific heat,
Using (4.58) and (4.61) to evaluate F(H= 0,T ) = r ( M ( H = O), T ) yields
so that
and the mean field theooqr yiierds a discontinuity In the specific heat but no singularity, Because of the possibility of an analytic: background term in the specific heat, the critical behavior is defined by the relation
and the mean-geld value of the critical exponent is Q! = 0. All the preceding definitions of critical exponents and the mean-field values are tabulated fn Table 4.3 at the end of this chapter for reference. All these mean field results for lsing spins are straightforwar9y generalized t o an n-component C? (n)spin system near the critical point where S(#, Eq. (4.50b). may be expanded in powers of 4, As shown; in Prabfenr 4.3, the? Legendre transform wpanded to fouvlh order with an irrelevant additive constant omitted is
a$).
For uniform magnetization t;iii = 6 we obtain:
and in zero magnetic Field, the equation of slate 5s
Again, we see that there: "I a swancl-order transition at
4.3 MEAN FIELD THEORY
203
The shape of I'(%)for a two-component spin system i s a surface of revolution suggested by the dashed line in Fig. 4,3, For T > I',, r(&) has a single paraboloid minimum at 6%f.= O whereas for T T", the minimum is a circular valtey with radius
-
Far larger numbers of components, the minimum i s a s p h m in higher dimension with radius given by Eq. (4.77). Since ( G ) (T, - T ) ' / ~we . again obtain the critical exponent B = $ . The carretation function now depends on the lattice sites and spin components and is given by
and the burier transform for a translationally invariant state with uniform magnetization r;3r is
The direction of r;ii is defined as the longitudinal axis, and all other perpendicular directions are calted transverse directions. Abave T,, fi = O and the correlation function and susceptibility are isotropic and equal to the king result:
Belw T,, using Eq, 64.69).we obtain qualitatively different results in the longitudinal and transverse directions. In the iongitudinat direction, where rna = fG1.
204
ORDER PARAMETERS A N 0 B R O K E N SYMMETRY
whereas in the transverse direction, where rna = 0,
Thus, for any number n of spin components, including ta -- l, the behavior in the direction of the magnetization "t identical, In addition, however, for la 2 2, there are components perpendicular to the direction of tr3r which behave totally digerently. Since the minimum of the Gibbs free energy i s degenerate in these other directions, it costs negligible energy to create excitations in these directions, and the resulting proliferation of spin waves "I responsible for the divergence of the suscept"lility and the correlation length,
APPLICNION TO GENERAL SYSTEMS The mean field theory may be applied quite generafly to any system for which the local order parameter has been identified. Let us consider the general partation function
~ ~ ~ : l + J~'lu(S1 ~ ~ & ~ s * where fq;) denotes the microscopic degrees of freedom, p(%,~f is the microscopic focal order parameter, U ( z )is the external field which couples to the order parameter, and $ d z represents a spatial integral for continuous systems or a sum over lattice sites far crystals, The basic idea is t o express Z as a functional integraf over a field whlcf? Is an order parameter
The integration variable f(z) in this equation is an order parameter, since by virtue of its linear coupling to U(%),its thermal average is equal to that of the microscopic order parameter
) defined where in an obvious notation, the thermal averages ( d z , S ) ) and ( f ( t )are by Eqs. 14-84] and (4.851, respectively, Hence, application of the stationary phase approximation to Eq. 14.85) directly yields the desired form of a lree energy as ct function of ran order parameter. Any linear transform of the microscopic order parameter is also an order parameter in the Landau sense, because it vanishes in one phase and is non-zero in the other. Thus, the action in Eq, (4.85) may haw a slightly more general coupling tern d z d y U ( z ) M ( z ,y)g(y). Recall. for example, the functional integral for Ising spins. Eqo. (4.52, 4.53)
I
4.3 MEAN FIELD TWEQRV
205
h which case the order parameter (B was linearly related t o the microscopic order parameter by 4i = *
CJii{Si,
Note that in addition t o being an acceptable order parameter in the tandau sense, In this case has the physicat interpretatjon of a mean molecular field, That is, gives with the mean wlue (S,.) the potential Men at site i as a result of interactions of each of the surrounding spins, Clearly, changing the variable of Integration in Eq. (4.85) by an arMtrary lfnear transforrnatlon would nat change the physical result, and we could just as well use the variable s JG'+~,in which case X i = (Si). Thus far, in the c a s of spin systems, we have seen an example of a sp~46iCic technique, the use of an aulciliary Beid, to obtain a functional integral for the partition function. However, in general, it is straightforward to obtain the functional integral Eq. (4,85) from Eq, (4.84) by introducing the constraint / ( E ) = p(z, qi) by use of the identity
+i
r
(4.89)
with the result
where
(4.90b)
Even in cases for which Tff(s)] is too complicated t o evaluate exactly, it may be and used to study the behavior in the region of the critical expanded in powers of f (l;) point where f [ z ) is srnal, Let us illustrate thls method for the fiqu"r-gas transition, The grand partition function for a classical gas, with two-body potential v(?) In an external one-body potential U(?)is
(4.91~) The momenta fli can be integrated out, yielding:
where X
is, the
fugaidty, given tsy:
206
ORDER PARAMETERS A N D BROKEPI SYMMETRY
As discussed in the introduction. a natural order parameter in the liquid gas phase transition i s the dendty of the Ruid (or rather the difference p~ pc). Defining the density as
-
c N
p(?) =
&(F-
(4.92)
7%)
i=t
and using the Integral representation of the G-fumctian
we write the partition function of the system as:
Regarded as an integral over p(?). this i s precisely of the desired form. Eq. (4.85). However, in the present case, because the action is quadratic in p(r), one may proceed even further by integrating out the field pfrf to obtain
(4.95) where v-'(?) is the inverse of v(?) and we have redefined #(r) to be equal to $+(F). We obsewe that the result Is completely analogous to the lsing spin case, written in Eqs. (4.53) and (4.87). The field d(?) plays the role of 4<, U(?)corresponds
j dor e-@#tvl
replaces 2 chp#ie and #(F") = to the magnetic field, the potential term e dDr' v(?6(@-Ti)) = ( v ( i - 3,)) is the mean field a t point r generated by the equilibr'rum distribution of the surrounding particles, The liquid gas transition Is studied IR more detaii in Problem 4.4. Before proceeding; beyond the mean geld theory, several observations should be made. The first i s that the Landau theory discussed in Section 4.1, which was originally introduced on a purely phenomenological basis, follows naturally fvom mean field theory, Recall that the Landau function was defined as a function of the global order parameter with the property that its minimum specifies the state of the system. When the partition function is written in the form Eq. (4.85) as a functional integral over an order parameter f (F) and the stationary phase approximation produces a spat'ialfy uniform solution, the minimum of the exponent F ($(F)) / d~ f ( 3 ) ~ ( 2 ) ifies the state of the system, Hence, the exponent of the functional Integral provides a
i')(~g,
xi
+
4.4 FLUCTUATIONS
207
microscopic definition of the Landau function, and the criterion for the va1"rity of the tandau theory is the valid"ry of the stationary phase approximation. Note that since an analytic function of the order parameter is also an order parameter, there is considerable freedom to define alternative Landau functions, For example, "r the case of lsing spins, both S(4i), Eq, (4.521, and P ( q ) : Eq. (4.57) are equivalent Landau functions and are related by the transformation = th(84i)= B(4Q). Since these two order parameters are proportional near the. critical point, they necessarily have the same critical behavior. The previous discussion of the symmetry properties of the tandart function applies both to the Legendre transform and to thdt exponent of the functional integral, and is frequently useful in identifying the allawed forms of their series expansions. For example, the (7 ( n ) symmetry of the ra-component spin model implies that onfy povvers of ??a2 may appear as in Eq. (4.75). In contrast, there are no exact symmetries for the order parameter 4 = Pfirluid - Pgasin the liquid gas phase transition, and in fact odd terms in gl enter the expansion after fourth order. The second observation about the mean field theory is the remarkable absence of any dependence of the critical indices of the spin models we considered on the spatial dimension D,the number of spin components n, or even the detaifs of the coupling matrix Jii. These features only affect the actual value of the critical temperature, The critical exponents for the nearest neighbor O(n)model with any n or D are identical to those of the infinite-range modet for which the stationary phase approximation is exact. When we examine corrections to the stationary phase approximation, that is fluctuations around the mean field, we wilt see that n, D,and Jii may play an important role. In some cases, such as the infinite range model or high enough D,we willf see that the fluctuations do not ageet the critical behavior and that the mean field result is eexcct. At the other extreme, for cases such as the lsing model in 1 dimension or the s - y model in 2 dimensions, we wilt see that fluctuations completely after the mean-field result. Hence, we now grocrted to cafcutate the contributions of fluctuations,
+
LANOAU GfrMZBURG THEORY AND DIMENSlONAL ANALYSIS We have just seen that the Landau Function corresponds to the expansion of the exponent OF a functional integral over an order parameter for the special case of a spatially uniform order parameter. In studyhng Ructuations, we wit1 be interested in the more general case of a spatially dependent local order parameter, and the correspondhng expansion of the exponent in terms consistent with the symmetry of the system yields the tandau-Ginzburg functional. Suppose we have a problem with local -4[r), and with order parameter dfr), possessing symmetry with respect to d(r) translation and rotation invariance, In addition to the even powers of + ( r ) discussed previously in connection with the Landau function, the action may contain invariant derivative couplings of the fields. The iowcst-order derivative term consistent with the symmetries is ( a 4 ( r ) ) 2 ,i n which case the Landau-Ginzburg functional in zero field would have the form -+
ORDER PARAMETERS AND BROKEN SYMMETRY
208
For a uniform order parameter +(r) =. m, we recover the familiar Landau form and the ( ~ + ( r ) ) 'term accounts for the additional energy required if the order parameter is non-rrnifarrrr. Just as only a limited number of powers of $ are required to study the criticaf behavior, it is shown in Problem 4.2 that terms involving higher derivatives of 4 or higher powers of (V41 are not necessary to study the critical behavior. tn general, the bandau Ginzburg functional can always be obtained by expanding the exponent of a functional integral over an order parameter, and we will now iltustrate how the form Eq, (4.96)i s obtained for the lsing model in D dhensions. Expanding the lsing action, Eq, (4.52) t o fourth order in zero fieid yields
-
-
Since we will be interested in dimensions, note that in units such that A = c = ks = 1, ,B L. J -- f and -. Hi so that PSi is dimensionless as required. Defining a continuum field $(%a)= 4%.replacing sums by integrals according to C, aD+& = dDrJ(?), and calculating the first term in a Taylor series expansion of J - I by noting
+i
l
h
so that
we obtain the result
It is conventioonaf to rescale the fields such that the coeff-ic'rentof the gradient term i s
$. so we define 4 =
W here
and we have used II", = 2JD.
'124 and thus obtain the form Eq. (4.96)
4.4 FLUCTUATIONS
209
!t i s ,inportant to appreciate the physical motivation for taking the continuum limtt. Although the lsing problem on a lattice has a fundamental microscopic length scale, the lattice spacing a. we have seen that the behavior near the critical point is characterized by fluctuations and a correlation length which become arbitrarily large. Thus.'for the long-wavelength excitations of the system which dominate the physics. the microscopic length a is irrelevant, and the correlation length determined by the coefficients in Eq. (4.100) is the physically relevant length scale. At the critical point. where the correlation length diverges, the system has no characteristic length scale and is expected to behave as a scale invariant system. The dominance of the long wavelength, or infrared, behavior of the action near the critical point, illustrated here for the lsing model. is completely general. Far example. in the liquid-gas phase transition, the correlation length becomes arbitrarily large compared to the microscopic length scale associated with the molecular potential v ( r ) ,and the potential only affects the magnitude of the coefficients in the expansion of the exponent of Eq. (4.94). By the same argument. the critical behavior of a quantum system is also dominated by length scales arbitrarily large relative to the microscopic quantum scale, and quantum effects only contribute to the numerical values of coefficients. Having established that the critical behavior is governed by a Landau Ginzburg functional of the form Eq. (4.96). we now examine the role of the spatial dimension D by dimensional analysis. Denoting the dimensions of a quantity by square brackets.
we find from the term
I dDr ( ~ ~ ( r )that )?
Similarly. the term ro $ dDr+2(r)yields
and the term
'j dDrqJ4(r)gives
consistent with the specific lsing results. Eq. (4.100). Recasting the action in dimensionless form by the transformation
the partition function for the system may be written
210
O R D E R PARAMETERS AND BROKEN S Y M M E T R Y
W here
Recall that the critical point occurs when the quadratic term ro(T")#a changes sign, so that as seen in Eq, (4.4) for the general case and Eq. (4.1QOb) for the lsing model, we may write rg, E 2i(T - Te) (4.104) so that the ekctivf! perturktion parameter for the quartic term in Eq, (4.103) is
This resuit clearly displays the rote of the spatial d h e n s b n D in governing the behavior of the functional integral Eq. (9.103af for the partition function. We distinguish two qualitatively different cases. Far D 4, g 4 O as 21 --+Tc and a perturbative expansion in g around the mean field result becomes hcreasingly accurate as I' approaches Tc. Hence, the mean field accurately describes the critical behavior. For .D r 4, g -4 cm as T --, r", and perturbation theory breaks down wlthin some range of I",, Although mean field theory may still be accurate away from the critical point, it will give the wrong critkal behavior, The case L3 = 4 is marginal and requires special care. In general, however, one should not expect mean field theory t o apply for D = 4. Thus, we have identified the upper cr'rticaf dimension, d,, discussed at the beginning of Section 4.3, which specifies the dimension above which mean field theory is valid around I',. For the action (4.96). we have just shown that $, -; 4. For other systems which admit other invariants in the action, dhensional analysis wilf yield different results, f o r example, as shown i n Problem 4.2, a cubic term in the action implies d, = 6. For L3 < c&, one can determine the temperature region, called the Ginzburg region, i n which one expects the mean field theory t o begin to break down by the condition (AT) w hick yields g li, Using Eq, (4.105).we write I = U@ 6
v
>
For subsequent reference, using the explicit parameters (4,100) for the lsing model we find
J .
f4.106b)
The size of the Ginzburg region may be estimated far physical systems by finding =. a combination of observabtes which yield Eq, (4.106a). Fifst, note that using given in Eq. (4.961, the mean field approximation to the free energy below T, is F = VTZ2(T T,)2/(2uo)so that the discontinuity in the specific heat per unit trofume a t Tcanalogous to Eq, (4.73) i s
-
4.4 FLUCTUATIONS
233
-
The correlation length for g[#]is calculated in mean field theory as in Eqr. (4.63 4.68) using G-l f z, yj oc from which it follows that Gm'(@) oc q2+ii(T-T.) and
-
.
= [Z(T T , ) ] - ' / ~ Since . mean field theory is only valid away from the critical point, we will estimate the correlation length well away from the critical point IT.-TI = O (T,) as follows
- [BT.]-~I~
.
Combining Eqs. (4.106a. 4.107a and 4.107b). we obtain the result of Ginzburg (1960)
This dependence on the correlation length scale accounts for the dramatic diFfemnce between the size of the Glnzburg region for the superfluid transition in liquid Helium and for the superconducting transition in metals. In both cases, &C i s of the order of several j o u l e l e d 'K and T", i s of the order of several " K . However, the characteriftic range of intermolecular ~orreiationsin liquid Helium is of the order of several A yielding A T 1'K whereas for a superconductor the correlation length is the size of a Cooper pair 1032 SQ that AT 10-15" K .
-
We wilt now explicitly calculate the Ieadbng order correction to the mean field theory for the s"rmple case of the fsing model, and defer until the next section the additional complications arising from continuous symmetry, Using the Ising action S given i n Eq, (4.52). for which
and "tncluding an ctxpticit expansbon parameter 8 ta keep track of orders in the stationaryphase approximation, the stationary phase result including quadratic fluctuat"lns is
Here
with
(4.109~)
212
ORDER PARAMETERS AND BROKEN S Y M M E T R Y
the stationary solution satisfies the mean field equations
and we wilt ignore the overall multiplicative constant C = To leading order in
$, the magnetization is given by
and as shown In Problem 4.5 the Legendre transfarm is
.
& i i ~ ~ ( I - ~ ~ ) ~ i j ] The equation of state is:
(4. XXX)
and the susceptlbiiity in the high temperature region, where m; ==: 0 , is given by
The sum is simplified by taking the Fourier transform
where J ($1 i s the Fouiler transform of Jij defined by Eq. (4.63) and with the convention that the lattice spacing n = 1. the integral $ d D p is over the Brillouin zone R c g, < R. The critical temperature is defined by the divergence aF the susceptibility so
Hence. to leading order in
#
4.4 FLUCTUATIONS
213
We thus observe that the effect of fluctuations is t o decrease the critical temperature = 2DJ. This downward shift in Tc is reasonable below the mean field result. ~f~ physically because fluctuations tend t o disorder the system. around Tc. we subtract Eq. (4.113a) from Eq. (4.112b) In order to expand with the result
Since the critical region is dominated by the low ir fluctuations, we expand f ( a ) to second order in as in Eq. (4.64). To order $, we obtain the low q behavior
a
From this result we observe that the
/9
$ correction to X-1
at Tc for
D = 4 contains
and is thus logarithmically divergent. For any other dimension, we make the change of variable
T - Tc
(4.115b)
and the susceptibility becomes:
This result displays the role of the dimension D observed previously from dimensional analysis. If D > 4. then the corrections t o X-' are finite. and do not modify the dominant singular behavior of X . In particular, the critical exponent 7 remains 1. It can be shown that this result persists to all orders in $. diverge around Tc. and therefore dominate For D 5 4, the $ corrections to the critical behavior. The critical regime is said t o be fluctuation dominated. The higher order terms in $ become more and more divergent around Tc.and the mean field expansion is inapplicable. The critical behavior in this region may be understood using the renormalization group. An introduction is presented by Huang (1986) and more extensive treatments are given by Ma (1976) and by Pfeuty and Toulouse (1975). From Eq. (4.115~).we can estimate the size of the Ginzburg region by calculating the value of T Tc such that the $ term equals 1, with the result
i
-
which is consistent with the rough estimate (4.106b). Before leaving the Ising model, several additional comments are in order. In two dimensions, it may be solved exactly, and as an alternative to the lengthy original derivation of Onsager (1944). a simple derivation using Grassmann variables is presented in
214
QROER PARAMETERS AND BROKEN SYMMETRY
Probtem 4.8. One should also note that there are physkal systems such as If2 Cop4 which are accurately described by the two-dimensional Ising model: the anisotropic ionic fieid allows only two orlentations for the magnetic C o ions and the anisotropic exchange interactions between C o ions via intermediate ions are essentially confined within planes. tn three dimensions, the critical behavior is appro6nrately known numer'rcalty, and in Table 4.3 at the end of the chapter, the critical exponents for D = 2 and 3 arc3 tabuifat& to show how they approach the mean field results as D "Increases, Calculation of the leading order fluctuation contfibutlons for n-component spins 'is similar t o that we have just presented for tsing spins, except for the existence of a zero ts"tgenvalue in the fluctuation matrix associated with symmetry with respect t o continuous global spin rotations. Hence, we now consider the general problem of continuous symmetry.
We. now consider the Ructuations around the mean fietd solution of a functional integral of the form:
(4.517a) where the action has a continuous invariance group 9:
The mean field equation:
-SS
6 # , ( ~ )-
has degenerate solutions, since if J,(z) is a particular solution, gJ,(z) is also a solution. Since is a continuous group, its elements may be parameterized by a continuous variable, 6, which may be real, cornpiex, or a multicomponent vector, and we may write 9= ) (4.1l&) and S de(z) = d.($, 8)
To compute the quadratic fluctuations, we must diagonalize the operator:
In particular, we must identify the zero eigenvalues, and treat these modes explicitty, Note that if +,(z,Bo) is a particular mean field solution to Eq. (4.117). then is an eigenvedor of Eq. (4.119) with zero eigenvalue since:
The last line follows since
= O for all Bo.
with zero eigenvalues correspond to the Goldstone These eigenmodes is the angle modes we have discussed previously, For example, in the z - y mode!, corresponds to a mode in which the specifying the magnetization direction, and magnetization moves around the circular minrmum of the potential as sketched in Fig, 4.3. Since in field theoly these modes correspond to zero mass pa&ides, Goldstone modes are also referred to as massless modes, The number of independent Goldstone modes is the order of the remaining symmetry group of the system 'in the ordered phase, For example, far the Q ( n )spin model, the symmetry guoup of the ordered phase is O (n- l ) , and there are (n- I) Goldstone modes, VVr; wit! now calculate the quadratic csrrections by evaluating the partition function
where A(%,y) is defined in (4.119). Since A has a zero eigenvaiue for the mode fluctuations of C# into a component proportional to orthogonal to Hence we define the function:
%.
, it is useful to separate the
% and the remaining components
so that fluctuations orthogonal to the zero mode %(g, go) are defined by the condition
This constraint may be included in the Functional integral for the partition Function by inserting the identity
fy@o)6 (f (80)) = 1
(4.123)
in Eq, (4.r21f to obtain
0t$fzfdBof '(@a)&( f ( @ ~e)-)t l ~ ~ ~ ~ ( ~ ~ S ) - ~ ~ ( ~ . @ ~ I A ,~ ~ ~ I I (4.124) Using the Fourier representation of the &-function. the identity (4.123) can be rewdtten as
216
ORDER PARAMETERS AND BROKEN SYMMETRY
We make the change of variable in (4.124) to &tain
x e- t
l
+(X)
= 4(z)
-
#.(S,
go) and substitute (4.1231
$
b d ~ + ( = ) ~ ( = . ~ ) @ ( ~ ) + d.%(r.@o)@(z) io
(4.126a) The term linear in q5 in Eq. (4,126a) vanishes t>y parity since simultaneously changing $J into and a into -a leaves the exponent invarhnl while changing the dgn of the tinear term. Therefore,
-+
To evaluate the remaining Gauss'ranIntegral in which A has a zero eigenvatue, we define a new operator
1%)
where s i s s positive infinitesimal number which is eventually taken to zero. denotes the zero made, and AL is the projection of A on the subspace orthogonal to the zero mode, In block matrix form, A, and A may be written
The operator A, has a finite inverse, and the Gaussian integral (4.126b) with A
replaced by A, can be performed to obtain
From Eq, f4.527), we see that
det A, = ~ d eAA t is the zero mode.
and that since
dzdy
@ "
860
A;' (z,y)
(4,128b)
4.4 FLUCTUATIONS
2117
The integral over ar may now be performed, yielding:
far the quadratic fluctuations fs replaced by the
Thus, the usual factor result
&
The factor is the usual factor associated with each eigenvalue in a Gaursian integral, and the symbol I d s o denotes the volume integral over the invariance group
9.
It is shown in Problem 4.6 that for the general case of a vector field J(z) in which the mean-field (E, g) has an invailance group parameterized by the vector (8&, 8.), there are n Goldstone modes. and the partition function is given by
3,
..
g,
These condderations atso apply to the case of mean-field sotutisns which are localized En space and time, where the zero modes are associated with space and time translation, We will see in Chapter 7 how one sums over all possible combinations of such locshizedl solutions to obtain physical results.
ONE-LOOP CORRECTIONS FOR THE .X - Y MODEL To illustrate the physlcal efFijct of zero modes, we now evaluate the contribut"rns rJf fluctuations for the two-component X - Y model, Recall that the X - V model has two component spins, that is spins on the unit circle, at each lattice site. the same argument used to arrive at Eq, (4.100) far lsing spins or, alternatively, by genexai symmetry considerations, the tandau-Ginxburg form of the partition function is
by
where and are the two components of the vector The mean field equation i s
4.
2113
ORDER PARAMETERS AND BROKEN SYMMETRY
Below T,, we know that ro is negative. and
4%is given by:
We wlll parameterize the degenerate mean field solutions as fo!lows:
e.
where B. is a real parameter in the interval [Q,2r] and = The invarithe volume of which i s ance group is the group of two-dimensional rotations 50(2), dBo = 2 ~ Let . us compute the quadratic fluctuations around the solution B. = O and denote the fluctuations relative t o ? ,,(Q) as
1:"
4
Retaining quadratic terms in
q. the partition function becomes:
or in I;ouder representation,
(4.135b)
In the language of fleld theory, given a lunctlonat integral of the form (4.135a). the
6
-2
square of the mass of the particle represented by the field is the coeficient of $9 in the Lagrangian. There-fon, in Eq, (4,135a) we see that the longitudinal fluctuations. whereas the transverse represented by "t(r correspond to a mode of mass Ructuations represented by 4%have zero mass, This zero mass mode, resulting from the broken continuous symmetry is the previously mention& Goldstone mode. The quadratic fluctuation operator A(% may be written in block matrix form
The longitudinal modes, corresponding to fluctuations in the direction of the magnetization, and the transverse modes, corresponding to fluctuations perpendicular to
4.4 FLUCTUATIONS
longr t udinai
219
transverse
Fig. 8.7
Skatch o t the spattar dtstrlbution of maznetlzatlan for lonpltudlnal and transverse modes prapagating In the direction k.
the magnetization, are decoupled. The longitudinal mode given by
and its eigenvalue XL are
The transverse mode, $ire known as the spin-wave mode, and its dgenvalues are given
AT = 2% .
(4.1316)
The spin waves have an eigenvaiue spectrum which goes t o zero and correspond t o the Goldstsne mode associated with the broken continuous symmetry. In contrast, because ra 5 D below Td.the longitudinal moder have strictly positive eigenvalues. Since (4) is proportional to the magnetization, one can associate these tongitttbinal and transverse modes with the fluctuations In the magnetization sketched in Fig. 4.7, Were, one observes that addSng fluctuations in the direction of the magnetization 3i;r changes Its magnitude and thus casts finite energy in the long wavelength limit whereas adding transverse fluctuations leaves the magnitude fixed and c a t s no energy in the long wavelength !"tit. It is now straightforward t o calculate the partition function, applying the general result for continuous symmetry. (4.130). Using the parameterization (4.135)
so that
Finally, we write [det AA]-$ as - i e x p ( t r in AA] using the diagonal form (4.137b) and note that the phase space factor elhminates the C = O divergence for D 2 1, so the partition functian may be written
220
ORDER PARAMETERS AND BROKEN S Y M M E T R Y
LOWER CRITICAL BlMENSlQN The z - y model illustrates one of the ways in which the mean field approximation may completely break down in sufficiently low dimension. Let _us consider the correlation function far transverse fluctuations. The external field H couples linearly to the order parameter so in the representation (4.135).the coupling of the transverse field t o the fluctuation field may be written d D z H a ( z )$ a ( ~ ) . Hence, the transverse correlation function is
4,
where we have used the Fourier representations (4.135b) and (4.531b). The convergence of the transverse correlation function depends on the dimension D. For more than two dimensions, D 2, the integral (4.140) converges at k = Q. The Goidstone modes generate finite fluctuatians which tend to disorder the system, but the phase space factor dDk renders their effect finite, fn contrast, for D 5 2, the integral diverges, and the Goldstone modes eompretely disorder the system, This result is the essence of the Mermin Wagner theomm (1966), which states that for an a-vector model, with n 2, the tawer critical dimension is 2. That is, for 13 2, spin waves destroy the long range order of the magnetizatlon, and the mean field prediction of magnetic order is csmpletely incorrect. VVe can understand the lower critlcal dimension for magnetic systems somewhat mare general'ly by physical arguments based on domain walis, This approach not only reproduces the spin wave instabiiity just identified for vector spins, but also identifies the relevant mechanism for lsfng spins. Whenever a system has degenerate states of broken symmetry, one must consider configurations In which the system breaks up into domains of digevent states separated by domain walls. Since in the vicinity of the walls, spins are nat optimatry aligned, domain walls increase the internal energy. However, because there are so many alternative positions for the walls, they may also significantly increase the entropy, Depending upon the competition between internal energy and entropy in the free energy, the system wilt either remain magnetically ordered or spontaneously break up into domains. Let us heuristicafly estimate the entropy and internal energy of a plane domain in a D-dimensional system wit22 linear dimension I;, Since the plane may be located at any position along any of the D axes, the number of possible positions is so the entropy behaves as
>
<
S-DInf;..
(4.141)
The internal energy may be estimated by examking the interactions between successive layers of spins as sketched in Fig. 4.8. For lsing spins, a domain extends over only one lattice spacing, and the cost in energy relative to a uniformly magnetized state is proportional to the number of spins at the surface of the domain wall
4.4 FLUCTUATIONS
Ising spins
Fig, 4.8
Skslches of damaln walls for
221,
Vector spins
lslng splns and far vector splns,
For vector spins, the orientation of the spin may change from B = 0.t o B = B. over a domain A layers thick. Assuming the change in angle between sulface energy between the layers calculated by summing
-Jij
sites in one layer and adjacent sites on the next layer is .GD-' (1 summing over the A layers and expanding cos for large A.
5
- cos 2).Therefore.
Udng (4.141) and (4.142). the domain wall free energy in the case of lsing spins is
- T In L .
f 4.144)
If D 5 1, the entropy dominates, the free energy is negative, and the system will spontaneously break up into domains, F=or II > X, at sufficiently low T the Internal energy dominates yielding a positive free energy and a state of uniform magnetic order is stable with respect to domain fsmatbn, Tkius, for lsing spins, the tower critical dimension is i. For vectlx spins, we note that the domain wall energy (4,143) decreases with increasing waif thickness A. Since A. is limited by the length L of the system, we have
If we allow the wall thickness t o approach the size of the system, its possible positions become restricted and the entropy estimate (4,141) becomes an overestimate
Hence, we have the following inequality for the free energy
tf D r 2, we are assured that betow some fin%@T , the free energy is positive and an srdered magnetized state is stable, Although the present inequafirty cannot distinguish whether the disordered case arises at D = 2 or lower, it is clear physically that the ddscalized domain wall is essentially a spin wave. The domain wall argument i s thus consistent with the spin wave calculation (4.540) which showed that the lower critical dimensbn is 2.
222
ORDER PARAMETERS AND BRQKEN S Y M M E T R Y
THE ANDERSON-HIGGS MECHANISM The zero-mass Gotdstone modes we have been discussing associated with degenerate broken symmetry states only occur for finite range interactions. For example, as long as the interaction is neglighble beyond some finite number of lattice spacings, spin naves as sketched In Figg,4.8 will have zero energy in the long wavelength 'limit. However, fur infinite range interactions, spins with some finite relative rotation will produce a positive excitation energy at any distance, so the energy at zero wave number will be finite and the mode becomes massive, Thus, it is not surprising that systems with electromagnetic interactions and having broken symmetry do not have zero mass Goldstone modes. What is remarkable, however, is the structure which emerges for a theory with spontaneously broken sy mmetry w hich also possesses a local gauge invariance, This phenomenon was originally discovered by Anderson (1958) in the context of an electromagnetic field in a superconductor, and was extended t o mare general gauge field theories by Anderson (1963)and Higgs ($9641, W will ilfustrate the essential idea for a charged scafar field coupled to the etectromagnetic field, From the discussion in Section 4.2, one may regard the scalar field as the order parameter for a suberconductor and consider the present model as a Landau Ginzburg description of an electio&agnetic field in a superconductor, To facilitate the treatment of gauge invariance, it is convenient to extend our previous discussion of spatiaHy dependent local fields to include space and time dependence on an equivalent basis, and t o use the covariant natation of relativistic field theory. Vlle will use the conventions z@= (so,z" zz%, s" = {t, 2, y, it'), 2& = g ~ t ~ 1 = goo = --gl1 = -g2% = -933, and 3, = As in the original presentation of path integrals, we will consider the path integral as an integral over the Lagrangian wvitten in terms of the fetds, and to study fluctuations, we will be concerned with expanding the Lagrangian to second order in the fluctuations around the stationary solution, To appreciate the structure of the "rnteracting theory, first consider separate, noninteracting scalar and electromagnetic *fields. The charged scafar field is described by the real and imaginary components of a complex field, or equivalently by +*(S) and # ( X ) . The Lagrangian contains the kinetic term and the potential terms include -V#*V4 and the usual quadratic plus quartic potynomlal needed far spontaneous sy mmetvy breaking:
h.
l ~ ~
4'4
The mean field equatbn from varying 4* is
Mote that since L3 contains the negative of the potential, uo must be positive and 0, in which case the minimum energy the broken symmetry case corresponds to configurations are given by
\#la=--- v0
U0
.
EC$;
(4,146~)
4,4 FLUCTUATIONS
To parameterize fluctuations parallel and perpendicular to the minimum, we write and +'(z) in terms of two real fields F(%) and q(z)
223
4(z)
Expanding f4.14621) to second order in the fluctuations
so that the equations of motion are
+
+ +
rnZ = -EZ jjZ mZ = Q, The Fouiler transform of (9.148~)has the form -p@p, so that -2ro plays the role of the mass squared, Thus, as expected from our previous treatment, of two component fields, we have one massive mode, g , perpendicular to the circular minhmu of the potential and one massless mode, t, parallel to the minimum, The tagrangian for the eleceromagnetic potential A@has the familiar form
1 l( A ) - -4 F"""FP, where =f
(4.149a)
a@Ay $"A@
and variation of A@yields the free space Maxwefl equations:
In particular, F7.E = V-8 = 0 or in momentum space z.E = Z-B = 0 so there are two independent transverse massless modes of the electromagnetic field, For subsequent reference, these degrees of freedom and the associated masses are tabulated in Table
4.2. The Lagrangian for Interacting scalar and electromagnetic fields is
where we have used Eqs. (4,146af and (4.149a). and dpis coupled to C$ via the minimal coupling (8,+id,),The tagrangian is invariant under local gauge transformations
ORDER PARAMETERS AND BROKEN SYMMETRY
224
Non Interacting
Scalar
Symmetric Phase
Scalar
Broken symmetry phase
Tabfe 4-2
I Scalar
q
c
+,fb*
1
-2r0
I
0
2
!
Q
Summary of the degrees of freedam .for a charged scalar field ancl
elsctromagnetlc Pleld.
as wefl as global gauge transformations in which 51 is canstant. The mean tieid equations are obtained by variation with respect to Ql*, with the results
4, and A@,
For ro > Q, the statkonary solution for C# with mhnirnum potentirtf is q5 = #i= *Q, and linearizing Eqs, (4,1521 for infinitesimai fluctuations in (6, and A, yields
c,
Thus, as tabufated in Tabfe 4.1, the symmetric phase of the system has two massive modes for the scafar field and two massless transverse modes far the electromagnetic
field, Q, the stationary solutions with minimum potential correspond to the For ro broken symmetry fields satisfying Eq. (4.146~) and A, = 0. To parameterize fluctuations parallel and perpendicular to the minimum in the potential. we again use Eqs. (4.147) t o write 4(z) and 4'(z) in terms of two real fields E(z) and q(z). To evaluate the Lagrangian in terms of these new fields, it is convenient to make the gauge
4.4 FLUCTUATIONS
transformation (4.151) with
225
= f(z) to obtain
@(S)
Note that the electromagnetic field tensor F"*" is unchanged by the transformation from A@ to
[email protected] terms of the new fields q and A p . the Lagrangian (4.150) becomes 1 4 EpY +ll)(a, +ieA,1(4~+ rj) - ro(do + - 2(+0 + q)4 - -P (4.155) yield so that variation with respect to rj and A, combined with Eq. (4.146~)for
e = (a.
U0
-ierit)(4o
the fallowing Iinearized equations for infinitesimal ftuctuations of the fields
Since d,d,FW@ -'- 0. Eq. (4.156bf implies the Lorentt gauge condition
and (4.1SQb) may be rewritten
From the equations of motion (4,156), we observe two important results, which are atso tabugatecl in Table 4.2. a
The mode c(z), which in the non-interacting case was the massless component of the scalar field corresponding t o fluctuations along the minimum of the potential, no longer appears explicitly in the action. Instead, it has been subsumed into the field j ( z ) by the gauge transformation, Eq. (4.154). The massless Goldstone node is said to have been ""gauged away'" Only the massive node q of scalar field remains, and its mass is ana8ected by the coupling to the electromagnetic field,
*
The electromagnetic field J@(z) diRers from the free-field solution to Maxwell's equation in two crucial respects. By Eq. (4.156d)it has a mass mZ = - 2 e Z z .
A@
Furthermore, the four components have a single constraint, the Lorentz gauge condition (4,156~) so that there are three independent massive modes in contrast to the two transverse massless modes for Naxweftk equations. Thus, the massless Goldstone mode of the scalar field f has been egeetively replaced by the massive longitudhat component of the electromagnetic field,
226
ORDER PARAMETERS AND BROKEN S Y M M E T R Y
The Anderson-Higgs m ~ h a n i s moccurs in diverse physical systems. The example wet have discussed agplhs directly to a superconductor in an etectromagnetic field. The zero mass excitations of a neutral super~onductingFermi gas become longitudinal plasma modes of finite mass when the gas is charged, One physical manifestation of the finite mass is the Meissener egect, in which an externalfy applied magnetic field can only penetrate a depth equal to the inverse mass into a bulk superconductar. In the Weinberg-Salam theory of electvomagnetlc and weak interaclions using non-Abelian gauge theory, ail the masses of the gauge fields are generated fmrn as yet unobserved WSggs fields by this mechanism.
Table 4.3 Definltlons of crltlcal exponents and their values for spln systems. The mean field rftsutts, whlch are exact for Z) rf for a n y number of spln components, are compared wlth appraxlmata numsrfcaf values for lslng spins In 3 dlmenslons (LeGulllou and Ztnn-Sustln, 29.85) and the exact lslng values In 2 dlmcanslons,
>
PROBLEMS FOR CHAPTER 4
The first two prabbms treat phenomeaoXogica3 Larrdaa t h e ~ e : Probbm 1, explores the effect of a sixth order t e r n in the Landm iirnctioa and Problem 2 marnines the egeet of higher powers and grildientaa in the fiandau-Ginrburg functhnal, ProbXemrs 3 zurd 5 fill in detaib of defivations omitted in the text for the mean field mlutiaa to fhe O(n) spin model and the on*loop comections to the Ising model, re~pectively. Problem 4 treats the liquid-gas p b a ~ eLraasi-Lion in the ca~onicaXerrsembfe, and demonstrates haw the same physkal results emerge as in the grmd canonical ensemble, The next two problem deal with comections to the mean field approximation in the presence of Piero ntadeg. Problem B generali~es
PROBLEMS FOR CHAPTER 4
227
the one-loop corrections presented in the text to the case of N .era modes and Problem 7 shows how the general perturbation expansion is obtained. To complement the mean-field results developed throughout the chapter, Problem 8 presents an efficient and elegant exact solution of the twpdimensional king model using Grmsmann vasiables. PROBLEM 4.1 LainBau Theory Conaider the Landau function including a sixth order contribution
Note, in comparing with Eq. (4.41, for simplicity we have set c l = -1, d2 = 1 and bo = 1by appropriately scaling h, t , and M and we assume M couples to h line&. a) Analyse the generic f o m of L in the br - T plane at H = O identifying regions of single, double, and vasious clases of triple minima. b) Find the lines of first-order and second-order phase transitions at X = 0. c)
Locate the tricritkal point and calculate the critical exponents.
PROBLEM 4.2 Lsndsu-GOnzburg Theory Consider a Landtau-Ginzburg theor~r,with the funcfional:
a) By using dimensional analysis, show that there is a critical dirnensiolr d, == 2rl ( r - 2) such that for d > d,, mean field theory is valid, where= it, is iwal"2d neas Te for d 5 d,. b) -Nrite the rnean field equations for r 3, and calculat;e the critical exponents, c) Show, using dimensional analysis, that if L contains higher derivatives of C# or higher pourers of (V#)these terms become negligible as T" -* a",,
>
PROBLEM 4.3 The Classical O(n) Spin Madel Considw the classkal O (n)spin model defined in Eq, (4.50) for general numbem of csxnponenLnr n. To study the rnean gelcl theoq close to the critical poine, expand ln($ d ~ 6 ( $ ~ - l ) e @ a i .through ~) fourth order in 4 and Legendre transform to obtain Q. (4.75a). Explain how this resulk follows directly from symmetv mguments, Cdculaee all, the critical exponents, PROBt E M 4.4 The Liquid-Gas Phase Transition Consider a cfassical gas of pasticfes interacting via a twebody potentid v(?) subject to an external potenlial U(r), The grand canonical partition function is given by Eq. (4.91a).
a) Wride the mean field equation by applying the statianaq-phse approximation to the functional integral (4.95). In the a;b~eneeof an external patential Cr(F), %sunning periodic b o u n d q conditions, one can seek a constant solution,
ORDER PARAMETERS AND BROKEN S Y M M E T R Y
228
4(F') = cPo.
Write the equation for q%oin terms of the volume integral of bhe intttraction V. = d3r v(I). bf Show that if l$ > the mem-field solution is unique, whereas if V. < 0, there exhts a critical temperature T, below which the system can undergo a phase transition frorn a liquid phase to a gas p h w . Compute -the critical -temperature, and the critical exponents. c) Using methods similar for those used to derive Eq. (4.951, show that the canonical partition function. for a system of k" particles is:
cl)
p=
l
Show that the conat& mean field solution is given by
5 is the particle density.
=:
pV;t where
e) Calculate the presswe P and the chemical potentid p aa a function of p. Show that when V. c 0, there exiets a temperrtLure 1"", below which there are two values of p, ctexroted by and p&, for which the presswe is identical:
Pkc) = Pfpt). f ) The condition far coexistence of two phzbses is the equality of their pressure and their chemicd potential. Using pc; and p& frorn part e), show that the condition @(pG) -"" @ ( p & ) is identical to the mean-&lid equation derived in the grand canonical ewe,
PROBLEM 4.5; Pedorm the Legendre transform for the fsing model including one-loop corrections to obtain Q. (4,150). The algebra is greatly simplified by expanding consistently to first order in and avoiding explicit evaluation of the quantity which caneeh out of the final result.
:
PROBLEM 4.6 One-Loop Corrections In the Presence of Zero Modes Consider the pxtition function
in the presence of a continuous symmetrg for which the mean field equations
have degenerate solutions 4.(z, 8') parameteriaed by an n-component vector 8 = {B1, 82 B,). Expansion of the integral (1) around a specific mean Geld 4,(z, g) as in Eq. (4.122) yields
...
PROBLEMS FOR CHAPTER 4
229
where
By t & b g derivatives of equation (2) with respect to Bi, show that there we n Goldstone mode9 a)
b) As in Section 4.4, we will separate the transverse and longitudinal fluctuations. If we define n functions fi (B'):
then the ttuctuations orthagonal to the Goldstone modes satisfy fi (8) = 0, i = 1, n. Show that the constraints fd(8) = O may be included in the functiond integral by incladkg the following multidimensionail generalisation of the idenkity Eq. (4.123) in the padition functian
c) Introducing an given by:
field h(%),&hawthat the pastikion function f 3) ib3
where
(41
%
are not necessarily orthonormal, let 4) Since the Goldstone modes (++(r),i = 1, a) denote an orthonormal basis of the eero eigenspace of A and
define
n
AE
=
E El+i)(+i[ -4-
.&L
i=l
maIogous t.o Eq, (4.125a). Replaircing A by A, in show that when goesr to eero
(4, and performing the (14 integfal,
ORDER PARAMETERS AND BROKEN SYMMETRY
230
where (f[g) denotes e)
dz f (z)g(z). Pedoming the {Ai) integrals in (51, show that
and thus
f)
Show that in the case of a vector field J(z) thia ~ g u m e nyields t Eq. (4.131). 4
PROBLEM 4.7 Perturbation Theory in the Presence of Zera Modes; In order to perform a perkwbakion expansion, it is sufficient to be able to calculate Gaussian integrak, using the identity
Lee us define
and 84, =
R is a eero eigenvalue of A(%,g). We regularire A as:
and integrate over transversle fluetuakions by introducing in Q. (If the identity
where
34
dz $(%,go)
(-d(z) +#C(Z,@@)) -
PROBLEMS FOR CHAPTER 4
232
where
and
= j(s) b)
li>84, (z) (a+e(a4e)as,-
Show that:
where we have used the fact that
in terms of
e formal pedurbation ex-
tial of an operakor acting on l ( j ) , at j = 0, pansion of Z ZLB the ex Evaluate the firsit order term of this perturbation expansion. PROBLEM 4 3 The Twc)-DEmenrbonaltsfng Model This probhm autlint?as the solutbn of the tw+dimensional fakg model using a method due to Samuel (1989). Consider am king model an a q u m e ladtice of s k e H = (2Mz 4- 1) X (%Mg l), wifh ~ e a r e s fneighbar intcaraction J, and periodic boundw conditions. The partikion. function of the system ia:
+
where C{s denotes a sum over all spin configurations S, = &l, the sum runs 71 over all latdtee sites, and 2, and a", me unit vectors along the horiraoataf. and vertical bonds, a) Using the fact that we are dealing with king spins (S = &l),show that:
where
K -- th@J . E o m this identity, show that
232
ORDER PARAMETERS AND BROKEN S Y M M E T R Y
b) The high temperature expnaioa of Z is obtaimd by expanding in powers of K , U ~ i n gthe properties that:
show that t; ia, the sum of all, clased polygons joining adjacent sites of the lattice, not connected and possibly selfcintersecting. The might wsociaded with a graph h obtaiaed in the follawing way: there is a factor Ir: per bond, and there is an overall factor I", which is the numb= of ways in which the palygon can be drawn on tiha XatLice?. S h m that graphs I, 2, 3 below conk&bute to t;, and vaphs 4 and 5 do not.
Show that the contributions of the first three graphs are $1 = N K 4 , h = $ N ( N 4)K8,and 23 = NKa2 c) The essential idea of thk method is to wr;ite a Grassmann functional intee g r d which, when expanded, generates all the ~graplte;of 2' with the comect weights, Since each bond belongs to eero or one polygon, it is natural Lo try do asrhsclciate Grassmann variables with bonds. Hence, we will let JI!" denote a creation operator denote a creation operator for for a horilrontal bond between F and i a,, $iV a vertical bond between ? and 3 -+ a,, and define the comespomdhg ann&ilation We associate with each of these operators the Grassmann operators +3 and We will now prove the following formula: variables rlfh g:"
+
+:.
.
n,
where dp = dr):hdrl,hdr):hdq: by showing that it generates the same set of as in (b) with the same weights. Fmt, show that (2) can be rewritten as
where the brackets ( ) are defined by:
PROBLEMS FOR CHAPTER 4
233
In order to evaluate (31, we can use Wick's theorem, with the Cmrrtian weight given in (4). Show that the only non-vanishing contractions are (r):hqFt) = srr, (v:@V ; # ) = &-rf There are four t e m s in (3) that can be represented W corners:
-
and two t e m s that; can be represexlted as bonds;
Show by expandiag (3) that the only non-vanishing graphs are the polygons discussed in [b). Finally, evaluate the factors associated with each graph. In pmticular, check that all e~raphscome with ar pius sign and have the same weights as those of the Ising model. d) In order to evaluate (21, it ier convenient to diagonalillre the quadratic form by going Lo Fourier reprmentation, Us~ingthe following Faitrrier @@fie@:
r.
where k = (kl,kz), kl = &g, t = (-4,. .,+MJ),show that:
.
kz = f i t
with s =
(-M,,...,+Ms) and
The quadratk form in ( 5 ) is not yet quite diagonalized due to the couplings of g to -&. Show that it can be rewritben W : e)
ORDER PARAMETERS AND BROKEN SYMMETRY
234
Re+te
g in terms of the new vmiablies
and show that the quadratic form is diagonal in g a) Evaluate 1, In the thermodynamic limit, &, MW -. +W and the surnmations over g are replaced by integrals: %. Show that the -r N fmas enerw isl given by the Onsager rt?~uXt:
$2
PP fV
=.:
-In 2-
-
-h [ c h 2 ( 2 @ ~ )sh(2PJ)(cos kl +cos kz)1 . (6)
g) The wgument of the In in (B) is minimal fi7f kl = k2 = 0. Show that far kl -- k.2, the wgurncsnt of the h isl p o ~ i t k e ,and that it vanishes when the temperature T reaches a critkal value T,. C_alculate T,. All the non-analyticities of F, if any, thus corn? from the vicinity of k = 6 and T = T,. By expanding the casineg; in (6) around k: = 0, and rescafing appropriately, show that when T goes to T,,the leding non-ana'lyticity of F h= the behavior:
Show that the specific beat has a lagarithmic divergence, and thus the critic4 exponent
a! vanishes.
CHAPTER 5 GREEN" FUNCTIONS
In Chapter 2, Green's functions emerged as natural quantities t o characterize many-particle systems, Qn the one hand, they are defined as thermal or ground state averages of t'ime-ordered products of operators which are directly calculable i n perturbation theory, and on the other hand they may be related easily to expeimental obsenrables. Thus, although any paeicular obsenrable could be calculated directly using the techniques of Chapters 2 and 3, it Is worthwhite to develop the general properties of Green" functions which are applicable to a broad range of phenomena and are widely utilized in the literature.
The n-bady real-time Green" function In the Grand Canonical Ensemble was defined in Eq, (2.21) as the thermal trace of a time-ordeved product of creation and annihilation operators in the Weiisenberg representation:
To simplify and unify the notation, It is convenient to replace operartors a f ( t ) in the a-representation by field operators defined in Eq. (1.89) $ ( z , t ) I ei(ir-@B)rd(z)e-i(B-at
and to denote the thermal average by brackets:
Note, foflowing the convention established in SmLTon 1,1, any internat degrees of freedom such as spin or Osospin, which may have been explicitly included in the label a,are now implicitly Tnctuded in x. It will; often be desirable to compress the notatbn stili further, reptacing the arguments {x,t,) by n:
Similarly, using field operators in the imaginary-time Helsenberg representation, Eq. (2.231, af ( r ) -+ 4(z, r ) = e(8-'&lr$(z)e-(8-'A)r the thermal Green's function defined In Eq. 12.22) may be written
where the brackets again denote the thermal average, Mote that in this notation realtime and thermal Green" functions are distinguished by arguments G and r , respectively, Finally, the zero-temperature Green" function is defined as the ground state expectation value:
t o emphasize its similarity "t structure to the finite temperature Green" function, we wilt also write it
~ c ~ j ( z. - ~z,t,iz\t:s.. t ~ ,
m
.
