Renormalization: An Introduction (Theoretical and Mathematical Physics)

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Manfred Salmhofer

Renormalization An Introduction

With 24 Figures

Springer

Prof. Dr. Manfred Salmhofer Mathematik ETH-Zentrum CH-8092 Zürich, Switzerland

Editors

Roger Balian

Nicolai Reshetikhin

CEA Service de Physique Théorique de Saclay F-91191 Gif-sur-Yvette, France

Department of Mathematics University of California Berkeley, CA 94720-3840, USA

Wolf Beiglbiick

Herbert Spohn

Institut ftir Angewandte Mathematik Universitüt Heidelberg Im Neuenheimer Feld 294 D-69120 Heidelberg, Germany

Theoretische Physik Ludwig-Maximilians- Universitüt München TheresienstraBe 37 D-80333 Mtinchen, Germany

Harald Grosse

Walter Thirring

Institut für Theoretische Physik Universitat Wien Boltzmanngasse 5 A-1090 Wien, Austria

Institut für Theoretische Physik Universitüt Wien Boltzmanngasse 5 A-1090 Wien, Austria

Elliott H. Lieb Jadwin Hall Princeton University, P. 0. Box 708 Princeton, NJ 08544-0708, USA

Library of Congress Cataloging-in-Publication Data Salmhofer, Manfred, 1964— Renormalization: an introduction / Manfred Salmhofer. p. cm . — (Texts and monographs in physics, ISSN 0172-5998) Includes bibliographical references and index. ISBN 3-540-64666-3 (alk. paper) I. Renormalization (Physics) I. Title. 11. Series. QC174.17.R46S35 1998 530.1'43—dc2 I 98-40042 CIP

ISSN 0172-5998 ISBN 3-540-64666-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Veriag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1999 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready copy from the author using a Springer TEX macro package Cover design: design & production GmbH, Heidelberg SPIN: 10652045 55/3144-5 4 3 2 10 - Printed on acid-free paper

Preface

Why another book on the renormalization of field theory? This book aims to contribute to the bridging of the gap between the treatments of renormalization in physics courses and the mathematically rigorous approach. It provides a simple but rigorous introduction to perturbative renormalization, and, in doing so, also equips the reader with some basic techniques which are a prerequisite for studying renormalization nonperturbatively. Beside these technical issues, it also contains a proof of renormalizability of 0 4 theory in d _ < 4 dimensions and a discussion of renormalization for systems with a Fermi surface, which are realistic models for electrons in metals. Like the two courses on which it is based, the book is intended to be easily accessible to mathematics and physics students from the third year on, and after going through it, one should be able to start reading the current literature on the subject, in particular on nonperturbative renormalization. Chapter 1 provides a brief motivation for studying quantum theory by functional integrals, as well as the setup. In Chap. 2, the techniques of Gaussian integration and Feynman graph expansions are introduced. I then give simple proofs of basic results, such as the theorem that the logarithm of the generating functional is a sum of values of connected Feynman graphs. In Chap. 3, the Wilson renormalization flow is defined, and perturbative renormalizability of 0 4 theory in d < 4 dimensions is proven using a renormalization group differential equation. The Feynman graph expansion of Chap. 2 is the explicit solution to this equation, but the analysis of the differential equation leads to a very simple renormalizability proof. It also brings out clearly that the renormalization subtractions really amount to a change of boundary conditions. In Chap. 4, a similar renormalization flow is applied to an infrared problem, that of many-fermion systems. Using the method of overlapping loops, I determine the leading contributions to the renormalization group flow to all orders of perturbation theory. This leading behaviour is then calculated in the simplest cases and its physical implications are discussed heuristically. Readers with a little experience in field theory can read Chap. 4 independently of Chaps. 1-3. I would like to thank Horst Kniirrer for the suggestion to write this text and for many valuable comments. I also thank Ldsz16 Erd6s and Pirmin Lemberger for reading carefully through Chaps. 1-3 and for suggesting var-

VI

Preface

ious improvements. Finally, I would like to thank Joel Feldman and Eugene Trubowitz for the collaboration which brought about many of the results about many-fermion systems described in Chap. 4 and for many discussions, Christian Lang and Erhard Seiler for teaching me a good part of what I know about quantum field theory, and Volker Bach, Georg Keller, Christoph Kopper, Walter Metzner, Edwin Langmann, and Christian Wieczerkowski for discussions. A saying attributed to Einstein goes 'Everything should be made as simple as possible, but not simpler.' I have made the proofs as clear and concise as I could without oversimplifying matters or discarding important details. Certainly, not everything here is as simple as it could possibly be, but I hope that it is at least readable.

Zurich, September 1998

Manfred Salmhofer

Table of Contents

1.

Field Theory 1.1 A Motivation for Path Integrals 1.2 Gaussian Integrals and Random Variables 1.2.1 Preliminaries 1.2.2 Gaussian Integrals in Finitely Many Variables 1.2.3 The Covariance Splitting Formula 1.3 Field Theory on a Lattice

1.4 Free Fields 1.5 Properties of the Free Covariance 1.6 Problems With and Without Cutoffs

1 6 6 7 10 11 12 15 17 19 21 23

Techniques 2.1 Integration by Parts 2.2 Wick Ordering 2.2.1 Definition and Main Properties 2.2.2 Further Results 2.3 Evaluation of Gaussian Integrals 2.3.1 Labelled Feynman Graphs 2.3.2 Symmetry Factors and Topological Feynman Graphs 2.3.3 Motivations for Taking the Logarithm 2.4 Polymer Systems 2.4.1 Preparation: Graphs and Partitions 2.4.2 The Logarithm of the Polymer Partition Function 2.5 The Effective Action and Connected Graphs 2.5.1 Definition and Semigroup Property 2.5.2 Derivation of the Graphical Representation 2.5.3 Result and Discussion 2.6 Graphical Representations: Conclusions

27 27 28 28 30 33 33 38 39 41 42 44 47 47 49 55 57

1.3.1 Discretization 1.3.2 The Partition Function and Correlations 1.3.3 The Ising Model

2.

1

VIII 3.

Table of Contents

The Renormalization Group

3.1 A Cutoff in Momentum Space 3.2 The Semigroup Structure of Renormalization 3.3 The Renormalization Group Equation 3.3.1 The Functional Form 3.3.2 The Component Form 3.4 The Structure of the RG equation 3.4.1 The Graphical Representation 3.4.2 The Relation to the Feynman Graph Expansion 3.4.3 The Continuum Limit at Fixed Ao 3.5 Differential Inequalities 3.6 Two Dimensions 3.6.1 Boundedness 3.6.2 01 3.6.3 Convergence 3.7 Three Dimensions 3.7.1 Power Counting for the Truncated Equation 3.7.2 Renormalization: A Change of Boundary Conditions 3.7.3 Renormalized 01 3.8 Four Dimensions 3.8.1 Counterterms in Second Order 3.8.2 Power Counting (Skeleton Flow) 3.8.3 The Boundary Conditions for Renormalization 3.8.4 Renormalized 0 4 Theory 3.9 The RG Flow in the Ladder Approximation 4.

The Fermi Surface Problem

4.1 Physical Motivation 4.2 Many-Fermion Systems on a Lattice 4.2.1 The Hamiltonian 4.2.2 The Grand Canonical Ensemble 4.2.3 The Fermi Gas 4.2.4 The Functional Integral Representation 4.2.5 RG Flow: Energy Scales 4.2.6 Model Assumptions 4.2.7 The Physical Significance of the Assumptions 4.2.8 The Role of the Initial Energy Scale 4.3 The Renormalization Group Differential Equation 4.3.1 The Effective Action 4.3.2 The RG Equation 4.3.3 The Component R.GE in Fourier Space 4.4 Power Counting for Skeletons 4.4.1 Bounds for the Infinite-Volume Propagator 4.4.2 Sup Norm Estimates 4.4.3 Estimates in L1 Norm

63 64 65 68 68 69 72 72 74 74 75 78 78 80 82 85 86 88 91 99 99 102 103 105 109 113 113 114 115 118 119 123 125 126 128 129 130 130 131 136 137 137 139 142

Table of Contents The Four-Point Function 4.5.1 Motivation 4.5.2 The Parquet Four-Point Function 4.5.3 The One-Loop Volume Bound 4.5.4 The Particle—Particle Flow 4.5.5 The Particle—Hole Flow 4.5.6 The Combined Flow 4.6 Improved Power Counting 4.6.1 Overlapping Loops 4.6.2 Volume Improvement from Overlapping Loops 4.6.3 Volume Improvement in the RGE 4.6.4 Bounds on the Non-Ladder Skeletons 4.6.5 The Derivatives of the Skeleton Selfenergy 4.7 Renormalization Subtractions 4.7.1 Motivation; the Counterterm 4.7.2 Full Amputation 4.7.3 Bounds for a Truncation 4.7.4 The Meaning of K 4.8 Conclusion 4.8.1 Summary 4.8.2 A Fermi Liquid Criterion 4.8.3 How the Curvature Sets a Scale 4.5

IX 145 146 147 149 151 155 157 157 157 160 161 162 165 167 167 168 170 173 175 175 176 178

A. Appendix to Chapters 1-3 A.1 A Topology on the Ring of Formal Power Series A.2 Fourier Transformation A.3 Properties of the Boson Propagator A.4 Wick Reordering for Bosons A.5 The Lower Bound for the Sunset Graph

181 181 181 184 185 189

B. Appendix to Chapter 4 B.1 Fermionic Fock Space B.2 Calculus on Grassmann Algebras B.3 Grassmann Gaussian Integrals B.4 Gram's Inequality; Bounds for Gaussian Integrals B.5 Grassmann Integrals for Fock Space Traces B.5.1 Delta Functions and Integral Kernels B.5.2 The Formula for the Trace B.5.3 The Time Continuum Limit B.5.4 Nambu Formalism B.5.5 Matsubara Frequencies B.6 Feynman Graph Expansions B.7 The Thermodynamic Limit in Perturbation Theory

191 191 192 196 198 201 202 204 205 212 213 214 217

X

Table of Contents

B.8 Volume Improvement Bounds B.8.1 The One-Loop Volume Bound B.8.2 The Two-Loop Volume Bound

220 220 222

References

227

Index

229

1. Field Theory

1.1 A Motivation for Path Integrals To motivate the study of field theory via the functional integral, we review the path integral of quantum mechanics, as discovered by Dirac [1] and Feynman [2]. We shall be brief and refer to the literature for many details because a number of excellent accounts (e.g. [3 ] ) deal with this topic in depth (and because these details will not be required for understanding the rest of the text). The presentation given here is taken from [3], it is included for the convenience of the reader and because, even though beautiful mathematical work has given a mathematically rigorous foundation for the functional integrals of quantum mechanics [3] and quantum field theory [6, 7, 5], many people still seem to believe that this is not the case. In fact, however, the Hamiltonian formalism of quantum field theory is so singular in more than two dimensions that the path integral seems a much better starting point. The classical Hamiltonian

Hc i =

p2

m

+ V(x)

becomes a quantum-mechanical Hamiltonian h2

H = — — 2i + V(x) 2m

(1.2)

under the replacement p —> hli 010x. States of spinless particles on Rd are described by vectors Iii E ii = L2 (Rd , c) , with the interpretation that for any measurable R c R d , fR itp(t, x) I 2 d d x i s the probability for finding the particle at time t in the region R. Given an initial state 00 in the domain D(H) of the Hamilitonian H at t = 0, the state at later times t > 0 is determined by the Schrödinger equation

alp at

ih— = HO

(1.3)

with initial condition 0(0, x) = 00 (x). If V is such that H is self-adjoint, then the time evolution t v.4 exp(—itH), t E R, is a strongly continuous semigroup, and for an 00 E 9-i, the state at time t> 0 is given by

2

1. Field Theory

= (e—itHlh,lpo)(x)

V(t, x)

= f dy K(t;x,Y) 00(Y)•

(1.4)

Rd

In Dirac's notation,

K (t; x, y) = (x IC iti I lh I y).

( 1 - 5)

Feynman's path integral formula is a rewriting of K in terms of an integral over paths, obtained from the Trotter product formula. Lemma 1.1. (Trotter product formula) If A and B are bounded opera-

tors, then e lt+B =

Lim

(eA/N eBIN )N

N—k>3

(1.6)

5

with the limit in the operator norm. This proof is taken from [3]. Let C = exp((A + B)/N) and D = eA lN eB IN . We have to show that IICN — DN II —> 0 as N > oc. By telescoping,

Proof.

—

N-1 cN _ DN =

Ls Ck(C D)DN -1— k .

(1.7)

k=0

Using the product inequality of the operator norm and

max{IIC11,11D11} <

e(11A11+11B11)/N ,

(1.8)

we get N-1

IICN — Div !' 5-

E ink lic pH IIDIIN-1- k

(1.9)

k=0

5- IIC —

DM N e(11A11+11B11)(N-1)/N.

(1.10)

Expanding the exponentials, we see that C— D starts with the term of order N-2 . More precisely

C— D =

E m>2

N' ((A + B)m m!

E

m

m!

„,AmiBmii

(1.11)

mi+m"=m

so, taking norms and using the product and triangle inequality, we get

II' — Dil hence

I cN

< N-2 e (11 ,4 11+11B11)/N (I1All + IIBI1) 2 ,

DI

< N -1 eliAll+rIBII

(1 1A l + 11B11) 2 .

(1.12) (1.13)

N

1.1 A Motivation for Path Integrals

3

The Hamiltonian H given by (1.2) is unbounded, so the Lemma does not apply to it. It is proven in [3] that in the physically interesting cases, the limit exists in the strong sense, that is, if both sides are applied to a vector 0 E 7-1, the resulting vector converges in 7-1 as N —> co. More precisely [3]: —if A and B are self-adjoint and A + B is essentially self-adjoint on D(A) n D (B), then ei(A+B) = s—lim ( e iA/N e iB/N) N (1.14) N-00

—if A and B are bounded below, then s—lim (e— A/N e—B/N)N .

(1.15)

N—oo

(see also [4], Sect. X.11). These are the physically relevant cases because both the time evolution e —itil and the heat kernel C-011 are covered. The free time evolution has the kernel

K o(t; x , y) = (x 1 e— itHo/h 1 y) = ( In 127riht) d/2

eimlx—y12/2ht

,

(1.16)

and ([4], Sect. IX.7)

(e_itHo/h0) (x) = f dy Ko(t; x,Y)0(Y),

(1.17)

Rd

where fRd dx is understood as a limit limm_ oo Sect. IX.2). Because

(x I e —iAt/ h Ho e — iAtlh V

fi xi
I y) = Ko(21t; x, y) e —iAt/h V(y)

,

(1.18)

we have (e

—12it/h (Ho+V)o) (x0) = lim

N—oo

m Nd/2 f

( 27rih„At)

dxi .. dx N R dN

e i/h SN(xo,...,xt) 2/1(XN)

(1.19)

with ut = tI N and N

M (

(x k — xk_ l

2 ) — 17 (X0) .

(1.20)

k=1

A classical particle in a potential V moving along a path X (t) has the action t m 2 k(s) 2 — V(X(s))) . S(X) =-- f ds (—

(1.21)

1. Field Theory

4

For a polygonal path X (P) given by xo,

, xN, one gets

(Xk - Xk-1) 2

S(X (P) ) =

(

2

k=1 kJ/1r

2it

(XO, • • • XN; t).

I

ds V(X(s))) (1.22)

Thus one may interpret the right hand side of (1.19) as an integral over all polygonal paths, with SN approximating the classical action associated to each path. One would expect that in the limit N > oc, the set of all polygonal paths becomes the set of all paths, that SN(X (P)) 5(X), and that [1, 2] —

(e —it(H 0 +vvh.0) (x0 )

f eis(x)/h(x (t))dX

(1.23)

sa.0 where Q the set of all paths X : [0, t] > Rd with X(0) = xo and dX is a measure on Q. This is Feynman's path integral formula. One of its most beautiful consequences is that the variational condition for the classical solution (5 (1.24) S(X) I (5X = appears as the stationary phase approximation to the integral, which should be valid in the limit h > 0. The problem is, unfortunately, that the measure dX does not exist, and that therefore, the path integral formula makes mathematical sense only at finite N. Thus all manipulations have to be done before taking the limit, which is messy and removes the elegance of the approach. A concrete indication that the limit dX does not exist is that the prefactor in (1.19) diverges as N oc. However, if one takes e-t instead of eitH , the exponential of the kinetic term decays exponentially and can be included in the measure: replacing it by 7, that is, going to imaginary time, one gets instead of Ko the kernel —

—

P(7,

Y) =

2/2kr

L7r(tr )d/2

(1.25)

Since (X

I C AT lh 110e-AT/h V

y) CAT/h V(y) I y) = P(21t; x 7

(1.26)

we have ( e Ar/h —

(1 o + v) o) (x 0) _■

m =

Nd/2

f

I iln „ ,, N—).00 (z7rnzar) IVN —sE e (x 0,.••,xN)/h 0 (xN )

dX1 ...dXN (1.27)

1.1

A Motivation for Path Integrals

5

with 20" = TiN and N

SE(X0,

E

m ,AT — ... , x; T) = 2 k=1

Since

f

(

Xk

— Xk_i

2iT

2 ) + V(Xk)

dy p ( r; x, y) = 1,

)

(1.28)

(1.29)

Rd

the Gaussian measure defined by N ( dpV0 ) (X) = H exp k=1

TT/ (Xk —Xk-1)2)

2h

2■ T

1 a x i . . . dxN

(1.30)

is normalized, i.e. f die ° ) (X) = 1, independently of N. And indeed, one can show that the limit N —> co of dig) exists as a measure on the set of all paths starting at xo . It is the conditional Wiener measure dpxo (X) describing Brownian motion, that is, the analogue of free quantum mechanical motion, but with the Schrbdinger equation replaced by the heat equation. For suitable [4], (1.27) converges as N —> co to the V, e.g. V E LP n.L', where p > (convergent) integral

g-

— 1 V(X(r))dr

(e —S11 0) (X0) =

f diixo (X)e 0

0 (X()).

(1.31)

17 00

Note that the improper integral f dx exp(ix 2 ) is convergent, but not absolutely convergent, which prevents a similar procedure in the real-time case. In fact, complex-valued Gaussian measures on infinite-dimensional spaces exist only under very restricted conditions, e.g. if the imaginary part of the covariance is relatively trace class to the real part. For determining spectral properties of H, e—SH is as good as eitil . Moreover, the path integral representation for the partition function for the canconical ensemble (in a finite volume) tre —'81-1 is obvious from (1.31). The Feynman - Kac formula (1.31) has many applications in quantum mechanics

[31In quantum field theory, one is not dealing with a single particle, but with infinitely many particles, because one has to account for the creation and annihilation of particles. One can formally write down a Hamiltonian, but it becomes very difficult to give a mathematical definition of it. We shall simply define the theory by the functional integral. The lesson from the quantum mechanical case is that free particles are described by a Gaussian measure. We shall therefore put the quadratic part of the action into the measure before taking the continuum limit that defines the field theory.

6

1. Field Theory

In the Minkowski spacetime, there are further problems: because of the indefinite metric, the kinetic term is not semibounded. For this reason, one studies the Euclidean version of the theory, i.e. the theory with imaginary time, and then tries to do the analytical continuation to real time. A historical introduction and physical motivations for doing quantum field theory, as well as a thorough development of the theory is, e.g., in the recent book by Weinberg [8].

1.2 Gaussian Integrals and Random Variables The fields in quantum field theory can be considered as random variables associated to a Gaussian measure that is perturbed by an interaction term. The Gaussian measure determines the properties of free particles. In this section, we give the elementary definitions and properties of Gaussian random variables indexed by a finite set. This is of relevance for quantum field theory because we shall define a regularized field theory on the lattice with finitely many points and then study the continuum limit. 1.2.1 Preliminaries All identities for Gaussian integration are based on the following two elementary formulas.

Remark 1.2. Let a > O. Then 00

L. e -2 " dx = f

(1.32)

and for all b E C

1(b) =

J.

e- i x2+bs dx = e

b2/2a

(1.33)

b2 1 I 2 + 2a Proof. Since --c2lx2 + bx = —1(x — i) a

1 (b) =--- e b2 /2 ' f dx C a ( x—b /a)2 /2 .

(1.34)

R

For b E JR, a shift in the integration variable proves (1.33). For ,t3 = 1m b 5L 0, the shift gives

1(b) =...,_ eb2 12a f dx e— a ( x— Wa)2 /2 . R

Since 1 e -a(x-ii3/a) 2 /21 = e -ax 2 /2 e 02 /2a ,

(1.35)

1.2 Gaussian Integrals and Random Variables

f Isi>/,

dx e-a(x -is/a) 2 / 2 < es2/2a

f

dx e—as2 / 2

---)

L-00

0,

7

(1.36)

lx1>i,

so it suffices to show that

L f dx e __

2 /2

---3 f dx e—as2 / 2 . L—oo

(1.37)

111

—L

Since the integrand is analytic in x, we may deform the contour to consist of a piece [—L, Li] + if3/a and two vertical pieces, going from ±L to ±L + i$/a. The contribution from [ — L, L ] + iiVa equals f LI, e —ax2 /2 dx, which converges to the desired result as L —> oo. The contribution from the two vertical pieces vanishes as L —> cx) because the integrand is exponentially small in L and the integration path is of fixed length 1/3/al. M

1.2.2 Gaussian Integrals in Finitely Many Variables

From now on, let N E N, A E MN(R), A = At > 0 (i.e., all eigenvalues of A are positive) and denote C = A -1 . Denote the scalar product on RN by

N

(0,0

=Eozoi

(1.38)

i=1

and

dNO

= 110i . • • dON -

Remark 1.3.

f e -1("4') dN O = (27) 1- det A-4 = det (270 i and for all

(1.39)

J E CN -105 ,A0)+G74)dNo = det (27C) 4 el (Jicv) . e f

Proof. Since A is symmetric, there is a V E

SO(N)

(1.40)

such that

V-1 AV = D = diagfai, • • • ,aN}

(1.41)

where ai are the eigenvalues of A. Since A is strictly positive, ai > 0 for all i E {1, ... , N } . Let V) = VO, then

(0 ) AO = (0 ) DO)

N = Eaioi2 i.i

(1.42)

8

1. Field Theory

and dNO = 1 detVidNO = eV). Thus by (1.32)

f

_1 E aop2

( 0 A ck ) d N ,

f

e ll) e i-i

N

f dIP i.i. 11

=

_ Similarly, (J, 0) = (J,V

-1

N

/2 7 I i V ai

la. 2

(1.43)

(27) 4 1 = det (27C)i. det A) i (

0) = (V J, 0), so, with b = V J E CN , and by

(1.33)

f dN o

f di/'e_

=

2 1P2 +bilP

i=1

= i=1

\i 22 eh i /2ai /

E

(det 27C) ei (J7cJ)

ai

(1.44)

because

z_s 2a. = i=i 2

— 2

(b,D -1 b) = — (J,17-1 D -1 VJ) = 2

(J,CJ).

(1.45)

•

Definition 1.4. Let C E GLN(R), C > O. The measure

dpc (0) = (det 27C)-i

(0,c -1 0) dN

(1.46)

is called the Gaussian measure on RN with mean zero and covariance matrix C. By definition, pc has total mass equal to one, i.e. f dpc (0) = 1. By Remark

1.3,

f dttc (0) ei( ") =

(1.47)

holds for all J E CN . This Fourier transform is often called the characteristic function of the measure pc. All moments of the measure pc can be calculated using (1.47) as polynomials in the covariance C, e.g.

f

cipcmo102 =

,2

, j2

I

(9 2

L aJiaJ2 =

f 0, (0)

e(J)(15) ] J=0

ei.(

_11=0

(1.48)

1.2 Gaussian Integrals and Random Variables

9

We shall see much more of that when calculating more complicated moments of pc as sums of values of Feynman graphs. Equation (1.47) already suggests that the limit N —> co may be taken under suitable conditions on C and J, e.g. that J is in the form domain of C. The existence of moments of pc in that limit then depends on further properties of C; this will be central in the discussion of renormalization. A natural interpretation from the probabilistic point of view (as well as that of field theory) is to regard not only the measure dpc as interesting, but also the variables 01, - - • , ON. The random variables 01, - - - , ON distributed according to (1.46) are called Gaussian random variables indexed by {1, ... , N}, that is, Gaussian fields. The right hand side of (1.47) is defined for C > 0 as well (in fact, for any C E MN(C)). As we shall show now, the f dpc on the left hand side of (1.47) is a measure for any C > 0 (i.e., where some of the eigenvalues vanish), but not for C G 0. To see this, let us go back to the one-dimensional case where the matrix C is a number e > 0, and where dpc (q5) =-- dOi gc (q51 ) with density

17.rc e—x2/2c. gc,(x) . \/..._

(1.49)

ji, gc(x)dx =

1 and gc (x) --- 0 for all x 0 0, gc is an approximate c—al identity, i.e. gc —> 8 in Si(R). Thus Since

dPc (0)

c, -->+0

00)4-

(1.50)

Similarly, the vanishing of an eigenvalue of C in the N-dimensional case simply implies that the measure contains a 8 function that restricts to the linear subspace orthogonal to the corresponding eigenvector. In this way, the measure dpc has a natural meaning also for C > 0, which we adopt from now on. It is natural because dpc for C > 0 can always be obtained by taking E -> 0+ of a measure dace with Ce > 0 for E > 0. With this definition, (1.47) holds for all C > 0, with a genuine measure pc on the left hand side. We shall need nonnegative covariances all the time in what follows because we shall impose strict cutoffs in momentum space. Finally, note that for C with some negative eigenvalues, the left hand side of (1.47) diverges: although for any C E GL N (JR) the linear functional

Lc(F) = (det 27- C ) f dN 0e 4( "15) F(0)

(1.51)

can be defined on a suitably restricted space of functions, such as CO° (RN), the inequality

Lc(F) < const Moo

(1.52)

holds only if C > 0. So there is no measure associated to the linear functional Lc if C has negative eigenvalues.

10

1. Field Theory

1.2.3 The Covariance Splitting Formula Theorem 1.5. Let Ci C2 E MN(R), G = Cif for i E {1,2 and C2 > O. Define C = C1+C2. Then for all F E L l ( PC)7 }

f dPc 6) f d 2 (02) F(01 + 02) f apc1 (01) f apc2(0 - 00F(o).

f diiC(0)F(0)

i(

>

a,

ttc

(1.53)

In other words, if C = C1+C2, the Gaussian random variable 0 is a sum of two independent Gaussian random variables, q5 = çSi + q52, and the Gaussian measure factors, dpcM = dpc i (Mdpc2 (02) •

Proof Let F(0) = ei (J,0 ). Then by (1.47)

f

dpc (0) e i J4) =

e —lp,c1,0-4(J,c2J)

(

= f dlic (01) ei(j 41) f =

f

dpc2 (02) e i(J'02) (1.54)

dPci ((bi) f dttc2 (02) ei(J,01+02)

so (1.53) holds for this case. Since a measure is uniquely defined by its Fourier transform, the measures dpc and dpc, * dpc2 are identical, therefore (1.53) holds.

Remark 1.6. One can derive (1.53) from (1.54) also more explicitly. By the explicit formula (1.54), both sides are analytic in J. Thus we may take derivatives with respect to J. If F is a polynomial, F(0) = P(0), with P (0)

=EE

(1.55)

where the inner sum runs over all sequences (ii, we have

f

dPc (0)

P(0) e i(J4)

, ji,) with jk E {1,..., N},

= f dpc P(.10

ei(J

= P(To ) f dtic ei(j'°) where the notation means that every 0i1 gets replaced by this becomes

(1.56)

k. Using (1.54),

1.3 Field Theory on a Lattice

a =

f

11

f apc,(01) f dlic2(o2)ei(J ,0 1+0 2)

2) . dlic1( 01 )f dpc2( 02) P ( 01 + çi,2 )ei ( -1,0 1 +0

(1.57)

The exchange of integration and differentiation with respect to J is justified because all integrals are still absolutely convergent. For the same reason, the limit J 0 exists, and it implies that (1.53) holds for all polynomials F. More general F can be approximated by polynomials. This technique of differentiating with respect to the source terms J will be a Leitmotiv in most of the technical discussion about Feynman graph representations.

Definition 1.7. Let F be a finite set, Irl = N, and let F = {0 : F —> 11} (then F RN). For 0,0 E F let (ç5, 1/)j-' = Es Er (15(x)0(x), let C be a linear map on F such that (0,C0) > 0 for all 0 0 0, and let A be its inverse. The Gaussian measure on F is defined by

d pc (0) -= (det 27C)

(10(x),

(4) 'A`b)

(1.58)

xEr Remark 1.8. If v : 11,

, n1 —> I' is bijective, d(ç5) can be written as (15 (v(0)Am(i)v(i)0(v(i)) N

dpc (0) = (det 27(C,( i

Hdo,(i) .

)) e ij1

(1.59)

i=1

dpc (0) is well-defined because all factors occuring in its definition are basis independent: the right hand side of (1.59) does not depend on v.

1.3 Field Theory on a Lattice We want to define a regularized field theory corresponding to the Euclidean action iii 2 =

fEdd d X

fEd

GVO(X)2

ddX [0(X)

0(X) 2 4 (X ) 4 )

+ 7710 2 )0X) + A O(X) 4 1

(1 . 60 )

where, in going from the first to the second line, we have assumed that 0 vanishes at infinity so fast that there are no boundary terms - this is justified because we have to assume enough decay of 0 and its derivatives to make the integral for S converge.

12

1. Field Theory

1.3.1 Discretizat ion We use the lattice regularization only as a method to define the field theory. To analyze perturbation theory, we shall later take the continuum limit at a fixed ultraviolet cutoff in momentum space, to restore the Euclidean symmetries of the theory. The advantage of using the lattice is for us that we can use ordinary derivatives instead of functional derivatives (the latter can easily be defined in the continuum, but we shall not need that here and it loads the presentation with unnecessary discussions). However, the importance of the lattice theory goes well beyond the purpose for which it is used here. Up to now, the lattice regularization provides the only nonperturbative regularization of functional integrals that works independently of the coupling strength. Also, to this day it remains the only nonperturbative regularization that does not break gauge-invariance. We discretize space and time and introduce a finite volume. For convenience in using the Fourier transformation, we introduce the finite volume as a torus of sidelength L. We take a lattice of spacing E> 0, and define

F = re ,i, = Ezd 1 Lzd

(1.61)

where e > 0 is to be small, L» 1 and f', E N. When no confusion can arise, we simply denote FE ,L, by F. A particular choice of coordinates is to write F as

Ix

L

E EZ d : -2 < xk <

L ,,„

— 2

VIC E

{1, ... ,d}}

(1.62)

with periodic boundary conditions. The number of points of this lattice is L

Ife,L.

I = (. ) d .

(1.63)

In the end, we are interested in the field theory on Rd which means that we want to take the thermodynamic limit L —> oc, and the continuum limit E -) O. The theory where only E -> 0 is taken is a continuum field theory in a finite volume, the theory where only L —> oc is taken is a system of statistical mechanics. Since the number of lattice points goes to infinity in both of these limits by (1.63), both limits are systems with an infinite number of degrees of freedom, and hence interesting by themselves. The theory of critical phenomena, i.e. of the long-distance behaviour of the correlations of an infinite-volume system at fixed E is closely related to studying the continuum limit of field theory [14]. The continuum derivative of a field 0 : Rd --> R is discretized by

40(x) = ( 1 (0(x + eek) — 0(x))) E

(1.64) kE11,...41

where ek is the unit vector in k-direction, and the discretized action is

1.3 Field Theory on a Lattice 2 1 S r (0) = Ed (— (d e 0(1)) 2 ± -1( 27 4 . (5) 2 + A(x) 4 ) z, 2 2

13 (1.65)

where A> 0 and m> O. m is the mass and A the self-coupling. In all detail, SF

(0) = Ed

E ( .2x Er

If

'(0,(s

+ Ee k) — 0 (x)) 2 + *r P(s)2 + 4(44) .

k=1

0

0 r : F —3

F, OF

=.--

' is obtained by restricting (I) Ir, then

(1.66) E C ( /LZd ) to the lattice

--S (0) . ( 1.67) E–H) Many other choices for S have the same property, so the lattice action is not uniquely determined by the continuum action one wants to regularize. Whether different actions give the same continuum limit for the correlation functions of the statistical mechanical system given by S (defined in the next section) is a much more difficult question. We now streamline the notation to avoid too many E's and sums. We introduce the notation

151 FE ,z,

(4) Ts , L)

fF dx F(x) = E F(x), Ed

(1.68)

x Er

which is suggested by the convergence dx F(x) —+

E–>o

J

f

dx F(x)

(1.69)

RdiLzd

L

for any F E Cl(Rd/LZd, C), and define a scalar product on CF by

(q5, 7P) r = f dx 0(x)0(x).

(1.70)

F The backward derivative de*, defined by

(c/:(0) k (x) = –El (0(x –

Ee k

) — OW),

k E {1, ... ,c/}

(1.71)

is the adjoint of the forward derivative de :

(0, deOr = (c1:0,0)r -

(1.72)

Equation (1.72) is proven by summation by parts. There are no boundary terms because we chose the lattice as a torus. Introducing the lattice Laplacian A ,---- –4de (1.73) we can sum by parts in the sum for Sr, and get

14

1. Field Theory

Sr(40 ==

e d xer

2

dx =

0(x) (( —a + mo 2 )0)(x) + A0(x) 4 )

(1.74)

(0, ( — A ± rn0 2 )0)r + À f dx0(x) 4 .

The action of Laplacian is, more explicitly, d

1 20(X) — A0)(X) E =(

eek) 4 6(x

(1.75)

AO)r = (cle0, dE(15)r > 0,

(1.76)

k=1

Since for all

0, (0,

—

—d is a nonnegative operator, just like its continuum counterpart. Every linear operator on Cr is, when represented in a basis, simply a matrix in Mi r' (C). Choosing the orthonormal basis (Ex ) x er by Ex(Y) r (x , y), where (1.77) SF(X,Y) = E —d (5xy, we have 0(x) =

(Ex Or and

f

dy Ey (x)Ey (x`) = Sr(x, x 1 ),

SO

(A0)(x)

=

f

(1.78)

dy A(x , y)(P(x)

(1.79)

with the matrix representation (or "integral kernel")

A(x, y) = (Ex , AE y )

(1.80)

of A. In particular, the unit matrix has kernel 1(x, y) = (51 (x , y). The corresponding distributional kernel in the continuum is, of course, 5(x y). If A is invertible, the kernel of the inverse satisfies -,

—

dz A(x, z)(A -1 )(z ,y) =

Sr (x, y).

(1.81)

We also define a derivative

a

6 J(x)

-

a J(x)

(1.82)

so that J(x)

J(y) = 5(X, y)-

(1.83)

1.3 Field Theory on a Lattice

15

This also agrees with the formal continuum convention

6.1(x)

(1.84)

.1 (Y) = 5 (x — y).

With this, we have

6 e (J,(19». 6.1(x)

J(x) e(J'`15) r

(1.85)

-

The Laplacian has the kernel d

—

(x, y)

=

e —d-2 E fq kz, uzy

(5x+Eek,y

(1.86)

(5x Eek,Y) —

k=1 d

E -2

E

(26,(x,

y) — 6r (x + eek,y) — 6r(x — eek,Y))

k=1

(in the continuum, the distributional kernel would be (

-

1 6)(x

y)).