~",t',) = ( - i ) " ( ~ $ ( z l t *~ $(z,t,)$'(z',t',) )
- q(z\t\)) (5.3b)
Note that finite and zero-temperature Green's functions are distinguished by the use of $ and G,and that the brackets denote a thermal trace or ground state expectation value, respectively, Also, the inclusion of the factor (-i)" in Eqs. (5,lf and (53) is not a universal convention: whereas it is used by Abrikosov, Gorkov and Dzaloshinski (ISS3),Doniach end Sondheimer (1974). Fetter and VVaSecka fd971), and Kadanoff and Efaym (1962),nu factor of i is used by Thouliess f1912), and Brown (1972) uses Physkclly, the real-time Green" functions at finite and zero temperature have an obvious interpretation: they describe the propagation of disturbances created by injecting particles at the space-time points ( z l i t l ) * (z,t,) and removing them at (zitg - .{zkt",). Note also that the definitions are quite general, applying ta finiteor irrfinite systems, with no requlrement of translational invariance. Since one-particle Green's functions will be studied in detail in subsequent sections, it is useful to introduce several auxilhary definitions, For notational convenience, the superscrbt (n) will usually be omitted on one-particle Green" functions. The two time orders in real-time Green" functions at zero or finite temperature are distinguished by the definition
so that
1') = ($(t)tjf (l')) and iG'(1jl')
=c($~(lt)~(l))
where g is the usual factor denoting 1 for Basons and -1for Fermions. These quantities are combined to define retarded and advanced Green" functions as foliovvs
Analogous quantities are defined at finite temperature by rephcing G by (5.4- 5.6)
in Eqs.
5.1 INTRODUCTION
237
EVALUATION OF OBSERVABLES Although Green's functions were introduced because of their physical connection t o response functions. they contain sufficient information t o evaluate all ground state observables or thermal averages. Since the ground state expectation value of an nbody operator requires the matrix element of n creation operators t o the left of n destruction operators. it may be obtained directly from the n-particle Green's function by evaluating all the times associated with creation operators at time t+ infinitesimally later than the time t associated with the destruction operators:
A general one-body operator d = d z d x t 6 ( X ' , x ) ~ ) t ( x ' ) $ ( x ) thus has the expectation value
(0)= ii
.
d x d z ' ~( z ' x ) G ( x ~ ~ x ' ~ + )
(5.8)
For example. using Eq. (1.96). the kinetic energy operator corresponds to M that
6 ( X ' , X ) = -&;;6(d, x ) V j
where the explicit spin sum is written in the last line for the case of particles with spin. Similarly. for the spin density at position i * ' . V ( z t , z ) = 6(i' - i " ) 6 ( i i')i?,t, and
-
For any system with a time-independent Hamiltonian. the Fourier frequency transform may be written G ( z t lx't') = G ( x , X'; t
so that
(0)= is
/
- t') =
dzdxtO ( X ' ,
X)
1: /
--@(X,
S'; w ) e - ' ~ ( ' - " )
& e i w q ~ ( x , X'; W ) 27r
(5.12)
where rl is a positive infinitesimal. Finally, i f the system is also translationally invariant, it may be useful to take the Fourier momentum transform. being careful to carry the spin dependence separately if necessary:
238
GREEN'S FUNCTIONS
The kinetic energy may then be rewritten
FinlW-temperature results analogous to Eqs, (5.8- 5,141 are abtained stralghtforwardty using thermal Green's function, The only changes are elimination of the factor i !R the relation dzdz' 0(z'z)
(zrlzlr+)
(5.15)
and mplacement of the Fouder hquency integral in Eq. [fi.il)by the Fourier series Eq, (2,129). The ground state energy B@,internal energy U, and grand potendal iZ arts of partkutar interest. Although for a Wamilton"rn eorttaining two-body forces, they can be calcutated dfrectty from the two-body Green's function, they may also be obtained from the one-body Green" function by using the Heisenberg equations af motion. Using the field operator commutation relations, Eq. (1.881 and the commutator identity (5.143) B&]- = B],6 - $BlA,&l,
[A,
the Heisenberg equation for
$(S,
[A,
t ) yields
where we have used the symmetry of v(z, zt) and T, denotes the differential operator for the Mnetic energy used above. Thus,
The graund state energy may be written
or for translationally invariant systems
Analogous relations are obtained for the internal energy ~t finite temperature using thermal Green's functions and the Heisenberg equation - &$(S, t ) = [$(S,r ) , &-p&I with the result
The connection between the expressions for .E& Eq. f5,20f, and the perturbation expansion of Chapter 3 i s not evident because of the presence of the time derivatives or w Factors, To obtain a direct connection to diagrams, it is useful to recall the defin"ilion of the self-energy, Eq,(3.38)
-
Thus, the combination (i& T ) G appearing in (V) as a result of the equation of motion can be rewritten (Gw% Z)G = 1 66. Since we have diagram expansions for both and G they can easily be combined for the complete result. However, when (T) is added t o obtain &, Eq. (5.20)no longer has the desired simple form. A convenient technique, originally due to Pauli, to generate an expression far Ea of the desired form is to consider a continuousfy parameterized set of Hamiltonians
+
having eigenvalues Eo(X). eigenfunctions rYo(A). Green's functions G(", and selfenergies C(*).Using the familiar relation &E@(A) = ($o(X){V[+o(X)) (which follows Hl+o) ($01 ( & H ) /$o) ($ol~$l$o) and from evali~ating&EO(A) = (&($Q/) noting the first and third terms yield E ~ ( x ) & ( $ ~ ~=+O~since ) is normalized). the difirence between the energy of the fully Interacting system E. =. E o ( 1 ) and the energy of the noninteracting system WO= Eo(0) may be written
+
dX
s2
i i C
($0 (A)
+
/AV/$a(A))
d z d z ' ~ *(z, z'; w ) G* (z' ,z;w ) e'""
.
140
GREEN'S FUNCTIONS
For a transtatbnally invadant system
d3kdw
C C;,
(k,w ) ~ $ , , ( kw)eiYq ,
.
(5.24b)
@pf
Analogous tlxpressions for the Grand potenthat are obtained in terms of thermal Green's functions using the relation &Q(A) = ( V ) proved in Problem 2.11 and the definition of the self energy Eq. (2.178) (note the sign)
Note that the price: of obtaining direct contact with diagrams Is the additional parametric integral over the coupling strength X. It is an instructive to exercise to combine the diagrams for G and 9 to obtain the and the details are given in Problem 5.1, The familiar expansion for R-R. or &-h, trace ZA9*is represented by a closed graph composed of g* connecting to the two exrernai points of CA. An nth order contribution is obtained from an mth order term of C h o mbined with an (n m)" order term of and the parametric integral yields a factor ,'A" = It is seen in Problem 5.1 that a given graph in $92 - $"lo arises from combinations of many bits and pieces of C& and g*. Thus. the Grand potential is an example of a quanltity which may often be more easily calculated directty rather than by using Green's functions, In addition to bdng inconvenient, use of the one-particle Green" function nay also be dangerous since a seemingly innocuous approximation having little eRer;t on one-particle properties may have a large uncontrolled e f i c t on f2. Physically, this refiects the fact that the one-particle Green's function is directly related t o single-parjcicle propagation in the many body system, instead of the twobody correlations to whkh h . and flo may be strongly sensitive. The principal results of this section are summarized in Table 53.
jl
5.
-
Singfe-particle Green's functions have impovtant anatytic properties which follow from generat p~inciples. We will establish the saiient properties of zero-temperature. real-time fin"ie temperature, and thermal Green" functions in turn.
ZERO TEMPERATURE GREEN" SFUNCTIQNS The spectral representation for the zero-temperature Green's functions (lehmann, 1954) is obtained by inserting a complete set of eigenstates between the field operators in the definition of G(zt
5.2 ANALYTIC PROPERTIES
Table 5.2,
242
Summary of results for zero temperature and tlintte-temperature
Green" Functions,
Mote that the creation and annihflation operators only connect the N-paeicle ground state to (N+I)-particle states and (N--1)-particle states, respectively, The Heisenberg field operators acting on eigenstates !$,M) with energies E? yield the explicit timedependence c - ~ ( ~ : " - ~ @ ) ( ~ - " ) for the first term in Eq. (5.26) and c-i(E:-x-E@)(g'-g) for the second term, Using Eq, (3.301,the FouPier frequenq transform of the Green's functtons is
The analytic behaviar of G [ @ has ) thus been clearly isoiated and is sketch& in Fig. 5.1. For each eigenstate )+I:' of the (N l)-particle system. there is a pole in the lower half plane at E ~ + '- EO with residue ($Q $(z)l$:+x) (+f+ there is a pole In the up er half plane: at ene t(z)1~-~)($f-'1$(~)1+~ The ) . poles in the lower half plane begin - E@and atend to +m and those fn the upper half plane extend . ~ f - ' Because . it is often convenient to deal with from -m up to pN = E functions which are analytic in the upper or lower ha@plane, it is fwquently useful t o use the retarded and advanced Green's functions, Eq. (5.6). The analytic structure for
+
-
G (U) X X
X
X X X I X X X X X
G ~ ( ~ J )
(W)
K k X X X X X X X
X X X X X X X X
X X X X X
fN*t)
Fig. 5.1
Potss of
G ( @ ) aR(w). . and GA(u)In the complex w
plane.
GC)and
shown in Fig. 5.1 immediately follows from that of G(@)by noting that GR(t - t') contains only B(t - t') like the (N+ l)-particle contdbution to G(t - t') and GA(t- t') contain B(t' - t ) like the (N- l)-particle contilbution. For natationa! sllltpficity, we ill present the remaining results for a translationally invariant, infinite system. It is important t o note, however, that all these results have. obvious counterparts for finite systems. Since = p ( N ) = p in the limit of large systems, It is conven'rent t o measure ail energies relative t o the chemical potential and we define
E:"
- & = E:+'
EN+I 0
+E
~ + X
- E0
= E:+' m
and
Writing the field operator in momentum representation $ ( S ) = C, &e'L''bk and of total momentum f k. Fourier transformation of the Green's writing eigenstates function Eq. (5.21) to momentum space yields
where we have. ROW indicated the advanced and retarded cases as well for completeness, The two simplifrcati~nsfor infinite sqrstents are positive definite residues, representing the probability of finding a: or okl$@) in eigenstates of the (H 1)-or ( N - 1)particle systems. respectively, and a common starting point p for the ( N 1)-and ( N - l)-particle poles. For real w . Eq. (5.29) shol~sthat G, GR,and GA are related as follows C" (W)' = C"(@) (5.30)
+
+
5.2 ANALYTIC PROPERTIES
243
and
or equivalently
Since the poles of G ( k , w ) become arbitrarily closely spaced i n a large system. only averages can be measured and it is useful t o define spectral weight functions
and
p(k, w ) = e(w)p+(k,w ) - s@(-w)P- (h, -W)
g
In terms of these weight functions,
and
l
GR( k , W ' )
From the relation
& = P$
i d ( w ) . where P is the principal part. it follows that dw'
p(k,wt)
so that the Green's functions satisfy the dispersion relation
244
GREEN'S FttPdCTtONS
Fig. 5.2
Schslmatle representation of a (?,p) or (e,e"pf
reaction.
The commutation relations of the creation and annittitation operators give rise to a sum rule! for the spectral weight
Combined with Eqs, (5.33). this sum rule establishes the high hqueney behavior of the Green's functions:
h' p+(k, U') - $p- (k,W ' ) 2a
W
1 = --o
,
(5.37)
Experimentally, the spectral wdght function is accessible through semi-inclusive aperimentsi. Consider, for example, a ( 7 , p ) or (e,e") reaction on a nucleus, as sketched in Fig, 5.2 in which a reat or vivltual photon of momentum g is absorbed by a nucleus in its ground state and a proton is ejected, In tha finat state, only the qected proton having momentum p = k q is detected and the rest of the state is unresotvd. tn the impulw approximation, (in which the interactions of the ejected particles are neglect4 the proton knocked out by the photon must have had an initial momentum of k, so a proton of momentum k has been removed and the cross section is
+
D
= 2 n x I($:-'
+
a&l$o)la6(~f-' E, - Eo - E,)
(5.38)
= P - ( ~ , E+, p N --E@) By varying the kinematics and iactudlng wrrections to the impulse approximation, much has Been learned about the behaviar of the spectral weights (Frullani and Wtougq, 1984). An interesting energy weighted sum rule for p-(k,w) has been derived by Koltun (1972) and is derived in Problem 5.2.
FINITE TEMPERATURE GREEN" FUNCTIONS We now turn our attention to the analfiic properties of finite-temperature Green's functions. Instead of regarding thermal and real-time Green's functions. Eq. (5.1- 5.2).
Domains of complex t - t' plane in whlch Q'(zt[zet') and Fig. 5.3 $<(ztlz't') may be contlnuad. By the perlodlclty of Q, the domalns are rePQrtBQindeflnftely tn the tnaglnary t dlractlon. as distinct entities, Ft is useful to consider $(ztlztt)) as a function in the complex t -t" plane. Under the change of' variables it -+ 7 ,
so that
Thus, a singte function of a complex time variable specifies the real tine Green's function alang the real-$-axis and the thermal Green's functions along the imaginary t-axis. The real-time Green's function, whlch describes the physical response of a system, may thereby be obtained from the thermat Green" function, which is cafcutable by the perturbation theory described in Chapter 2, by straightforward anrrfytfc continuation, To see the analytic structure of Q in the complex t-plane, we write
+ B($' - t)S<(~tls't)
-
$(ztlztt()= @(t tt)$'(~tl~'t'l)
(5.40a)
where
g'
--i (zt12'tt) = -
Z
-
-
i
t
-
-
(
-
-
-
1 tlit (4
In order for the thermodynamic traces to converge, 9% and can only be continued into regions i n which the factors multiplying &' - p& in the exponentials must be positive for the partition funchave negative real parts. (Note that .& tian t o exist), Hence, Q> ft t') can be eontlnued if Re (6 - i(t - t'))) r O and Re (i(t - tt)))3 O, Similarly, t') can be continued If Re ( B i(t -- t'))) > Q and Re(i(t - C?)) < 0,Thus, 9' and Q< may be continued from the real axis in the following domains, which are also sketched In Fig. 5.3.
-
-
+
246
GREEN'S FUNCTIONS
We have already shown in Chapter 2 that thermal Green" functions for the noninteracting system are pdodlc or antiperiodic with period @. Thus, the interacting Q constructed from them must also be periodic or antiperiodic when the imaginary part of the argument is shifted by /3, and W confirm that GCft--- t') is related to G>($- t' - i p ) appropriately as follows, using Eq. (5.40) and the cyclic property of the trace:
This periodicity is consistent with the domain of continuation shown in Fig. 5.3, since the discontinuity between G> and G< adjacent to the real axis must be repeated again along the .t. -- t1 =.: f ip axes. Ta study the analytic properties in the w plane, we insert a complete set of states t o =tract the explicit time dependence as in the zero-temperature case and Fsourier transform. As btsfore, the singularity structure is the same for finite and transtationalty Enlrarlant systems, and we only treat the translationally invariant case here for notational convenfencc3, Wdtlng the trace and completeness relation using a complete set of dgenstates (I$,)) with ail number of particles
(5.43) Extracting the time dependence as In Eq. (5.26). Fourier transforming t o fwquency and momentum space as i n Eqs. (5.27) and (5.29) and treating the advanced and retarded cases fn the same way, we ~ b f a l n
Note that we have used the fact that the only non-vanishing contributions ads@for states in which the number of pavtictes in state n, .Em, is one larger than the number in m, Nm. Since the W o terms in Eq, (5.48) include all N, .= Nm 1, they have a
+
more symmetrical form than the conesponding terms in Eq. (5.2.4) which involve only f and N -- 1. Thus, we may combine them to write the spectral wdght function as
1V
+
Qne observes that this expression efietiveiy rduces to Eq. (5,325) in the zero temperature limit as foftows. Far w r O the first term dominates and denoting by the state with minimum E, - pN,,-,, C, ~ - ~ ( ~ ~ - ' ~ " ) l ( $ , l o : / $ m ) ~ = ' (&lo~~$,,)~~. Analogously, for w < 0. the second term dominates and yields the factor l(+m0 1+%0 1%. Using Eq. (5.45). the spectral representation of gR and gA may be written in the same form as the zero-temperature case, Eq. (5.33b).
#
14
The real and imaginary paes obtained from Eq, (5.44) and (5.46) may be expressed in the fallwing form using the identity 1 ~ ~ e "=f taah l ~
Hence, we obtain the dispersion refations
and the relation between 9. gR. and
+
GA
where nfw p) is the fttlrnitiar occupation probability, Eq. (2.75bf. The last result reflects the fact that the poles in the upper end tower half plane overlap at finite
temperature and only in the zero temperature Iimlt does one recover the simple structure of non-overlapping poles reflected in Eq. (5.346):
+ +
Note in making this comparison that a t finite temperature the combination w p occurred in Eq. (5.44) where only the frequency occurred at zero temperature in Eq. (5.27). Denoting the zero-temperature convention for the frequency as = w p. we confirm that the factors B(fw) = B (f(wo - p)) agree with Eq. (5.31b). The sum rule for p(k, w ) is obtained as in Eq. (5.36).
so that the gnite temperature Green's functions have the high frequency behavior
Finatiy, we refate the thermal Greenesfunction to the real-time Green" function through its spectral representation. Since gfsc,zt; T - r') is periodic or antiperiodic on the interval (0,8). we expand in a Fourier series with Matsubara frequencies w, =
F
defined in Eq, (3.129). Inserting a complete set of states "t $[zrlzrO).Eq, (5.2). and evaluating the Fourier transform on the interval O < r c as in Eq. (2.131) yields
Comparison with Eq. (5.46) shows that -$(k,w,), gR(k,w,) and gA(k,w,) are given by the same complex function specified by the wdght p(k,w) evaluated along the imaginary axis at the discrete Matsubara frequencies or infinitesimally above ar below the real axis. Given the positions of the singularities of gR and gA sketched in Fig. 5.4, it is clear that -$(W,) calculated in perturbation theory is to be continued In the upper half plane to determine gR and in the lower half plane to determine gA. From S R ( w ) and g A . $(W) is specified by Eg. (5.49). Although perturbation theory only specifies $(wR) at a discrete set of points, the continuation is unique because of the at infinity. requirement that g ( w ) Although we will subsequently present a detail& example of haw this analytic continuation works in Section 5.5, we conclude this present section with the simple
-
5.3 PHYSICAL C Q N I E N T OF THE SELF ENERGY
249
l
In the G O ~ P ~ GWX P B ~ ~ B Fig. 5.4 Contlnuatlon of the function from polnts w = iw, along the Imaginary axfs where It equals -$(W,) to points inflnlteslmatly above the real axis where it ylelds gR(w) and below the real axls whsre it gives gA(M). example of the Green's function for a non-interacting system. Equating the nonInteracting thermal: Green" functions, Eq, (2,131bfto the spectral representation Eq. (5.53). we obtain -X (5.54a) k n
- (ek - CL)
from which it fofifows that
These manipulations are equivalent to simply replacing iw, in the thermal Green's function by ortiq in the real-tlme Green's function (and including the overall minus sign from our conventions). Since there i s only a single pole, there is'no distinction between continuation from above! or below the real axis. BY Cartson's theorem, other functions, -f- sin. @W which coincide with .the thermal or Green's function a t the points iw, are ruled out because they do not converge as at large W , Thus, shifting the frequency by p to coincide with the aers-temperature convention and using Eq. (5.40) we obtain
consistent with the zero temperature limit, Eq,(3,31)
5 3 PHYSICAL CONTENT OF THE SELF ENERGY At this point, I t is appropriate ta complement the treatment of the formal properties of Green" functicrns by consideritlg the physical cantent of a specific illustrative case: the
[email protected];eGreen" function for Fermions at zero temperature, Since
250
GREEN" FUNCTIONS
Dyson's equation, Eq.(2.178) expresses the difference between the non-interacting and interacting Green" ffunetions in terms of the self-energy, I=
l
=w
(5.56)
- Er + iesgn(w)
all the many-body physics is contained in G and E t is most convenient t o study G(k,w) directfy. Recall the convention from Chapter 3 that all energies and frequencies are defined relative t o the Fermi energy e F . Also note once again that we assume translational invariance not of physical necessity but rather to simplify the notation by rendering equations diagonal in momentum space. The essential features of C arise in the first two orders of perturbatbon theory, so we shall consider the approximate GreenPsfunction defintsd by the second arder self-energy
1 w -rk
- ~ l ( k- )~ z ( k , w )
g
(5.58)
The first-order self-energy is
where, according to our standard conventions, the curly brackets denote an antisymmetrized matrix element, upper case letters denote occupied momentum states, lower case letters denote unoccupied momentum states, and the Greek letters (with the exception of w which always denotes frequency) Indicate an unrestricted momentum which may be above or below kF. Because the ekenfunctions in a translationalfy invariant system are plane waives, Cl(a) coincides with the ldartree b c k potential, Eq, (2.180). Mote that since the Wartree-Fock potential i s instantaneous, it has no frequency dependence. The Cull generality of the structure in the self-energy arises in second order, for which we have previiousty evaluated the cantributlons of diagrams having twa-particle one-hole intermediate states and two-hole one-particle intermediate states. Eqs. (3.58 3.59);
-
Because these and aft higher order diagrams have finite extent in time, they have explicit frequency dependence. Both terms in Cz have an infinite number of poles. The denominator of z 2 p z h vanishes when w = e, eb --- c ~ j gwhich requires w positive corresponding to an energy greater than e p . Similarly. ~~~~p has poles when w = m + c ~ e. corresponding to w 0,the finite negative and energies less than CF. Since Irn C Z p l h O and Im C Z h ' p imaginary parts of C%replace the infinitesimal displacement i q s.$n(@) required in the non-interacting Green's function.
+
5.3 P M Y S I W L CONTENT OF THE SELF ENERGY
25%
+
Fig. 5.5 Sketch of e, & ( a ) 4- x n ( a , w ) as a funaton of w. The dashed vsrtlcal asymptotes d ~ n ~ the t e poles of Cz(a,w) and the dots tndfcats the grzrphlcal soiutlczns fer the positions af the poles In Ga(a,w).
W now consider the pores in Gs(or, W ) . By our genera! arguments in Section 5.2, the poles represent the dgenstates o f the interacting N 4- I, particle system and the residues specify their overlap with ail$@).If the system behaves as non-interacting particles, there will be a single pole with unit strength as In the case of ~ ~ ( a , ~ ) As the system becomes more and more strongly Interacting, the strength will become fragmented between more and mare complicated states, subject only to the sum rule that the integrated strength remains 1. To analyze the pole structure of G z ( a , o ) , it OS convenient t a pedorm the graphical construction shown in Fig. 5.5, wheuc?!E , +Cg (a) Ea(&,w) is sketched as a function of w . We will first consider states above the Fermi sea. For every value w = ea et, LB at which ~ T ' ~ ( a , whas ) a pole, a vertical asymptote is drawn i n Fig. 5.5. and the function r, Cl(o) ~ ? " ( a , w ) must smoothly decrease from +m t o -W between even pair of asymprotes as shown. The conditbn for a pole in G2(a, W) that w = e, Cl(a) ~ r ' ~ ( a , wis) represented on the graph by the intersection of c, C l ( a ) f ~ y " ( a r , w ) with the straight line at 45@. Having appreciated the general structure, it is instructive t o consider a schematic exampie. Instead of the infinite number of poles discussed above, we will assume Ela(@) has only two poles
+
-
+
+
+ +
+
+
and study the potes and residues of
vv here far notational convenience = e,+Z1 and we have suppressed the i q . Further, we ~ I l assume f that El i s above .Eo, E2 is below E. and that the residues in C2 are very small, satisfying the conditions
Fig. 5.6
Graphical solutlon for the, poles of
Gt(u) and G%(@) In Eas.
(5.62)
and (5.64).
For subsequent reference, since Al and Az are very weak so that C2fw) is in some sense small, we first neglect Zz(w)entirety and perform the graphicat construction for
in pavt (a) of Fig, 5.6. Since & has no W-dependence, the graph is structureless and a singte pole of unit residue occurs at RQ.In part (b) of Fig, 5.6, the graphical construction is repeated far Ca(u). Far away from El and E&, &+E2(@) appr~ache~ the horizontal line sketch& in part (a), and it is only very close to the singularities El and that the curve diverges to &W. Instead of the single pole a t E@in case fa), we now have three poles: ~1 very close to El, wz close to E%,and wo close t o .Eo. VVe now expand Ga(o) around each of the three poles w, which are solutions t o the equation (6.65) S = 0, 1, 2 W , =. Eg Z2(w*)
.
+
Near the pole U,, W
- C2(w.) - (W -w,)C;(W,) = (U - U&)(1 - c:(@.))
- E o - C 2 ( w ) M W - - E.
(5.66.)
so that Gz(u)@S
1 (1 - Z$(w,)]
1 *
02 - W ,
(5,G@&)
Far our schematic model. the Thus, in genera! the residue of each pole is assumption (5.63) makes it easy to evaluate the residue for each pole, First, note using Eqs. (5.65) and (5.63) that the shifts w. - E, are negligible relative to the energy spacing and
5.3 PHYSICAL C O N fENT OF THE SELF ENERGY
253
To leading order in 6, the resldues are
and
Now, the full physical eff"et;rc of switching on the matrix elements Ai is evident. Without this coupling, the system has a pole with unit residue corresponding t o the propagation of a single particle, as sketched in Fig. 5.6a. After switching on the "iteraction, Fig. 5,6B,the strength is now fragmented between three potes. The system still possesses a fundamental excitation near the odginal energy E@,but now the energy is shifted slightly t o wo and the residue I - C:, is less than one. This ch like a single-pa~icle pole i s called the quasiparticte pole: It still behave excitation, but is no longer a true particle pole because! of the medium modifications, Condstent with the sum ruta, Eq. (5,36), the strength which has been removed From the quasSplartlcfe pole has been distdbuted to the two new poles, with strength going t o the pole at wi. These poles represent mare complicated excitations of the many-paetcle mttdium, such as two-particle, one hole states. With this schematic model as an introduction, it is now appropriate t o return t o the general expression for the seeond-order self-energy, Eq. (S,6a), which has both two-particle one-hole and two-hole one-particle contributions. The residue of the quasiparticle pole correspondkg t o the state a, assuming a weak interaction v , is given by
1
~{ablvl~~)~"
w l - i ' ( W e+ t&& f g - e a - c b ) B
(5.68)
Thus, as shown in Fig. 5.7, the strength associated with a particle state is depletd by couplhng both t o two-particle one-hale states above the Fermi surface and to twohole one-particle states below the Fermi surface. In the limit of a continuum, these two-particle one-hole and two-hole oneparticle states yield a smooth background in addliaon to the simple quasipa~icieexcitations, Having introduced quasipart'ides, It is natural t o ask if there is any regime in which they provide a useful and accurate descflption of a physicaf system or whether
254
GREENS FFUNCTiONS
Fig, 5.7
Fragmentatton of strength ai the qLIaSlpsdICl@pole.
they always decay so qtrickfy to more complicated states that they are of no practical significance. In fact, LandauVsFermi liquid theovy is based upon the quasiparticle picture and as shown in Chapter 6 can provide an exact description of physicai Fermion systems in an appropriate limit* The essential pdnt can be seen simply by calculating the lifetime from the imaginaq part of I=2 in Eg. (5.60). Since by the usual argument with outgoing wave boundary conditions. a state with complex energy E = ER - i$ has lifetime r = f. the lifetime for a quasiparticle state a evaluated at the quasiparticle pofe e, above the Fermi energy, i.e., e, > 0, is ghven by
Mote that if we had considered a hole state, the two-hole one-paflicfe component of Ga would have contributed instead of the Wo-particle one-hole camponent. When the energy 6, is close to the k r m i energy E F , phase space restrictions on the sum over a, b. and B severely limit the contributions to $. With the convention of measuring energies relative to &F, (E& C Q and e,, E,, E& 3 O so that the energy Thus, neither e, nor e~ conservation condition On Eq. (5.701 is e , = leaf $. frb1-tmay be greater than E,, Hence, letting p(r) denote the density of states and defining the maximum values of p[e) and ((arBIv lab11 for O c,, a&, tegl < as , ,p and V=,, we obtain the bound
.
5 .v:,
,.,,,e:. No matter how strong the two-body fnteracthn, as tong as Its matrix elements and the density of states remain finite in the vicinity of the Fermi surface. is therefore bounded by a constant times E:. A completely analogous argument holds for a quasi-hole. So in general, as the energy c of a quasiparticcle or quasi-hole excitation approaches the Fermi Energy EI;., the lifetime increases as (5.72) T ( E ) O: IC- ~ ~ 1 - l ~
!
Thus, under very general conditions, a strongly-interacting many-fermion system will always have a domain suRciently close to the Fermi surface in which quasiparticler have arbitrarily long lifetimes and are the appropriate degrees of freedom to describe the system. Note that nowhere in this argument have we invoked momentum conservation. as in most common derivations, so this result is clearly applicable to finite systems and non-transitationally invariant systems.
5.3 PHYSICAL CONTENT O F THE SELF ENERGY
255
EFFECTIVE MASSES We now consider the effect of the energy and momentum dependence of the selfenergy on quasiparticfe propagation In an interacting Fermi system. Whereas the analysis pertains t o a variety of interesting physical systems, such as a atom in liquid 3We or a low energy nucleon propagating in a nucleus, we will illustrate the major points for the case of a nucleon in transtationatly invariant nuclear matter, From the dispersion retation defining the quasipafiicle pole
the 4ensi.t-y of states may be calculated as follows
It is o&en convenient t o subsume the complicated egect of the medium on a particular
process Into a suitably defined egective mass, The density of states may thus be expressed
-dr= - ik: rik m"
Notkg that m* ltwlf is the product offactors associated with the energy and momentum dependence of C ( k ,W ) it i s uwful t o define the additional efictive masses m, and rnk as follows (Jeukenne, Lejeune, and Mahaux 1976)
and t o study m, and mk separately. Note that the factor is just the residue of the quasiparticte pole discussed in the 1ast section. The mass reflects the spatial nonlocality of C, and may be understood qualitatively by considering the non-locality of the exchange term of the Hartree-Fock potential. As s h w n in Problem. 5.3, the general result for the exchange terms assuming a central potential vfr) and spin degeneracy ZS+l may be evaluated with plane wave states t o obtain
256
GREEN'S FUNCTIONS
Fig. 5.8 ERectItra; mass rnk In nuclear matter, Sketch (a) shows the two lactars contrfbutlng to the Fourfer transform cf C;xch(k) In Eq. (5.77). The
resulting slle~flvsmass
% = (1 + F g)-' is shown in (b).
Note that a central potential contributes t o the exchange term with the opposite sign If the potential is state-dependent. then different and a strength reduced by combhations of partiat waves contribute to the dtrect and exchange terms, In part'rcufar, if we consider nucleons with two internal degrees of freedom, spin and isospin. and assume even partial waves interact with a potenthalt uev,, and odd partial waves interact with the potential %dd. then, from Problem 5.3, the exchange term is
h.
Since the nucleon-nucleon interaction is strongly attractive in even padiaf waves and
weakly repulske in odd partial waves, the efictive potential
contributing to the exchange integral is strikingly diferent than in the state-independent case, Both contributions to vemh(r)are attractive, so that the net attraction from the exchange term Is larger than from the direct term, apd the quafitative behavior of ~ , ~ ( r and ) the Slater density are sketched in Fig. 5.8a. Since is given by the Fourier transform product of these two factors, its momentum ancf when k b.the dependence is obvious. At low 1,Elk) Is strongly attra is a positive decreasing charamristic scale in the in
zFh(k)
2
has the behavior sketched in Fig. 5.8b. = function of k, so that Note that at low momentum, the spatiaf nonlocality reduces m to roughly half ctf the bare mass m. The mass m, reflects the nonfocality of T: in time, Since the Hartree-Fock contrabution to E= is instantaneous, the leading contribution to m, arises from Ca. Eq.
5.3 PHYSICAL CONTENT
Fig. 5.9
ERecttve masses m, and
QF THE SELF ENERGY
257
m* In nuclear matter.
(5.60). The qualitative behavior of this contribution may be understood by the following xhematic argument (Bertsch and Kuo 1968). Represent the sum over all two-particle E, and onchole states by a single average state with exdtation energy E , eb - c~ t an egectlve caupting matrix e f e ~ e n V:
+
Similady, represent the sum over two-hole one-particle states by a single average state, and fuetter assume that near the Fermi sufiace particle and hole states are symmetric. Then, E A EB - C;, -Ea and
+
Thus, both terms in Xz become mare negattve as folfowing enhancement in me at the Fermi surFarze.
E
is Increased fram 0,yielding the
This schematic analysis is too crude to calcufate m, away from the Fermi surface, but detailed calculations fjeukenne, Lejeune, and Mahaux, 1976) yield the behwlor graphed in Fig, 5,9a. The enhancement is very Iarge a t the Fermi sudace, of the order 50%, and falls off significantly away from the Fermi surface, The combined effect of m, and rnk in the total eFective mass m* measured in the densi9 of shtes has the structure shown h Fig. 5,9b. Note that; b~ausa!mh is so small, on the average m" is significantly less than m, Mowever, near the brmi surFace, the peak in n k brings m' neady up to m. Although the product of m, and mk appears in the dendty of states, other observable~depend on m, and rnk separately, Consider, for example, the mean free path, which Is caleulabd by specifyfng a real energly E, and solving the dispersion relation (5.74) for a complex k, Denoting the real and imaginary parts of T: by U and W, Is is given by:
258
GREEN" SF-UNCTIONS
Since the imaginarly part W is small, it is suf"licient to expand to first order In W about the teroth-order solution h @wen by
+ ikr and expanding Eq. (5.81) to first order, we obtain
Witing k 5
with the result
Since the attenuation factor for a complex wave vector is t,bz em2hr, the mean free path is
k.
-
and is thus proportional to Simitarly, the lifetime of a quasipmicle excitation is obtained by specifyirrg a real k and scltving far the complex energy. The zeroth wder equation is
and writing
L
= CR - ?$we obtain
with the solution
5.3 PHYSICAL CONTENT
O F THE SELF ENERGY
259
Thus, the lifetime T = $ is proportional to me. The two results for X and I' in Egs. C5.85) and (5.88) are consistent since h and F' are related through the group velocity v:
tt
~ E ~ ~ k ~ k~ m= m dk m* m rnk m,
"
"
"
"
-
(5.89)
"
~
'
The eflective masses play a quantitatively significant rote in determining the mean free path of a nucleon in the nuclear medium and serve to resolve a long-standing dlscrclpancy with experhent, The mean free path in a nucteus of neutrons in the m e r w range 50 -. 150 MeV may be determined from the amplitude of shape resonances in total neutron sicattering cross sections, in this analog of the atomic Ramsauer efict, interference between the incident and transmitted wave can only be obwwed if the mean free path is long enoagh for a neutron to pass through the nuclear medium, and In this way one measures A Bfm for neutrons in this energy regime (Bohr and Mottelson, 1969). which ignores the Pauli The naive classical estimate of a mean free path h = principle Is far toa tow, with the average nucleon-nucleon cross section at 100 MeV F = 5.5fm2 and nuclear density p = 0,16fm-~yietding X = 1.1fm. The simplest approximation to W =. XmC is obtained from Eq, (5.70)by rrsplacing matrix elements of the potential (kB@lab) by the free! spa- Pmatrix (kBjTjab) measured experimentally and replacing the energies c. in the medium by the free space energies
2:
Using this estimate for W and ignoring the egective mass En Eq. (5.85) yields X =. 3fm. still far short of the experimental result of fifm. Thus, simply linduding the Pauti principle through the restriction ab 3 kp, B kp is insiilficient. b4owever. this formula omits two eRective mass factors. The 6-function in (5.90) with free propagators simply includes the free Fermi gas density of states. Since the 6function in I=, Ecl, [5.70),contain the energies in the medium, WT should bt! multiplied by to include the density of states in the medium. including the additional factor of mk from Eq. (5.85). the correct T-matrix expression for the mean free path is
5
-&
Thus, the medium dependence reflected in the two eRectiue mass factors, each of the is crucial to understanding the nucleon mean free path (Negele and order of Vazaki 1981, Fantoni, Friman and Pandharipande 1981).
OPTICAL POTENTIAL Another impofiant physical propevty of the self-energy is the fact that it specifies the optical potent'rat for the scatteving of a particle fmm a composite system made up of identical particles (Bell and Squires 1959).
Consider the elastic -* :3:t%brir.i; L an electron from an atom or a nucteon from a nucleus. Since the compc,are system is requ'lred to remain in its ground state, the asymptotic scattering state nay be described by a wave function #(r)depending only on the relative eoordhaate between the composite system and the scatter& particle. By definition, the optical potential is a one-particle potential producing phaw shifts identically equaI to those produced In +(r)for the full many-body problem. Note that since any phase-shift equiwlent potential is satisfactory, the optical potential is not unique. The essential point in relating the optical potential t o the seif-energy is the obsewation that the wave funetion can be written in terms of the one-particle Gren*s function. Since G(rtlrft)) is the amplitude for adding a particle to a system at r't' and detecting it at rt, it is cleat that we should be able to express the scattering wavefunction fn Lerms of G, Ta be precise, let I*) be the ground state of the N-body composite system and pick the zero of the energy scate such that Its energy i s zero. Then a scattering state may be generated by creating a particle at some point v' far away from the system at time tt and projecting onto a specific energy E by integrating over initial times (t'
- t) + B(t - t')c-'H(t-")+t(r')
Since the optical model wave function is the amplitude for obserrring one parSicle at r and all other particles in the ground state jk), we obtain
- t35;(rpg)$t(r'st') - B($' - t)jt(rt,tt)$(r, = c - i E t i ~ ( ~r, F')
,
in the second line, we have used the fact that the &"Itt- t) term in f5.92) does not 0)=. 0 and have inseed exponentiats next to l&) because contribute because (0 El[+O) = 0. The second time order in the third line required to obtain the Green"s function could be inserted becaure (Ol$t(rf, t') = ($(rl,t')10))~= O since r' is far from the target and there are thus no parlictes for to annihilate. For convenience, we will evrtluate the wave function at time 14 = O and drop the factor d m that #(r) = G(& r, rf). Consider, for reference, the problem in which the interacting Mamittonian H is replaced by & = T U, where U is a one-body potentlal (such as the Hartree-Fock potential) producing a first approximation to the localized N-body target. Then, by the previous argument, scattering from the onebody potential E f is described by the wave function +o(r) = Go(& r, r'). Substituting +(r) and #@(v)in the Dyson equation G == Go $- GoCG,we may write
+
+
dr"drttt
GO(E;rl r") C(& vttr i"')#(rot) .
(5.94a)
5.3 PHYSICAL CONTENT O F THE SELF ENERGY
261
Note from the speetrai representation for C&. Eq, (5.291, that for the positive- energies relevant to the scattering problem the denominator in the (ET i If particle term never wnishes so that Go(E) may be replaced by the retarded Green's function G ~ ( E ) having E a'q in botfi terms. Thus, we may rewdte Eq. (5,Qda) in the form
+
I#+)
where and 14 ); denote the scattering wave function for the interacting and noninteracting problems 4(r) and #@(r) and E+ z E+iq. Because 14 ): is the scattering wave Functfon for the potential U", It satisfies the Lippmann-Schwinger quatfon
where ido) denotes an incident plane wave. Substituting 14;) from Eq. (5.94b3 into (5.95) we obtain the desired Lippmann-Schwinger equation for l#+)
+
so that the optical potential is U C. W hen the self-energy is expanded in H- HQ= v(ri - r i ) (I,the first order term containing --U exactly cancels U and the first feu time-ordered c-ontributions to the optical potential are (af
(b)
? xij
fcf
(a1
-
gel
Diagram (a) describes propagatioa in the Wartree-Fock mean field and (b) represents the amplitude for coupling to a two-particle one-hole state and propagating in that state rather than in the single-particle state, Diagram (c) expresses the fact that in the Interacting system, two normally occupied states A and El may be virtualy excited to states e and k, thus blocking the addition of a particle in state k to the system, fn contrast to other approaches to multiple scattering theory in which antisymmetry is either neglected or put In laboriously by hand, the self-energy systematically inctudes its eRects through terms such as this, Diagrams (d) and (e) are representative of an infinite class of terms in which U and the HartreeFock potential enter with opposite signs. As usual, it Is advantageous to cancel such krms identically by choosing U to be the Hartree-Fock potential, Because one-parllcle irreducibitity is defined in terms of Feynman diagrams rather than theordered diagrams, both of: the fairtowing time-ordered diagrams are excluded from the optical potential (A)
(8)
Whereas it is quite plausible that fA) represents a time history which wilt be generat& when dlagram (b) of (5.97) h iterated i n the Lippmann-Schwinger quation, one might niriwty be tempt& to regard (B) as a valid four-particle three-hde contribution t o the optteal potenttaf. Clearly, the derivation shows it must not be included, which is associated with the fact that the tippman Schwinger equation has no projector onto states above the Eltsrmii sudace and thus generates propagation In hole states as welt as partick states.
THE RESPONSE FUNCTION Section 2.1 show& how experimental obsewables could be expressed in terms of msponse functions, which we wit! now write Fn terms of t-vvo-patsicfe Green" functions. T0 obtain the product of two one-body operators, we must consider a twa-particle Green" function in which the creation operators art! ewluated infinitesimally later than Eke annihitation operators and we define the density-density ccouelatlon function and its retarded counterpafl as
As in Sectbn 5.2, a spectral reprewntation is obtained by inserting a complete set of states and Fcourier transforming. Far brevity, we will only write the zero temperature results, and leave the analogous finite temperature expressions as a straightforward aercise:
where the density operator is wiltten +t(zl)$(zl) using
we obtain
zi a ( z l ) .
In momentum space.
5.4 LINEAR RESPONSE
263
.
The essential difference relative to Eg (5.27) is the presence of N-particle intermediate states rather than states with 1V =t: 1. part"ldes, W that the density-density response fwnction contains information about the eigenstatcts of the N-particle qstem itself. horn Eq. @*%W).it is evident how to calculate the observables disclrssed in Section 2.1. First B is calculated in an apprapvjate approximation using the pefiurbation theory we have derived for the Green's function G%.A response function. Eq. (2.16). is then evaluated by calculating BE from the relations
Analogous expressions can. of course. be wiltten with an arbitrary one-body operator replacing the density operator. Similarly, the inclusive scattering cross section. Eg. (2.19) can be evaluated from the imaginary part of fT, since only the first term in Eg. (S.dQa) or (5,102) contributes for positive W :
Because of its direct re! ation ta experiment, the imadnary part of the response function is often referred to as the dynamic structure factor:
Finally, it is convenient and conventional to define the density Wuctuathon operatar
and k s c m r a t i o n functions
Mote that DR = IDA! since (3) does not contvjbute t o the commutator and that @(p,W) = -2Im D(k,W ) C ' ( ~ ) because (p) does not connect t o any excited states
in (5.104). The fluctuation-dissipation theorem is particularly clear in this language. since transport coe-fFicientsare spedfied by the expectat'ron value of products of ftuctuation operators. The advantage of Eq. (5.106a)is the fact that its diagram expansion contains afl liked diaqams which connect the density operators at polnts 1 and 2, Taken by itself. 2) contains linked diagrams of two generic types. those which connect the operators at 1and 2 and those which do not:
a(l,
264
GREEN" FFUNICTlONS
Feynman dlagrams for the response function D. Actlon of the denslfy opentors p(1) and p(2) are denoted by and whkh In momentum space may be regarded as tnjectlnp momentum at (1) and removing It at (2). Note that In addlrlon to the dlagrams shown, the cclntrtbotlons through sscond order also lnciude Haarm-Fock self-energy tnsenlons on the propagators In diagrams (a) and (b),
Rg, 5.10
/
a
/
Thie disconnected pieces in Eq. (5,107) simply correspond to all diagrams in the product (@(1))@[2)) which are precisely subtracted off in D(f, 2). Feynman diagrams mrresponding to the first few orders of perturbakbn theory for D[l, 2) are s h w n in Fig. 5.10, The dlagram rules for D follow Zmmecfiately from the definition Eq. lfi.991 and the zeretemperaturt? Feynman rules of Section 4.1. Recall that i2& has propagators iCo for each line and the overall factor ( - i ) ' ( - ~ (-1)"s )~ where r is the number of interactions, nh is the number of internal closed loops and (-l)P is the sign of the permutation of the external legs, For the identity permutation in which 1 connects to lit and 2 connmts to 2@the diagram for D has two additional closed toops retatkve to G%. For at1 other permutations, connecting 1 to 2@ and 2 ta 2' creates one additional closed loop in D relative to G%,Thus, associating a minus sign with every closed loop in D correctly accounts for both the factors (--x)~ and (-1)"'. To include direct and =change Interactions on an equal footing as in Fig. fi.jtO, we wHll use Hugenholtz diagrams and include the factor for equivalent liner. Including the explidt iin the definition (5.99). the overall factor for a diagram contributing to B is (-i)'+'(-l)":& where r is the number of interactions, nk is the total number of dosed toops in a diagram when it is drawn with direct matrix elements, and a, is the number of qulvaIent lines. The propagators in the diagram are 4Go and the interactions are antiqmmetrized matrix elements (kllczfv[bk r ) . Additional insight into the physical content of the response function is obtained by wnsidering the set of time-ordered diagrams corresponding to each kynman diagram. k r convenience, we will treat a transfadonally-invariant system and enumerate a complete set of multi-particle, multi-hole states.. Let us write the amplitude for prczducing each Intermediate state by Injecting momentum q into the ground state as a sum of ) ~ ~ by all possible time-ordered diagrams, so that the matrix element I ( $ ~ I $ I J ~is~ given pairs of one diagram from the set connected with the adjoint of another diagram from the same set, For example, the top tine of Fig. 5.11 shows four tlmemhistodes leading
&
5.4 LINEAR RESPONSE
265
Fig. 5.11 Tine-orderad diagrams for amptltudes contributing to the rssponse function. to a one-particle one-hole state. If diagram (A) is connected to (A*). which is the same diagram drawn upside down. we obtain one time order of diagram (a) of Fig. 5.10 in which a one-particle one-hole excitation is first created and then destrayd by the action of g. In the language of scattering theory, diagrams (B) and (C)would ba called finat-state and initia! state interaction corrections, respectively, Note how the cross terms between {A, B. C) and { A* ~t C*) produce various time orders of the "chain" diagrams (a), (b), and (c), so that initial and finat state interaction effects become totally entangled In the response function, Simitariy, for the two-parEicle twohole states in the second line. combining (E) with (Ft) yields particular time-orders (d) and (g), but diagram (g) is also obtained by combining (D) and (At). Thus, it is clear that the Feynman diagram expansion of the response function provides an economical description of a compticatd process. The interplay of initial and finat state interactions O s treat& consistently and the conbinatoric of obtaining the same contribution from products of distinct amplitudes Is automaticatly summarized by the rule that all symmetry factors are one for any Green's function. The ffexibility ta treat the response function in terms of time-ordered diagrams as well as Feynman dkagrams may be exploited in specific appticatians, such as studying the scaling behavior of the response function at high momentum transfer (see Problem 4)-
RANDOM PHASE APPROXIMATION Any practical calculation of the response Function must truncate the infinite dfagram expansion. One alternative to simply stopping at some sptrcifrc order of perturbation theory is to sum an appropriate infinite series of diagrams. Summing the set of all chain diagrams, the first thrm dements of which are shown in [a) (c) of Fig, 5.10 yield the so-called random phase approximation or RPA, (Actually, there are two distinct approximations whkh are cornrnonly referred to as the RPA: the present treatment of the response funcfcion and the summation of the ring diagram contributions, Eqs. (2.113) (2,115) to the ground state correlation energy. The RPA correlation energy is treated in Problems 5,s and 5.7,) As a prelude to summing chains. Id us first calculate a dngle link, diagram fa) of Fig. 50, which corresponds to the response functhon Do for a non-interacting system. For simptlcirtj(, we will treat a translationally invariant system in momentum space and assume spin-112 Fermions, Noting that diagram [a) has a closed loop, no interactions,
-
-
GREEN% FUNCTIONS
2615
no quivalent lines and two propagators iGo, we obtain the contdbution
Performing the momentum integrals in Eq, (5.108) with the zero-temperature occupation numbers nk =. @(hp - Jkr)as outlined Fn ProMem 5.6 yields: the Lindhard function (iindhard, 19515)
Im&fqto)
(b. 10%)
=.
where
The imaginary part, which specifies the inclusiw cross wctlon, Eq, (5.104],for a naninteracting Fermi gas has a very simple physical interpretation: it ilmply-counts the ways an occupied state [kl kF can be scattered to an unoccupied state ik+(jl > k~ For p > 2 k ~ . by transferring momentum 3 and energy w = eh+, - = @ the k r m i sphere conditions are satisfied for all Fe hp, sa as sketched in Fig, 5.12a. are obtained for 1;i.l = kF and g aligned with or the extremal values w =
+ g.
&
&
opposite t o 8. The maximum occurs for g perpendicular to corresponding to w = , < kp can contvibute in this case. h r q 2 k ~ the since all transverse values of result is sketched in Fig. 5.12b. Whereas at large W, one has the same parabola as in
5,4 LlNEAR RESPONSE
Fig. 5.12 and q
fmaglnaxy part of the response functlon & ( k , o ) for q
< 2kF
> 2kp
267
(a)
(h).