—

1.3.2 The Partition Function and Correlations With Dr0 = .., Ti xer d0(x), the functional integral with sources J : F is defined as (1.87) Zr,x(J) = f D rO e - sr(0)+GIMr. This integral converges if A> 0 and mo > 0, or if A> 0 and mo 2 = /./ E R is arbitrary. Expectation values of observables 0, that is, functions of the field 0, are defined by

( 0 ) = Zr,1),(0)

f Dr° e -sr(°) 0(0).

(1.88)

Only those 0 are admitted for which this integral converges; this includes, e.g., all polynomials in the fields, i.e., all moments of the measure with density Definition 1.9. The correlation functions, also called Green functions, are

(0( X 1) • • • (15 (X n)) =

1

Zrdk (0)

(1.89)

f DrO e—Sr(ç6) 110(X i) . k=1

These moments can be calculated as derivatives of Zr, x (J) with respect to J, via

(0(xi) • 0(xn))

[

6n 1 Zr x(J)] Zr,x(J) 6J(z i )...6.1(x n ) '

. J=0

(1.90)

1. Field Theory

16

Thus the partition function Zr, À (J) with source terms J contains all the information about the correlation functions. The derivatives 6n

[6.1(x 1 )...(5.1(x 7,)

log Zr x(J)1

'

(1.91) J=0

are the connected correlation functions. In Chap. 2, we shall prove that the connected correlation functions are sums of values of connected Feynman graphs. The significance of the Green functions is that the S-matrix can be constructed from them by the LSZ-formulas [9]. Note that in (1.88), we are integrating over all field configurations 0 : F R in the same way as we were integrating over all polygonal paths in the quantum-mechanical case. In the limit E --- 0, there is no reason to assume that these configurations arise from Cl fields 0 on i 'd/LZd. On the contrary, the typical field configurations are not Cl, just as the typical path contributing to the Wiener integral is not C'. A simple geometrical interpretation of the 0 field in d = 2 is that of the height of a surface, so Zr (J) describes a statistical mechanical ensemble of random surfaces. The probability density is given by the Boltzmann weight e - s/Z, so the action S effectively appears as an energy function. The form of S shows that large values of (40) 2 , of m0 2 02 , and of A04 are supressed. Note that, if e 0, the scale of smoothness, which is E, goes to zero, so there is no reason to expect the random surfaces to be Cl in the continuum limit. The particle physics interpretation of Z is that of a massive field theory with a repulsive self-interaction A > 0. For A = 0, the particles are free. Note that while the measure Dr0 has no limit as i F I oc, the Gaussian measure obtained by combining the quadratic part of the action with Dr0, and normalizing suitably, has a limit. This is a natural procedure from the physical point of view because the propagation of free particles is then given by the covariance of the Gaussian measure. The lattice Laplacian generates a random walk ensemble, which is yet another way of viewing the statistical mechanical system. Important nonperturbative results, like triviality of 04 in d> 5, are based on the random walk representation [10, 11]. In that language, m0 2 is the killing rate and A > 0 means that the random walks interact: they repel each other. For m0 2 > 0, —A + m0 2 > m0 2 > 0, so the Gaussian integral converges. Since A > 0, the 04 term only improves the convergence. The case m o = 0 (massless field) can also be defined (one has to take care of the zero modes of A, the "torons" (=constant configurations). This is no problem on but the limit L oc becomes more difficult in that case. The slow decay of (—A) -1 (x, y) in d = 2 causes the Mermin-Wagner theorem [15]. The limit E 0 and L oc does not exist for d = 2: there is no massless field in d = 2 [16]. We shall keep m o > 0 when analyzing perturbative field theory. --

1.3 Field Theory on a Lattice

17

1.3.3 The Ising Model

We show that the Ising model is a limit of the statistical mechanical system defined by our lattice field theory. The local term in the action is drawn in Fig. 1.1.

(a)

(b)

(c)

Fig. 1.1. The 04-potential for different values of mo 2 . (a) mo 2 > 0, (b) mo 2 =-- 0, (c) mo 2 < O.

Suppose that m0 2 = p < 0, A> 0, and let e = 1 for simplicity. Since 2

(1.92)

/±)2 _ 0 4A 7 16A

2

the local term in the action is of the form drawn in Fig. 1.1(c). We use

1 - - E(0(x + ek) - 0(x)) 2 2 x )k

. E 0 (x ) (t (x ± ek ) - d E 0(,)2 x,k

(1.93)

X

to rewrite the action in the exponent of the integrand for Zr (J) as

-Sr(4) =

E 0 (x) 0 (x + ek) -D( 122 + d)0(x) 2 - A0(x) 4 ) x

x,k

2 (kil+2d) 2 ( 4 (x) 2 ± ti + 2d) 16A 4A ±

I rl .

(1.94)

Zr(J) = Ki n f H ( c10(s ) e.1(x)0(x)--À(0(x) 2 -a2 ) 2 ) H eo(x)0(2,±ek)

(1.95)

(1)(x)0(s + ek) - A

= x,k

E x

If p + 2d < 0,

xEr

x,k

with

lp + 2di (1.96) K = exp( (11. + 2d)2), a = 16A 4A . The last factor in the integrand suppresses values of 14)(x)I far away from a. In the limit A oo, a kept fixed, the last factor becomes proportional to a sum of (5 functions because it is peaked at ±a, and because

18

1. Field Theory

f e _ot(02 _ a2,idO

f e-2a2

-ct)2

f e -2a2

2

(1.97)

2a2A. Therefore, the rescaled partition function

Zr (j) = Alf112 Z(J)

(1.98)

is a constant times

E

(J) =

eE

e

0(x)0(x+ek)

(1.99)

00E-q-ct,a}

Rescaling the fields, s(x) =

, and the sources, H(x) = as the partition function of the Ising model Zif,(H)

E

=

e--0E(8)

, we recognize (1.100)

sOE, { -1,+1 }

with the interaction

E(s) =

— E s(x)s(x

+ ek) —

H(X)S(X))

(1.101)

x ,k at inverse temperature

fl

IP(A) + 2d1 ' 4A )'-'°°

- a2- h im

(1.102)

The source term now appears as a spatially varying magnetic field H. The interaction is between nearest neighbours, and it is ferromagnetic, i.e. equal spins have bigger Boltzmann weight e0 s(x)s(x+ek) than opposite neighbouring spins. For this reason, the Laplacian is also called ferromagnetic. The quantities that can be put in relation to physical measurements are the correlation functions defined in Definition 1.9, i.e. the functions (F(s))

= 1

E

F(s)e- '3E(8) .

(1.103)

`j

For instance, (si) is the local magnetization and E y ((S x S y )— (S x )(S y )) is the magnetic susceptibility of the spin system. This is not the place to go into details about the Ising model and statistical mechanics, since there are many excellent references in the literature [17, 18, 19, 20, 21, 22]. We only mention very briefly some of the main results. The behaviour of the Ising model at H = 0 depends on 0: for small 3 (high temperature), the spins are disordered, and (s s) 0 exponentially fast as Ix-YI oo. For large 0,

1.4 Free Fields

(sz sy )

-Ix -y I -HD°

u(0) vanishes at a critical point

( (7 3 ) 2 0 0-

19

(1.104)

fie .

Note that for )3 > fie , the correlation functions depend on the boundary conditions. With our choice, we pick the symmetric phase where (si ) = 0. Had we chosen boundary conditions that favour spin alignment, such as the condition that all spins on the boundary are +1, we could have achieved (si ) = o- (0) so that

(s z sy ) - (s x )(sy )

--lx-Y1->00

0,

(1.105)

again, exponentially fast for 13 > fie . Only at )3 = 0,, one has a power law decay. It is at this point that one has to look for a continuum limit if one wants to construct a continuum theory without a momentum space cutoff directly 0 while keeping a finite from a lattice theory. As mentioned, we shall take E cutoff Ao in momentum space, and then study the limit Ao oc directly in the continuum (which is a different limit).

1.4 Free Fields For A = 0, the integral is Gaussian, and this corresponds to the case of free (noninteracting) particles. The free action is diagonal in Fourier space. We give only the main definitions of the Fourier transform here; details about the Fourier transform are in Appendix A.2. The dual lattice to F is F* = 2 /r zcii 2: zd . We define p - x = pi x i + .. . + ,.. 7 and 6(p,q) = Ld6pq . The Fourier Pdxd, fr. dPP(p) = IrdE pEr* P(P) transform of f : F

C is f : r*

C, given by

f(p) = f dx e -i" f (x). r The inverse relation is

f(x) = f - dp ei" r

i(P).

We denote the scalar product for functions i4, 7:b E Cr* by

(îko'k)r. = fdP (i5 (P)1 (P)

-

(1.108)

Fourier transformation is unitary: (0, Tp) r . = (0, T,b)r. The integral kernel of a linear operator transforms as

Â(p,q) = f dx f dy e -i"A(x, y)e i". r r

(1.109)

20

1. Field Theory

By(1.86) and (1.109), the Fourier transform of the kernel of the lattice Laplacian is

- .3(p, q) = 6r. (p, q) with

(1.110)

d

2

DE (p)

(p)

E(1

(1.111)

cos(epk )).

-

k=1

That is, the Laplacian is diagonal in the Fourier basis. Thus

(_ ± mo 2 ) 03, _ 6r. (A q) m In the limit E

(_ A rno 2)

0,

DE (p)

+1 DE (p) = 6r* (p, q) (p).

(1.112)

p2 , so

Lid(5_

1 (7 , q)

Pq

1 p2

rno 2

= r. (p, q) 0(p).

(1.113)

The quadratic form defined by the action is

=

dP (^b(P) (m0 2 + D (1)N(P)

fr *

=f

dp (7)(-p) (m 0 2 + De (pN(p)

(1.114)

Jr*

because q5(x) E

Vx implies

0(p) = ç( - p). Thus, the partition function is

Z r,À (J) = Z r » (0) f dpc r (0) e--À

0(x)4dx4-(J,(P)r

(1.115)

where the normalization constant of the Gaussian measure

Zr,0(0) = det(27rCr) 1/2

1

( (

k

-

Ed

L"

mo 2

± DE (p)

)1/2

(1.116)

pEr*

is unimportant because it drops out in the quotient defining the correlation functions. We can therefore redefine Zr,À(J) such that it does not contain that constant, and get

Zr,x(J) = f dpc, (0) e - w(0) + (J,O)r with V(0) = f 0(x) 4 dx The quadratic part of the action, which defines the covariance Cr of the Gaussian measure, describes free particles that propagate from x to y (graphically later denoted by a line connecting x and y). The interaction term describes the influence of particles on one another; for the 04 theory one may imagine it as an interaction between two particles, denoted by a vertex with four legs. The interaction is local. This is, in absence of other fields, the only way to maintain Lorentz invariance in the real-time theory.

1.5 Properties of the Free Covariance

21

1.5 Properties of the Free Covariance The properties of CF (X, y) in the limits L oo and E 0 are central to the question why renormalization is necessary. For e > 0 and L < oo,

fr. dp rno2eip(x-Y) ± imp) .

CF(X, y) =

(1.118)

For mo > 0, the denominator is always nonzero, so the summand is continuous in p on 1 1 (1 / E7rZd and one can take the limit L oo, in which the Riemann sum fr . dp becomes the integral T27 f8 dp where B = In other words, the integral is taken over [- 71e , 5-, ) d . Thus 1eiP(x - Y) Jim C(x,y) = (27)d Lddp , DE (p) rn o 2 -1-

The limit 6

( 1.119)

0, and is given by the integral

0 exists if x - y

1 C(x, y) = in_\d kh -11 1

ddp fRd

eiP( x - Y ) m 2 +p2 '

(1.120)

One can calculate C(x, y) in terms of a modified Bessel function d

C(X,y) = (27 ) —d / 2 (

774 ) 2 1 Kii _1(rnix — YI) -

IX — yl

(1.121)

In particular, for d = 3, 1

C(x, y) =

47rIx

e -inolx-Y1 .

(1.122)

- YI

Most properties of C can, however, be found without any explicit formulas. For instance, the exponential decay as Ix - yl oc follows directly from the analyticity of 1/(p2 -P 77102) for Ii < m. By translation and rotation invariance, C(x, y) is a function of ix - yl. Scaling out the factor mo gives ddp e ip(x- y) C(x, y) = f (2 70d rn0 2 ±p2 = rno d-20d(rno ix _ yo with

Cd(t) =

Lemma

1.10. (i) For (ii) For t > 1,

all t>

I

ddp eiPl t

(270d 1+p2

(1.123)

(1.124)

O:

Od(t) < const t- d;1 e-t

(1.125)

22

1. Field Theory

(iff) For t < 1,

> 33 if d> if d = 2

(d— 2)

Cd(t) = const { 'f log t The bounds become sharp as t

(1.126)

O. The constant depends only on d.

Proof. This is a standard proof [6]. For convenience of the reader, it is reproduced in Appendix A.3. •

Note that C(x, x) does not exist: there is a singularity in the propagator at short distances. This is the usual behaviour of a Green function (cf. the quantum mechanical propagator), but also the reason why renormalization is necessary. For large Ix — Y I, I C(x, 01 < const e - mo 1 x -Y 1 . If mo were zero, this would be replaced by a power law decay, and one would have an infrared problem as well. For E > 0, there is no singularity, but the propagator grows as 6 —. O. Remark 1.11. The lattice propagator obeys Cr e,L (X,

as e

X)

for d = 2 for d > 3

log E

r'd const

(1.127)

O. For d = 2, the constant depends on mo > O.

Proof Because

p2

2 Ez

(1 — cos(pe)) >

2 7r

we have

J

p2 _F ddm P 02

Hzi , if id

f m0

[__ t , lild

ddp

ddp 2 ± DE (p)

f [

m0 2 ±

p2 '

Z le , !ld 1E

Because [

:E 1 C

7 7r 6

1

d

{P : IPI < "\--Lr } E

6

(1.130)

we have

fP {7)415- Z76 }

dd p < 2 ± m0 2 —

Cr e,L (X,

X)

<

J

{P:iPl<

ddp

mo 2 ± 3 7 11 2 .

(1.131)

}

Both the lower and the upper bound behave in the same way, namely as given in (1.127) (see also (1.137)). •

1.6 Problems With and Without Cutoffs

23

1.6 Problems With and Without Cutoffs To see why the short-distance singularity of the continuum covariance is a problem, consider (0(s) 2 )F,À=0 . This function diverges for c 0,

f di.Icy (0) 0(X) 2 = C r (X ,

X)

—+

.

(1.132)

So, the second moment (0(x)0(y)) = C(s, y) has a singularity at s = y (coinciding points). This by itself would not be a problem. The problem is that these singularities cause divergences in the continuum limit of the partition function, and of the correlations, defined above. For instance, the integral

f dpc, (0) 0(x) 4 = 3 Cr(s, x) 2

(1.133)

occurs in the expansion f CIP, C r (0) e

À f 0(x) 4 dx — 1 + A f dx f dtic0) r q5( (x) 4 + ... r — r

(1.134)

already in first order, and it diverges as e 0. There are two ways out of this problem - renormalize, or never remove the cutoff. Chaps. 2 and 3 are concerned with renormalization. In the remainder of this section, we briefly discuss why keeping a cutoff is a problem for the particle physics interpretation of the model, and then motivate Wick ordering. We don't know if the world is a lattice or not, but on measurable scales, it is Lorentz invariant. A lattice model is not even rotation invariant, so one has to take e 0 to get back Euclidean invariance. But one does not have to take a lattice to regularize the propagator. One could, instead, also take the following continuum limit with a rotation-invariant cutoff Ao . Let K E C'(1R-0/- , [0, 1]) obey

K (x) =

Ii1 if 0 < x 0 if x > 4, <1

(1.135)

K ' (x) < 0 for s E (1, 4), and let A0 » 1. Define 1 f dd eiP(x - Y) K( p2 Co,A. (x, Y) — ( 27)d P p2 +m 2 ‘ AgP

(1.136)

Then Co,A„, (s, y) has no short-distance singularity because, as we shall show in (3.7), for Ao > 2, I Co,A0 (0, 0) I < const

{ log(A0 ) A0 -2

if d = 2 if d> 3.

(1.137)

The constant depends on mo , and grows like I log Mo I as m o 0 in d = 2, and it is independent of mo for d? 3. Hence all moments of the measure with this

24

1. Field Theory

covariance are finite (but Ao-dependent). Moreover, Co,A0(x, y) = 0 A0(1 1 - Y i) is rotation invariant. One could also have taken a Pauli-Villars cutoff (1.138)

p2 + Ag

p2 + mo2

instead of p2+M 1 O 2 •• this removes the singularity, at least for d < 3, and it is a less restrictive cutoff. But all these cutoffs violate physical positivity: a Euclidean invariant field theory with a genuine Hilbert space (i.e., where the scalar product is positive definite, as it should be), the two-point function has the Kallen-Lehmann representation f 00

=

Jo

dp(p) ti 2

+ 1 p2

(1.1 3 9)

where dp(p) is a positive measure. But then the two-point function cannot decay faster than 1/p2 for II oc. Thus any cutoff that makes 0 integrable leads to problems with physical positivity. We therefore have to remove the cutoff in the end. Another reason for not keeping Ao is that such a momentum space is not gauge invariant. Since there seems to be no way of smoothing the propagator without losing some physical requirement, one can only try to change the interaction. The first step, which suffices to overcome the above divergence, but not others, is to do Wick ordering. It means replacing the monomials in the action by their Wick ordered counterparts (which will be defined in the next chapter), for instance replacing 0(x) 2 by

- Cr (x, x).

(1.140)

Then, by definition,

f :,(x) 2 :dc,..(,) , 0

(1.141)

0. Similarly, one replaces 0(x) 4 even though f 0(x) 2 c1p,c,(0) diverges as e by (1.142) : 0(x) 4 := 0(x) 4 - 6 Cr(s,x)0(x) 2 + 3 Cr(x,x) 2 in the interaction term, which then satisfies

f : (x) 4 : dc() = 0.

(1.143)

For the moment we just note that the problems with the limit E —} 0 in (1.134) disappear to first order, i.e. the first order expansion in A of the redefined dlicr (0) e-À fr,:o(x)4:dx±(J,0), (1.144) Zr,À(J) = f

1.6 Problems With and Without Cutoffs

25

is bounded uniformly in E. There is no claim that this cures higher order divergences in general, and it doesn't. For E> 0, let us write this out as

Zr,),(J) --

1

:

Zr 0(0) --A fdx(4)(x) 4 -6Cr(x,x)0(x) 2 +3Cr(x,x) 2 )

By the lattice translation invariance, we have added the E-dependent term

6 ACF(0, 0)

f

Cr(x,x)

(1.145)

= Cr(0, 0) is a constant, so

dx 0(x) 2 — 3 A Cr(0, 0) 2

f

f

dx

(1.146)

r

to the action. The second of these terms is just a constant, which can be absorbed in a redefinition of Z. The first term is a correction of the mass term —6ACF (0, 0), 0 by (1.127). The interpretation of this term is the which diverges as E main point of field-theoretic renormalization: the extra term does not change the functional form of the action Sr (0), but only the value of the bare mass, that is, the mass parameter in the action, from m 1, 2 to mo 2 -6ACr (0,0). The bare mass is not observable because the particles are not free. The observed mass contains the interaction effects. There is, as we shall see, a one-to-one relationship between the bare and the observed mass, so one may fix the latter by experiment. This means that it is fixed as the value of a correlation function at a particular momentum. The simplest choice is

f dx (0(0)0(x)) = m102 .

(1.147)

Adjusting everything this way, one has removed the divergences in first order perturbation theory. The additional terms in Sr are local, so there is no problem with Lorentz invariance, once the correlation functions have been shown to converge. If one can absorb all divergent terms into a redefinition of finitely many parameters of the action (preferably, only few of them), such as mo2 and A, to any order in the expansion, then the theory is called perturbatively renormalizable. In such a theory, a finite number of parameters have to be fixed by experiment. Once this is done, one can predict the results of other experiments, by renorrnalized perturbation theory, in which no divergences appear, and compare them to measurements. Of course, the above discussion has not really proved anything of that kind. The divergences may reappear in higher orders, in particular because the terms added to make the first order finite may actually cause infinities in higher orders. It requires more insight into the structure of field theory to see that this is not so and to understand how renormalization really works (also, to say it again, Wick ordering is not sufficient for d> 2).

26

1. Field Theory

It will turn out that Wilson's form of the renormalization group, (see Chap. 3), provides a different point of view of this subtraction procedure, as a change of initial conditions for the sequence of effective actions. In particular, it can be formulated without any explicit reference to the bare parameters of the action, that is, to the very high energy behaviour, once the form of the action at that energy is determined. It relates only quantities at energy scales that are independent of, and much smaller than, the cutoff. In the next two chapters, we shall develop the formalism and prove some basic results about perturbative renormalization, that is, statements about the Green functions that hold in a formal power series expansion in A. The question about convergence or divergence of the expansion in A for the 0 4 theory is much harder and is discussed here only in a simplified case, namely the flow generated by "bubble" graphs, which leads to the so-called Landau pole in four dimensions (see the last section of Chap. 3).

2. Techniques

To get a deeper understanding, as well as the ability to do field theoretic calculations, we need to learn the standard techniques of Gaussian integration. Not all of this will be needed in Chap. 3, but these standard techniques are a prerequisite for reading most of the current literature in field theory. Most of this chapter is concerned with Feynman graph techniques. Emphasis is put on defining Feynman graphs in a simple but precise way and on giving an unambiguous meaning to the "sum over values of Feynman graphs". Although every statement about Feynman graphs can in principle be restated without a reference to a picture, their value for thinking about field-theoretic problems goes far beyond mnemotechnics. We shall see, for instance, that some combinatorial factors are very easily - and safely - calculated with the help of Feynman graphs. Needless to say, they are also the central tool of the detailed calculations that are the basis for the great success of perturbative quantum field theory in particle physics.

2.1 Integration by

Parts

Let A be a positive operator on RF and C its inverse. Recall that (50(x) 6 00(x) • Since fr dz C(x, z)A(z, y)=-- E -dSxy , and *OM = E -d 6xy , we have -

0(x) = f

dy Os, y)

6

1

(2.1)

SO

dy C(x, y) (5 ci("Or 5 0(Y) With this we get by integration by parts 0(x) e -

i (q5 'Aq5)r = -

Jrr

(2.2)

f Dr 0 F(0) 0(X) e - fr dy C(x, y) f Dr 0 F(0) 6 06

(Y)

1 6 dy C(x, y) f Dr q e- 1("49) F(0). 60(Y)

(2.3)

28

2. Techniques

Inserting the normalization, we get the integration by parts formula

f dpc, (0) 0(x)F (0) = f clY C(x, y) f dPc(0)

5F

(2.4)

r

2.2 Wick Ordering We have already seen that Wick ordering can cancel some divergences in the continuum limit. Since Wick ordering alone does not remove all divergences, this may not seem a big deal (and one can do renormalization without Wick ordering). However, Wick ordering is natural, because it simply corresponds to introducing orthogonal polynomials associated to the measure dpc given by the weight function e - i (0,A0 r , and it leads to substantial simplifications both in the Feynman graph representation and the renormalization group equation.

2.2.1 Definition and Main Properties Let

C = Cr be a nonnegative symmetric operator on Cr. For WF(- 1, A) (

J: F

= e i(JMr -#- 1(.1,C .1)r .

C, let (2.5)

Definition 2.1. Let Pr be the algebra of polynomials in (0(x)),, E r. Wick ordering is the C-linear map flc :Pr Pr that takes the following values on the monomials: (2.6) r2c(1) = 1 ,

and for n > 1 and z 1 ,...,x,, E F (not necessarily distinct) n 1

Q c ((z) • - - 0(xn)) = [H

6

k=1 1

r (J, 0)]

Ik)

(2.7)

• J=0

Remark 2.2. flc is well-defined because for all permutations r E Sn

rIc (401) - - - 0(x,)) =--- f 2c (0(x n (i )) • • • 0(x r(n))) •

(2.8)

The term of highest degree in the polynomial f2c(0(x i )...0(x n )) is 0(x 1 )

• • • (15 (xn)

-

Proof. Let 7r E S. Since commute,

6 =E d a and since the partial derivatives

r2 c (0(x It (l )) .. - «x,.)) =

1-112-r

6

1

11=11 i 6 J(xir(k))

14)r(j'd

45x0 vvr ( j, 0) 1

.i.o

[pi li 6.

i J=0 QC ((XXI) • - . 0(Xn)) -

(2.9)

2.2 Wick Ordering

29

Thus f2c (0(xt) . . . (x)) is well-defined on the set of all monomials, a subset of which is a basis of the linear space Pr. f2c therefore defines a unique Clinear map from Pr to Pr. • Usually, the Wick ordered monomial flc(0(x 1 ) ... 0(x,)) is denoted by : (1)(s 1 ) ... q(s) :. In the following, it will be important to keep track of what covariance C is used for Wick ordering, therefore we use the notation Lk, • Of course, a Wick ordered monomial is a polynomial, and not a monomial, in the fields.

Theorem 2.3. Define, as a formal power series in J, c°

flc (e i(J' (P) r)

in

(2.10)

n=0

Then

(2.11)

Wr(J,0) = fk (e i(j)(15)r ) • Let a: F C be defined by n

a(x) =

E aor(x,sk) = { 60 - dak

k=1

if x = xk otherwise.

(2.12)

Then n

1

f/c (q5(xl) - - - 0(sn)) = [ (11 i k=1

a

ei(a4)r+i(a,Ca)r]

oak)

(2.13) cy=0

Proof. By Taylor expansion in J, and by definition of flc,

Wr (J, 0) — 1 n 1

7 = n>1 E7

II J(xlc) xi E

k=1 n

,...,x n Er

[ n

II aJPx,) VVF(J, 0)

k=1

= V 1 H Axic) fdx,...dxn n! n>1 k=1

[ft k=1

6 j (6x k

J=0

) VVF(J, 0) J=0

n

=

n E ni1 11 J (xk ) f dx 1 . . dxn in flc (11 4(xk)) n>1

= L_., 7.7.

k=1

k=1

fk ((J, (15)r')

n>1

= f2c (e i(jM r) — 1.

(2.14)

The last equality holds because flc is continuous in the formal power series topology (see Appendix A.1). (2.13) is obvious from the definition if one sets J(s) = e- da(x). •

30

2. Techniques

We shall use the suggestive notation flc (ei( ") r) for Wr (J, (to) since the power series is convergent for all 0 and J. The next theorem contains an alternative formula for the Wick ordered monomials that will be useful in Chap. 3. Theorem 2.4. Let

1 (5 1 ( 6 6 (5 ) C(x,y)6 o , C Tt) = - f dx f dy •ic. = -2 T 2 60(x) 0(Y) r r

(2.15)

Then Qc (46(xi)...0(x n )) = C A' 0(xi) ...0(xn).

(2.16)

In particular, if C depends differentiably on a parameter t, then

0 — Qc (0(xl) . • .0(sn)) = Ot

aAc

at

(-lc (0(xi) . . . cto(x n )) .

(2.17)

Proof. Consider C as a formal parameter. Then for any power series f (z) = Ek fk Zk

PAO ei(J'°' = f Gi (iJ ,C iJ)r) eiGIMP ,

(2.18)

e —Ac ei( " )r = el () r +1(JM r 7

(2.19)

SO

from which (2.16) follows by definition of flc, because derivatives with respect to J commute with Ac. If C depends differentiably on a parameter t, (2.17) follows by taking a derivative of (2.16) because Ac commutes with V-. •

2.2.2 Further Results

Section 2.2.1 contains all the statements about Wick ordering that we shall need for doing calculations. For completeness, and to verify that Wick ordering indeed gives the results stated in Sect. 1.6, we derive here the recursion relation and prove the orthogonality relations. Theorem 2.5. The Wick ordered monomials satisfy the recursion relation

(n+1 QC

(P(Sk))

=

n flc (11 0(xk)) Oxn+i) k=1

k=1

n

n

_ E c(sk ,xn+i )

f/c

k=1

(

11

(xi)

(2.20)

ig

Moreover,

f dttc(0) Qc (e i(jMr) = 1,

(2.21)

31

2.2 Wick Ordering

for

all n > 1,

f dpc(0) f 2c(0(xl) ...0(xn)) = 0,

(2.22)

and if n 1 n2 , n2

f (Lac (0) f2c ill (

flc

15 (S k))

(

Proof.

k=1

110(M))

=0.

(2.23)

(

Since

(5

'1

i 6J( sn 4-1 )

Wr (J, 0) = 0(xn-1-1)

_if

dx C(x n±i , x)J(x) Wr(J, 0),

r

(2.24)

(2.20) follows if we can show that n

. E 6 r (x , x k ) flc

1 6 11 i 6 j(xo J(x) Wr(J, 0)] n

[

k=1

J=0

( n

)

II q5(xi) z=i tore

k=1

(2.25) By the product rule, every derivative can act on J(/) or on Wr(J, 0). If ,y on J(/), we get 6(x,/ k ), and all other derivatives must act on (5. t5 acts Wr . This gives the sum on the right side of (2.25). If no derivative acts on J(x), evaluating at J =--- 0 gives zero, so (2.25) holds. (2.21) is obvious from doing the Gaussian integral: by (1.47),

f

cl A c ( 0 )

tk ( e v, o) r )

--72-

r f dc(q5) e i(J,49 )r e i(J,C J)

(2.26)

= 1.

Equation (2.22) follows from (2.21) by expansion in ni f CitiC(0)QC0) (H 40ric k=1

= li n

F(a, 13) = =

=

1=1

di i c f e

(n2

H 1=1

) O(Y1)

li k fi ai3i F(a, 0)]

[ k=1

where

J. By (2.13),

(2.27) a=0=0

( 0) ei(a4- 0,0P e i(a,Ca)r+i(0,CP)r

(a+13,C(a+0)) ,-.4 - i(a,Ca) r+ i(0 ,C 0) r

(2.28)

32

2. Techniques

Expanding the exponential, we see that every term has the same degree in a of as in O. Therefore taking derivatives and evaluating at a = fi = 0 gives zero if n i n 2 . • The recursion relation gives

0(40c(0(x)) - C(x,x)flc(1) 0(x) 2 - C (x , x)

=

fk(0(x) 2 )

(2.29)

and

A7(011 4)

ox) ,Qc(o(s) 3 )-3C(x,$)flc(0(x) 2 ) = q5(x) (0(x)flc (95(x) 2 ) - 2C(x, x) f c (0(x)))

=

- 3C(x, x) ,Qc (0(x) 2 ) = 0(x)4 - 6C (x , x)0(x) 2 + 3C (x, x) 2 ,

(2.30)

which are the identities used in the Sect. 1.6. The integration by parts formula gives

f

dPC(0)

n+1

Ec(xk,in+,) f d,c(0) 11 0(xi)

k=1

k=1

11 (too

(2.31)

z=1

zok

which is similar to the definition of the Wick ordered monomial, only that the terms that get added when one does integration by parts get subtracted in the Wick monomial. In one variable, the Wick ordered monomials are scaled Hermite polynomials: Calling (*I) = 0, C(xi, xi)_nflc c ,oneoigets Q(l) = 1, rk (0) = 0, and =C f-2c(on) (on+i) = j-) c for n > 1, n-1 ). Calling Pn ( 0) = flc (On), this is the recursion relation P

1 (5) = 0-Pn (0) - n C Pn-1(0) •

Thus

C n/2 \

Pn(çb) =

iin(f2-0)

(2.32)

(2.33)

where Ho (x) = 1, I-11 (x) = 2x, H1

(x)

= 2x H n (x)

-

2n Hn _ i (x)

(2.34)

are the usual Hermite polynomials. The generating function of the Hn is _ 82 == e 2xs

co

n=0

lin(X)sn.

n!

(2.35)

2.3 Evaluation of Gaussian Integrals

33

2.3 Evaluation of Gaussian Integrals 2.3.1 Labelled Feynman Graphs

In this section we evaluate Gaussian integrals of the form

( A,

, vp; 1p) = f dc(Ø)

(2.36)

Vk ± H k=1

where Vk (0) = f

.f

clxink vk(x l ,... ,smk )0(x i )... 4)(x mk )

(2.37)

and

11 q=1

I (V1 • • • ,Vpill)) = f dPC (0)

C (Vq(4 ) 0))*

(2.38)

I

We shall do the function in detail; the changes in the representation for Î will then be obvious. The result is going to be a sum of values of "Feynman graphs", which will be defined in detail below. However, we briefly indicate now how these graphs are going to arise, to make it easier to follow the derivation below. We shall visualize every vk as a vertex shown in Fig. 2.1, with mk external, numbered, legs. These legs will be joined to an external field IP or to another vertex by a line, which has an associated factor C(x i , x i ).

(k,m c l)

(k,2 (k,1)

(k,mk)

w

Fig. 2.1. A fully labelled vertex

A precise definition will follow below at the end of the derivation, but we shall discuss the graphical significance of various objects already when we introduce them. We introduce the notation Xk =

k ,1 • • •

Sk,mk ))

f dX k =

r

dzk1, • • •

r

dsk , mk

(2.39)

and k)

(2.40)

V,,(0) = f dX kV (X k)(1) Xk •

(2.41)

k

= (I)(X

AX k

C ,i) • • • (

so that

34

2. Techniques

The vk are assumed to be symmetric under permutations vk (xi, .. • , Sink) = Vk(Sii(1), • • • ,Sir(m),))

VT- E Smk ,

(2.42)

because any nonsymmetric part of vk would cancel in (2.41). Also, we define the set of legs of vertices determined by V1 , ... Vp as

Bo(p; mi • • Onp) = {(k,ik)

k E { 1, . • •

iP}iik E { 1, ...,rnk}}

(2.43)

on B0 by

and a partial ordering

(k , i)

(1, i) .#> k < 1.

(2.44)

Note that B0 specifies the set of vertices completely: every vertex has a number k E {1, ... ,p}, and every one of its legs is numbered as well. We now drop the arguments of B0, keeping in mind that it depends on p and m l , . . . , my . For 00Bc B0 let Lo (B) = {(b, b') =

b,

E B , b 11}

(2.45)

{((k,i), (i,j)) : (k , i) E B , (1, j) E B and k < l}.

We shall see that LO (B ') is the set of all internal lines that can be formed by pairing legs of vertices. Obviously, /(V1, , Vp; 0) is linear in every Vk• Since (Xk)k = (zb)bEB03

i(v1

, ..•

vp;

=

f II

dXb

ft

v(Xk)

Axi, • • • ,xp;//))

(2.46)

OC((4) 0) Xj0 •

(2.47)

k=1

bEB0

with J(Xl,

Xp;IP) = f dPC(0) k=1

Since the Xi will be fixed in the following, we write J(0) = J(Xl, • • • X,,; /'). We now use (2.13) in the form Mk

{ Qc ((o, + o xo = -.--....

i

n

1

U

(2.48)

ii •i aa k,j 11c (eir) j=1

) with

a (k) = 0 k,1 6F(x,xk,1) +

+ ak,mk 6 r(s,xk,mk)•

(2.49)

Note that even if the s k , r are not distinct, the a's are; this is a typical example of the combinatorial method of "giving identities" to identical objects that appear several times. Doing this for all k E { 1, ... ,p}, we arrive at

la

=

i act f

bEBo

b

flc(e,)]

dp,c(0) k=1

(2.50) aO

VbEBo

2.3 Evaluation of Gaussian Integrals

35

,0)) = ei(a (k) ,0)+I(0,(0 ,ca (k) ) , we get

Since Qc(e

P

1 a )0

J(0)=[(ll

P

(k) ) 1, \-L, (a (k),, a

7 Oat,

i Eca(k),o+o i

f Cl/IC(0)e

k=1

k=1

(2.51)

J a=0

b E Bo

By (1.40), the Gaussian integral is

f

i t(01(k/ i40-FIP) Cliic(0)e k= 1

i t(

p

P

) — 1( Ea (k), C E a (t) )

= e k=1

e

k=1

1=1

(2.52)

.