&
1x1
p+
- the simultaneous requirements 4 k~ and 81 2 k~ care (a), at w = further restrict the phase space, yieidfng the linear betravlor shown far lower M. MIF! may now sum the sequence of chain diagrams contributing t o the response function by writing an integml equation which iterates the addition of a single link. Let ~ ~ ~ p,( kElk2 k l q, k2;W) denote the particle-hole Green's function obtained by separating the pairs of creatbon and annihilatbon operators which correspond t o the externai density operator In I)(q, W ) :
+
+
with the propeirty
Were and throughout this section where no ambiguity wilil arise, we abbreviate momentum integrals and spin sums by C, = $ and ~eglectthe normalization volume 21 which cancels out of all obsewabtes, Chain diagrams are summed by the following integral equation which yields the particle-hole Green's functions in the randorn-phase approximation
268
GREEN" FUNCTIONS
The first term on the right-hand side is the paelcle-bole Grwnk function far the noninteracting system, and by definition, it is identical to Do in Eq. (5.108)except for the absence of the momentum sum The direct and exehange terms arc?written wpficitfy sa that one may absewe that the direct term has the foftowing factars tn : (--i)for the interaction. (-1) for the addition of a addition t o the potential and clomd loop and iGo for each of twa propagators. Since these factors are Just those in Do,the equation may be written
Eke
QRPA
("E + ~ r q , k r f S +
+ e k , - ek,+# l
k3;~)
o
,
,
f
- 0- nkthx+g
%.v
w $. ekl, - ~
-
l r ~ iq + ~
(5.11%) This equation clearly has the general structure of the integral equation discussed in Section 2.4 and in fact tzorresponds to the Bethe Salpeter equation in the parlicie-hole channel with the ve&ex functbn approximated by the bare interaction. We will return to the more general case in Chapter 6 in cconnection with Landau Fermi tilquid theory. The direct matrlx element in Eq. (5,192b) only depends on the momentum transfer Q. ( k , k l q l u f k f q, kl) == ET(-q) whereas the exchange mwtrix element iifkt3 - kr) depends on the internal momenta. Hence, when it i s physically just'tfied to negle-et the exchange term, the RPA equatkon bclcomes a simple algebraic esquation. There are several circumstances under which the exchange term maJl be neglected. First, consider the Coulomb potential, for which the dSrect and exchange terms are
+
respectively. For small g, the conditions that -t- ql > kp and lic;l K kF so the direct term generally dominates the exchange or vke versa force fklf 1
.
term by the factor A second exampie is use of the high-spin twhntque to treat Bosons, in which case the direct tern dominates the exchange term by the factor N. A specific application to the correlation energy of the dilute Sose gas is gjven in Problem 5.7, Finatly, in any system such that the momentum dependence of the potential is negligible between 0 and 2kp, the exchange term may be effectively included with the direct term. An example is the nsponse function for the one-dimensional S-function interactbn fn Problem 5.8. The sum of all direct RPA chain diagrams is abtained by multiplying Eq, (5,112b) by r7(q), summing over k, and dropping a l exchange terms, with the result
where we have denoted the partkk-hole G r e n 'S functions for the non-interacting system, which 1s the first term in Eq. (5,112b). by
5.4 LINEAR RESPONSE
269
Fig. 5.13 Graphical solutlon for the RPA modes. The vertlcal asvtwtotes correspond to the particle-hole excltatlons W = ek+, E, and the curves show the qualltatlve behavlor of $(g)Do(q,w) for the repulsive and attractlve potenttals, respectively.
-
Slnce the poles of Gpb (or D which is obtained by summing over kl and k2) give the excited states of the system, in the random-phase approximation the excited states occur at W such that (5.115) ~(rr)Do(q,w) = 1 To understand the structure of Eq. (5.115). it is useful t o consider the graphical solutions sketched in Fig. 5.13. Since &(~,w) is symmetric in w (see Problem 5.6). it suffices to consider positive w. For every value of g inside the Fermi sea such that %+G is outside the Fermi sea, the first term in Eq. (5.114a) has a pole at W = ek+, - e k . For each E in the sum contributing to Do(q,w), asymptotes are plotted in Fig. 5.13. and Do(q,w) is a decreasing function between each pair of asymptotes. Thus, for a repulsive potential the solution to (5.115) is as sketched in the left part of Fig. 5.13. A11 except one of the solutions are trapped between states in the particle-hole continuum. and the remaining colfective state is pushed above the highest state. Similarly, for an attractive potential, the colfective state i s brought down below the lowest particle-hole state. In order t o study the collective RPA modes in several physical systems at low g, it is useful to note the following behavior of Do(q,w) from Eq. (5.109)
&kg
fixed *(-l+glnlzl) s=~fixed W
w fixed sfrxed>X s fixed < 1
(5.1 l6a)
.
As the first example, consider the electron gas. Because of the
$ behavior of the
Fig. 5.14
Comparison
of plasma ascltlatlan and zero sound mode wlth the
parltlcla-hole continuum.
Coulomb force, we must take the limit in Eq. (5,116a) with finite o,so that
yielding the familiar plasma frequency (see Prablem 5.9)
The collective RPA made, or plasman, is a fundamental excitation of the electron gas which is readily obsemable experimentally, For example, if one measures the energy loss of etectrons in solids, peaks are obwwed at multiples of W,. Since typical Fermi energies in metals are of the order er 5 - 20eV, the plasma frequency is in the 7 -- 27eV. The spstrum of particle-hole exckations within the extremal range U, -+ is compared with the plasma frequency in Fig. 5.14. Note that values w = since ~ 3 pis comparable t o c p . the ptasma mode is welt-isolated from the padcfe-hole continuum and thus daesnt decay directly through onapa~icteone-hole states. The leading correction to W, which we have not calculated here. is proportional t o p2 In contrast to the case of the repulsive Coufamb potential, many physical systems have scattering channels in which the interaction is attractive at the Fermi surlace, Tn which case the collective mode is il~weredin energy, Consider, far example the case of liquid % e,! On the average, the interaction i s repulsive at the Fermi surface since It is the short-range repulsive care of the two-body potential whkh keeps the ilquid from collapstng t o higher density. However, although there is neglkgible explkcit spin dependence in the underlying potential, because two "~e atoms in a spin triplet state must be spatially antisymmetrk, they feel the hard core much less strongly than two atoms in it singlet state, and the interactbon in the triplet channel is averall attractive. Thus, the spin-dependence tends t o drive the system toward a ferrornagnetic state and manifests itself in a low-energy collective RPA mode known as a paramagnon. In the limit Sn which the attraction between 13ke spins were strong enough to drive the system ferrornagnetic, this mode would occur at zera energy. Although it is not pushed down that far, it makes a significant contribution to the self-energy, In addition t o the
-&
5.4 LINEAR RESPONSE
271
Martree-Fock and second order diagrams discussed In SectEon 5.3, RPA graphs such
make a dramatic contribution t o the effective mass. In the pressure range of 0.3 to 27 atm.. rang& from 3.1 t o 5.8, corresponding to the fact that in order to move a single $He atom through the medium, one must also drag along a large cluster of atoms wSth the same spin. A second example is the case of pion condensation in neutron or nuclear matter at high density (occurring for example in neutron stars). Whereas the 9He-9He interaction is attractive in the a - a channel, by vSrtue of the spin-lsospin coupling of plons t o nucleons and the resuttf ng strong tensor forces, the nucleon-nucleon interaction at the Fermi surface is stmngiy attracJive in the o .cr 7 7 channel, giving rise to low energy collective excitations with the quantum numbers of the pion, Although uncttrta'inties in the behavior of strong interactions at high densi& makes quantitative predictions unceutain, this mode is predicted t o have zero energy and thus t o lead to pion condensation at a density several times the: density of nuclear matter. We now consider finite-range repulsive potentials, for which the low-q behavior of the coltective mode Fs compfeteiy different than for the Coulomb potential, Since non-Coulombic potentials are less singular than $ at low q, the dispersion relation GB = 1 cannot be satisfied far a finite w , so u + Q as q -+ 0, Thus, modes which go t o zero at large wavelength are the generat rule, and the finite energy modes such as the plasmon are excepdonal eases associated with the infinite range of the force. Note that we Rave already seen an example af the role of the force range in cansidering spin waves. Since orienting dtflemnt spin domains in different directions costs no energy except at the domain boundaries, if the force range is finite, the egect of the boundaries bwornes negligible in the tong wavelength Ihmit and the energy of such a Goldstone mode must go t o zero. Only if the: force range is infinite do the boundary contributions remain finite, yielding a finite excitation energy. To simplify the discussion of the collective mode, which is called zero sound, let us consider the case in which G(g) is independent af q at small g and define
We now take the small q limit of the dispersion relation holding s fixed and greater than 1 to obtain real fundampedf solutions
272
GREEN'S FUNCTIONS
where s defined in Eq. (5.116) is the ratio of the speed of the zero sound mode, c0 = t;o the Fermi velocity, uf-,
t.
Note that soiurions with s > 1d s t for all positive J, that Is, far all repulsive Znteractions. For large f . (where . the. simple RPA presented here is an inadquite approximation) we may expand In
so that
In the limit of smatt f , a approaches 1exponen-kiatty
and the velocity of zero sound is therefore close to, but slightly above, the FermS velocity
k~
(5.123)
cam-. m
1. The zero sound mode is sketched in part (b) of Fig, 5.24. Note that the fact a places the mode above the particle-hate continuum and prevents its direct decay through coup!ing to one-particle, one-hafa states, b r comparison, compare this velocity with ordkary thermodynamic sound in a nearly non-Interacting krmf gas in cl dimendons in which. there: are sufFicient interactions to produce tharmodynamic equitibrium, but these interactions contribute negligiMy to the energy per pa&icie, Noting that the Fermi gas energy per partkcle in d dimensions is and p kg. the usual thermodynamic arguments yield the velocfty of thermodynamic, or first sound*
&&
Thus. in a weakly interacting system in three dimensions. e l is approximately %c,. Fuuthermore, at least the velodty suggests that the geometry of zero sound may bit more one-dimensional than three dimensional, and we will now develop an appropriate language to warnine the geometry of zero sound.
*
+
For an infinitesimal change in density, the continuity equation requires &6p poV v = O and Newton's equation of motion requires m p o g = -VP. Hence, the wave equation is &&p = & o ~ P + ( 6~ ~~= ) g ~ and c2~ = L m a. 6&P ~
-
&
5.4 LINEAR RESPOMSE
273
Since zero sound OS a coherent superposition of particle-hole excltations near the Ferm'l sudace, we would like t o slmultanftously examine the behavlor in momentum space and coordinate space, If we were dealing with a classicat system, it would be natural t a study Its cfassieali dfstdbution function as a function of coordinates and momenta. For a quantal system, the best analog of the cfassicai distribution functlon is the Wlgner function and for our present purposes it is mast useful to study the Wbgner transform of the one-body density matrix:
In addltion to yielding the? famfflar distribution functOon In the cfassical limit, its moments have the fotlowing properties associated with the classical Itistr'lbution function:
and rimilady for higher moments. Essentially the only property f (P, R) doer not share with the classical dfstfibution function is posltivity, and this deficiency wilt not a E ~ t the present argument, Mow, let @us use the RPA response lfunction t o look at the respons-tl of the W'lgner transform of the one-body density operator to a weak perturbation SU(qf p, (ta)which couples t o the zero sound mode. Using Eq. (2.15) and omitting the standard details in gafng fram the retarded t o time-ordered response function, which are irrelevant to the present point,
NoEing that the zero sound pole arises from the second term in Eq. (5.113) and that the only dependence on kl enters through ~ i ' ( k p, ~ W), , substitution of the RPA response
274
GREEN" FUNCTIONS
Fig. 5.15
Phase space
.
contrlbutlnp to
$
T h e csnt;ers of the
two solid rphens are dtrplayed f relaflve to the dashed Ferrnl sphere. Every
vwtar and
in the upper shadad reglon yfelds
a - g Indde.
+
outside the dashed sphere
function in Eg, 15.127) yields
5
For small #, the range of $which satisfy the conditions that $f be outside the Fermi sphere while p F 8 be inside can be calculated easily from the geometrical construction shown in Fig. 5.15. The shaded region defining the vectors j3 such that j? f lies lies inside have radial thickness qcose outside the dashed Fermi sphere while where B is the angte between 3 and ij, Since the p's am all confined to be close to - fp"jfqcoo38 . Finally. the Fermi surface, we may write the phase space fac wilting the denominator as W ep- f - E@+ + = m m and writing the explicit tf me-dependence, we obtain the desired result.
+
This result is also obtahed using Landau Fermi Liquid Theory in Chapter 6. The factor -specifies the change in the distribution function as a function of the anf~lebetween F and Since s is close to 1, the magnitude of the peak at 19 = O
a.
5.4 LfftlEAR RESPONSE
Eera sound
275
Tkemac8yaannic mund
Fig, 5.16 The dlstrlbutton funettan f (p, a) for zero sound and therrmodjrnamk sound. The dlqulllbrlum Ferrnl sphere Is shown at each spatial po~ltlon by the dashed Hne and the sol44 llns denotes the momentum dlstflbutlon of the propaaatlng mode.
is much larger than at B = R, s-o the volume of the Fermi sphere and thus the local density changes. Figure 5.16 compares the evolution of the distribution function for ZWQ sound with that of thermodynamic sound. The factor ei(@*R-we)propagates the fluctuation of the zero sound distribution function in the f direction at speed v0 = 8%. Snapshots are shown of the momentum distributian at spatial points R separated by a half wavelength. The origin Es select4 such that the density is a maximum at R == O, so the Fermi sphere distorts at successive displacements of by adding or subtracting % *:.,: The distribution is thus highly anisotropic and is concentrated in the forward direction. In contrast, the Fermi sphere for thermodynamic sound just dilates and contracts spherically s;ymmetricctIly, To the extend to which the zero sound distofi'ion is concentrated in the longitudinal dkection and the transverse radius of the fermi surface fs unaEectel2, the mode is more like one-dimensional thermodynamk sound than three-dimensional thermodynamic sound, consistent with Eq, (5.124). Zero sound and thermody namie sound thus represeot two opposite extremes, Zero sound is built out of quasirparticles, sa at finite temperatures the frequency must be high enough that the quasipartlctes do not decay, that is, that collisions do not dominate, Quasiparticles travelling in the direction are thus not scattered in the transverse directions and the mode can remain essentially one-dimensional. At zero temperature, for which we have performed our calcu!ation, the lifgtlme becannes infinite as quasipaeicles approach the %rmi surface, so zero sound propagates at aff frequencies. In contrast, thermodynamic sound requires local quilibrium. Hence, the frequency must
276
GREEN" SFLIPICTIQFIS
be low enough that collisions dominate and quasiparticfes and collective modes decay. Longltudlnait momentum is equilibrated wkh transverse momentum, so that the Fermi sudace is necessarity isotropic and the mode is fully three-dimensionaI, In liquid w e , one may experimentatly observe the transition from zero sound to thermodynamic sound by varying the frequency and the temperature between collision-free and collisian-domrnatd regimes,
MATRIX FORM QC: RPA Additional physical insight into the RPA may be obtained by deriving it from the tiime-dependent Hartree-bck approximation, dls~ussedin Problem (5,10), To make contact wOth this and ather approaches, it is useful to write the RPA equations in their conventionai matrk form, To display their fult generality, instead of working in momentum space, we wilt work in a genera! basis with our usual convention of unococcupied states {A, B and arbitrary states {or, B cupied states {a,6 . Rewriting Eq. (5.112) so as to distinguish the two possible cases GRP*(aAlaB) and GRP*(~ala@)we obtain the two equations e),
S).
v ) .
Wkh the definitions
these equations may be written in the matrix form
The poles of G are given by the eigenvalues of the generalized eigenvalue problem (note the minus sign an the right-hand side):
where X(Y)is a c ~ l u m nof elements X:;) labeled by particle-hole labels { b , B). Using the properties established in Problem (5.11). the particle-hole Green's function may be
5-4 LtNEAR RESPONSE
277
written as a sum over all positive energy modes
Since this is a spectral representation for the RPA response function of the same form as Eq. (5.200) for the exact response function. the X ( Y )and Y(Y) should corresp~nd t o matrix elements of between RPA approximates to the excited states, denoted ($FA). and the RPA ground state. denoted Mating that the upper component ( b ~ l a p involves ) the matrix element (a; abai a#) and GRP*(BB conesponds t o (@:aB a! a@).it follows that
(GPA).
If I$pA) were just the non-interacting ground state, akab would annihilate it yielding Y(")= 0. However. from Problems (5.5) and (5.10)it is clear that the RPA ground state contains mult~particle-holeadmixtures whkh praduce a non-vanishing Y ( Y )amplitude. Equations (5.134- 5.135) thus relate the two distinct random phase approximations alluded to eavlier for the response function and for indiviidual eigenstates.
S U M RULES AND EXAMPLES We conclude this swtion by stating several sum rules satisfied by the response function and presenting iflustrative physical examples, in aaition to dispersion refations of the form of (5.35) which follow directly from the spectral representation of D,the imaginary part of the response function sadsfies two important sum rules. The first fofiows From Integrating out the enerw conserving S-function and ushg completeness and the field operator commutation retatisn
-
($@I(P(zl) (P(z~)))Iuln) (+ml(B(f~2)
(P(~IJP(zZ)) == -44.1,
- (B(~z))l#a)O)
- (~(5~1)) (~(22))
+ S(z1 - 22)P(Z1)]
~ 2 )
(5.136) where g(;el, xz) is the Webody correlation function $0)-
In momentum space, where g(sl,za) = g(zl
- z2)
($0
IP^(z~)lJlo) (JloIB(zz)l+o)o)
(5.137)
Fig. 5.17
sketch of cross ssttan for knalastlc neutron scattering from llqutd based on the data of Stlrtlng e t cuf (1976). Solld contours denote the expertmental cross sections, the long clashed curve tndtcates the Farm1 pas maxlmum, &,;he short dashed curves show where the Fsrml Qas response
%k
goes t o zero. blocking seglns,
and the dotted llne denotes the point at whkh Paul1
Since Im D(Q,w ) is directly measurable in inclusive scattering. Eq.. (5.138) shows that one may measure the Fourier transform of the two-body correlation function by simply integrating over w e An energy weighted sum rule is derived in Problem 5.13 by evaluating the expectation value of the double commutator ( [ [ H , ~ , ~ j -with , ~ ) the result
A vailety of useful related sum rules may be obtained by choosing other operators in the double commutator instead of For example. one may take the electric dipole traniltion operator and relate the integral of the dipole transition strength for a finite nucleus to the charge radius (plus corrections for velocity dependent and isospin dependent terms). This gives a total sum with which to compare the strength of an individual transition. The giant dipole resonance is "giant" because its strength exhausts moz;t; of the sum rule.
F,.
5.4 LINEAR RESPONSE
279
Fig. 5.18 Ltquld structure functton $(g) and we-body corwlatlon funcOon g(r) for tlquld 4 ~ e .The left-hand graph compares X-ray scattering data of Wallack 11972) and Rubkoff et at, (1979) (crosses) and neutron scattrarlng data of Svensson et d. (3980) (clrclas) w1th the structurs functlon obtalnad from Monte Carto cakutatfons (solld curves) af Katos et d. (3981). The rlght-hand graph shows the two-body corrslatton fu'unctlsn corresponding t o ttrls S(q). Experimental measurements of response functions for several physical systems are presented in Figs. 5-17 5.19, Fig. 5.17 shows a contour plot in the (g, w ) plane of the cross section for inelastic scattering of neutrons from liquid $ ~ ate 0.63'K (Stkfing e t al.. 1976). Since the neutron-$He potential is eEectively a Cfunction on the scale of Angstroms relevant to the experiment, the crass section is directly proportional to Im D(p,w). Using the value of rCF = 0.186 -l specified by the ground state density. and zeros k expected from Do(q,W ) are indicated by dashed the maximum lines, Although the data has the qualitative structure of Do,it differs quantitatively in d&aif. In particular, one sees clear evidence of a large enhancement In the_ eRectlve mass. The experimental peak in the region of g 2kF occurs at -. x sso that m* -. Sm, The structure Factor for liquid 4He, S(q)defined in Eq. (5.138). has been measured both by X-ray scattering [Hatlock, 1972; RobkoE e t al. 1979) and neutron scattering [Svensson e t &I, 1980)Typical data are shown in Fig. 5.18, and are observed to agree quite well with each other and with the results obtained from Monte Carfo sotution of the zero-temperature ground state of 'He (Kalos e t al. 1981) using the Aziz (1979) potential, As discussed fn Chapter 8, the Monte Cario solution provides an esscnt!aIty exact solution for the ground state of N particles in a box with periodic boundary condkions, so the only error in the structure function is the departure from linearity a t vew tow q associated with the finite box size, One thereby has an exceedingly pre-
-
&
&
280
GREEN" FUNCTIONS
Fig, 5.19 lnclustva electron scattering from atomic nuetet. The lnclusrve cross sections; as a funcflon of energy goss W for ''C at lncldsnt energy = 198.6MaV and scattarlng angle @ = 135' In (a) correspond to momentum transfer q tass than 2kE. whereas the E == 500 MraV data a t f9 == 60' shown Cn (b) and " ' ~ b In (c) eorraspood to momentum transfer greater for than 2kF, Ths solid curves correspond ta the Fermi gas response functions described In tha text, Tha law energy data Is frflm Lelss and Taytor (1953) and ths hlgh energy data from Monlzr e l al. (1971).
cise measurement of the two-body correlation function of a dense, strongly Interacting quantum liquid, also shown in Fig, 5.18. One observes in particular the strong suppnssion of the wave function at short distance due to the strangly repulsive core of the H e H e potential. Note that the strength of the interaction betwrten the constituents in no way aficts the applicability of the linear response analysis: all that matters is that the coupling of the external probe, in this case the X-ray or neutrons, i s weak. A final example is inelastic electron scattering from atomic nuclei, shown in Fig. 5.19. A good approximation to the response may be obtained u s i ~ gthe response function for a non-interacting k r m i gas corrected for the medium dependence of the nucleon seff-energy. Recall from Fig, 5-12 that when the momentum transfer g > 2 k ~ , there is no Pauli blacking and Im&(q,w) is an inverted parabola with center at w = and width In contrast, when q 2kF, Pauti blacking yields linear
&
.
dependence below w = $f
- &. This qualitative digerence is observed in the low and
5.5 MAGNETIC SUSCEPTIBILITY 8 F A FERMI GAS
285
high momentum transfer data for lSC in (a) and (b) and the high momentum transfer data for "O'Pb in (c). Note that the excess cross section at high w corresponds to the inetastic processes in which: pions or A resclnances are produced. In Section 5.3, we saw that the nuclear self-energy was strongly momentum dependent, with a nuctean within the Fermi sea having a strong attraction and nucleons welt above the Fermi sea having much weaker attraction. This effect may be included apprwimatefy in the response function either by introducing an eRective mass m* or a binding correction (T representing the average diEerence between the self-energy of a nucleon in and outside the Fermi sea, in either case, the data spwify two parameters, kii. and either m* or . The l3C data of Leiss and Taylor (!g601 in (a) are fit with kF = 1.19fm-' and m*/m = 0.7 (Moniz 1969). consistent with the value m*/m discussed in Section 5.3 and the fact that the average Fermi momentum in s small, surFace dominated is somewhat lower than the value kF ==. 1.31 On fm"' Sn bulk nuetear nucleus like matter. The high momentum transfer data for IaC in (b) is fit with kF = 1.14fm-' and F = 25MeV (IVloniz e t al., 1971) consistent with the previous value of kp and the fact that the average of the observed single particle energies is of the order of 25 MeV. Finally, the same analysis of the 20BPbdata yields kF =: 1,36fm"l, i n satisfactory agreement with bulk matter, an8 = 84 MeV consistent with the average single-particle e n e w in 208PbeThus, the Fermi gas response cor~ttctedfor the known behaviov of the self-energy gives a simple semi-quantitative physical description of the nuclear response function.
5.5 MAGNETIC SUSCEPTIBILITY OF A FERMI GAS The magnetic srascept"tbitity of a non-interacting Fermi gas is an instructive, itlustrathe application of many of the ideas pl-esented in this chapter, Although the problem is i n some aspects trivial, b ~ a u s eit is an exactly solvabte problem with onebody interactions, it clearly demonstrates the dmilarities and differences between zero and finite temperat6res and between static and dynamic response. In addition, it will demonstrate some of the techniques involved in practicaf calcutations. The specific model to be solved is a system of non-interacting spin-1/2 Fermions coupled to a constant external magnetic field in the P direction
where a,p denote spin indices. p~ is the magnetic moment and is a Pauli spin matr"rx. VVE: will now calculate the susceptibility four diferent ways and compare the results,
STATIC SUSCEPT1SILITY AT ZERO TEMPERATURE
The magnetk susceptiMlity is
GREEN'S FUNCTIONS
282
where. using Eq. (5.12). the magnetization may be written in terms of the one-particle Green's function
(5.142) ln a one-body external potential, the Dyson equation is
where we have explicitly written spin indices. Because V 7s constant in space and diagonal in spin, G is diagonaf in spin and momentum. Using the familiar result G;' = G-I V, Eq. (2.178). we obtain for a = A 1
+
Hence
since the contribution of the two poles in the upper half plane cancel, and the formalism = 0. has handed us the unphysical result that X =
STATIC SUSCEPTIBILITY AT FINITE TEMPERATURE Let us see if the finite temperature theory is any different. Replacing Go by and the factor i by (-1) in (M) (see Table 5.1) we obtain 1
iw,
- (rk - p - poH)
-
l
iw,,
$0
1
- ( e k - p + mN)
(5.146) The standard techniques for performing frequency sums such as that in Eq. (5.146) is to perform a contour integral with an integrand deliberately constructed to have poles at W, and residues equal t o the summand. In the present case, we will calculate etvs rl
E W ~ by evaluating
5.5 MAGNETIC SCfSCEPTIB1LI"T"YOF A FERMI GAS
283
Fig, 5.20 tntegratlon contour and poles af Eq. (5,147) for evaluating a frequency sum. along the contour at infinity shown in Fig. 5.20. The integrand has poles at w = z and at the Matsubara frequencies iw, = $ (2% l)k.As w -+ W. the integrand decreases
+
,U(*-@)
so for a positive infinitesimal Q. as 7 and as w -, -m it decreases as the contdbution along the contour C vanishes.. Equating the sum of residues to zwa, we obtain the desfrd resuft:
where, following the conventian EQ. (2,755). we have included the chemical potential p in the Fermi occupation grobaMtity n(ef, Using this result to calcutzrte the nagneti-
zatlon and taking the weak field tlrnit
The zero temperature limlt is evaluated using
%@(c,
- s) = -&(E - v) and
changing the integration vanable to energy k = (2msr)'/' with the result
where p is the densky. On the basis of the general digussion in S ~ t l s n3.3, it is simple t o understand why the finite-temperature theory gives the correct result for the Paufi paramagnetism of a Fermi gas while the zero temperature calculatian produced X = O. The zero temperature theory finds the- lowest dgenstrate with the same; symmetry as the noninteracting system. Since the non-interacting system brad equal Fermi spheres for spin
284
GREENS SUNCTIQNS
up and dawn, the perturbation H could not arter this symmetvy and thus produced no magnetization. In contrast, the trace in the finite temperature thwuy has access to ail symmetries and thus produces the asymmetric population of the Fermi spheres reflected in Eq. (5,ld9). In the high-temperature limit. Curie's law is recovered by noting that n(e) or @-BC so that = -fin and the momentum integral in Eq. (5.149) simply yields with the result . , .
9
-gp,
DYNAMIC SUSCEPTIBlttTV AT ZERO TEMPERATURE From Eq, (2.251, the reoponse of the magnetization m&to an infinitesimal external magnetic field -w&H& is given by the response function ([0(1)*@(2)1). Hence, the dynamic suscept'ibil"ry, which is defined as the response of M to an infinitesimal fielid of frequency w and wave vector Q is given by
where the spin-spin response function is given by
The last Line follows because G is diagonat in spin so that the spin trace simpIy yidds a factor of 2 equal to the spin factor already included in Do. Hence. using Eq. (5.ii6a) the tong wavelength limit of the zero frquency susceptibilirty is (S, m4)
Thus, although we are working at zero bmperature, linear response theory allows the external field ta break the symmetry of the non-interacting ground state, so the proper Pautl paramagnetism Is again obtain&,
OVNANltC SUSCEPTIBILITY AT FINITE TEMPERATURE Finally, although there i s by now no question as to the outcome, we wiil show how the finite temperature rcrsponse function produces the correct result. All of the results for linear response ttteary in Section 5.11 have finite temperature counterparts given by
PROBLEMS FOR CHAPTER 5
285
the correspondence in Table. 5.1. Since the spin trace is trivial as before, the finite temperature response function is given by
In the last line. the summand has been rewritten as the difference of two frequency sums which may be evaluated using Eq. (5.148). Whereas the original frequency sum converged because of the dependence, the sums in the last line are logarithmically
2
divergent and a convergence factor frequency sums,
ekm9
has therefore been inserted. Performing the
and continuing to the real axis as in Fig. 5.4,
Thus, in the limit
W
= 0, k -r 0
as before. The methods for treating frequency sums and analytic continuation iHustrated in this very simple example are completely general and may be applied straightforwardly to any finite temperature calculation.
PROBLEMS FOR CHAPTER 5
The problems for this chapter elaborate and apply many of the basic ideas presented in the text. Problem I, relates the Green's function expression for the
286
GREEN'S FUNCTIONS
p m d potentid to the pertwbatio~a p w s i o ~in Chapter 2. Sum rules mentioned
in the text are d h v d in hobhms 2 4 11, hoblem 3 explorm the stmedare of the ~ c h m g eh m contributhg to the momentum, dependence of the se&energ~r, The relation between the abmctwe frxnction at high momentum transfer m d the grouad state momentum distGbu&ioais derivd in Problem: 5, Problem 12 expIora mad tmncatiom of the MartH-Schwhger Green" function fiierwchy to obtain; * ations do &heone pmticle Greeasa ftxaction, The remaining problem deal wifh vmiom w p e ~ t of s the RPA, Problem 6 oatlinear the derivatian of dke Lindhwd function8 quoted hthe #ex&md its hportance k e m p b ~ h e dby an *, The &round stage RE"A come1a;dion enerw is derived in Prob lem 6 and appUed L-o the dilate Base gas hProblem 7. The fam%w onadimerrsiond S-function sysfern pmvi&sr a s h p l e pedagogicd exampb of haw dhe RlPA reveals physicd bstabilitieg, m d i s dso emphasked by an *, Problem 9 ouklines the classical dedvation of the p1~mab.oscillakionwhich W= obt&n& asing the RPA in. the text. Fhdly, Probbm 10 o u t h w the d6vation of the timedependene Eartree Fa& appradmation and the quivaleltce to the RPA in &hesmaH-mplitude limit md hoblem 11 e s t a b b h ~the propdim of $he ItPA eigenatata used &hetwt.
PRQBkEM 5.2 Evaluation clrf the Grand Potential from the One-Parttclrt Green's knctfon Show that Ekg. (5.25) reprodaca the Wugegholts diagram expansio~for df R. developd in Ghaptm 2 though thhd order. F ~ s L ,note that $ dzdz'dr'Ck (zrlz'r') gA(z'r' lzr') or its Fourictr transfom
--
d9k &CA (k,W ) gA(k,W ) corresponds to the diagram
in which the two
external points of C we connected by 9. Thus, the problem is to expand C h d gXconrrktentb t b a g h thkd order an& show that eilch linhd diapam in Fig, 2.3 is obtaked with the proper eombi;xlatofial faetor. Using Dymnsa equation, s h m that, to third order
Use the fwt $hat;the d - e n w a is obtabed from mputated onepartick hedacible Grmn%ffnncfion dia and m the Bugeafiolts raltzls that Green" functions have no symmetry facton and a faetor of $ for eaeb equivalent pair of Lines to show that thaagh thwd order
PROBLEMS FOR Ct.IAPTER 5
28'7
Finally, substitute Eq. (2) into Eq. (I), collect all equivalent diagrams through third and thus order, for each diagram with n interactions include the factor $ from YcAgAreproduces the contributions of the linked diagrams in Fig. show that 2.3.
9,
I
PROBLEM $2 Energy Sum Rule for the Spectroll Function For a system interacting via two-body forces, Eq. (5.20b) for the energy and Eq. (5.33s) yield
By completing the w coatour in the upper half plane, show that the energy per unit, volume is given by
where (E mftwwes the energy nip from Lfre bottom of the fermi sea insteax3, of dawnward from p. In Mhematic form, this sum rule may be written E. = E ) where the brwkets denote weighting with respect to the spectral futnction, Thk form, which is exwt, h reminiscenG of the reault obtJned in Problem 1.6 in &theE-I&reeFock Recover this old, reetult by evaluating approximation E =
(g+
-' +
- E. c) in the special case in which Slater determinant and I$:-') is an (N pwtkle Slater deteminiant obtained by removing one pwticfe in orbit n from Thus, the sum rufe may be viewed as an exact genwafisation of the familiaa:Hartre* Pack energy expres~ion,
p-(k,p-(E) = C, (($r-lIakI
14,)is an N-p&icle pound
PROlElkEM 5.3 To understand the nonfocdity af the Hartm+Fock pokntid and self-energy, it is instmctive to evaluate the exchange t e r n in a translationally i n v h a a t system, a) Consider the sum 4; (z)#@(y) appearing in the HartreeFock exchange term in Problem 1.6. (Recall + ( S ) includes internal degrees of freedom such ss spin and buspin). Perform the sum over all namaliaeit plane wave spatid wave functions in a F e m i gw to obtain the Slater mked density
where
288
GREEN'S FUNCTIONS
and the total degeneracy associated with internal degrees of freedom ( 2 s + 1) specifies the number of Fermi spheres which must be filled t o obtain the total density p. Explain physically the value a t 1s - yf = 0 and why kF sets the scale at which the mixed density gom to sero. b) Ueing the result in part (a), evaluate the two-body denaity m(rl,r2) E 6(rl-ri)b(r2-ri)) in a Fermi g= with degeneracy (2S-i-1). Sketch pa(rl -rz) and note that the Slater density describes the &Fermihole" in the vicinity of r1 = ra required by the Pauli-principle. Noting that the normalisation condition on pz is $ dsraM(rl ,ra) = ( N - l)p(rt), show that the 'Fermi hole" excludes one particle. c) Assame two different forces act in even and odd partial waves. Since even and odd partial waves are spatially symmetric and antisymmetric, a convenient representation is
(xGj
where the exchange operator P, exchanges the spatial coordinates of two particles. Now consider nucleons with single-particle states characterid by spatial wave functions U, (rs), spin wave function X, (i)and isospin wave function X , ( i ) . Note f'r
G,( r i ) ~ , ,(1)xTI(1)U,:,(r2)xff3(2fXV,(2) ( f i ) ~ a g
( 1 ) ~ (1)ucx3 n ( r ~ ) (~z ), ~~ ~(2) :,
.
Show that in a Fermi gas of such wave functions
Note that the exchange potential, referred to in Eq, (5.78) reduces to the familiar result iu(r) for a central potential.
-
PROBLEM 5.4 Scaling Behavior of the Dynamic Structure Factor and Measurement of the Momentum Distribution At high momentum transfer g, the liquid structure factor for a non-interacting system measures the longitudinal momentum distribution of the ground state. Physically, this is obvious because it just counts the ways a momentum k may be knocked out of the system by transferring a momentum q so large that it is assumed that the state k + q must be unoccupied. Formally this may be written using Eg. (5.108)
PROBLEMS FOR CHAPTER 5
289
where the longitudinal momentum distdbution of the nominteracting system (including 2 spin 8tat;e-s for ewh EGf itl
the scaling vmiable is defined Y
Q --- -2 (1
zI
kll ia the component of k in the direction of p^, and denotes the two perpendicular components, The qucsstion is whether the structure fnnction for the inkeracting system i a shifarfy related ta t he momentum distAlauf ion of the inferacting system. a) Analyse a generaf time-ardered diagrm in which mmentum q is injected at time O m d removed ae time 2-" in the Emit a% q --, oo with y held h e d . Awnme for the present that the tws..bad;y potential is arnooth with zs Fourier Lransfom v(qf which decrewes exponentiaHy in. the g --, oo firnit. btablish the following propertiw: i) Unhsg G inject& at time O can ffow solely bbongh p&icle propagate= to be r e m o d at t h e T", the diagram vanishes at least exponentiallly h q+ ii) Ewh global denorninstar between time O m d time F ! has the farm E=@--
C particles
ep+Crh+iS holes
iii) Euh enerw denminator outside the range (0, 2') jhirs no g dependence, iv) The q-depenhnce of any timeordered diagram with n kteractiona in the range (0, T)and any number of interactions outside (OPT) is
h
v) Hence,
show that the most general contributions of order
f
ta
are of the generic farm
Relate these diagrams to the moat generd contributions to the momerrlum dist~batiomaxld thus ashow
bj Haw consider the ewe of s potentid with an infinitely hard core, U w the opticd theorem to show that the irnaginq pmt of the faward; seaetrering
290
GREEN'S FUNCTIONS
amplitude wows b e w l y with g. Givrt two exmples of diagrams which would be of order $ for smooth potential. but are of order $ for hard carea. Thus, although exhibits waEng by dependi-ng only on y, tbig scaling function is not the momentum distribution, e) Finally, consider practical aperimeats on finite nuclei rtjnd liquid helium, Experimentally, the total croa8 section far nuclean-nucleon scatterkg is approximately constank for 0.4 pb,,,, 5 1.7@c?Ve What; dow this say about nneasaGng the momentum distribution for experiments with q in the range of 1. GeV? E the helium-helium interaction i9 described by the Lennmd-Jonc?~potential, isr id practical to meware the momentum disd~butionfor fiquid helium? Some further dataifs may be found in Weinskein and Negeie (1982).
<
PROBLEM 5.5 RPA CorrelatZon Energy To see the relation of the ring diagram contributions to &heground state energy discussed in connection with Eqs, (2.113 - 2.115) to the RPA response fuhctbn, it is useful t o repeat the ground state energy derivation in Eqs. (5.23- 5.24) writing (V) in terns of D h a t e d of 6 . Note, by the Fermion comrnutaif ion reladions, we may write
and thus show in a tranalationdly invirsiant hsyatern
where (X)) and ' D denote the ground state and response function for the Hamiltonim Eq. (5.23) and l+@) is the non-interacting ground state lybo(0)). Hence, and &fierelation EO - wo = use Eq. (5.113) in the form = Do
+
Expand thisl rmult in powew of o and note that it generates the direct ring diagrams with the factor in &ss. ((2.112 - 2.115). (There is no overcounting problem in second order becztuse exchange diagrams have been neglected throughout.)
&
PROBLEM 5,@* Lfndttard hnction Evaluate tbc?m o m d u m integfals in the response hnction for a non-interacting Fermi gas, Eq. (5.108), to obtain h.(5.109). To calculate the real part, i t is convenient to change variables in the second term of Eg. (5.108) by defining L' =
PROBLEMS FOR CHAPTER 5
292
so that identical @-functionsappear in b_okh terns. The imaginary part is s i m ~ l y proportional fbe pharre space such that k -ia- outside a Fermi sphere while k ia inside a F e m i sphere and energy is conserved, = Thus, Im Do may be calculated geometrically by computing the area of the circle in the plane defined by k.p^+ 8 = msuch that iq+kl > k F and k < kF.
2 +&
PROBLEM 5.7 RPA Correlirtfon Energy far Dftute EIostt Gas Uskg $he RPA cornelation energy exprwion from Problem 6.5, Eq. (P), sum d X the ring diagrms for the dilute Bose gas treafed in Problem 3.3. The integrals can be pedormed analyticdly and should reproduce the exact leding order result.
PROBLEM 5.8' RPA tnstabfttty af Untform Gas; In Probbm. 1.0 the unXorm gm wfuki~nfor Fermions interacting with repulsive &-function forces in one dimension was shown to have a binding energy per pwticfe about 10%Xesa khan a collection of clusters of (2S+l ) Fermions in the ground state. This snggests that the gatil pbwe is unstable against .fluctuations. Note, as men for example in Problem 3.4, that perturbation theory for the ground state encrrgy does not reveal insdabilities, so insted it is uacl!fuf to exmine the rmponse furrction, If all the RPA modes have r e d positive frequencies, the gystern is stable with respect to all infinilesimal fluctuations, If at some q, there is a made with o = 0, the mode is seff-sustaining: k costs no energy to create il. In regions of q for which o is complex, the mode will v o w exponentially and $he system is nnstable. a Cdcufatcl! the RPA rmponm fanetion to semch far instabilities at all q and p, Show that W q -,Q, the long wavelength slabitity c ~ l e r i o nis identical to that d e ~ v in d Problem 1.7 and applid akeardy in Problem 1,9d. b) Determime the range of q for which &heequilibrium density gm is unstable. Compae the wavelength of the instabifitiw with. the she of a ground skate cluster of ( 2 3 I) pwiclear and explain this r a u l t physically.
+
PROBLEM 5,9 ClassScaC Plasma Oscilrations Show that the RPA mode (5,158) in an e k t m n gas corresponds to the pfasma frequency deGved claasicdly. Linewipte the confinuity equation,
arrd equation of mofion
ta bading order ijxz the density f2uctuation Sp = p - p0 and velocity, Combine these? resulks wiLb the Mawefl equation
to obdain the result
with frequency w =
m
.
PRO8t E M 5.20 mm-Oapendrmt W artre@-kckAppraxlmation a) Using the equation of motion for $(S,t)$ Eg. (5.171, derive the following exact equat;ion sf motion for at1]
This is the first of an infinite heirarchy of equations (Martin and Schwinger, 1959) d a t i n g the evolatioa of G(@)to ~ ( ,ad e ~~ ( ~ -)l ) Now, make the time-dependent H e r e e - h c k (TDISF) approdmation by assuming that the two particles propagate independently in G('), that is
to obtain
where the
TDHF single p&icle Hamiltaaian is
zuzd the one-body den~itymat*
is defifted
Combining $heequation for G with its &joint, the TDHF equation macy equiwdenkly be &tten purely in terns of the onebody densie matrh
b) For a Slater determinant, p(z,S') = CA4A( S )fA (2'). Thus, show that Eq, (l) is equivalent h the eingbgarticle equation of motion
T k equadion will be obtained by using the time-dependent vuiat;iosal principle in Problem 7.5. c) Recover the RPA by considering infinitesimal TDHF fluctuations around the H&ree-Fock ground state ,(Goldstone and Cattfried, 1959). Let p0 denote the static d e n h e matrts and p1 the infinitesimal tluctu&ians in the density matrix,
PROBLEMS FOR
CHAPTER 5
293
The &egc?ndictrrceof the TDHF f-fmiltonian on p is indicated explicitly by w ~ t i n g Eq. 12) in the form h = T W[p]
+
and the stakie HIF b a i s is defined
so that, taking mat^ elements in the HF bwirt
Findly, note that the most general infinitesimal change in the ground state wave function may Be written ;~$ga s ~ p e r p o ~ i t i oofn p&icEe-hole t3-xci$alions so that an excitation of frequency w is written in the Hermitian form c-'U' p!e'Ut with p1 only having particle-hole matrix elements, and thus obtain Eq. (5.133) with X,n = ( ~ l p f l a and ) YaA= (alptlA) Show that the TDHF single particle wave functions for this mode have the form
+
PROBLEM 5.11 Propertier of RPA Eigenfunctlons Derive the following p r o p erties of the RPA eigenfunctiona defined by Eq. (5.133). ) t j compare a) Orthogonality. Lefemultiply Eq. (5.133) by ( ~ ( ' ) t ~ ( @and with the adjoint equations to show
has energy
E(W)
then
has energy -E(@)*,
t to normalise eigenvectors by the condition
have the same norm,
294
GREENS FUNCTIONS
tene~a,Let an w b i k q =tar , Vere
F be expanded in eigenstateca E =
the campbden~(~.l relation
by applybg it to F. (i) Gmen'g hnetion. Show that t b Green's function
satisfies the equation (5.132). Thus,derive Q. (5.134) (with the appropriate f i g ) .
PROBLEM 5,12 Gteen" knctIon Summatlons of Ladder Dlagrama. Multiply the first equation in the M&in-Schwinger Green's function hiermchy, Eq. (1)of Prablern 5.10 by 660 obtain the exaet equ&ion
w h m 5+ denotes a positive infinitesimal t h e added to L&, v contahs a I-function in t h e , rrnd repeated coordinates are summed. &X diagrams, &hismay be written
Inzated af making the Har&ree-Pock appro~mafionas in Problem 5.10, consider mare general sunrmratians defined by the equation
which may be repr~endeddiagfamxnatieally as follows:
Note that in these diagrams the dotted lines are not propagators but just indicate haw the vwians pmts af the diagram me connected up, whereits the solid lines are really Go propagators,
PROBLEMS FOR CHAPTER 5
295
We will deaf with two approximation8 for h:
and
a) Treat the two equations as a system of closed equations and solve for the self energy C in the Aoo approximations. Retain t e r m sufficient to obtain contributione to the ground state energy through third oder in part b, below. Recall f: sdisfies the equation:
b) From G(') and C, obtain the expansion for the ground state energy through third order. Compare with the Goldstone expansion, (Eere you will need to remember that Green" functions include all re1acLive time orderings and both. pmdic1e and hole propagators. You will need to be v e e caseful with hetars.) Skate explieiLly which graphs are counted eomwtly, incomectfy, and omitted. c) Now1make contat with Bmecknar theory (see Eq. (3.64)) by determhing w h i d ladder graph are included in the doe and All approximations to all orders, on k ee%siesctsLh the low density limit where waphs with two independent hole lines dominate tho-se with any higher number of hole lines. For two-hole line waphs, da hli a d hoo comespond exactly to any pregcripkion. for the patentid in h e c k n e r theory? What is the essentiJ difference at higher density (morethan two hole lines)? PROBLEM 5.13 Energy Weighted Sum Rules Derive the sum rule Eq, (5.139) by evaluating the dollble commutaea show that [ r f , ~ ~=] tor [ [ X , ~ p ] , p - e ~ First* show [P,&] = 0. a$a,+, from which it follows tha = $aka,. a complete set of states in the matrix element [[H,pq],P-,J) to obtain the sum mle.
-c,
CHAPTER 6
THE LANDAU THEORY OF FERMI LlQUlDS The general methods presented in previous chapters to calculate observables in Fermi or Bose systems have been based on perturbation theory. In the case of strongly interacting systems, even when physically motivated resummations have been made. it i s dimcult to aswss the reliability of the resulting approximations. Hence. in this chapter we will present a completely different approach developed by Landau (1956. 1957.1958) for Fermi liquids which provides exact relations between certain observable quantities* The domain of validity of the Landau theory is restricted to phenomena which involve excitations very close to the Fermi surface. In this domain. the fundamental degrees of freedom of the system are quasiparticles which interact with a quasiparticle interaction f(k,k'). Although the theory does not specify any properties of the strongly interacting ground state, it describes small departures from the ground state and response functions. It thus allows one to express a number of observabfes such as the spc?cific heat, magnetic susceptibility, sound velocity, zero sound velocity, and transport coeficients in terms of the Enteraction f (k,k". This function can either be parameter'rzed phenomenologlca(ty By fitting several parameters to experiment, or else it can be calculated mtcrascopically, in which case the theory becomes exact. The chapter is organized as fotlows, tn the first sectlon we present the basic assumptions of the theory in a heuristic and phenomenofogicai manner and in the next section wt? show how to calculate thermodynamical observables with this set of assumptions. We emphasize that the arguments in these sections wlll not be rigorously deduced from general principles; rather, at key points assumptions or postulates will dmply be introducd following Landau's orighal development. For those who wish to pursue this approach to tandau theory in further detail, the review of Baym and Pethick (1976) Is pafliculariy clear and pedagogical. The last section presents the! microscopic justification of the theory* This microscopic understanding both prov"rcfes a clear foundation for the postulates of the earlier sections and establishes the relation of the phenomenological interaction f(k, ic") to the two-body vertex function.
6.1 QOAStPARTlCtES AM0 THEIR INTERACTIONS Let us consider a uniform gas of spin-$ Fermions, containing N particles in a volume V , If the Fermions are not "interacting, the ground state of the system consists of a Fermi sea of plane waves of momenta kF. The Fermi momentum kF is related to the density by:
The totat energy of the system i s given by
6.1 QUASIPARTICLES AND TMElR INTERACTIONS
where n(g) is the occupation number of the plane wave state
297
(g).