By the symmetry of C, (ci), Ca(') ) = (a(l) ,Ca(k) ), so

P ca") iDa(k),o), -E(a(k) , 1 a . e k=1 J(0) = [(H — —) e k
(2.53)

bE/30

Note that the Wick ordering factor has cancelled the k = 1 terms in the sum. This is the reason why after Wick ordering, no two legs of a single vertex will be joined to form a line that connects this vertex to itself (a "self—contraction"). P

We rewrite E i(a(k),

to =

k=1

E iab 11)(sb) and bEBo

_ E(c,(k),c.(/)) = k<1

E

iabC (Sb ) Sbi

(2.54)

)iab, •

(b,b1EL0( 130)

Note that the sum in (2.54) is the reason why L0 (B0 ) was defined with the partial ordering --<. It arises in a natural way from the restriction k < 1, which in turn is caused by Wick ordering and by the symmetry of C. Since we evaluate at a = 0 to get J, only a polynomial linear in every ab contributes to J. Thus we may replace the exponential by a polynomial, and iab by ab everywhere, so that i TT

J(4')

=

1-1 ' bEBo

abc(x,„. boa b, E abg ixb)+ E n ,..., , e bEB 0 (b,b, }EL0030) aab )

a )) — H (1+ abo(zo) [(II H bEB0 aa

;

bEB0

(1+ abC(sb,x0abi)]

(b,bi )E

a=0

.

(2.55)

L o(Bo)

It is convenient to use the following lemma.

Lemma 2.6. Let n > 1 and f and g be polynomials in e = (el) ... ) en) Then

[f (;)e g( e)]

c.-0

=

[g ( 2ae) f

(el

c.0

.

(2.56)

36

2. Techniques

Proof. The identity is linear in f and g, so it suffices to verify it for monomials. For monomials, it is obvious. • By Lemma 2.6,

J(0) =

[exp (

E ,P(xb) aaab ± E *cfcsb,zo act

bEB0

=[

II

H )

(1 ± Oaba C (X1) ) S b i ) Oaa b i )

(ti,b 1 )E Lo(Bo)

bEB0

ab

I

(2.57) cc---,0

H ( 1+ 0(xo 0+) H ad cx=0 • bEB0

bEB0

We now apply the identity

H ar,

Il (1 + at.) = E

(2.58)

NM rEN

rEM

which holds for any finite family (a r ) rE m of commuting objects, to (2.57), take the derivatives and evaluate at zero. Since every ab appears only once in H bEBoab, every derivative can act at most once. If we apply

II

a

,

(

bE.130

i+cso — aab

)

=E

H

, IP(Xb)

ECB0 bEE

a

(2.59)

— 0ab

first, this leaves FL EA, \E ab for the other derivatives to act on. Because we evaluate at a = 0, all of these factors must be used up by derivatives (that is, all non-external legs of the vertices must be used to make internal lines of the graph), and we obtain

Alp) = E 11 Csb) J

(2.60)

ECB0 bEE

with

E

J =

{

H

aaab c(sb,z049c,,a b,

= E n Ip(xo EcB0

bEE

H

ada.0

bEB0\E

LCLo(Bo\E) (b,b')EL

E

fl

c(sb,sbi)

(2.61)

LEL(B o \E) (b,w)EL

where L E L(B) if L = {(b,b l ) E Lo(B) : IrilL and 7r2IL are bijective}.

(2.62)

Here 71-1 and 71-2 are the projections on the first and second components of a pair (b,b1 ): ri (b,b1 ) = b and T-2 (b, b') = b' . In other words, if L E L(B), then every e E B0 appears exactly once as a first entry b or a second entry h' in a pair (b, b') E L. The procedure of differentiating with respect to the ab and setting the a's equal to zero has the following natural graphical interpretation.

2.3 Evaluation of Gaussian Integrals

37

—The labelled vertex drawn in Fig. 2.1 represents the monomial vk(zk,i, • • sk,,,,) ak,1 • • • ak,mk , that is, leg number (k, 1) corresponds to the factor ak,i. —The action of1P(xk i )-d— is to make leg (k, i) an external leg of the graph: the factor aki is replaced by 1P(xki ). —The differential operator e;--0, C(s ki , removes the factors aki and . Accordingly, the legs (k, i) and (1, j) are joined to form a line of the graph. —The restriction that k < 1 following from the definition of L0 (B) means that only legs of distinct vertices are joined to form lines. —Evaluation at a = 0 implies that all terms that still contain a factor ab vanish. Thus all legs that are not assigned external fields must be paired to form internal lines of the graph. Examples are given in the next subsection. The precise definition of a Feynman graph is as follows. Definition 2.7. Let B o = Bo (p;m 1 ,... ,mx,), as defined in (2.43), and for O B C B o , let Lo (B) be defined as in (2.45) and L(B) be defined as in (2.62). Let E c Bo . A (labelled) Feynman graph for the vertex set determined by B o , and with external legs in E, is an element L E ,C(B 0 \ E).

Remark 2.8. By this definition, we have identifed a Feynman graph with its set L of internal lines. This is unambiguous since the vertices have already been fixed by specifying B o . The Feynman graphs are called labelled because every vertex and every leg of each vertex is numbered, and hence distinguishable from every other one. We specify "internal" lines because we sometimes also say external lines for the external legs (people not used to graphs having 'external legs' can think of the endpoint of the external leg as a vertex with incidence number one; this is unambiguous because such vertices do not occur as internal ones in our models). Note, however, that the external legs do not count as lines of the Feynman graph under Definition 2.7. This is a natural convention, as we shall see later on. Using the permutation symmetry (2.42) of the vk we obtain the following representation for AO). Theorem 2.9. Let I(0) = 1(171, ,V,0) be given by (2.38). Then

.40) =

E f del • • • dem ml) • • • 4)(m) Um (C; ci, . • • , em) m=orm

with rh =

>m

and

(2.63)

38

2. Techniques

v.

1

Um(C; 61, • • ,6m)

fr

111,•••,71m

E

111)k (Xk) k=1

dsb ll Bo\E c(x,,sb,)

(2.64)

LEL(Bo\E) (b,bs)EL

where the sum over iii,. , i E B0 is restricted such that the ri 's are pairwise distinct, E = { n' , . ohn } , and where x n, = for all 1 E {1, . . , TO. The Urn depend also on p and , VP.

Remark 2.10. If we had not used Wick ordering, we would have had an extra term f1

(2.65)

exP

k=1 aa which generates self-contractions of the vertices.

2.3.2 Symmetry Factors and Topological Feynman Graphs As an example, consider the integral =

f citt0(0) Qc(0(x) 4)Qc(0(0 4 ).

(2.66)

The graphical rules tell us that we have to join the legs of the two vertices shown in Fig. 2.2 in all possible ways to get the graphs. For instance, one can have the Feynman graphs in Fig. 2.3, with = {((1, 1), (2, 2)), ((1, 2), (2, 1)), ((1, 3), (2, 4)), ((1, 4), (2, 3))1 L2 = {((1,1), (2, 2)), ((1,2), (2,4)), ((1, 3), (2,1)), ((1, 4), (2, 3))}.(2.67)

(1,2)

(1,3)

(2,2)

(2,3)

(1,1)

(1,4)

(2,1)

(2,4)

Fig. 2.2. Two labelled vertices The value of each such graph is C(x, y) 4 , so to get the result we now have to count the number of Feynman graphs that one can make with these two vertices. There are four ways to join leg number (1,1) to any leg of vertex

2.3 Evaluation of Gaussian Integrals

Fig. 2.3. Two labelled Feynman graphs, corresponding to L1 and

39

L2

2. After that, there are three ways of joining (1,2) to any remaining leg of vertex 2, etc., so there are 4! different labelled graphs contributing to J. Thus J(, y) = 4! C(s, y) 4 .

(2.68)

Generalizing this, we can also write the Feynman graph representation in terms of Feynman graphs (see Fig. 2.4) where the legs of the vertices are no longer labelled. Then one has to include the correct symmetry factors, like the 4! in (2.68). Unlabelled Feynman graphs are also called topological Feynman graphs. If a labelled Feynman graph G' has been obtained from another labelled Feynman graph G by a reordering of the legs of a vertex, then G and G' give rise to the same topological Feynman graph. For instance, the two graphs drawn in Fig. 2.3 give rise to the same topological graph, drawn in Fig. 2.4. For small graphs, it is trivial to calculate the symmetry factors and thus to pass from one to the other representation. For large graphs, doing this can get rather involved. For the following graphical considerations, in particular the proof that taking the logarithm of the the partition function removes all disconnected Feynman graphs, working with labelled graphs is advantageous because every labelled Feynman graph appears exactly once in the sum and one does not have to think about any symmetry factors. 2.3.3 Motivations for Taking the Logarithm

The integral (2.66) occurs in the second order term of the expansion for

40

2. Techniques

Fig. 2.4. The topological Feynman graph corresponding to the graphs given by L1 and L2.

2r . fdp c (4)) e A2 = 1 + T

=1+

fr

dx f 2c (q5(x) 4 )

dx fr dy f dpc(0)Qc (0(x) 4 ) (20(4)(0 4 ) + O (A 4 )

12 2

4! f dx f dy CF(x, y) 4 + OW). F

(2.69)

F

Since CF depends only on z — y, and since fi-, dx = Ld, 2r =1+ A2 4! L d f dy C F (0, S) 4 ± OW). 2 r

(2.70)

The factor Ld was to be expected: in a statistical mechanical system, 2r . exp(L d fF) usually holds ("the free energy is a bulk quantity"), meaning that limL_÷ 00 fr,,,= f, : the free energy density stays finite in the thermodynamic limit. A proOf that this is indeed the case requires decay assumptions on the interaction and is not trivial. We shall do this proof in the next sections. One reason why this is nontrivial is that from fourth order on, disconnected graphs appear in the expansion for Z r . This gives rise to terms of order (Ld) 2 and even higher powers of Ld in higher orders, which is of course consistent with what one gets when expanding eL dfr , but it is clear that one has to prove finiteness of lim 0() fr . This proof is not by expansion in L d, of course, and one also knows from cluster expansions that the corrections to the bulk behaviour are not O(L) but more complicated functions of L [25]. By taking the logarithm, we shall be able to prove finiteness of the limit of the free energy density. The other problem is the limit e --4 0, in which the question of the convergence of fr dx CF (O , x) 4 for c --* 0 arises. If we take the continuum limit e -+ 0 naively, replacing the Riemann sum by the integral fR, dx C(0, x) 4 , and recall that

IC (0, x)i < const e —mixl

for II > 1,

(2.71)

we see that the integral over Rd \ Ix : II < 1} converges because m o > 0. But since

2.4 Polymer Systems

0 < C(0, x)

—

I ,I (d-2)

const { ' I log(x) I

if d> 3 if d = 2

41

(2.72)

as x -4 0, the integral

1 fix 1 <1

ddx C(0, x)4

dr r d -1

fo = fo dr r7 -3d r...,

r -40-2)

1

(2.73)

diverges for all d> 3. Only for d = 2, 1

ddx C(0,x) 4 — f rdri log rI 4
fi

A more complicated, but still elementary, calculation shows that the limit E -4 0 of the well-defined quantity fr ds Cr(0,x) 4 is indeed divergent for d> 3, and convergent for d = 2. Thus Wick ordering has not removed all divergences, at least if d > 3. To remove these divergences, we shall have to analyze the singularities in more detail. For this purpose, it is inconvenient to deal with connected and disconnected graphs at the same time. The most efficient way to proceed is to use Wilson's effective action, defined by

G erff(0) = log

f dp er,(0)e - w( 4)+0)

(2/5)

for V(0) bounded below. We shall prove in the next sections that Gerff (0) is the generating functional for the connected, amputated Green functions, and that Gerff (0) = Ldgerff (0) where gerff (0) has a finite limit as L -4 oc. The renormalization group is then an analysis where CF is split into many pieces, according to their distance from the singularity. The RG flow determines how singularities in Z and in the Green functions generated by geff build up in the limit E -4 O.

2.4 Polymer Systems In this section, we define the notion of an abstract polymer system and its partition function, and show that the logarithm of the partition function gets contributions from connected graphs only. This statement is of great use in statistical mechanics. It can be used to show that the free energy density indeed has a thermodynamic limit if the interaction decays sufficiently fast, and it is also useful to show exponential decay of correlations. The presentation in this section closely follows that in Erhard Seller's book [20].

42

2. Techniques

2.4.1 Preparation: Graphs and Partitions Let M c N be a finite set and

G(M) = P({(i,j) EMxM:i< j})

(2.76)

where P(A) denotes the set of all subsets of A. In other words, every G E Q(M) is a set of ordered pairs (i, j) EMXM with i < j. Every G E g(M) defines a graph with vertex set M by the definition that the line between k E M and 1 E M is a line of the graph if (k, 1) E G. We identify the graphical object with its set of lines G E G(M) by the following Definition 2.11. A graph G is an element G E

g(m). A subset H of G is

called a subgraph of G. For example, let M = {1,2, 4,5, 6, 7, 8} and G = 1(1,2), (1, 4), (2,5), (6, 8)}. The graph G is drawn in Fig. 2.5.

Fig. 2.5. An example of a graph In graph theory, a graph on M is usually defined as a set of unordered pairs {i, j} where i < j. This definition is obviously equivalent to the one given above. The notion of a graph is not the same as that of a Feynman graph because in a graph, two vertices can be joined by at most one line. The graph theoretical notation is, unfortunately, not completely standardized. In the nomenclature of [24], Feynman graphs would be called multigraphs and Feynman graphs without Wick ordering pseudo-multigraphs. In the graphical representation, the concept that a graph can be disconnected (as in the example drawn in Fig. 2.5) or connected is obvious. If G = 0, G is totally disconnected, i.e. it has I MI connected components, each consisting of one vertex i E M. Intuitively, a graph is connected if any two vertices i E M and j E M can be joined by a walk over lines of G. More formally, one defines for M = {m i , ... , mk} with m 1 < ... < mk that G E g(M) is connected if there is a permutation 7F E S(M) such that {(mi,m2), (m2, m3), • • • ) (mk-1, mk)} C {(r(i),r(j)) : (i,j) E G} .

(2.77)

An example for such a permutation is drawn in Fig. 2.6. We denote the set of connected graphs on M by G(M). If G E g(m) is disconnected, G induces a partition p(G) of M by grouping together the vertices in the connected components of G. More precisely:

2.4 Polymer Systems

(_. 1245

43

1111-...-111--411—AP

1425

4125

Fig. 2.6. Permuting a connected graph Definition 2.12. P is a partition of M 0 if P =

k E N and for all 1 E {1,...,k}:ONz C M, and Ni denote the set of partitions of M by P(M).

Nk}) with Nk = M. We

If M is finite, 'F'(M) is finite as well, because all N1 0. Lemma 2.13. p: g(m) -+P(M),G fore

E

p(G) is a surjective mapping, there-

E

F(G) =

E

F(G).

(2.78)

PEP(M) G:p(G)=P

GEg(m)

Proof. p is a well-defined map because the set IN1 , , NO, and not the sequence (N1 , , NO, appears in the definition of the partition (otherwise the correspondence wouldn't be unique because for example both {NI , N2 , . , Nk } and {N2 , Nk} are generated by the same G). Surjectivity is obvious. Remark 2.14. Let M be finite and G E g(M). If p(G) = On, 1N1, • • • then

G=

(2.79)

Gk k=1

with Gk E çc (Nk) for all k, and GIN, = Gk = {(i l i)EG:iENk and j E Nk}.

(2.80)

Moreover,

E PEP(m)

1M1

F(P) =

E m=1

J.

F(m,{N i ,.. ., Nm }).

771.

(2.81)

cm N 1 U...

6 N nt =.114

The is there because we now sum over all sequences (N1 , of all sets {N1 , , Nm }. Lemma 2.15. Let A c N be finite and nonempty. Then

,N) instead

44

2. Techniques

11 (1 + = E H ijE A

aii =

GEG(A) (i,j)EG

E 11 aii ) k=1 GEgc(NkHiMEG

Nk otvk

(2.82)

Ni U... U Pl tn =A

Proof. The

first equation is a consequence of the multinomial theorem 11 (1 + ar ) =

E 11 as

rER

SCR rES

with R = {(i, j) EAxA:i< j}. By Lemma 2.13 and Remark 2.14

EHaij = GEG(A) (i,j)EG iAi =

-11 i

E

E

Ni U... U N„,=A

G:p(G)= (m,{Ni,.••,Nm})

E rrt! IAI ,J.

E in m=1

PEP(A)

aii

(i,j)EG ni

m=1

=

H

E

>2

II

k=1 (iMEGINk

m

ri (

E

,

Ni U... U N„,=A

k=1

E GkEGc(Nk

11 )

(2.83)

ai i )

(i,j)EGh

• 2.4.2 The Logarithm of the Polymer Partition Function

ro

Definition 2.16. Let ro be a finite set, and be a symmetric relation on that is antireflesive, i.e. V-y E /70, -y Then we call the elements ey E ro . Let A be a commutative Cpolymers, and -y and 7' compatible if 7 algebra and a : a(7) a map. a(7) is called the activity of 7. * A , A set r c ro is called compatible if V-y, E r : -y The partition function of the polymer system is the finite sum

ro

—

Zro (a) =

E rcro

r compatible

a( -y) ^rEF

=l+ E rcro rot

H a(7).

(2.84)

-rEr

r compatible

The name originates from chemistry. Polymers are long chain molecules with a hard-core property: no two polymers can occupy the same region of space. This disjointness is an example for a compatibility relation, and it is obvious that any polymer must be incompatible to itself. Polymer systems arise in a natural way in expansions of the partition functions of statistical mechanics. For instance, the high-temperature expansion of

2.4 Polymer Systems

45

the Ising model can be brought in the form of a polymer expansion, with the polymers being connected clusters of bonds between nearest-neighbour sites on the lattice. Similarly, the low-temperature expansion of the Ising model can be rewritten that way, but with Bloch walls as the polymers [17, 18, 20]. One can make the compatibility constraint in (2.84) more explicit by introducing [20] geY , 7

')

ry 76 7'

01

={

(2.85)

otherwise

and u(71,...,ryn)= H (i±g(eYWYj)).

(2.86)

1
Since

1 for all i, j: U(71 ' • • . ' 71.1) = {

(2.87)

otherwise,

0

the constraint in (2.84) is enforced by including a factor U('"Yi, • • . ,7„,) in the summand. Thus Z ro (z) is the polynomial n

00

1 Zro (a) ,i+E w E n=1

-yi , • • • ,'),.

_ ,ryi / u(

/ rYn)

II

a('yk)

(2.88)

k=1

Ero

with u(-yi, • . • ,

H g ( wri ) . 0 EG ({1 , ... , n} ) (i,i) EG E

ryn) =

(2.89)

Theorem 2.17. Let A be a formal power series ring over C, and the a(7)

all have no constant part. log Zro (a) =

n

oo ,t--% 1

7 7 n=1

E -yi ,. • • ,^tri

uc(71,...,-rn)

Erb

11 a (rYk)

(2.90)

k=1

with the Ursell function uc(yl, • • • , 7.) =

E H g(en, 7.1) GE Gc ({1,... ,n}) (i,i)EG

(2.91)

for n >1 and U0(71) = 1 if n = 1 . 1. The only change in going from Zro to log Zro is that the 1 is left out and U is replaced by U. Also, the difference between U and and Uc is that in (2.89), the sum runs over all graphs G E Q({1,. that in (2.91), the sum runs over all connected graphs G E Ç({1,.. . , n } ). In this sense, taking the logarithm replaces graphs by connected graphs. The analogous statement for Feynman graphs will be proven in the next section.

Remark 2.18.

. . , nn

46

2.

Techniques

,ryn ) is symmetric under permutations.

2. Uc (yi,

Uc(en , • • •

VT- E Sn .

, eYn) = UceTr(1), • • • , l fir(n))

(2.92)

0 only if {en, .. • i lin} is compatible . By way of contrast, Uc (ryi, ,ryn ) = 0 if there are two nonempty sets A,B C {71, )7 such that A — B and A U B = • • ,'"Yril• 4. A trivial polymer system is given by ro = {1}, a(1) = a. Then Z = 1 + a 3. U(71, • • • ,l'n)

}

and (-1)n n log a .

E c<3

Z =

(2.93)

n=1

Proof. Denote n = {1, ... ,n). By Lemma 2.15, U(rn,

1 E— m! m=1

• • •

E NI. U...

=n

U

H k=1

(2.94)

um-yr)rENk)

We now use

II(

ria(-yr)= r=1

(2.95)

a(N))

k=1 sENk

and get 00

00

1 1 Zro (a)=1+E 771,,E 7 7, n=m

m=1

where

E Ni U...

H ik(Nk)

U Nm =n

cm) = E uc(('yr)rEm) H

(2.96)

k=1

(2.97)

a(Yr).

rEM

Since U, and H rEM a(ryr ) are permutation invariant, 0(M) = 17)(IMI). Thus we may rewrite the sum over ,Nm by first summing over , Nm with INk I = nk fixed. ni = INin I, and then over Then Zro (a) = 1+

771

1 ,t--, °° 1

2 7 77 ! 2_, 77, m=1

n=m

E 71

[J

E

(ni)

k=1

1,-.onnt > 1 n1+•••+nm=n

1. (2.98)

Nm =n

IN1

1=ni

Vt

The last sum gives a multinomial factor,

1

=

n! n i !...n m !'

(2.99)

N1U ...0 Nm =n INI I=n 1 Vi

The n! cancels, so the summand is independent of n. Thus the sum over n can be done, and it simply removes the constraint ni + + nm = n. Thus

2.5 The Effective Action and Connected Graphs

47

00

E m!1 E IIm m 1 4_ E 1 (v, O(n))

Z 0 (a) = 1 +

771=1

ni,...,n m >11c=---1 co

oo

=

m--z--1

= exp

m!

t c n=1

(2.100)

r) 1 ..k.

,

-

n=1

n! (2.101)

n!

Looking at the definitions, we see that 0(n) =0({1,

, n}).

•

2.5 The Effective Action and Connected Graphs 2.5.1 Definition and Semigroup Property

C > 0 be a positive linear operator on Cr , and V be a polynomial in the fields. Assume that either V is bounded below and A > 0,

Definition 2.19. Let

or that A is a formal parameter. The effective action for the interaction -AV

and the propagator C is e Gt" (1P 'C' —AV) = f dp,c(0) e —M4(°4-11') •

(2.102)

Under the first assumption, the integral for the effective action is convergent for all A > 0. If A is a formal parameter, the effective action is a well-defined formal power series in A for any polynomial V. For a nonnegative covariance, C > 0, the effective action is defined by a limiting procedure. For the graphical representation derived below, it does not make any difference whether the covariance is strictly positive or nonnegative because the values of Feynman graphs are polynomials in C. We now discuss the relation of the effective action to the correlation functions and then motivate why it is a convenient object to study. Shifting the integration variable and recalling the definition of the Gaussian measure, we have

e G e ffet1)C

--AV)

f dpc (4) -) = (det(2T-C)) -1 f Drip e- i (4)- Ib ic-1(45-1b)) r -m4( q5)

e

'

' '

dye (o)e- m7 (0) +

'0,0)r .

(2.103)

Comparing this to (1.117), we see that apart from the explicit prefactor, this is the usual formula for the partition function with source terms, only that

48

2. Techniques

the source terms are J = C -1 0. This modification implies that the functions are amputated, i.e., that in the Feynman graph representation, no propagators are assigned to the external lines of the graph. Thus we have 1 Geff ( P, C, -AV) = -- OP, C -1 0)F + log Zr,x(c --1 0). 2

(2.104)

We show in this section that the Feynman graph representation of Geff contains only connected graphs. For this reason, Geff (0, C, -AV) is called the generating functional for the connected, amputated Green functions. Equation (2.104) then implies that only connected (but unamputated) Feynman graphs contribute to log Z F, À(J), unamputated meaning that propagators are associated to external lines. Thus log Zr,À(J) is the generating functional for the connected Green functions. By (1.90), Zr,x(J), and thus also log Zr,x(J) and Geff,) determine all moments (correlation functions) of the measure dpc(0) exp(-AV(0)). Thus if one knows Geff,1 one can obtain all correlation functions. Let us now consider the case where C = C1 + C2. By Theorem 1.5, e aeff

(iPtei +c 2 ,-w)

=

f dPc2 (02) f dtici (01)

.

f dpc2 (02) eGeff (4)2 -"P '

e-W(01-1-02+0)

(2.105)

Thus, integrating out the 01 field associated to the covariance C 1 generates an effective interaction G eff (02, C 1 , -AV) for the remaining 02 -field, which simply replaces V in the exponent. One can rewrite (2.105) as Geff(0,C1 + C2 ) — AV) = Geff OP )C21 Gee') C1) - AV));

(2.106)

this is the semigroup property of the effective actions. If the covariance C is a sum of n terms, (2.107) C = Ci + ... + Cn with Ck > 0 for all

k

E { 1,..., n}, the field 0 is a sum

(2.108) of n independent Gaussian fields. Successive integration over 0 1 , ... , On generates a sequence of n + 1 effective actions G( °), ...G(n), where GO) = -AV is the initial interaction and for k E {1,...,n},

0)(0) = G eff OP , Ck , 0 -1) ) = log f

dpc, (0k )eG(k 1) OP±q5k )

(2.109)

is obtained from 0 -1 ) by integrating over the field Ok. This is the implementation of the idea of "integrating out fluctuations", originally due to Wilson. From the probabilistic point of view, one may regard the G(k) as random variables, and the sequence (0), ... G(n)) is a martingale by (2.109).

2.5 The Effective Action and Connected Graphs

49

Wilson's idea behind the splitting was to use this to approach the singularity in many little steps. This means that although the continuum propagator is singular, one can write it as an infinite sum of regular covariances, so that every finite partial sum is bounded and the singularity arises only in the limit. Posing a cutoff simply means retaining only a finite number of terms in the sum. Removal of the cutoff means taking the limit to an infinite sum. The strategy of Wilson's renormalization group is to study how the singularities build up, and how they can be prevented by adjusting the initial condition V, as the cutoff is removed. Most of modern renormalization is done that way [27, 28, 44, 45, 46, 47]. This is the subject of Chap. 3, where we consider a variant of this method (originally also due to Wilson and Wegner [14, 23]) where one writes the covariance as an integral instead of a sum. The effective action has an expansion in the fields; more precisely, in every fixed order of the original coupling A, Geff is a polynomial in the fields. The coefficient functions Gmr have two interpretations: they are the (connected and amputated) Green functions for the interaction —AV and the propagator C1 obtained from integration over 01 , and they are also the vertex functions for the interaction of the field 02. Thus, in the integral over 02 7 these vertex functions are again joined by propagators to form Feynman graphs. It is the combination of these complementary interpretations that makes the effective action such a useful object. Also, one sees why it is natural that the graphs are amputated: The legs of the vertices generated by Geff (4)2, AV) are paired with propagators C2 in the next integration step, so it would be inconsistent to associate a propagator C1 to them in the definition of the value of the graph (this is also the reason for the convention of not counting external legs as lines of the Feynman graph: in the graphical expansions of effective actions, propagators are assigned only to the internal lines of the Feynman graph, but not to the external legs).

2.5.2 Derivation of the Graphical Representation We shall study Gaff in perturbation theory, that is, by a formal expansion in A. So we write 00

Ge ff(, C, —AV) =

E Ar Ge( li op, c, v) )

:

r=1

with

G(erd OP) C, V) =

ar [ 4,7 ., log f dpc(45)e w(0+0 ]),=0.

In this section, we derive the Feynman graph representation for G? V). We shall again use Wick ordering, i.e. use V(0) = f2c (7(0)). Note that when using (2.105) to integrate over 01 only, the covariance with respect to which we Wick order is in general different from the one in the measure: typically,

50

2. Techniques

one Wick orders with respect to C = C1 + C2 and integrates with respect to C1 only. Since for any f that is n times differentiable at zero

a

(2.112)

(49A )n f (As)

A=0

and

an aA n

0A1

f (Aixi + . . . + Ansn)

=

Sk )f (n) (0) (II k=1

(2.113)

we have

a

= [ aAi an aAn f (Aix]. + . . . + A x )] xi=0,xi=xvi . (2.114)

(— OA r f (As) Thus

ar

Ge( rj (p,

C1 , V)

= [— log f dpci (0) e Av(("41x=o =

(2.115)

ST (V, . , V;

where V occurs r times in E T and

,vp; ci,o) =

sT

[

0A 1

(2.116)

log f dpel (0)

aP

•

We may regard A1, , Ap as formal variables since they are only used to do combinatorics. Note that E T (Vi, • - • , Vp; C1, 1p) is linear in every Vk since it is , Vp , and a p'th derivative is a multilinear the p'th derivative applied to V1 , map. We assume that every Vk is a polynomial, i.e.

14(0)

=

E Mk=7. fr

mk

mk

Mk

dxk v k(mo, k xk,i, mk

,xk,mk),(2,

11 0(Xk,i)

•

i=1

z=---1

(2.117)

This assumption is natural because at the end, the effective action will appear as exactly such a polynomial as well. We now calculate ET in a similar way as we calculated /(Vi , , vp, 0) in Sect. 2.3. We use the linearity of Qc to write

mk Vk(0) =

E f m k =1

dsk,1 .. • dsk,m,v k

(sk,i, • • • ,xk,mk)

rink

[i- mk

amk

aa k , 1

aa k,m k

pc (ei(a(k) 4) )1

a---0

(2.118)

2.5 The Effective Action and Connected Graphs

with

51

mk

= E ak,i6r(s, sk,i).

a(k)(s)

(2.119)

i=1

The important point is now that E T (Vi , ... , vp ; c1,0) is linear in every Vk , so we may take out all sums and the derivatives with respect to the a variables. Denoting again

Bo = {(k,ik) : k E IL.- .)/ibik E {1,...,mk}}

(2.120)

and Xk = (Zkj,... ,Zk imk ), we get

ET (171 , ..., Vp; C1,10

=

(2.121) P

E

11 dsb 11 viink)(xo

f

mi ,...m p

bE .130

n aaab E(a,C,C1,0) [bE Bo

k=1

a=0

with

E(a, c, cl op) = ET(r2c(ei(a (1)4 ), .. . , f2c (ei(a(p) '') Let

(2.122)

P Z I = f Cithe i (0) eXP (

E Ak eifiCRa(k),0-1-01)

(2.123)

k=1

then E(a, C, Ci , 0) = [ wti

o..p

log Z1 A=0 .

(2.124)

The integral for Z' converges for all (A 1 ,.. ., Ai,) E CP because le 1( "5) I = 1 . So, in Z', we may regard the A's again as complex, not just formal variables. This would not have been the case if we had not put an i in the exponent. However, we are only interested in 8A, 8 78),, log Z at A = 0, so we need only a polynomial of degree 1 in every A. Thus

E(a, C, C170) = [

aP aAl • • • +94

log Z] X=0

(2.125)

where P Z = f dILC 1 (0)

H(1+

Ak flc(ei(a(k) 4+16)))

k=1

=

E

11 Ar f dpc. 1 (0) H Sk(ei( a(k)444) ). (2.126)

MC{1,...,0 rEM

kEM

Inserting the definition of Qc and doing the Gaussian integral, we get

Z=

E

(

11

MC{1,...,p} rEM

--1 Ar ei(a(rl'IP)+I(a(r),Ca(r)))e

E

k,lEm

(a (k)

'C

i a (1) ) .

(2.127)

52

2. Techniques

Collecting the diagonal terms and using C = C1 +

fl Cr

E

Z =

mcli,...,plrEM

=

we get

e _wk),cia(1) )

(2.128)

i k k,i<Em

with Cr

C2,

,c2 a(r) ) •

Ar ei(a (r) M+4(

(2.129)

We now employ the standard trick of writing

= 1 + bki with

bid

(2.130)

= e—( a(k) ica (1)) — 1. With this,

E

Z =

oc H

Mc{1,...,p} kEM

(2.131)

(i±bk/).

k,1EXI

Using Lemma 2.15, we get

Z =1+

E

mc{,moo ..... Pl

1mi , H Ck E kEM

E N 1 ,....N m

m=1

N,

#0 1=1

GEgc(N1) (i,j)EG

6... 6 N k=11.1

bii )

(2.132) It may seem at this point that we are just redoing all calculations that led to the formula for the logarithm of a polymer partition function. That is true, but it is exactly here that we cannot proceed as before because the and Ck all depend on the a and lk variables, so they are certainly not going to produce permutation invariant objects. But we can still rewrite Z as a polymer partition function: using FLEA/ Ck = Hlm-1 (HrENI Cr) and denoting

a(Nk) =

11

E

ck

11

bii

(2.133)

rENk GEGe(Nk) (iME°

we have

Z=

1+ EP 772.

(2.134)

m=1

Ni....,Nm#.

N1 U... U

Here, we have joined the sum over M C {1, ... ,p} and the one over

N1 , , Arm with the constraint N1

6...ÙNm = M

into a single sum with

the constraint N1 U ...0 Nm C Note that by definition (2.129) of the Ck, the A-dependence of a(N) is

a(N)

Ar . rEN

(2.135)

2.5 The Effective Action and Connected Graphs

53

We now recognize that Z is the partition function of the following polymer system: F0 is the set of nonempty subsets of {1, ....p},

0

is a polymer .4#, N

N C {1, ....p}

N N'

N n N' = 0.

(2.136)

11 a(Ni).

(2.137)

By Theorem 2.17, oo

log Z =

--n,1

n=1

E

, Nn )

1=1

Ni,...,NnC{1,•••,P}

At this point a big simplification arises: only n = 1 contributes to (2.125), because of the following. For n > 2, U,(Ni, • • • , Nn ) = 0 unless there is a k E {1, . , n} such that N1 and Nk are incompatible, i.e. N1 n Nk 0 0. But by (2.135), a(Ni)a(Nk) contains (FLEN, Ar)(rh ENk Ar ), in which Ar appears quadratically for every r E N1 n Nk. Thus, taking only one derivative at and evaluating at Ar = 0 gives zero. So only the term n = 1 contributes, and

[

log Z 1 A=0

aA l

aP

aA,

aAp Nicht-zi a)}

a(N1 )1

(

2.138)

A=0

since Ue (N1) = 1. Since a(Ni ) contains FL ENI Ar , and no additional factors of A8 , only N1 = {1, ... ,p} contributes to (2.138), and we obtain P Ai)

=

(2.139)

a({ 1 )• • • 4)})

ri

i=1 P = Il (e0 (k) M -FiW k) ,C2 a (k) )) (

11 ( e -cia(i)) 0

E

GEgc({1,...,131)(imEG

k=1

We have now again reached the point where we only have to use Lemma 2.6 and differentiate, to complete the derivation of the representation by Feynman graphs. Before doing that, let us compare (2.139) to the result we had for the analogous object in the equation (2.53) for J(Xi, • • • , Xp; ti))• There we had C2 = O, SO C = C1, and by the first equality in Lemma 2.15, e

=

k1

k

rie .(k),,,,),1(1 , e —(a(k),c1c,(0) k=1

(2.140)

k<1

E k=1

1)

11

GEQ({1,...,29}) (i,j)EG

(e -(a(k),c i a( 0 )

1).