If an arbitrary weak external field is coupled to the system, the net effctct will be a variation of the occupation numbers, and the corresponding variation of the total energy can be written: 12
Further. is 9 e field is very weak, it can only excite states close to the Fermi sudace. so that 6n(A;)will be sharply peaked amund kP. Let us now adiabatically turn on the interaction between the particles. A normal Fermi liquid is defined as a system in which the non-interacthng ground state evolves into the interacting ground state and there is a one-to-one correspondence between the bare particle states of the original system and the dressed or quasiparticle states of the interacting system, A quasiparticte state with )k+j > kF is defined as the state obtained from a non-interacting Fermi sea plus a plane waw tk,) by switching on the interaction. and simikrly, a quasihole state is obtained by starting with a Fermi sea with a hote in state Ikh). Note that in order for a system to be a normat Fermi liquid, switching on the Interaction must not produce bound states. For exampie, in a superconductor, the noninteracting ground state does not evolve into the &CS ground state and the formation of Cooper pairs destroys the one-to-one correspondence between non-interacting states and quasipa&icfe states, Throughout this chapter, we will tacitly assume a normal state, dther by vivtue of a totally reputshe Interaction or by treating temperatures above the superfluid or superconducting transltlon temperatures. Normal Fermi liquids t o which the theary is commonly applied include llquid w e , the, electron gas in metals, and nuclear matter. We have seen In Eq, (5.72) that the quasiparticfe Lifetime is proportiwal t o (a - c ~ ) ' . so that even in a normal Fermi system, quasiparticles far away from the Fermi sudace are Df-defined, unstable states. If one tried to prepare them by turning on the interaction slowly enough t o be adiabatic, they would decay before the process was complete. Hence, the Landau theory is only appf'lcabte to tow-lying exdted states of the system, which are made of superpositions of quasipaeicle excitations close to the Fermi surface, In particular, it can describe neither the ground state energy, whfch would require summation over all occuplred states in the sea, nor highly excited states. Let us temporailly suppress spin and denote the quasiparticle energy by e i and the interaction energy of quasiparticles of momentum 2 and by f (g, g'). If we apply a weak perturb_atii to the system which takes it away from iis ground state, it inducer a change 6n(k) in the occupation number of quasiparticte L, and tandau postulated that the change in the total energy of the o;ystem is dven by
298
THE LAPIDAU THEORY O F FERMI LIQUIDS
The energy of quasiparticle g, when it is surrounded by other quasiparticles. is given by:
The interaction f (g, g') of quasiparticles
and it i s thus symmetric in
and
g and 2'
is given by
g'
f ( E s 2) = f(if g)
'
Although the Landau theory is much more general than the Wartree-Fock approximation, it Is instructive t o note that Eq, (6.5)is of the same form as the total Hartree Fock energy where r: i s the Knetic energy, eg is the self-consistent single-particle energy and f f (g, g') is the antisymmetrized matrix element of the bare two-body interaction. Since quasipartic!es are adiabaticafly evolved from Fermions, they obey FermiDirac statistics and the distributioon functions at finite temperature is gken by the usual Fermi distribution 4
where p is the chemical potential. At zera temperature, we recover a step function distribution, and 2 non-zero temperature, Eq. (6.8) becomes a set of self-consistent equations since e ( k ) depends on n(z) through (6.6). VVe now extend the notation t o include spin. Formulas (6.5) to (6.8) are unchanged. except that g should be replaced by ( 2 , ~where ) a is the spin projection on a gaantization axis and the sums over are replaced by C,=,,/, In the absence
xz.
of a magnetic field. by symmetry r(i,o)is independent of o, so that r(g,o) = a(g). Similarly. f (grr, gtd)can depend only on the product o .d . and may be parametrized
with the relation
Note that here cr denotes a spin $ matrix so that a Pauli matrix would be 2 ~ . A crucial irrnplification occurs for the translationatty invariant systems. Since Landau theory is restricted to phenomena Involving only quasiparticles close to the Fermi surface. f (k,k t ) only enters observables with the two vectors k and kt on the
6.2 OBSERVABLE PROPERTIES OF A NORMAL FERMI LltQLtID
Fermi surface. Thus, f will depend only on the angle B between convenient t o expand it on the basis of Legendre polynomials as:
and
299
g. and it is
(B.lOj
The orthogonality relation for the Legendre polynomials:
Pj, (COS B) PLg (cos B)d (cos 8 ) = 6 ~ ~ e
(6.11)
implies that
decrease sufficiently In practical applications, the legendre coefficients ft and rapidly with L that truncation after several terms yields an adequate phenomenology. In this case, a small number af empirical coeffidents descrkes a much larger number of experimental observables and the theory has non-trivia! physkcal cantent and predictive power. For metals with non-spherical Fermi surfaces or Gnite fystems, k') depends on many more parameters and the theory is correspo~dingtyless povve~rt!.
JP,
6.2 OBSERVABLE PROPERTIES OF A NORMAL FERMI LIQUff) We shall now catcutate a number of observable propertIcts of it normal Fermi liquid in terms of the effective interaction f (k,k'), assuming the validity of Eqs, (6.5- 6.6). The microscopic justification of the theory and the relation of f (k,k') tto the bare interactbon wilt be deferred to the following section.
EQUILIBRIUM PROPERTIES
A fundamental parameter af the theory is the e k t i v e mass, As In Eq. 15.74). m" is defined in terms of the density of states or group velocity a t the Fermi surface
By the definition of a quasipartlcfe as the eigenstate obtained from a non-interacting Ftlrmi sea with a particle is state fk), E@ must be the chemical potential p and k~ expanding in the neighborfioad of the Fermi surface
3-00
THE LAPdOAU THEORY O F FERMI LlQUlDS
Specific Meat The specific heat at constant volume Is defined
A change in temperature induces a change in occupation numbers, so that cv is given by
A t low temperature. the sum f (g, gf)&n(z')in Eq. (6.6) is of order TZ so that the leading contribution to ev may be obtained by replacing rg by c; and converting the integral over L to an integral over c: using Eq. (6.13). The standard method for ea!eufating the resulting integral using the low temperature expansion for the derivgitive of the Fermi function
is reviewed In Problem 6.1, The result t;o lowest order in T is
Physically, dnce the egective mass is proportional to the density of stateeatthe Fermi surface, we expect the specific heat to depend on m*, The result, Eq, (6.17).in fact shows that It depends on no other parameter of the theory. That is, to leading order in T the specif-ic heat is equal to that of non-interacting Fermions of mass m* and thus provides a direct experimental measurement of m*. For liquid 3He, the egective mass ranges from roughly 3m a t zero pressure to over 5m at 21 atm, where the primary experimental uncertahty Sn measuring the specif c heat arises from uncertainties in the temperature scale, These results as weir as subsequent experimental llandau paramare taburated in Table 6.1 a t the end of the chapler. So-called eters for liquid l ~effective 'heavy Fermion systems" such as CeCuaSi2. UPts. UzZnlr, and t e ~have masses m*/m of the order of 1ClZ- 1CI9 (see Stewart. 1984). Erecttve Mass Having seen that; m* may be measured experimentally, we now relate it t;o the quasiparticle interaction f (k,k'). It is clear that if one postulates that Eqs. (6.5- 6.6) hold in any Galilean frame. the different relative contributions of E; and f(k,V ) in digerent frames will give constraints relating c i and f(k, h'). Problem 6.2 explicitly uses Gafitean invariance to refate m* to f . Here, we present Landzsuesoriginal argument which equates the momentum of a unit volume of the Liquid to the mass flow.
6.2 OBSERVABLE PROPERTIES OF A NORMAL
FERMI
LIQUID
305,
The momentum per unit volume Is given by
9,
On the other hand. since the velocity of a guasipartiele is and the number of quasiparticles equals the number of particles, the total momentum can also be expressed as:
where m is the bare mass of a particle. Equating (6.18a) and (6.18b). we have:
Since (6.19) is an identity, we may take a functional derivative with respect to
?(g, o)
which we rewrite:
In (6.201, we have used the usual replacement of the sum At zero temperature U)
ag
= -.F b(kF
xzby an integral V
- k)
&$. (6.21)
where k is the unit vector along E. Thus. using (6.13) and (6.20) and taking g on the Fermi surface, we obtain:
_
Finally. denoting by 6 the angle between obtain
1
1
kF
_ -- ;;;;+ W 1 m*
=-+-ft
RF 3x2
4"
.
/
d (cos 8 )
and
kr, and using Eqs. (6.10) - (6.12)
C + 4g (fi.
t
.'(L)
PL(COS
we
8 ) cos B (6.22b)
302
THE LANDAU THEORY OF FERMI LIQUIDS
Because thls and all subsequent integrals over j(k,kt)are evaluated at the Fermi surface and thus are ultimately multiplied by the density of states at the Fermi surface. it is conventional t o define new Legendre expansion coefficients which include the density of states. Using Eq, (6.13b). the density of states at the Fermi surface is
and we define the normalized expansion coeficients*
With these definitions. we see from Eq. (6.22) that the effective mass directly specifies the L = 1component of the spin independent quasiparticle interaction
As is also evident in Problem 6.2, we observe from Eq. (6.22) that the current Because of the interaction with all associated with a quasiparticle is not simply the other particles in the medium. there is another piece coming from the interaction j which represents the other particles dragged along with the quasiparticle. or the backflow of other particles around the quasiparticle. CompresstbilTty end Sound Velocity The compressibility X of a tiquid characterizes the change of pressure with volume according t o the standard definition?
5.
5.
where V is the volume of the system and the density is p = The velocity of thermodynamic sound. c l , provides a convenient experimental means of measuring X , since by Eq. (5.124)
*
A more modern notation for Ft is F' and for & is F f f ,where the S and a denote symmetric and antisymmetric, spin combinations. Note also that our definition of Z differs by a factor of 4 from some of the early literature.
t Frequently, the compressibifity of nuclear matter is expressed in terms of the compression modulus. K H k$& A t equilibrium. =
(5).
9.
6.9 OBSERVABLE PROPERTIES OF A NORMAL FERMi LIQUID
303
From the observation in Ffg. 5.16 that thermodynamic sound corresponds to a sphere icalty symmetric change in the occupation of states at the Fermi suufaca, it is clear physlcaily that X and c% will depend on a spherically symmetric average of f (k,kt) and We now derive the precise relation. thus on the Landau parameter Because the free energy is extensive], we] may write it in terms of a free energy per unit volume as follows:
F ( l f : Y , N )= 'V f
and thus
Since Et Is convenient to tlxpress the finat result in terms of the chemical potential, we use Eq. (6.28) t o obtain aF d p=----
aN
- -f(T,p) dp
so that
In order t o calculate
%, we note that
and thus, using Eq. (6.6).
Using the relations p =
from &(ht- kr) from Eq. (6.3) and (6,24), we obtain
Eq. (6.1).
Ef(8,g,g')
g = 9from Eq. (6.13). = fa =
from which it follows from Eqs, (6.32)and (6.27)that
from Eqs. (6.12) and
304
THE LANDAU THEORY O F FERMI LIQUIDS
As anticipated, the compressibility depends upon &, coming from the spherical average of f ( B ) , as well as F%,arising from the effective mass. Note that the compressibility becomes infinite, that is, the system becomes unstable against density oscillations, -1. T T s is a specific example of the general stability criteria f;"L > when E-(2L 1). and & > -(2L 1) derived in Problem 6.3. Magnetic Suscoptib"r1lty In contrast ta the previous properties which were independent of the Fermian spin, we now consider the magnetization which is induced by an external magnetic field, The magnetic interaction energy of a Fermion with an external field H is -@?H,where for a point particle with no anomalous moment, the gyromagnetic ratio is 7 = and cr = f$ is the spin projection along H. Hence. in the presence of an external field H. the energies of the sptn particles are lowered and the energies of the spin particles are raised. causing the equilibrium occupation of the spin states to be states and thus producing net magnetization. larger than the spin To calculate the induced magnedzation, we need to calculate how n(a), the density of particles with spin projection a,changes in the presence of a weak external field H , Since the total number of particles is fixed, we may write
+
+
2
+
-5
+i
&
Since the chemical potential of spin state cr, p @ ) , depends on bath n(c) and nf -a), the equilibrium condition p b ) =F poses a constraint which we will use to specify An, The chemical potential For each spin is equal to the quasiparticle energy at the corresponding Fermi sudace P(@) =: (b (4, p ) (a.3~) where, in the presence of an external field, the quasiparticle energy is
Since both k F ( o ) and the distdbution function n(k,o) are specified by the density p(o-). p(rr) is a function of p(@) and For a weak external field, we may expand the chemical potential to first order around the resuft p0 at H == 0.
The derivatives oF the chemical potential p(@) with respect to the densities for each spin state are evaluated as in Eq. (6.34).where we now distinguish the individual contributions of each spin population
5.2 OBSERVABLE PROPERTIES
O F A NORMAL FERMl LIQUID
305
Using Eqs, ( 6 . 3 s 6.40), the equilibrium condition p(@)= @(--L provides the desired 2x2 r H = =(If kFm*
ZojAp
*
Using Eqs. 66.36) and (6.41) the magnetization density i s
From which it foIIows that the magnetic susceptibility is
As in the case of the compressibility, the L = O Legendre coeCfricient of the quasiparticte interaction enters becaus aF the spherically symmetric deformation of the Fermi sphere. However. since the spin sphere expands while the spin -$ sphere decreases, the digerence, Za, between the interactions of like and unlike spins contributes rather than the sum PO,Again, we see a specific example of the general stability critsrbn > (2L+- 1) of Problem 6.3. tn liquid %e, as tabulated in Tabte 6.1, 2& ranges between -0.67at O atrn and -0.76 a t 27" atm indicating that the strong spin dependence of the egective interaction makes the system nearly unstable with respect to ferromagnetic ordering. The physical origin of this dependence is simple. By antisymmetry, the spatial wave function of a spin singlet, is S-waw and appreciably samples the shortrange repulsive core af the He-He potential, whereas the spatial wave function of a spin tviplet is P-wave and experiences much less repulsion,
+$J
NQNEQUlLIBRIUM PROPERTIES AND COLLECTIVE MODES The dynamics of a Fermi liquid close to clquil"lrium is governed by a Boltzmann equation for the quasiparticle distribution, Two fundamental assumptions are required. The first is that we consider only long wavelength, low-energy excitations. Then, instead of considering the quantum Wigner distribution function introduced in Section 5.4, we may treat the n2n-equilibrium distribution function at position i as a clasricai distribution function n(k, i", $1. The second major assumption is that the local quasiparticle energy at the position 7, r(g, ?), plays the role of the quasiparticle Hamiltonian, so that
de(k, +.= ak
k=;:---a ~ ( kr ,) ar
frj,a4) '
Then, the Boltzmann equation is obtained by equating the total time derivative af n(g, 7, t ) to a quasiparticle collision integral i ( n )
THE LANDAU THEORY OF FERMI LIQUIDS
306
Since we will be interested in small deviations from equilibrium, we expand n and around their equilibrium values as follows
E:
Expanding Eq. (6.45) to first order in 6n. we obtain
d -&n(k, F', t ) +
at
2 a6n ano 1 +-m* ar ak V ,
a
f ( k , k') &bn(k', r, t ) = I(n)
(6.47a)
where throughout this section where no ambiguity arises. spin sums will be understood to be included in sums aver k . Using
the tinearlzed Boltzmann equation is written
Conservation Laws Using the relation
summing (6.45) over momentum and spin, and noting the sum over the collision integral vanishes yields
where the current is defined
Linearization using Eqs. (6.46) and integration by parts yields
6.2 OBSERVABLE PROPERTIES O F A NORMAL FERMI LtQUlD
307
In particular,if 6n corresponds to the distribution function for a single quasiparticle of momentum k, we recover the result, Eq. (6.22a) for the quasiparticfe current including the drag term
The equation of momentum conservation is obtained by multiplying the Boftzmann equation (6.45) by ki. integrating over momentum and spin using (6.491, and noting that the collision integra! vanishes by momentum conservation
Integrating the last term by parts and using the fundamental relation Eq. (6.6) that = we may write
fir,
Hence, the momentum conservation equation may be written
where the momentum flux tensor is given by
The law of conservation of energy is derived analogously by multiplying (6.45) by r(k,r), using Eq. (6.49). and noting that the integral of the collision term vanishes by energy conservation
Integrating the fast term by parts and using Eq. (6.6) yields
3Q8
THE LAFIDAU THEORY
OF FERMI LtQUfDS
W here
Linear'rzing Eq. (6.55b) using Eqs. f6.41i) and integrating by parts yields
S Q =~
-c 1
(k)ra(k)kj- - anozox - -m*
k
ski
V
E f (k,k')&n(kt) k,
Like the current, the energy current has a drag term accounting for the energy flow of the particle interacting with a quasiparticle. Zero Sound The zero sound mode dlscursed in Section 5.1 fn connection wlth the RPA llnear response theory was originally discovered by Landau (1957) as a cottr~lctiveoscillation of the Fermfliiiquid, In this section we will show how this mode emerges naturally as a solution t o the linearized Boltzmann equation. Recalt that Eq. (6.27) for thermodynamic sound relies on the hypothesis of local equilibrium and thus requires that the saund frequency v be smatt compared to the inverse collision time v << !. By the general phase space argument used in Eq. (5.72) for the quasiparticle lifetime. we know that the collision time is of order r -. (s s~)"""'. Finally, since the quasiparticles excited at temperature T have energies ( ~ ---€JP) k kT, it foffows that ordinary sound propagates for
-
and thus cannot propagate for sufficiently low temperatures. At temperatures sufficiently low that ordinary sound ceases to propagate because collisions are negligible, we may neglect the collision term I(n) in the Boltznrann equation and solve for the relevant cotlective modes in the collisionless regime. We will seek a solution of the farm
in which case the linearized Boltzmann equation
(B
Jr - U ) 4g
+
Eq. (6.48) with f(n) =: O becomes
1 J~6(p ekJV
C f (k,k t ) 4 b ~= O k'
where
=
6.Since bk is proportional to
and using the fact that we obtain
1%
= VF =
- er), it is convenient t o define
&(p
3 for momenta restricted to the Fermi surface.
5 2 OBSERVABLE PROPERTIES O F A NORMAL FERMI LtQUlD
309
Note that physically, uk i s the displacement of the Fermi surface at momentum k since. by Eqs, (63.51)
Since momenta are restricted to the Fermi surface. %(g) only depends on the spin rr and the direction of g. which we denote by polar coordinates Cl = ( 8 , 4 ) with the polar axis in the direction of As in Eq. (5.121). we also define the ratio of the velocity of the mode to the quasiparticle velocity a t the Fermi surface.
a.
so that (s
Eq, (6.58)
becomes
- cos B ) %(R,Q) = cos B
(6.6l a ) where, as in Eq, (6,24),F includes the density of states a t the Fermi surface
Bone denotes the angle between and g', and we have explicitly displayed the spin dependence for future reference. Let us first consider the case in which the Fermi surfaces of both spin states: oscillate in phase, in which case Eq. (6.61a) simplifies t o (e
--
cw 8)u(0) = CM 8
C
eao,)u(nf)
F~P~(CQS
L
IVlzrgngthe further assumption that the quasiparticle interaction is independent of angle simgtifies the rtquadon still Further
The solution has the form
320
THE LAPtDAU THEORY
OF FERMI LIQUIDS
where
Substituting the solution f6,53a) in (S.63b) and allwing for the possibility that s m y have an infin"redmai positive imaginary part, we obtain the condition
which is identical t o Eq. (5,120). As we argued In Chapter 5 , in order to obtain a real solution to [6,64l,corresponding to an undamped mode, we must haw s > L which requires P > 0. As F runs from O to m, 8 acquires afl values greater than 1, with the limiting behavlor
(6.65) Reliation (6.63a) shows that the Fermi surface for the longitudinal zero sound mode is deformed into the sudace of revolution sketched in Fkg.5,165,with the elongated end pointlng in the ;g directian. Let us now consider a more generat case in which all multipoles fii are included, but we still assume both spin states oscillate in phase and that u ( 0 ) is independent of the azimuthal angle 4. Then, Eq, (6,62a) becames
As shown in Problem 6.4, expanding ~ ( 8 In ) Legendre polynomials
U(@)=
C
PL(cos 8)
U&
L
and using the addition theorem in Eq, f5.66a) yields the coupled equation
where
Using the explicit results of Problem 6.4
6.2 OBSERVABLE PROPERTIES
QF A NORMAL FERMI LtQUtO
339,
we find for the case in which only Fa and fi"l are non-zero the simultaneous equation
which have a solution when a satisfies
Vet another solution arises if we allow for azimuthal dependence. Let us keep only the first two terms in the Legendre expansion
and assume a circutarly polarized solution
Using this form of F and u in Eq, fCi.f52a) yields the solution =
cos 6 sin 8 C# s cos@
-
where d (cos et) sin 8'
W(@"
.
Note that in contrast t o the forward peaked zero sound mode in Eq. (6.63a). this mode has the largest deformation at B -- f . Substituting (6.69a) in 16.69b) yields the equation for the propagation velocity of the mode
In order for this mode to propagate, we must have s the limgting behavior i s given by
1 which rgquires FI > 63, and
Zero sound has been abwrved irr liquid 3He at 0.32 atm wi.h a velocity relative to thermodynamic sound clif = 0.035f .a03 (Abel. Pndeson and Wheatlay.
312
THE LANDAU THEORY OF FERMI LIQUIDS
1966). Assuming this ratio of ueloc'rties changes slovvfy with pressure, we will compare it with the prediction of Fermi ihquld thwry at 0.28 atm, where the Landau parameters
from measurement of the specific heat and thermodpamic sound (Wheatley. 19%) are Fa = 10.71 and lPX = 6.25, Fmm Eqs. (6-35) and f6.50), the measured value of corresponds to S = 3.60 f 0.01. Using only the single multipole .Po in Eq. 16.64) yields a s = 2.05, in poor agreement with experiment, Including both Fa and J'g in Eq. (4.67~) yields s = 3.60,in perhaps fortuitously good agreement considering the omission af still higher multipoles. Although Table 6.1 shows that f i increases significantly above six at high pressure, the drcularly polarized mode has not yet been observed. Spin Waver Waving explored in detail zero sound modes in which both spin states oscillate in phase, it is strakhtflorward t o atend the analysis t o spin waves in which the spins o~~clftate out of phaw, Spin waves are obtained from solutions of the general equations (6,tilaf of the! form (~~72) U,(Q, Q) = U , (Q)%@
7
.
SubstBtuting this arasatg in Eq. (G.Sla), using Eq. (6.6lb)and performing the spin sums yields the equation far u,fhl.)
and ~ ( 6 1 )replaced This result is of the same form as Eq, (6.62a) with .FLreplaced by by U, (n), so thag all the solutions of the previous sectban apply to spin modes as well. In the case of liquid 3 ~ e however, , the relevant Lztndau parameters shown in Table 6.l are negative so that there are no real solutions to the equations (6.H) or (6.67~). SIncat a is not real, the spin wave modes are damped and do not propagate. A genera! theorem by Mermin (19-67) discussect In Problem 6.5 shows that at least one of these two possibilities, spin waves or density waves, must always propagate in a Fermi liquid. In the case of' nuetear systems, In addition t o spin degrees of freedom, there is also isospin, Thus, the preceding analysis is generalized to Inetude terms of the form G@osBnnt)r ;it amd G"(cosBaat)a . 87 . i? in the quasigerticle interaction and one obtains a rich variety of spin, isospin, and spin-lsaspin modes. Electromagnetic probes couple to the protan much more strongly than to the neutrons in nuclei, and thus strongly excite iso:spin modes, of which the giant dipole resonance Is an outstanding example. At energies of several hundred MeV, (p,n) reactions, in which a proton is scattered from a nucleus and transfers one unit of charge to produce an outgoing neutron, produce sp in-isospin excitations, QF particular interest in dense nuclear and neutron star matter is the spin-isospln mode with the quanit?um numbers of the pion, As the density increagses, the energy of the mode decreases until at a critical density the mode becomes self-sustaining and gives rise to pion condensatisn [see, for example, Midgal, 1978). To conclude t h k section, let us emphaskze again that Landau theory describes the properties aF the lowest excited states of a normal k r m l tiquid in terms of two functions ff8) and p(@) which parameterize quasiparticte interactions on the krmi surf;tce. The ~~eflficients of the expansion of these functions in Legendre potynom'rats
6,s MICROSCOPIC FOUNDATIQN
313
can be related to simple thermodynamic observables, and Landau theory can be used as a phenomenological theory where the coeflicients &,fir -.-and & , Z l , . . . , are measured by simple experiments, tn the next section, we show that the quasiparticfe interaction is related t o the twoparticle irreducible ve@ex function taken on the Fermi surface. that is, to the forward scattering amplitude of two quasi-particles on the Fermi surface, and we show how it can be computed From the microscopic interaction,
The microscopic foundation of Fermi flquid theory at zero temperature and the definition of the quasiparticte interaction may be seen by studying the exact integral equation for the two-body wrtex function, The basic idea is sirrrple. Me know from the study of Green's functions that the poles of ~ ( and~ thus 1 of the vertex function l'(2) describe the excited states of the N-body and N k 2 - b o d y systems. By restricting our attention to the relevant poles of the N-body system and anafyzing the structure of the exact integral equation for the vertex function in the vicinity of these pales, we will obtain an intcsgral equation for the fundamental excitatians of precisely the same form as Eq. f6,56a) in the last scrction describing zero sound and collective excitatians, From this correspondence, we wilt be abfe to relate the quas@article interaction t o the vertex function at the kvmi surface in the particle-hale channel, normalized by the residue of the quasipartlcie pole, In addition, we will also be able to identify the quasirparticIe forvvard seaicering amplitude, and use the restriction posed by antkymmetry to obtain a sum rule on the tandau parameters. Since all the essendal ideas are clearly present& in the original paper by Lrtndau ($9581, we will fallow his development and notation.
RELATION OF THE QUASIPARTICLE [NTERACTION AND VERTEX FUNCTION Using the notation of earlier chapters, we will write the one-particle Green" function as (6.74)
where is the ground state and 1 and 2 denote space, tlwre, and spin coordinates. For notational simptfcity, we will suppress spin sums where convenient, and reinstitute them at the very end. Jn a translationaify invariant system in the absence of a spindependent external field, the momentum and frequency transform may be written
where
P denotes ($,U).Using the results of Chapter 6.G(P)may be written
Expanding C(@,u)around the Fermi momentum kF and Fermi energy using Eq. (5.74). and using the fact that the imaginary part of Z: goes to -91 sgn fw - p ) as one
334
THE LANRAU THEORY OF FERMI LIQUIDS
approaches the Fermi sudace from abow or below, where q i s a poskive Infinitesimal, we obtain
where VF = and we use Landau's notation for the residue at the quasiparticle pole, a = (l - %)-l. As in Section 2.4 far thermal Green" functions, the real t"re Wo-body Greenes function ~ ( ~ ~ ( 1 , 2 1= 3 4(+a) ~ ( 4 ( 2 ) $ ( 2 ) $ ~ ( 3 ) 4 ~ ( 4 ) ) (6.78) is related to the vertex function r(2)(1,2 3,4) by the relation
which may be written diagrammaticalfy
Recall from Section 2.4 that l'(2)is the sum of all one-particle irreducible amputated diagrams with Wo incoming and two outgoing legs, The factor of i in Eq. (6.79a) for real time follows from the dhagram rules of Table 3.1. Shce the first order contribution to i a ~ i nis)i j' GG v GG, in order for rf2) to reduce to v in lowest order, it must enter Eq, (6.79a) with a factor of i , The Fourier transform of Eq, (6,79a) is
From the conservation of energy and momentum at each vertex, we know that both G@) P1 -@S -P4 and with the definition
and I"(2) are proportional to ( 2 ~ ) ' s
+P,
6.3 MICROSCOPIC FOUNDATION
Fig. 6 , l
Firsc and second order graphs cantrlbutlng to
315
r.
Eq. (6.79a) becomes
As Is the case for the Green" function, the vertex part 'I is antisymmetric with respect to the exchange of two particles, By the Eehman representation arguments of Chapter 5, the vertex function like has poles that correspond to states of systems with different particle number depending un the time ordering of the field operators. For example. the time ordering $1$2(/)4$i defines the paflicle-particle channet and has holes i n the energy val=lablew l + w g corresponding to ( N 3- 2)-particle states. In contrast, the particle-hole channel is defined by the time ordeilng and has poles in the energy variabfe ws - w l corresponding to N-particle states. Just as we restdcted our attention to the particle-hole channel in Chapter 5 to study Npartide excited states in linear response theory, t o make cantact with Lanciau" tttreary of low-tying collective excitations in the N-particle system, we will examine the taw lying excitations in the partkct-hafe channel by studying the poles of I' in the variable 0 3 - W $ . Because QF the restrfction to low excitlatbon energy and tong wavelength, we need to consider the vertex function for smati values of ws w l and - g,. Thus, we shall consider the vertrsx functbn for
r(P1,Pz/p3,P4),
-
and define
(6.8fb) where R = (E,w) is a small four-vector. This corresponds to nearly forward scattering. The first and second-order graphs contdbiuting to I' are drawn In Fig, 6.1 in the Mugenhoiltz representation, The first order graph (a) is just the antisymmetrizd matrix
316
THE LANDAU THEORY OF FERMI LIQUIDS
- +
element v(g) - v(B1 g) and each of the second order graphs (b). (c) and (d) involve a loop integral over the four-vector with the propagators indicated in Fig. 6.1. With arbitrary PI and P?, graphs (b) and (c) are well-behaved at K = 0. and for small 8, we may just set R = 0 in the propagator. In diagram (d), however. -+ 0,the poles in the propagator G(B)G(R coalesce, and this diagram as requires special care. To calculate P, we must sum the entire perturkation series. Let us denote by f' the part of the function I' which is not singular at X = 3. and adopt the notation
a
+ a)
By analyzing a general diagram, it is seen that f is the sum of all the graphs of l?, whkch cannot be decomposed into two components connected only by two propagators &%ring in four-momenta by K , Graphs fa), fb) and ( c ) of Fig. 6,sare the lo\rvest order cont~butionst o p. By the construction of F. it follows that the entire series for integral equatbn
I'
is generated by the
where the double lines represent full one-particle Greents functbotls, Far example, graph (d) of Fig. 6 3 is obtained by the second iteration of this integral equation with graph (a) for Since f is regular a t R = 0. we may evaluate it at R = O and write the integral equation in the form
P.
Using Eq. (6.77) and writing
0 = (3, c),
the product of propagators in (6.83b)i s
+ Q) = - P - V F ( -~ ka~ +)i q sgn(a -- p ) ) a
G(~)G(R
(6
X
(6.84a)
When g and w go to zero, the ringularitier of this product approach a = p and p = kF. and we may therefore approximate it by the form
where 4(Q) corresponds to the non-singular principal value contribution. The coefficient A(#), which depends on the angle B between and $. may be evaluated by
6.3 MICROSCOPIC FOUNDATION
317
+ 0) over 0 leading to
integrating G(@G(R
(B. 84c)
-
Let us first perform the integral over e. We note that at the poles sgn(e P ) = sgn(g kF) and sgn(c W p) = sgn((E iji k ~ ) Thus, . ifthe quantities @ k ~ and /k+ql kp have the same sign. the two poles of (6.84~)lie in the same half plane so that tho integration contour may be closed in the other half plane yielding zero. Therefore, the only contribution arises when q kF and 4 kp have opposite signs. We first treat the case that 6 = qkeose is positive. Noting that
-
-
+ -
+ -
la + -
-
a
we observe that q
- kp and I&+ - kF have opposite signs if
so that
.
kF-kcos8
Closing the e integral in one half plane and evaluating the residue yields
Similarly. for the case cos B opposite signs is
< 0 . the conditions fw
(g
- kp) and [g+
q+kcosB-kp
- k ~ to have (6.87)
which again yields the result Eq. (6.86). Thus, the product of propagators may finally be written
where 4 denotes a unit vector in the direction of becomes
a and the integral equation (6.83b)
)
THE LANDAU THEORY O F FERMi LIQUIDS
318
W here
d 4 =~deq2dqdh2 and df3 i s the element of solid angle in the q^ direction. We are now in a position to study the singularity of I'at = 0. In order t o take this limit, because of the kernel of (6.89a). we must specify how the w -+ 0
R
and W
g --+ 0 limits are taken,
+0
since they do not commute. Let us first consider the case
and define
rw(P1, P2)= wlime 0 k-0 lim r(P1,P2;R) .
(6.90)
This limit requires that momentum be strictly conserved while allowing for small energy transfer. and we shall see that this limit corresponds to the quasipartiele excitations composing a collective excitation. The kernel goes to zero in this limit
so the integral equation becomes
which we rewrite in the obvious symbolic notation
We may solve for
f
In terms of
rWas follows
Similarly, we may rewrite Eq. (6.89a) in the symbolic form
and substitution of Eq. (6.93) yields
which in explicit integral form is written
6-3 MICROSCOPIC FOUNDATION
A second limit of interest is the case f
-t
319
0. and we define
This limit requires that energy be strictly conserved, and thus pertains to the physical scattering of two quasiparticles. In this limit. the kernel of (6.89a) approaches a constant I. * =-(6.96) WE"
so that
Using Eq. (6.93) as before. we may eliminate f: from this result and obtain the following relation between the two limits:
2
rL(P,,P,) = rw(P,,Pz) --
447%
(2wP @F
rw(P,,B)rk(Q, P2) .
(6.976)
With these results, we are now in a position t o study the poles of I'(P1, P 2 ; K) correspondkg to the tow lying, long wavelength excited states of the system. Since we are interested in the lowest tying excited states of the system, in addition to taking the limit as 2 goes to zero, we shall restrict and Pz to the Fermi surface and use the notation
where $l and hp are unit vectars, From Eq. (6.921,we obsewe that P'" is a regular function. and therefore when k goes t o zero, it i s negligible compared t o l? which becomes infinite. Thus, l'""' may be neglected in Eq. f6.94c1, and we obtain
B
only enter Eq. (6.99) as parameters and by Eq. (6.981,the only Note that Pa and variable in PI is the unit vector $1, so we may rewrite Eq. (6.99) as the eigenvalue equation
Finally, introducing the function
THE LANDAU THEORY OF FERMI LIQUIDS
320
Fig, 6.2 The geometry aX the momenta In the vertex lunctfon (a) detinlng t h e quaslpaxtlcla Interaction !s shown In (b), and dividing through by kvF yields
which, when the suppressed spin sums are restored, is identical with Eq. (6.6laf for the cot!ective oscillations of the system In Fermi liquid theory, In adbltion to justifying the fundamental equation for the zero sound modes, the derivation pravides a microscapic definition for the quadparticte interaction
The geometry of the momenta defining rW(p", 3) is sketched in Fig. 6.2. The vectors p' and B + g are on the Fermi surface in the direction $, a and + E are on the Fermi surface In the direction 4, W know Further from our restriction to the pzlrticcle hole
a
channet, as rtiiflrrcted in propagator pofes on opposite sides of the real axis, that one of the mornenta in each direction is a pzs~iclestate and the other is a hole state, Thus, the vertex function simply describes the scattering of a particle hole pair at one point on the Fermi surface t o any other part. The two factors of a, the residue of the quasipartlcle pole, in Eq. (6.100) represent the amplitude far the intermediate state i n the integrat equation t o be in the simple two-quasipart"rc1e state which has a pole, rather than in some complicated non-singular background configuration. Finally, let us consider the physical scattering amplitude for forward scattering of quasiparticles on the Fermi surface. Whereas represents virtual excitations with small energy transfer, descilbes the physical forward scatteilng amplitude (gl, & -+ in which collisions involve changes of momentum with strictly conserved energy. Using Eqs. (6.97b) and (6.100) and displaying the spin dependence explicitly we may =press Tk in terms of f as follows
s2)
rL
rW
where ail momenta are oa the Fermi surface with direction denoted by unit vector F,. Just as we haw parametrized the spin dependence, Included the density of states at
6.3 M1CROSCOPIC FOUNDATION
321
the Fermi surface, and Legendre expanded the quasiparticle interaction f m* kF -p-f ($v,fu')
= F (9,p") =
where cos6 = fi
+ 4rr
Q'@($,
fit)
C FcPe(eos6 ) + 4~ C ZePe U'
(COS6)
fit, we write the physical scattering amplitude Fk as
Equation (6.101) then becomes
from which we may solve for the tegendre coefficients
One useful application of these results for the scattering amplitude is t o obtain the physical constraint on the Landau parameters implied by the Pauli pri'nciple. By antisymmetry, on* can show that the following limit of the scattering amplitude for particles with parallel spin must vanish
which, from Eqs. (6.102b) and (6.103b) yiefds the Landau sum rulet
The limit in Eq. (6.104a) i s crucial. In addition t o I'k(po,p,o'). IY (pu, pc') also vanishes by antisymmetry. from which one might erroneously conclude an additional sum rule CL(FL ZL)= 0 which i s inconsistent with Eg. (6.104b). Because of the singularities i n the propagators in (6.83a) one must carefully calculate the limit of
+
as PI -+ W
= 0,
4 which
as shown by Mermin (1967) vanishes when f = 0 but not when
THE LANCIAU THEORY OF FERMI LIQUIDS
322
CALCULATION Q f THE QUASIPARTICLE INTERACTIORI Having seen that the _quasiparticle interaction corresponds to t_he particle-hole vertex function r(PI,9%; K) in the limit -+ 0 with ~ P Iand I = k F e it is instructive to see how the same set of diagrams arises from taking the second functional derivative of the energy with respect to the occupation numbers. To see how the essential features arise, let us consider the following tow order contributions to the ground state energy
E
The first and second order contributions may be written explicitly as
E=
C k2 + 1 C ( I 2/v112) n i n z 1
+
1
-
12
C ' Z ~ + E ~ , S1 ~ + E"12'v134"2 ~ nln2(l-ns)(l-nr) (kf 4- ki - kg - ki)
1234
(6,105b)
2m
where we have used the antisymmetrized matrix elements and factors zppropdate for Hugcnholtz diagrams. (1) denotes the momentum and spin variables f k l , ~ l ) ,and n l is the occupation number B(kF kl) for D -"^ &l.VVe thus obtain
-
which may be represented d2rtgrammat"rcally as
6.3 MICRQSCOQIC FOUNDATION
323
fable 6.1 Fsrrnl llquld parameters far llquld "e, The left column of values for each pressure Is taken from Wheatlsy" scompllatlon (1975). The right column In each case Is taken from Greywall (1983)who obtains dlverant; vatuas of the specific haat and thus m* from uslng a dtverant temperature scare, and th8 Other parameters are scaied aecordlngly. The estimates of gl,are obtatnad from the forward seattsrlng sum rule (6.104b) assurrtfng at! parameters wtth & > vanlsh. is to break Clearly, for a general diagram, the eqect of differentiation with respect to open any propagator which we draw as a paeicte-hole pair enterlng from the bottom of the diagram and differentiation with respect to n2; breaks a second propagator which we draw as a pa&-licle-hole pair entering from the top, The usual time-ordered diagram rules produce the correct energy denom'lnators, S ~ G @ between the Initial and final times of the original propagator, there fs a single propagator of the same momentum in the same direction, and outside this time interval the particle and hole energies cancel* To see how the factor a correspondhng to the renormalization of the quasiparticfe pole arises, consider the ccntribution from the third order diagram in Eq. (Gi.105a) In which differentiation breaks open propagators 2 and 5:
The result is just the bare interaction (TZ(vliZ) multiplied by the lowest order contribution t o the renormatlzation of the quaslparticle pole, Eq, (5.66). Since the overall sign of the factor multiplying v Is negative because of the krmion loop and the matrix eiement and energy denominator are squared, the renormalizat"rn reduces the strength as it must phydcatly. By systematically extending this atnaiysis, one can see are generated by how the full vertex function and renormalization factor (1differentiation of the energy. In practical microscopic calculations of the Landau parameters, one typically performs Infinite summations of diagrams, For example, one may use the G-matrix to sum all two-body ladder diagrams, and make a self-consistent definitian of the single-particle energies 'rn terms of this G-matrix. In the low density limit, as shown in Problem 6.6 it Is possible to evaluate the second order contdbution in Eq, (6.106) analryticalty t o
g)-'
324
THE LANDAU THEQRV OF FERMI LIQUIDS
obtain a claoed form expression (Abrikosw and Khalatnikov, 1957) in terms of the aperimental scattering length. This chapter has only presented a few of the basic ideas and elementary results of Fermi liquid theory, and there is a substantial literature on additional aspects of the theory, A clear introduction to the treatment of transporE coe@eients is given by Baym and Pethick (1976,Chapter 1). Attempts have been made to calculate tandau parameters both in tiquid helium and in nuclear matter [Feenberg, 1969; DickhoR. 1983). The theory we have described for normal Fermi systems has also been extended to a variety of other systems, including solutions of We in fE3aym and Pethick, 19-16, Chapter 21, superffuids (teggett, 1966 and 1968), finite Fermi systems (Migdal, 196323,and relativistic systems (Baym and Chin, 1976).
PROBLEMS FOR CHAPTER 6
The: first four problems a l a a r a t e results di~cussed:in the text: the derivation of the specific heat, an alternative evaluation of the egective n n ~ us s i ~ gGaEilean invmiance, the stabaity criteria for the Landau parameters, and derivation, of the generaf formula for the velocity of sera sound, Problems 2 and 3 w e pwkiculmly important and are designated with &E *, Problem 5 proves that there must alwztys be a sera sound mode, either in. the form of a, denssiky olseiIIation or a spin. oseiflation. Problem 8 evaluates the q ~ a i p w t i c l einteraction in second order perturbatioar theand from it recovem the well-known rwult for the binding energy of s dilute Femi gas. The last problem specialkefs the general resultsj of Fermi liquid theory to one spatial dimension and shows that $he sound velocity is uniquely specified by the spedfic heat. PROBLEM 6.1 SpedPic Meat of a fermi Liquid a) Consider the following integral
where f (e) is Fermi fitnetion
and g(€) e+ym 0. Show
PROBLEMS FOR CHAPTER 6
325
where ~ ( n r k ) the Riexnaon S-fanctian
To obtain this result, define G(€) =
$I,
g(r)de and integrate by parts
Than expand G(E) as a Taylor series mound p, expand the Fermi function as a geomedric series
iend integrate term by term. From the re~ufdfor I, obtain Q. (6.16). b) Use the result in (a) to calculate the specific heat aE a non-interacting Fermi gm. F\Or a fixed chemical patenkial, p, calculate the density and internal energy at temperature T",
Show that t a keep the density constant, the chemical potential at tempmature If mwt, be CL
Uairrg dhk temperature dependence of g , show
c) N a w evaluate the integrd Q, (6.15) for a Fefmi liquid two ways, Using = (6.13b), show as above that to maintain constant density and evaluake
326
THE LAMDAU THEORY OF FERMl LIQUIDS
to obtain Eq. (6.17). An easier nay is t o note that since must integrate to sero, we may shift by an w b i t r w coneCant so Q. (6.15)maJr be rewritten
&
No&* that the change in chemicd potential no longer contntbutes do leatling order a d &g. (6.17)is ~bfilined$mectly; PROBLEM Q,2* Gwtfliaan Invzlrfsnca and the EEectfve Mass Obtain the relation between m* and l"f, Eq. (6.25) by using GaMean invmiance. Consider the energy in a new frame moving at a velocity relative to the original frame, explain why the energy should a) Since the Bnid has been given a velocity inmase by p& relative to the original energy. b) NW, e&uI&e the energy change using
5
5,
+
&E=
ri&n(k) k
a
f (k,k t ) ~ ~ ( ~ ) ~ n ( ~ ' )
LR"
rind Eq. (6,28),where bn(k) is the difference between the occupation numbera of the original F e m i sphere and one whose origin iaphifted by q. Show that the fimt and the second tern yields p& Equating parts a) and b) yields the desked result. Yet anothczr instmcdhe derivalion, is given Zr)r Bayrn and Pethick (1975).
9.
PROBLEM 6.3" Starbfttty Crltcrtol Show that in order for the ground state energy to be a minimum rather than simply stationary, the Landau parameter must aatiafy the conditions F& > -(2L+1) and > -12t 2). Pkst, pmameteriae a general distortion of the Fermi surface by a pofa angle and spin-dependent Fermi momentum kr (8, o), so that the distribution function may be written n(k, o) = B(kF(B,o) - k). Then, to second order,
+
where
The first order change in E - p vanishes by stationarity and the second order change must be positive for stability. Show that the second order change is
PROBLEMS FOR CHAPTER 6
Expand 6kF (B,
U) = CkL(G)PL(cos 8) and
which is positive only if
327
use the addition theorem to obtain
FL and Ztare greater than -(2L + 1).
Zero Sound Mode for General f (k, k') Legendre expand the sero sound solution U(@)= ULPL(coa9) and use the addition theorem to derive Eq. (6.66~) PROBLEM 6.4
CL
W here
Use the properties of Legendre functions to show
/ d" 2
),(_ P?-,
X - S
P,.
(X)
= -~LL'
2L+ 1
(S)
Q L (S)
where the Legendre function of the second kind QL(s) may be written
and
PM-~ ( ~ ) P L - M5 2 1 L=O Thus, obtain Eq. (6.67a) for DLL. and Eq. (6.67~)for sero sound assuming non-zero F. and Fl.
.
PROBLEM 6.5
Existence of Zero Sound Modes
The object of this problem is to prove that in the collisionless regime, a Fermi liquid must have at least one zero sound mode, either in the form of a density oscillation or a spin oscillation. The ingredients for the proof are the following four relations, which we rewrite in a compact operator notation: The kinetic equation, (6.61a)
328
THE LANDAU THEORY OF FERMI LIQUIDS
"She relation of the fomard scattering amplitude to the quwiparticle inter=tioa, Eq,(8,103s~)
The atability condition from Problem 6.3
Tbe vanishing of the forward scattering amplitude, Eq. (6.104b)
Note that each relation (l- 3) h w its counterpart for spin modes with F --, #F and
B -+c, a)
&gin by observing tbak (4) requires thak either B(1) or C(1) be positive
(Weexclude the csrrse of an accidenltal degenerwy B@).= C(1) = O which cauld be eliminated by a slight chaage in pressure,) Hence, we need to show that if B(1) > 0, $hen relations 1 - 3 imply an unidarnped eero sound mode. Identical reasoning would then requke an undamped spin mode if C(1) > 0. b) Define
o
(l
+ F)%
and show LhaG &ations (If and (2) imply that
where
cos e If#sz +B 9
Therefore, there will be aa nndarrrped sera sound mode if H,, has the eigenvalue 1 for same real s betweerr l and m, c) Use relations (2) and (3) to show that the maximum eigenvalue of B , and thus of Hm,is leas than l, Therefore, if we can show th& at aome finite s > 1 the maximum eige~valaeof W, h greater than 1, by continuity there must be a value of s for which the eigenvalue equals 2 and Lhere exkts a zero sound mode, d) Note that the following variational expression provides a bound on the largest eigenvalue X,
PROBLEMS FOR CHAPTER 6
329
where
Sin= B ( l ) > 0, one can find a poLti&iveBa and a B. sufficiently small th& B(zf > .Bo r O for cos@@ < z < 1. Therefore, use the folEowing trial function
with A fixed by the nomalleation conelilioa (6) to show
Thus, s h w that there exists an s such that X, r X , e) Xn $he c a e of both sph and isospin degrees of h d m , what can you say &ouf the existeace of sera souad?
PROBLEM 5.6 Second Order Perturbation Theory for the Ditute Fermi Gas The object of this problem is to calculate properties of a dilute F e m i gm using the second order expans3ion &, ffa,lO6), A8 in PmbXem 3.3 far the d h e e Base gm, we will msurne thak the density is gufficiently low thati the range of the interaction is z~hortcompaed, with &hewavcthngtb, The potent i d may, therefore, be approkrnatd by a &-function in coordinate q a c e or a constant in momentum apace V ( r ) = ObS(r) and in lawe~forder, U is related to the s-wave scattering bngt;h. a by
Cafcufate the cscirttefing bngth to second order in pttrtwbation theoq, remembering thoughout the problem th& with sero raage farces only opposite spins intwact. Although your expression is hfiaite, the infinite t e m a will cancel oud when second arder obsemables me exprewed in terns of a, Evaluate the wmipartkle Interwtian udng Q. (6,106)and show
Note that although tlrk expression hw a logarithaic singrrliul.i%yad B impart ant for intepals of f (B) with regular functions.
=. S ,
id is not
330
THE. LANDAU TWEQRV O F FERMI LfQUtDS
Tha~i,obtain the results
and
Finafiy8use the relation 6 2 = - -P
a~
m dp
which follows from Eqs, (6,2q and (6.32) to find the total energy
This may also be obtained by evaluating Lea and Yang (1957).
h.(6.105) and agrees with the result of
PROBLEM 6.7 Landau Theory "I One Spattaf Dimensiaon Considler a Fermi liquid in one spalid dimension, in which, erne the F e m i sadace is comprised of two points, k t== =fikF.The quwipwticle iateraction in any spin channel is characterised by tiwo values, f (kfi"cr, kFo" anand f(kpa, --kpa" aaad il is convenient t o establish contact wibh aur previous thret?-dimensional notation by defining f (kg,k'o') = f (k,k') 43 - B'+ (k,k f )
+
31
F { ; ) = N(Oli ( f ( k ~k , ~+ )f ( k ~-,k ~ ) ) 1.
1= N(OJZ
(4(k~,
+ + ( k ~ -,b J )
6
where the one-dimensional density of states is E and o represents some internal symmetry such m spin or isospin with degenerwy g. Note thak %hesymmetric and antisymmetric combinations .Fo and Pi correspond to the Rmt two terms in the Legendre expansion (6.10) since P{ ;l (1) = l and P{ ;l (-l) = f l. a) Derive the following results for the effective mass
specific heat
sound velocily
PROBLEMS FOR CHAPTER 6
331
and fomard scatterbipzg sum rule
b) Now, consider the s3peeial ease of no internal symmetry, The only two parametem (3if the theofy fi"o and .Flthen are related by $he foward sc&tering sum rules, so that the specific heatc is uniquely specified by the sound velocity. Show
where c ( o ) and C?) denote the sound velocity and specific heat of the noninterabeding system, c) Express c and Ci/ in terms of appropriate derivabhes of the grand potential. Q, Using the first-order perfurbation Lheozy. result for Q from Chapter 2, that, in oae dirnen~ionthe resalt in p& (bj is satisfied to first order in the potential. Is there an analogous relation between e and CV in first order for high= dimensiorrs?
CHAPTER 7 FURTHER DEVELOPMENT OF FUNCTtONAL INTEGRALS Thus far, fundionaf Integral methods have been used to derive more eficientty, elegantly, or physically, results which could also be obtalned using other ttrchn'iques. This chapter and the next address a range of problems for which functional integrals provide unique fnsfghts and approximat9ons. tn the first section of this chapter, we wiil show how to generate a variety of afternatlve functional integrals for the evolution operator w h'ich give rise to digerent physical approximations in the stationary-phase approximation. In Section 2, we show how to pedarm the (general saddle point approximation around a static mean field solution and derive an expansion in terms of the fundamental ~"rrationalexcitations of the system. tn subsequent sections, we shaw h w time-dependent mean field solutions may be used to calculate transgtion amplitudes between specific states, approximate quantum eigenstates of large amplitude wlfective motlon, study banier penetration and spontaneaus fission, and evaluate the asymptotic behavior of far@ orders of perturbation theory.
7.1 REPRESENTATCOMS OF THE EVOLUTlQN OPERATOR Dlgerent functional rntegrail represntatlons for the evolution operator produce digerent physical approximatbons On the stationary phase approximatisn. For warnpie, the Feynman path integral obtained by inserting complete sets of coordinate states yidds the classical equations of motion whereas the functional Integral derived from inserting Bason coherent states produce the Wartree equations of motion for the Bose condensate wave function. Thus, in this section we will explore the f r d o m available in formulating functknal integrals for the evolution operator and how to exploit this freedom in making physical approximations. To foeus the discussion, we will consider Fermion systems. The modifications to treat Bosons are straightforward.
THE AUXILIARY FIELD Consider a system of Fermions with Wamiltonian:
where Ha is a one-body operamr
and V" is a twebody interaction:
We wkh to evaluate a matrix element of the evolution operator between two
7.1 REPRESENTATIONS OF THE EVOLUTION OPERATOR
333
It is usdul to rearrange the Wamiltontan as follows:
where
a., ,=.:a
,
and VaS,& = (aPtvfrlif * The unphysicat =If-interaction twm group& with the kinetic energy in K arises from anticommuting the creation and annihilatian operators in the normat-ordered twobody interaaion, V , and will ultimately cancel aut of physical observabtes. W now express the evot ution operator containing the two-body Marnilt;onian as an Integrat over one-body evatution operators using a transfsrmalion introduced by Stratonovkh (1957) and Mubbard (1358). This transformation is simpty an operator tbrm of the familiar Gausdan identity. The non-commutitivity af the two terms in H Is handled in the usual way by dividing the Integral Itf G ) in N slices of length e= To order E%, we obtain
'*,
-
Tcr simplify notation. we denote m = fay), n == (86).and v,@,& = Vmn. Then, using the Gaussian integral (5.P79).we obtain:
-9 C,,.
L:
- -- -
6 v
(detc~;:)~/l
fr.5a)
or alternatively, changing the integration variable by defining d,=: V g :;rn.
(7.56) tt is often convenient to go back and forth betwen forms (7.5a) and
(7.55), and we
wlll do so Freelry throughout this chapter. Since this transformation has to be pel.formed at each time step, we intraduce a time index k in er, and write
334
where
FURTHER DEVELOPMENT OF FUMCTIONAt INTEGRALS
U
is a normaHzat;ion constant, given by
Note that since Eq. (7.4) is valid t o order e2, we expect Eq. (7.6) to be valid to order N r" Since we take the limit e --+ 0, it is natural t o repface the time index k by a continuous wdabte t , which runs between t g and t r , and the sums over time by integrals. With obvious notation we write the evolution operator
where there is an implicit summation over repeated indices. The time-ordered exponential E r the continllous generalization of the product over ib. in (7,Ga) and the measure is defined by:
\Ne wilt amit the single particle indices a whenever it will cause na ambiguity an8 rdnstate them for specific equations, h r mample, we wilt use the abbreviated notation
with ~f(gf,ti)
= (*f
-i
(K+@(~)@)P
Ik)
(7.8b)
Thus, the evolution operator is the functional integral over an auxiliary field o of the evolution operator for a one- body time-dependent Hamiltonian h: ( t ) ,defined by
and weighted by a Gausshan factor. As we will subsequently verify, the superscript H in Eqs. (7.8) indicates that in the stationary phase approximation. the single particle Hamiltonian represents evolution In the Xartree mean field, from (7.6). we see that the a,@($)are independent integration vadabtes, and thus the trajectory o,@(t) is not a continuous function. This differs from the case of the Feynman path integral, in which the kinetic energy term in the action. d t i 2 , forcer at! relevant trajectories to be continuous, but non-diferentiable, l e t us also note that
l
7.1 REPRESENTATIONS O F THE EVOLUTfON OPERATOR
335
the form (7.7a) of the Hamiltonian implies that a(t) is a classical field conjugate to up^, and similarly i n Eq. (7.5a). a is a fiefd conjugate to fi. There is a large amount of arbitrariness in the choice of the reanangement of the Hamiltonian. For example, had we written:
with
1
Ra* = Tab + 5
Vabbb
(7.9b)
we would have obtained an alternative functional integral:
with
w,'(tl,
h) = ($g lT[c
-c
f'y d ( a a p - q l &
(g)r,p&)bab
] I&)
(7. 10b)
where the superscript F denotes evolutlan in the exchange or Fock term of the mean field. Similarly, writing
W here
= aiai and As7 = a-,=&
(7.11b)
one obtains:
W here
and the integration is over pairs of complex fields X:@($) and xap(t) In this case. as denoted by the superscript P . xZb(t) and xab(t) play the role of cfassical pairing fields. Although each of the functional integrals (7.8). (7.10). and (7.12) represents the exact evolution operator and reproduces the correct perturbation expansion for all observables, in the stationary phase approximation, they produce the Hartree, Fock. or pairing mean fidd, respectively. As shown in Problem 7.1, there is additional fiexibility in representing the evolution operator by taking combinations of these rearrangements of the Hamiltonian, which may be exploited to obtain Hartree-Fock or Hartree-FockBogotiubov mean fields. In physical applications, the choice of the auxiliary field representation should be guided by the physics of the problem. For simplicity, in derivations in subsequent sections, we will use the Hartree form. (7.8).
336
FURTHER DEVELOPMENT OF FUNCTIONAL tNTEGRAL%
IR addftlon to this arlbitrariness in breaking up the Wamiittonian, there 1s further freedom in formulating the auxiliary field Fundional integral (Kerman, Levit and Troudet, 19821, In the limit when t: --,0,Eq. (7.5) may be written
The term linear in Q on the right hand side integrates to rers, since it is odd and the quadratic term integrates to the contribution of order E of the left hand side. The: quadrat-rc term Is actually of wder e because the dominant values of B with the Gaussian weighting factor eifcV@ are of order r-'12. The f a d that the linear term in er integrates to zero altows one to modify its coefficient arbitrarily, Therefore, let us condder the equimtent expression
mm
'mm
where W is arbitrary. Although Egs. ( i 1 3 ) and (7.14a) are identical when the integral over a OS pedormed exactly, they diFFer completely in the stationary phase approximation, As shown in Probiern 2, the one-body evotution operator for a continuous stationary audliary field er(#) is
so that ane may specify the mean field at wilt by the chaice of W. In particular, the yields the Hartree-Fock mean field. Alternatively, one chdlce = v,bTd may choose W to be a Gmatrix, corresponding to the sum of ladder diagrams, or some other physically motivated effective interaction. In contrast to (7.14a). where o E - ' / ~ , the stationary solution ate) i s independent of e so the quadratic term does not contribute to (7.14b) in the continuum limit. Beyond the stationary phase level. the theory does not have a well-defined continuum limit and as shown in Problem 7.2. one must use discrete time slices w"rh finite c.
-
OVERCOMPLETE SETS OF STATES An alternative and powerful method for generating functional integrals is to use overcomplete sets of states. A set af states is overcomplete i f it redundant@generates the Witbert space U - We shall assume that the set of states is conrinuous, and denote it by {(p)).Thus, from its definition. there exists a measure dp(p) on the set {IF)) such that any state 121) of the Hlbert space can be decomposed as
7.1 REPRESENTATIONS O F THE EVOLUTION Q P E W T O R
337
The overcompleteness impties that the decomposition [7,15) is not unique. Examples of overcomplete sets of states are the Boson and Fermion coherent states introduced in Chapter I. Fmm (7,151. Et follows that there exkts a decomposition af wiQ:
and this is the bask formula to construct functional integrals, As "t the case of Bason or RrmSon coherent states, we will assume that the set (!p)).is parmetrized by a compkx variable p, and the adjoint set ({pl) by the canjugate variable p*, The measure must be a function of p* and p and we shall write*
By repeated insertion of Eq. (7.16bf in the Trotter Formula
where c = the matrix element of the evolution operator between two states Ipi} and lpi)which betong to the overcomplete set is given by:
If the measure "y(p*(p) decreases fast enough when lp"p1goes to infinity, it Is legitimate to reexponentiate each matrix element in ( K M )using:
where we use the notation:
S By making a gauge transformation p* -+ e*-"p*. measure may depend only on the combination p*p,
p
-"
e"ap, ome sees that the
338
FURTHER DEVELOPMENT OF FUNCTIONAL INTEGRALS
Using the definition 164k) = rewrite (7.18) as:
- I&-l)
and using continuum notation, we may
W here
Es the functional integral measure, and
is the action. Mote that as in the coherent state functional integrals discussed in Chapter 2, the continuum notation is only a convenient shorthand for the original discrete expressions, The trajectories have no reason to be diRerentlabte or even continuous, and thus, the time derivative only represents a finite difference. = Also, although and M, are integrated over. and p;, are held fixed. the vailables Let us note that when expanding (7.20a) using the stationary-phax method, we shall seek stationary solutions of the action (7.20c), VVe shall use only solutions which are eSther constant or diffcrrentiable with respect t o time, In that case, Eq. (7,20c) becomes:
and the stadonary-phase approximatbon thus reduces to minimizing this action with respect ta the wave functions jyst) and (pet. To iliustrate this method and provide the background for the use of Slater determinants for the many-Fermion problem, it is useful to examine several simple examples. The simplest application of' this general formalism 1s in the case of Boson or Fermion coherent states. As shown in Problem 7.3, it is easy to see that Eq. (7.26) reduces to the coherent state results obtained in Section 22, WIE? next write the one-body problem as a functional integral over an overcamplete set of wave functions, The set of all square integrable wave-functions is identical to the total Wifbert space, and it is thus an overcomplete set. The closure relation in the Hilb-ert space is given by
339
7.1 REPRESENTATIONS O F THE EVOLUTlON OPERATOR
where the measure D(V,$)is equal to constant chosen so that
+ n,
d+*(jt)d+(jt) and W is a nmalization
The dorun relation is easily proven by taking a matrix element of Eg. (7.22a) between (E) and I@). and noting that with the normalization (7.22b). the contraction b(q$*(j)is equal to 6(8 3). Therefore, equation (7.20) yidds
-
where
PP (+* (P*t), +(g, t)) =
n
{e-(*(t)i*'t"(tb(t)
l+(t))~(tt*, $1)
(7.236)
ti
.
As in the previous case, this relation is proved by calculating the matrix element of (7.24) between the antisymmetrized states (xl z ~andl lyl yM), and showing that it b equal to det ((xi (yj)). Since it is often useful to consider Slater determinants composed of orthonormal wave functions, one may use the alternative closure relation
...
...
where U is a nomtafired constant. The details of the derivation of Eqs. (7.24a.b) are outtined in Probtem 7.4. Using (7.24). the functional integral becomes
W$i =
[
PP (P*, P) clet ( ( ~ a~~ ~j ))l ( t eiSf
340
FURTHER DEVELBPMEPfT OF FUNCTIONAL INTEGRALS
B8
~P~P*,P,=
n:
@
- Id.'pacs"'l'~(p: ( t ) ,p. ($1)det (p, ( t )lpfl( t ) ) (7.25b)
a=l L j < t < t j
and S is given in 67.20~3. In the case of continuous trajectories, the action (7-21) becomes:
It Is shown In Problem 7.5 that the stationarity of S with respect to the Slater determinant [p1 yields the time-dependent Hartree-Fock equations oF motion. Con-
...v M )
sequently, stationary phase evaluation of (7.25a) leads to the time-dependent HaPtreeFock approximation, and provides a simple, strakgfitforuvardway to consistently include direct and e x c h a n ~terms in a mean field approximation.
7.2 GROUND STATE PROPERTIES FOR FINITE SYSTEMS We wilt now perform the camplete stationary phase expansion around localized time-independent solutions. For simplicity, we will use the functional integral (7.7) in terms of the auxiliary @-field, which will yield the Wartree approximation and a loop expansion in terms of Nartree phonons. It is possible ta do the same catccllations using functional integrals over Slater determinants, fq. (7.25). which yield an expansion around the Wartree-%ck solution, This is cumbersome, due to the lack of a continuum Ilmit, and is discussed in Problem 7.6, We will also defer for the moment the important physical question of the vatldfty of the stationary phase expansion and the nature of the small parameter In which one is expanding. After seeing the structure of the resulting theol?g, we wit! return to this question in Section 7.4.
THE RESOLVENT OPERATOR All the informatian about the dgenvalues of a Hamiltonian H and the matrix elements of an operator Q In eigenstates of H" i s contained in the resolvent operator:
where q is positive infinitesimal. Denoting by Em,we see. that:
1%)
the eigenvectors of L-f with energies
7.2 GROUND STATE PROPERTIES FOR FINITE SYSTEMS
342
Thus. the eigenstates of H are poles of Go(E). and the matrix elements To evaluate the ground state energy of a finite Fermion system, we may choose 0 = 1 and cafculate
(@nIOIQn) are the corresponding residues.
where we have used the Fourier transform (3.31b). We now express the evolution operator in terms of a functional integral over an auxiliary field. For notational simplicity, it is convenient to use the form (7.5a) in which U-l occurs in the Gaussian factor and no U occurs in the one-body operator. Thus, we write
W here
and
It is convenient t o evaluate trU, in the basis of eigenfunctions of U ., Let us denote this basis by keeping in mind that it depends on a. Since U , is the evolution operator of a time-dependent one-body Hamiltonian, its eigenstates are Slater determinants. composed of singre-particle eigenstetes of U,. We will denote by (lek)) the set of orthonormal single-particle eigenstates of U :,
and an N-body eigenstate of
U' is given
by:
N
with eigenvalue e - ' x i - 1 ahi('*T). In the N-particle space, the unit operator 1and U , may be represented as:
and
342
FURTHER DEVELOPMENT OF FUNCTtONAL INTEGRALS
By its definition (7.28)
U, is a unitary operator:
so its eigenvatues have unit modulus. and the ak(t?,T)are real. Using this basis. the resotvent in the space of N particles is
and the resolvent in the fuH Fock space is obtained by summing over all N. This is the basic formula that we shall use in the folfowing sections.
STATIC HARTREE APPROXIMATION In applying the stationary phase approximation to the (a)integral in (7.32). one should consider general time-dependent stationary solutions c ~ , ( t ) .In this section. we will consider the simplest case of time-independent solutions, denoted GO. The stationarity condition is
For a timeindependent
m. Eq, (7.29a)
becomes:
so that (pk)is an eigenstate of h, with eigenvalues
F:
Taking a derivative of this equation with respect t o QO. and projecting on the left with (pkf.we obtain
which. combined with the adjoint of (7.349. yields
Using the stationarity equation (1.33) and the definition of h.. Eg. (7.28d) we obtain N
7.2 GROUND STATE PRQPERTiES FOR FINITE SYSTEMS
343
or In the (Q representation
This result s h w s that Q Is the Warlree mean field generat4 by the Slater determinant !pLa pkM). The altwnative quantity b = v-"@ introduced in Eq. (7.5b) conerponds to the density. Note that for each set of occupled states (kg kN) appearing In the sum (7.321, we obtain a different stationaly o. From Eq. (7.34133. we see that the satisly the: self-cons'isknt Haflree: equations single-pavticte wave Functions
...
..-
where the quatltlty Eh
"C=
(m, T)
fZ+ is time-independent and plays tire role of the single-partide energy. Note the presene of the ?V(O), which shifts all single-particle energies, but does not modify the waveFunctions. Using these results, the stationary phascl: approximation to the resolvent, Eq, (7.321, is
(7.38) In this static approximation, the total energy is thus:
Using the single-particle equatlon (7.12a), with the notation [k) lpk),we see that
so that the rotat energy Is:
S44
FURTHER DEVELOPMENT OF FUNCTIONAL, INTEGRALS
EquatSon (7.39~)is the familiar Wartree approximation to the total energy, and arises from the fact that the factor oov-'oa properly conected the overcounting of the potential energy which occurrd in the sum cki. Since one can obtain static selfconsistent solutions for any set of occupieci states (b., kN), we see that G ( E ) has poles at the Hartree ground state and all multiparticle-multihafe acitatbns relative t o it, Let us note at this stage, that if we had used a functional integral representation on the sat of Slater determinants, as in Problem 7.5, the stationary-phase approximatton would have yielded the Wartree-bck, rather than Haeree: approximation, Using the Wartree form of the auxifiavy field, the exchange term tn the enerlSy is one of the corrections included in the quadratic fluctuations which we will now evaluate,
l
xi
.
In order t o calculak the corrections to the stationary Martree contribution, it Is convenient t o evaluate the trace of U, in a fixed basis of Slater determinants where { h ) denotes the set of $uantum numbers of the occupied single-particle states, Since the basis is fixtsd and Independent of s,we can no longer use Eq. ('1.32). but rather must use (7.28b)i which becomes
G(E)i=
C G{L)(E)
(7.4Oaf
W here
& f $f:;,
d@@@-'@+ln(@{rjl~m~*(r))
M
.(7.40b)
The expansfan must be perfarmed for each set of quantum numbers { b ) and we will show how it works around one particular state To optimize the expansion for this specific state we define the fixed basis to be Slater determinants composed of eigenstates of the Hartuee Wamiltonian in whkh 00 is degned self-consistently for the specific occupation numbers {k). Thus. is a self-consistent determinant of Wartree dgenknctions which we shall denote as Note that for all the other occupation numbers {P))which we are not trying to calcu/ate, the Slater determinant IP(kl))is not a self-consistent Hartree wave function and the expansion is not optimal. W make the change of variable
(@d.
where oo(r)is the Hartree mean field corresponding t o the term {k) that we are expanding. The contribution independent of' generates EH,the terms linear in q vanish, and we obtain
7.2 CRBUND STATE PROPERTIES FOR FINITE SYSTEMS
345
where the notation istands for (rr,tr). The q-integral is rewritten as: @-
$S
/:""&dr's(rt)r-' frt,r't')q(r'tt)+l'(v)
(7.42a)
where
and
Using the definition Eq. (7.28c.d) of U, and the result in Chapter 2 that the derivative
oT the logarithm of the evolution operator generates connected Green" ffunctions, we abtain
= (-i)n(@~ ITU,@(f*+)p^(l)
.. P ( ~ ) I * H ) .
where the Heisenberg representstion density operator is defined i(r% t) =
,ithe,
6, -ithe,
(I,@ (g,$) = ,-iThfla
(7.436) (7.43~)
h,, is the Wartree Hamittonian, and c denotes a connected apectation value. W@now evaluate the quadratic fluctuatiaons given by
The derivative appearing i n IT" may be evaluated as follows, using Wlck"s theorem in the Slater determinant !gN)
346
FURTHER OEVELQPMENT O F FUNCTIONAL INTEGRALS
where $t(l) is a creation operator at (rg,tl) in the Heisenberg representation of the Mai&ree Warnittonian Jt (1) = Cih~ota j t (,l),-ih.,
Note that in the limiting case of ctquai time,
t, ,
$1= t1 - e . Eq, (7.45a)
Denoting by G(I, l') the Hartme Green's function
and by
Do the Ijartree particle-hole
we see that iftl
propagator:
+ti:
and the quadratic fluctuations, Eq. (7.44) may be written
Using the identZty detA = expftr ln A) we write
The nthterm in the ttxponent has the exptfcit form
and can be represented by the diagram:
(7.456) becomes equal t o
1.2 GROUND STATE PROPERTIES FOR FINITE SYSTEMS
947
This contribution is the nth order ring diagram that enters the RPB approximation. Note that the factor Is just the symmetry factor of this unlabelled ring diagram as discuswd in Section 2.3. The n =. 2. term, which Involves equal time propagators, 1s calculated as fot!ows using Eq, (7,45c]
&
The n = 1 term thus maks two important corrections. It removes the unphysical reif-energy $ v ( ~ )from the operator K. and it provides the exchange energy. which was absent at the Hafirm tevd. The sum of all the ring dhagrarrts far n 2 2 i s the usual expression for the RPA energy with direct ring Qagrams.
- g). Mote that the overall contribution to the energy whert? V denotcls 6 ( t - ~9)vfr of an nth order ring diagram is i"* fiGiGo)n,in agreement with the factor (-i)- (-l)nr specified i n Table 3.1 since the number of loops is nc = n and the symmetry factor is S =. 2n. Collecting together Eqs. (7,4'1), f7.48) and (7.49). we see that when quadratic curredions are included, the new poles have energy:
or "I terms of dhagrams
This result begi~st o reveal the physical content of the expansion. Physically, we expect the fluctuations around the mean field to describe the fundamental collective excitations of a system, such as giant dipole, quadrupole, or octupofe vibratiorrs On a finite nucleus. Furthermore, \from the study of response functions in Chapter 5, we know that the RPA phonan is the leading approximation t o collective excitations. Thus,
348
FURTHER OEVELOPNEFJT OF FUNCTlONAL INTEGRALS
it Is physically reasonable that the principal e8ect of the quadratic corre-ctions is t o include the contribution ts the ground state enera of a single loop composed of an RP&
phonon. In higher order, we shall see that the eFFects of multiple eollwtfve excitations are included. Since the tzxpslnsion Is systematic and equivalent t o perturbation tbeov, along the way all of the other physics besides coltective eftects OS also built In. We have seen two concrete examples in second order: the $ v ( ~ )term and the exchange term, Note that in the present Formutation, since the stationary solution is the Mavtrw approximation, even though the direct and exchange terms appear in the energy, the propagators are Xartree propagators and RPA phonons haw only direct ring diagrams. For some systems, this is physically Inappropriate. For example, in the case of nuclear forces, the exchanm term in the potential Is much more attractive than the $Oreet term and the Wartree approximation with a realistic egectlve interactian does not even produce bhding. In this case, It ts neeessavy t o use a functional integral representation which yieitds the Martretl-Fack appraxirnation at the stationary phase levet and sums direct and exchange ring diagrams, One way to do this using a functional integral over Slater determinants is discuss-ed in Problem 7.6.
TW E LOOP EXPANSION
W now evaluate the cowwtions beyond quadratic order in (7.42) by perturbation theorlr, Using (7.43). Y ( q ) may be written
where! ( ), again denotes the connected wpectation value in the Hartree state. Expanding the titxpectation value by means of Wick's theorem, we note that, the completely linked contributions correspond t o closed loops in which $t(nl) at one point nl is contracted with $(m) at one of the (n- 1) remaining points m.$l(*) is contracted with (Ir(ns)at one of the (n-2) remaining points ns and so on until +f(n,) at the last point n, is contracted with $(al). These (n - l)!contractions contr2bute eglratly to Eq. (7.51)" and using the Haeree propagator G(m, n) = - i ( ~ $ ( m ) $ (m)) t as before. Y (q) may be rewdtten
Diagrammaticatly, V"(q)may be represented by the vertices
where a solid line denotes a Hart;ree propagator and a wavy line denotes an varlat>le. Fhnlly, to perform the pevturbationi expansion, we need t o compute the qt propagators, that is, the contractton of two rj variables, Taking the quadratic term of Eq.
7.2 GROUND STATE PROPERTtES FOR FlNlTE SYSTEMS
349
(7.42a), the cantraction Is defined
The relation l?"' = -$V-'[l- V Do]. Eq. (7.46~).may be inverted and expanded as fotluws l? = i(X - If D@)-'V
Mle recognize VRpA as the prapagator of the RPA phonon introduced in Section 5.4 and its expansion (7.5k) may be representd diagramaticalljr as f o l k s
The diagram expansion for Z is now straightfomard to generate. Expanding c V ( q ) yields any number of dosed hrmian loops of the form (7.52~)with three or more wavy lines emanating from each loop. The Gausrian integral Dq c- $ J d ~ d ~ ~ [ i ) r - ~ [ connsts these wavy lines with propagators ikPA in all pasiFibie ways. By the link& cluster theorem, only the comptetdy tlnked diagrams contribute to the total @new* Thus 6 E(@ + h Linked diagrams (7.54)
2 tT
"r-oo
.
>
Each krmlon loop has n S w a q Ilnes and cantributes a hctor +C(12)~(29). C ((n lln). The total number of wavy lines *must be even and these tines are Joined with ppagatws iIrRpnfnnt)in all possible ways such that the entire diagram is linked. For a diagram with m Fermion loops, there is a factor Tkc?csxpansion is organized according to the number of WPlA phonons. That is, the "loops" of the loop apansion are the wavy phonon propagators, not the closed Fermion propagators. Far example, at the two phonon lml, there are two distinct graphs*
..
-
5
Expansion of the RPA phonon propagatw in these graphs yields an infinite seguence of diagrams in the bare interaction of the fotlawfng genedc farm
@*:o;; $-a-.
Q-:Q--
I)
t - if-
~ ~
350
FURTHER DEVELOPMENT OF FUNCTIONAL INTEGRALS
S'lnzHerliy, the thrw distinct threphonon graphs are the following
The final result Is an =act reainangement of perturbation theoiry in which the basic elernene are drwsed Harlrw particles and RPA phonons. Such an e#ective field theory in pafticle and vibrational degres of frwdom Is useful in understanding the iow-lying states of nuclei, for example, where the fundamental exdtations are singte part;kcte and vibcational modes (Reinhardt, 1978)" 7b understand the loop expansion in detail, it Is tnstructlve to derive the precise conespondenctz- between the diagrams contributing to Eq, (7,541 and the expansion of the ground state energy in kynman diagrams as outlined in Problem 7.7, To conclude this sitlctfon, it Is important to note that the stationery phase expansion we have presenbd here for the ground state may also be carried out a t finite: temperature. Problem 7.8 outlines the evaluation of the partition functian at finite temperature and shws how the ground state properties are obtain& in the zero temperature limlt.
7.3 TRANSITION AMPLITUDES In contrast to the treatment of the ground state in the prevlous section, the stationaw phase approxinat9on to an arbitrary transition amplitude is in general timedependent. We will now show how a time-dependent; mean field approxlmat'ion is obtain& for general transition amplitudes and $-matrix elements, Ear Brevity, we treat only the auxiliary field path Integral hare following closely the work of tevit (f980)and present a detailed =ample udng coherent states in Problem 7.9, Using the form of the Mubberd Straitonovich transFormation in fl,fibf, the transiand final state (@$l may be written tion matrix element between an initial state
W here
and the effsttve action is degned
Note that as in Eg. (7.81, we have suppressed the spatial labels and for example
The stationary values of the auxiliary field a,fs, t f are obtained from the condition = 0,with the result
This result i s a self-consistent quation for a., The solution o, is a matrix element of 1 taken between the states (I,o(t,ti)lgd) and (gf IU,, ( t l ,t ) . where the evolution operator U ,, defining these states evolves IfPi) and ( g t ( in the mean field specified by oe. We thus have a form of mean field approximation in which the one-body mean field depends upon the initial and final states defining the transftion. If the initial and final states are momentum eigenstates,
then If oo(z,t ) is a soliutiotz, a,(z
a,(z
- or,t ) = m
,-iQa
-- a, 5;)
is also a solution:
, i P a ~ (z),-iPa,iPap^(z, t)e-iBa c i P ~(,),-iBa,iPo ~ cr;?
@E-
( q f l e - i P a ,iPau
@a
(%)@-iPa
(*rIUbc(~--'~)P(z- a,t)Ue,(r-- a)lWi)
1%)
1%)
(gIIue(s - a)lQi)
(7.S%)
Hence, summation over all stationary solutions yields momentum consewation
Energy eonservatfon follows analogously from t h e translation invadance, Since the self-consistent rioiution a,(s,L)is not newssarify real, the evolution operator n e d not be unitary. Hence, it is o f k n convenient to make an alternative stationary phase approximation by varying only the phase of Eq. (7.57a). that is. wrying ReS,ft.fa),with the result
Now eonsidw the irnpoPZant special case in which the initial and final states are Slater determinants, which we write in terms of singte-particle wave functions as
352
FURTHER DEVELOPMENT OF FUNCTIONAL INTEGRALS
If we define solutions to the time-dependent SchrBdinger equation in the mean field generated by the real a, of (7.60)
with the boundary conditions @k(~,ti)=&(~)
$k(~~tf)z6~(~)
(7.6kc)
then
Hence, the stationarity condition gives ment in the mixed basis {$){$l
%(S,
t ) as a simpte determinantal matrix ete-
where the denominator is easily seen to be time-independent. It Is instructive to see the relation of this general stationary-phase approximation result t o the time-dependent Wartree approximation (which is the analog of timedependent Martree Fock with the Waflree form of the auxilkary field). Obsewe that with o;: real, the stationary-phase approximation
has modulus .f 1 and the equality only obtains if the final determinant equals f(+k(z,tf))) multiplied by an irrelevant phase factor. If the final determinant is equal to that obtained by evolving the initial determinant with a,($),then the two sets defined in (7.61) coincide. By (7.63). c,(z, t ) = of wave functions { # k ) and C, t)lZ so that the determinant evolves with the time-dependent Hartree equation. Thus, 'if we consider the evolution of any determinant in the stat'ronary-phase approximation, the most probable final state is found by solving the time-dependent NavLvee equation. The transitbon t o any other less likely final state may be found by the general result (7.63). and it is physically clear that the mean field o,(z, t) which best approximates the transition must depend upon the final state.
I#k(~,
{h)
S-MATRIX ELEMENTS The basic idea of how this approach can be applied to the calculation af S-matrix elements Is illustrated by the calculation of the response of a many-Fermion system to an external potential (Alhassid and Koonin. 1981). The S-matrix may be written as the limit of the following evof rrtlon operator
7.4 COLLECTIVE EXCITATIONS AND TUNNELIN G
353
where G denotes evolution under the many-body Hamiltonian f 7.1) and Udenotes emlution under the many-body Hamiltonian plus an external field @ = 858 Wag(t)baB where W ( t ) -.. 0 as t -+ &W. Each evolution operator may be written rn twms uf a functional integral over an auxiliary field in the usual way with the result
and
One may view (7.66) as an auxiliary field evolution aperator of the usual form with a time variable whfch runs from 0 to --T without W, from -T to +T with W, and from 2' to O without W , Requlving statfonarity of the phase of (7.66a) yidds the stationary solution, analogous to (7,60),
denotes any one of of,o,or 04 and p inserts j3 into the corresponding interval (-CO), respectively at time t , Since the physical S-matrix is independent of T as 3" --,m, time reversal Invariant i f W(r,tj has time reversal symmetry, and unitary, one may ask which of these properties is also reained by the stationary phase approximation. To see that (f ,457)is independent of T , note for times -t prior to the interaction pedod. (-g) = ~ ~ ( - t ) Is a solution and evolution by prior to the interaction time 1s precisely compensated by l&. Hence, once --If is earlier than the interaction period, it does not enter the: stalionary approximation to the S-matrix. Similady, the mean-8etd solution may be verified to be time reversal invariant if W has time reversal symmetry. However, because the mean field solutiion satisfies a difFerent non-linear sdf-consistent equation For every final state (%l, the approximation for each final sta& Ts obtained by a diFferent evolution operator U,.U@eU,; and the result in general is not unitary. f Formulation of a complete scattering theory inuolves additional technical complications which do nat arise Fn consider'ing the response to an external potential, One must define asymptotk cchand states for the target, projstiile and all the reaction products, and prqject then onto states of specified center of mass momentum. These and other aspects of scattering theory are addressed by Reinhardt (1982).
where
(O,T),(T,-Q, or
We now consider general eigenstates of a many-body system in which application of the stationary-phase approximation to the resolvent operator leads to tlmedependent periodic mean field equations. In Section 7.2, we have already seen the
354
FURTHER DEVELOPMEhtT QF FUNCTfQNAL INTEGRALS
special case of static Wautree solutions which are trivially periodic and describe the ground state and multiple paflicfe-hole excitations relative to Ft. Waving Zntroducd the idea of self-consistent mean field quattons specifying Itime-dependent solutions for specific trandtlan matrix efements in the preceding section, we witl now develop the period'tc mean fieid equaltans descriKng quantum efgenstates of large amplitude collective motion and spontanwus decay of a metastable state fdlowing the approach of Levit, Negele and Paltiel flB0a.b). To Zliustrate the basic ideas, it is useful t o begin with the Eiimpte problem af quantum mechanics in a one-dimensional potentiaat,
E m M P L E WITH ONE DECREE OF FREEDOM We will calculate the eigenstate in a general potential V ( q ) using the resolwnt operator introduced in Eq. (7.28). The resolvent operator may be written as a f"eynman path integral as follows
where
Ss the classlcai action and the tr;ljt?ctory satisfies the boundary condltians
The exact eigenstates are specified by the poles of 6 1 E ) when the thrw integrals over T , g, and q(t) are perform& exactly, Mere, we consider the approximation obtained by evaluating each of these three integrals in the stat-ionary-phase irpproxintation. Application of the stationary-phastc3 approximation to the functional integral over q ( t ) yields (glCsk(t)llq) A e'SI~a(')i where q,(fl') is the solution to the classical Euler-Lagrange equation
subject to the boundary condition (7.68~). The factor A denotes the result of integrating the quadratic fluctuations around q,(t) and its explicit f m is not required for the present argument. To apply the stationary-phase approximation to the integral over q. note that S[q.] depends upon q through the end points of q,lt). Since the dwivartive of the action with respect to the end point yields the momentum, stationarity of S[q.] requires
7.4 COLLECTLVE: EXGITATBONSAND TlJNNELING
355
Fig. 7.1
Sketch of potsnttal V ( q ) wlth a slngla mlnlmum and the pertE between cizlsslcal turning polnts ql and g2 which contrtbutes In the stationary-phase appraxtmatton, odlc trajectory at energy
Thus, both q and q' are equal at the end points and since the classicaf equation of motion is second order, the trajectory is periodic. Hence, the resotvent may be revvritten in the form OC7 ~ T A@e"ET+@SCTl ? E -..a (7.71)
S(q
where denotes the dassicali action for a periodic trajectory of period T and an additional quadratic fluctuation factor has been included in A", The quant'lty W [ T )zz ET S ( T ) appearing fn this and subsequent results is called the reduced action, The derivative of the classical action S(T) with respect t o the final time T yields the negative of the classical energy. Thus, application of the stationary phase approximation t o the remaining integral aver T leads to the condition
-+
where E.(T) i s the energy of a classical peilodic orbit of period T. TAe final stationary phase result obtained by summing over ati periods Tm which yield classical periodic solutions with energy E Is
where A:, denotes the factor obtained by integrating the quadratic fluctuations around the stationary point Tm, The aructure of the final result, Eq. (7.731, is particularly simple for the case of a potential Vfq) with a sEngle minimm as sketched in Fig. 7,1. The classical pedodie trajectoty may bei visualized in terms of the motion of a padcle on a curve Vjq) in a uniform gravitational field. When it is released from position ql it executes periodic clasdcaf motion between the turning points ql and (12 as shown* tfdng energy conservation t o express $$ i n terms of E-V(q), the fundamental period for a trajectory with energy E is
FURTHER DEVELOPMENT OF FUNCTIONAL INTEGRALS
356
Figg,7.2 Sketches of a double we13 potential (a), and the Inverted potential obtained by contlnuafton to fmagtnary tlms (b), The traleccorles contrlbutlng tn the stationary-phase approxlmatlon In the classlcatty allowed and forbidden ragtons are shown In (a) and (b), rsspectlvely. Any integer mult'lple of T(E) atso satisffes the stationailty condition, so that Tm mT(E). 1C is conwnient t o write the reduced action in (7.73) in the following form
. ;
where
If we temporarily ignore the dependence of the quadratic fluctuation factors multiple m, Eq. (7.73) yields the follwing series
ALL on the
which has pales for all energies satisfying the quantitation condition
As s h w n by Gutzwiller (1918), the factors A' turn out to contribute the same magnitude for each m and a phase of for each classical turning point. so their inclusion simply changes the minus sign in the demoninator of Eg (7.76a) to a plus and thereby yields tha! m r w t Bohr-Sommedetd quantization rule
5
(2n
+
l)lr =.: VV
(T(E))=.
(7.7&)
The lowest state In the well thus acquires Its proper zero-pslnt energy in this approximation.
7.4 COLLECTIVE EXCITATIONS AND TUNPIELING
557
An important new feature arises in the analogous treatment of the double-well potential sketched in Fig. 7.2, General application of the saddle-point approximation to the time integral in Eq, (7.71) requires summation over isolated stationary points in the campiex I' plane, Whereas trajectories in classically altowed regions yield real stationay paints Tmas discuss& above, we will SW that classically forMdden regions introduce complex statiionarlg points, To be specific, we will treat an enera E betow the barrier hdght E1 and consider the three regions f, I!, and Ill indicated in Fig. 7.h. The analysis of periodic traectories in the classically allowed regions t and tlt is the same as for the single well consider4 abave, We will denote the fundamental periods given by Eq. (7,741 with the appropriate end points by and RxI and the corresponding reduced actions given by Eq. (7.75) by and WxI. In the classicalfy forbidden region !l, the existence of a classkaal periodic solution with purely "taginary period may be understood by conthnuation of the clasdeal equation of motion t o imaginary time, Reptaclng (it) by r , Eq, (1.69b) may be rewritten
Since continuation of this second-order equation introduces an overall minus sign, a picturesque way to visualize the dationary solutions is t o conslder the classical solutions in the inveeed potential sketch& in Fig. 72b. Calculation of the stationary solution in region II corresponding to one classical asciliation from qz t o q3 aad back and repet"rtjon of the steps performed above for real solutions give rise t o the period
and a contvibution t o the resolvent of
e-Ir(rfg.
where
QS
{2m [V (p)
- E])'l" dq
.
(7.78b)
2
A general periodic trajectory i n the double well of Fig. 7.2a is thus composed of any number of closed orbits in each of the three regions connected in any order, so that the stationary phase approximation to the resalvent is
The technical question of phases in the quadratic conection factw A;, is addressed by Bender e t al. (1978), with the result that A;, is a constant times ( - I ) ~ + ' + ~ . Calculating the sum over .all trajectories beginning in any of the three regions and containing all combination of cycles in each re@onyields
358
FURTHER DEVELOPMENT OF FUNCTIONAL INTEGRALS
To understand the physical mntent of this result for general double well, it is instructive to examine two specific special cases. We first consider the symmetric double well, for which = WIII, and will recover the familiar WKB expressbn for the spl"rtfng of nearly degenepate even- and odd-parity states. in lowest approximation, i f the central barrhr were very high, the problem would reduce ta a degenerate pair of single-well problems, yietdhng degenerate solutians satisfying Eq. f 7 . 7 6 ~ ) :
where E:') denotes the zeroth approximation to the ntheigenstate. This result follows imnedizrtefy from Eq, (7.80). since in the case of a high barrier, VV;I is vey large, smal, and the demoninator reduces to double poles at rendering e-m"exponendally In next approximation. we may write E, = FP,+ &E, and expand the energy E&@). condition for a pale in Eq, f 7.80)
to first order in E",. Using Eq. (7.81). noting from Eqs, (7.74) and (1.75) that
and observing that C - ~ ~ ~ ( ~ :is)second-order AE, small, we obtain
which yields the familiar WKB result for the energy splitting
Thus, In this application, the physics of tunneling in the ctass!calIy Forbidden region, has been described by the periodic imaginary time salutions corresponding to cfasskal motion in the "tnveded well. A second example is the decay of a metastable state. In thSs case, the right well is distorttd to extend to the edge of an arbitrarily large narmatization box as sketched in
359
7.4 COLLECTIVE EXCITATIONS AND TONNELING
Fig. 7.3 Sketchss of a potcantlat for spondlng fnvsnad pQtanEfal (b),
a metastable state! (a) and the! carre-
Fig. 7.3a. tnstead of calcutating the resolvent, it ts convenient to evaluate the smoothed level dendty (Batlan and Block, $974)
where the finite width 7 is smaller than any physical width in the problem but larger than the level spacing in the normalkation box, In this ease, we obtain periodic stationary salutbns in region 1 as before, and in lowest approximation these yield the result Eq. (7,7&) for the energies of the quasistable states, Also, as in the case of the symmetric potential, periodic Imaa;inary-dme trajectaries are obtain& En regfon It, corresponding to solution of the classical etquations of mation in the Inverted potential sketched in Fig. 7.3b. The role of periadlc solutions in region ftf Is quite dltRFerent In the present problem. Since
~ not contribute in the limit as the length sf the normalization and 7 is finite, e - ' e ~ daes box, A, goes to infinity. Thus. the smoothed density of states has poles at compfex energies @ dEN satisfying
+
1+C
=C - ~ ~ ~ ( ~ O + ~ ~ . )
d ~ g ( ~ O + ~ ~ n )
and expansion to first order in AEmas behrr: yields
vv here Near E-, the level density Is therefore proportional to
(7.88.a)
FURTHER DEVELOPMENT O F FUNCTIQNAL INTEGRALS
360
so that F, 1s the inverse. lifetime of the netastable state. 7"0 within the factor 2, whkh is prwumably corrwted by a careful evaluation of at! corrections t o the stationary phase approxfmatiion. Eq. (7.88~) is recognized as the familiar WKB formula for tunneling dczeay of a metastabte state, These dmple examples from one-dimensional quantum mechanics illustrate the essential features which arise En the many-body problem. Quantized eigenstates in a ctassicatly altowd region are given by periodic solutions to the real-time statbonaryphase equations subject t o a Bohr-Sommttdeld quantlzation condition. Tunfieling In classicaliy forbidden regions is descdbed by perjodk catutions in imaginary time which may btr, thought of as ctassical sotutions in an lnvePted potential, Such lmadnary time solutions for many degrees of freedom were first Intuoduced by Langer (1969) in the treatment of bubbie Format'lon and have been used extensively in field theory (see for example Polyakov, 5977 and Coleman, 1977) where solutions connecting degenerate vacua are called "instantons" and the solution describing the decay of a metastabte state Is called a "baunce"
.
EtGENSTATES QF LARGE AMPLtTUDE COLLECTIVE MOTiON To generalize the calculation of dgenstates in a one-dimensional potential to the case of the many-Fermlon problem, we wit1 use the form of the resolvent and notation given in Egs. (7.28) and (P-32):
where @)rl...kx) is
h,lt) = (If + ~ ( t )fi ) the Slater determinant of eigenstates
{I$k))
I*/%
Tc-i
S,.-
dehe(t)
Equation (7.89~)is equivalent to the boundary-value problem
with the boundary conditions
af l!.&) satisfying
7.4 COLLECTIVE EXCITATIONS AND TtfNNELtMG
361
Defining new single-particle functions vvlth a phase factor removed f7.91a)
(r, t) izi e ' ) ' * ( " * T ) ~ k (r, t ) we abtaln an equivalent boundary value problem
with the periodk boundaw condition
The stationary solullon for the auxlllavy field cro(z, t ) is obtained by requiring that the exponent in the resolvent. Eq. (7.89a) be stationary with respect to variation of o(z, t )
"t a form sufificiently generat to In order ta evaluate the functional derivative apply to both real and imaginary time, we introduce a biorthogonal basis {&k4L)where the dr are solutions to (7.91b) and the J k are determined by the adjoint operator
+Qrk -6#k(r,g) = - T4k(r, as'".
tl
with the normalization convention
Note that from Eqs. (7.91b) and (7.93a) that = ark and dr Jk(r, t)4*(r, t ) is time with respect independent and thus equal ta unity. %king the derivative of Eq, (-7.91b) to o(r', t) and prqecting onto &, we obtain
which yields the desired functional derivative
362
FURTHER DEVELOPMENT OF FUNGT10FJAL INTEGRALS
Mrnce, from
Eq. (7-921, we obtain
This result for the stationary solution oo(r,t ) is the many-fermion counterpart of the pefiodic classical solution obtained ER the statbonary phase approx1mat"tn t-o the onedimensional path integral. Since and $ are periodic. rro is periodic. The form of with periodic then follows hom Floquet's theorem. As in solution $ = e-'a'/T+ the analogous case of Bloches theorem far crystals, in which periodic functions are multiplied by expanentials containing a quasimomentum, the quantity ar/T has the physical interpretation of a quasienergy. For real time, t , it follows from (7.91b). (7.93a) and (7.95a) that &(r, t ) = Qk (r,t ) and Zh == ak. Hence. m(r,t) has the familiar Form of the Martree mean field
+
and the ($ik
(r",
+
E ) )satisfy the self-consistent couplrtd equation
with the bounday conditions that q6 vanish on the spatial boundaries and be periodic in time with perbd 27. One particular solution t o these eqoatlons is, of course, the static Martree solution discuss& in Section 7.2. We now consider the approximate eigenstates given by the poles of the. resolvent when it is evaluated in rht? stationary phase approximatbon. Using the statbnavy solution oo(t)in Eq. (7.89a). and ignoring quadratic corrections far simplicity. we may write
where dt oO(t)v-'cO(t) -
E ax(o0,T )
17.97b)
k
...
hrr each set of occupation numbers (kg km) the integral over period T has the same structure as Eq. (7.71) for the one-dimensional example, and we obtain poles by applying the stationary-phase approximation to the T integral as before. Stationarity with respect t o ";l' yields
.LE=--
d$(a, Tj al"
(7.98) *
363
7.4 COLLECTIVE EXCITATIONS AND TUNNELING
The derivative of ak(o0,T ) with respect t o T is conveniently evaluated by rescaling the time variable according t o t = qT and writing the eigenvalue equations as
- T ( R+ o(r, q ) ) ]4 t ( r , q ) = -&k(c, T ) d k ( r , ~ ) Then. taking the derivative of both sides with respect to T . and projecting with 4; (r,V ) as before yields
dq o ~ ( q ) v - ~ c ~ just ( q ) subtracts ] off the overcounting of the The term $ [$T potential energy as in Eq. ( 7 . 3 9 ~for ) the static case, so that after rescaling back from t o t. stationarity with respect t o T yields the condition
where
is the Hartree Hamiltonian density
Using the equations of motion for 4k and 4;. it is easy t o show that the Hartree energy. EH = X(+;, d k ) is conserved (see Problem 7.5). Equation (7.101a) setting E equal t o the Hartree energy of the periodic mean field solution i n the many-Fermion case i s analogous t o Eq. (7.72) for the Feynman path integral requiring that E equal the energy of the classical periodic solution. Finally, using the expression for E in Eq. (7.101) we find that
since ET cancels all the terms in S ( T ) except those involving the time derivatives of single particle wave functions. From this point on, derivation of the quantization condition follows precisely as in the one-dimensional example. Eqs. (7.73) (7.76). Denoting T ( E ) as the fundamental period which gives rise to periodic time-dependent Hartree solutions with energy E. stationary points i n Eq. (7.97) are obtained for all
-
364
FURTHER DEVELOPMENT O F FUNCTIONAL INTEGRALS
integral multiples. mT(E), with reduced action WfmTfE)]= mVY[T(E)],The same geometric wries, Eq. (7.76a). arises, yielding the quantization condition
The similarity of the quantlzation conditions in the example with one degree of freedom, Eg. (7.76b) and the many-Fermian problem, fq. I7.103) is particularly clear when one observes, as shown in Problem 7.5, that the time-dependent Martree Fock equations are of Hannlttonlan form with z k zi playing the role of the momentum conjugate to so that C, f d t ~ h is 4 ~a natural generalization of f dtp4. At the SPA level, quantum states: of large-amplitude collective mation are fully specified by the self-canshtant equations for periodic time-dependent Wartree solutions, Eqs. [7.%), and the wantization condition, Eq, (7.103). In general, Eqs. (7.96) speclfy time-dependent oscillations havf ng amplitudes which depend continuously upon T,and the quantization condition singles out a discrete set of amplitudes and energies corresponding to quantum eigenstates. The theory represents the simplest availabIe approximation that has all the physical elements of a fundamental theory of quantum wflective modon. AI! the degrees of freedom of the one-body density are accessible, and the dynamical equations and quantization condition arise with no further preseriptions concerning cotlective variables, inertial parameters, or quantization pracedure. Furthermore, corrections to the SPA may in principle be systematically evaluated. Two special limits of these time-dependent equations should be noted, Timeindependent solutions to Eqs. (7.96) correspond to n = O in Eq. (7.103). so the static Hartree states discussed previously are automallcally included. As shawn in Prsblem 7.10, for Infinitesimal periodic fluctuations about the static Wartree solutions, the time-dependent quantized theory reduces to the familiar random-phase approximation
(RPA) .