54

2. Techniques

Comparing this to (2.139) (with C2 = 0), we see that the only difference is that in (2.140), the sum over G runs only over connected graphs instead of all graphs. Since

e—(a(k),ca(0) —1 = —(a (k) ,Ca(1) ) + 0(a4 ),

(2.141)

the vertices k and 1 must be joined by a line whenever a line (k, 1) appears in G because we differentiate and evaluate at a = 0. It should now be obvious that only connected Feynman graphs contribute to ET (Vi, . • • , Vi,; C, 0), where we define:

Definition 2.20. Let B C Bo(P,mi, • • • ,mp). A Feynman graph L is connected if

E

L(B)

T (L) = {(k , 1) : 1 < k < l < p, there are i E { 1, .. • , mk} and j E {1, ... , mil such that ((k,i), (1, j)) E L} (2.142) satisfies T(L) E Ç({1,.. . ,p}). We denote the set of all connected L by L s (B).

E

L(B)

This definition corresponds to the intuition one has about connected graphs; it also implies the existence of a spanning tree for any connected Feynman graph. Remark 2.21. Note also that there are no additional combinatorial factors. The effect of taking the logarithm on the graphical sum is only that the disconnected graphs are deleted, nothing else. This is because only the term with n = 1 in the sum (2.137) contributes to E(a, C, C1 , 0) (the terms with n > 2 would give extra signs and combinatorial factors by definition of the higher Ursell functions Us ). If one is used to writing the expansion in terms of topological Feynman graphs and the associated symmetry factors, this means that taking the logarithm does not change the symmetry factors; its only effect is to remove the disconnected Feynman graphs. The procedure to derive the Feynman graph representation is now identical to the one used for /(Vi , ... , Vp ; 0). The case C1 = C, C2 = 0 has just been discussed. We briefly discuss the more important case C = C1 + C2, with C2 0. It is more important because we have assumed that the starting interaction V is Wick ordered. To iterate the semigroup equation for the effective action (this generates the renormalization group), we need to express the result in Wick ordered form as well. We only have to Wick order the function in (2.139) that depends on tp with respect to C2. This gives P

i(E a(k),0) e = e k1

P

E Ik1 .--1.

P

i E (a(k),o) / k=1 = nc2 V

- Eca(k),c2c,(0)

) e k<1

.

(2.143) Thus in that case

2.5 The Effective Action and Connected Graphs Ecia(k),c2ia(0)

E (a(k),IP)

E(a, C,

,

=

e k
(2c2 (e k=i

(e GEgc({1,...,0)

55

lia(1), _ 1).

(2.144)

(0)EG

When one now does the a-derivatives to get the sum over Feynman graphs, the graphs can have either C1 or C2 on their lines. Usually the Ci lines (corresponding to the integrated fields ch) are called the hard lines and the C2 -lines (coming from writing the result in terms of Wick ordered polynomials) are called the soft lines or Wick lines. It is obvious from (2.144) and the above discussion of connectedness that only those combinations of hard and soft lines occur for which the Feynman graph is still connected by the hard lines alone.

2.5.3 Result and Discussion The resulting representation of the effective action as a sum over values of Feynman graphs is as follows. Recall that by (2.110) and (2.115), it suffices to represent ET (Vi , , Vp , C;1,b) as a sum over values of Feynman graphs. We give both the expansion in terms of ordinary and in terms of Wick ordered monomials. Theorem 2.22. Let Vi , ...,Vp be given by (2.117), where C = C1 and C2 = 0, and let iit = Ern, — 2(p — 1). Then m

E Efll dck 0(0c)k(cf, ) mP

(2.145)

m=o k=.1

where C = ((i, • • • 'Cm) and 1 m!

E

Val(GL(C,()).

(2.146)

LE4c(B0\ all distinct

Here, the Feynman graph GL is identified with its set of lines L, and

Val(GL(C,

=f H

dxb H vk(Xk)

bEB0\E

where xb, = Ci for all 1 E {1, and tc depend on v1 , . . . , vp.

k=1

,

H

c(x,,xbi)

(2.147)

I=(b,b9EL

m} and E = { bi,

, b,}. Both V al(GL)

56

2. Techniques We shall refer to this more briefly as

tz(C, (1, • • • , (m) =

E V al(G)

(2.148)

G

where the sum is over all connected Feynman graphs with vertices v1 , , vp has mk external legs), m external legs (the sum over all b1, , b„, is just(vk the sum over all choices of external legs), and a set of internal lines L E Lc. The restriction on the sum over m is there because the Feynman graph has to be connected, so it must at least be a connected tree. A tree with p vertices has p 1 internal lines. Every internal line binds two legs b, h' E B0 of the vertices it connects. The number of available lines is IB0 1 = m1 mp . By these considerations the total number of external legs is at most that of a connected tree: —

= IBOI

2 14 I BOI — 2 (p -1 ) = »Li

— 2(p— 1).

(2.149)

i=1

Theorem 2.23. Let V1,. ,V p be given by (2.117), where C = Ci + C2, and let m= E mi - 2(p —1). Then

sT 071 ,...,vp,c;

E E

(2.150)

f dci • •

d(mr2c 2 (0(6) .0((m))Vm(C)

E Leca

V alG P ( ) (Ci, C2, () • E h:L—q1,21

t711 ,...,Mp M=-0

with

Vn

„,((")

=

1 m!

E all distinct

(Bo\

{

,...,b m

(2.151)

The value of the labelled graph OP) is

V al(01,h) )(Ci, C2,0 =

f H

dxb

bEBO \ E

H

H

vk(xk,i;

k=1

C h(i) (X b 7 X b' )

. • . 7

xk,mk

) (2.152)

1=(b,b9EL

if kh) is connected by lines with h(l) = 1, and V al(k h) )(Ci,C2,() = 0 otherwise. Here E = {bi, ,bm } and xb, = ( Vl E {1, ..., m}, and the

labelled graph G(Lh) is defined as the pair (L, h) where L E L e (Bo) is the set {1, 2} is the map that determines which lines get a factor of lines and h : L C1 and which ones get a factor C2.

2.6 Graphical

57

Representations: Conclusions

The graph must be connected by C1 ("hard") lines because those lines came from factors exp((a (k) , Cia(I) )) — 1 = 0(a2 ), whereas the C2 ("soft") lines came from exp((a (k) , C2a (I) )) = 1 ± 0(a2 ) lines. For iterating the renormalization map that relates effective actions at different scales, the representation given in Theorem 2.23 is more important because the result is written in the same form as the input, that is, (2.150) is an expansion in Wick monomials in the same way as (2.117). The covariance with respect to which Wick ordering is done also matches the semigroup property of the flow of effective actions: Wick ordering is always done with respect to the unintegrated covariance. Once the fields corresponding to Ci are integrated over, one has to expand Geff (02, Ch —AV) in Wick ordered monomials with respect to the covariance C2 of the fields 02 that are not yet integrated. If C is a sum of n terms, the sum of all unintegrated covariances enters into Wick ordering at every step in the sequence. is the generating functional for the connected, amputated Green functions. This means the following. Geff is a formal power series in the coupling A. The coefficient of A' is a polynomial in the fields whose coefficients are given by a sum over connected Feynman graphs, as proven above. The graphs are called amputated because by the graphical rules, there are no propagators associated to the external legs. It was discussed above that this is the natural convention for calculating effective actions.

Remark 2.24.

Ge ff

Remark 2.25. It is not essential for the results derived here that we are dealing with bosons. The generalization to fermions is given in Appendix B.6.

2.6 Graphical Representations: Conclusions The above representations by sums over values of Feynman graphs have many applications. We shall discuss only the most basic ones. We can summarize the calculations of the last section as follows. As a formal power series in A, g(0, C1, C2, V) = log f dttc, (0) CA0c(V(0+10) oo . v, r=1

(—AY fil(r)

f H. m T-r

m=0 k=1

(2.153)

m

dx k ( 2 c2( H 95(xk))Gm,r(xi,

•••,

xrn)

k=1

with

Gm ,r (x i , . .. , x m ) = EVal(T) r

(2.154)

where the sum is over all connected, amputated Feynman graphs with m external lines and r interaction vertices, and with hard and soft lines. One result is already built into the above formula: for a given r, m cannot exceed a certain 77/(r), because of the following. If the initial interaction V

58

2. Techniques

is of degree mv in 0, then the maximal number of legs of vertices that can occur in r'th order is I/30 1 = nwr. By the formula 1E1= IB01— 2 ILI

(2.155)

which holds because the graphs contributing to Gr,„,,, are connected (and which we have already used in (2.149)), we have th(r) <mvr — 2(r — 1) .

(2.156)

So, e.g. for V(0) = fr 0(x) 4 dx, we have ih(r) <4r — 2(r — 1) = 2(r + 1).

(2.157)

Also, the sum over r starts at r = 1, since for A = 0, the logarithm gives zero because f cific (0) = 1. The third observation is that G m,r is invariant under any transformation that leaves fr dx, C(x, z`), and all vk (x i , ... ,x mk ) invariant. This symmetry can also be deduced from the representation of g by a Gaussian integral, but it also holds termwise in the sum (2.154), i.e. separately for the value of every graph. In particular, in our lattice field theory, C is invariant under translations, C(x + a, y + b) = C (x y)

Va E f.

(2.1 5 8)

If the same holds for all vk, i.e. if v k (x i + a, ... , x m , + a) = vk (x i , ...,xmk ) then Val G (C, 61, • • , 6m) (which depends on vi, as well:

Va E 1"'

(2.159)

, vi,) has this property

Val G L(C, 61 + a, • • 6m + a) = Val G L(C, 61, • Gri)

(2.160)

for all a E f. Using Fourier transformation, we can rewrite g as a polynomial in the Ø(p). Noting that the Wick ordering formula for (lc ((pi ) )(p m )) follows directly from the formula for the generating functional nc(e i(aMr ) = Oc(e i(") r*) =

(2.161)

we have

c2 v

g(0,

,

oo

= E (--)or r!

r=0

with

fik(r) f

m=0

(2.162)

)

171

H dP c 2 ( k=1

(Pk k=1

Om,r (Pi

• • • Pm)

2.6 Graphical Representations: Conclusions

59

1TL

Om,r (P1, • - •

,

Pm)

=

H dxk eiPkxk Gra

f

,r

(XI , • • • , 5 m ).

(2.163)

k=1

For any translation invariant function F(xi, .. • , 57-n),

P (p1, • • '

,

II dsk 171

pm)

f

=

k=1 1TL

f II

=

dXk e iP kxk FA X2 — sl. , • - . , X rn

—

X1)

k.------ 1

=

dx l e ix l (P ' + ' -± ' ) P(P2, • - • , pm)

= 6 r.(Pi + • • . +pm , 0) fl(p2 , ... ,pm ) with

(2.164)

m

P(P2,.

.,pm) =

f ri dek e iP"k F (0 , 621 • • • 1 &n)•

(2.165)

k=1

If F is permutation invariant (as in the case for vk and G,,,,,,) then permutation invariant in the sense that -P(P71- ( 2) 3 . • • 7 Pr(m)) = - ii. (P2, - • •

where p i is defined as pi left hand side. Thus

=—

Eirn± 2 pi

, Prrt)

VT. E

Sm

P is also (2.166)

whenever it appears as P(k) on the

dm,r (P1, • • • , Pm) = L d6P1±...+pm,0 0- m,r(P2, • • • , PM) •

(

2.167)

We shall show in the following that if C (s , y) decays fast enough for large dr (x , y), and if an analogous statement holds for the vk, then an ,r converges to a smooth function as L oo. Here di-' (s, y) is the distance on the torus. We shall actually show this for the more general graphs GL encountered in ET (Vi , . .. , V,.; 0). Let T(G) be the Fourier transform of V al(kh) ), then, setting pb, = qi and relabelling the integration variables, we have

(2.168)

f/ (Gih) )(41, • • • , qm) =

f

H dxb 11 bE Bo

r e iPbXb

bEE

ITV... k (v .3(_ _ k), k=1

H

Ch(1)(X b, Xbi).

1=0,69EL

Now insert Ch(i)(Xb, Xbr) = f

then

dpi eiP' (xb -xv ) 0 h(t)(P1) ,

(2.169)

60

2.

Techniques

(G 1( ,h) )( q i ,

••

qm )

f

=

dxb bE B0

r

TTLk

TI („ ( xk, 11 e k=1

f

(dpi h(i) (pi))

(I)

g n ) ix(k.i) EIELuE - (k,=)` 1

k

(2.170)

i=1

where E( 0) = 1 if 1 = (b, (k, i)) for some b -‹ (k, i) or 1 = b E E, E(k,o = —1 if 1 = ((k, i), b') for some b' >. (k, 0 , and 6 i) = 0 otherwise. E I) is related to the incidence matrix of the graph. This is just a formal way of saying that eixk , sPi occurs if 1 joins leg number j of vertex k to a leg of vertex number k', with k' > k, etc. We can now do the x-integrals and get -

(Gih) )(q1 , • • • , qm) =

f

dPi

Oh(i)(PI)

tEL

with Pie = (A k),

y ( 1)

—

(2.171)

k=--1

as above,

the p

(k)

rif,k(pk)

,(b) + z_.,-(0) Pb

tEL

(2.172)

bE E

(note that by definition, only one term in this sum is nonzero). Less formally, but as precisely, one may say that the momentum pi that "flows" in line 1 = (b, b') is the momentum exiting from leg b (hence -FA k) if b = (k, 0) of vertex k, and it is the momentum entering leg b' of vertex lc' (hence —A k) exits from that leg of vertex k'). Thus the signs are due to the convention that every momentum is counted as exiting the vertex.

Theorem 2.26. Let Ch(x,y) be defined as in (2.169), with Fourier transform Ohm E C 8 (Rd I 2tr Zd ,C) and assume that for all vertices k E = Ld (

(k) 0 ic(A k) • • - , p$), ,

with fik E C8 ((Rd PilZ ( )mk -1 ,C). Then, for all E C Bo (with 1E1 = m) and all L E Gc(B0 E) : If m = 0 (E = 0), then

1-7(G) = Lde ) (L, C1, C2, V1, • - vi,) and limL,,, c, gih) g( h) exists. If m > 0

(2.174)

(E 0), then

fT(G.,h) )(qi, • • • , qm.) = L d8q1+...+q„,,0 -;'2: h) (q2, • • • qm)

(2.175)

with E C8 ((Rd /27rZ) m-1 ,C), and 7.4,:h) and its derivatives up to order s converge to functions 14,1,;?• E C 8 ((Rd /27rZ) m-1 ,C) as L oo. 1. If Oh(i) and the Ok are analytic in a strip around the real -(L,h) axis, rynir is also analytic in every D in the same strip.

Remark 2.27.

2.6 Graphical Representations: Conclusions

61

2. For the covariance given by 0(p) = (m0 2 +De (p)) -1 , all functions depend on E and they diverge as E O. They also depend on mo, and it is essential in the following proof that mo 2 > O. Proof. We do an induction in the number of vertices r of the graph. Recall that

1-7 (c4,h) )(qi, • • • qm)

f

HdpiOh (i) (pi)

11

t EL

k=1

70 ;i3k (Pk),

L d4529(1 k)

(2.176)

where, as before, Pk = (Ak) • • • Wdk) • If r = 1, all legs of the vertex vi are external legs of the graph, and f/'- (4) )(qi , . , q 1 ) = f)(qi , , qm ,), so the claim holds by the hypotheses of the theorem. Let r > 2, and assume the statement to be proven for all r' < r. There is a vertex k of GLh) such that the subgraph G', obtained by deleting the vertex k and all lines that end at k, is still connected. Without loss of generality, we may assume that k = 1. Let L1 be the set of lines 1 E L joining to this vertex. The graph is connected, so L 1 0 O. See also Fig. 2.7.

Fig. 2.7. The decomposition of G in terms of the vertex vi and the subgraph Then

17 ( 40)(qi, • . •

,qm) = f II dp,oh(,),b(pof (Gq.,\ LELi

1 )(-P1 )

(2.177)

where Pi, = — (Pi)a if (a, b) E L1. Denote 13' = ' ,). The number of , pm external legs m' of G' must of course be at least one since otherwise G' could not be joined to vertex 1. The inductive hypothesis applies to G', so

=

6p/I +...+0 ".Ytn i1

(A)

6p ( 1 ) +...„(1 )

(P21) • • •

(2.178)

• • •

By our hypotheses,

ii(p(1 1) ,

•,

)=

where i3 E C 8 , and 5,1 converges to a

C8 function

)

as L oo.

(2.179)

62

2. Techniques

Suppose first, that (as drawn Fig. 2.7) some legs of vi and some external legs of G' are external legs of G. Then (p i , . . ,pm ' ,) contains those qm . that enter G', and ) contains those qm that enter vertex 1, and therefore the product of the two 6's is L'1.5

P1 1

(1)

r dx

L..

Pm'

0=

rdx

rdx P1

(1) n •

(2.180)

I P7711 1`;

If no legs of vertex 1 are external legs of the graph G, then (pti , . . , pm ` f) = (1) so (2.180) still holds. Since pi(1) m mi +p = IELi E pj + Q where Q = Em qm the sum running only over those external moments for which qm is associated to an external leg of vertex 1 (Q = 0 if vertex 1 joins only to internal lines). ,

1-7* (Gi(Lh) )(D,

(2.181)

qm )

= fiELi

H dpi

Oh) (pi) L d 6

E

,r-1( 131 ).

IELi

If m = 0, (Sql+ ...+0 is replaced by 1 and Q by 0 in this formula. Remove the summation fr. = Ird E pi I Er. of one line /1 E L1 by the delta function, then (2.182) (G )(qi , , qm ) = L d (Sql +.„-EOW(q2) ...34m) 'i,h) where

W(q2,

.,q) =

r !EL I

dp,-61(,)(pi))0h(,)(—

Q) '51Tre,r—i (P i )

1 0ii

(2.183) with qi = qk. This is the approximation, by a Riemann sum, of an integral over the bounded region [-71- /E, r/E]d(i Lli -1 ) . The integrand is e Zd bounded because Ohm E Cs (Rd/ zd , c\) for all t and because Rd P2-r is compact, and the same holds for ;y" by the inductive hypothesis. By the dominated convergence theorem, the limit L oo is finite and (by repeating this argument after taking up to s derivatives) C8 in qi, qm- • At this point, we have reached the stage that we know the preliminaries for reading at least part of the current literature on field theory without too much trouble, for instance [26, 27, 28, 44, 45, 46, 47, 58, 59]. In the next chapter, we introduce a renormalization group differential equation that allows us to analyze the renormalization problem in a very efficient way.

3. The Renormalization Group

We now turn to the problem of renormalizing our field theory to all orders in perturbation theory. To this end, we shall define a flow of effective actions, by defining an effective action that depends on a scale parameter A. The differential equation obtained by applying 8/8A to the functional integral for the effective action is the renormalization group (RG) equation. The graphical expansion derived in Chap. 2 provides an explicit solution of the RG equation. However, the differential equation itself is very useful for deriving bounds for the solutions without using the details of the graphical representation. We shall learn more about the connection between the two approaches as we develop the theory. The main ideas for the renormalizability proofs using the RG differential equation are due to Polchinski [29]. His proof, which uses a non-Wick—ordered form of the equation, was further simplified in [31] and then extended to gauge theories by Keller and Kopper [32]; a brief but very readable review of that work appeared in [34]. Here, we shall use the Wick—ordered form of the differential equation which was introduced soon after Polchinski's work by Wieczerkowski [30]. Using this form of the equation makes the inductive proofs still simpler. The renormalization group is in truth not a group, but a semigroup, namely the semigroup of effective actions introduced in Sect. 2.5. It is only a semigroup because Gaussian integrals are defined only for nonnegative covariances. Thus integrating over fields with covariance C > 0, which is a convolution with the Gaussian measure dpc, cannot be undone by another convolution with the measure associated to —C because the latter does not exist. If a 04 term is present, one may argue that the integral may still be convergent for C with negative eigenvalues, but then one has to prove that the sign of the 04 coupling does not change when integration is done, which is not obvious. When considering formal power series expansions of the effective action, the argument that the 04 -term controls the growth of the quadratic part does not apply anyway).

64

3. The Renormalization Group

3.1 A Cutoff in Momentum Space To motivate the choice of flow parameter, recall that the cause of the divergences was the singularity of C(x, y) = lirnE ,0 Cre. ,, (x, y) at x = y. This singularity is due to the slow decay of the propagator in momentum space: Oe (P)

=

1

—4' E --+o

mo 2 + D 6 (A

1 p2 ± mo 2

= 6 l (19)'

(3.1)

In particular, 0 V Ll (Rd) and thus C(x, x) is divergent. In principle, we could now study how the singularity builds up as E O. This is, however, messy, because the theory on a lattice has no rotation symmetry. Instead, we shall take a continuum limit at a fixed momentum space cutoff Ao > 1, which is possible on the level of perturbation theory as a formal power series, and then renormalize directly in the continuum. As mentioned, the main motivation for introducing a lattice in the present setting was to give clean definitions of Gaussian integrals and objects like without going through easy but lengthy (and in the end, irrelevant) functional analytical details of defining functional derivatives in the continuum. Let K E C"(RZIF , [0,11) obey

4

(3.2)

K (x) = {9

and K ' (x) < O for x E (1, 4). For convenience, we may choose K such that

ii/Cii00

5_ 1.

Let

ei1L(x-11) K( = f dp 0 m 2 + D e (p) r.

(3.3) p2 — -

).

(3.4)

The function K (p 2 I Ag) cuts off large values of p because, by definition of K, it vanishes whenever II > 2A0 . In the thermodynamic limit we get

Co 1; 0' ') (x , y) =

2 ddp eiP(x–Y) P (27r)d mo 2 + D,(p) K ( A8 )

(3.5)

which converges for all z, y, in particular for x = y by the dominated convergence theorem because the integrand converges pointwise and is bounded by the function --LANA, which is integrable. Thus ino -

II)

e rL Co,A 0 = Ern lim Co ,Ào

6-+0 L--+oo

exists and is a bounded function. For x ,---- y,

(3.6)

3.2 The Semigroup Structure of Renormalization

=

Co,no (x, x)

f

1 (27r )d 00

=

65

K -2 -) ddp p 2 ± ° m 0 2 ( A

d-1

r

2

Ad f r2 ±rno2 K — A2 )dr

(3.7)

o with Ad =

. Thus co

(1

Co,A 0 (x,x)

=

r r2 ±

Ad

, dr +

r d-1

72-)dr) f r 2 ± mo 2 K( All 1

0 2Ao

< Ad(v(rn 0) + f rd-3 dr) 1 log 2A0 < Adv(m q5) + Ad { 1 ( 7 &Id-2 d-2

\''''11)

d=2 d > 3.

(3.8)

This bound gives the correct behaviour for large Ao . The function v(m) is uniformly bounded in m 1 for all d > 3, but not for d = 2 (however, mo > is fixed). The continuum propagator CO 3A0 (x, y) is rotation invariant because a rotation R(x — y) can be undone by a change in the integration variable from p to R -1 p since p2 and the measure ddp are rotation invariant. Before taking the continuum limit E 0 of the effective action at fixed Ao, we derive the RG equation.

3.2 The Semigroup Structure of Renormalization Let 0 < A1
(x , y) — d r) (x , y)

Cno (x, y) =

2 P

e iP(x - Y)

f dP mo 2 ±

/

(p)

AS

Its continuum limit is the inverse Fourier transform of p2 p2 119./10 1(P) ))• - 2 ) K ( s)). 2 (IQ A0 p9

" mo

(3.9)

(3.10)

Note that '

11,A0 (p) > 0

Vp E Rd

(3.11)

because A < Ao , hence i2y < , and because K is a decreasing function.

66

3. The Renormalization Group

For A = A0, 6A,A o (p) = 0 for all p. A plays the role of an infrared cutoff, or, alternatively, that of a varying energy scale. We define the effective action for a given initial interaction —AV0°) as e

g(ccno,-w(A0))

f dpc(r) (0)e-w(A0)(0+o)

(3.12)

A,A0

Here, we shall allow V to depend on Ao, since we have already seen in low orders that otherwise the limit Ao oo, which is our main interest, will not exist. V does not depend on A. We now come to one of the main points of renormalization using the effective action, namely the semigroup property. It is this property that allows us to separate large and small energy scales in a conceptually clear and technically convenient way. Let V(A0) be a polynomial in the fields with Cc° kernels (coefficients) As shown in Chap. 2, g has the formal power series expansion in — Av(A0))

A 1-71 (r)

00

f

=EE r=---1

rri=0 (v

.

M

11 dpk (5,.

(pi

(3.13)

+pm ,0)

k=1

G,/, ,! Ar'A ° (P2, • • • , Pm) Qco,A 041)•• • (kPm )). The cutoff Ao < oo removes the short-distance singularity of C AA) = limo 1',A0 , that is, e'A,A 0 is a C' function of p, uniformly in e (but not in Ao ,C'5 which is kept fixed at the moment). By Theorem 2.26, G ,A° converges uniformly to a Cy" function GiVr° as L —> oo, and as e —> O. The coefficients GA!A,..° have two complementary interpretations:

(i) GAm'A,..° is the order r (in A) connected, and amputated m-point function with infrared cutoff A and ultraviolett cutoff 2A0. This follows from the graphical representation derived in Chap. 2. The ultraviolet cutoff is 2A0 because values 1/31 > 2A0 are forbidden in all loop integrals. Similarly, the propagators vanish when II
732

1 6A,no(P) =

p2

mq52

p2

K( Ar2 ) — IC( A72)

(P)

p2

K' T ) 6

,n 2 Al2

(3.14)

,A0 (P)

and CA,A 1 > 0, CAI , A0 > 0 because A < A'. So, by the covariance splitting formula,

g(4), C

— AV (A°) ) =

Ç(,

C A,A 1

C Al ,A 0 , — AV (A°) ))

(3.15)

3.2 The Semigroup Structure of Renormalization

67

holds (for details, see (2.105)). This identity is the semigroup property. Together with the graphical representation, it implies that the coefficient functions G° also appear as effective interaction vertices for a field theory with Ultraviolet cutoff A' (and an infrared cutoff A < A'). Note that the semigroup property also shows that it is natural that amputated Green functions appear as effective vertices: in the graphical representation of the G; in terms of the G0 the propagator CAA , pairs of legs of the vertices defined by the G 1° are joined to lines associated to propagators CA,Ai Thus it is natural that no propagators CAI , A 0 are associated to these legs of vertices. Note also that the Wick ordering with respect to CA A ', as implied by the representation given in Theorem 2.23, is also natural in this context: Wick ordering is always done with respect to the covariance of the fields that have not yet been integrated over. For A = Ao, CA O ,A 0 = 0, and the effective action indeed consists only of the vertices originally present in V. For A = 0, all fields have been integrated over, thus one has a theory without infrared cutoff, only with the ultraviolet cutoff Ao. ,

.

This has the following significance for the problem of perturbative renormalization, which we may now formulate. We say that the theory is perturbatively renormalizable if we can adjust a finite number of parameters in V(A0) such that for all m and r, the continuum Green functions GM° have finite limits as Ao —> oo. This is physically sensible because the can be used to construct the S-matrix and hence to predict the outcome of experiments. If only a finite number of parameters has to be fixed, the theory remains predictive. Moreover, the details of the interaction at very high energies A o —> oo cannot be observed, so changing V( A0) is allowed. Remark 3.1. The importance of the semigroup property is that it allows us to separate low and high energy scales in a clear way: Suppose we know that "° has a finite limit as Ao oo. Then the same holds for G°5 GA 171 5 T m,Ar° because g(0)CO,A01 - AV (11°) ) = g(01CO,A'

AV (A°) )).

(3.16)

In other words, once we know that the effective vertices G0 for the fields with energy scales below A 1 are finite, the same holds for GM.° simply because they are given by finite-dimensional Feynman integrals over IpI < 2A1 and with bounded integrands. In fact, once the limit Ao —> 00 has been taken in G."°, any reference to Ao is removed, so the limiting effective action G m il l,.A° = GAml-;:l° contains all the information about the theory at energies below A 1 . In more physical jargon, the effective action represents all the data that are accessible to us at low energies. We show finiteness of GAm1 ;(1 0 for fixed A 1 > 0 by use of the RG differential equation.

3. The Renormalization Group

68

3.3 The Renormalization Group Equation 3.3.1 The Functional Form The RG equation is obtained by taking a derivative of g(Ø, CAA) , — ) V (A° )) with respect to A. To do this efficiently, we introduce sources J, (J(x))xEr and write eg(0,cAA0,-w) =

f dpcAA0(x) e

-XV (x+95) (3.17)

= [e —AV (4'67) f d liCAA 0 (X)

J=0

{e -Wqr) e G1,0) e (.1,cAitc, .7)

This identity holds in a formal power series expansion in A because V is a polynomial, so in every order in A, the differential operator is just a polynomial in k 6 J Differentiating with respect to A, we get

a eg (

= e _ w ( h ) e (J,0) e i(J,CA A 0 .0 _1 (j acAA 0 J)]

OA -

2

an

. J=0 (3.18)

Rewriting

1

°CAA

-2-(j'

an

1. 6 acAA 0 6 ) e (.1,0) , ° J) e(j4) = 2( 50' an 60

(3.19)

and taking out the 0-derivatives, we get

a

1 = o y, e g(0,c„A

an -

6 acAA 0 (5 aA 60)eg(0,cAA0,v).

2 (4 '

(3.20)

This is a heat equation for eg:

a gA (o) _ a2LA,A0 gA (o) — an e an e with

=

1, 6

6

(3.21)

(3.22)

the Laplacian associated to CA,A0 in field space. In the formal continuum limit, 2iA,A 0 becomes a functional Laplacian. For E > 0 and L < oo, the number of field variables is finite, and therefore 2iA,A0 is an ordinary Laplacian in finitely many (but very many) variables (0(x)) sEre ,,, . Doing the derivatives, we get 1 6g acAA0 6 g .92iA (3.23) an = an' A ° g 2 ( JO' an so ) •

a0

3.3 The Renormalization Group Equation

69

Remark 3.2. It is obvious that the form of this equation does not depend on the nature of the parameter with respect to which we differentiate. That is, as long as only the covariance, but not the the initial interaction, depends on a parameter t, application of -g-i will lead to an equation of the same form as (3.23). Remark

3.3.

e g(o,cA,A o ,— A V)

= e ,AaA,A0

(3.24)

Proof. The right hand side of (3.24) also satisfies the heat equation (3.21) because 2icA , 40 and its derivative with respect to A commute. Since 2ic40 ,40 = 0, both functions coincide at A = Ao. By the uniqueness of the solution to the heat equation, they must be equal for all A. • We know already that the solutions to this differential equation are given by (3.13), but the differential equation itself will be very useful for proving bounds for the GA' A0. MO"

3.3.2 The Component Form We now derive the component form of the RG equation, that is, the corresponding equation for the Gf,?1,.,A °, simply by inserting the expansion (3.13) into (3.23). We begin by motivating how the flow equation will look like in terms of the functions af,,°. If we denote the associated polynomial graphically by the vertex shown in Fig. 3.1, where every leg carries a field variable 0(x), then the two terms on the right hand side of (3.23) can be visualized as in Fig. 3.2, where the slash on the internal line indicates that it carries propagator an CA,Ao•

Fig. 3.1. The vertex associated to Gnm'rn ° At this point, there are two different ways to proceed: one can expand in the ordinary monomial basis (RA) ... Ckpm ), as done in Polchinski's original work [29] and in [31], or one can expand in Wick ordered monomials, as done in [30]. We follow the second route and use (3.13). This removes the linear

70

3. The Renormalization Group

Fig. 3.2. The right hand side of the non-Wick-ordered RG equation

containing the self-contraction, and thereby simplifies the inductive proofs (as we shall see). The details of the calculation are similar to those of the derivation of the Feynman graph expansion, only a little simpler. We show why the term in (3.23) that is linear in g is removed, and defer the details of rearrangement of flc îk(p i ))(2c (n •- (qi )) occuring in the quadratic term to the appendix, where we also give the formulas in position space. will produce two terms, one It is evident from (3.13) that a derivative appears, and one where the Wick ordered monomial gets differwhere entiated. This second term cancels the first term in (3.23): by Theorem 2.4, A and because aAco,A = —aAcA,A0 implies A -A—o,A = term, i.e. the one

w

((pi) ...(pm))

— aAA0,A 9CO 3 A(s(PI) • • • j) (Pm))

(3.25)

aAAA,A0 0co,A(41) • •

When multiplied with (Sr. (p i + .. . + pm, 0)GAm' ,Ar° (P2 , . • , pm) and integrated over Pi,.. ,pm , this gives exactly the same as the first term on the right hand side of (3.23). Thus, the flow equation (3.23) now reads rit(r)

00

m

E E f H dki (Sr. (ki + .. . + km, 0) nco,A (;( 1e1), • • • , Ar

(k m ))

r=1 m=0 j=i

• aA GL'i^.A0(k 2 ,..., km)

g ( (56(/) ,

,( r) ög, ,A0

If we rewrite the right hand side in the same form as the left hand side, , «k m )) to get the component we can compare coefficients of fleo A ( form of the RG equation. When comparing coefficients, we have to remember that (lc° A (j)(k 1 ), . , «k m )) is invariant under permutations of {1, , m}, so only the symmetric part of the coefficients is nonzero. This means that we have to symmetrize the coefficients. For a function F of (k2, , km ), let

3.3 The Renormalization Group Equation 1

(SmF)(k2

,•••,

km)

=

71 (3.27)

E Fuc,r(2),•••,kr(m))

In ' rEsm m

E ki wherever it appears

where k1 = —

as kir ( i) for some 1 E {2, ... ,m}.

j=2

In Appendix A.4, it is shown that, as a formal power series in A, (.., (5g ,

aAc (r,A) (5g. , )r. =

00

- A

50

° 60

(3.28) 7.1=12

where Q(rr,A,A0)

f

f 11 -rr

La

kw)

dki 5 r. (E ki, 0) Qri'llriA° (IS) f2 Co ,A i=1

i=1

m•=0

m

m

in

fn(r) ..i.‘ _ V-•

(11 (Ici))

i=1

(3.29)

with Is = (k2, . . . , km ) E (Rd) m 1

(3.30)

and mt, (p, le) G,°7.,, (—p, Qm rArA ° (ls) = f dK nir f dwA(p) aACAA0 (po) GA

ID

.

(3.31)

In this expression, the integral f dkrar is written for a weighted sum

f

dkmr (m i , m", r', r", 1) F(m l , m", r', r", 1)

E

,

T., +T."-, r i > l,r fi >1

(3.32)

km , ,,,,, i F(m.' ,7fl",r',r",1)

E (m',m",1)EMrirdini

where Ni 7, r i , In

=

{(ne, m", /) : 1> 0, m` E {1, . . . ,

m" E {1, . . . , fil (r" )} , m` + m" = m + 2 1 + 2}

(3.33)

and where the weight kmi mui is the combinatorial factor kmi mili = m' m" 1!

— 1) (m' — 1) (m" ) . / ) /

(3.34)

Moreover, we use the notations p .

(pi, • • • ,P1)

E (R1 1

Ill

k

(—Ek 3 ,k2,... ,km , --1-1) 8=2 k" = (kne _i,... , km ) .

(3.35)

72

3. The Renormalizat ion Group

and po is given by m'-- 1--1 PO

—

pi —

I

E

=-Epi +

j=1

j=1

m

m'---1-1

j=2

E

kg,.