BARRIER PENETRATION AND SPONTANEOUS FISSION in the case of a singre particle in a potential, the stathnary-phase solution in a classicafty forbidden region was given by the dassical equation of motion in imaginary time and corresponded to cIassical motion in the invert;ed welt, This sectbn presents the anatogous theory for the many-Fermion probiem and applies it to the example: of spontaneous fission. First, consider the stationary solution for the auxiliary field in the case of purely imaginary time, which we write in terms of a real variable r ZE it. Equations (7.91b) and (7.93a) for the biorthogonal basis yield
7.4 COLLECTIVE EXCtrATlaNS AND TUNNELING
365
(r, f)* = 4r (r, -f). Denoting the imaginary time from which it follows that solution 4k(rIr) E #k (F, f) and using Eq. (7.95a) for c0 we may write the selfconistent periodic equations in imaginary time as follows
where Jr(r, 7) satisfy the periodic mndition
and the orthonormalky relation
Several ohsewations should be made concerning these equations. First, although this self-consistent eigenvalue problem is not Hermitian, it has a set of real solutions with real eigenvaiues, and general solutions differ from these reat sdutions Iby the trivial phase factors c'(2wsfl)r and eigenvalue shifts i(2nnlT) where n is any integer. The real solution? will yield the "bouncew solution governing spontaneous fission. Second. note that if +(t, r) is real, the combination 4r(r, -7)4&(1;7) enterlng oo(r,7) may be wiltten in terms of the original function $r (P, t = :) as dr(r,t*)*4b(r,t ) , so that it indeed represents analytic continuation of the real time expression (7.95b) to complex time. In addition, observe that although Jk(r,r) and Jr(r,-r) grow and decay exponentially in time, the combination Jk(r, -r)dk(r,r) remains normalized so that oo retains the physical interpretation of a mean field. Finally, note that the static Hartree solutions are valid r-independent solutions of these equations and in Fact, the bounce solution will approach the static solution asymptotOcally at large t"re, To establish contact with the example of a single particle in an inverted potential, we must express the action for both a single particle and for many fermions in anaifogous Haniftoniian form, First, consider the Lagrangian for a particle in one dimension with a position-dependent mass:
Mate that although our prevlous one-dimensional example had a constant mass term, the more generat form in (7.106) is useful in identifying the structure of the many-body problem. In the usual way, the momentum is p = $$= m ( g ) i and the Hamiltonian is p2 V(p). Thus, the action S = dt L may be written in the H = p i -- l = g) Wamittonian form
+
3Cj@
FURTHER DEVELOPMENT OF FUNCTIONAL INTEGRALS
$1
The action S(+*, = ,f dt(+l which under variation li& - HI$1 . yields the time-dependent Hartre-Fock equat~on.is derived in Eq. (7.21) and discussed in Problem 7.5. The analogous action which yields the time-dependent Hartree equation tn real time is
To obtain a form analogous to Eq. (7.107). we must perform a canonical transformation from the variables i+; and JIk to new canonical variables and momenta whrch are timeeven and time-odd, respectively, Let us choose the time origln at a cfassiical turning point and define $k(r,
$1 S
(7.109a)
where p&, analogous to a coordinate, Es rea! and time-even
and
corresponding to a momentum, is real and time-odd
Transforming the N a ~ r e eaction (7,108) to these new variables yields
where
is the Waeree energy, Eq.. (7.101b). far the time-even single-particle wave function The action & ( X , P). Eq. (7.1lOa). is precisely of the fwm of $(p, g) in Eq. (7.107). where corresponds to the coordinate q. ~k replaces the momentum p. = fi is the coordinate-dependent mass , and the Hartree energy W ( f i , V ( p ) plays the role of the potential, The potential V ( p ) provides a useful way to visualize the timedependent Martree problem. Think of the Wartree energy surface in the multidimendonat space of all determinants composed of time-even wave functions. This surface has local minima at at! the stable Martree solutbons, and in general, for an energy abom that of' a local mlnirnum, the configuration wilt evolve in some classically allowed domain around the minimum, A convenient way to characterbe the gross features of this surface is to
{a)* &a
7.4 COLLECTIVE EXCITATIONS AND TUNNELtPtG
367
Fig. 7.4 sketch of the constrained Hartreo energy E(Q) as a functlon of quadrugole moment far a tlsslle nuclaus, At any value of' e), E(&) Is deCIned as a mlnlntum of the WarErm energy, Eg. (7,109b), 'ln ths space of tlme-@Ven determtnaints havlng quadrupole moment 9.
evaluate the constrained energy of deformation surface defined by minimizing VCp) with respect to all determinants satisfying one or more canstraints. A familiar example to whieh we will return in discussitng nuclear fission is the constrain4 energy curve as a function of deformation sketched in Fig. 7.4. The ground state of a deformed, fissile A-pa&icfe nucleus has a ground state quadrupole moment Qz. tf an external field i s applied to Increase the quadrupoie moment, the energy increases since the cost in supface energy fs greater than the savings in Couiomb energy. For any Q > there exists some minimum value obtain& Iby searching over all determinants of timeeven wave functions having total quadrupole moment 9.and solid curve E ( g ) is the envelope of at! such minima. Beyond the saddle pohnt Q2.which is a stationay point of the action, further increase in the deformation can decrease the Coulomb energy more than it increases the sagace energy, Eventually, at very large Q, the odginaf nucleus breaks into two separated fragments with A12 particles, with an energy lower than the originat energy BE. The regions I, Ifand fll of this figure correspond to those in Fig. 7.3b. Introducing more constraints would generalize Fig. 7.4 to a multidimensional surface in which one coutd visualize the projected motion. Consider now the changes which occur when Eqs, (7,100- 7.10i) are continued to 'imaginary time, We define
and
in terms of which
368
FURTHER DEVELOPMENT O F FUNCT10FdAL INTEGRALS
Wt-itingthe actbn for the imaginary-the Hartrw quations
and transforming to the new variables
3 and as before yiefds
where Slnce the owratl sign of S does not affect the Euler-lagrange: equation, we note that the only difference between the real-time equations of motion for ( x k , p k ) from Eq, (7.110) and the imaginay-time equation of motion for {gk,&) from (7.113) is that V ( p ) and Y ( 3 enter with opposite signs. Thus, the multldimenslonat energy surface ?(p) and the conshined project'ion B(Q)sktatched in Fig, 7.2 are simply inve~ed as In the one-dimensional example, and even in the many-Fermlon problem, we may continue ta think of the imaginary-time solutions as periodic trajectories in an fnvtlrted potentia!, One last obst?wationfadtitatd by the transformation to the ( p , X ) representation is the meaning of the quasiperiodic boundary condition $r (r g) = c-'a* +k (r,'% Eq. (7.90b), and the corresponding condition in imaginary ttne, Using the sym~ = --ak so that metry pmperties. Eq. (7.109). this condition implies 2 2 (r, V x k (r, f = 0. Hence, the kinetic term in the Hamiltonian. ( ~ ~ wnishes r ) ~ . and $ is a classical turning point. With this understanding of the structure of the real and imaginary time periodic solutions, the stationary-phase approximation t o the spontaneous fission of a nucleus is neady analogous to the treatment of the fifetime of a metastable state in the onedimensional potential, Fig. ?.Id. Recall that in calculating the smoothed level density, Eq, (7.%), trajwtorles in regian Ill did not cantribute, and summation of periodic trajectories in regions I and l! yielded the WKB result, Eq. (7.88~). in which the penetrability was gfven by the reduced actbn of the tritjectafy in region if having an enerw equat to that of an eigenstate In region I.Thus, in principle, we should sum all trajedodes in regions I and II of the multidimensional analog of Fig. 7.4. The essential elements in describing fission are the so-called "bounce" whtutions, the peilodic self-consistent solutions to the imaginary-time equations Eq. (7.105a). Like the real-time sohtions discussed in the preceding sstion, the periodic $01 utions in region fl of the inverted wefl of Figg.7.4 depend continuausty on the period T. As T" Increases, the amplitude grows and the energy decreases until as T W, the energy approaches the static cartree energy EH and the solution at the; classics! turning point approaches the statlc Harlree sotution. In contrast to the simple one-dimensional
S),
5)
5) g
7.4 COLLECTIVE EXGlv;4TiOPIS AND TUNNELIEJG
569
Fig. 7.5
Baunce sorurlon for a flsslonlng one-dtmenslrrnal nuclear model. Parr (a) shows density profiles a t evenly spaced time intervals from r = to 0. Part
(b) shows the four
cllstfnct spatial wave fitnctlons at ttme r
-- -5S
and a t r = Q.
example, in general there are several distinct, well-separated bounce solutions carresponding to symmetric fission, asymmetric fission, and more complicated many-body breakup. Each such solution evolves from the Hartree ground state through a saddle point to some distinct configuration at the boundary of the classicrhfy attowed regime, and alt solutions invotving any combination of these traject-ories should be summed in the stationary-phase approxhation. To help vistratize the bounce, the solution for symmetric fission of a model nuclear system in one spatial dimension is shown in Fig. 7.5. The sequence of density profiles p(s,r) .= 4k(z, - ~ ] g ) ~ ( z ,shows r) how the to system evolves from the Hartree solution for the original nucleus at time r = two nearly separated symmetric fragments at r -- 0, Note that although the continuous wriabie 7 has nothing to do with the physFcal time (it is Just a formal variable in the path integral t o deal with the non-commutivity of T and V) the sequence of densities through which the system evolves corresponds to the most probable fission path for a given channe! and is physically significant, Let us now sum all the trajectories at the Wartree energy, BM,for which all the bounce: solutions in region tl join into the static Wartree sol,uPion in region I, Denoting the reduced action 'in region l, Eq, (7.102). by Wr and defining the imaginargr-time counterpart in a specHic decay channel a by
E,
-5
we obtain the foliowing csntrit.lution of ail stationary salutions to the resolvent
370
FURTHER QEVELOPMEMT QF FUNCTIONAL INTEGRALS.
If analogous solutions In regions 1 and fl could be join& infinitesimally above ERE., then by the arguments leading to Eq, (7.88~). one could obtain the total width as a sum of parlcial decay widths = (7.r lea)
C rc.1
W here
Since the joining problem has been solved only at the Hartree Fock energy, the present and the premultiplying factor must be treatment yields only the penetrability c;""', deriW by other means. A derivation of the premultiplying factor using the dilute instanton gas approximation is outlined in Problem 7.11, It fs instructive t o note how the essentiaf physical aspects of the I'lssion problem enter the stationary-phase approximation ta the lifetime. Both the gross competition betwetrn volume, surlace, and Coulomb energies and quantitatively significant singleparlticle sherl e @ ~ tare s inctuditlcl through the evofution af a detwminantal Hartree-Fock wave function, The bouna solution determines both the relevant collative degrws of freedom for the most probable fission path and the corresponding conjugate mornenta. Finally, the competition between alternative decay channels, such as symmetric or asymmetric fission, is manifested 'in the dfstinct wlutions governing each partial width, Cfearty' this functional integral approach to tunneting problems is quite general and has been applld to a variety of physical probtems, ranging from bubble formathn in a pharc transition (Langer 1977) to the structun and decay of the vacuum (Polyakov 1977. Coleman 1977).
CONCEPTUAL QUESTIONS Whereas the applications of functional integrals in this chapter are physicatfy and intuitively appeal1ng, they are subject to limitations which require comment. The most salhet problem is the absence, except in specificaity constructed models, of an expiick small parameter in which to generate! an asymptotic expansion. Recall that the Feynman path integral has an explicit factor of multiplying the action and thus generates an asymptotk expansion in powers of h, Unfortunately, the Havlree or EtartreFock action obtained using an auxll'rary field, caherent states, or an overcomplete set of Slater determinants contabs h In the kinetic energy as well as an overail muftiplkative factor, Hence, to take advantage of the physics of the mean field, one must reflnquish a strict wmicIasslcal expansion in powers of h, A second alternative is t o generate a TIN expansion. One may imagine a class of theories which diffczr from one another by their interaction strength and the number N of internal d e p e s of freedom. In ceHain speclat cases, rquiring that the class of theodes has a sensiMe limit for iarge N spedfies the N dependence of the interaction strength and atlows one to rescale the integration variables such that an explicit factor of N will multiply the action. The resulting formal expansion in powers of 1 / N may then be useful if the physical Hamiltonian embodies a suflciently large number of
7.3 LARGE ORDERS O F PERTURBATION THEORY
3'1%
internal degrees of freedom. A detail& uample is discussed in Problem 7.12. In the case of nudear physlcs, where there are four spin-lrospin degrees of freedom. one might naively invoke a %INexpansion with N = 1. Unfortunately, the spin and isospin dependence of the forcer is m large that the argument is inapplicable. For uample. the leading contribution to the potential energy. the direct Hafirm potential, is consideraMy weaker than the Fock exchange tm, which formally should be a factw of 1/N smaller. Similarly. in S(I(N) nondbelian gauge field theories. where N is the number of colors. there ir a significant question as to the rele~nceof the large N limit to the physical a s e N = S, The lack of an ucplicit upanrlon parameter does not necessarily preclude a p plication of thc stationary-phasc approximation. Indeed. the functional integrals we evsluate m;ry well possess saddle points with very large ~ c o n dderivatives in the directions of steepest descent. giving rise to useful and accurate low-order approximations. For example, in the case of very colfectiw states, it is physically plausible that many particles pa&icipate in notion characterid by! the appropriate colfectlve wriable, and that the action for thls variabh is thus multipiid by a suitably large constant reflecting this coltectivity, However, in this case, there is as y& no quantitative measure: of the accuracy of the expansion. A swond aspect ass-ociaited with the lack of agandon parameter 1s the fredom to wdte a variev of digerent exact expressions, each of which yields a digerent lowestorder approximation. In the a b ~ n c eof a formal expansion parameter, one must be guided by the phydcs of the problem, and for example choose a formutation which includes the physical!y relevant combinations of dirwt, exchange, and pairing mean field contributions. Sfmllarfy, when the presence of strong short-range c~rrelations rquires that mean-field t h w q be formutatd in terms of some egective interaction such as a G-matrlx, it is essential to rcrforrnulatethe functional integral in terms of the e@ect!ve Enteraction instead of the bare interaction. Sfnce the choice of the hnct'lanat integrat repre%entatlondtstermines the stafllng paint for a systematic rtxpansion. it is analogous to the choice of the d=omposition of .R into r non-interacting HirmSItonian Ha defining the unperturbed basis and the residual tnteract'iort V in which one wpands perturbativefy, A finat question concerns the covect counting of quantum states when one considers expandons around all stationary sotutions. The proMem is illustrated by considering the RPA flmit* S m e RPA states eollrrspond to nearly pure singfe part;lcle-hde excitations which are alsa $enerate;dby a static Wartreefock solution with an apprcrpriate set of occupation numbers. Thus. in this case, the same physical state is approximated both by a static self-consistent solution and a time-dependent periDdic isalution. In general, there thus exists the problem of avercounting the physical quantum states of a system by considering all stationaq-phase solutions. Since substantial mathematicat difficulties arise in attempting to add the contdbulions of distfnct stationary solutions, one must again be guided by pkydcal considerations In selecting the form of aationavy solutions with whick to apprDximate a partScutar physicaf state,
7.5 LARGE ORDERS QE PERTURBATION THEORY The final application of functiclnal interal methods tase shalt preRnt is their use as a tool to study the behavior of targe orders of perturbation thmry, The mdvatian
FURTHER DEVELOPMENT QF FUNCTIONAL INTEGRALS
372
for studying the brshavior of large orders is the conviction that deeper understanding of the nature of the asymptotic behavior wilt lead to the devetopment of new methods to extract physics from the divegent series, We wilt briefly dEscuss two such possibilities, Borel summation and Pad4 approxSmants. Historically, Bender and VVu (1969. 1971. g9721 first calculated the asymptotic expansion of the energy of the anharmonic harmonic oscillator as a function of the coupling constant udng the WKB method, which has no natural extension t o probl h s with large numbers of degrees of freedom. Subsequently. Lipatov (1976.1977) develop& a functional integral method which is directly applicable to many-body theory and field theory and which has been applied t o severai mamples by Brbtin e t al. (1977). We will illustrate the basic ideas on the simple Integral introduced in Section 2.1 where we first discussed asymptotic expansions. Then, we will discuss Borel summadon and finally use these methods to evaluate the asymgtotie befiavior of the anharmonic osdltator, Because the developments i n thls secdon are particularly formal, we prewnt only the essential ideas in the text amd relegate many of the mathematical dcrta"rs to the problems, STUDY OF A SIMPLE INTEGRAL, Consider again the Integral of Section 2.1, Eq. (2.24)
which corresponds to the classFcal partidon functbn of a particle in a qoartic potential. Physically, we have already seen that Z(g) has an essential singularity a t g = 0, Mathematically. Z(g) is analytic in the complex g plane with a cut along the negative real axis. The asymptotic expansion of Z(g) in powers of g obtained in Eq. (2.26) is
where
, to write the asymptotic beUsing Stirlinges formula k! ~ P + * Cit-is~useful havior of Zk in the following equivalent form f'or future reference M
As noted in Chapter 2. the physical origin of the divergence of this alternating series is the fact that the number of contractions or dbagrams at each order grows like k!. In preparation for the treatment of functional integrals, we now show how to obtain this asymptotic behavior of & by applying the stationary-phase approximation
7.5 LARGE ORDERS
O F PERTUREzATlOM THEORY
t o the integral for I(g). Expanding the exponential c- f the integration variable z = fig yields
373
in Eq. (7.117) and rescaling
where
y" - 2 1 ~ 3. (7, l X9b) A(%)= 2 tlence, as k -4oo,the asymptotic expression for ZI, is obtained by the stationary-phase approximation. The stationarity condition
has two stable saddle points ye = =t2
with curvature 4
=.b-+=.-T=12.
(7.120c)
Sts
Adding the stationary contribution and quadratic fluctuations of these two saddfe points yields
in agreement with Problem 7.13.
Eq. (7.118~).Higher order terms in powers of
are evaluated in
BORE& SUMMATION As Sn the preceding example of a simple integral, one frequently encounters divergent series of the form
which divere because some combinatorial factor makes iTk: grow like k!. Hence, it is useful to consider the Borel transform in which the IctQterm in the original series is divided by k! 00
nr
374
FURTHER DEVELOPMENT QF FUPICTtONAC I N T E G M L S
If B(s) converges for all vatues of o, one may then extract a finite result from the divergent series (7,522) by the inverse Bard transform
Obviously, if the ari@nalseries convwges,
yields the cawect result sfnca
A simple a m p l e of- how Borel summation works for a divergent series Is provided by the power mnes for T?; for which Zk= Then
and
which 1s the cwrwt rmutt. hrthttr mathematical aspects of Boret summation are glwn by Hardy (1948). Physically, the obvious question is whether the finite result extracted from the divergent perturbation series by Boref summation Is the coneet physical solution, and one must argue on physical grounds that pathofogical terms such as e- f with mnishing duivatives are excluded. Let us now use Borel summation to rum the leading contributions of high orders of penurbatlon theory. Denoting the asymptotic coeEcient In Eq. (7.118d) by 5 ,we wtsk to sum the divergent series
Hence, we consider the Borel transfwm
for which the Inverse transform Is
7.5 LARGE ORDERS OF PERTURBAT10E) THEORY
375
A s ~ m p t o t kexpansion ol" %(g) for g = 0.1 and 0.02, The dashed ltnas lndlcate the magnitude of the residual error R, = d tn Flg. 2.2 and the ecfld tlnss den@ using the Sorel sum In f q. (1,81).
Fig. 7.6
One practical way to use this result Is t a approximate Zlgf by evaluating a finite number of orders of the asymptotic expansion exactly and then use g($)t o approximate the sum of all the remaining orders, Thus, we may define the approximsnt
z(")
(g) yields several orders of magnitude imAs seen in Fig. 7.6. the approximant provement relative: t o the first n terms of the asymptotic expansion, Note that the series (7.128a) has a singularity at g = and thus has a finite radius of convergence about the origin. Hence, the Borel transform, whkh involves an integral on the intewst 10,+mf,required the analytic continuation in Eq. (7.128b). For cases in which the analytic continuation eannat be done by tnspect:iczn, one may alternatively use a conformal transformation. In the present example, Eq. (7.128b) shows there Is a cut along ) - as, -114f. Hence, we may use the transformation
-t
376
FURTHER DEVELOPMENT O F FUNCTIONAL INTEGRALS
Fig, 7.7 Sketch of ths mapping al; the cut g plans lnta the unit circle using the transfarmarlon, Eq. (7.131). t o map the cut plane into the interior of the unit circle as shown in Fig, 7.7. The Borel transform
is thus analytic in the unit circle of 7,By pehrming the inverse transformation, we obtaln
(l.133) where, by construction, the integrand has naw bee@continued to the entire cut plane. The general procdure far Bore! summatbon of a divergent sedes thus consists of three steps: evaluation of the Boret transform of the series, analytic cantinuation to the interval 10,W [ .and inverse transformation,
THE AMHARMONIC OSCILtaTOR Now consider the quantum anharmonic osci:illator with the Hamiltonian
and the expansion of the ground state energy in powers of g
We will evaluate the ground state energy by calculating the normalized partition function
and taking the zero temperature limit of the free energy
7,s LARGE ORDERS
O F PERTURBATIQPJ THEORY
377
The Feynman path integral for $(g) is
where the measure is normalized such that 210) = 1. Ex anding the exponential in powers of g and ~scatingthe integration variable s + kz yields the expansion far the partition function
P
c m
z(g)=
zkdi
(7.138~)
R=@
where
and the measure is normalized such that the integral of the quadratic term in the exponent Ss 5.
As in the example of the one-dimensional intqral, the stationary-phase approximatton yields the asymptotic apansion of for large k, The stationarity condition for the exponent is
with the periodlc boundary conditions
Recall from Eq. (7.70) that variation of the end point yields periodicity of the momentum and tlrus k. This equation may be dmpl'rfied by the change of variable:
and
f 7.140b) with the result
-Q1+Q-49=~
FURTHER DEVELOPMENT QF FUNCTIONAL INTEGRALS
378
Fig, 7.8 Sketch of the eaffectlve poeentlal for wblch stationary sctutlons fer the actlan carraspand to classIcaf perlodlc tr@ectorles, The bounce trajectory Is lndlcatad by the cfcrttsd Ilns, with
.
9 ( 4 ) =(l(-$, ; i(Ql= d.(-$) (7.1416) t is useful to visuatize the periodic stationary solutions by As In Swtion 7.4, S regarding the stationarity equation as the cfassicat equation of mation of a particle in an eRective potential. Hence, we write
where
1 l v,a(n) = -p2+
The eflFective potential is sketched in Fig. 7.8. Note that its relation t o the physical potential in the path integral in Eq. (7.137) Is slightly diffwent from the simple inversion encountered On Fig, 7.2. In the present case, the quadratic term has the usual sign change associated with a Euctldean path integral, The quartic term, however, comes from the logarithm in (7.138bj and is positive irrespective of the sign of g. Hence, whereas the physical potential has a single minimum, the eF1Feetive potential is a double wel. Using the fact that the energy is a constant of the motion (as may be verified by cailcufaeing d E / d t and using Eq. f7.141a))
the period of a periodic solution may be written as in Eq, (7,lrC)
where q* ( E ) are the classical turning points at energy E. We now seek the statlionary solutions which provide the daminant asymptotic contributions to Zk as p -+ cm,As shown in Problem 7.24, the trivial constant solution q = fL is unstable and thus does not correspond to a minimum of the action. The relevant solution for our present purposes is the bounce solution atready considered in Fig. 7.3 for the decay of a metastable state, As s h w n In Problem 7.15, in the low temperature limit, -. cm,the periodic trajectory In Eq. (7.144) approaches the zero energy solution exponentially
E
g-8
Eir-*m
7.5 LARGE ORDERS
O F PERTURBAT10N TNEQRV
379
The zereenergy bounce solution has the analytic form
where is an arbitrary parameter specifying the time at which the trajectory reaches the dassical turning point A. This degeneracy of the bounce with respect t o transiation In 'rlme wilt produce a zero mode in the quadratic fluctuation matrix which we wilt subsequently treat udng the techniques of Chapter 4. H o b that in addition t o a single bounce, there are additional stationary solutions: eorrespond'lng t o n well-separatd bounces in which the trajectory runs fmrn O t o A and 0 t o B n times. Slnce the action S,,of an n-bounce trajectory "i s times the ztcthn of a single bounce 8%. the contributions to ifkof %-bounce trajectories are subdominant for targe k An c-'"S. m e-"~ (7,147)
zk
C R
k-m
so we only need t o consider the single bounce trajectory, Eq. (7,146). Mote, however, that when the dilute instantom gas approximation Fs made in Problem 7.11 t o calculate the splitting i n a double well or tifetirne of a metastable states, the ~oeficientshave ribdependence A, and the complete series will be summed t o obtain an exponential. The action far the single bounce (7.146) can be calculated using Eq. (7.138b), the change of variables ff ,140). and the identities
..&
with the result that the stationary contribution to ifkis
As in the case of the simple ittttstrative integral, the asymptotic behavior is given iby an alternating series with eoefFicients which grow as k!, and is therefore Bore! summable, Fluctuation Contributions
+
tn order t o evaluate the quadratic fluctuations, we write %(l) = ss(2) qft), expand the action in (7.238b) t o second order around the stationavgr bounce traje~tory
and evaluate the 8uctuation integral ("I$5 la)
580
FURTHER DEVELOPMENT QF FUNCTIONAL INTEGRALS
whare
As previously noted, because of the degeneracy with respect t o transla-t,iorr of the bounce in time, A has st zero eimnvalue assachated with the made
which may easily be verified by evaluating $ dt' A@ - C') the zero mode using Eq. (4.130)and obtain
. Hence, we praject out
where AA denotes the determinant of A in the subspace orthogonaf to its zero e@a;enfundion, There are various techniques for evafuating determinants, one of which i s explained In detail in Problem 7,16, The result of combining the stationary csntribution of Eq. (7./49) with the quadratic fluctuat'rans is
Finally, the expansion far the gmund state energy is obtained by etxpanding the togarlthm in Eq, (7,136b) as foIIows
where the coefFicient of $C is given by
1.5 LARGE ORDERS 0 F PERTURBATION THEORY
382
Since Zkgrows as k!,the dominant term in Eh is Zkand thus, using Eq. (7.154).the asymptotic behavior of E;;, is given by
This result is identical to the WKB result of Bender and Wu. Further details of the instanton or bounce calculations for this and related problems are given by Zinn-Justin
(1984). We conctude thfs section by briefly considering the aiternatives avaltabfe to approximate the energy when one knows exactly a finite number of coemcients E, of the peeurbation ctxpansion. W have already considered two options, One is to eatculate the finite sum C;=,E n d as in the case of the illustrative one-dimensional integral and terminate this sum when the contributions begin to increase. The other is to approximate at! coemeients beyond k = N by the leading contribution given in Eq. (7.154) and use Borel summation to sun these contributions to all orders as in Fig. 7.6, Another alternative is to use Pad6 approximants to represent EO(g).The [N;M ] Pad4 approximant to a function f (z) = a,zm i s defined by the ratio of polyno-
* qiNfM! ~ i are polynomials of degrees N and M, respectively, such where ~ l ~ and in powers of that the first M+ N coeRcients in the expansion of f f N t M j (S)and f ( l ) z agree; that is,
For example, the [l,
11
Pad4 appraxlmant to ez Is given by
A general discussion of Pad4 approximants i s given by Baker (1965). To "rlustrate the eReetiveness of Pad4 approximants in extracting the energy from a finite number of terms of a zflvctrgent perturbation series, consider the energy of the anharmonic oscillator for g = 2. Using the exact eoefFiclents given by Sender and VVu (1969).the first few terms in the expansion of the energy, Eq. f7,155a), arc! given by
and the energy for g = 2 is Ea(2) = 0.696175-... For g as large as 2, the magnitude of the correction at each order increases, so the optimal asymptotic approximation i s obtained at first order. The relative error at each order is shown in the first column of
382
FURTHER DEVELOPMENT O F FUNCTIONAL INTEGRALS
Table 7.2 Reiattve error In tihrge approxlmatlons ta the ground state energy of the anhzrrmonlc osclllatar wlth g = 2, The first column shows the sum of the first N orders of the psrturbatlon serles. ths sscond presents the [N,N] Pad4 approxlmant, and the thlrd shcrws the PattB approxlmant of the Borel transferm deflnsd tn Eq, (7,161b). In each case, the magnlructa of the rekattve Is tabulated. Table 7.1, from which we obsewe that the minimum error is 26% amcl conclude that simple summation of the asymptotic s eries is not quantitatively useful, in contrast, Loeffel e t d. (1969) have proved that the [N, N ]diagonal Pad4 approximants converge t o the ground state energy and have shown that even low values of 1IJ give accurate approximations. The relative errars for their results with g = 2 are shown in the second column of Table 7.1, An even better resummation is obtained by evaluating the Etoret transform
approximating it by the [R, N]Pad4 approximant. ~ inverse Borel transform
[ ~( g )*, and ~ then l petforming the
The nsults obtained by Graffi e t al. (liJ7Q) are shown in the third column of Table 7.1. Note that E ! ' ~ ~requires only four coeMcients of the original perturbation series and yields a relative error less than one half percent and that. the error may be reduced to the order of 10-~.Atthough realistic systems have yet to be analyzed t o the same =tent as the anharmonic oscillator, these resummation techniques may be of more genera! utility 'in extracting the physical content of d'ivergent peuturbation expansions.
PROBLEMS FOR CHAPTER 7
a ~ E field w The h 1 &woprobbxns show how the f r d o m in htrodmcing macy be exploi$ed t o obtain, a physically relevanb me= field approSmation. Prob b m 3 and 4 derive prape&ies af wscompb$e aeks of aka&- cited in khe kext;. Prob1em 5 elucidata the akmctwe of the thedependent Emtree Fock approxim&ion M a, c l w i c a l field kheoq. Since tinne dependent, mefield t h e a q iar the physied
PROBLEMS FOR CHAPTER 7
383
fonadaticm anderlying the statio~ary-phaseexpansions in thie chapter, thak problem is particularly important and is therefore accentuated with an *. Corrections to $he startionq mlutian a& s m tempmatwe: are t r e a t 4 h Pmblems 6 and 7 and Pmblem 8 outbeer the &ationw-phw ation to the pwtition fmctiun at finite tempmatwe mative to the treatmen&of trwition mpEtudes in field, Problem 9 shows how to use coherent states, the generd t h m q of collective =citations and tunaeK~gin Section 7.4 to andyticdly solvable cma: the RPA md dmmling in a qoartic potential. The wrpansion in Problem 12 and isr of suEcient ms treat, aspects of fhe mymfiotic; hpo&ance to m s i t am *. The n a t behaavior of Iacsge: ode= of p&wba$ion thwv which were not d e ~ v d in. Sectioa 7.5. Findly, Problem 16 shows how to evalu-zlfe detemhmts in one dhension ntid for cdeufsting gnctu-adions snck abll in M, andfiicdy. This pmblem is (7.154)and in Pmblem 11 and $herefore is also emphasised with an '. PROBLEM 1.1 AuxCfEary Ffetd Representations. This pmbLm aplorm the freedam go take combinations of the decompositiun of the Hamiltanim (7.21, (7.91, a d (7.11) tto obtain physicd canrbinatioas a f fhe Basrtree, Foek and p s t & i o n q p h w appro~miation. a) FoElowiag the wgurnengs in the text for the Ilastrect case, show that stat i o n ~ i t yof
1e4s to a me= field which coatacizla the Fock exchange tern and no dkect t Wdte &hermalt in coordkate? repremntatia~for a spin-dependent potential
where denota the spin compoaent for state a. Shilwly, vary ao analogous expressian to show that (7.12) yields a g&ng mean fieILL. bf to obtain the W Fock mean field by replackg im (7.2) by the antkymmetri~edcombination GapT& = $ v - ~ ~ show ~ that ~ whereas this givvedl rur; e x a t repraentation of the evolutlioai operiitor, the mean field ia one haE the numal H&ree?-Fock me- field. c) Sappose vvs d e w p o s e the two body interaction into the sum of direct and acha-nge:terns il @D $(1) $
~
~
~
~
a d requse that the Eartree potentid generated by 0" vmhh and the exchange potential gemrated by vmkh
Show Chat the mean field is the: H&r-Pock potentid, the sdationv phme eaergy fiastreeFo& enefw, the s&-enerw t in K vahhes, =B the! n = 1 ion, Q. (7.481,vanhhes.
~
384
FURTHER DEVELOPMENT OF FUNCTIONAL INTEGRALS
A simple way to implement &s, (I) and (2) for spin saturated systems, that is, systems in which comesgonding single particie states we occupied with spin up and apin down Fermionsr, is to use spin dgebra. For a central potential u(rl - rz) ~ l h mth& the following decomposition h= the d e ~ k e dpropertiee
where PS is the apin exchange operator and denotes a Pauli matrix. When the potent;i;tl is written in this t o m , it is clear that the aaxifiaq field may be expressed in terms of four independent spatial functions
Show that the quations far the spatial components a; me identical so there are only turo independent functions which generate the direct and exchange components of the mean field. Since this ixltrodactioa of spin dependence may a p p e a wtificial, it is instructive to consider the relation between an a u x i l i q field functional integral for nucleoxls interacting via a, static twa-body potential and a field theory of rnesons coupled to nucleons. Note that the integral Dr ~ - ' " ~ b +:"U" looks like a path integral for a scalar field coupled to the hrmion field p^ = $t$ with the free scalar field action c r w , Thus, it is natural fo mgociate vacrious components of the cf field with vaiaus meson fields. Since the cr. V coupling of pions to nucleons generates a (71 contribution to the nuclear potential, the pion field is naturally associated with the exchange potential v E . Similarly, the direct potential vD is associated with scalar meson exchange, cl) FinalXy; show that the functional integral
l
produces the Hartrew F'ock-Bogoliubov mean field for a spin-satur ated system interacting with a central potential in the s t a t i o n q p h a e approximation, where
PROBLEM 7.2 Atternatfvc AuxifEary Field, This problem examines the afternative auxiliary field formulation obtained by writing the evolution operator as a
PROBLEMS FOR CHAPTER I
385
sf the fom (7.11a):
whwa m fay) aad we have incladd the one-body operator K. a) Write the stationary-phw eqaations for a matrix element ($le-*TI$) with h i t e e, Take the 1hi-t r -* 0 e w e f a y and show that the adatioctq mfution <(it) sa6kfie8
appm~mationas in Cdcalste the w a n d state enerm in the st~tioioaim~ph;lrse Smtbn 7.2 and a b t h the: Her-Fock ation for the choice Wa8,&=z v a @ ~ S %a679 ""
b) Cdcnlate the quactratk %othe static rrsfnkionfsr finite r, Show direct and exchmge Ang diagrams. that the b i t of this exprwiom ad with wdkmnre&*ed mak& efRemember the mmcoanting pro ements r d k n d in eonnctcfka wi&6 (2,124) and node how the present with it, PROBLEM 7 3 ehrarcsmptata Sett of Gaharcnt Stator, In the cme af B m n or Fernion cohaend stakats, show k m h, (7.2%) redace@to the fixnctioaalintewal reprwntakion of C h q t e r 2.
PROBLEM 7.4 Closure Ralatfons for Slatat Olctermtnantr. a) Daive Eq. (7.24a) by &&kg its matrix element behveen' (gl 91. g#), and usimg Wick' m in the f o m
..
...znrl and
P;(~)v?I(Y)bii6fs yf b) Prove Q. (7.24b) as follows. Write the Cfunctiona aa
.
-and show that; the madrk dement of the left-hand side bkwwn ( z s. .s~ l and to fyr m) is
...
where P, P', we pemudalioms of IV elements, Show thact this exprasiain iz3 propofiional to de&6(5ciq)and +ts an exprwion for the propai&ionalit.y constant. An alternative deGvatia~is given by Blairal&and Orland (1981).
9436
FURTHER DEVELOPMENT QF FUFdGTlOFaAL ENTEGRALS
PROBLEM 7.5" VarlatConeel Derivation of the Time-Dependent HattreeFOck Equartfono. Consider the aetian
...
where 14% 4M)is a Slater det inmt of adhonormaf wave fuactioas, s) Show &hillthe action may be remikten
where the H&reeFock Ifilimiltonian density is
R
(+@($F,#[$l1
By v&ation with rtrsgect to #t, and Q),, o b t G the Euler-Lagrztnge equations
Nate that these equations are of Bam2tonim f o m wi&h 4, and t"4: playing the role of conjugate coordinates and momenta. Write out explicitly to obtain the &hedependentEartree-Fa& equation of makion for 4, (r,t ) bj S h that the werlap mat* of the shgle-parLiele- wave faactions ia a co~~tm of tthe motion
Show that the t h e dependence of any opcitrator dependence may be written
O
having no inb~nrsict h e
Hence, coaclude that the th+dependent BwtreeFoclr energy ( W ) = X is con=med. Aka, show Ghat any one-body opwakor which c~mmuteswith H i s con=wed c) In applying the stationary-phase approximation to the functional integral (7.25a) with the orthonomal measure (7.25b1, we may enforce the orthonomality conatrdne by ktroducing Lagrange multipEem and v w h g the modified action
PROBLEMS FOR CHAPTER 1
387
Whte the resulting equations of motion and the initid conditions satisfied by 4 and 4". Now considler the special case in which (4,<) are a set of orthoaormal wave functions and are the wave functions obtained by evolving {bail by t he time-dependent, HwtreeFock equalions a-F (a) for time ZT. S h w what if we set Aii(L) P= Of then d -(4, (4I4a = 0
ad
an4 the wave functions remain orthonorrnal in time, Thus, conclude that for these b o u n d q conditions, the s t a t i o n q phwe approximation yields the time-dependent H&ret?-Fock equalions. d) Write the resolvent (1.28a) using the functional integral in (b) and look solutions, Show that these solutions for time-independent orthonomd stationsatisfy the static Hartree-Fock equ&ions and yield golea in the rwolvent at the H&ree-Fock energies. PROBLEM 7.6; Quadratic Correctlonas For Slater Oetermlnent Pethi Integrals. In Probbm 7.5, it was 8hown thak calculating the resolvent using the Slater determinaat path integral (7.25a) yielda the Eartree Fock energy in the atationq-phaere acpproximation. Now, calculate the qua&atic Iluctuatians wound the s t a t i o n q solution and obt&n the dkect a d =change ring diagrams with Hartree-Fock p r o p agators, Note that i& shplest to work with orthonamal determhantal wave functions. PROBLEM 7.7 The Loop Expansion and kynman Dhagramro. To undemtand the Xoap expansion in detdl, it is useful to show explicitly haw the diapams contributing to &. (7.54) reprodurn all the signs and h t o m occurring in the s t a ~ d a s d Feyaman diagram expansion for the ground stage energy; a) Fke~e,emurnmate the lhked Feynmaa diagrams contributing do the gro-and state energy through third order. The sera temperat,ure diapam rules me =mm& r i a d in Table 3.1 and the diagrms and ~ynnmrttryfactam we given in Figures 2.4 and 2.5, b) Collect the diagrams from E(') of the loop expansion, Fig. 7.1, and expand out the RPA propagakoru3 in (7,55a) and (7.56) to identify all the terms through third order. Cmefully evaluake the facctom mrsociated with each term and show that the rwults in (a) are exactly reproduced, PROBLEM 7.8 Stationary Phase Apptoxlmation at Finite Temperature. Cow sider a wand canonical ensemble of Femions, with chemical. potexlkial p, at ternpealure I". The grand partikian function of the system is given by:
a) Using an auxiUw field to eliminate the quartie term, show that:
388
FURTHER DEVELOPMENT OF FUNCTIONAL INTEGRALS
where the d e t e m b a n t is evalmated on B, and
a,
let of antiperiodic functions, of anfiptsriod
b) We will perform a stationary phase expansion, considering only timeindepeflident solutions U. Show that for a kkme-independent U,,the aetion is g h n
where E, are the eigenvaluee of fhe equation:
and &hep, we nomalioed to one. c) Shaw that the stationarity condition on t w e Bartree equations:
reduces to the finite tempera-
where f, is the occupation number of state a:
Not;@the additionat sseff-consistency requkememts which occur a t finite temperature. d) Assuming that the sysLem is transfzttionally invmiant, we make the ansartz that, plane wavtls are Hastree r;lsfutiorts, VerZy this assumption, by showing that
1 9(i)= -%@
is a solution, where V. = 1d3rv(7), and p is the average density of particles. Calculate the s t a t i o n q action So, e) We wan6 to evaluate the RPA cameeticllns to the Hartfee approximation, Show that the comtributioion of the quadrabic @uctu.iheionsis given by e
-$In dct ( & ( r - r f ) 6 ( t - t ' ) - S d f .(F-r)~o(n,r't')~o(r't'.rt))
where Go is the Hartree prapagator, given by i3 dt
fi2 2m
- . - - - . - V ~ - ~ + ~ V ~Go(rtlr'tt)= 6(r
- rP)&(t- t')
PROBLEMS FOR CHAPTER 7
389
wit h antiperiodic b o u n d q conditions, f) In order to evaluate the quadratic auetuationa, one has to diagonaliee the operatar:
Show thaf A is periodic in t h e , wibh period F o ~ e sr e ~ m: e ~
8, and
that it can be expanded in a
where
with W, =
9 -".adAr
where
= eg,
- E Z - ~ , . Symbolically, this equation may be written
F(Z, g')
and
= f(%') -
f(z - g') It
pc(&, it;W,) = -iwm - Akkf
*
Using the faclorisation: show that
the operatar A being defined with p e ~ o d i cb o u n d w candi&ioionta, g) We now evaluate the determinane det A
==
det (& - A - V F ) det - A)
(g
by diagosaliskg the opersltom in the aumera;tor and demanimator, Let ifv) be the eigeduaetions of the aperator h+VF wibb eigenvalue Ep, Write down explicitly the eigenvalue equation for l&), These sre the temperature dependent RPA equa6ions. - (h VF),with the Show that eiasSB Itp)i, W eipenvector of the operator proper boundary condition, and with eigenvalue E, Show that:
+
det A =
&
v*
+
390
FURTHER DEVELOPMEMT O F FUNCTtONAL INTEGRALS
Ushg Eu'ier's formula:
show thitL
det A =
n, sink n(k,r#) (er - er.)
'
h) Give a physical interpreta,tion of t h k expresasion in terns of h caatars. Show that these expressions redwe to the usnd RPA in the limit; when j9 g0ea t o inSnity. PROBLEM 7.9 Transition AmplStude for the Forced Harmonic Oscillator. An alternat ivct to the treatment of transition amplitudes in Section 7 3 Its to use coherent states, The bansic idea L anstraked by &her-cme of the forced h d e ~ r i b e dby the Hamildonian
u s h g the cohere~tsf atet? of Chap~er1 a
d coherent atate path integral of Chapter 2, ahow that the transition m a t r k element between an inikial hmmonic oscglator eigenstate h)and ffnal st&e Inj) may be written
where
Now, evaluate the integrals over S$* and 4 in the atationkly-phase approximation. For t between and tf, derive the equations of motisn
and analytic solntions
PROBLEMS FOR CHAPTER 7"
39%
where
F=
c'@'f7(t') dt'
*
S h m t h d &ation&ty of the action with rapecct to (bj. and
$i+
ykXd the condition
The four conditions (H)fully aigeclEy the stationary solutions. Note that, because we have applied the sta&litanw-pheapproximation, ifr* need not be the complex conjugate of 4. Show that for (&< RI < (G+ l ~ \ ) % ,there are two whwew outi~ide&ha& donnaia there which in conjugate s t a t i o n q t~aIuGio~8 axe only apo~entially G o~iUator example h repretrentarlive of the genThe structme of $h eral strue$ure of the anabgouar coherent state path inlsep* for the mmy-body problem. Time dependent mean field equations of the I o m i#,.(t)= % evolve the ( # h ) from ti to t f , aa irsl (I), the: qumtkation condition for the End state relates {h($, 1) to {f (9)) I in the timedependent mean field equations -44; = evolve the C#;) from t g to t; M in ( 2 ) and finally the quantisitikion on the initid st&e cloclea the loop by relating (&lk.(@)) $0 (4i(ti)) BS in (4).
&
PROBLEM 7.10 RPA Llmft for Qurrntfzedi Efgenotarfes. Show that the smallamplitude limit of this theoq for eigemtata of eollectiw motion reproduces the Rmdom PBwe AppraAmation of Section 5.4 m foaawa. Wd&ea solakion to the self-comktent periodic mean field equations (7.98) which dig- hfi~it,t?hlimaliy from eh@stalic H&rw wave function. as follows,
where ex and $K(%)deaoke Rartrw eigeavaluea md eigenfunctiom, upper case and lower c a e hltiaLte fabefs denote occupied and uaoecupid stateis, respwtively, and the expansion coeBcienGs Cmrcare aammed ismdi. Show that ordhomomalilcy of the f #X (S, g)) requires
Wdte CmK in the conventional;f o m CmKE XmK
+ YGK'@''L
392
FURTHER DEVELOPMENT 6 F FUNCTIONAL INTEGRALS
a d ghaw that Enehsation of the equation. of motion yields the RPA mat*
equa-
tiom
X
--IY where AmM, n~ &M,
= (€m - ~)&rn,h~ $- (mHfVfMn)
f i= ~ (mnlV\MM)
.
Denoting the RPA eigenvalues and eigenvectors by wv and ( X ( & )Y(@)), how that the p e ~ o dw,T = 2% and that arx = eKT. Show that the quanthagion condition (7.103) yields the RP& nomalkation condition
and that the comeaponding energy af the state E63
Note that since both the quantkation integral and energy involve terns of O (C2) all tern8 through one must use the condition (2) in h,(1) and consistantly c o ( c a )"
PRQ5tEM T e l l Dilute Enstanton Gas Appraxhrnation. Thb problem develops an altmative to the treatment of tunneting in Section 7.4, a) Consider a symmetric double well V(%)of the form shown in Fig. 7.2 and Eet the positions of the minima be denoted &sea, The Feynman path ixltegral far the propagator i s expanded in the statioxrw-phase approximation. as
where Z(7)denotes a stationary solution. As in Section 7.5, in addition to the exact single-instanton solution for large 8, approximate stationary solutions are obtained by joining together sequences of N instantona and antiinstantons and assuming that they are well-separated (i.e. dilute). Show that the one-instant on contribution is
W here
wa = V"(&Z@), So is the action for the instanton, and det' is the determinant after
projection of the serocenergy translational mode.
PROBLEMS FQR CHAPTER 7
398
Sum ovw afl, numbem of instatone to obtain
By campmkg with the exact result;
ahaw that the two lowest eigenvaiues in the dilute iastanton gas appro~mationrwe
Campae this rssait wikh Q, (7.85), Note thab the treatment of the zero-point the two exprasions, The cXasi.cal action So h defind for a energy. is diEerent trajeckaq from the bottom of one well to the battorn of the other well, where= A 2 W 2 ( c ) is the action for a trajectory joining the claasieal turning points at Bnite enerm. Also note the digersrrtce in premuX#iplying factom. b) Now, consider the specific case of a partkle of ma%s m in the potential V(z) = (2' 1)'. Show that the exwt instanton solutions are given by ~ ( t = ) and K = One method t a n h l t where 7 = and that So = g for evaluating the ratio of deteminmts h K is presented in Problem 7.16. c) N m use the same method t s trezlt the case of fi~sionwing an awirliw field path i n t e p d , Show tftae the contribution of a single bounce to the evolution operator itJ
-
6
3(6)"'.
where
. as denote the det' Cfenotf?~the determinant omitting the ttrero eigenvalme, a ~ and static Haxtree and t h e dependent bounce mlutions, r~pectivefg,a; ia a fwdor o b tained from projecting out fhe sera mode, the factor 112 only one-haE of the Gaumiaa p e a in appliying the stee to the variable comesponding t o the negative sigenvalue of ixnilginq due to the single negative eigenvalne, Finally, sum over all numbms of instantoas to obtain.
PROBLEM 7.12* VfU Expansion for the One-Dimnolonal 6-hnction Qrsbfern. To explore the nature of the expansion parameter when the bsdationw-ghwe
FURTHER DEVELOPMENT OF FUNCTIONAL INTEGRALS
394
*
ation iss applid do a, functional integral over an rruxiliq field, it is inR stmetive to rmxmine the 1IN expasion dkcumd in Problem 3.5, a) Conaider the trace of the evohtion operator
I=)
+
where o denotes one of the (25 1) spin projections and, for the moment, E, denotes a complex Beld far a Boaon coherent state. Show that by introducing ;hn aw3iasy freld, perfoming the Gauaim iategsd aver and racaling the coupling constant snd auxiliary field according to 8 E gN and d (z, t ) = o(z, t ) / N that
e:fa,
Note: that shce N appem as aa expficit m~ltipficativefactor ia the exponent, the atationq-phrise approximat;ian y S d s an expansion in 1 / M , det A = A;' det A to show that stationarity of the b) Use the relation mponeat in (2) yields
&
corrections with diag~amsin Identify the s t a t i o n ~ ycontribution and leading (7.50b). Note that cl~sificationof diagrams according to N C - l , where C is the number of eltosttd paticle propagators and I is the number of inter;aetlisns, fogaws immediately from the fact that the conpling constant sc&s as M-% and the Green's funedions scales M N. are c) Repeat the prweding analysis for Femions, in whieh cG r a m a n n vmiablm. Hate thibfr mina8 signs asisct both from the Girumian i n t e grd md from the definition of the Green's ffutnctliona. d) Finally, consider the case of a general finiterange potential. Formally = vaB-rsNand repeat the analysis in Eqs. (2) and (3). Whereas the define argument was exact for the &-functionbwaam it bad no length scale, to make the 1/N expansion for a finiterange interaction one must consider a hypothetical class of d h e o ~ win which the i n t e r ~ t i o nstrengkh vwiea with N and have a physical reason to believe that the limit as N -4 cm is meaningful.
([:c,)
PROBLEM 7,53 dorrt?ctlons to the Leading Behavior of Wfgh Orders of Perturbation Theory. Consider the integral
PROBLEMS FOR CHAPTER ' 7 3 9 5
which appears in Q. (7.119a)for the expansion of the simple illustrative integral. E x p a d the exponent wound the eaddlict point a, 2 and ahow that =L
zrc= ( - I ) ~ cIn~'L-L
@(sum of all connected graphs)
G
where the connected graphs have n-point vertices for aU n 2 3 with vahe m connected by propagators with value $. Thus, evaluate the corrections to Zk of order and
h.
PROBt EM 7.114 Consider the f-unctiond integrasl (7.138S)for the coeffchnt Zk: onic ogcalatoft Show in the expansion of the pwtition function far the anh z. = f2 l f l is a stationary mlution. By expanding around either solution, show that these wlutio~raare not minima of the actiom and we thw unstable.