(3.36)

j=1

Finally, ddpi „ I ) dwA (p = .1* y (27)d U0A

(3.37)

p, , k" , and po depend, of course, on (m, r, m', m", r', r", 1). We do not denote this dependence explicitly because we are soon going to take sup norms where the dependence of the G, on p, , k" , and po drops out. We have omitted the caret and the superscript F on C since we shall only use momentum space in the following, and because we shall take the continuum limit soon. With all this, the component form of the RG equation is (9A G

A0 (k2,

km)

=

sm Q 7/;:rA,A0 ( k2,

km ) ,

(3.38)

with cglryt,A0 given by (3.31) and S given by (3.27). Contrary to what one might believe at a first glance, the details of this equation and of its derivation are quite simple to understand, and it has a simple and convenient structure, which we explain in some detail in the next section. Also, we relate it to the Feynman graph representations derived in Chap. 2, which, as mentioned, provide explicit solutions to this equation.

3.4 The Structure of the RG equation 3.4.1 The Graphical Representation Visualizing Gr niAr A° as in Fig. 3.1, every term in the sum over r', r", m', m", and 1 has the graphical representation shown in Fig. 3.3. In other words, it is the value of a graph with only two vertices, with m' and in" legs, such that the number of external legs is m. The graph has 1 > 0 loops. The sum over rre and m" starts at 1 because otherwise ( I '(p) in (3.23) would have had nothing to act on. Thus Go,, does not appear on the right hand side of the flow equation. The graph is connected for all 1 > 0 because there is always the line carrying 8ACA,A0(p0) connecting the two vertices. If 1 = 0 it is a tree graph. If 1 = 1 it is a one-loop graph, usually called a bubble graph. Retaining only this term in the sum over 1 and integrating the RG equation gives the so-called ladder approximation, done in Sect. 3.9. A brief description of the procedure used in Appendix A.4 to derive the right hand side of the RG equation is as follows. The functional Laplacian generates the tree graph shown in Fig. 3.2, where both vertices correspond to

3.4 The Structure of the RG equation

73

Fig. 3.3. The right hand side of the Wick ordered RG equation coefficients of Wick-ordered monomials. There are 7n1 legs of the first vertex on which the first 0—derivative can act, and m" legs of the second vertex on which the second derivative can act. By the permutational symmetry of the Gmr , we may permute and thus get a factor m'm" (this argument involving the permutational symmetry will be applied also in the following to derive the combinatorial factor by counting the number of graphs). We now have to reorder the product of two Wick ordered monomials; this gives a sum over Wick monomials, indexed by /. In every term, all possible self-contractions of the new vertex associated to the entire graph are subtracted, hence appear as soft lines. Since both vertices already correspond to Wick-ordered monomials, all soft lines must join vertex 1 to vertex 2. If / > 0 is the number of Wick lines, there are r ' ï 1 \) possibilities to choose a subset of / legs of vertex 1, and (mu -1 t ) possibilities to choose a subset of / legs of vertex 2, to form pairs. Given these legs, there are /! ways of pairing them together. This explains the combinatorial factor. The remaining m legs are the external legs of the graph. Counting legs and internal lines gives the constraint in' + 7n" = 771 + 2 1 + 2. In Fig. 3.4, we show how the flow equation looks for small values of m and even interactions.

A_0_ =_0÷0_

d_ dA

-

+

Fig. 3.4. The RG equation for m = 2 and m = 4

+

74

3. The Renormahzation Group

3.4.2 The Relation to the Feynman Graph Expansion The RG equation is a differential equation for the kernels of the effective action, for which we derived a representation in terms of values of connected Feynman graphs in Chap. 2. Thus the sum over values of connected Feynman graphs is a solution to the RG equation. It is a consequence of the Feynman graph representation that fh(r) <

max (fii(r') + iii(r") - 2).

— r'+r"=r

(3.39)

Conversely, using the RG equation, one can derive both (3.39) and give an inductive proof in r that the Feynman graph representation holds by transforming the RG equation to its equivalent integral equation by an integration over A,

f

A0 nA,A0 _ n.A.0,A0 j... 'mr — ' mr

1

A

.

'-i°

nsA 0 ''''C mr 1

(3.40)

A

inserting the definition of Q, and then iterating this integral equation in the way familiar from generating series solutions to integral equations. We shall not go into the details of this here, but only note that when regarded in that way, it is almost obvious that only connected Feynman graphs contribute to the sum because the graph in Fig. 3.3 is connected, and the iteration thus produces only connected graphs. When renormalizing field theory using Feynman graphs, one often starts by proving that graphs without two-legged subgraphs, or for d = 4, graphs without two- and four-legged subgraphs are finite without renormalization. We shall do the same using the RG equation because there is a truncation of the RG equation which produces these partial sums. To get the sum over all graphs without two-legged subgraphs, one omits all terms with m' = 2 or m" = 2 in the sum f dkn,r in (3.31). Obviously, then, no two-legged subgraphs can appear, since the only way they could have been produced in the iteration has been removed. To get the sum over all graphs without nontrivial four-legged subgraphs (i.e. fourlegged subgraphs with more than one vertex), one omits all terms with m' = 4 and r' > 1, and all those with m" = 4 and r" > 1.

3.4.3 The Continuum Limit at Fixed Ao Thus, the functional form gets translated into a system of differential equations, with the following structure. The system is labelled by (m, r) where, as shown in the representation by Feynman graphs, m 1 and m" > 1, and r' < r and r" < r, since r' + r" = r and r' > 1, r" > 1. Thus the m = 0 term never appears on the RHS. Since we are interested in the m > 1 terms to get the correlation functions, we may restrict to in > 1 in the equation. As proven in Theorem 2.26, af.nV,A0

3.5 Differential Inequalities

75

converges as L oo, so Q mr converges as well, and both are C' functions of A and Ao . Moreover, the limit e 0 exists because C(F) A, A0 (p) converges uniformly to the bounded function p2 CA,A0(P) =

as e have

p2

1

(3.41)

( K( A - 2) — K(A2)) p2 +mo2

0, and for fixed Ao , all integrals are over a finite region only. Thus we

Theorem 3.4. The limits L œ and E -? 0 of and the limiting functions satisfy

aAG° (k2,

GL,! 7

A,A .

° and (4,rA 'A ° exist,

(3.42)

, km) =-- Sm Q m A 'rli° (k2 • • • km).

An explicit solution to this system of equations is given by the expansion in Feynman graphs. If the initial interaction VA0 has rotation invariant kernels, the GAm,rA ° are rotation invariant, i.e. VR E 80(d)

(3.43)

km) = ajr,1 ° (Rk2, • • • , Rkm).

Gm A 'rA° (k2,

Remark 3.5. If V A ° is even in 0, i.e. m

VA° (0) = V.‘ m=2

f H dpi Jr. (E Pi, 0 ) f4711

.1°)

i=1

714

(P2 • • •

ikpi.))

Pm) (2 CO3A0

i=1

i=1

(3.44) with Vm = 0 if m is odd, then for all A,and all r> 1, GAm,rA ° is even in if), i.e. Gm A,rA ° = 0 if m is odd. If '410) is Ck in all its arguments, the same holds for G,. IF f) m is even in p, that is, if 0210) is invariant under pi pi, the same holds for Gmr . —

Remark 3.6. The properties stated in Theorem 3.4 could also have been proven by induction using the flow equation only, without considering all details of the Feynman graph expansion.

3.5 Differential Inequalities We know now that for a fixed UV cutoff Ao , the RG equation

aAG Am'rA° (k2, • •

km) = 1SmC2m AtrA0 (k2, • • • , km)

(3.45)

has a smooth and Euclidean invariant solution if the starting interaction is smooth and Euclidean invariant (since we have taken the limits e 0 and L oo, we now drop the superscript 1'). This is the case for V A ° = Q 0 , 40 (Vo) with

76

3. The Renormalization Group

vo = f o z) 4 dx

(3.46)

because the kernel in momentum space is simply (3.47)

f)4(P2,P3,P4) = 1-

We also know that we only have to show that the functions GAmIr,A0 are bounded in compact regions because the Feynman graph expansion then gives us G kir° in terms of CO 3 A1 and the GAO. Definition 3.7. For a bounded function Fm :R( 711-1 )d

C,

/et

AA,, I (Fm ) = sup{IFm (P2, • • • , Pm) I : IN I _ < max{A, 0 Vk E {2, . . . , m }} . (3.48)

the sup norm of Fm on a ball whose radius depends on A and n (so it is not a norm for functions on Rd). Obviously, AA,, (Fm) < AN 07 / (Fm ) whenever A < A' and n < nt. To get a differential inequality, we calculate Aytro ‘) . Thus, in (3.31), we take a supremum over all external momenta A2AA (c2m ki over 0 < lkil < max{A, 0. The integration variables pl, ... ,pi must be in the support of CO,A , SO 'pi I < 2A must hold (this is the reason why we took A2A, n and not AA,,), and we can thus estimate AA, (.) is

IGA mArc, sup (pi, 0
AAr 0 pi, ki, • • • ,km, _1_01 < A.2A, 77 (G m

• • • ,

).

(3.49)

We also bound (3.50) laAcA,A0(p0)1 c IlaAcA,A01100 then the integral factors into 1 independent integrals over pi ,...,pi , each giving

Thus

f dp ro ,A(.01 = 11Co,A111.

(3 .51)

1 A

mAr° ) = — 112A(S Q AA° ) < —1 A2A (Q AA° ) A2A,n (aAG A ' I' m r '17 m n' — 2 2 1 < - f dicmr (m',m",r',r",/)110ACA,A 0 11 0c, (3.52) —2 ill/i/clr,/ ). (iro,Alii) i A2A,n(Gm Allir°, ) A2A,77(Gm

Lemma 3.8. There are constants K1 and K2 such that for all 3 < A < Ao 1

IlaAcA,A01100

1C1

(3.53

)

and IICO,A111

K2

{ Ad-2 for d> 3 log A for d = 2.

(3.54)

For d > 3, K2 depends only on the dimension d. For d = 2, K2 depends also on mo, and K2 —> 00 if m4, 0.

3.5 Differential Inequalities

77

Proof. By (3.10) 2p2

I p2

1 aACA,A0(1 3 ) = -AT IC (A 2 ) p2 +m 2 1

SO

2 ,7111(1 1100

110AcA,A0 1100

< —

(3.55)

Ki

(3.56)

A3

with Ki = 211106. By (3.3), K1 = 2. 2

P 2 )p2 IICO,A ll 1 = f d dPIC( A

1 + mo2 _ 3, 1/3 1 d 3

1 ddp p 2 + mo 2 '

f

(3.57)

ate

is integrable, so we may estimate IPId-1

(3.58)

p2 + mo 2 <- IPI d-3

in the integrand, to get (3.57), with a constant that depends only on the dimension. For d = 2, 2A IICO,Aill <

27r

2A

f p2P+dP mo2 = 7r log Vp2 +m2 1 0 0

<

4A2 1 7T log(1 + 2727 02 ) 7

<

K2 1 log Al

where K2 now depends on

mo.

As

mo

(3.59)

0, K2

r`-'

'log mol.

•

By (3.52), we thus have Lemma 3.9. For all d? 2,

(3.60)

A2A,n(aAGAmAr° )

1 K1

AlliTS) A2A,,i(G m A ;14.,,) . < - — f diçmr (IICO,A111) 1 A2A,n(G m 2 A3

We are going to use these bounds in a Taylor expansion: A. GA/to Mr

=

G A0A0 _ Mr

f

GsAc, as —a as Mr

A

A0

= Gm A°rA° - f dS A

sQsA0 m mr •

(3.61)

78

3. The Renormahzation Group

Inserting (3.60), we get a recursive inequality for the A2A m (aAG AInAr° ). The effective action at scale Ao is simply the initial interaction: GA°A° mr

(3.62)

17(A°) mr

where 14Ar0) appears in the polynomial 00

V (A°) (0) =

E Ar

M(r)

f dpi ...dpm (27) d6(pi + . . +

Pm)

r=1 m=1 174 • ° (p2 , • • - Pm) r2co,A 0 *PI) - • • ((pm)).

(3.63)

For the unrenormalized 04 -theory, V( A° ) (0) = —A f 04 0, so

VIAr 0) (P2

- 9 Pm

)=

(3.64)

r 1 6m4

We thus see that the initial interaction provides the boundary condition for integrating the RG equation, which, in terms of the GAmAr°, is a system of first order differential equations in A. In every fixed order r in perturbation theory, the system consists of finitely many equations because Gmr = 0 if in > Tr/H.

3.6 Two Dimensions We start the investigation of the renormalizability with the two-dimensional case. We shall first show that the GAm'rA ° remain bounded in the limit Ao oo (boundedness), and then that they converge.

3.6.1 Boundedness Applying A2A, 71 to (3.61), and using (3.5), (3.52), and the specific behaviour of liCo,A Iii in two dimensions, we get AAr0 ) < A2A,71(V10°) A2A, 71 (Gm

1 — 2

f drçm r (m', m", r", 1) K 1 1(1 2

slor, m f ds s -3 (log(G A2s,ri )A 2s

m s/ r,i ) (Gy

(3.65)

A

for r > 2 and

coAc, ml

_ v(A0) mi

for all A

(3.66)

for r = 1. If the kernels VIAr°) are all bounded, the simplest ansatz one can make is A2A,n (GAmAr° ) < const (3.67)

3.6 Two Dimensions

for all (m, r) and all A E [A1, Ac]. This ansatz works in d

79

=-- 2 because

Ao

f ds s -3 (log s)l < const < oo

(3.68)

A

(recall that we assumed Ai > 3). The interaction does not even have to be of 4)4 type. Any semibounded polynomial in the fields will do.

1410) satisfies 141,10) = 0 for odd m and that all m and r and all Ao . Then for all r > 1 and all

Theorem 3.10. Assume that

AA 0 ,71 (14A,») = vim- for m> 1, all A o > A1 and all A

E P 1 ,A01,

A2A,71 (Gm AAr o ) < 9trir

(3.69)

(and G 0 = 0 if m is odd). In other words, all Green functions are uniformly bounded in momentum space uniformly in Ao . There can be no infinities as

Ao

oo .

Proof. Induction in r. For r = 1, the statement holds by (3.66). So assume r> 2, and (3.69) to hold for all r'
1

< Vmr + — 2

f

dKinr (m', m", T.', r", 1) If 1 If Ao

f ds (log s) 1 s3

.

(3.70)

A

Since / 1 e -I <

1!, Ao

f

i (log s)' f ds_ (logs3 s)< 1 sup dss2 < 1!

A

Ao

s>1

—

S

(3.71)

A

and the theorem follows if we put g,

1

= v, + 2- f dtçmr (m',m", r', r", 1) Ki. Al gmirigmurn 1!

This is a recursive definition of gmr-

(3.72) •

Already in this simple example, we see some typical features - logarithms may produce factorial growth, and they, as well as Kins mu r , , determine the precise growth behaviour of g, in m and r. The dependence of gmr on r is important for the question of convergence of the series in A (or its resummalions). For d = 2, and for 4)4 -theory, the 1! from the logarithm is, however, not really there, and it is instructive for an understanding of the boundary conditions to discuss this point further.

3. The Renormahzation Group

80

3.6.2 414

Let us consider 4 theory, where

V, (,A7.°) =

(3.73)

m4 ri

and look at the r = 2 flow. 1A) this order, we have for nri, = 6

with I' = ih /P2

(3.74)

= — 9AcAA 0 (P)

aAGg° (P2, • • -

-- 14 -- p5. Integrating from A to Ao gives

42/100)2, • •

(3.75)

= CAA 0 (P) + 176(2A°) •

By (3.73), Vg° ) = 0, so

const IIGV ° 00 < —A2

(3.76)

because the infrared cutoff A restricts 'Pi to 'Pi > A so that p2+17no 2 < Note that (3.73) was crucial for this argument. The contribution to the m = 4 term is

f

q)dp.

COA(P)aAcAA 0 (p

(3.77)

At q = 0, this is, by supp K' C [1, 4),

f

1

COA(p)aAcAA0(p)dp1

f p2 M

2P21 p2 / 7/ 1 2 A3 I KI I. 1

1 p 2 1_ -r

Mo 2 dP

2A

2

1

f p dp (p2 A3 1/000 1

mo2) 2P

(3.78)

A 2A

22 A3 IIK'lloo f P = A3

iIK'

log2.

A

There is no log A term because the lower limit of the p-integral is A. In any case, a single logarithm does not create factorials. Similarly, as for m = 6, the r = 3,m = 8 tree graph requires two propagators, CA A. and is therefore 0(A -4 ). This leads us to the ansatz for the Green functions. Theorem 3.11. For Vt(A n r0) = -8m441 (Y 4 -theory) in two dimensions, there are rymr > 0 such that, for all A> 3 and for all q

A2 A, Ti ( GAmAr0 ) <

'YntrA -(rn-4) rYtnr

77/ > 771 <

4 2.

(3.79)

3.6 Two Dimensions

81

Proof. We have already verified the cases r = 1 and r = 2. To do the inductive step, we may insert (3.79) on the right side of the flow equation, since r'
(log s) 1

)A2s m (Gsmi`iri, ).

(3.80)

X < f ds (log s)17m,rvYmur" s -3— (m1 -4)—(m" —4)

(3.81)

X = f ds

s3

A

For m' > 4 and m" > 4, Ao

A

The triple (m', m", 1) contributes to OAG AmAr° only if m + 2 1 + 2 = m' + m". Inserting this, we have Ao

X

(m+21-1-2) (log s)1 < 7m' r' 7m" r" f ds 55— A Ao

=

/M I T.'

7m" r"

log s \I ds s3-m( s 2f ) •

(3.82)

A

Since A > 3 was assumed, we have for m > 6 Ao

X<

eYm'r'rYmnru

f

ds 53—In <

1 m—4

4 4m

.11 -

I

(3.83)

.

ri

A

For m = 4 and m' >4, m" >4, 21 = m' + m" — 6 > 2, so 1 > 1, and Ao

ds s -

X < 7,n,r,rymn A

i( logs 2 ,/

) <

Ao

r,, f ds s-

2 ( log s )1

A

(3.84)

7m' r' 7m" r" -A-- •

The restriction 1 > 1 if 2 V { rn', m"} means graphically that if a four-legged graph is 1-reducible, i.e. if it gets disconnected when one line is removed, this line must join a two-legged piece to one with four legs (if in = 4 and 1 = 0, = 2 and m" = 4 or vice versa). There remains the case m' = 2 or in it = 2. Without loss of generality, let m' = 2. Then m" = in + 2 1 . If m > 4, Ao

X < f as,

(log s)t eymy7m,,,,,s-(m"-4) s3

(3.85)

82

3. The Renormalization Group

so the power of s that appears is -3 + 4 - (m + 2 1 ) = 1 - m - 2 1 , and

X

<

Ao f

si _mt lOg \I

'Ynt' r' lim"r"

1 7m'r'''Ym"r"

s2

m - 2

A-(111-2)

A

1

m-2

A-(m-4).

(3.86)

There remains m = 2, with the cases

m' = m" = 2, 1 = 0 (ii) mf > 4 and m" > 4, 2/ = m' + m" - 4 > 4, so / > 4 In case (i), we insert the inductive hypothesis into (3.80), to get Ao

X < f dS 3-3 727.'72r"

Ç

1

72r" A

2.

(3.87)

A

In case (ii), we may use (3.81) and replace m' + m" = 2 1 + 4, to get A0

Ao

X < f ds(log s)1.95- (21+4) = A Ao

f ds (log s) 2

ds(log

8)28 _3 1 10g S \1-2 s2

A

S —3 <

const

1 -

(3.88)

.

A

A

Since

A2A, n (GAmAr0 ) < A2A,ri(Ver 0) ) +

1

Ktni mfr.! X,

(3.89)

r'-Er"=r (m , ,m",I)

ye»

the A-behaviour of the statement follows from = 0 for m > 6 and the bounds for X. As before, the inequality for the constants is then satisfied by an appropriate recursive definition of the 7,nr . •

Remark 3.12. The -yrnr are independent of ?I. 3.6.3 Convergence The above theorems do not imply that the GAmAr0 converge as Ao oo. One might fear that, while remaining bounded, they oscillate strongly for Ao large. It is, however, a simple consequence of the representation in terms of Feynman graphs that the limit exists for the 04 theory. The argument is as follows. GA mAr° is a finite sum over values of Feynman graphs, so it suffices to consider

3.6 Two Dimensions

83

every graph separately. The vertex functions are Ao-independent (in fact, they are 1). The only 210-dependence appears in the propagators Cn/0 (p ) which are nonnegative. So the integrand is nonnegative and increasing in Ao , and the same holds for the integral. Since it is bounded, it must converge. This argument will not apply in higher dimensions, and it depends on monotonicity properties that do not hold in more general situations. Also, it does not give a rate of convergence. Convergence of the GAn,A7. 0 as Ao oo can be shown using the RG equation. The main idea is the same in all dimensions, and we sketch it briefly here. A °rA° =174,Ar°) , By (3.45) and the boundary condition G m GA mAr o

so (abbreviating

A-0

-

by

= v100 r

— 1 Sm

2

f

A

ds CAJ,,1,? ,

(3.90)

a) Ao

aA0GAmAro

aA074,Aco — — S r

m (QA morAo

2

f dsaA0Q:71tir°)•

(3.91)

A

Since aA0 CAA 0

0)

2 2 rz. p2 \ P ) — A3 11 i k A2 / mo 2 \

(3.92)

p2 '

we have at A = Ao 2 re., p2 3A0cAA003)1A.A0 =

k Ag O

p2 mo 2

p2

(3.93)

so the Q rA°-term is nonzero. In 3A 0 Q.mAr0 the derivative t9A 0 can act on GIn sA,?r,, as well as on aA„CAA 0 (p). But by definition of cAA0,

G s/0 , or r

for all p.

3A 0 3AcAA 0 (p) = 0

Taking

A2A,71

(3.94)

of (3.91), we get

A2A,71 0A0 G AmAr°

)

< A2A,r, (aA0

f clKmr

v,,(nAr° ) )

(nli

M" 7

A°rA° ) 1 A2A,ri (Q m

r ' r" , 1) R m Aji1;7)zi , i

( 3 . 95 )

with Ao

DA/1.0

f

dS IlasCs,Ao

/ ',SA° \ A / 1100 11CO3s11 11A2s,n k U m'rs ) 112 s,nk U 0 17 mirrit ).

(3.96)

84

3. The Renormalization Group

R takes into account all terms because the sum is invariant under exchange of M I and m". The first term on the right hand side of (3.95) is the contribution from the explicit A0 -dependence of the initial interaction. Since no counterterms are required in two dimensions, V(A0) will typically be independent of Ao , or decay with Ao , so this term will be small. The second term is the influence of the scale Ao on 3A0 G. Under the stated hypothesis, the factor determining its behaviour is aAc,moiA.A0 7ts. This factor also determines the decay of the full G because the last term is covered by the inductive hypothesis. Since aA„GAmA„or„ appears linearly in there, it will reproduce the inserted behaviour for 0A 0 G Arnii,r°1 also for 3A0 G0 .

oo, Theorem 3.13. Assume that V4A,0) converges to a finite limit as A o such that A2A0,17( 14Ar°) ) < vmr holds for all Ao, and that V4Ar0) = 0 for m odd and A2A 0 ,n ( 817-,V4Ar 0) ) < '` .1 for all m, r and Ao .

For d = 2, the 51mr such that

GA mAr °

converge as A o oo. More precisely, there are constants

7Ymr , AS

"-oil&

a m ) k —u aA o r

A2 A

(3.97)

For the 04 theory (V2- 0) = 5m48r1), there are lynx such that A2A,n

.1 log Ao ,-Tmr ) < A3 Ifmr

a nA.A.0

f

0A0

(3.98)

for all m > 1 and r > 1. Proof. For d = 2, we use (3.59) and (3.56) to get Ao 1717AnAvr0n,,

f ds s -3 (log

A

A23 ,17 (G

(3.99)

)A28 , 71 (0A 0 G m s t„ )

and A2 A

,n (Q m A orA 0 ‘

)

<

K1 .K AV (log Ao ) 111.2A° 07 (

r° ) )A2 Ao ,77

)

(3.100)

If r = 1, the statement follows from the hypotheses on Vril,Ar° . Let r> 2, and the statement be proven for all r'
1

{ — vmi r, Vm" r" 2

) <

EE

WMT

Ag

r'±r"=r (m' ,m" ,1)

Ao

(log MI

Ag

f ds A

mr,

K1

s3

(log s)lgmir'

'5/miir /I

As

}•

(3.101)

3.7 Three Dimensions

85

The integral converges, and the coefficient of IF is bounded, so with the usual recursive definition of 9m,, the theorem follows. • Part (2) is left as an exercise to the reader.

3.7 Three Dimensions Wick ordering does not remove all ultraviolet divergences in three dimensions. The value W2(k) = 42 3! (270 -6

f d3pd3 q CO,A 0 (p) Co,A 0 (q) Co,A 0 (p + q — k) (3.102)

of the second order "sunset" graph, drawn in Fig. 3.5, grows like log Ao and thus diverges as Ao oo. We show that there is a logarithmic divergence by proving a similar statement for the integral IAA 0 (k)= f d3p d3 q COA (p) COA (q)

aA cAA, (p + q — k)

(3.103)

which contributes to Q2A2A °(k) (the integral of which gives the full second order two-point function).

Fig. 3.5. The second order "sunset" graph

Remark 3.14. There are constants Q2 > Qi > 0 such that for all A > m, L\I ,. Q2 < LiAo (0) m, Ao Ao

f dA /AA° (0) ? Q 1 log T

(3.105)

Ai

and

Ao Ao

f dA II/AA01100 Q2 log Ai . A'

(3.106)

86

3. The Renormalization Group

Proof. Estimating OACAA„ by its sup norm (as usual), we get /AA() (k) < I ICOAII? IlaAcAA0 II.

K1 = KA (K2A) 2 2,7

-Al

for all k, so the upper bound in (3.104), and (3.106), hold with Q2 = The lower bound in (3.104) is proven in Appendix A.5.

(3.107)

K1 K3. •

Of course, this graph will appear as a subgraph in many others, and so its divergence will cause all Green functions to diverge - and we have not even looked at larger graphs yet. In a graphical analysis, one can postpone this problem by considering only graphs without two-legged insertions. We can do the same in the flow equation by truncating the sums on the right hand side so that mi = 2 and m" = 2 do not occur. It follows from the graphical representation of the flow equation that this removes two-legged subgraphs from the sum over graphs that solves this truncated flow. We shall derive bounds for the so obtained functions dAmAr° and turn back to the full problem later. One can apply other truncations, which provide a lot of insight, to the flow equation. One can, e.g., also remove the four-legged insertions from the right hand side, or remove all terms except for 1 = 0 from the sum over 1 this truncation of the flow produces tree graphs only. Keeping 1 = 1 only does not lead to one-loop graphs, but to sums over all graphs where every line is part of one loop only, the non-overlapping graphs. In these truncated systems, the flow of the coupling constants can often be calculated (see Sect. 3.9). -

3.7.1 Power Counting for the Truncated Equation We now denote the modification where the sums over m' and m" both start at 4 by f dk mr . By (3.54), 'rub < K2A for d = 3, so by (3.60),

(aiiA G AA.0 ) < _1 fA2,r‘ diz_ m

2K1

-Al r A3

AA° AA. A2A, n (G rn t r i )A2A m (G m li r li ).

(3.108)

Motivated by the results in d = 2, we try the ansatz A2A

07 (am AAro )<

const Aa -l'in

(3.109)

with b> 0. To match the A-behaviour we must then have A1-3-1-a-bm' -1-a-bm"

< Aa-bm-1

_

A

(3.110)

where the -1 in the exponent on the right hand side is there because we want to get il a-brn after integration over A, to satisfy the flow inequality. Equating the exponents and using that m' + m" = M + 21 + 2, we see that in drops out of this equation. There remains

/(1 - 2b) + a - 2 - 2b < 0.

(3.111)

3.7 Three Dimensions

87

By appropriate choice of m', m", r' and r", 1 can be made arbitrarily large, so for this inequality to hold, we must have b> . In fact, this inequality would suggest that the larger b, the better. However, we also have to satisfy an induction beginning for the low orders, and this restricts b in a dimension-independent way: (3.111) has to hold for 1 = 0, i.e. for tree graphs, and the scaling behaviour of values of the graphs is 21 -2n where n is the number of internal lines in the graph. i ICAAo II This bound is independent of dimensions since there are no loop integrations involved. In low orders, the tree graphs are

r=2:

m _ 6, Aa-6 1

f•-, A-2 (n = 1)

r=3:

m

-

_ 8, Aa-8 1

A-4 (n = 2)

(3.112)

From this, we get b = 1, a = 4. This simple tree-level-estimate holds for all graphs produced by the truncated flow.

= -8m48,1 , the unrenormalized truncated flow obtained by removing two-legged insertions satisfies

Theorem 3.15.

In 04 theory for d = 3, i.e. v

AA,.° ) A2A,n(G m

_< {

Ao 2: 1A0g4—m gm

0)

m>4 m = 2.

(3.113)

Proof. r = 1 is trivial. Again, we look at

Ao

X = f ds A28,71

(Gsmilor, ‘ A )1123,71(Gsmt„ )

(3.114)

A

with r' + r" = r, but with min{mi, m"} > 4 because of the truncation. By the inductive hypothesis

Ao

Ao X

f

< gm'e gnoe, ds

5 -3 +1 +4 — rn'± 4— m"

= gmiri

A

f ds A

(3.115) Let m > 6, then 3 - m - / <3 - m < -3, so integration proves the claim. For m = 4, 21 = m' + m" - m - 2 > 4 + 4 - 4 - 2 > 2, so / > 1, and thus Ao 3 - m - 1 < 2 - m < -2, and f s -2 ds < A -1 <m 1 . A

For m = 2, 2 1 = m' + m" -4 > 4, so 1 > 2, and

Ao

Ao

Ao Ao f ds s 1-1 < f ds s -1 = log Ti < log 2,. A

(3.116)

A

The constants gmr are again fixed recursively.

•

88

3. The Renormalization Group

Remark 3.16. Closer inspection of this proof shows that (i) for r > 2, what we have proven for the four-point function is A2A, 77 (G141rA0 ) < g4r

11 -1

(3.117)

so the 0(A° ) term is only there in first order. (ii) for in = 2, any contribution from 1> 3 is convergent, so the only graphs that can be divergent are of type shown in Fig. 3.5, with the vertices denoting four-point functions. But as we just saw in (i), whenever r' > 1, there is an extra decay factor A -1 , so the only graph whose value can oo is G2A2A0 . We shall show in the following diverge in the limit Ao that, indeed, only G2A2A ° needs to be renormalized in d = 3, and we state the better bounds below for the untruncated flow. 3.7.2 Renormalization: A Change of Boundary Conditions

Although we have just argued that in d = 3, the value of only one graph is divergent, we keep the following discussion a little bit more general, to prepare for the four-dimensional case. The full argument leading to the logarithmic behaviour of G2ArA0 is: integrate the RG equation 0Aqr 0 = 1Q2ArA ° from A to Ao, and take A2A,n. This gives

' 2r

Ao

Ao (2./1A0 f,,\ = _1 f ds Q8A 0 (”) "2r

IF

VP)

2r

2

A

A.

AoAo — (G2r

A2A (GA2rA°)<

ds QV ° ).

2

A

Up to now, we had the boundary condition G2Ari° = V2(7j,10) = 0 (i.e. only a 04 and the integral over Q grows like log Ao. We will now try to changeterm), the boundary conditions, i.e. choose G2Ar° A0 (p) such that it cancels the log Ao term in Q. The simplest condition one can put is V2(rA 0 (p) = (5tif.A 0

(3.120)

where Ji4u1° ) is chosen as a function of Ao to cancel the terms in G2A71.1° (0) that grow with A o , e.g. (SW' ) — log Ao . That is, the action now contains a mass counterterm 00 r 8n (Ao) f d3p 0(13) oFp) . (3.121)

A E r=1

r

(270 3

"

But how, and why, should that work? After all, we are only subtracting a constant from a function, and it is not guaranteed that there might not be

89

3.7 Three Dimensions

any divergences in derivatives of that function as well. However, if we take a derivative

a akay

a „ 22A- ki = f dpi dp2C0A (pi ) COA(p2) — oituAA 0 (-.731 — P2 + k) A

aka

(3.122) we see because of

a

a

AAA0( k + 19) = aka aka ac

2 (

(k + p) 2

(k + p) 2 A2 ) (k + p) 2 + mo2 )

A3' ( —K

that wc,:k- oAuAA01100 < and hence A2A,71(

4

a

AA

const A4

(3.123)

(3.124)

< const A2 A -4

ok a--.- Q22 °) —

(3.125)

Integrating 8 D G GI2T° = é Qg° , we see that the derivative of G 121121° remains finite. Inserting the Taylor expansion '

1

(42A° (tk)

Qgo (k) = Qgo (0) + f dt o

(3.126)

into the flow equation and integrating we get Ao

A° (k) — G

Gg° (k) — Ggo (0) = G

A ° (0) -

f ds f dt k • V Q° (tk). A

0

(3.127) =Nitvio) 4.po (k) G0 (0) _ Since qTA°(k) A2A, n (G Ao

A0 (k) - G 1

A° (0 ))

f ds A.2A, n (f dt k VIC4 121° (tk))

A

0

A0

f ds ./4.2A, n (VQ 1210 Ao < r, const f ds s 2 < const 17 Ai

(3.128)

A

A 2A2A,-,\ Thus, once limit°, G0 ku) is made finite by a subtraction, the function G 2A2A0 (p) is bounded for all p uniformly in Ao . This feature is typical for three dimensions. In d = 4, a mass counterterm is insufficient and one also needs

90

3. The Renormalization Group

a wave function renormalization to get all derivatives up to order 2 finite. Note also that an 77 appears in the bound. Because of the Taylor expansion, factors of n will appear in the renormalized bounds. Graphically, one can denote the mass counterterms 84110) by --0—. They thus appear as new, two-legged vertices in the perturbation expansion. Now, a typical problem in renormalization arises: we have subtracted a divergence by adding a counterterm. This counterterm appears as a new interaction vertex everywhere in graphs, and, since it grows with Ao, it might actually spoil the convergence of more complicated graphs. Moreover, how does one put such counterterms consistently to all orders? Enter Wilson's RG, which arranges things in a way that counterterms appear only in combination with the divergence they are to cure. We shall see now that this is particularly simple in the flow equation because the counterterms do not even appear there, so that the RG equation relates only quantities at the same scale A. Instead, the counterterms appear only as an initial condition for the flow. Moreover, they can be determined by a renormalization condition imposed on particular Green functions at the final value A 1 of the flow. Thus the only place where they are needed is the statement that they can be chosen such as to make G 1° finite. Suppose we demand that G2Ar1A0

(0) =

8prii

for all

r>2

(3.129)

where ( 8$R) ) r >1 is a given sequence which is independent of Ao. (the superscript R only indicates that this is a renormalization condition; R is not a variable). Then (3.128) implies finiteness of G2Ar' A0 (k) for all k. This boundary condition determines the mass counterterms at Ao uniquely in the following sense. Lemma 3.17. There exists a unique sequence (8/4-110) )r>2 such that the effective action GAmlrA0 at scale A 1 obtained from the initial interaction

{ —1 for r = 1 and m = 4 ( vo) 81. `1°) for m = 2 and r > 2 mr = for all other m and r 0

(3.130)

satisfies (3.129). Remark 3.18. In fact, a renormalization condition like (3.129) is necessary to fix the counterterms since the requirement that a term growing in Ao is cancelled leaves the freedom of adding finite constants.