PROBLEM 7,115 Urn Q, (7.144)ta how that the energy of the bounce wlution onic osciIlator appmaches Eero M e-B when @ --+ m. First, evaluate Then, rescale the integration variabb in the turning p h t s qA(E) a9 z and take the Emit E -+ 0- to show that, Eq, (7.114)according La q =
PROBLEM 7,16* Ewluatton csf Determinants, The determinmts x i s h g from iuctuations wound a stationary soIutlian h oae dbensioa may be evaluated =aIyticaUy. The method practrtd in this problem fol1aws that of Colemaa (1971). The Redholm deteminant of an operabr A($,tq b isthe product of the eigenvdnes of A ~ormafirtedso as to render it finite, We w U evduate det A by considering the quantity det(A - X) a(x)E det (Ao - X) where A. is a reference operator. a) Fkst, conbsider the ewe of ;a ht-order digerential operatar
with pe~odicor aadiperiodic Isoundq conditions. Let $%[tj and X, be the aigenfunctioion~and eigenvdua of'A
w i ~ hbounda~ycondition
391;i
FURTHER DEVELOPMENT OF FLIQSCTtONALINTEGRALS
+Lo)
and denote the corresponding eigedunctions for A. by and X;". As usual, g is +l for the periodic caBe and --l for a~tiperiodicb o u n d q cmditions, i j Show that D(X) is a xneramorphic function with simplk?z-eros at X = X, and simple poles at X = X?' and that D(&) ---+ 1 except along the positive real lxt--.m
axis, "ii) Now consider the ratio
where
$A
(L) is the solution of the: equation
with the b o u n d w conditiion
and $ r l ( t )is the carresponding solution for Ao. Note that $ ~ ( tis) an eigenfunction with eigenvahe X d e n (g) = c. Thus, show th& A(&) a meromophic function with the same poles and %erasas D(Xf and thiht A(X) 1
#
ixt--.c=
except along the positive real axis. Thus, conclude that ia an analytic function that goes to X. sts X -+oo in m y dkection except along the positive r e d =is and therefore is q u a 1 to I. so that
iii) Observe that the first order equation for Jlr(t) may be integrated directly to sa that ($1=.: e - j': (j obtain
Also, calcularte the ratio of determinitnts by expficitly evaluaking the products of eigenvalues. Show that the eigenvalues may be written
PROBLEMS FOR
CHAPTER
397
7)
where the integw and halflinteger values come~pondto g = 3. and --l, reBpectively: Evatuate the product41 using the Eulw fomulas
The fact that the result obtained in &hisway digers from Q. (2) by a multiplicative constant demonstrates the need to define the regularisation of the Redholm. determinant carefulEy. One option is to replace the differential equation by a difference equation with N mesh points and take the limit as N + m. b) Now, treat the case of a second order differential operator with sero boundary conditions at f
!.
22
The eigerrfunctions of A satisfy
with b o u n d l v caaditionrs
and we define the function
(g) satisfykg
with boundary. anditions
With analogous definitions for A(@),we now compare D(X) defined in Eq. (1) with the ratio
a(x)== By argumenfs analogous to those in parts i)and ii) of a), show that G1(X) and thus
=;
&(X)
FURTHER OEVELQPMENT OF FUNCTIONAL INTEGRALS
398
c)
function
Next, treat the caae in which the second-order operator A has an eigenwith Bero eigenvalue
and in whliCh Ag h a a a ~ u c k~ e r oeigenvdae. The quantity of intereast irJ
det AA D="-----net A@
where AA is thre projection of the operator A in tbe subgpi~ceorthogonal to thti sera eigearnode. i) Show that
ii) To evalnate the limit in which X goes to .em, we expand lasJ
( t ) in powers of X
foll~ws: 4bx ($1= $0 ($1 + Xtp(t)
*
Show that:
w l r m p($)zretirsfi-ea the equakion:
with b o n n d q conditions
d) Finally*coasider the cwe in which p-* +m. fa. thk ease, the eigenstatbtes of A we defined aar solulions of the equation:
PROBLEMS FOR CHAPTER 7
399
with modified boundary conditions
for bound states and
for outgoing wave scattering states. The quantity W is defined by
Consider the wavefunction $A (t) defined by
with boundary conditions
(t)
t+fw
=i.
where we use the convention that il Show that
where we also assume that Wo(t) goes to W when ftl goes to infinity. Note that +(O) now appeass in the numerator and $ in the denominator since the boundary conditions were fixed at t = +W. ii) Show that in the case of a sero eigenvalue, the result for becomes
zt::",2
D = det AA -
m -
det A.
where pft) satisfies the equation:
with boundary condition
+PP'((--.-.))
CHAPTER 8 STOCHASTiC METHODS Functional integrals reduce the quantum many-body problem t o quadrature. All that remains t o calculate the observable properties of physical systems is evaluation of a very complicated multidimensional integral. In situations in which analytical methods such as the stationary-phase approximation or perturbation theory are uncantrolled or impractical. Monte Carlo evaluation of path integrals is a powerful and appealing afternative, !t provides precise calculations of physical obsewables which are exact within controtlable systematic and stochastic sampting errors, This chapter describes the range of techniques available for the stoehastie evaluation of path integrals and haw this Rexibility may be efeetivety exploited t o incorporate one's understanding of the underlying physics in the method.
8.1 MONTE CARLO EVALUATION OF INTEGRALS Wt? will eventually need t a evaluate path integrals of the form $ &.d7jr) L - ~ ( ~ ) O ( Pwhere ) P/') denotes the coordinate i of the ia particle on the rkhtime slice or integrals over fields of the form nijr,&{iL e-5(@)0(o)where c,!$is a Field defined on a spacetime lattice. It is easy to see that conventional quadrature methads appropriate for integrals in few dimensions are completely useless far such problems. First, consider the number of points at which the integrand must he sampled on a single time slice for a system like the nucleus a08PB(or equivalently, a drop of liquid Helium cantaining several hundred particles), Since the rms radius is 7 frn rand the two-body potential varies significantly on the scale of 0.2 fm, one requires at least 100 mesh points in each carteslan direction for each coordinate, so that the integrand must be evaluated at ( 1 0 0 )=~10'~~ '~ ~ ~ points. ~ Even at the fastest computer speeds of the order of 10' evaluations per second or 10" evaluations per year, this corresponds to years in a universe whose age is only 2 x 101° years. Similarly, simply tabulating the integrand at 1OOO values per page in a 1006 page book wdghing 1Kg would requiw a mass of 10'~" Q whereas the mass of our galaxy is only of the order 4 x 10~'Kg. Clearly, even ignoring the need to treat multiple time slices. the situation would be hopeless 'if physics required sampling of the integrand in this degree of detaH. (Note that this same counting indicates that solution of the many-body Schradinger equation as a finite difference equatfon requires diagonafitatlon of a X f01248 matrix, The essence of the problem is the unfavcrrabte way in which the accuracy of conventional quadrature formulae depends on the number of points at which the Integrand is sampled En high dimensions, Consider, for example, Simpson's rule in one dimension. Since the integrand is tiocalty appr~ximatebquadratically and odd terms vanish, the leading error in a single interval (-h, h) is $_hhdz z4 = O(h6).I f the total interval is divided into N mesh points, h -- $ and the total error is of order Nh5 Similarly, in two dimensions. the error in a single square cell is ~_h, dz $!h dy z4 = O (hB)
- h.
8.1 MONTE CARLO EVALUATION QF INTEGRALS
401.
and in d-dimensions, the error per cell is O(h4+d).If the integrand in d-dimensions is and the total error is of order Nh4+&= 0 ( sampled at N points. then h -Thus, as the dimension d becomes large the error fatts off hcreasingly slowly with N , Indeed, it is the fact that N must vary as to keep the error fixed which led to the impracticaf estimate for 208P6with d = 3 X 208. The basic idea of the Monte Carlo method is to sample the integral statistically, so that independent of the dimension of the integral, the sampling errors decrease as where N is the number of points at which the integrand is sampled. Note that there is no reason in principle why the evaluation of physical observables should require Far example, the incrediibte level of detail cited in the quadrature example for 20"pb, ln evaluating the potential energy, one only needs t o know the two-body eorretatlion function accurately within the range of the two-body potential. The detailed behavlor af distant particles is totally irrelevant and even ifcalculated to high precision, identically cancels out of the numerator and denominator in calculating ($lvl$)/($l$). Hence. the fact that we will use valuer of N immensely smaller than the value (~onst)~ required to fully specify the many-particle wave function poses no problem in principle.
CENTRAL LIMIT THEOREM The Monte Carto method for evaluating an integral is based on the familiar central limit theorem. Consider an integral of the following general form
where P is a vector in d-dimensions and P(%)is a probabiiity dktribution satisfying the condition
P(%)2 0
Note that there is "rnfinite freedom, which we will subsequently expfoit, to decompose the integrand of any multid'tmensional integrat in the form of Eq, (8.21, VVe will now try to approximate X by forming the average of N independent samples of the probability distribution P(%)
where throughout this chapter, the notation EieP(3) indicates that the variable E< is sampled accordhng to the functbon P(B). In order to use such an average as a controlled approximathn for I , we need to know the probability distributian for the variable X , and especialty how this distr"rut"ton behaves for large N . To simplify notation, it is convenient to define the mean value of a function g(P) with respect t o the d"ltrlbut"rn B(3) as f o l i w s
(8,k)
402
STOCHASTIC METHODS
By definiticrn, ( f )=~I , The: probability for obtaining a particular value X when each 2 is distributed according to P(%)may be written as follows:
Using the integral representation for the &-function. the probability distribution for
X
may be written
where
For a specified value of X, we will perform the X integral using the stativary phase approximation and denote the stationary value of X for a given X as X(Z1"). The stationarity condition = O yields
IX(x)
which implicitly defines ;\(X). including the quadratic correction f (i(~)), the probability distdbution for large N i s
'To display the X-dependence of the exponent explkittly, we first find the extrema
= 0. Note that there is only one where we have used the stationarity condition solution X(X) = D for which by Eq. (8.7) X = ( f ) p . Expanding t o second order
around this point
g.
P(X)thus where we have diEerentiated Eq. (8.7) with respect t o X t o evaluate has a single maximum at X = ( f I p and is monotone decreasing away from this point with curvature specified by the variance ( f a ) p - (f):. In the limit as N -+ m. higher terms in the expansion of the exponent contribute negligibly to the tails of the distribution. In the sense of distributions, P (X) thus approaches a normal distribution
Equation (8.11)is the desired result which provides the foundation for the Monte Carlo method, No matter what the distribution P($).the function f (31, or the dimension d . for large N the average Eq, (8.3)becomes normally distributed about I with standard #. Thus. a general integral may be approximated deviation o = [
and the variance may be evaluated by the usuai unbiased estimate*
Note that the factor to remove the bias is irrelevant for large N so that one efFectively uses Eq. (8.12) to evaluate both (fa)p and ( f ) ; .
IMPORTANCE SAMPLING Although in principle, Eq. (8.12)i s applicable to any multi-dimensional integral which could arise In many-particle systems, i t s practical utility is governed by the sbnal to noise ratio. The ratio of the vau"lance ta the mean must be sufficiently small that the fractional error may be controlled with a feasible sample size, For some problems in
*
is clear from the extreme case of two random variables T h e origin of the factor z and y having mean zero and standard dewbation L., The naive estimate of the variance ;(X" $1 - (?)l = kzy has mean value $oZ instead of f12.
$+ $ -
404
STOCMASTIC METHODS
which the integrand has nearly cancelfing podtive and negatke cantributians which are separately very large in magnitude, the Monte Carlo method becomes very difficult to apply. Such dficulties will be encountered subsequently for Fermbans in more than one dimension. For problems with dominantly positive integrands, however, the freedom to decompose the integrand into factors f(%) and P(sj may be effectively exploited to reduce the variance. Selection of P ( z ) to reduce the variance is known as importance sanpihng. The basic idea is most simply demonstrated for a one-dlmensknal integral with a non-negative integrand. Integrals with bat h positive and negathe contd butlons can in the same way as the diferrtnce of two non-negative integrals as long as be treated a one integral clearly dominates, and the generalization ta higher dimensbn i s obvious. Consider the integral
which has been wrkten in the form of Eq, (8.1) for an arbitrary probability distribution P(%).By Eq, (8.121, it may be approximated
The questian, then, is what we can do ta reduce the variance
9,
Obviously. the best choice is to use P(%)= for which case the variance is zero and each term in the average Is simply 1. This optimal choice is only of academic interest, since ifwe knew I and were able to sample f ( z ) , we wouldn't be in the business of calculating the integral In the first place. However, it suggests the extremely useful and practical strategy of choosing P(%)t o be a simple function w hick i s easy to sample and which is as similar as possible t o f ( S ) , The quantitative utility of such a choice "r iilustrated by a simple numerical example, Consider the function sketched in Fig. 8.1
for which
1
e-2
f(r)ds: .= -- 0,418 e-l
.
8.1 MONTE CARLO EVALUATION O F INTEGRALS
Sketch of the example functlon f ( z ) = tlnczar lmportanca sampllng functlon (dashed tine.)
Fig. 8.1
405
(solld curve) and a
If P(%)i s taken t o be a uniform distribution on the interval (0, $1, then the vailance = l(f(S)- I ) ~ leads ~ z to the statistical error
-
Thus, the sample size must be of the order N 4.7 x 10%to obtain 1%accuracy. Now. suppose we use a simple linear probability distribution P(%)= 22. which we will show in the next section is easy to sample. As seen In Fig, 8.1, although a linear distribution does not reproduce f (S) in detail, it does eRFectively concentrate the points in the rcrgion of' the most '"important" contributions, thus fulfilling the intent of impoutancc? - I)' 22dz leads t o a sampling. With this choice, the variance c;=,, = sampling error
(e
-
which requires only of the order N 155 samples t a obtain the same 1%accuracy. In this one-dimensional example, introducdon of even the simplest impautance sampl"rng reduced the sample size for a given accuracy by a factor of In higher dimensions. where a progressively smaller fracthon of the phase space contributes siignificantiy, the advantages grow dramatically. For instance, if the previous example Fs generalized t o a d-dimensional separable problem
h.
406
STOCXASTIC METHODS
then the variance Is
so that the ratio of the variance for the d-dimensional problem with importance sampling o"(d)p,as t o that without importance sampling oa(d)p,l is
Even for a modest dimension d = $0,the reduction of the variance and the necessary sample size by 8 orders of magnitude becomes quite impressive.
8.2 SAMPLING TECHNIQUES Application of the Monte Carlo Method requires the generation of statistically independent variables { 3 ) distributed according t o a specified probaMlity distribution P($).Because of the diversity of distrlbutlons which may arise in many-body problems, it is useful to s u m y a variety of sampling technlques.
SAMPLING SIMPLE FUNCTIONS The fundamental building block for all sampling methods is a random number generator for varia btes uniformly distributed on the interval (0,1). Computers can, at best, generate a sequence of pseudorandom numbers which are determined by a suficiently complicated rule that they appear to be random for all practicat purposes, One common algorithm: implemented on most computers is the linear congruential method which generates a sequence of pseudorandom numbers according t a the rule
When a and c are apprapdately chosen, this generates a non-recurring sequence of length m where m i s the largest integer the computer can represent (i.e., m = 291 - 1 for a 32-bit word). Although this algorithm may be adequate for many purposes if optima! values of a and c are used, it Fs important to realjire that sequential values of zi have definite correlations, To see these corretations geometrically, let k sequentla! points define a point in a k-dimensional space: that is. it = {zl,Q,...,z r ) , T2 = ( z ~ + sk.t.2,. ~, z Z k )and so an, It may be shown that these points lie on a sequence of at most mljk k I dimensional planes[Press et. a1..1985). In three dimensions. a t best, there are 291/3 = 1290 planes for a 32-bit word and only 216/' = 32 planes for a 16-bit word, In practice, algorithms provided with commercial machines are often
..
-
8.2 SAMPLING TECHNIQUES
407
Correlation betwean the trlplets of points = a pmudorandom number generator. The value of the third coordinate is representad by the Intanslty of the paint.
Fig,
8.2
( z ~5~3, R + r r ~ g r cQW8ratBd +2) by
less than optima!. Figure 8.2 shows the 15 planes generated by the randam number generator orr the IMB P C G d ane notorbus randam number generator on an eartier main-frame computer produced only 11 planes. It should be obvious that sampling the integrand of a three-dimensional integral which is hbghty peaked between these planes uvoutd give totally erroneous results, One eRective cure for such correlations is to stare pseudorandom numbers in a look-up table and a particularly emcient procedure is given by Knurhl191J1). For the subsequent discussion, it will be assumed that a satisfactory random number generator is avaUable t o provide uniformly distr"lbuted random numbers. Given a uniform distribution of random numbers, It is easy to sample any dktribution specified by an integrable function by dmpfe change of variables, tf we define
then
g = P(%)and we may write
Thus, when the variable y is sampled uniformly, the inverse function z(y) is distributed accovdlng to the desired probability distuibutbrr P(a). Far instance, the example used
*
This figure i s generated by rweating the foliowing BASIC instructkons under DOS C=Q"RRID: PSET(X,V),C.
2.0: X==320"RNO: Y=ILOO%RND:
408
STOCWASTIC METHODS
t o illustrate importance sampling required
so that sampling values of y uniformly distributed on (0, 1) yields values of z = distrlbuted with P ( z ) = 2%on (0,1) Gawssian distributed random numbers arts required for many applications and thus provide a practkal exampre, * Since the integral of a Gaussian "r an error function, straightforward transformation of variables leads to the inconvenient prescriptian of catcuIating r = erf-'(9) with y uniformly distributed. it is more convenient t o use the Box-Mutter method and calculate a palr of Gaussian distributed variables z and y using
I
g-
g)
polar coordinates B and p. Since dzdy cf ( z pg ) = d@d(c- f ( p , 81,if B 'Ls uniform on (0, 2 ~ and ) w is uniform on (0,l),then p == has distt.ibution e-e; and z = pcos 19 and y = psin 8 are Gaussian distributed. Another common alternative used for Gaussians is to invoke the central limit theorem and use the sum of 12 random variables distributed un;ihrmty on the interval (0, l). Since each variable has mean
(2)=
5 and variance o2 = f,'(z-
12
$I2dz=
h,the sum s = i = l zi - 6
has
mean 0,variance 1, and is approximately Gaussian, A final elementary method for sampling simple functions is von Neumann rejection iltustrated in Fig. 8.3. Let Ffs)be a positive function everywhere greater than or equal to a probability distribution P(%)to be sampled, If points are generated uniformly in the plane below F ( $ ) and only those which are atso below B(z) are accepted, then the accepted points will be distributed according to P(%).Operationally, a random variable zg is selected with probability proportional to F(%) (i.e.. P (z) = ~ ( z ) / F ( % ) )and a random number ( uniformly distrlbutcsd an the interval ( O , l ) is generated. The value zi is accepted if P ( z ~ ) / F ( z>~ (: ) otherwise it is rejected. Clearly, this procedure is only eflficient i f one can find a function F ( % )which can be emciently sampled and is close enough t o B ( z ) that a large fraction of the points are accepted,
MARKOV PROCESSES
.
A Markov chain of variables (z('), d2), dS)..) i s generated by a rule which element z("+') solely on the specifies the probability distribution for the (n basis of the nabelement ~ ( " 1 . We will use the notation that P ( z -+ y) denotes the probabitity to reach starting from a;, tn this section, we will be interested in the probability distribution of elements of a Markov chain and, in particular, will seek algorithms such that the distribution of the z's converges t o a specified distribution e"'"SIs) (The notation e-S(") is used t o emphasize our eventual interest in sampling the action and the cfistributian is assumed to be normal2red.) Although in general zln)may be a vector with an arbitrary number of components, vector notation will be suppressed W t~eneverpossibte. For a Markov process to sample e-S(Z), it is sufficient to require that the rule .P(z + have eventual access to every configuration in the system and satisfy the microrevers!biiity condition
+
8.2 SAMPLING TECHNIQUES
409
Geometrical Illustration of \ran Netrnann reJectfon. Polnts are rJected In the shaded area and accepted In the onshaded area,
Fig. 8.3
is an The proof proceeds in two steps. The first step is t o show that equilibrium solution: that is, if z is distributed according t o c-S(') at one step of a Markov process then the distribution for the next step P(y) = dz e-S(")~(z4 y) is c-S(~). This property followsimmediately from microreversibility and the normalization condition dz P(@--t z] = 1:
I
dz e-'(")~(z
-r
y)
(8.28) ,
. C .
- ,-S(ar)
.
,
The recond step is to show that the distribution converges t ~ ~ e - ~ ( ' )A. useful measure of the deviation b e ~ e e ntwo distributions &(%) and p~(a) is given dzlPA(z) P8(2) , If the distribution at one step of the Markov process is by M(%), its deviation from equilibrium i s Dold = f dzlM(z) - e-'(')( and the distribution at the next step P(y) = $ d z M(%)P(% 4 Y) has deviation Dnew = dyl dz M(z)P ( z 4 y) - e-S(@) 1. It follows that the deviation is non increasing:
= c-S(l)) or if some conSince the strict equality only occurs in equilibrium (M(%) figurations are not accessible to P(% + y) (which i s excluded by hypothesis). the
deviation decreases with each Markov step and the distri butlan approaches the desired equilibrium solution In the case of N degrees of freedom, for which each element of the Markov chain . 2g1}.one has the choice of using a microreversible is a vector 3 (4 zz algorithm to evolve the entire vector or evolving each of the 1V^ variables separately. In the latter case, whlch is usually the most practical, if variable ~4 is being updated, all other variables zj, j @ C: are held fixed and the microrevers"tbitity condition is
{ZP))%p1,..
The degree of freedom to be updated, %g, may be selected sequentialty or randomly and in either case the general"tations of the equilibrium and convergence proofs, Eq, (8.28) and (8.29) are straightfaforward. There are several alternative microreversible algorithms P(%-+ y) for generating M a r k chains. One of the simplest is the s*called heat bath method P(%---t = c-S(v) for which the result is independent of the previous variable. For a single variable. the method i s of no practical significance since it assumes we can atready sample S(y), Hawever, for a system with many degrees of frecsdon and nearly tocat action, such as spins on lattice with nearest-ndghbor Interactions, it may well be possibite to sample with all S'S fixed. If values of y distributed according to e-S(**.*"-l*"*"+l*..s~) W regard S(2) as @-FT[$), this corresponds to fixing all the neia;hbors of yi at the specified values 2i and letting yg come t a thermal equilibrium with a "heat bath" at inverse temperature p, By the general proof, sequentially solving the one-body problem for a shngte variable g,: in a heat bath with Ets neighbars fix& generates the equilibrium safution for the full many-body problem, The methad 'of Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller (1953), cammoniy called the Metropolis method, is similar in spirit to van Neumann rejection, and like the latter, applies to an arbitrary distribution. VVf! wlll first state the rule for P@ -. 3) and then demonstrate that is is microreversible. A mare general algorithm is discussed in Problem 8.1, Given 2, a tentative value ?ZT is generated by some convenient symmetilcal probability distribution F@ -+ = ~ ( 355). This ~ tentative value is PT with prabability rni l, ems(zF) /ems(z] is set equal t o accepted, that is. awise, it is rejected, that is. 3 is set equal to P. in other words, Ef the change reduces the action, it is atways accepted; ifit increases the action, it is accepted with probability emAS. One simple implementation of this. rule is t o reject jtr i f e-g(3F)/e-s(')
p)
(-6, i).
+
8.2 SAMPLING TEGHNfQUES
413
iE is accepted with probability c - S ( @ / ~ - S ( @ ) . Since F is symmetric, the ratio of probabilitks satisfies detailed balance! as follows:
<
is analogous. Verification for the case e-S(Bl It is useful to compare the heat bath algorithm with M iterations of the Metropolis method for updating one vaviable x i of a many-body problem with at1 other variables z j , i f i fixed. For a single step of the Metropolis algorithm. M = 1. a new variable y cannot dEer from z; by more khan Az. In the limit, M -4 W , the distributijon of y approaches e - 5 ( " ~ , * . " * - ~ * ~ ~ " i + 1 . * . " ~and ) the limit of a large number of Metropolis updates of a singie site becomes equivalent to a single application of the heat bath afgorithm. Hence, i f the effort required For one! heat bath update Is less than for several Metropolis steps, the heat bath is preferable. A final microreversibfe alternative is integration of the! Langevin equation, Cons"ter the variables in the action to be a function of a formal continuous variable r, not to be confused with physical time or inverse temperature, and let them satisfy the following partial diflerential equation
where I" is an arbitrary scale. parameter and & ( r )denotes a set of Gaussian distributed random variables which are independent at each e" and T and have variance In practice, the continuum Langevln equation i s replaced by a discrete finite digerence equation ~ i ( 7 , , +=~zi) (7,)I- bqr , so that the variable zi") = z ~ ( T , )corresponds to the nthelement in a Markov chainind the probability distribution for
The rule for the Markov chain is that the probability to go from i to j is equal to the probability for the noise fi to equal + P E , In the limit 4 . -+ 0,it is straightforward t o see that the evolution is m'reroreversibie:
An obvious advantage of this method is its simpliciity and generalkly, A potential disadvantage Is the fact that any finite digerence approximation to Eq, (8.32) wili lead t o errors of some order in 67 in the distribution of 2, In sgttciaf cases, such as field theory and crklcal phenomena, these errors may correspond ta renormalization of bare coupling constants and introduction af irrelevant variables and thus be of no consequence(sec?, for example. Batrouni et, aE,,i985).
NEUMAFtlN-UlAM MATRIX INVERSION A special problem frequently arising in stochasti~solutions of many-body problems is sampling of the inverse of a matrix. For example, the Green's Function Monte Carto method for determining the ground state wave functions requlres repeated sampihng of P(r) = (rl&Irt). Similarly. when Grassmann variables have been integrated out of a many-Fermion problem, evaluatbn of the change of the resulting efictive acdon when Boson or auxiliary field variables are updated requires calculation of In Det M =
&
M;'.
Hence, it is useful to define a random walk which will evaluate the inverse of a matrix, As a prelude t o matrix Inversion, consider evaluation of the ~r~~power of a matrix, T N .as arises in repeated application of the transfer matrix or calculation of ( c - E H ) ~ . Let be decomposed into the product of a transition probability and a residual weight as follows: Tij E Pii Wii (8.36a) with
C&i=I. P c j k O . 9"
Befine an N-step random walk on the domain of integers labelling the basis a' -+ ia 4 iS 4 .. -+ i N such that the probability of going from state k: + t! is Pke and for each walk define the score
.
..
Since the probabiity of the score is I ) ; i 2 f i 3i a , value of the score is the desired matrix element of
i N eit
folfow~that the mean
(TNIij
The Neumann-Ufam method of emluating the inverse of a matrix is a simple variant of this procedure (Forsythe and Leibler. 1950). Let the matrix M be scaled and shifted if necessary such that M = 1 T with the magnitudes of all the eigenvalues 03 2'". To of T < 1. The inverse is then given by the von Neumann series M-' =
-
sum the series,
zki s decomposed slightly dlfierently:
n=O
8-2 SAMPLING TECHNIQUES
43.3
with transition probabilities Pij ktween states .s' and j defined such that
and w&h a probability far stopping at site i defined by
A random walk of indefinite length Is now defined such that the probawlity of going from state k -+E is PbLand the probability of termkaadng the walk at point ?c is P2. Gisr a walk of k steps beghning at i , the score i s defined
...
The probabilky of attaining this score is PiioP61;s PiL-tiLPikSO that the mean value of all scores of length k is ( T ~and) the ~ mean ~ value of all scores of an lengths is M1 k=O
Variations of this basic method are discussed by Kuti and Pofonyi (1982) and in Problem 8.2,
MtCROCANQNECAL METHODS The microcanonicat Monte Carlo method and the method of molecular dynamics utilize the ideas of classicat otatistlcal mechanics in the canonieat ensemble to sample an arbitrary distribution, One can understand the microcananical Monte Carb method (Creutt,l983) for by regarding S(%) ar the energy of a classampling an arbitrary distribution c-@'(", sicaf system for the canfiguratlan 3 = (zi), imagine that this system is in equilibrium with a thermal reservoir and consider the microcanonical partition function for the combination of the system plus resemir
where ER is the energy of the reservoir, E: is the fixed total energy of the system plus reservoir, and the zero of the energy scale has been defined such that ER 2 0,Since the only role of the resewair is to accept or provide energy to conserve the totat energy E, it is very easy to sample Z, As in the case of the Metropolis algorithm, let all components = zi+C where is uniformly of 3 be held fixed except zi, and define the trial moves and = ER S(sl,. Zj*.sN) $ ( g 1 sN). distilbuted on This trial move i s accepted if > 0 , since it corresponds to an accessible state of the system plus reservoir, and is rdected if G < 0. since the move would not
(-9,v)
G g
+
.
zr
.
-
...ZT ...
correspond to an accessible state. By the usual arguments of statistical mechanics, if the average value of BR is much larger than the average value of the enerl;yy of the system, the accepted moves sample e-flS(') for the system and the distilbution of energy in the reservsir is P(ER) cr e " b B ~ ,Since E rather than j3 is specifled in the algorithm, Ff must be evaluated explicitly. For a continuous spectrum, this is most easily accomplished by calculating f? = : otherwise, the exponential e - p E ~may be evafuated. Motczcuiar dynamics uses s"rBar ideas from stat"rticat mechanics ta sample an arbitrav distribution, but digers conceptually By invoking ergodicity rather than a random number pnerator to sample representative accessible states (Callaway and Rahmann, 1982). To understand the use of molecular dynamics to sample the distribution e - @ s ( g ) sit is useful to think of S(%)as the classical potential energy for canonical coordinates (zi), to introduce a canonically conjugate! momenta (4) and to define a classical Hamiltonian H = +S($). Then. the mean value in the canonical ensemble of an observable O f f ) depending only on the coordinates (Q) yields pecisely the desired average of 0(3) with respect to the distribution of e-PS(%)
4 xi
Next, one uses the fact that for an extensive system In the thevm~dynamicslimit, the mean values of" an observable i n the canonical and mlcrocanonicaf ensemble become equal. That is. the integral dFd2 may be carried out on the hypersurface of states with constant energy E =: Hfp,8). Finally, one considers the ctassicai Hamiltonian equations of motion for 2 and in a formal time variable 2 (not to be confused with physical time) and invokes ergadidty to let a representatiw sample of variables jl and with constant energy be generated by the classical trajectories $(I), i j f ~ Thus, ). this Full chain of reasoning yields
I
a
where the trajectory Z(7) is determined by integrating the equation of motion
with an initial condition correspondhg to energy E, The name molecular dynamics is motivated by the direct analogy of this procedure to the method which has been
8.2 SAMPLING TECHNIQUES
915
extensively used to samplle the classical partition function for molecular system by evolving classical equations of motion, In practice, as in the case of the Langevin equation, a discrete finite digerence approximation to the digerential equation, Eq. (8.45). is evolved yielding a discrete set of coordinates z'(") E 3(r,). The similarity between the molecular dynamics and Langevin methods and advantages of combining: them are discussed in Problem 8.3. A simple variant of this method (Polonyi and Wyfd, 5983) is especially useful for sampling an e8ective action of the form det M(%) in which a Fermion determinant arises from integrating out Grassmann variables deed( e-('M(a)( and the remaining functional integral is over Boson coordinates or an auxiliary fieid denoted by 2. In add"rt"lonto the formal momenta pli conjugate t o introduced above, additional cananicat variables Pi and X$ are introduced to define the Hamittonian
S
The corresponding Lagrangian yields the classicat equation af matlon in the formal variable r
Following the previous reasoning, the average values of an observable OEP) may be sampled using the solution to Eq. (8.47). f (?l,
Using the equipartition theorem, the temperature lijr may be calculated conveniently by evaluating the kinetic energy
where Me is the number of degrees of freedom associated with the coordinate 3. Note that this treatment may be generalized straightforwardly ta the case of a complex ~ 2' P z;zi ~ in the matrix M with real determinant by using ~ ( M ~ ( Z ) M ( Z ) ) w
+
i
Hamittanian and that in the case of lattice gauge theory, local gauge symmetry of the variabfe 3 reduces the number of degrees of freedom NEr! to be used in evaluating P.
434%
STOGHASTIG METHODS
8.3 EVALUATION OF ONE-PARTICLE PATH INTEGRAL The stochastic twhniques described above may be applied to many-body problems in a variety of ways, The principal intellectual chaftenge is to exploit the freedom and t o utilise physical insight t o Incorporate as much of the essential physics of the problem as possible into the stochastic algorithm. There are three main categories of choices to be made: the obsevvabies to evaluate, whether to sampfe the complete action or solve an initial value problem, and the h r m of functional integral to calculate, The first two sets of choices may be iltustratcsd most simply by the Feynrnan path integral for a single degree of freedom discussed in this section, in addition, even this simple example illustrates two OF the primary limitations of stochastic methods: the need to calculate a dominantly positive Integral and the need to independently sampfe the entire range of configurations
In order to sample dominantly positive functions, we must evaluate Euclidean rather than real time path integrals, One abvlous quantity to sample is the trace of the evolution operator C, (rile-BHin) which provides all thermodynamic information at inverse temperature 'The ground state expectation value of an operator O may be calculated by taking the limit in which $ is small compared to the excitation energy of the first excited stated:
a,
The imaginary-time response function:
may be used to extract energies and transition matrk elements for low lying excited states. However, it is not practical in general to anaiytica'ily continue the decaying exponenthais In imaginary time into oscillatory functions in real time to extract much useful information a bout the physical real time response function, instead of calculating the trace, it is often preferable to evaluate a specific matrix element of @-BN which is optimal for calcut&ion of a specific observable. Given an approximate wave function 6,. the ground state energy may be written
8.3 EVALUATION OF ONE-PARTICLE PATH INTEGRAL.
417
Clearly. the more physics that is incorporated in 4,. the less work c-BX must do to filter out excited state components, Note that the ability t o calculate the ground state energy may be used to evatuate the scattering phase shifts for a finite-range potential by placing an infinite wall at an arbitrary position z, in the ~txteriorregion, Since E specifies the phase shiFt S, E(x,) determines 6(k). Similarly, the apeclation value of a general operator O way be evaluated using g,l either exactly
or perturbatively assuming
I#o)
---
I#a)
is small
Fhatly, the splitting AE between the lowest two nearly degenerate states in a double well separated by a high barrier may be determined by evaluating the ratio
where and (BR are wave functions localized in left and right welts, respectively. ((1\4L)/(014L))z = For the simple case of a symmetric well with the definition a
=
((~I#R)/(QI#R))~
from which L1E may be extracted easily,
SAMPLING
THE:ACTION
One straightforward
way to evaluate the obsemables is to sampte e-S(sa.*.s~) as a probability distdbution t o be used to calculate the appropviatc mean value of operators, Far the trace of a local operator
Ps(3) = e-S('l*.."")/
xsL,..sN N
O(Z). Eq. (8.49).S ( 2 ) =
C g(z, -
nz=l
+ tV(z,)
time slices are equivalent by cyclic symmetry and interval
O
with zo
= z~
so that all
may be averaged over the entire
For a matrix element of the form Eq. (8.52a). S(%)contains In +In 4bcr(sr) and the operator must be evaluated only at the midpoint. O(zjy?). Because this approach requins sampfing the global trajectory at all times and we must consider the tlrnit t e 0,it is useful t o insure that the leading error in the path
418
STOCHASIIC METHODS
integral is of order r2 rather than 0 (c) or even O (&). We have already seen in Problem 2.4 that for finite potentbats the symmetric decomposition e-a = 6- f V e - " ~ is accurate to O (e2). By regrouping terms in products of and a local operator Q(%), it follows that Eq. (8.55) is also accurate to O(eZ). In the case of a hard wall boundary wndition, simply putting an infinite potential into the action introduces an error of O (J;)as shown in Problem 8.4. Hence, to implement a hard wall boundary condition a t position z = a, the path integral should be written using states which are manifestly odd about z = a, with the result
This antisymmetrization just subtracts off the Green's function corresponding to the first image across the boundary and one can explicitly verify for the first odd state of the harmonic oscillatar that the leading error is C? (e2) when the first image is included (Problem 8.4). Non-local operators, such as the kinetic energy operator are less convenient to evaluate than local operators. As shown in Problem 8.5, two straightforward ways to evaluate the kinetic energy are to evaluate i%) or ~ c - ' ~ ~ ~ z ~ [zmml) ) ( z with ~ ~ the ~ c results - ~ ~
and
The factor e - ~ ( 2 ~ - ' m - ~ ) zin the action makes the naive second difference diverge in 2
9
&+
the limit r -. O. -+ 0(1).reflecting the fact that the dominant trajectories in the path integral are non-diflerentiabfe. Both the remaining finite term in Eq. (8.57a) and the split-difirence in (8.51b) am subject to large variance, so it is preferabre t o use the virial theorem to replace (T) by a local expresshng, if is an eigenstate of H defined on the domain [a,b] then for any
I+,)
As long as +L vanishes on at least one endpoint, ."c0may be selected to cancel the boundary term at the other endpoint and (T)= $((g - z o ) ~ ' ( z ) ) . As mentioned previousty, one standard way of sampfifig the action P,(zl,% a . . .zN) is t o sequentially make a tentative change in each of the zi's in turn by setting zT = zi (A and to accept or reject the change using the Metropolis algorithm, However, other alternatkve are passible and sometimes desirable. For
+
8.3 EVALUATION QF QME-P"ARTIG/E PATH lPdTEGRAL
419
example, one may introduce a correlated addithe trial change for the entire trajectory
That is, one adds ta or subtracts from each coordinate a [physicalfy motivated) global change fi defined on each of the tlme slices. Another alternative is a multiplicative move of the form
which can scale the overall. trajectory to a larger or smaller spatial =tent, Note that any sequence of reversible moves may be overed in any order to fadlitate equilibrium and complete sampling of the entire space.
Instead of sampling the global action, it is often preferable to calculate matrix elements of the form ( + , I c - B ~ ~ # ~ ) appearing in the observables (8.51) (8.53) by solving an initial-value problem, Starting with an initial ensemble of points cfistributed according to 44%) (which must be positive), the ensemble is evofwd by successive applications of e-CB t o obtain an ensemble distributed according to $ ( S ) = e - @ H + b ( z ) which is then used to calculate the overtap (@,l$). This alternative is particularly useful for evaluating ground state properties since instead of having to retain an entire time history over many time slices, an ensemble of configurations is simply refined by emex the required number of times. The method has the additional advantage that it can conveniently Incorporate physical understanding of the nature of the solution. The simplest method for evaluation of a matrix element of the Eucfidean evolution operator as an initial value problem for an ensemble of coordinates is analogous to Eqs. (8.36-8.37) for evaluation of TN.Let us decompose the infinitesimal evolution operator into the product of a Gaussian probab"rity and a residual weight as follows:
-
where
so that
so that
420
STOGWASTIC METHODS
Fig. 8,4 Inlriai-value random walk for the ground state wave functlon $(z) In the potential V*(z).tn this example, an lnltlat ensemble o f five polnts Is sampted whlch Is nore delocallzed than $(Scf. A t each r step, the paints from $,(S) undergo both dlnuslon and rcsptlcatlon, wlith points being ramaved outside the classical turning points and created In the ctassfcatly allowed reglon, B y the final time, the comblnatlan of dtfluslon and repllcatl~nytstds the dlstrlbutlon represntatlve of $(a;) as shown. This product of probabilities and weights is evaluated as follows. First, XI is randomly selected according t o the distribution function 4b(a;t), which may be chosen to be positive, and the temporary value of the score i s defined to be W(sl). Gjven sl,2 2 is chosen to be Gaussian distributed about zlaccording to the probabiliQ P(lsz,sl)and the score is multiplied by W ( z 2 ) ,This procedure is repeated for all n and finally z, Is chosen to be Gaussian distributed about z,-l and the scare is multiplied by g,l(z,). For an ensemble of such calculations. each score #X(,), W(zi) is obtained with probability P ( s ~ +z ~~ ), # ~ ( z so ~ ) that the average value of the score for a large ensemble of random samples approaches (4, le-B(H-Ell+a). The statistical accuracy of this basic method may be improved greatly by replicating points at each step wilh probability proportional to W fz,) instead af accumulating the weight M/"@,) in the score. The calculation may then be viewed as a diRusion as sketched process with a source-sink term W ( Z ) and diffusion term P(z,, in Fig. 8.4 An Initial ensemble of points ( z i ) distributed accordlng t o 4&) is first
D:=;'
H:=;'
8.3 EVALUATION
O F ONE-PARTICLE PATH INTEGRAL
423,
difFused by the Gaussian P(zz,zl). In regions where V(z) > E, W ( z i ) l and the point zi is deleted from the ensemble with probability l- W(%').When W(%')> 1the point z ' is always replicated [W (z )f times (where [U;"]denotes the gveatest integer in W) and with probability W (d)- W (zi)] it i s replicated one additional time. In each successive step, points diguse according to zLml)and are replicated according to W (z;). Thus, elements of the ensemble are created in regions of attractive V E and deleted in regions of repulsive V E such that the final ensemble of points (S,) is distributed according t o P(z,+l, %,)W (z,)4(zl) and (4, [ e - b ( H - E ) 1 4 b ) is given by the average value of +,(I) evaluated with the ensemble {S,). Whereas the first method retains ensembles having products of weights W ( z i ) which may vary over many orders of magnitude, with corresponding loss of statktical accuracy, in the replication method each member of the final ensemble (z,) contributes with the same webht. In practice, the value of E is selected t o maintain a constant average ensemble size and yields an independent evaluation of the ground state energy. To show how such an initial value Monte Carlo calculation is actually carded out i n practice, a schematic outfine of the pr'lncipal steps corresponding to the process depicted i n Fig, 8.4 is shown in Fig. 8.5. The method may be improved still further by guiding the random walk using an approximate trial wave function Q I ( x , T ) which contains as much of the essential physics as may be understood at the outset, Instead of evolving the solution to the imaginary-time Schr'tidinger quation
~(zk,
-
-.-
n:=,
consider instead evolution of the product
As derived in Problem 8,6, the equation of motion for f ( z , 7) i s
which includes a driR term
$&
and a source-sink term
The infinitesimal evolution operator for $ ( B , .r)$(z, as shown in Problem 8.6 has the matrix element
where
and
4. 6'.and ( H -
g)$are evaluated at z,-1.
7) is
4(&,T- + ~
) e - ~ ~
422
STOCHASTIC METHODS
Replicate or delete according t o weight
W (gi) t o obtain
coordinates zlrn") (rejection with uniform random number) Adjust E; if necessary t o maintain
generation
R
Fig. 8.5 Schsmatlc rapresentatlon ot an lnltlal value Monte Carlo calculation of the ground State energy. T h e prtnclpal steps in generatlna 3Gi dlstrfbuted according t o $(z) and calculating F =
art?shown in the upper dragram.
E
as a functlon of generatfon. The lower graph shows the schamatlc form of tnttlai equlllbratlon t o the ground state requfres ME Inltlal stags, Thereafter, essentially Independent samples are obtalned by calcuiatlng i!?svrary kfI generattons.
To understand the role of the drift and source terms, it is instructive to campare sampling the ground state wave functions using the unguided walk. Eq. (8.62), and using the guided walk. Eq. (8.65). i n the special case that the trial function is constructed from the exact ground state wave function and energy Eo. 4(z, r ) = cEor$o(z). Whereas even in equilibrium, points are continually added and deleted in the unguidad
8.3 EVALttATlON
O F ONE-PARTLGLE PATH INTEGRAL
423
-
walk by the source-sink term ( V ( $ ) E ) leading t o a large variance for rapidly varying potentials, the source-sink term in the guided walk
vanishest yielding
$
which guides variance. The Gaussian difision term is shifted by the drift term & members af the ensemble away from regions where the wave function ts small towad where it is largest. fn this special case. f =. #(S,t)+(z,r ) is just the ground state density ($o(z))a. Even when the wave function is not known exactly, t o the extent t o which an approximate 4 incorporates much of the essential physics. the evolution is guided by that portion of the physics through the drift term and the stochastic treatment of the source term is only required to treat the remnant of the physics which is left out of the trial function, In the extreme case i n which nothing is known about the wave functbon, a constant, trial function reproduces the unguided walk. Ratios af matrix elements such as those in Eqs. (8.51) and (8.52b) have a particularly convenient form for guided walks, since
By the previous arguments, in the case of a general matrix element (4,/e-@Xl#b), the optimal guiding function is the solution t o the time-reversed equation of motion with the initiat condition d,(z) at fi (see for example Pollack and Ceperley 1984).
The initiat value random watk is closely related t o the Green's Function Monte Carlo Method which is described in detail by Ceperley and Kalos (1919). Whereas path integral Mante Carto projects out the graund state by successive applications (which of the filter e - L ( X - E ) , the Green's function method iterates the filter -may be thought of as an integral over exponentials c-'(H-E)dr). Importance sampling is included in the same way as above by filtering f = q!+ using Eq. (8,641. In practical caiculalions, the two methods are roughly comparable. Sampling the inverse by some interactive technique such as the Neumann Ufam method entails more computation than a single application of e-'(N-E), but the entire calculation does not need to be repeated several times t o obtain the E -+ 8 !"lit.
Tunneling problems ittustrate some potential pitfalls of the stochastic methods we have discussed and physicafly motivated remedies. Consider the spl"ltinp; between the twa nearly degenerate lowest states in the symmetric double well
Fig. 8.6 Tun~sHngtrajectories in a daubis well, A porz-Ion of a tr4actory contatnlng two Instantons connecting the left and right wells ts shown In (a), and (b) compares a classlcat Instanton (solid curve) with the average over many Monte Carlo histories (error bars). where the spatial coordinate has been scafed to yield minima a t z = &l, the energy has been scafed such that the barrier height is 1, and the penetrability i s controlled by the one remaining parameter m, As discussed in Chapter 7, in the stationary phase approxirnatbn the leading contribution to tunneling proper-ties is given by the instanton solution which satisfies the classical equations of motion in the inverted wet!, Using the results: of Problem 7.11 for the symmetric well in Eq, (8.681,the instanton solution is
with action
Thus, we expect a general trajectory to be composed of segments fluctuating around each minimum, connected by trajectories across the cIassical/y forbidden region which fluctuate around the classicaf instanton. Figure 8.6 shows a segment of a typical Monte-Carlo trajectory connecting the two minima and demonstrates that it has the expected instanton behavior, Given n periiodic trajectory locafized i n the left well, consider the problem of seqclentiatfy updating the configuration with the Metropolis algorithm to randomly sample the action, When the time step is so large that it is comparable t o the width of an
8.3 EVALUATION OF ONE-PARTICLE PATH INTEGRAL
425
F&. 8.7 Sketch of two tnstanton trial move showtng the result of addlng a tlassical two-instantan configuration to a trajectory orfgtnally tocallzed in one wall. an instanton-antb-fnstanton pair can be created by moving a single instanton, E fit coordinate S, from one minimum t o the ather. The change in the kinetic contdbutlan to the action AS -= 4 G , is then comparable t o the action for the instanton N 3.8fi. Thus, a pair will be created directly with roughly the correct ,, has crossed, there is no additional penafty for subsequent probabilky, and once z coordinates to crass. When E +c
9
G(ri-
+
-
Q) tanh ri) and the positions of the configuration fi = tanh instanton T(I, and antiinstantan are picked randomly. The cost in action for such a global change is dkstuibuted around the classicaf result, thus assuring that the correct equilibvium distribution of instantons is built up. Alternately ofifedng sequenthat and global updates thus ePFectively sums tocaf fluctuations and samples alt instanton sectors. Note that ofFering an unphyslcai globail move will not bias the result because it
426
STOCWASTlC METHODS
will never be accepted: the only danger is omitting physical moves which leave part of the space unsampled, If a physical observable i s dominated by some specific number of instantons, statistics may be Improved by importance sampling
vf)
Mere is any function which counts the number of instantons. It does not need t o be absotutely precise, since as long as it correctly counts a significant portion of the configurathns in the desired sector it will appropriately paputate that sector. The same pbysical information can be incorporated in a guided random waik by approopriate choice of the guiding function +(s,rf. For orientatian, it Is useful to note that ifone wishes to sample the ground state wave function beginning with an ensembte of points in the right wefl, one should definitely not use the ground state wave Function will inhibit rather than encourage tunneling to the other well. since the drift term Heuristically, it is clear that something like i s required in the forbidden region to force the ensemble across. The optimal choice 4(z, r ) = c ( ~ - @ ) ~ 4always , depends on the specific final state 4, appearing in the matrix element (&,lc-@Hl+s) being evaluated. Problem 8.7 presents a detailed example of how points are migrated across the barrier and collected in the region of maximal statbticat importance for the matrix element. Note that in contrast to sampling the action, where an unphyskal colieetive move will not be accepted, an Inappropriate trial function vvilll migrate the ensemble in the wrong direction, If the trial function becomes too unphysicat, the guided random walk may became a biased random walk, The basic ideas described here far the symmetric double well are applicable to a variety of physical tunneling problems, ranging from the lifetime of many-particle metastable state described in Problem 8.8 to the contribution of the ground state of cyclic interchanges of rings of atoms. energy of solid
5
8.4 M A N Y PARTICLE SYSTEMS The stochastlc methods developed in the preceding section may be applied t o a variety of many particle systems, In addition to all the freedom encountered in treating ane degree of freedom, we also have the possibilkty of utilizing atternathe functional integrals for the evolution operator.
PATH INTEGRAL IN 600RDINATE REPRESEPJ1A"F"IQN One obvious approach ta the many-particle prsblem is t o try to generalize the methods of Section 8-3 to a Feynman path integrali for many particles. We begin by considering the initial value random walk to calculate the ground state of a many-Efoson system. Recall that for a single particle in a potential, the filter e - B H ( p t s ) acts in the space of all one particle states to select out the ground state, Furthermore, the ground state wave function may be chosen positive so that $(c) may be sampled as a probability distilbution Eq. (8.67) and the diRusion equation for $. Eq. (8.63) yields a positive solution. In the many-particle case, e - f l H ( ~ z s . . ~s ~l ** - * z l )acts in the full space of M-particle states, including ail totaily symmetric states, totally antisymmetric slates,
8.4 MANY M R T I C L E SYSTEMS
427
and states of mixed symmetry to select the lowest energy state. For ordinary systems interaction with Coulomb, atomb. or nuclear forces, the low kinetic energy associated with spatial symmetry causes the lowest eigenstate of H to be totally symmetric with respect to particle interchange. so that e-BR will automatically project out the Boson ground state, Thus, for a system like liquid 4He, the ground state may be sampled by repeated appiication of the infinitesimal evolution operatar
..
to a positive! trial function dT (sl . tN),Because of the strong short-range repulsion and long-range attraction of the Helium-Helium atomic potential, allowing v(yi yi) to serve as a source-sink term introduces prshibitively large variance. Since, as ernphasized in the discussion in Section 8.1, importance sampling becomes especially important in high dimensions, it is essential to use a guided random with a trial function 4(zl...zn) Which has physicatly reasonable short-range correlations, One conf(zi zi) where f is determined venient possibility is the Jastrow form $(%)= variationalry, As derived "t Problem 8.6,the infinitesimal evoluthn operator for a guided random walk in the many-particle case is
-
nigj
-
where the source term is
the drift term is
and the eflective time step is
In contrast to the case of Bosons, formidable problems are encountered in formulathng the "Initial value random walk for many-Fermion syshms. The essentiat difficulty is the fact that e-@W#T filters out the lowest state of any symmetry, so that at large times the component of the lowest antisymmetric state e-Bg*($A14T) i s exponent"tat1y suppressed relative to the component of the towest symmetric state e-BEs($s i4T)[+s). At some point in the calculation, it i s necessary to project onto antisymmetric states. If this pria;jection couid be done exactly, there would be no problem, tf it is only done stochasticalfy, there is competition between the exponential growth of the symmetric noise -- c @ ( ~ * - ~ oand ) the stochastic projection error which only decreases as
h.