A lrA° Proof. Suppose we have a theory with vertices given by (3.130). Then Gm is a sum over values of Feynman graphs with four legged and two-legged vertices, such that the orders of the vertices add up to r. This fixes in particular G2Ari A0 (0). For instance

3.7 AiAo (0 )

Three Dimensions

= 0,

G2A.A o (0)

91 (3.131)

f

Ao (pi ) CA, Ao (p2) CA, Ao ( —pl — p2)( 3 pid3 P2 + (5P(2A°) •

It is obvious that this equation can be solved for 8/.4110) if G A° (0) is given. We now proceed by induction in r. In the induction step, we may assume that the 6/4A,°) have already been fixed for all r' < r, in terms of the G 2A,r1,A0 (0) with r' < r. In order r, we have G2ArI A0 (0) = 8/ ( `10) + a sum over values of graphs with two legs and at least two vertices. In this sum, only 8/4j,i0) with • r'
Remark 3.19. There is no first order counterterm because we expand in Wick ordered monomials. 3.7.3 Renormalized 4):

Before stating the theorem about renormalized 04 in d = 3, we recall the standard notation for multiindices. For m > 1, a multiindex a is an m • dtuple a = (ai,... ,ûm ) E ( 4) m ,

ai =

• • • lai,d)

We denote

(3.132)

m d

I ûIIûlI+.+IûmI = and

ûj

z =1

m

a! = !

am ! =

(3.133)

j =1

d

HIT au! .

(3.134)

i.1 j.1 We write a' + a" = a if a'ii + aiiii = au for all 1, j. Note that a' + a" = a =

la'l +

= lal.

(3.135)

Leibniz's rule, Da (fi • • • fp)

a! =

a(1)!

.û(P)!

Da(l)fi ...D'(P)fp,

(3.136)

is proven by induction. AAr° are C0° in the external momenta We shall show inductively that all Gm if the same holds for the initial interaction vISIAr o . For this we note that CAA° E C"(R d ,R) and

3. The Renormalization Group

92

DctaACAAo(P)

-

(3.137)

Da (; Kt( A P22 ) p2 +P2m0 2 2

E

mo 2

P2

a!

A3 ce±crii =a aqa"!

Da' K i (

)Da (1

A2

p2 mo2 ).

By the support properties of K, D ' K'() = 0 for all a' unless IPI E [A, 2/1], Moreover 1 if a" = 0 (3.138) K (a" ) 1Da" ( 1 p27Y14)2 2 )1 < M if la"I 1 A 2+1a" {

SO

IDaaAcAA0(p)1‹

A K31+(a121 „

(3.139)

< Ipl < 2 A)

(A

with a combinatorial constant K1 (which depends on mo). We also use the following lemma to reduce the number of cases to go through as much as possible. Lemma 3.20. Let m,m 1 ,m" E N, / E No such that m" + m" = m + 21+2. Define

D(m) = min{4 — m, 0 } E(m) = max{ 4 — m, 0 } .

and

(3.140)

Let 6 E R. Then

D(m 1 )+D(m")+16-3 = D(m)-1—(2-6)1+E(m)—E(m 1 )—E(m"). (3.141) In particular, if m > 4,

D(mi ) + D(m") +16 D(m) + E(m) = 4

Proof.

—

—

3 < D(m)

—

1

—

(2

—

m, so

D(mi) + E(mi ) + D(m") + E(m") — 16 — 3 = 4 m' + 4 = 5 16 — (m' + m") = 5 16 — (m + 21+ 2) = 1 + 4 m (2 8)1 = D(m) — 1 — (2 6)1 + E(m) —

—

—

—

—

—

(3.142)

6)1.

—

m"

—

16 — 3

(3.143)

—

—

• Theorem 3.21. Let d = 3, 8pR,2 E R and 04 with a second-order mass counterterm be given by

V2°) = - 67-1 6m4 where 8,14i1° ) is fixed by the condition

G .1.21A0 (0) _

6r2 8

e

3.7 Three Dimensions Then all

alrinAr° with m odd G 71,1° (pi , p2,

93

are identically zero,

(3.146)

for all pi, A, m,

= —8m 4

the GAmAr0 for m even, r > 2 are Cc° functions of the external momenta (Pi, • • . ,pm—i) and for any multiindes a there are constants grnri al, depending all 77 > A1 , all m > 2, and all r> 1, on 614 , such that for all A E [AI,

A2A, n (D'

ilAro ) Gm

 1 m = 2, r> 2, a = m = 2,r > 2, lal > 1 or m = 4, r > 2 m > 6.

I log A

A-1 log A 01(G A rnAr° J <

AD(

) — Ia l

(3.148)

2. In Fig. 3.6, the scaling bounds for GAmAr° as a function of A are sketched (a) for the truncated unrenormalized Green functions, (b) for the renormalized Green functions. The change of boundary conditions has only shifted the two-point function, by fixing a prescribed value at A i instead of at Ao. The scaling behaviour itself does not change because renormalization does not change the decay properties of the propagator. 3. We shall see in the proof that the boundary condition Wilri °) = 0 for m > 6 is important for these bounds, to ensure that A2A, n (GAmAr0 ) AD(m) — I'l for m > 6. It can be relaxed slightly to a condition 14,Ar° ) A0D(m) . This is useful for the so-called "improvement", i.e. speeding up the convergence of the GAmAr° to their limits Gr as Ao cx) by adjusting the boundary condition [30]. AAr. = Da Qm AAro By Leibniz's rule Proof. By the flow equation, DaaAGm

E

f dKmr f dwA (12)

a l +all +cell

Dcv"

a! aga"!am!

D ai GA ntlior°,

(p, k i )

(3.1 49)

k") Dam aAcAA0(p0)

so, by (3.139), A2 A,n( Da Q A mAr °)

f d,,mrilcoAg

E

a l +ce, ±am =a

A2A,7 1 (.13''

AA,roi , Gm

)

a! Ki (am ) aqa"!a"' ! A 3+1c," I

A2A, n (Dcv" GA mjilt°rif ).

(3.150)

This bound holds in any dimension. Specializing to d = 3, we have liCoA Iii < K2 A, so

94

3. The Renormalization Group

A2A,T) (Da

Qm AAro ) \

a!

E

<

a l± a ll+ a lll =c,

A2A,17(Dclj G

aga"!a" !

f

clKmr ICI (a'").KA -3— lani1 +1

AA,roi \ rn

) A2A,,? (Dc'n G

AAno ). rn

(3.151)

We do the usual induction in r, using (3.148) and GAmAr° E Cm as inductive hypothesis. The case r = 1 is trivial by (3.146). We do r = 2 first. GAmA2° 0 unless m < 6; the boundary conditions are (Ao) (Ao) V6 = V42 = 2 G i A° (0) =

(3.152)

0

64.

The lower order Green functions on the right side of (3.151) must have r' = r" = 1, so m' = m" = 4, and by (3.146), a' = a" = 0, and A2 A (D ' QAml0 )

<

f dicm K (a)10 -3-1 a 1+1 i

2

.

(3.153)

The summands m', m", / in f dKin2 satisfy m' + m" = m + 2 1 + 2, so / = ..(6 — m). For m = 4 and m = 6 and for m = 2, lal > 1, we integrate the flow equation from A to Ao and use (3.152), to get A0

Ki(a)K 23-7 constfds 8 -1 ' 1-7

A2A,, i (D'GAnti° )

A

Ki (a)K23-7 const A — i al — 't1 +1

(3.154)

which satisfies (3.148). For r = 2, a = 0, we integrate from A1 to Ao, to get A

IG1212A0 (A i

16141 + f ds Ic4P(p) I A1 A

f 7ds

< 164 I + IC1 1q const

(3.155)

A,

A

< 16141+ KIM log — . AI Since A > 3, log A > 1, so we can bound this sum by g220 log A, with the same constant g220. Let r > 3, and assume (3.148) for all r' < r. We integrate the flow equation from A to Ao and use IdAr°) = 0 for all m (since r > 3), and the inductive hypothesis (3.148), to get A2A m (13'

G AAro) m < f clitmr

E

cx 1 -1-ce"-Fai" =a

a! aqa"!am!

K1 (a") .Kg m , eke I gm" r's I a" I XAAo

(3.156)

3.7 Three Dimensions

95

(a)

454'1

Fig. 3.6. Scaling bounds: (a) unrenormalized; (b) renormalized. The scaling behaviour is unchanged because it depends on the propagator. The boundary condition is changed under renormalization.

with XA Ao

= XAA0(m%m"7 1/ 7) )r",ial)ia"kia t"I) A0 . f ds 8 -3+1-1&" I Ornirs la' I (9 )

(3.157)

A

where log s s—a

onira(s)

s'a (log s) 6-0 3-1—a

=

84—m—a {

m = r = 2, a = 0 M = r = 2, laj > 1 m = 2, r > 2 m = 4, r > 2 M > 6.

( 3. 15 8)

We can write this more compactly as

Ornra(s)

= S p(m)—a—rnir (log s )5m25.0

(3.159)

where rmr = 6m2(1 — 67.2) — 6m4(1 —

41 ) .

(3.160)

It is clear from (3.156) that if we can show that XAA0 has the A-behaviour of (3.148), then the Theorem is proven because 9mr 1,1 can be determined

96

3. The Renormalization Group

recursively. We verify this now by going through all cases of (m', m", r', r", 1) that can occur in f ditm.,.. Recall in particular that m' + m" = m + 2 1 + 2, and that r' + r" = r > 3, so one of r' and r" has to be larger than 1.

1. Let 2 fl {rre , m"}. By (3.159), the integrand for

XAA 0

is

7 = -3 + / - Ian + D(m 1 ) WI +D(m") — 3+1+ D(me) + D(rn") — lal — ./"' = and where r = rmy + F,,,,, ,,, > O. 1.1 m > 4: By (3.142) (with 6 = 1)), -

7 < D(m) — 1 — 1 — lal — r SO

-

.9 7 with lal -

r (3.161)

= 3 — m — lal — 1 — F

(3.162)

Ao X AA 0

< f ds 33- m- lal -l-r

(3.163)

A

For m > 6, we bound this using 1 > 0 and F > 0 by

A0

X A lt o < f d8 3 3- "H a l < A

1 A4-m- l a 1 4 l — a + l M

(3.164)

so (3.148) holds. Let m = 4. Then 2 1 = mid-mu-m-2 > 4+4-4-2 = 2, so 1 > 1, and

A0

XAA 0 < f d88 3-4-1- lal

(3.165)

- 1 + lal

A

also implies (3.148). 1.2 m = 2. By (3.141) (with E(m) = 2, D(m) = 0, and 6 = 1), the integrand is .9 7 with

7=1

-

1

-

lal - E(me) - E(m")

-

F.

(3.166)

In the present case, we have assumed m' > 4 and m" > 4, so E(mt) = E(m") = 0, and (3.167) 7 <1- (I' + 1 ).

lai

Counting the numbers of external legs,

2/ --= m' + m" - 2 - 2 = in` + m" - 4

(3.168)

so 1 --= 2 if in' = in" = 4, and 1 > 3 otherwise. Now r' + r" = r > 3, so r' = 2 or r" = 2, which implies that for in' = in" = 4, F < 1 must hold.

3.7 Three Dimensions

Thus, in all cases,

F + 1 > 3, and therefore Ao

XAA0

97

<

f

Ao d881-1a1—(r+0 < dss —(2+14 A

A

1

<

1+ lal

(3.169)

A-1-1 ° 1

which proves (3.148) for all la > 0, even without the logarithm. 2. m' = 2 or m" = 2. Since everything is symmetric under exchange (m', r', a') 44 (m", r", a"), we may assume m' = 2, m" > 2. The sum over 1 in f dttm, runs from 1 = 0 to 1 = min{ml ,m"} — 1, so m' = 2 implies 1 E {0, 1}. Thus there are the cases 2.1 m' = 2,m" = m, 1 = 0 (tree graph) 2.2 mi = 2,m" = m + 2, / = 1 (one loop) 2.1 Using (3.159) and Fm,,, > 0 and rm„,." > 0, we get Ao

XAA 0 < f ds .9

m" ) (log s)

(3.170)

A

where 0 = 2 if m" = 2, and 0 = 1 if m" > 4. If m" > 4, Ao XAA 0 <

f ds s l— m" -1 ° 1 1ogS A

<

Ao

log t

sup

t>i

f

t

ds 82— m" Hal

A

1

< M"

(3.171)

3 — m" — icti < AD(m) - 1 0 1 -1 /1

.

+ lai — 3

If m" = 2, Ao XAA o

< f ds 8 -3-1 ° 1 (log s) 2 A

i

(log t) 2

t

A0

f ds 8 -2-1 ° 1 < 2A-1-1 ° 1 . (3.172) A

Both these bounds are better than the one stated in (3.148). The next case is one from which one gets the logarithm.

98

3. The Renormalization Group 2.2

Since m" --= m + 2, m" > 4. The exponent of -3 +

-- -2+4- (m+ 2 ) - la' - r = - m - lai - r

1 +D(m") - ial - r

so, since

s is

r > 0,

A0 XAA0 = f ,ds 8 —m— l a l log s.

(3.173)

A

For m > 4, A0

XAA 0

< i

t f ds s l-m- l'i

(3.174)

t f

A

11 A2— m— I'l < m + lal - 2 m + lal - 2

—lial -2 )ilp(m •

For m = 2, then m" = 4. We distinguish between further cases. If r" > 2, then F > r42 = 1, so A0 xAA0

<

dss -3- kal logs

(3.175)

A

< _

A0 log t sup — d5 8 2 H i t f A

If r" = 1, the function GAmt„ is the bare vertex function, which does not depend on momentum at all. Moreover, no other factor of the integra,nd in this contribution to %trA° depends on the external momentum to either - this is most easily seen in the graphical representation shown in Fig. 3.7. Thus any momentum derivative D' of this function vanishes, and hence fulfills (3.148). So there remains only the case r' = 1,a -= O. There we have

00

Ao

XAAo

<

f

ds S -2 log s < 7d5 ds-2.9log s A

A

=

00 log A f ds log A — < 2 -A + f 32 A

(3.176)

A

which is the bound stated in (3.148).

•

3.8 Four Dimensions

99

Fig. 3.7. A generalized tadpole graph

Remark 3.23. It is instructive to look how the logarithm from the second order graph proliferates to all orders: In third order, only r' = 2 makes such a logarithm in the graph drawn in Fig. 3.8(a). The value of this graph now reappears in the fourth order graph 3.8(b) etc. So the logarithms come from the so-called generalized tadpole graphs.

(a)

(b)

Fig. 3.8. Some of the graphs producing the logarithms in d (13) Fourth order

= 3. (a)

Third order

3.8 Four Dimensions In four dimensions, there is an infinite number of divergent values of graphs, not just one as in three dimensions, because the theory is just renormalizable. However, the scaling ansatzes to bound the Gmr will actually be a little simpler as in d = 3, and so the proofs will be shorter. 3.8.1 Counterterms in Second Order

For d= 4,

iroA Ili < K2 A2

(3.177)

and the right hand side gives the correct behaviour of liCoA Ili at large A. In second order (r = 2) of unrenormalized perturbation theory, we thus have the following results, corresponding to the graphs in Fig. 3.9. For m = 6,

100

> /< >

3. The Renormalization Group

<

-e--

Fig. 3.9. Second order graphs m

frIAA0 \< const

nito \ const

.A2A,17 ll-r62

A2A,fill'e62 / —‹ A3

For m = 4, AA o 0 ''P 42

so

-

—

const

f d4

p CoA(P)

a 4 cAA0 (p + q)

T",. A2 Ki

//I AA° \

A2A,n kW42 / < C011St

n.211

I — A2 •

const

- = A A3

(3.178)

(3.179) (3.180)

and Ao

ds A0 f const — = const log — . A 8

A2Am (G0) <

(3.181)

A

Similarly for m = 2, Ao A2A

,n (G2A2A0 )

<

f ds 8-3

( .9 2 ) 2 = 1 2 — (A

2°

— A 2 ).

(3.182)

A

Thus, in four dimensions, the four point function diverges like log Ao and the two-point function even like A4 as Ao oo. With the same bounds applied to derivatives of these functions, we would get for la' = 1 A2A,,i (D'G 2A°) < const / log Ao for lai = 2 i for la' > 3. 7

Ao

(3.183)

By analogy to the three-dimensional case, we may expect to need four types of counterterms — one for the four—point function, and three for the two— point function, to make GA22A0 (0), as well as the first two derivatives of GA22A°, converge as Ao —> oo. However, if the initial interaction 14,Ar 0 ) is even in p, i.e. invariant under p —p, one needs only three counterterms because this symmetry poses automatically a renormalization condition on G2A2A 0 : since CAA ° ( -1)) = CAA ° (P)) V4Ar°) being differentiable and even in p implies G2Arrio (_p) = GA 2rrio (p)

and therefore

( a

kV , ii. G2ArA0 ) (0) = 0 .

(3.184) (3.185)

101

3.8 Four Dimensions

The usual Taylor expansion 1

d '9

A A

G 27! ° (p)

A A°

t9

= (— 27! api,G

)(0) -1-

Ep, f dt api,ap,, G AA 2r ° (tp) '92

(3.186)

o

then implies that once the logarithmic divergence in the second derivative of G2ArA° has been removed by renormalization, the first derivative is automatically finite, by (3.185) with A = A l . We shall start with an initial interaction that is rotation-invariant, i.e. for all R E SO(d), V4Ar 0) (pi 1

. ..

1 pnl, 1 ) = 174Ar 0 ) (Rpl 1 . . . 1 Rpm 1 ) .

( 3.187)

This invariance, together with the invariance CAA° (RP) = CAA0 (P) for all R, implies that there is a function g : Ri; —it such that Gi24rAO (p)

9(p2)

(3.188)

Differentiating this equation, we get

aom G AA. (p) yir

=

2pi, g I092 )

2r

32

AA

appapi,

G2r ° ( P)

=

2, 91 (1)2 ) + 4P11Pv g it (P2 )

(3.189)

which restricts the matrix structure of the second derivative of G2ArA° to

32 Pis

,AA t ,,,,\ Pu'2r O li-') — 611v gl(P2 ) +POI, 92(1'2

).

(3.190)

In particular, we see that to get a finite second derivative at p = 0, it suffices to put a counterterm proportional to öpp. In summary, the second order analysis and symmetry considerations suggest that we need three counterterms, to make Ge° (0, 0, 0), G A0 (0) and gi (0) finite. In other words, we need terms of the form a constant times

04 , 02 , and p2 02

(3.191)

to renormalize. These are exactly the terms that were in the original action.

102

3. The Renormalization Group

3.8.2 Power Counting (Skeleton Flow) Inserting (3.177) into (3.150), we get A2A, n(D

aQm AAro\) <

f clit mr

al ar vi cerr(

tini

./‹.' Ki (a" t ) A -3 +21- lam I

."4.2A,n (Dai GAmirir°, ) A2A,tj (Da" GAnPr r, Again, we consider a truncated flow obtained by a restriction on m' and m" in the sum occuring in f dn,, and as before, we remove all those m' and m" corresponding to the above divergences. That is, we leave out all terms with tre = 2 or m" = 2, and those where m' = 4 and r' > 1 or where m" = 4 and r" > 1. This removes two- or four-legged proper (and nontrivial, i.e. consisting of more than one four-legged vertex) subgraphs from the graphical —AA0 expansion. Denoting the solution to the truncated equation by G mr 1 we make the ansatz that for all m > 6, il2A,17 (D a

(3.193)

GA mAr° ) < gmr A4- m- I'l.

This bound is obvious for r = 1 and m > 6, since GAmAi° = 0 for those m. For m = 4 and r = 1, it holds because of the initial condition V4(111°) = —1. Because of the truncation, other terms do not occur on the right hand side of the RG equation. We now proceed inductively in r. Since m' + m" = m + 2 1 + 2, A0 f

ds 8- 3+21- lam I A2A,ri

(Daiont sAior, , ) A2Am (Da"

Ontst„)

A

Ao

< gm/r' gm"r"

f ds 3-

3+2/+4-m 1 +4-m n -loci

A

Ao

= grai r, gm u ri,

f ds 3 5' 24' 1

(3.194)

A

A4- m - I'l m±la -4

grniri gm"r"

5 log A0 m = 4, a = 0 or A024' 1 m = 2, lal < 1. m+

la I

?

m = 2, lal = 2

We have thus proven the following lemma. Lemma 3.24. In unrenormalized, truncated (skeleton) irk 4 , i.e. with

V4Ar° ) (pi , p2,p3) = —6 ri 45m4

3.8 Four Dimensions

103

and with the truncation m' > 6 and m" > 6 in f dKrnr, there are constants grnr i cd such that

A2A,n(D

(:,,n1 AAr

m + lal > 5 log A0 m lal = 4 m = 2, lal < 1.

o

(3.195)

Remark 3.25. It is no contradiction to calculate the behaviour of alr and omit all these terms on the right side of the RG equation. This is clear operationally, since this simply corresponds to a particular modification of the RG equation. In terms of the expansion in Feynman graphs, air is the order r contribution to the four-point function from all those graphs that do not contain two— or four—legged subgraphs. Thus, leaving out these terms on the right hand side only removes the four-legged subgraphs that contain more than one vertex, and thus its feedback for the other Green functions; this truncation does not make the four-point function zero.

Remark 3.26. In contrast to d = 3, there is no extra decay 8— / in the integral A2 produces a 21 in the exponent instead of an in (3.194), because IAA Iii 1. In other words, the power counting becomes independent of 1. For d > 5, (d A d-2 1 so there remains a factor 8 -4) 1 in the integral. The theory I I COA is then perturbatively non-renormalizable.

3.8.3 The Boundary Conditions for Renormalization Recall that the original action of the model was 1

(V) 2sf(

—if)

02 + A04 )d4 s

(3.196)

2

and that we had put the quadratic terms into the Gaussian measure and (Ao ) the 04 term as the initial interaction Vmr = 6r 1 &nil (corresponding to e —xf 04 dx,. ) To incorporate the counterterms, we write an action Sb

= f

m(b 2 +74) 2 02 _ +45)004)d4x

(3.197)

where Z —1, Smo 2 and SA are the formal power series in A, starting at order A2 , i.e. , r > 2. In other words, we put GA m0rA0

= vr(i,Ar0) =

if m > 6 or m is odd

G A° (1)11P27P3) = V4(rA ° n P27 P3) =

qi

? A°

(p)

= V2(71.10)

= b '° p2

This means that as formal power series,

co

arA0

if r = 1 if r > 2 r > 2.

(3.198)

104

3. The Renormalization Group

smo 2

s

00 E brA° Ar

2

Z— 1 2

E

r=2

co

_Ao r r A

(3.199)

r=2 co

6A

E

CrA° Ar .

r=2

Although Z — 1 and 6m4, 2 multiply quadratic terms in 10, these terms are not put into the measure, but left as interaction terms. The formal series start at r = 2 because of Wick ordering. More explicitly, we are now expanding

f

diLC (0) e-AV(A°1(°41P)

(3.200)

in A, where V (A ° ) is the formal power series V(A0)

=

1 )(v0)2 + 57710202) + (A + 6)44)) nco , A0 (f clEy(Z — 1

(3.201) To fix the counterterm, we impose renormalization conditions at A = Al. They are

WI? A ° (0,0,0) =

—1

G141:A0 (0, 0, 0) = Cr qi! A ° (0) = ar

a2 apt ap

GA1A0)(0) =

24,143

Vr > 2 Vr > 2

(3.202)

Vr > 2.

The following lemma states that the boundary conditions (3.202) fix the counterterms uniquely to all orders in A, provided the form of the action is fixed by (3.198). Lemma 3.27. Let (arR b be a sequence in1;13 . There is exactly one (ar A o ,brAo C 0 r) > 2 such that the effective action Ç A ' A ° at scale A1 sequence obtained from the initial condition (3 .198) has expansion coefficients GA,T,17.A° obeying (3.202). ,

Proof. The proof is a trivial generalization of that of Lemma 3.17.

•

Remark 3.28. By Wick ordering,

Gi41? )10 (Pi7P2IP3) = —1 -<=>. G 4111)11° (P17P2,P3) = —1.

(3.203)

3.8 Four Dimensions

105

The renormalization conditions can be thought of as being fixed by experiment. For instance, setting crR = 0 Vr > 2, i.e. , imposing 00

GpA0(0,0,0) = EArG4A,1^0(0,0,0) =

—A

(3.204)

r=1

means fixing the renormalized coupling constant A by the four-point function at (0, 0, 0) (which can be related to scattering amplitudes). The bare coupling constant 00 Ab = A + SA = A +

E crAo A

(3.205)

r=2

will be seen to diverge for Ao oo. As we shall discuss later, one nice point about the RG equation is that it allows us to write all quantities in terms of renormalized parameters, so that the bare parameters never appear explicitly. 3.8.4 Renormalized 0 4 Theory

We define renormalized 0 4 theory as the sequence of effective actions Gm AAro in d = 4, satisfying

GorA l 0 = vr(,,Ar 0 ) = o

for

m>6

Or

m

odd,

and satisfying the renormalization condition (3.202). By Lemma 3.27, the initial interaction V,S,Ar°) is fixed completely by this to be given by (3.198), and in particular, V,;,,Ar° ) is C' in all momentum variables, invariant under Euclidean rotations, and even in p. Consequently, the GA„,Ar° have the same properties for all A E [0, Ao]• Theorem 3.29. In renormalized 04 theory, for any multiindex a, all i> 1, all m > 1 and r > 1, there are constants gmficd > 0 and Ent,. E No such that for all A> A1 > 3,

A2A,n (D "

Gm AAro )
(3.206)

Remark 3.30. This bound is far from optimal, as far as the coefficients and the powers of the logarithm are concerned. A better bound which tracks down the polynomial behaviour in log A in more detail is necessary for good bounds of the gmr i cd as functions of r. m Proof. Recall 8AD0GmAAro = 22,,sm D'' AAro 7 (3.150), and (3.192). Again, we use induction in r; the case r = 1 is trivial. r> 2: We insert the inductive hypothesis to get

A2A, n (D OE Q AmAr0 )

<

a!

E

E atIcelaml K (a i")

K21 gm / r e l a, !gm,/ r it i ce'

( m r 7no 71) A21-3— I am I +4—m1 -1-4—m"

(log A)E m r,./ -FE,,,,u r f; .

(3.207)

106

3. The Renormalization Group

Using m' + m" = m + 2/ + 2, lail +

+

= lai, we have

A2A,n (D' QAmAro ) < qmriai A3—m- 1 ' 1 (log A) Èmr -Érnr =

max (Emi r, +

m l ,r1 ,n 11 ,r 11 ri

(3.208)

Em u rn).

+T."

"Irrelevant terms": m+ lal > 5: Then D'VlAr°) = 0 (for m = 2, this holds because DŒp2 = 0 for lai > 3) AO

Da GAmAr° (k) = D'114,Ar° ) (k) —

(3.209) A

The first term is zero. Applying A2A 07 , its monotonicity in A, and (3.208), we get A0 A2Am (Da GAmAr°

f ds s3—(m+1 ' 1) (log S) Èmr

.

(3.210)

A

Since this converges for Ao —> oc, it is 00

qmricki f d88 3—(m+ l a l ) (logs) Èmr.

(3.211)

A

Using that for w> 2, E > 0

f A' (log A) E dA = —E! A-011-1)V‘

(log A)P p!

1

(w

1)E-4- 1 — p ,

(3.212)

p=0

we have A2Am

<

(Da Gm AAr 0

nr (log Ay a A4 —m—lal Èt qmriale

m + 'al — 4

(log A) E-mr

(3.213) 1

(m + lai — 4)Emr — p+1 Mr•

•

(the last inequality is in fact only a very crude bound, concerning the combinatorial behaviour. One has to retain the polynomial in the logs to get the Borel summability). It is now obvious that for m + lai > 5, the constants gmr i ed and E, can be defined recursively. For m = 4, we define Emr = Émr + 1. For m = 2, we define E2r = -t2r + 1. Marginal and relevant terms: m + lal < 4

3.8 Four Dimensions

107

1. m = 4, a = O. Because of A

qrA°

AO 7

0) —

W4,! A0 (0, 0, 0) = f dAt aA,Gti Ao (o,o,o)

(3.214)

Ai

and the bound for .42A, 77 (Q A,,4r9, we have A

IG '11,1.10 (0, 0 , 0 )1 < i crRi + f ds gniro 8 3-4 (log s ).84r Ai A

<

ds ICril l + (10g 11) É4r qmr f — Al s

<

lcrR l + (log A) Ê4r +1 qmro ,

cis = log fii- < log f because Eiji 7

< log A.

(3.215)

By Taylor expansion, (3.216)

GtA° (Pi /P2/ P3) — 47.43 (0,0,0) = 1

= f dt (pi Vpi + p2Vp 2 + P3 Vp3 ) GtA° (tP1, tP2 1 tP3). o G rAo ) < g4r i A4-4-1 (log A)E4r 1 By (3.213), and since A2A,,I(D1 1 c rR 1

II

A2A,n (GtA° )

+

(log A) Ê4r+1

+

3 max{2, 1 } g4, 1 (log A) E4r

(3.217)

< gmr (log A) E4r . Note that Eir was already fixed previously to be > E4,. + 1. 2. m = 2, lal = 2: t 02

G itito\

l apo ap,, 2r

to\ =

R.

t

82

G A,A 0\to\

/

A 82 r pAo ( m 4. f ds 12 ap op,„ "C 2r k`-')

(3.218)

Al

SO A a2 G A A°

apti Op y \ /

( 82 Q .) A0

< — 1 26pv b rR I + f ds —21 A2,,,i A1

ds s f

Op t, epu

A

<

1 2 b1 -1- q2r2

-3-2-2

(log 3 )B2r

A1

<

I 2brR I

+ q2r2

(log A) É21+ 1

(3.219)

108

3. The Renormalization Group (the rotational symmetry of G2 is only essential if one poses the renormalization condition at some 13 0; this can be done [31]). Thus

o

2

op,iaa pi,

AA G° 2r

1

2

a

■

(p) =

G AAO (0) + f dt N 2r

03

,--,AA 0

at,avae u2r ( tp),

(3.220)

0

and (D2G

/21rAo <

I rR lo I + q2,2(log A) 22r+ 1 I

+max{2A,n} A2A,n (D3 qrA0)

(3.221)

Since .A2A, n (D 3 GV 0 ) <92r3A423 (log A) E2r, this is

< 21 brR + q2r2 (log A) É2r+ 1

max{ 2,

g2r3 (log

A) E2r .

3. m = 2,1al = 1: By the invariance of 141,A,0) under p (÷ 0 qrA0 )(0) = 0. So = a (IAA() — 2r 11-')

f

dt pz,

02

—

GO ( .1 ,\

2r k`l-'1

(3.222) p, we have

(3.223)

0

and

A2 A m (D I qrA° ) < max{2A,71}

(D 2 qrA° )

< max{2A,71} 92 r2 A 4-2-2 (log A) E2r (3.224) < 92r1 A (log A) E2r where we have defined 92ri = max{2, -} g2r2. 4. m = 2, lal = 0: The by-now standard procedure gives

A `-7 2r

f ds aAGL,Ao (0)

0) = G Ao (0) (2A,! k

(3.225)

A1

A laR

IGV° <

I

— (log I + J ds q2,0 s 32

I

)s E2r

A2

q2 ro

—(log E27. 2

(p) = qrA ° (0) + f dt pP ap m %-r 2r ll'P)\ 0

(3.226)

(3.227)

3.9 The RG Flow in the Ladder Approximation

109

SO

I

A2A ,n (G 17,A0 ) < qrA° (0)1 +

Since A2A, n (D 1 G

max{2A, 71} A2A,r7 (D i qrA° ).

(3.228)

1,11°) < g2ri A4-2-1 (log A) Enr 7

< mo (0) i + max{2, 1} 92r1 A2 (log A) E2r. .42A,n (q.A0 ) _

(3.229)

Defining g2ro suitably, we have proven the theorem.

• Remark 3.31. 1. The proof is a little lengthy, but the arguments are all similar — a combination of power counting and Taylor expansion using the renormalization conditions. 2. Applying a l io to the RG equation, one can prove in a similar way that A2A

,nk ail°

(log Ao ) E° GAAo, < h mr / _ mr /16

(3.230)

co exists. This proof is a relatively straightforward so that the limit Ao generalization of the one given for d = 2 with application of renormalization conditions, so we omit it here.

3.9 The RG Flow in the Ladder Approximation We have shown that 04 is renormalizable in perturbation theory, that is, that the renormalized Green functions exist as formal power series in the bare coupling A. We have not discussed convergence of the series in A. With the flow equation methods, one can show 71-bounds, so that the perturbation theory is locally Borel summable [33]. One should, actually, not expect convergence since even the one—dimensional integral fR e—Ax4—x2 dx is not analytic in A in a disk around zero (in fact, it diverges for ReA < 0). One may still ask if the perturbation series is Borel summable, i.e. if the Borel transform is maybe analytic in a strip around the real axis, not just in a disk around zero. This has been proven for d < 3, but in d = 4 it is believed not to be true. In the following, we shall give a heuristic argument why there is a problem. In the course of this argument, we shall see another aspect of renormalization, namely that of running coupling constants. We have proven that the limit Ao oo is well-defined as a formal power series. Now we study what a simple truncation can tell us about convergence. Instead of the exact flow, we consider an approximate flow. The first approximation we shall make is to discard the two-legged Green functions. Then the lowest terms contributing to aAG^rA0 are those with m' = m" = 4. The second approximation is to discard all m` > 6 (or mll > 6) terms. This gives

110

3. The Renormalization Group

(3.231)

aitqrA0 (k) =

E

M S4

f

ddP COA (p)aAcAA0(p+ki+k2)

r' +r" =r G1417i., 0 (ki ) k27P) Gitilt ( — P) kl +k2 +

p, k3 )

(with a combinatorial constant M) so when summing up the series again and defining 00

G141A° (A, il)

,E

1 Gi,47/.10

(10 A

(3.232)

r=1

we get the integral equation aAGpo (A;

li) = M f ddp COA(p) 8,1cAA0(p+k1+k2)

GP10(A ; ki, k2,P) G"° (A; — A k1 + k2 + p, k3 )

(3.233)

(see Fig. 3.10).

—k-k-k 1 23 Fig. 3.10. The bubble graph

Recalling that aAcAA0(p)= -19A COA (P)

(3.234)

and taking the limit Ao oo, we get aA Gi41(A ; k)=A1 f copc0A(p)(-3Ac0A)(p+ k1 + k2)

Gi4i (A; k1, k2,P) G(A; —P, ki + k2 + P, k3).

(3.235)

For the boundary condition that we imposed,

G`411 (A;0,0,0) =

—

A,

(3.236)

the coupling we expand in is now the renormalized coupling at scale A1. (Note that by our definition, y (A) > 0, since we want to have convergent integrals, i.e. the 04 term has the "right" sign). If we had imposed the condition (3.236) at a different scale A2 > A1, we would have got the same Green functions, but expressed in terms of a different renormalized coupling. Note that in (3.235),

3.9 The RG Flow in the Ladder Approximation

111

Ao is removed: the RG equation relates only the Green functions at scales A, without any reference to the cutoff scale. The semigroup property of the flow implies that 00

A2 = Al +

E /37-A1 r=2

(3.237)

where the Or are given as sums over Feynman graphs with propagators CA1)123 as discussed. In other words, if we define a scale-dependent coupling v (A) by defining v(A i ) by (3.236) and setting

u(A) = —G141 (A; 0,0,0),

(3.238)

and approximate the G`41 in (3.235) by v(A) (as suggested by the power counting behaviour of derivatives) then the relation between all these different choices of A is given by

aAv(A) = 132 (A) v (A) 2 .