To project onto the antisymmetric space, we write the infinitesimal evolution operator between antisymmetric coordinate states
where The problem now arises from the minus signs in the determinant Det M, To see the egect of these signs most cfearty, consider the special case of a two-krmion system. which when w ~ t t e nin relative coordinates corresponds to solving for the lowest odd state in a symmetric potential, The antisymmetrized evolution operator in this case, with refative coordinate 2 = PI -- Pz and reduced mass p, is
As sketched in Fig, 8.8, one may now view the evolution as the diffusion of both positive and negative points, The kinetic contribution to the propagator for a point at y to z is composed of two parts as shown in Fig. 83a. Gausshn diffusion about the point y with no change in sign and Gaussian diasion about the pdnt -g with the sign of the point changed. This sum is the exact odd state propagator shown by the dashed tine. The sequence of configurations obtained by starting with an arbitrary odd wave function and diffusing both positive and negative populaticzns is shown in Fig, 8.8b. Both the positive and ntitgative solutions separately approach the symmetric ground state and the sum, which corresponds to first odd state, becomes exgonent'lally suppressed as shown by the dotted line in the bottam graph* In the special case of one spatlat dimension, the sign cancellation problem may < z~ with the be circumvented by working in the ordered subspace sl ~2 boundary condition that the wave function vanish whenever any two adjacent coordinates coincide, The wave function in this sector is positive and the complete wave function may be obtained from It by antisymmetry, in the two partkct emample, this corresponds to evolution in the domain of podtlve relative coordinate where the wave function may be chosen positive and the kinetic propagator is given by the dashed line in Fig. 8.8a. Note that one may group the factor potential term and view it as a repulsive ""ptential" VV,arising from the Paul1 prindple. Spin cancellations for Fermions in more than one dimension, however, cannot be avoided. In two or more? dimensions, the configuration obtained in a single step with a slgn change by interchanging particles a" and j may atso be reached in a series of steps with no sign change in which i and g' simply circle around each other, This digerence between one and higher dimensions reflects the fundamental digerence in
.. .
8.4 MANY PARTICLE S Y S T E M S
429
Fig. 8.8
Lowest add state In a symmetric potential, The odd-state ktnstlc propagator 1s shown In (a) and a sequence of config~ratlonsobtalned by dt(l'ttslon by Eq, (8.73) Is shown In (bf, how nodal surfaces and the surfaies defined by antisymmetry depend on dimension. In d dimension, the condition y$ (F1,F 2 . . ,Fm) = 0 defines a dra - L dimensional nodal surf"ace, whereas antisymmetry specifies constraints of the form Ti = 7;: corresponding to (dn - d ) dimendonal surfaces, Thus, for c% = 2, the dimensions of these sudaces coincide, and antisymmetry alone can cornpletefy specify the nodal surFace of the wave function, For higher dhension, however, the nodal sudaces have higher dimension ( d z - 1) than the surfaces specifid by antisymmetry (Qn- n), so that antisymmetry alone cannot specify the nodal surfaces, For example, for two particles in two dimensions wit h relative coordinate 7, antisymmetry of the relative wave function $(if = -$(-F) only specifies a single point $(0) = 0, whereas the nodal surface is a tine in the s; y plane passing through the origin. Hence, .For d > 1 antisymmetry alone cannot uniquely specify a sector in which the wave function is posktive. Whereas there i s no compfete solution to the sign problem for Fermbns in more than one dimension, there are two useful strategies, One possibility is to fix the nodes by selecting the best possible trial wave function and to perform a gu"red random walk in which no particles are permitted to cross the nodal surfaces defined by the trial function. Within each region bounded by nodat suufaces, the problem is egectively Bosonic, and the result is a variational calculation and upper bound an the energy corresponding to the best possible wave function with the specified nodes, A second possibility is to perfarm a transient estimate starting with the best fixed-node distribution and releasing the fixed-node restrictions. As paints cross nodes, canceilations willi give rise t o statistkal errors which increase exponenthatry with evolutbn time while the system retaxes to its ground state. The early stages of the relaxation may then be calculated by using sumciently large sample size. The same symmetry considerations encountered in the initial value difFusion prob-
-
430
STOCHASTIC METHODS
lem for many particle systems also arise when the global action is sampled. The unrestricted trace Tk (Oe-pH) for a many-body system would average the observable O over all Boson states, all Fermion states, and all unphysical mixed-symmetry states. Hence, the only physical use of the easily evaluated unprojected trace is in the limit p -) oo in which case it projects out the Boson ground state. To calculate true thermal averages of proper statistics, one must sample an action in which one or more sets of symmetrized or antisymmetrized states have been inserted. Using Eq. (8.72) at each time step yields
for Bosons or Fermions, respectively, where Mnis the matrix defined in Eq. (8.72) for the nth time slice, X: = S?, and N normalizes the probability. Since the permanent in Eq. (8.74) is positive, thermal Boson averages may be sampled straightforwardly in any dimension using the technique developed in Section 8.3. For Fermions. negative contributions of the determinant may also be avoided in the special case of one dimension by working in the ordered subspace as before. Finally, before considering alternative functional integrals for the many-body evolution operator, it is useful to note that the path integral Monte Carlo approach illustrated above for the case of N particles in coordinate representation may be applied to a variety of other physical systems with many degrees of freedom. For example, although the most common way of solving lattice gauge theories is to sample the global tagmngian action using the Metropolis algorithm, one may also treat the Hamiltonian form of lattice gauge theory as a many-body problem. The degrees of freedom are the elements of a group defined in each of the finks of a lattice in cl spatial dimensions (see. for example. Creutz. 1983). In the simple case of U(1). corresponding to quantum electrodynamics, a group element at a link originating at site n in direction k may be written U = e i A k ( " ) and the resulting Harniltonian is
where B is the sum of the link variables around an elementary plaquette corresponding to the lattice curl (8.75b) Bi(n)= c ijk [Ak ( n ej) Ak (n)]
+ -
(n')l = ibkktb,,t
imply Ek(n)= -i,&l. and the commutation relations [Ak(n), Note that conventional electrodynamics is recovered when the lattice spacing -+ 0 and the *Ba term from the cosine dominates. Discrete U(1) lattice gauge theory thus looks just like a many-body problem with coordinates xi replaced by compact variables 0 < Ak(n) < 27r and a four-body potential connecting links on common plaquettes. The only difference relative to the evolution operator (8.70) arises from the kinetic term for the compact variable A. which follows-by using a complete set of periodic states (Aim} = eimA and the Poisson sum formula
8.4 MANY PARTICLE SYSTEMS
43%
fn the limit E -+ Q, the contrttbutions of the peViodic 'images outside the fundamental interval [0,2~] are exponentially suppressed, and the evolution for A is precisely the familiar Gaussian difision encounter& in the many-body problem, The infinitesimal evolution operator for non-Abellan groups has analogous structure. For example, in the case of SU(2). group elements may be written in the form II = a01 i C:=, airi where the ri denote Pauli matrices and the real coeficientr as satisfy C,!=,a; = 1 and are thus constrained to the sudace of a three-sphere, In the limit E --,0,the kinetic term becomes ewhere AS is the distance between U(Z) and U(ii') and produces Gaussran diRusion on the three-sphere, The generalization of the potential term In (8.11'5) is the sum over all plaquettes of ReTr(UfL"UU)where the product of' the U's is taken around each plaquette. Note that this term stltl has the form of a four-body potent-"ral in the compact variables and that Re(eiAeiAeiAeiA) reproduces the cos B potential. (Eq. (8.75)) for U ( 1 ) . Further details are provided by Chin e t a[, (1984, $9851. Vet another example of this path integral Monte Carlo approach is the case of spin systems djscussed subsequently, En which a random walk. in the space of continuous coordinates zi is replaced by a random walk in the space of discrete spin projecthns m.
+
~* .
It is natural to ask whether any of the functional Integral representations of the many-body evolution operator using fields are preferable t o the particle coordinate path intelyrat for stochastic calculations. A prdori, it appears counterproductive ta resort t o field variables instead of particle coordinates, shcc continuum fields must be approximated on a discrete mesh, The field representatltion is thus no longer exact in princbpte, and in arder to obtain reasonable accuracy, the number of degrees of freedom required is immensely larger than the number of particte coordinates, Recall, for example, the case of 208Pb at the beginning of Section 8.1,which for even moderate lattice with 1 0 ' degrees of freedom to descilbe the accuracy would require a (100)~ physics of 6324 physical degrees of freedom, Hence, a functional integral over fields will onfy be useful when it circumvents an othewise insurmountable problem, such as the Fermian sign problem in mare than one d"rension. Consweer First the possibility of sampling the action for the coherent state partition function Eq. (2,601
where 4, is a complw variable or Grassmann number and a denotes the point of a space-time lattice and any internal quantum numbers. For Bosons, the "ttegrand is unavoidably complex, since even the one-body piece 4; (& - p T ) ~is w~n Wermitian, and one is thus confronted with the usual sign problem in sampling the real part, Note that arthough the coherent slate path integral for Bosons i s farmafly similar
+
t o the field theory of a scalar field, only the fatter is suitable for Monte Carlo sampling since the actbon for a scalar field may be written in terms of a real field (b and contains the Hermitian contribution d,(& For Fermions. practical methods have not been developed for Monte Carlo integration over Grassmann vaPiables. Recall, for example, that a matrix representation of Grassmann numbers w"rh ra generators requires X Zn matulces.. Thus, in order t o obtain a suitable functional integral over field variables 1603 either Basons or Fermions, it is necessary to introduce an auxilhry field and integrate out the complex ar Grassmann variables 4, as in Chapter (7).
Note that in order for the Gaussian htegral over cr to converge, a15 the eigenvalues of u must be negative so E t may be necessavy to introduce a constant or diagonal shift in
the potential (see Sugiyama and Koonin, i1)86). The essential question is when the resulthng determinant is positive so that the action may be sampled without sign cancellations, Although positivlity is nag assured in general, the integrand is positive?for one important class of problems: those in which the potential is independent of an internal symmetry having an even number of states, Consider the case of Eermlons with a spin Independent interaction and write the space, time, and spin labels included in a explicitly 4, -- 4rn(zr7 ) . Then the interaction is $ dz dzt C, 4;(z)b,(z) u(z - 2') C,. &L,(zt)4,8 ( S ' ) , only a single spin-independent auxiliary field is required and
D (c)[Det M(C)]'~"
e-'(@)
where
+
Since Fermions have half-integer spins, only even powers 2S I of the determinant arise and the inlegrand is positive, The same argument applies for any other internal
symmetries such as isaspin, eolor, or flavor and ground state properties for even numbers of Bosons may be obtained by treating them as Fermiions with spin degeneracy M as in Section 3.1. Once the partition function has been expressed as an integral over a,Eq, (8.791, observabies are evaluated as the sum of contractions using Wick's theorem in the usual way (sea Eq, (2.84)):
Note that even though [Det ~ ( o ) ] ' ~ ems(@) " is positive. the signs gP arising in Wickas theorem may cause uncontrollable cancellation for some operatars. The biggest practical problems in applying Eq. (8.80) are encountered in evaluating Det M and M-'. The quantity ~ e t ~ ( ~ ) e - ~may ( " be ) sampled directly using the microcanonical method, Eq, p.48).or the exponent In Det, M ( a ) - $(Q) may be updated M . The inverses arising in matrix using the relation &in Det M ( o ) = elements, Eq. (8.80) or in updating the actbn may be evaluated by conventional sparse matrix techniques such as the conjugate gradient met hod, or by stochastic methods such as the Neumann-Utam method (8.61) or sampling a Gaussian integraf d(4le-f Q~M.I~,, M,;' = d { # ) e - t h M * ~ @+a4r/ j Another closefy related method is to use the operator form of the Hubbard Stratonovich transformation and replace the trace by a matrix element of Slater determinants, (Sugiyama and Koonin, 1985). Thus, for example, the gmund state energy is calcutaled by evaluating
E
I
E. = lin
CI-..m
where
and l@) denotes an N-particle Slater determinant. Since U , corresponds to evolution in a time-dependent one-body field specified by cr(z,~), U,jl;b) is a Slater determinant and (O\&l&) is a determinant of overlaps between the single-particle wave functions in (41 and U,14), Again for spin-independent interactions, one obtains the overlap determinant to the (2s -tllthpower.
8.5 SPIN SYSTEMS AND LATTICE FERMlOMS Additional interesting possibillties arise in the stochastic solution of quantum spin systems on a lattice and of closely related lattice Fermion models, The magnetic properties of solids may be described by a quantum spin Hamiltonian since the exchange interaction between electrons gives rise to an eRFective spin-spin interaction (see. for example Ashcroft and Mermin. 1976). For illustrative purposes. it suffices t o consider the simple case of the Weisenberg Hamiltonian for spin one-half with nearest neighbor interactions
where poskllve S corresponds t o ferramagnetic coupling and the spin one-half operator is wiltten in terms of the Pauli matrices Si = {Q:, or, of). Extension of the basic ideas to anisotropic couplings, external fields, and higher spin is generally straightforward, and further details may be found in the review by Dettaedt and Lagendijk (1985). The connection in one dimension between lattkce spin Hamiltonians and lattice ermion models, such as the Hubbard model (1963,1964) is seen by noticing that the raising and lowering operators
behave nearly like Fernions. They have Fermlon commrrtat'ton relations on sites
but have Bosonic commutations retations between sites [ai a!l = [nisaj] = [af a!] = O 2
r 3
.
(8.83~)
True Fermion operatars may be constructed from the a's via the Jordan Wigner transformation which inserts the requis2e minus signs by csunting the number of occupied sites to the left of the it" site [see Problem 11). Thus, for example, we may rewrite a lattice Fermion model as a spin $ model as follows
In general, except for technical details such as treating the Jordan Wigner phases with periodic boundary conditions one may pass fi.eety between the spin and Fermian language, and it i s often usefut t o think of the spin-projection representation as an occupation number representation.
8.5 SPIN SYSTEMS A M 0 LATTICE FERNrONS
435
One alteunathe t o evaluate the evolution operator for spin systems is to use the Trotter formula as we have for other path integrals (Suzuki, 1996). To break up the evolution operator into a sequenw of infinitesimal steps which art? particularly easy t o evaluate, "I i s useful to decompose the Hamittonian into sums of non-overlapping eel! Mamittonians. To illustrate the decomposition, which may be performed for any lattice in any dimension, we consider a one-dimensional chain and break up M into nearest neighbor interactions beginning on odd and even sites
X{even odd = -J
gi .$;+I .
(l:
is written in the uslrat way that when complete sets of intermediate states are inserted, the evolution operator has the .Form
The
infinitesimal
C
operator
2 tl-#H~dde-asereae-*Wodd.
50
The essential advantage of this result is that since Boddand He,, are sums of commuting operators each of the matrix efements of @-"add and e - e H ~breaks ~ ~ ~up into products of independent two-spin matrix elements. b r example (MS
le-eHewem
/M')=
*
a
sewn
( m ~ m ~leLJSi'Ss+x +l im: m:+,
)
8.878
where ( M )denotes the set of spin projections on the sites (rn1m2... m u ) , The sequentlai action af He,,, and Haad i s sketched in Fig. 8.9, Since the total spin projection is conserved, as emphasized by Hirsch e t al. (1981). it is convenient ta view the evolution in terms of world tines representing the trajectories sf the up spins or equivarentfy, lattice Fermions occupation numbers. Using the familiar identity retating to the spin exchange operator .Pi S;
and the fact
.PZ= 1, the infinitesimal
evolution operator Eq. (8.87) may be written
436
STOCEIASTIC
METHODS
Fig. 8.9 Checkerboard decamposltlon of In oncr. space dlmenston, The shaded regions Indicate sltas connected by He,e, and Hodd. The softd world Ilnes denote an@posslble trajectory for the up spins under the sequential action af Hevenand H o d d fn a small portion of the space-time lattlce. Because a t each tlme step the system evotvas elthsr under K,,, or Eoddr the world lines are constrained ta the shaded regions.
tsJ
osh 2
O O
rS + sink ---Ei 2
9
cosh sinh 6f
sinh %f cash
9
0
O
For the Heisenberg ferromagnet, J > 0, ail the matrix elements are positive and the inkiaf-value random walk far any dimension lattice proceeds like the coordinate space path integral for the many-Boson problem with the world fines for the up spins corresponding t o the trajectories for particles. The antiferrornagnet naively appears to have s k t ~problems analogous t o the manyFermion problem because of the negative of-diagonal matrix eiements appearing h Eq. (8,891 Tor J < Q. However, a cure is suggested by the fact that classical magnets and ferromagnets on bipartite lattices are equivalent. By definition, a bipartite tattlce is composed of two sublattices such that all the nearest neighbors of a site on one sublattice belong t o the other subfattice and vice versa, Common examples include the linear chain. square, cubic, and body-centered cubic lattices. Classically, if all Fe spins on one subfattice are mapped into new vadables with a sign change SA --,-SA while the
8.5 SPtN S Y S T E M S AND LATTICE FERMLONS
cisi ssi
437
$;Bis,.
signs are unchanged on the other lattice SB -+ #B. then J -+ -J Quantum meehanicafly, the ferromagnet and antlferromagnet are fundarnentalfy different and therehre cannot be mapped exactly into each other. However, we are still free to choose d'rflferent bases on the two sublattices for our awn convenience, The appropriate transformation for our present purpose is to rotate all states on one sublattice by 180° around the z-axis = e'p'slm). The transformation of the spin -+ -ay and os -+ Q" so that aperetors under this operation is as -., -czs, ifi -, -i?i .8i 2a:o; and the troublesome off-diagonal minus signs are removed. The ferromagnetic and antiferromagnetic cases are conveniently combined by defining the matrix
I&)
+
y r o o Q1
which corresponds t o the matrix elements of Pi$ in the original basis for 71 == 1 and the matrix elements in the transformed basis for q = -1. Since = 1, we may write
where == 1 for a ferrornagnet and q -- -1 for an antikrromagnet. The explicit q dependence af the diagonal matrix etements contains the physical digerence between quantum ferromagnets and antiferramagnets. Note that in contrast to the Fermion sign problem which was cured only in one spatial dimension, the sublattice transformation has efiminated minus signs for bipartite lattkes of any spatial dhmensian. For other antiferromagnetic probfems, however, such as the triangular lattice which is important because it is the simplest example of frustrated spins and the problem of ciclic exchanges in solid 'He. the minus signs cannot be removed completely. When the trace of the evolution operator i s evaluated instead of a spec"tfic matdx element as in Eq. (8,873, one must again address the prohiem oF trapping in a restricted subset of at! spin states. Sequeneal updating of inndivlduat spins onfy samples states of fixed total spin projection (or total Fermion number) since lir commutes with S,. In order to sampie the Full space, microreverslble global moves must also Be included which offer the possiglity of adding or removing an entire world line. For Fermions with periodic boundary conditions, which may be represented by world tines on a torus, one must simHariy allow for transitions between configurations of diflertlrnt winding number n, where n is the number of times a point on a world tine must circle the torus before returning to its original position.
SPECIAL METHODS FOR SPtFJS
A method by Hanrcomb (1962)exploits the special properties of spin algebra to avoid the Finite E errors inherent in path integrals and represents the thermal trace as an exact average. Using Eq, (8,881, the MamiltonIan may be rewritten as Ii = Si . S j = .Pii ~ N where ~ J Pij permeates the spin states at i apd j and Nb is the total number of bonds. To calculate the partition function, one may omit the constant i N b J and expand E - @ H as a power series with the result
-Jx(ij)
+
.
l,, , Nb denotes each nearest-neighbor pair and Cm E denotes a particular sequence of n permutation operators, The Monte Carlo problem then corresponds t o a random walk in the sample space of permutations (G;, n .= 1, m) and thermal averages are given by
where
m$
(Pml,P,
=
...P,)
( 0 )=
C,,c, Q fca)n(cm) C,,,. n(cm1
where
and
The spin traces of permutation operators are straightforward t o evaluate, and for ferromagnets. II(C,) is a positive probability. A random walk to sample EfC,) may be defined by a sequence of steps in which a permutation i s nicroreversibly added to or deleted from the sequence C,. For example, following Lyklema (1982) one may define a trial move for which the probability for adding a permutation is
F (C,-. C,+I) =
if(,)
Nb n + l
&
is the probability of choosing a particular permutation P,. is the probwhere ability of choosing a particular position in the sequence C, and f(n) is a distribution chosen t o specify the probabiiity of adding a pernlutation to a sequence of length n. The corresponding probabiiity for removing a permutation is
i s the probability of choosing the specific permutation to remove. Using where the generalized Metropolis algorithm of Problem 8.1, the process is microreversible if i s accepted with probabljity the trial move to configuration
8.5 SPIN S Y S T E M S AND LATTICE FERMIONS
439
For an antiferromagnet. Eq. (8.93) is unsuitable as it stands, since each odd term in n is negative giving rise t o sign cancellations which cannot be handled stochastically except at high temperatures. Thus. following Lee e t al. (1984). it is useful to rearrange the series using the identity (8.95a) p-. '3 - l = hij h:j
-
In the representation of Eq. (8.90)
h=
I*
o o q o q D Oj 0 0 0 0
from which we note that hij interchanges spins if i and j have opposite spin states and yields zero for identical spin states and that the minus sign of h may be removed for a biparticle lattice, Ignoring irrelevant constants, the partition function for J < 0 may be expanded as follows
where 0 = h:$ or -hij. Since only closed loops with an even number of h. *S contribute or equivalentty by virtue of the sublattice transformation, all Cars yield positive contributions and the evaluation of thermal expectation values and the random walk in the space of operator sequences C, proceeds as before. Another novel feature of spin and lattice Fermion systems is the possibility of using a discrete rather than continuous auxiliary field. If some operator A can only assume a finite set of discrete values, it is always possible t o replace the integral over a continuous auxiliary variable in the Hubbard Stratanovich transformation e t b A z = & ~ d o e - f " ' + ~ ~ ~ by a discrete sum. For example, consider an operator A which can only assume the three values {-1,0,1), i n which case a single sum over an lsing variable suffices
W here
This discrete lsing auxiliary field has been used by Hirsch (1983. 1984) to replace the two-body interaction Untnl in the Hubbard model by a sum over one-body fields. Using the fact that this occupation number for spin up and spin down particles, nt
+
and nl, are 1or 0,the operator A may be chosen t o be (mr - nl) or (nf ni - 1). yielding the following real identities for positive and negative coupling. respectively.
and
When the Fermion trace is taken over the one-body operator. the resulting lsing action has much less phase space than the continuous Gaussian action and is more convenient t o sample. The basic stochastic methods presented in this chapter are currently being applied to a broad range of quantum systems involving many degrees of freedom, including both many-particle systems and field theories. Although it is impractkal t o provide an exhaustive list of references, the following list provides a starting point for exploring applications t o specific systems, Because of the absence of sign problems for Bosons, Monte Carlo studies of liquid and solid 4He have been the most definitive, and representative results are given by Kalos e t al. (1974). Whitlock et al. (1979)and Kalos e t al. (1981). Extensive results are becoming available for a variety of many-Fermion systems lncIuding molecules (Anderson, 1981: Moskowitz e t al., 1982: and Reynolds e t al., 1982)' the electron gas (Ceperley and Alder, 19801, llquld and solEd Wydmgen (Csperley and Alder, 5981). liquid 3He (Lee et al.. 1981) and fight nuclei (Zabolitzky and Kalos. 1981: Zabolitzky e t al. 1982). Applications t o spin systems and modefs such as the Hubbard model are cited in the review by DeRaedt and Lagendljk (1985). Because of the simpfifications which arise for krmions in one spatial dimendon, particular attention has been devoted t o one-dimensional models such as a potential model for nuclei (Negete, 1979),mesonnucleon field theory (Serot e t al. 1983). a quark model of hadronic matter (Horowitz e t al. 1985). and the Gross-Neveu model (Hirsch e t al.. 1982). Lattice gauge theory has been studled both in the Mamiltonian form described in this chapter (Weys and Stump. 1983, 1985: Chin e$ al., 1984, 1985) and in Lagrangian form as reviewed, for example. by Creutz (1983) and Kogut (1985). Recent references to these and other topics may be found in the conference proceedings edited by Gubernatis e t al. (1985).
PROBLEMS FOR CHAPTER 8
The moet crucial problem for leaning to apply the ideas dmeloged in t h k chapter is Problem 9, which leads one thraugh the salient practical pfoblems encountered in an actual calculation and then requires a real numerical Monte Carlo calculation. The remaining problems derive, apply, or generaIiae results pfesented in the text,
PROBLEMS FOR CHAPTER 8
441
PROBLEM 8,l. Gt?rreralfred Metropolis Atgortthm The standwd algorithm of Metropolis e t al described in Section 8.2 may be generalised in several ways, Let F(3 -+ iT) be any distribution, not necessarily symmetrical, and a) define
Show thag if a tentative step ET is generated by F($ -+ Z T ) and accepted with probability min (1, q ( P --r i&)), then the evolution satisfies the mkrorevemibility condition (8.27). Note that this algorithm reduces to the standard method if F is symmetric. b)
S h w that acceptance with probability
is also microreversible.
PROBLEM 8-2 Varlant of the Maurnann-lffem Method Show that when M"' is calculated by evaluating the mean value (Sij), Eq. (8.40), the variance is ((Sij i j P;' where Xi, = Thus, the method has two problems: eaeh walk only contributes to a single mzLtrk element of the inverse and the variance increases inversely with the stppping probability. A useful alternative i s to define a separate score Sii for each j as-folXows, At eaeh step k when the walk has progressed to state i k ,the product of all the residue &aia ..Q,- is added to gi iii,. Thus, the stop probability is removed from the score although id still affects the length of the walk and each walk contribulers to many matrix elements. Show that ($;$) = MC'.
.
PROEIL E M 8.3 Comparison of tangevin and Molecular Dynamics Methods Write the lowest order finite diRerence spproximations to the Langevin equation, Eq. (8.32) with = so that (8.34) has unit standard deviation and the molecular dynamics equation of motion, Eq. (8.45), to obtain
9
I as z?'~ = zl - - (&T)'2 dz;
+ ZI (%l" - S;-')
(Molecular dynamics)
.
Thus observe that the stochastic noise enters in the same way as the velocity fi 2Ar
'
Comment on the efffciency of the two methods in sampling the local space and in sampling the global space, Thus, explain the potential advantages of aiternizting between the two evolution algorithms, A specific example is given by &gut (3985). PROBLEM 8.4 Olscrete Time Errors a) To see that the symmetric decomposition c- * T ~ - t v e -gT really yields leading errors of B ( c 2 ) and asymmetric choices yield O(C), find the eigenfunctions of U v ( z t ~S f - &(S-@)'*[V@' +f'-q)sal, Since it is clear that a quadratic form will satisfy $ dy U, (z, Y ) $ ( ~ ) = e-LA$(z),one straightfaward method i s to try a general quadratic h r m and ta solve for the parameters. Verify dhat for the symmetric case q = you obtain a harmonic oscillator wave function C"" with energy f
442
STOCWASTIC METWOOS
.''l Note t113t if ( e - L T ~ - e V ) n is used as a Glter to obtain the ground state wave function, an extra order of E accuracy is obtained if it is multiplied by ewhenever it is sampled for an observable. b) Now consider an o~cillatorpotential with an infinite wall at the origin, i.e, V ( % )= 5 z Z z > 0, and V ( % = ) +m z 2 0. Show that simply adding an infinite potential for 1; < O in the infinitesimal evolution operator yields errors of 0 (fie c) Use the propagator with images, Eq. (8.565) to solve for the first odd eigenstate of the harmonic oscillator as in part a) with q = $ and show this hard wall boundary condition yields C1 (c2) accuracy.
PROBLEM 8.5 Evaluation of Kintttlc Energy a) Derive eqs, (8,49a,b) for (T)by inserting wstys:
and (z,[e-:Ve-eTe-*Vfl(z,-l)(z,-l I$e-*Ve-eTc-fV
into the path integral two (z, le- f V e - L T $ Z ~ - tV/znel)
\z,-~). What is the order
of the leading emor in c in each case? b) Derive the continuum form of the virial theorem, Eq. (8.58), being careful to allow for non-Bermitian boundwy conditions. In the special case that $/ vvanislres at the upper and lower limits, note that (T) = ; { ( S - zo)Vt(z)) for any choice of zo. Show explicitly that (V') = 0 in this case. How should zo be clsosen to optimilie statistics for the case of the ground state of the Itwmonie oscillator? 6 ) I)&w the virial theorem for the discrete wolution operator as follows. Let 4, denote any eigenfunction of the discrete evolution operator e - ' 9 eeETe-' 5 . Hence, ( $ , l [ p i , e - ~ $ e - € ~ e - ' f 114~) = O go that the commutator is eero in the trace or ground sbate expectation value. Evaluate the commutator to obtain
Since to O(c2) the evolution operator may also be written e-:Tc-LVe-*T, show ) ) 0~ (,e 2 ) , where S is the action for the trace. that (T) = ($ Ci z ~ V ' ( ~ ~ to
PROBLEM 8.6 Guided Random Walks a) Using the Schriidinger equation in imaginary time for $(z, r ) , derive Eq. (8.64) for the evolution of the product f ( X , i)ZE #(z, r ) +(z, i ) . b) Derive the infinitesimal evolution operator for f, Eq. (8.65). Note that this equation has the form naively expected from the diffusion equation Eq. (8.641, except that E is replatced by er^ in the Gaussian, Om metlrod is to show
in the usual way and to expand in$(%) wound y being careful to retain all terms of order E and to normalizle the Gaussian appropriately. Another alternative is to use the Baker-Hausdorf identity eA+D = e A e D e - * ~ A * " ~ + . . . ~ on and note so that terms like must be retained.
PROBLEMS FOR CHAPTER 8
443
Qbsewe that in the cme of the hmmonic oscillator, d is a con~ltantcorresponding to a trivial rescaling of the time step. The role of E^ in practical calcula$ions is discussed Iby Moskowitz e t al. (1982). c) Derive the infinitesimal evolution operator far a many-Bomn sysbem with wave function $(zl ..S,, r ) and guiding function #(zl. ,z, 7 ) . Do any new complications arise in the many-particle c a e ?
.
..
~ ~understand R haw a timePROBLEM 8.1 Time-Dependent Guiding F L I R C ~To dependent guiding function, Eq, (8,64),guides a random walk, consider a ymmetric double well with a high B i t ~ e rat the origin a;nd minima a t fD/2. Let +,(z) be a n m o w Gaussian of width Az centered at; -D/2, let 4b(r)be a positive w8ve function localired in the right well, and consider the m a t r k element (4, a) Explain why the guiding function d(z, 7) = c ( v - B ) X 4 , is positive at all tiyes. Determine its behavior at times very close to and explain how the drift term $ forces the ensemble of points into the region of maximum statistkal importknee.
&
Consider 4 at early times such that /3 - r > for which It#(@ - 7 ) ) R+ 10)(0/4b) I l)(l14b)c-(@-')a where A and E2 denote the excitation energies of the first and second excited states and ID) and II)denote the ground state and first excited state. Sketch 4 for (p - r ) << % and for (p - r ) >> Note that 10) an& 11) correspond to the lowest eigenfunctionrs $5, and in the right-hand well sati~isfyi-ngeven and odd b o u n d q condikions at the origin, and LIhuss show ehitt $. (O)$L (Q) = 2mA $,(z) 4 .(z). Hence, evaluate the drift term at the origin and show
b)
+
fr.
$r
$2
Note that since the density at the origin (0) is much less than the average density, D+z(O) << 1 and the drift term will migrate the propulation from the right well to the left well in characteristic time much less than %
In summary, explain qualita(live1y the role of the guiding function in each of the three time domains: (B - 71, < (p - r ) $, and ( p - r ) < Define a locali~ed eigenstate 4; as the solution in a single well defined c) equal to V ( % )for z < 0 and equal to WO) for z > 0, Let 4.R be an analogsue eigenstate in the right welt, How does this analysis change if one calculates the overlap ($L used to determine the gap in Q. (8.53)1
<
-&-
&.
PROBLEM 8.8 tffetfme of it Metastable State Although in general it may be difficult to express a physical observable in real time in terms of quantities which can be evduated in i m a g i n q time, it is possible t;o do so for the lfitime of a
444
STOCHASTIC M E T H O D S
metastable state.
a) Use the asymptotic form of the wave function in the potential sketched in this problem to obtain the familiar Breit-Wigner phase shifi
6(k)
=t-
6,-t- k6-i- tan-'
r ko - .K:
G,
==" where k = E. is the energy of the metastable state, 7 = and b is the origin witft respect to which 6 is defined. Using the argument in Sect~on 8.3, explain how a hart3 wall boundary conditio~limrty be ixnposed at an arbitrwy positior~a in the exterior region and used to deterntilre Wlrat are the linlitations of tllis mettrod?
7"-
a,,
+b 4
Now a left well Vr.(z)= B(zb- z)V(z)+ B(z - zb)llband a right well VR( X ) = B ( z ~z)Vb+ B(z - zo)V(z)with ground state energies E& and ER. Show tllat to order a, comesponds to the value of a for wltieh EL I- ER arrd that b -4- a = Ltrn Hence show
c,
-
that is, tlre lifetilne is given by the energy splitting in a saltably defilied double well. Note that siltce &E (LJvIR)this result has the familiar Fermi golden rule form of a transition matrix elemextt squared tinles a density of states. PROBLEM 8.9* Monte Carto Celcu@atiortTo fully understand and appreciate the stoct~asticmethods described in this cl[raptrsr, it is necessary to solve a concrete
PROBLEMS FOR CHAPTER 8
445
problem
a) The principal practical problem in real calculations is establishing credible error estimates. To illustrate the essential i~sues,Monte Carlo results from studying a many-l"ermion system ia one dimension (Negele 1979,1985) are shown in the Bgure. To use &s. (8.12- 8.13) for the statistical emor, we must know how many steps of the fitropcrlis atgoritbrn or applications of arc? required to obtain statistically independent samples. Hence, if Oi denotes the it" sample of some observable (Q), it is useful to calculate the autocorrelation function,
Oi Oi+*. Figure (a) shows the autocorrelation functions where ((Oi Oi+k)) E for the enttrgy and density. Explain why these are different and haw often to sample each. for independent results. Xf a calculation had not yet equilibrated, how might this lack of equilibrium show up in the autocorrelation function? Suppose one uses Eq, (8.55) t a cafcutate an observrtBIe by averaging over the complete time mesh. Are observable8 on; adjacent time slices correlated? How would you take this into account in est ablishlng errors? b) Figure (bf shows the mean value of the energy obtained by s t a t i n g with our initial configuration, sampling the energy and variance every (m)'" update and averaging fn) such, samples to obtain the energy and error bars shown. At what point has the energy equilibrated? Are the emar bars past that point ststisticicalj.y sensible? Suggest several ways to check the equilibration and error estimates.
Figure (c) shows the energy as a function of time step sise AT c) for t h e e different calculations: EM denotes the virial theorem average (Cigj - zijvt(lzi - z i l ) v(lzi - ~ ~ 1 ) ) when ) the action Eq. (8.72) is sampled with the Metropolis method and ET and denote the trial energy, Eq. (8.51) and normali~ationenergy (8.61) obtained using a trial function composd of two-body Jastrow correlation factors xnultiplying a Slater determinant, Explain the expected dependence on AT for each energy and comment on the magnitude of the A7 dependence, If this characteristic scale of variation of u is of the order of 0.2 (in units where h = m = I), explain the value of AT at which you expect cansergeace to set in and comment on the numerical results, b) WiLh this preparation, c a r y out a numerical Monte Carlo calculation on simple model or physical system. For those inexperienced in numerical calculations, an ideal starting point is povided by the H - molecule. The calculation is described in detail in project 8 of the text by S, E. Koonin (1985), which aIso prwides a working program far use on persond camguters.
(41~~
+
PROBLEM 8.10 Par the case of many Fermions in one spatial dimension, show that in the ordered subspace, the determinant in Q, (8.72) ma,y. be a p p r o ~ m a t e d Det M
W
nN (1 -
Intepret this result in terms of images.
e - ~ ( r $ - z ~ - ~ ) ( ~ ~ - @ i - ~ ~ ) .
i=2
Determine the evolution operator analogous to Eq. (8.71) for a guided random walk,
PROBLEM 8,Zl Jordan-Wfgner Ransfarmatian a) Verify that the operators defined in Eq. (8.84) satisfy the Fermion anticommutation relations (ciscf) = sij, { c i P c i )= {cf ):: = 0. Use Eq. (8.83) and the algebra of the Pauli matrices to show that ci"a:ai is a rotation aperator and thus note that the operators c i and cf in Eq. (8.84) create kinks and antikinks in which all the &pinsto the left of site i we- rotated around the z a x i ~by &E. Does. a, spin problem of the form of Eq. (8.85) with periodic b o u n d q conditions transform into a Fermion problem with periodic boundaq conditions? z the one-dimensional &-function problem in Problem b) The Bethe a n ~ a t for 1.9 was actually made for the Hekenberg spin Hamiltonian H - J C, SiSi+l. Show that this may be rewritten in terns of Fermion opctrakors as
and take the continuum limit to obtain the second quantieed form of h.(1) of Problem 1.9. Utilise the possibility of rotating the spins on one sublattice to show that the ferromagnet corresponds to an at tractive &-function interaction and that the ant Zemamagnet corresponds to a repulsive S-function.
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INDEX Analytic continuatfon, 248-249 Anderson-NFggs mechanism, 222-1126 Anharmonic oscillator, 37G382 AnnTtrilatIon operators, 11-19 definition,
13
commutation relations, 13 Antiferrornagnet, 4 3 H 3 7 Asymptotic expansions of high orders of pe~urbationtheoy,
371-382 of integrate54-56.121 Autocorrelation function, 445 Auxiliary field, 186, 198, 332-336, 383-
385,431-433 discrete auxiliary field, 439-M0 Barrier penetration, 124, 366370 Bethe ansatz, 44 Bethe-Safgeter clquation, 268 Bohr-Sommerfeld quantization ruk, 356 Boltzmann quatian, 305-308 Boref summadan, 373-376 Bosons, 4 Bose clondensation, 126131.153,
193-194 Bose gas, 1fi9-171 coherent states, 20-25 normalized state a%, am),8, 12 symmetdzation operator, 5 syrnmetrized state jal, sac.. * a d ,6 wave function, 4 Bounce sotution, 360, 365,368,378-379 Broken symmetry, 1814-185.192
-
Central limit theorem, 405-463 Checkerboard decomposition, 435-438 Chemicat potential, 48, 164-166 Ctassical limit for Boson coherent states, 33 for harmonic oscillator, 39 far path Integral, 68 Closure relation Boson coherent states, 22, 38 Fermion coherent states, 31 manyrpaaicle states, 12 Coherent states, 20-.36,6H8,338
table of principal results, 37 Cornpressibility, 362 Conjugah field, 191 Contraction, 17-78, 80, 81, 121, 339 Cowelation functions correlation length, 201 density-density, 262 spin-spin, 200,203-2014 Creation operators, 11-19 commutation relations, 13 definit'ian, 11 Curie" law, 284 Density of states, %S5, 359 Diagram rules Bosons at '2 = 0, 15s157 density-density cowetation function,
264 expectation tcatulis of operator, 160-
101, 135,152 Fermions at 21 = 0, 146153 frequency representation, 95, 148 graund state energy, 147,150, 159 Hugenholtz diagrams, 90, 135 labeted Feynman diagrams, 81-82 momentum representstion, 94 self-energy, 114, 152-153, 161 Eable of, 154 tkermat Green" function, 103-109 unfabeled Feynman diagrams, 84 xero-tern perature Green's functian,
16&161 Diagrams amputated, 153 anomalous, 164,169 connected, (34, 99, 106 distinct, 80,98 irrdudblcs, 113 ring, 86,87,91, 171,265, 347 tlme-ordered, 157-1433. ill Dimensfonal analysis, 201)-211 Domain walts, 226221 Dyson equadon, 113,260,282 IERective mass, 255-259, 3 W 3 0 2 ,326 m,
255,257
INDEX
456
mk,
255-256
ERective potential, 108-111 Equifilrritrrn condktlions, 42 Equivalent pair of propagators, 89.91 Euclidean action, 65 Evaluation of determinants, 395-3% Faithful representation, 78 Fermi fiquid theoq, 29G331 collective adtatfons, 305-313 equiffbrium prope&!es, 29+305 microscopk foundation, 313-3214 normal fermi tiqu"rd,297 stability criteda, 326327 table of parameters for Ifquid 9He, 323 Fermions, 5 anl'isymmetrizatlon operator, 5 ant'isymmetrited state
lal,cuz, * *am),6 coherent states, 2S33 Ferml gas, 42-44,171, 329-.330 normalized state lair sal - a,). 8, 12 wave fundion 5 Field operators, 16 Fluctuations, 207-22.63 in the presena of continuous symmetry, 254-217 longitudinal and transverse, 21&-219 Fluctuation-dissipation theorem, 51, 263 Fock space, 15,16, 20, 22 Fourier transform, 93, 148, 212, 248 Functional integrals, 57-74, 332-389, +
232-231 application to barrier penetration,
366370 application to coflectiive excitations,
353-3M apptlcatfon to large orders of perturbation theory, 371-382 application to S-matrix eternents,
352-353 application to transitbn amplitudes,
35&351,39+391
;2c using auxiliary field, 332-336 using overcomplete sets of states,
33G340.385
Gaussian integrals, 33-37,59 complex variables, 34 Grassmann variables, 34-39 real variables, 33 Generating function, 76, 305-108, 142 Gibbs free energ, 182, 195 Gintburg region, 210, 213 Gotdstane wpansion, 159 Goldstone modes, 185, 215, 229 Grand canonical potential 0,48, '19, 97,
131,164-lH,240, 286 Grassmann algebra, 25-28 conjugation (Inwtutfon) , 26 derivative, 26 enerators, 25 integration, 27 Green% functions. 19--53, 235-285 advanced and retarded, 236 analytic properties, 24E249 connected, 505-108 dispersion relations, 243, 247 evaluation of ob~uvabtes,237-248 for non-tnteracting particles, 71-74.
f 46 high frequency behavior, 244, 248 integral equations for, 105-120 partide-hole, 267-268 real-time $cn)(t)* 53, 235 spwtrat representatbn, 24&244,247249 summation of [adder diagrams, 294-
295 table of results, 241 thermal g(")(,)* 5% 69-70.76. 102-
104,235,244-249 two-particle, 262-265 zero-temperature G(")($). 141. 237.
24&244 Havlree-Fock approximation basis for perturbation expansion, 162-
163 for self-energy, 250, 261, 287-288 from stationary-phase approximation, S35,342-344 relation to Fermi liquid theory, 298
time-dependent, 292-293,340, 362,
366,38+387 variational derivation, 41, 38G38rl Helsenberg representation, 50, 53, 345 Hifbert space, 5, 6, 11 Hotomorphic representation, 23 Hubbard model, 434 Hubbard-Stratonovich transhrmation,
333.350
Mean free path, 257-259 Mermin-Wagner theorem, 220 Microreversibi!Sty, 408,410 Monte Carlo method ewluatbn of integral evaluatioa of path int ptlng action, 417419 for auxiliary field, 431433 e path integral, 426far ma
431, Identical particles, 4-8 Importance sampling, 403406 Inclusive scattering, 51,263 electron scattering f'rom atonic nucl&,
280 neutron scattrsring from tiquld Helium,
278
for spin systems, 43-40 for tunneiing, 423,4 4 3 4 4 Grwn's Function Monte Gado, 423 guided random walk, 421423,442443,446 Wanscomb method, 438439 initial value random walk, 41%421
tnstantons, 360, 3Q2-393,424 Jordan-Wigner transformation, 134, 446
Neutron star matter, 43 Normal order, 19,58, 66 Number operator. 16
Landau function t(m, ET, T),180-584.
286,P91),20&207,227 Landau-Ginzburg functional, 207-21 1 Lattice gauge theory, 43%431 Legendre transform, 109, 195-197, 299,
202, 212,228 Lifetime of metastable state, 35%360,
443-444 tindhard function, 226, 269, 29G291 Linear response thwry, 262-281 Linkd cluster theorem, 9697,121, 135, 173 Lippmann-Schwinger quation, 261-2fi2 Local gauge invariance, 222-225 Loop expansion, 22Q-.131,21l-214,
228-229.348-350.387 Magnetic suscieptlbiitily, 182, 186,192,
200,203,212,281-285,304-305 Many-body operators, 9-10 second quantizect represent;a tion, 17-
19 Markay processes, 408-415 Matsubam frequencies, 248, 283 Mean-field theory, P62, 195-207, 222
Occupation numbczrs, 7, $4, 72, 297 Qne-dimensional problem with 6function forces exact ground state, 44 Ferrnl gas, 45-46 Harlree-Fock approximation, 45 Ip-expansion, 171-172,393-394 Optical potential, 25%262 Order parameters, 179-.181,185,190,
193-195.206205 table of, 193 Pad4 approximants, 381-382 Partition function Z", 48,63+6,68-74,
198,206 far non-interacting paeicles, 70-71 peeurbation expansion of, 74-10.4 Path integrals cakerent state, 6-8 kynman path integrat, 5743, 123-
$24,f32-134,136 HamiltonSan form, 62 imaginary time, 6346 Lagrangian form, 62
458
INDEX
PaulS paramagnetism, 283-284 Permanent, 8 Perturbation theory, 76104, at finite temperature, 74-104 at zero temperature, 138-163, l @ 169 Brittauin-Wignet., 172-174 in presnce of zero modes, 230 Raylelgh-Schrijdinger?173 zero temperature limit of, 1W-167 Phase transitions, critlal exponents, 191, 201-202 critical pdnts, 176, 178, 190 critical temperature, 176, 188, 200, 211-213 ferromagnetSe, 17+1?7, 193, 197-204 Landau-Ginzburg theory, 20r-211, 222, 227 Landau phenomenologicaf theory, 179184,227 liquid-gas, 178, 205, 227-228 lower cdtical dhension, 226-221 order of, 118 phase diagrams, 176179 superconducting, 193-195 supefiuid, 193-195 table of critical exponents, 226 upper critical dimenslon, 260 Pion condensation, 271, 312 Plasma oscillation, 270, 291 ProbaHlity distribution, 401 Quaslpsruticles energy, 298 interaction, 298-299, 320, 322-324 lifetime, 254, 258 quasipartket pole, 251-254 residue, 252 Quantum statistical mechanics, 47-49 Random phase approximation, 265-1272, 276-277.290.291. 293-294.344348, 349,391-392 Reaction matuix. 163, 171 Replica technique, 96,98, 142 Resolvent operator, 34+342
Response function, 49-52, 261-271, Sampling t~hnlques,4 Gausslan, 408 heat bath, 410 integrable functions, 407408 Langevln quatian, 411412, 441 Metropolis atgodthm, 410-.411, 441 microcanonicat Monte Carlo, 4P3 molecufar dynamics, 414415,441 Heumann-tflam matrix inversion, 412413,441 uniform distri butbn, 406 won Meurnann rejection, 408 Sehur" lemma, 22 Self-enera, 113-115,239, 24S262 Slater determinant, 8, 42, 33S340, 341, 345, 3W. 385 Sound modes thermodynamic sound, 272,275, 303304 zero sound, 27b276, 308-312,327329 Specific heat, 202, 210, 300, 32&325 Spin systems Weisenberg spins, 181, IsOng model, 181, 18S190, 198, 208, 221,231-234 O(n) model, 181, 197-204, 221,227 quantum spins, 434-4412 1; - y madet, 181, 217-220 Spin waves, 219, 312 Spontaneous Fission, 364-370 Spontaneous symmetry breaking, 107 Stationary-phase approximation derivation of central timlt theorem, 401-803 for anatyzing large orders of pstrlurbation theory, 373.376-381.394-395 for Fermions at fin&@temperature, 387-390 for kynman path htegral, 123-12.5 for infinite-range tsia-g model, 187-190 X2&131, 136 for integral, far large-amplitude collective motion, 3-364
for partition function, 1 2 6 1 3 1 for quantum eigenstates, 354-3611) for static Nartree approximation, 342344 far tunnefing, 357-359, 365-370 quadrsttlc corrwtions, 215, 345, 379381,387 Stirling" formula, 55 Structure factors, cfy namic S(q,W), 263, 288-290 static q g f , 277, 279 Sum rules, for Landau parameters, 321 Cor response function, 277-278,295 far spectra! weight of Green" ffunctfon, 244 287 for symmetry factors, 87, 95-92 Symmetry betar, 83-87 Thermodynamic limit, 48 Time-ordered product, 49, 6243, 132, 139 Pace of an operator Boson coherent states, 23 Fermion coherent states, 32 %otter formula, 337, 435 Variance;, 403404 V e ~ e xfunction, 110, 11%.115-120,313320 r" 319
FU, 328 Wick rotation, 64 Wfckrs thearem, 75-78, 121, 134-135, 137, 345 Wiener measure, 65 Wigner transform, 273-2'75