(3.239)

where the coefficient 02(A) = M f

ddpC0A(p)

aAcoA(p)

(3.240)

is the lowest order term in the beta function of the RG. Since

02 (A) =

Mf

p2 p2 2p2 1 ddp (p2 + mo2)2 ( A3 ) KI (A2 )K ( A2))

(3.241)

we have (again for A> m) oA d-5 < 02 (A)
(3.242)

So, to get the behaviour of u(A), we need to solve

19,0(A) = OA" v(A) 2 where 0 is a some constant. For d

< 4,

(3.243)

this integrates to

1 1 v(Ai) v(A2) _ ij(A4 _ Ar4)

(3.244)

with ,3- — -- 1 so — —fl 4—d

v(A 2 ) =

u(A i ) 1— v(A1 )/(A

4 — Ar 4 ) .

If at fixed A 1 , u(A 1 )/3-4-4 < 1, the limit A2

lim v(A) -=

A-40o

(3.245)

00 exists, and we have

v(A i ) 1 — V(AOSAr4

(3.246)

112

3. The Renormalization Group

(all this because d 4 < 0). However, for d = 4, the same calculation gives —

1

1

v(Ai)

v(A2)

SO

v(A2) = The limit A2

=0

A2

—

(3.247)

logAi

v(Ai)

(3.248)

1— v(A1 )(3 log 421, •

00 cannot be taken because v(A2) already has a pole at

1

A2 = A1 ev(A1)1,

(3.249)

the so-called Landau pole. This is an indication that the UV limit of the theory does not exist, since the scale-dependent coupling diverges at a finite scale already. The approximations involved cannot easily be justified (one would like to argue that the terms we left out are irrelevant by power counting, but they aren't really if one looks at the details because the theory is just renormalizable). Moreover, when increasing A2 in (3.248), one leaves the weak-coupling region where a perturbative analysis may be meaningful long before one gets to the singularity. The arguments for triviality known today are independent of, and go far beyond, the arguments given here [10, 11, 12, 13]. In the next chapter, we shall investigate a system where it can be proven by a more detailed "improved" power counting theorem that the ladder approximation indeed gives the leading contribution. The improvements to power counting arise there because the singularity is on a curved surface instead of at a point. Had we started with the "wrong" sign of the coupling (y(A) <0) we would have got asymptotic freedom. Similarly, the flow going into the infrared is asymptotically free: at fixed v(A2) > 0,

v(Ai) =

v(A2) -->. 1 + v (A2 ) log 42i, A i -40

O.

(3.250)

This behaviour is the basis for a nonperturbative construction [35, 36, 37] of infrared 01. There are several excellent accounts of nonperturbative renormalization with bosonic fields. Readers who want to continue in that direction should consult [7], [38, 39], or [40].

4. The Fermi Surface Problem

In this chapter, we consider an infrared problem with fermions, which is physically more realistic than the ultraviolet problem in 04 theory. In infrared problems, the perturbation expansion is singular because of the slow decay of the free propagator at large distances, not because of a short-distance singularity. We shall again restrict rigorous proofs to perturbation theory to all orders. Nonperturbative bounds for these systems, given in d = 1 in [59, 60] and in d = 2 dimensions in [55, 56, 61], are still the subject of ongoing research and thus at the moment beyond the scope of an introductory text. Still, the perturbative analysis done here, which is based on [45, 46, 47] and [53], justifies certain resummations of perturbation theory, which we then perform in order to discuss the onset of superconductivity. It also gives a precise relation between the Fermi surface geometry and the power counting behaviour of the Green functions, and thus shows why these systems behave differently in one and more than one dimension. In particular, basic regularity properties of the selfenergy are proven, and their role in distinguishing between Fermi liquid and Luttinger liquid behaviour is discussed briefly.

4.1 Physical Motivation The models that we shall analyze in the following describe electrons in a crystal, and we are particularly interested in the case where these systems show metallic behaviour. We shall not investigate the reasons why crystals exist at all. Given a system of nuclei and electrons, it is a basic problem to show that, at sufficiently low temperatures, they arrange to ions situated at the sites of a periodic crystal and remaining electrons that move in the field of these ions. To this day, there is no rigorous quantum mechanical proof that the continuous translation symmetry gets broken to a discrete one at low temperatures by the formation of a crystal. Kennedy and Lieb have proved a discrete version of such a crystallization theorem for the Falicov-Kimball model, which is a lattice model. They showed that the symmetry under the lattice translation group remains unbroken at high temperatures but gets broken down to a smaller group when the temperature is low enough [41]. This is up to now the

114

4. The Fermi Surface Problem

only rigorous result concerning crystallization derived from first principles in quantum mechanics. Once one assumes the existence of such a crystal as given, there is a standard way of obtaining a model Hamiltonian. In a first step, one usually neglects the (residual) interaction between those electrons that are not bound in the ions. The dynamics of such independent electrons in a static background of ions is given by a Schrödinger operator in a periodic potential. The effect of the periodic potential is that the relation between the (quasi)momentum p of an electron and its energy E is given by a number of band functions. For a mathematical periodic Schrödinger operator, the number of bands is infinite, but in a real crystal, only finitely many of them occur (if the energy of a particle gets very large, it will leave the crystal; if the temperature gets too large, the material will melt). To this "free part" of the Hamiltonian, one adds the interaction between the electrons. This interaction may be a screened Coulomb interaction. If the crystal is not taken to be static, the electrons also interact with the lattice vibrations . In the harmonic approximation, this is described by phonon interactions. These phonon interactions are responsible for superconductivity in ordinary superconductors. A popular, and in many cases realistic, approximation to the free Hamiltonian is the tight-binding approximation where, as a first approximation, one assumes the electrons to belong to a single atom as before, and then builds Bloch states from a superposition of these localized states in the natural way. One then gets a model with electrons that can occupy sites of a lattice, and hop from one site to another. Thus the kinetic energy takes the form of a hopping term for the electrons, with the hopping parameters given by overlap integrals [43]. The simplest such model is the Hubbard model, introduced about 40 years ago [42], in which the hopping is short-range and the potential is on-site. Many of the features of this model are still not completely understood, in particular in two dimensions.

4.2 Many-Fermion Systems on a Lattice Again, we take a finite lattice as our starting point. In fact, the crystals of solid state physics are always finite, so this is the right starting point. However, the applications we are interested in involve macroscopic pieces of crystal, that is, sidelengths L which are huge when measured in terms of the lattice spacing. For such systems, it is the correct idealization to look at the thermodynamic limit L oo. In the presence of symmetry breaking, some care is required in distinguishing between a large but finite and an infinite system. For the discussion of Fermi liquid properties, however, we may take the thermodynamic limit. Let E> 0 and let L E JR such that t- E N is a very large number, and let A = EZ d /LZd be the cubic lattice of spacing E and sidelength L. One need

4.2 Many-Fermion Systems on a Lattice

115

not take a cubic lattice - any other lattice of maximal rank will do. Again, A is a torus, that is, we use periodic boundary conditions for convenience. For the questions considered below, the boundary conditions do not matter. Let r = A x 1-1, 11 and 7-1 = L2 (F,C), with the scalar product

(f ig) = fd f g(e) = e d

E E

f (x, cr) g(x,

(4.1)

xEA crE{-1,1 }

The variable a- labels the spin of the electron in units of h/2. Since r is a finite set, 7-1 is simply a set of sequences of finite length Irk hence has dimension in I . The Hilbert space for a many-particle system is the fermionic Fock space Fr = 01,7_0 An 71 (see Appendix B.1). Because 7-1 is finite-dimensional, An = 0 if n > dim7-1, so Tr is finite-dimensional, too. For every vector f in the one-particle Hilbert space 7-1, there is a creation (annihilation) operator a* (f) (a(f)) acting on states of Fr, which has the effect of adding a(f) is conjugate linear, and (removing) a particle in state f. The map f f a* (f) is linear in f. The operators a(*) (f) obey the canonical anticommutation relations (CAR)

{a(f), a* (g)} = a(f)a* (g) + a* (g)a(f) = (f 1 g).

(4.2)

Let (xx, ( .)(x,,) Er be the ONB of 7-1 given by

(4.3)

Xx,0- (3r7 7) = Cd/26x,Y 6cr,T• We denote the corresponding creation and annihilation operators by

(4.4)

= a( *) (xx,,),

= they obey the CAR with the normalization fax,,, ax*, 6 is (x, a)(x,0-)En given A basis better suited for the continuum limit by (4.5)

5Cx,o- (Y) 7) = C d6x,y(50-,T The corresponding operators à

= a ( *) (ix,,)

satisfy

f ax , o. ax* , ,04 = e —d (5x,x 1 60- t ai = Sr (

, X1 )(50- 00-t .

(4.6)

4.2.1 The Hamiltonian The Hamiltonian of the system is H = Ho ± AV, where

1/0 = f d4" f dij T(4-3 71)Feei,” =

E edT(x, a ,y, 7)4,, ay,,

is quadratic in the creation and annihilation operators, A constant, and

E

(4.7)

R is the coupling

4. The Fermi Surface Problem

116

V=

f r

d

f

dr,

V(,77)FLZ c- ickl e-zn = x,y,cr,r

r

(4.8) is the interaction. The quadratic part of the Hamiltonian, 1/0 , describes hopping of electrons from position y, spin T, to position x, spin c r, with an amplitude EdT(x, a, y, 7 ) . The condition T (y , 71 X ) 0') = T(x, cr, y, 7-)

(4.9)

makes 1/0 hermitian. The interaction V couples the local densities nx,cr = and nyo- = ay*, T ayo-. Since, by the CAR, [nx,,, nyod = 0 for all x, y, a, and r, we assume that v(y, r, x, a) = v(x, o-,y , r). To have V hermitian, we require that v(x, a, y, r) is real. In the detailed analysis of the model, we shall only treat the case where both T and v are translation invariant, and where T is diagonal in, and v is independent of, the spin index. That is,

T(x, a, y, T) = ii,,T e(x- y),

e(—x) = e(x)

(4.10)

V(x, a, y,r) = v(x — y), v(—x) = v(x) ER.

(4.11)

and In this case, V can be written as

V

= E n(x)v(x — y)n(y)

(4.12)

x,y

where n(x) = nx, + +n,_ is the charge density operator at x. These assumptions imply that the model is translation invariant and that it has a global spin rotation invariance, i.e. H is invariant under the replacement a(x) i— U a(x), where ax,_ ) (4.13) a(x) = ( ax,± and U is an arbitrary SU(2) matrix. The symmetry is global because U is the same for all x. There are more complicated interactions that have this spin rotation inw(x, y)S(x) . S(y), with variance, for instance a spin-spin interaction E S(x) = ax"fax the local spin operator (ci is the vector of Pauli matrices). These cases can be dealt with in a similar way as we do here; we restrict to density-density interactions for simplicity. The discretized Laplacian corresponds to nearest-neighbour hopping, d

1 T (x , cr) 3 r) T) = 6,7,7 - —2- E (26,(x,y) - Sr (x, y + Eev) — Sr (x, y — Eev)) 6 v=1

(4.14)

4.2 Many-Fermion

Systems on a Lattice

117

with e,, the unit vector in I/ direction. The model on the unit lattice (E = 1) with H = tHo + 4 1 V — ( 112 + 2td)N, 1/0 given by the hopping amplitude (4.14), and v(x, y) vanishing for x 0 y is the Hubbard model with nearest-neighbour hopping t and on-site interaction U. The subtraction of the term proportional to the number operator N = Ex n(x) implies that Eaa +

Hhubb = — t

u

x,y,cr

nx,±nx,_.

(4.15)

X

Ix— y1= 1

Terms proportional to N can be subtracted from H without essential changes because, as discussed below, N commutes with H. With our assumptions on T, the quadratic part 1/0 of the Hamiltonian is diagonal in Fourier space. To make this explicit, we introduce the ONB (Vic,cr)(k,a -)EF* 7 where r* = A* x —1, 1} with

A* = given by

aLer zd,

7c1

(pk,o. (y r) = scro. eiky (EL)—d/2

and define

(4.16)

/ I

ck(* o.) -- a (*) ( k, ,.).

(4.17) (4.18)

Then {ck,,, } = 6k,k, Scr,a' • For taking the thermodynamic limit, the basis consisting of (13k,o- = Ld/2wk,0. is more natural; the corresponding creation and annihilation operators obey

}=

= 6A. (k,

(4.19)

e ikx (pk, o-

(4.20)

By Fourier transformation, x.,47

=L

"2

E

kEA* ikx By the (anti)linearity of the creand, similarly, = 11—d EkEA* ation and annihilation operators, they obey the Fourier transformation formula (4.20), with xx ,, replaced by 4,0. and wk,, replaced by Under the assumption (4.10),

1/0 =

E P,cr

(p)cp* , Œ c„„, = L— d

E 6(p)ap*,,ap,,

(4.21)

P,cr

where 6(p) = Ed Ex e(x)eiPx is real. The number operators fip,, = all commute, so they can be diagonalized simultaneously. The interpretation of 1/0 in Fourier space is simply that the total energy is the sum of the one-particle energies, where a particle in state (p, a) gets the energy 6(p). The discretized Laplacian has (p) = 1L (1 - cos(ept,)), which has the limit p2 as E 0.

118

4. The Fermi Surface Problem

4.2.2 The Grand Canonical Ensemble

The grand canonical ensemble corresponding to the Hamiltonian H, the inverse temperature 0 and the chemical potential p is given by the linear functional ( • ), where L(.7- , .T) C, 0 1— (0), 1

(0)

) / = Tt M. ( e11—AN)(9

(4.22)

with Z11 = Tr e'1 ", and N is the number operator. Because

N = f keqek =

E 7, =

(4.23) k,o-

X,IT

and because all the local number operators nx ,,,- commute, N commutes with V. By the Fourier representation of 1/0, and because all iik a commute, N also commutes with 1/0 . Thus [H, N] = O. The expected particle number is

given by

1 a (N) = p 3 log Z11.

(4.24)

, -,.- ,,

By (4.23), N is a sum of positive terms, so (N) > 0 by the following lemma. Lemma 4.1. For all 0 > 0 and all y E R, ZA is nonzero, so ( ) is well-

defined. Moreover,

( o K 1 loll

(4.25)

where 11011 denotes the operator norm of 0, and for all linear operators B on Jr, (4.26) (B* B) > 0. Proof. Tr has an ONB of eigenvectors 77,„ of H — AN. The eigenvalues cc, of H — ILN are real because H and N are selfadjoint and p is real. Thus Z11 = E (i ia I e— (3(H - 01) lia) =

E e—o'fa 1 oc, I

zA(B*B)

= E e—'36" I (qc,

(4.27)

a

a

z111(0)1

E e-03E- > 0,

07/0)1

E e—acc. 11011 = zit 11011,

(4.28)

E e—'3f c" IIB%11 2 ? 0.

(4.29)

and

I B * B77)1 =

a

Corollary 4.2. For all p E R, A (N) > O.

a

•

4.2 Many-Fermion Systems on a Lattice

119

Proof.

l(N) ap

a2

=

ap2

= =

log ZA =

a ZA

Tr (e -fl (H- A N) N)

(V + ( N2 ) = ((N — (N))* (N - (N)))

(4.30)

(N)) 2 )

0.

In the second line we used that (11) = 1. Thus (N) is an increasing function of p. This is not unexpected because of the term AN that p contributes to the energy. Typical observables that we want to study are the correlation functions (I1 ax* 0. 11 ay,r ). In particular, we are interested in the behaviour of the twopoint function By spin symmetry, we have (c4,0.ay,T ) = (5„(a;,,,a y,,) and, by (4.18), (4.20), and the linearity of ( ),

*

1".3c,cr u'Y,a/

= c d E e ik(x-y)(4. ,ack,a )

(4.31)

kEA*

because, by translation invariance, (c 0.cie ,u ) = other bases, the relation reads

ox* ,cray,c )

L -d

0 for k k'.

In terms of the

E

(4.32)

kEA*

4.2.3

The Fermi Gas

Ultimately, we want to study the system with a weak interaction (A small) by means of a perturbation expansion in A. It is thus natural to look at the case of no interaction (A = 0) first. For A = 0, the fermions are independent ("free"), and calculating the partition function and (ce1) is easy: we evaluate the trace for Z A as a sum over the ONB (OA )A c r. given by A = n(x,0)EA cf2 (we fix an ordering on r* and take the product in the ordering induced on A). Recall that 1/0 = >k,0. 6(k)iik,o. with fik,o- = qck,a• The iikosr fulfil the

relations

=0

,cr 7 qc; , 0-1] = Skki cr CZ, 0- •

— tbk,o- 7

,

Thus

e -fl(Ho-t,N)

_ H e -0E(k)fik,ff

( 4.33)

(4.34)

(k,cr)Er*

where

E(k) =ê(k) For (lc,

(k', o'), fik os,

-

p.

and cZ, ,0., commute, and fik,012 = 0, so

(4.35)

120

4. The Fermi Surface Problem

ii

=

e - )3( Ho - AN4A

e--)3E(k)fik,Œ 4,13

(k,cr)EA

H e-f3E(k)fik,,

=

. ri

e—flE(k)fik'' f2

ez ,o. (2 (k:1 vA

(4.36)

(k,a)EA and all factors in the last product acting on f2 anticommute. An expansion of the exponential and (4.33) imply that e -SE(k)fik, ff = 1 + (e-r3E(k) - 1)iik, c,-.

(4.37)

e—flE(k)fik '' 4 ,012 = 4 ,,f2 + (e -flE(k) - 1)4,04,012.

(4.38)

Therefore

By (4.33), nk,0rc12 =

50

e -)3(Ho-piv)

'

A

= H

e -f3E(k

A•

(4.39)

(k,cr)EA

This equation has the simple interpretation that for independent electrons, the state ' Cl l which contains particles in the states (k, ci), for (k, a) E A, has total energy E (k,o.)EA E(k). By the binomial theorem,

Z

A

(c4 1 e--0(110--AN4A) E Acr.

=

= E H e—I3E(k) (k,o-)EA = il (1 ±

(4.40)

ACT'

(k,cr)Er*

The number density is calculated in the same way: 1 Tr (e -0(110-AN) n- 13 " ) (flp,T) = _

Z zA (7-ip,,-) = E (IA I e--0(110-AN)/ip,74A). , 7

A

SO

(4.41) (4.42)

ACT*

Since iipo-f2 = 0, hp,4■ A = 0 unless (p, 7) E A. If (p 7 T) E A, then iip,74A = A • Thus -f3E(k) . e (4.43) A(fip,r) =

Z

E II

ACr* (kMEA

(pr)EA

The constraint in the sum that (p, T) must be in A can be enforced by a factor 7 .., (ka) vA Arp,r (k, a.) in the summand, where Arp,T (k ,a)

={0 1

if k = p and cr = T otherwise.

4.2 Many-Fermion Systems on a Lattice

121

Again, by the binomial theorem, ZA(lip,T)

c II A437-(kIr) H e -'3E(k) (k,cr)VA z' H (ArP,T(k, Gr) + e -flE(k) )

= V

ACT' (k,o-)EA

.

(4.45)

(k, o-)Er

TT

e

=

(I- + e-f3E(k))

(k,n œ)Er* (k,a)*(13,1

)

Dividing out Z A l we get

e —OE(p)

(fip,T ) = 1 + e- OE(P) = f o(E (P))

(4.46)

1 fo(E) = 1 + 0E

(4.47)

where

is the Fermi-Dirac distribution. It has the zero-temperature limit 1

lim fo(E) = {

/3-+ao

if E < 0 if E = 0

(4.48)

0 ifE>0.

The interpretation of this is that in the Fermi gas at T = 0, all states with energy E < 0 are filled and all states with E > 0 are empty. The set of p for which E(p) = 0 is the surface of the occupied region, the Fermi surface

S= p {

E

B: E(p) = 0}.

(4.49)

Here B = Rd PaZd is the first Brillouin zone for the infinite lattice. On a finite E lattice, also A* is a finite set, so {E(p) : p E A* } is finite, and S n A* 0 can happen only for finitely many p. The discontinuity of the zero-temperature Fermi-Dirac distribution at the Fermi surface is the characteristic feature of the Fermi gas; it is a direct consequence of the Pauli principle: the energy of different particles simply adds up, no state can be occupied twice, and thus in the ground state, only the states with lowest possible energies are filled. On the other hand, (4.31) implies that the position space two-point function is simply the inverse Fourier transform of the number density, so, in the infinite volume limit L oo,

(a * a

) = f ddk eik(x- Y)fo(E(k)).

f (27r)d

(4.50)

13

The discontinuity of the Fermi distribution at T = 0 implies a slow decay of the two-point function in infinite volume - if the two-point function were

122

4. The Fermi Surface Problem

in LI, its Fourier transform would be continuous by the Riemann-Lebesgue lemma. As an explicit example, we calculate the two-point function for d = 3 in the continuum limit e -> 0, where e(k) = k2 /27n: in infinite volume and at zero temperature (T 4. 0), with kF = N/I' WIT

(4,,ay

, Œ )

-

f

ddk ik oc 3,)

(27r) d e J pci
.

(4.51)

Because of rotational invariance, we may assume that x - y = Ix - yle i and introduce polar coordinates. This gives kF

(ax* to.ay,u )

ir

= 27r f r2 dr f dO sin Oeirlx - YI cos 0 o o lc,

=

47r I x - 3r i

=

f

r dr sin(rix - yl)

(4.52)

o

47rkF 47r - YI2 cos(kF lx yl) + lx - y13 sin(k F Ix - yl). ix

It is obvious from this form that the function is not in L 1 (113 ). As we shall see, this slow decay leads to divergences in perturbation theory, very roughly speaking in a similar way as the singularity at coinciding points led to the ultraviolet singularities in the ultraviolet 04 problem. However, the treatment of these singularities differs from the previous one: we will not be able to argue that putting counterterms to cure the divergences only changes the parameters in the action at an unobservable cutoff scale. In infrared renormalization, the inclusion of the counterterms changes the model, and it requires a separate argument to show that things can be arranged such that one gets back a statement about the original model. Note that at positive temperature, the Fermi distribution fo is C", so the two-point function decays exponentially in the distance. Although this decay is nonuniform in the temperature, and thus not useful for proving statements about the behaviour as a function of temperature, it means that in the model, there is a natural physical cutoff, namely positive temperature. Whenever one encounters an ill-defined quantity at zero temperature, the correct interpretation, if any, is to look at its positive-temperature counterpart and see if the limit exists. In fact, the physical definition of zero-temperature correlation functions is that one has to take the limit T -> 0 of the ones at positive temperature. If one takes x = y and sums over a in (4.31), one gets the local number density (nx ), which, by translation invariance, does not depend on x and hence equals the density p = 2L-d(N). By (4.50), we thus have that the

4.2 Many-Fermion Systems on a Lattice

123

filling factor, defined as the density divided by the maximal possible density in the band (hence proportional to p), is, in the temperature zero limit, equal to the d-dimensional volume enclosed by the Fermi surface. For instance, in the Hubbard model on the square lattice, the filling factor is v = L —d (N)/2 because the maximal number of electrons at each site is 2 by the Pauli principle.

4.2.4 The Functional Integral Representation We introduce the functional integral as a Grassma,nn integral, to set up a flow of effective actions similar to the bosonic case. The idea is to use the formula )n ) (4.53) Tr e—)311- = lim Tr ((1 _ g_H n n ÷ao -

and to rewrite the nth power and the trace in terms of Grassmann integrals. On a finite lattice, H is just a matrix, so the right hand side of (4.53) converges. The functional integral is discussed in Appendix B.5; here we just state the result. The correlation functions can be obtained as derivatives with oc of the Grassmann integral respect to source terms q of the limit n,Zn i.,L(71) = f dp, c (0)e — Av( 0 )+(fhiP)

(4.54)

with the following specifications. The fermionic fields 0 are labelled by X = (x, a, j) E 1-', where 1" is the set F = A x { -1, 1} x {1, 2}, where A = T x A, and (4.55) k E {0,...,n T — 1} T = Irk =

-4+ to:

is a discretization of the interval [-0/2, 0/2). We require for convenience that nr is even. The variable T is a discrete Euclidean time, a is the spin index and j is a "charge" index that distinguishes between fields ti) and V) that replace the creation and annihilation operators in the Grassmann integral, that is, for x = ( r, x) E A,

= 11)(x, a, 1)

and

tp, ( x) = 0(x, cr, 2).

(4.56)

The fields 0 have antiperiodic boundary conditions on T, that is, they are actually defined on a discrete torus of sidelength 20, with the restriction (4.57)

=-0 (T, x,a,i ).

One can rewrite the integral (4.54) for Zn,,L (7)) in terms of the product measure H (x,0.) clt/3(x, a) clz(x , a) times the exponential of the action 0/2

51

( )=

f

-0/2

dT

[MT),

497-0(T))r—HA(0(T),O(T)

)

] •

(4.58)

4. The Fermi Surface Problem

124

Here (Or), 07-0(7))r = Jr d6t(r, 6)04 (7, 6), the notation HA N-1(T) , 1P (7)) means the normal ordered polynomial of with a* replaced by t-i) and a replaced by tp. The integral over T is shorthand for a sum Er with ET = Ofra.,-, and 6, is a discretized derivative with respect to T. Details of this are discussed in Appendix B; here we shall only need the properties of the Gaussian measure and its covariance, which we summarize now. Because the Grassmann variables anticommute, the Grassmann bilinear form

H,

(i7, tp) =

ETET,

f dX71(X)11)(X) = f dr f dx EE q(7-, x, o-, j)0(7, x, cr , j) r

A

T

a

i

(4.59) is antisymmetric: (71, tp) = — (0,77). The `Grassmann Gaussian measure' is not really a measure but just a continuous linear functional. It is determined by its characteristic function

f

dfic (0) e(n 4) = e i (7) ,c,i) .

(4.60)

This looks formally identical to bosonic Gaussian integrals, but the crucial difference is that the Grassmann bilinear form is antisymmetric. Accordingly, the propagator C is a skew symmetric matrix,

C(X, X1 ) = —C(X 1 , X)

(4.61)

given by

C(X, X') = eii,6„, f dp eiP ('' ) e,

( ( — 1 ) ip)

(4.62)

A*

where El2 = - E21 = 1 ,

Ell

= E22 =

0, A* = Mni. x A*, with

M„, . { 71 (2n ± 1) : — Ly- < n <

}

(4.63)

is the dual to A, and for p = (w, p) E A*,

1

6 ' (w

'

13)

—

(4.64)

i6)

with CD =

1E1

(eiErw — 1),

(4.65)

•

and E the dispersion relation given by 1/0 (see (4.35)). (4.61) can be seen by transforming the summation variable to k = ( 1)j p in the integral, k x') = k(( 1)j x + ( 1)j z') is symmetric unalso runs over A*, and p(x der exchange of X and X'. The antisymmetry of C now follows from the antisymmetry of The interaction V is the Grassmann polynomial -

—

-

-

4.2 Many-Fermion Systems on a Lattice

V(0) =

f

dr Vit (

( T ), i)(7)),

125

(4.66)

that is, t(r) replaces a* and //)(7) replaces a. In the limit n, —Y 00, the time variable becomes continuous, 6.3 co, and the set of Matsubara frequencies becomes M = {(2l + 1) : 1 E Z}. The Grassmann integral then becomes infinite-dimensional. Although infinitedimensional Grassmann algebras and Grassmann Gaussian integrals are welldefined objects, we shall take the limit n, oc only at a point where we do not need to refer to Grassmann integrals any more. Thus no infinitedimensional integrals are needed here. Since M is an infinite set and because wi a 1, the sum E 1/w diverges, and correct limit nt,. —Y 00 of the propagator is the boundary value e iw0+ 6( ``' ' P)

(4.67)

-E(p)

This can also be seen more directly - the Fermi distribution (1+ efl E) -1 is periodic with period 270 in the imaginary part of E, and its poles are at iw, where w E M are the Matsubara frequencies. Fourier transformation (see Appendix B.5.5) implies that 1 eS E

= lirn

1 v,

e iew —

E•

(4.68) 1+

In the following, we study log Zn L, and the effective action, by perturbation theory in A. Perturbation theory for fermionic models is derived similarly to the bosonic case, but there are extra signs as compared to the bosonic case because of the anticommutation relations. It is proven in Appendix B.6 that these extra signs do not disrupt the polymer formulas, so that the theorems about connected functions extend to fermions. As mentioned, the singularity of the propagator in momentum space, related to the slow decay in position space, makes renormalization necessary. We introduce a renormalization flow in a way similar to the bosonic case. 4.2.5 RG Flow: Energy Scales

The infrared divergences are caused by the singularity of Ô for co = 0 and p E S. Therefore we introduce the renormalization group flow by a cutoff that forbids small energies. We label energy scales by a parameter t > 0, where the energy scale associated to t, Et =

(4.69)

is a decreasing function of t. co is a fixed energy scale, which, however, depends on the filling region one wants to study (this is discussed in Sect. 4.2.8). Since the cutoff was not present in the original model, we want to take the limit

126

4. The Fermi Surface Problem

t -> oc after rearranging everything suitably by renormalization. The cutoff is put in by replacing the propagator C by the t-dependent cutoff propagator Ct , which has the Fourier transform

1

Ô(k) = ic.7.) - E(k) x2 (Et -2 1i6.3 - E(k)1 2 ) .

(4.70)

Here xi and x2 are a partition of unity, xi + X2 = 1, Xi E C"(Rct,[ 0 , 1]), with if x < / (4.71) Xi(x) = { 10 if x > 1 _1 Xi (x) < 0 for all w E (t, 1), and 11Xilloo < 2. We shall also use a Wick ordering covariance Dt which is defined similarly, only that X2 is replaced by Xi in (4.70). We define the effective action at t > 0 as

g(t, 0) = log

f dPc,(x)

(4.72)

4.2.6 Model Assumptions We first state the assumptions and then discuss their significance and give some examples of models where they axe fulfilled. Let B be the dual of the infinite-volume limit of A*; for the cubic lattice A = EZd , B is the torus B =RdIalZd. E

1. Assumptions on the dispersion relation and the Fermi surface: There is k0 > 2 such that E E Ck°(B,R), and for all p E B,E(—p)= E(p) holds. The Fermi surface S = {p E B : E(p) = 0} has a nonempty interior I = {p E B : E(p) < 0}, and 0 is at the center of I. There is g 0 > 0 such that for all p on S, IVE(p)1 > go holds. The Fermi surface is a subset of an s-independent bounded region of momentum space (hence S is compact). S has positive curvature everywhere (thus I and I U S are strictly convex sets). There is V1 > 0 such that for all L and E

f

dk 11(1E(k)1 < 2)

Vi.

(4.73)

A*

The constant

Erna.= suPIE(P)I

(4.74)

PEB

is independent of e. 2. Assumptions on the interaction: The interaction f) is a Ck° function from R x B to C, all its derivatives up to order ko are bounded functions on B x R, f)(-po, p) = f)(po, p), and the limit Po oc of f) exists and is Ck° in p.

4.2 Many-Fermion Systems on a Lattice

127

A priori, the statement that E is symmetric under p > p does not look sensible on a torus. By the above we mean that there is a point p(°) such that for all p E B, E(R op) = E(p), where Ro denotes the point reflection at p (°). We then choose coordinates such that p (°) = 0, then the reflection symmetry condition simply reads E(p) = E(-p). The above hypotheses allow us to introduce natural coordinates in a neighbourhood of the Fermi surface: there is ro > 0 and a C2 -diffeomorphism ir from (- 2r, 2r0) x S d-1 to an open neighbourhood of the Fermi surface S in B, (p,0) 7r(p,0), such that -

-

2

E(7r(P,0)) = P and it9p 7r(P) 0)1 < — .

(4.75)

go

The coordinates p and 0 are easily constructed by first going to polar coordinates (r, 0) and then making a change of variables from r to p = E(p(r, 0)). The latter is regular by our assumptions on the Fermi surface. Coordinates (p, O) satisying (4.75) exist under much weaker assumptions on S (see [45], Lemma 2.1, and [46], Sect. 2.2). Let J(p,O) = I det r i (p, 0)1 be the Jacobian of this change of variables and denote Jo = sup J(p,0) Ipi<E0 8ESci-1

and J1 = sup f dO J(p,0). Ipl
(4.76)

The density of states N(E) in infinite volume is defined by

N(E) = f (2d7rP)d (5(E - E(p)).

(4.77)

B

With the above change of variables we see that

N(E) = (2 716 d f d0 J(E,0).

(4.78)

sd-1

Thus the Jacobian plays the role of a local density of states, and its integral over O is the density of states at energy E. So, under our assumptions, the density of states is a finite Ck0-1 function of the energy E in a fixed energy interval (-ro, ro) around the Fermi surface (E = 0). The energy co that sets the initial scale of the RG flow will be chosen smaller than or equal to ro , we shall discuss the role of co below. We can now also be more precise about the local integrability properties O. With the coordinates p and 0, we see of the propagator in the limit T that

f

ad+lp 1 (21r)a+ 1 lipo — E(P)1 .2

=f 42a fdE liPoN(E) /r - E1 *.

liPo—E(P)l
Introducing polar coordinates in the (E,p0) plane, we see that

(4.79)

128

4. The Fermi Surface Problem E

r dr „ 4, = f —v(r) ra o

(4.80)

with v(r) = JO r 21r `-2ir 1-9-N(E cos 0). Since v(0) 0, 4, diverges for all a > 2. The problems with unrenormalized perturbation theory come from graphs where, because of many two-legged insertions, high powers of C get integrated [44]. These contributions diverge for T 0.

4.2.7 The Physical Significance of the Assumptions The assumptions on the interaction mean that it has to decay fast enough in position space. It need not decay exponentially fast; a power law decay with a large enough power suffices to make all derivatives of ii to order ko bounded. Thus the analysis applies to the Hubbard model and all its generalizations obtained from making the interaction spread out a bit. It also applies to screened Coulomb interactions, but not to the Coulomb interaction itself, which has Fourier transform 1/k2 . Screening in Coulomb systems is in general not exponential. Another interesting case is that of interactions with acoustical phonons, which are po-dependent and of the form

p2 i3 (p°/ P) =po 2 ± c2p2 / C

2

(4.81)

where e, the velocity of sound, is a small constant. The i3 given by (4.81) is bounded, but its derivatives are not. Treating such interactions requires a slight generalization of the analysis done here, which is, however, not very difficult. The condition f)(—po, p) = 1. (p 0 , p) implies that the two-point function 12 (w, p) (defined below) is real for co = 0. The condition that E is Ck° is fulfilled for the band functions arising from Schrödinger operators with L2 potentials if the levels do not cross. It is fulfilled in lattice models if the hopping amplitudes decay sufficiently fast with the distance. For lattice models with finite-range hopping, the dispersion relation is even analytic in p. However, the dispersion relation itself gets changed by selfenergy effects, so one needs to do the analysis with assumptions that are as weak as possible. The geometric conditions on the Fermi surface S impose a restriction on the filling factor v = (N)/(2L d). They hold in all cases where the band function has a nondegenerate minimum and where the Fermi energy p is just above this minimum of the band. In that case, the interior of the Fermi surface is strictly convex because the level sets of a strictly convex function are strictly convex. If the quadratic form at the minimum is positive definite, the Fermi surface has positive curvature if p is just above the minimum. A typical example is the discretized Laplacian

4.2 Many-Fermion Systems on a Lattice

129

d

2 E(k) = t 7 E(1 - cowko) -11. E

(4.82)

v=1

m the electron mass, E(k) k2 /2m — p, the disperFor E 0 and t = sion relation of the jellium model, which satisfies the above hypotheses for all p 0 0 if large lki are cut off. For E = 1, one gets the tight-binding dispersion relation with hopping parameter t, which satisfies the above hypotheses if p 0 2td (half-filling). The bounds on the curvature lc of the Fermi surface and the lower bound go for the gradient of E depend on p. The Fermi surfaces of real metals can be much more complicated than the ones discussed here [43]. Part of the analysis can be extended to rather general Fermi surfaces (not necessarily convex or connected) [45]. 2 n 1 1 1

4.2.8 The Role of the Initial Energy Scale We assume that co < 1 and (for convenience in stating some bounds) that 0 is so large that Oco = I , > 6. With the units chosen in a natural way, i.e., with typical bandwidths of electron volts, this corresponds to temperatures T up to 1000 Kelvin if co is of order one, which seems a sufficient temperature range to study conduction in crystals. However, co is not an arbitrary parameter, and, as discussed in the following, it may need to be chosen rather small. If Co were equal to zero, the renormalization group flow for the effective action g(t) defined in (4.72) would have the initial condition g (0) = —AV. However, with our definition of Ct, Co 0 0, so g (0) differs from the original interaction. It already contains the integral over fields with momenta away from the Fermi surface. The focus on the fields with momenta near to S was motivated by the observation in (4.80) that singularities only come from the singularity on the Fermi surface. More generally, for the behaviour at long distances, only the behaviour near to S, i.e. at low energies is important. The vagueness of the notions "long distance" and "low energies" is evident. For a complete analysis of the system, it is important to be more precise about these issues — energy scales correspond to temperature scales, and values of the coupling constant for which a certain behaviour is proven also depends on these energy scales via constants. These issues axe, in the present setup, connected to the choice of the initial energy scale co. The above assumptions on S and on the behaviour of E on S are stable in the sense that the same conditions hold for the level sets S, = { p E B : E (p) = e} , if c < co and co is chosen small enough (depending on ro and go). The renormalization group flow for t > 0 takes place at energies ct < co . Although the integration over fields with energies > Eo cannot produce any singularities, it does determine the initial condition g(0) for the flow. This initial interaction may differ substantially from the original interaction of the model. For instance, even if the original interaction is repulsive, g(0) may have attractive parts and therefore induce superconductivity

130

4. The Fermi Surface Problem

at low temperatures. This is the basis for the Kohn-Luttinger effect; see the discussion in Sect. 4.5.4. If one gets near to half-filling in the Hubbard model, the scale co becomes smaller and smaller, so that the analysis done here applies only in lower and lower energy ranges, and the initial condition becomes more and more important. An analysis of the effects of approaching = 0 is possible, but will not be given here. Some heuristic remarks will follow in Sect. 4.8.

4.3 The Renormalization Group Differential Equation In this section we derive the RGDE for fermionic models. The special setup of the previous sections is not necessary: the RG equation can be formulated more generally on any finite-dimensional Grassmann algebra.

4.3.1 The Effective Action be a finite set, for a function X F(X) from to any linear space let fr dX F(X) = Er ExEr F(X), where Er > 0 is a constant, let (5(X, X') = er -1 (5xx , , and define the bilinear form (f, g) = fr dX f(X)g(X). Let A be the finite-dimensional Grassmann algebra generated by the generators (5A 8, 0-‘) ; recall that the fermionic = E -1 1 (0(X), X(X)777(X))xEr and let -7fryc derivatives anticommute. For t> 0 let C't be an invertible, antisymrnetric linear operator acting on functions defined on F, i.e. (Ct f)(X) = dXC t (X, X') f (X I ) with Let

Ct (X I , X) = —Ct (X, X').

(4.83)

Let C't be continuously differentiable in t; denote 49,4*-' = Ô. Let clfict be the linear functional (Grassmann Gaussian measure) with characteristic function

f

dpc, (0) e (n,r = e l(n, ct n)r

(4.84)

The integrals of arbitrary monomials are obtained from this formula by taking derivatives with respect to 71. The measure is normalized: f clitc t (0) =1. Let V(7,b) E A be an element of the even subalgebra and have no constant = AV(0). The effective action at t> 0 is part, A E C, and

mop)

g (t,) = log f dp ct (x)

(4.85)

Because the measure is normalized, eg(t4) = 1 for A = O. By the nilpotency of the Grassmann variables, eg (t ' ll') is a polynomial in A (the degree of which grows with Thus 0,0) is analytic in A for I AI < Ao (F).

in.

4.3 The Renormalization Group Differential Equation

131

4.3.2 The RG Equation The RGDE is obtained by taking a derivative with respect to t of the defining equation (4.85) of the effective action. The Boltzmann factor eG (t M is given by a convolution of the initial Boltzmann factor eG(°,0) with a Gaussian measure, just as the solution to a heat equation is the convolution of the initial function with the heat kernel. And indeed, the differential equation satisfied by the Boltzmann factor is simply the associated heat equation. Proposition 4.3. Let

1

f dX fdX'

F Then

r

6

scx)

a

Ct(X, X')

(4.86)

e go "0 )• = , Ûte go, ,,b)

(4.87)

eg(t 'lb) = eAct eg(°M.

(4.88)

`9i

a

and If g(0,0) is an element of the even subalgebra, then for all t > 0, 0, 0) is an element of the even subalge bra, and it satisfies the renormalization group equation

—a ol o = .Act g (t, 0) + 21 at

coop) A sg(t op)) SO

7

Lit

SO

.

(4.89)

Proof. For any F(0) E A, define F(4) by replacing every factor 11)(X) by in the polynomial expression for F. Then F(0) -- [F(h) e (no,b)r int---_o T( R-T60 (the derivatives also generate a finite-dimensional Grassmann algebra, so the expansion for F terminates at some power). Since Grassmann integration is a continuous operation, and by (4.84),

pr,

e g(t ' 0)

= [eg(° '* )

f dpot (x) e(71 ')c±'b)11 n=o

= [eG(0,*) e i(n,co)r e (no,b)ri i n.o For any formal power series f (z) =

(4.90)

E Axk,

f (ict) e ( ") r = f ( 21 (Th Ct TI)r)

(4.91)

so e I (7) , C' n)r+("b)r = et e (no,b ) r. Since .Act is bilinear in the derivatives, it commutes with all factors that depend only on 77 and can be taken out in front in (4.90). This implies (4.88). Since Lic, also commutes with (4.87) follows.

132

4. The Fermi Surface Problem

If g(0, 0) is an element of the even subalgebra, the same holds for g(t, by (4.88), since every application of ,Act removes two fields. Thus performing the derivatives with respect to 0 gives (4.89). • Let the Grassmann polynomials

gr(t,o)

be defined by

00

= E Argr (t) 0).

g(t,

(4.92)

r=1

Assume that C = Ct exists and let Dt = C Ct so that Dt is the covariance of the unintegrated fields. Again, we expand in the Wick ordered monomials —

f2Dt (1,b(X1)

. . . (Xp )) =

'

t 11)(X 1 ) ...

(Xp ).

ôt = —ii. (4.93)

They form a basis of the Grassmann algebra because the relation (4.93) can simply be inverted by multiplying by eApt , and they inherit the antisymmetry from the ordinary monomials: f2D,( 1P(Xx(1)) . • •

ti)(X/r(p)))

= E(7 )flpt(IP(Xi) • • • 1,b(Xp)).

The polynomial Ç7. (t, 0) has the representation (here fh(r)

gr (t)

=E

f d;_v_ Ginf(t

f2D,

X = (XI).

(11 cx0) .

(4.94) , Xn,)) (4.95)

k=1

m=0

where G mr (t I X) is the connected, amputated, Wick-ordered m-point Green function. Using Wick ordered monomials provides even greater simplifications for the fermionic models than for the 04 case. The Gm ,.(t I X 1 , ... ,X„,,,) are assumed to be totally antisymmetric, that Xir ( m )) = E (71- )G mr (t I X1 • • • X m) is, for all 7r E Sm, Gmr (t I X ir (1) because any part of G that is not antisymmetric would cancel in (4.95). Application of to (4.95) gives a sum of two terms since two factors depend on t. By (4.93) and Ôt = —be, the derivative of the Wick ordered monomial is if2D tt t (11k 11) (X0) = ZiCt f2D,(Ilko(x0). This implies that the term linear in g drops out of (4.89) by Wick ordering with respect to Dt , and (4.89) now reads

g

m(r)

f

,opt (11

m=o r,n

k=1

E

a G mr (t _ at

I X) = — Q 7.(t,t1)) 2

(4.96)

where Q,.(t, 0) is defined by (sg(t,11))

(50

_o

,

00

6g(toP)) = E Ar a7.(t ) 1,b). À.1

(4.97)

4.3 The Renormalization Group Differential Equation

133

Being an element of the Grassmann algebra, Qr (t,ip) has the representation

f

771 ( f )

Qr(t, 0) = E

dX_ Qinr(t

Ix) f2Dt

(

m=0 Tn,

Because g is in the even subalgebra and Ot =

H:, 11)(xk))

.

(4.98)

-A,

Qr(t,o) = -2-1 [((t,x),Ottip- g(top))] x=11, = — 12

(4.99)

rfitg(t,x)g(t,o)] x=,,b

- L -

with Ht = ( A7, D t 4). Upon inserting the expansion of g, we have to reorder expressions d the form m n 1. (4.100) P = ---Htf)D, flipt (Htp(m)

(11x(xo) k=1

i=1

By definition (4.93) of the Wick ordered monomials, n _A(0) 1 ,_ r-r m P = - -2 lite Dt il X(Xk) e D t H ip(Y1).

(4.101)

/=1

k=1

Here the superscripts on the Laplacians indicate with respect to which variables the derivatives are taken. All differential operators are bilinear, hence commute with one another. Moreover, ,6 A ( ) commutes with all polynomials in x. Thus 7, =

1 —

m

n

A 00_,6 (0) . D t Ht 11 X(Xk) e Dt

HIP

k=1

Since

(

(4.102)

Yi ) .

/=1

— ,4') = — ,6 V')

(4.103)

where 6 — Vi ± Vri, 1 Dt(4 +

A(x0P) _ 1 ( 6 Di

we get 1 _(x') ( gt Ht) Dt - e

m

(4.104) n

11 x (x,)

H

k=1

1=1

V) (11) -

(4.105)

Because

II x(xo

kEK

Howoi tEL

= f2Dt (Hltock) H V) ( Yi) ) x=lb

kEK

tEL

(4.106) because e — D t is and a linear differential operator, it now suffices to calculate the action of ellt on the monomial of V) and x. A cx,Ab)

134

4. The Fermi Surface Problem

Lemma 4.4. m

n

ell' rr 11 X(Xk) k=1

H V; ( Yi ) =

E(_1)m8+ 3 4.E1)

1=1

s>0

E

v(A)v(B)

ACm,BCn

Itil=1B1=a

detAB(Do

x(X) pflA

II 0(yq).

(4.107 )

qflB

Here m = {1, ... ,m}, 8

det AB (Dt ) = E e(7,)

H Dt(Xak,n, r(k ) ),

(4.108)

k=1

irES3

where e(7r) is the sign of the permutation 7r, and v(0) = 1, and for A = { a i ,... ,a,} with a l < < a„

v(A) = (-1)Ek

(4.109)

ak.

Proof. We abbreviate xk = x(Xk), h = 0(Y/), and 15k/ = Dt (Xk, Y1 ). Then m n

Ht

=

EE a

a

(4.110)

—• 19 01

k=1 1=1 aXk

The expansion of the exponential gives e

t=

E

H X°a 7=1 —a •

SCmXn(k,l)ES

(4.111)

alP1

k

A set S with s > 1 elements contributing a nonzero term to this sum is of the form S = f(ak,b7r(k)) k E {1, , as } c m with s }} , where A = al, al < < as and B = < bs , and where 7r is a ,b8 } C n with b1 < permutation of {1, , s}. The map S (A, B, 7r) is bijective. Thus

=

E 8>1

a A H uXai.

01/4 1r(k)

ACm,BCn rES s k=1

(4.112)

IA1=1B1=3

and we need to calculate its action on xi • • • Xml,b1 • • • 1,bn. All sign factors now come from reordering the Grossmann monomials: 8

H

8

S

8

( 1)8(8-10

TT

eXak athir(k)

TT

8 eXak

k=1

1=1

k=1

8 "bw(t)

s

=

s a IA axa k II aiPbt '

( ..... 1) 8(8-1)/2, (70 H

k=1

1=1

8-1 = (-1) 8(8-1)12 6(70

11

k=0

s-1 A

a

as—k

IT .91148_1 a

1=0

(4.113)

4.3 The Renormalization Group Differential Equation

and

8

Xi • • • Xm = v(A)(-1)8(8+)/2

xk

xa.

135

(4.114)

kA

k-=1

(the factors in the last product are ordered according to increasing k), and similarly for the 0. Commuting the product of derivatives with respect to with x i xm gives a factor (-1) 8m. There are no additional sign factors. • Lemma 4.4 and a comparison of the coefficients give the component form of the RGE

1 a yiGmr(t I Xi,— Xm) = AtlIQmr(t I Xi, .. Xm) where Am is the antisymmetrization operator

(Amf)(Xi, • and

1 E ,xm)= m! irES,n

6(7) f (X70), • • • Xir(n)),

Qmr (t I X) is given by the following expression.

Proposition 4.5. Qmi (t I X)

= 0, and for r > 2,

= f dtz, f dV f dW (--aat det Di(I.) (V ,W))

Q mr (t I X)

Ti

Gm i ri (t

I

ri

X1, V) Gm2r2(t

I it, x2),

(4.117)

where f diz, stands for the sum

f dtzmr F =

r17r2> 1

E

mi,r2, m2, i)

(4.118)

E.A11.r 1 r 2 n

with positive weights li,m1tn2 i = (T) ( 77), so that dizm, is a positive > 1, 1 < m1 < measure. Mrir2m is the set of (m i ,m 2 ,i) such that i _ rri(ri), 1 < m2 < rri(r2), mi + m 2 = m + 2i, and m 1 and m2 are even. Xi = (X1, ... ,X mi —j), X2 = (Xmi—i-1-1, • • • , Xm), V = ( 171, • • • ,Vi), W = (W1, • • • , Wi), 1I = (Wi, • • • WO, and 130i) (V,W) is the i x i matrix

(12T) (V , W))

ki =

Dt (Vk

One important feature of (4.117) that distinguishes it from bosons is that the determinant of the propagators appears in this equation. The Gram bound for this determinant improves the combinatorics by a factorial (see [53, 541).

136

4.

The Fermi Surface Problem

4.3.3 The Component RGE in Fourier Space Let A= TxA and A* = M„, x A*, as given in Sect. 4.2.4. Let I' = A x {-1, 1} x {1, 2} and f* = A* x {-1, 1} x {1, 2}. The Fourier transforms of the Gmr (t I -) are m

dmr(t

I Pi, • • • , Pm)

= f

II dzk e-i' x 'c

Gmr (t

I Xi ,— ,Xm.).

(4.119)

Am k=1

For K = (k, a, j) E I' let - K = (-k, a, 3 - j), and define br,(K, IC) = 5A* (k, ki)(5„, (5ii, . The Fourier transform of the propagator Dt is of the form 15(K, IC) = 61-,..(K, , K') b t (K 1 ), where b(K) = (-1)iDt((-1)ik).

(4.120)

This combination of signs implies that Dt(X,Y) = -D t (Y, X). The propagator Dt for the many-fermion system was given in Sect. 4.2.5. The Fourier transform of Q mr (t I .) is Qmf(t i ri, • • • ,Pm) = i

f dkmr f cl.K 1 ...d1Ci (

81)t(Ki)) et f

Ti rI Di Ki ) (

(4.121)

j=-2 Omiri(t

I -P(1) , Ki, • • • ,Ki) Om2r2(t I^ ' Ki, • • • ,r- K1,P (2) )•

Here f diimr = f dismrii!, P (1) = (P1,- - ,Pmi-i), and P(2) = (Pmi-i+i, ... mi +m2). By translation invariance in space x and time T,

6 mr( t I P) = 6A. (Pi + • • • + Pm,0)Imr(t I f)

(4.122)

with a totally antisymmetric function /mr (t I P) of P E Frn that satisfies

(1,...,1) • Vimr (t I P) = 0. (4.123) A priori, (4.122) only implies the existence of a function inar (t I P), defined for those P = (P1, . . • , P„,,) for which pi + ... + pm = 0. However, since

H = {Pi, • • • 'Pm : Pi + • • • + Pm = 0}. (4.124) is a linear subspace of (A*)m, one can simply extend f to a function on all space by defining I = f o Hi with //H the projection to the subspace H. Since //1/ is symmetric in all its arguments, I is totally antisymmetric. The product of the two deltas in (4.121) can be combined to cancel the delta in the relation between d and I, and to remove the integration over kl . Thus k1 is fixed as

k1 = —(k2 + . . . + ki + pi + . . . + Pmi i+1 ) —

(4.125)

4.4 Power Counting for Skeletons

137

and the RGE in Fourier space is &I,,r (t I P) = lAmanr (t I P), with

il Am

,,,,,.(t I P) =

f dii,,,, f dK2 . • • cliCi 1,,i1(—D.t(Ki)) ,E

1:

II NIC 8 /min (t I P (1) , K1, • • •

(4.126)

)

s=2 Itn 2 r2 (t

I"' Ki, • • • , — Ki , P (2) )

with IC1 = (ki , cri,ii), where k1 is given by (4.125). 4.4 Power Counting for Skeletons

We now proceed to analyze the field theory by the RGE in perturbation theory. To this end, we take the thermodynamic limit. The existence of the thermodynamic limit, as well as that of the limit n7. —> oo, is shown in Appendix B.7. There is no problem with singularities as long as the temperature is positive; taking L oo simplifies some of the technical arguments because the discrete momentum sums of the finite volume are replaced by integrals. 4.4.1 Bounds for the Infinite Volume Propagator -

In the thermodynamic limit, the spatial part p of momentum becomes continuous, p E B, and the set of Matsubara frequencies becomes M = { (2n +1) : n E Z}. It is convenient to take (po, p) E Rx5 and put all the 0-dependence into the integrand. To this end, we define the step function cos : R ME by

7 (2n + 1) if po E (271- n , 2— 71- (n + 1)]. w(po ) = — 0 0 0 For any continuous and integrable function f,

f —d2P: Awfoo))

= T31

E

f (w)

(4.127)

(4.128)

•

wEm

R

Moreover, sup

lcdfl(p0)

7r — po I = —

and

inf P (R)) I

PoER

0

PoER

=E 18

(4.129) '

> 1, and ws (—pi) ) = —ws (po) holds Lebesgue-almost everywhere, so that in integrals like (4.128), wo can be treated as an antisymmetric function. With this, the propagator now reads Di (p) = Ct(cdp(po),E(P)) with ICds (A ) I

Ct(z, Y) =

1 xi ( fT.2 (z2 + y2 )) y

IX —

(4.130)

138

4. The Fermi Surface Problem

(and et = co e—t ). In infinite volume, the restriction on the spatial part p of momentum improves the bounds on the propagator over the finite-volume ones used in Appendix B.7. in t and Cko in p. If t > log g-92, then Dt(p) = 0 for all p ER x B. For all multiindices a ENg with lai 0 such that

Lemma 4.6. Di is bounded, C

l

(4.131)

ac' 154)1 < Baft—i—lal 1 (liw(P0) — E(P)i ft)

Baft—i—lal 1 (1w00)0)1

ft) 1 (1E( 3)1

ft) •

Bo =4. For t < log g?,

f talaczA(P)1—< BafTial .(1E(p)i<

Et)

,

(4.132)

R

f

dd + l p

(....,\ I na L itn VI)) 1 -

(21r)d+1 lu

< 2 ./1 /3a 6t1— l a l ,

(4.133)

Rx.6

and

f

dd + 1 (2,r)d-ri I

(4.134)

IMP ) I < &li ft.

Rx./3

Proof. For Dt(p) to be nonzero, 1(.0430)1 < ft must hold. By (4.129), Dt = 0 for t > log . Since

L5(p) = —24-2 (iwo (po) + E(P)) Xi (ft 2 PO (PO

) 2

+ E(p) 2 ))

(4.135)

and Ilxill oo < 2 , 1 154)1 Clicdo(Pa) — E(P)I < et) . Thus Bo = 4. Derivatives with respect to p can act on E(p) or on x'1 , in which case they produce factors bounded by ft lIVE(P)1, so there is B a such that (4.131) holds. Inserting (4.131) into the integral in (4.132) gives

1'154) 1 <

dl f ÷: R

Bc61-1 ' 11(1 E (01 ft ) 43

with M13

= 7-11

In E Z: 12n+ II<

2 < —ft, 7r

—

which proves (4.132). This gives for the integral in (4.133)

J i

Rx B

± p ddl

1 n ii 1

I

\

(271-)d-F1 UaLjt kP 1

T.,

"

Daft ' a '

1E( < ft ) ddP 11 "(01 f (27r)d 13

(4.138)

4.4 Power Counting for Skeletons

139

For t > 0, ft G fo, so by a change of coordinates in the integral,

f

Et

ddp

(270d 11

(1E(P)I

ft)

f dp f d0J(p,0) = 2Jift

(4.139)

—Et

13

which implies (4.133). (4.134) follows by integration over t.

•

The geometrical interpretation of the integral in (4.139) is that of the volume of a d-dimensional neighbourhood 1?,(€t) = { p E 8 : 1E(p) 1 < ft } of thickness et of the Fermi surface, and the power counting bound is obtained by weighting the smallness of that volume — as well as that of the according 'volume' Ms of the frequencies, (4.137) — against the largeness of the function itself in that region. Careful estimates of more complicated volumes will be important for the analysis of the model.

4.4.2 Sup Norm Estimates We first prove a power counting bound in the sup norm I • 1 0 , where, for Fm : (Fim C, (4.140) 'Fmk) = s 11 p*),, IF(f)i• PE ( F Obviously, lAmFmlo < 'Fmk) . By Lemma 4.6,

1Qtnr(t)10 <

4 f dkp mrlItIli i— 1 11Alloolirmiri(t)10 lim2r2 ( t )10

G 128J1

f

dkmr ( 8 .11 et) " limiri (010 1 1m2r2 (t)10. (4.141)

Integrating the RGE, we have

t Ilmr(t)10

limr( 0)10 ± f ds Am Q,(s)

limr(0)1 0 +

1

o t

f ds IQmr(s)lo-

o (4.142)

o We shall now give bounds for the scaling behaviour of the /„(t), i.e. of their behaviour as functions of t. It is instructive to look at the case r = 2 first. Assuming that only a first order four-fermion interaction is present at t = 0, we have 1/, (0)1 0 = 6, 1 6m4y with some constant y > 0, whcih is finite because the interaction is bounded in momentum space, i.e. short range in position space). The restriction r1 + r2 = 2 in f dk, implies r1 = r2 = 1, so

140

4. The Fermi Surface Problem

I /7712(t) lo < 64J1

Jf

ds

f dk,(8Jift) i 2

V26 m14 6m24 •

(4.143)

o

The restriction m i +m2 = m+2i in f dk, leaves only the possibilities m = 6 and i = 1, m = 4 and i = 2, and m = 2 and i = 3. For m > 8, itn2 (t) = 0 for all t (in other words, 771(2) = 6). The nontrivial cases correspond to the graphs shown in Fig. 3.9. Inserting all the numerical constants in the definition of f dkmr and doing the integrals over s, we get

t 2 E0—l et 1 162 (010 < 2 (8v) 2 61 f ds es < 2(8v) o t

1142(01 0 < (96v) 2

f ds = (96v) 2 t

(4.144)

o t

1122(0 0 < (384J1v) 2 E0

f ds e— s < (384J10 2 f0. o

All three bounds give the correct behaviour of 1m2 (t) as a function of t; this can be seen for m = 6 from the explicit expression 1162(t)1 = ICt (p)1 (where p is some combination of the external momenta); for m <4, we shall discuss it in detail below. A further analysis of the tree graph (i = 1) contributions to 4„ suggests that for in > 6, 4, — e ( m--4) , that Li r — t' , and 12r — const . However, these bounds hold only for the skeleton functions, defined as follows. Definition 4.7. The skeleton functions fm,.(t) are defined as the solutions where in mf (t), f dkinr is replaced of the skeleton RGE Mm r (t) = 07,(t), by the weighted sum f dkmr in which all terms with m i = 2 or m2 = 2 are left out.

In the graphical expansion obtained by iterating the RG equation for the skeleton functions, the two-legged insertions that would require renormalization do not occur. Theorem 4.8. Assume that lImr(0)1 0 < a2,),. and let Imr (t) be the solution of the skeleton RGE. Then, for all m and r there are constants am ,. such that for all t > 0

limr(t)1 o <{ .12r (0 converges as t

a,.(1 + t)r_1el(m-4) a2r

oo, uniformly in 0.

ifm>4 if m = 2.

(4.145)

4.4 Power Counting for Skeletons

141

Proof. Induction on r, with (4.145) as the inductive hypothesis. For r = 1, there is nothing to prove because 0, 4 1 = O. Let r > 2. The inductive hypothesis applies to both factors I/mk rk leI on the RHS of the RGE. Since mi = 2 and m2 = 2 are excluded from the sum in f dkmr , we can insert the upper line of (4.145) for both factors. Because m1 + m2 = in + 2i, the exponent becomes independent of i, mi , and m2, and we get iQmr(t)10 with

_< amr( 1 + t) r-20(m-4)

1 Ni amr = J1 ez f Cliçrnr (8 th C) O a rn 1 rl aM2 1.2 •

(4.146)

(4.147)

All that remains is to use Ifmr(t)lo < Ifmr ( 0 )10 + 1 fot ds 10mr(5)10 and to do the s-integral. For in > 6, in — 4 > 0, so t

t

fo ds I 0mr(s)lo

amr

f ds (1 + s)r_ 2 ei (m -4) o

2a mr (1 + or-2 e (m-4) . 771 - 4'

<

(4.148)

The induction step for m > 6 is thus completed by noting that 1 < (1 + t) r-2 < (1 + t) r - i and defining a, recursively as am,. = aZ. + a,/(m— 4). If m =-4,

t

t

fo

ds 104r(s)lo <

a4r

o

a4r ds (1 + ).9r-2 < (1 + t)r-1 . (4.149) r—1

With the definition a4r = aV) + a4,./2(r — 1), (4.145) holds for m = 4. If m = 2,

t

J o

t ds 12r(S)lo

a2 r ds (1 o t < a 2r e ds (1 + o 00 < a2r e

(4.150)

da ar -2 e—u _ < a2re(r — 2)! o

and the appropriate recursive definition of a21. finishes the inductive step for m = 2 as well. In particular, 121. (t) is bounded uniformly in 0. By (4.146), I 8tf ' /2 (t)lo < I Q2r a 00. • 0 as t oo, so I2r (t) converges s t (t) lo

142

4. The Fermi Surface Problem

No attempt was made in the above proof to optimize the recursion for the coefficients, which would be necessary for a more careful discussion of the r dependence of the amf . However, the last bound (4.150) shows the potential danger of the powers of t: a factor tr produces an r! in the coefficients. If the coefficients grow factorially, the expansion in A is divergent for all A 0, and thus makes sense only as a formal power series. As one sees from r = 2 and by inspection of the above proof, these factors come from the four-point function. We shall investigate in more detail when and why these factors arise, and as we shall see, they are no artefacts — their occurrence is linked to physically important phenomena. If mk = 2 had not been left out on the right hand side of the RGE, the induction step would have failed because e (m- 4 ) = e—t for m = 2, but the two-point function is only bounded by a t-independent constant, which is et larger. Note that the derivative with respect to t of the two-point function vanishes as et, but the function itself does not because the integral starts at zero. Renormalization will change this boundary condition. Theorem 4.8 implies that for getting perturbation expansion termwise finite, it suffices to renormalize the two-point function. The above bounds only imply the convergence of the skeleton two-point function, but not of the m-point correlation functions for m > 4. Convergence of the latter cannot hold in sup norm because these functions indeed have singularities. Convergence can, however, be shown in an L 1 -norm, which implies that they are finite almost everywhere in momentum space. We do this in the next section. 4.4.3 Estimates in Ll Norm To give bounds that have a convergent behaviour also for the m-point functions with m > 6, we introduce a norm in which part of the P are integrated, and the supremum is taken over the remaining ones. For s E {1, , m} and C, let a function Fm : (Elm

IFm I. =

s

sup f H dPq (5(p1 + • • • + Pm)IFm(Pi • • • Pm)i•

max si=.

"

qES

(4.151) Because of the maximum over subsets, IAmFm I = IFm I. Conversely, if Fm satisfies 1Fm(P7)1 = iFm(f)I, where P (Pir(i), • • • , Pir ( m )) for Es (Pi, • • . , Pm ), then I Fm i s =

sup

f &PI .. . dP, (5(pi + . • • + Pm) IFm(E)1.

(4.152)

For s = 1, the norm is bounded by the sup norm,

< 4 sup IFm (P)I = 411mr lo _

(4.153)

4.4 Power Counting for Skeletons

143

(the factor 4 comes from the sum over ui and ii). To estimate the right hand side of the RGE, we reinstall the (5 functions,

to get

Om =

+ .. . + pm ) Qm r (t I 2 1)

i

f dk„„ f dKi ... dKibt (KO

11 Dt (Ki )

(4.154)

j=2 âmiri ( t

I P(1), K ) 61stn2r2 ( t I fs

K, P(2) )-

Here f dkm, = f dismri i!, and dmr(t f 12) = 6(pi + ... + pm )imr (t I P). In the maximum over sets S c {1,... ,

1. (Pq ) qEs is

m}, three

(4.155)

cases can arise:

a subsequence of P(1) . This can happen only if s < mi — 1

because i > 1. 2. same as (i), with 1 replaced by 2. 3. s i of the variables

(Pq) qEs

are arguments of /mi „ and s2 = s — si of

them are arguments of im,. Again, because j? 1, this can happen only if s i <m 1 — 1 and s2 < m2 — 1. In case (i), we use the momentum conservation 6 function in Om2r2 to fix k1 in terms of the other momenta, and bound 4bt(Ki) by its sup norm and im2r2 by lim2r210. By the antisymmetry of /m1r,, its arguments can be permuted such that (Pq ) qEs come first. Integrate over (Pq )qEs. The only factor depending on (Pq) qEs is now /miri . Taking a sup over all remaining momenta in /mini gives a contribution

Ti s) = 11(s < m i ) (

f dknirilDt111 2 14 1Vitiloolimiri(t)18 1 1m2r2(t)l0 (4.156)

where iiptIli = f dK Ibt (K)1. Case (ii) is similar, and by the symmetry off dk, under exchange 1 44 2, the contribution is again bounded by T1 (s). In case (iii), taking suprema after integrating over (Pq ) qE S, we get T2(si,s2) = l(si < mi A s2 < m2)

(4.157)

f d m r liptili z-l iiiitiiiiimirin si gm2r2(t )182 . The crucial difference to (4.156) is that both delta functions are used up in the definition of f • I sk and that therefore Ki is also integrated. Thus the L'-norm of the differentiated propagator appears instead of the sup norm. We thus have

4. The Fermi Surface Problem

144

lAmQmr (t) I „ < 27i (s) + .,,3 max 2 >i T2(si , s2).

(4.158)

By Lemma 4.6, and since B0 = 4,

(s) <

16, _ in(t)18 1 1ra2r2(t)lo ft m

(s < mi) f dk mr (8.1ift)i-1_

(4.159)

and T2 (Si s2)

11(Si <

A

(4.160)

< M2)

f dkmr (8Jift)iIImlrl

(0 181

14

121.2(01 3 2

Thus

< f dkmr (8J1E0) i e—ti max-{01 (s,t),92(8, t)} et1/mr (t) I — 8

(4.161)

with

(S t) =

1( .9 < m l ) *e2t 1-rmiri(t)1 8 11m2r2(t)lo

02 (. t) =

max

(Si

8 1 ,8 2 >1 8 1+ 8 2 =8

(4.162)

< mi A s2 < m2) 1 1 rain(t)1 81 1 1 ra2r2(t)I8 2 .

4.9. Let 11„(0)18 < b.;,° L, for all m, r, and s. For any 0 < a < 1, there are sequences (b,$ ) m ,r,„ and c, (depending on a) such that for all

Theorem

t 2 0, .Ymr (t)

IbMr8(1 e lnr

8

imr (t) converges in I

) t)-1 ei(M-8- 1

• I m to a function

if s < m if

imr

as t

s = m.

(4.163)

co, uniformly in /3.

Proof. Induction on r, with (4.163) as the inductive hypothesis. For r = 1, the statement follows from our hypotheses on the initial interaction /mf (0) because Q mi (t) = 0 for all t. Let r > 2. The inductive hypothesis applies to /miri and 1m2r2 . Let 1 < s < m. By the inductive hypothesis and Theorem 4.8, and because mi < s and mi > 4 and m2 > 4,

(s, t)

j 1,0yb Ji1E0

Similarly, si < mi and s2

(s, t)

< <

miris am2r2( 1 + t)r1-1 +r2-1 ei (m 1-8— a±m2-4)

b M1 r l sam 2 r 2 (1 + tr 2 ei (m—s—a) .

< m2 and the inductive

hypothesis imply that

max bmiri 81 bm2r282 (1 t)r_ 2 e i(m1 - 81 — a+m2 - 52 — (4 — ti

31,.2>i .1+82=8

). max brainsi bm2r282 (1 -I- t r 2 e1("1-8-121

81,82>i

.1+.2=.

(4.165)

4.5 The Four-Point Function

145

Thus

l ti imr (oi s < el (m-8- ') (1 + t )'

2 f dkmr(8J1f0)iMmrs

(4.166)

max bmirisi 1'm2r282 b

(4.167)

with Mmrs

=

max{

—iris am2r2)

:48322>=13

and therefore

11 mr (t)l s
)

2

. (4.168)

If s < m, then m-s-a >1-a > 0, so the integral over t' is bounded by 1a (1 + t) r_2 e (nI-8—a) . Since this quantity is at least one, the induction step is completed for in < s by defining

bmrs = ers + If

s = m, m - s - a f

f

= -a < 0, so the integral over

dt1(1 ty-2e-te

e2

(4.169)

dkmr (8theo) i M m,...

t'

is

dt (1 + e)r-2e-1(1-Ft1)

f

(4.170)

o 00

0

<

f0

La\ r--1 la )

dr T r-2 er i-

2

2)!

and the induction step is completed by defining Cmr

= b(°) mrm+

f

dkm,(8theo)iMmrm

e

(r - 2)! .

(4.171)

For finite 0, convergence is trivial because -41-E = 0 for t > log . For = oo, convergence follows from (4.166) because for s = m, the right hand side of (4.166) vanishes as t oo. •

4.5 The Four-Point Function In this section, we do some explicit calculations for a truncation of the RGDE. This truncation corresponds to the ladder approximation to the four-point function. The justification of this truncation will be discussed in Sect. 4.6.

146

4. The Fermi Surface Problem

4.5.1 Motivation

We have seen in the last section that the four-point function produced factors tr <

(log(fifo)) r (4.172)

in the estimates for all Green functions. The powers tr produced r factorials in the constants in the bounds for the Green functions, for instance in (4.150). If instead, one uses (4.172) to avoid the r factorial in (4.150), one gets bounds that depend on 0. It is very important to find out if these bounds are saturated: suppose that for some P, 0 a a2,r 12,r(tIP)1 .

where a2, r cc 71 const r, then the series tively,

12,r (tiP)1 o with a2,r < const r, the series temperature-dependent region

Er Ar12,r (t1P)

a a'2,r ( log(fifo)) r,

Er )r12

(4.173) diverges. If, alterna(4.174)

(t1P) converges at most in a

I A log (0f0 ) I < const ,

(4.175)

and a similar statement holds for all other Green functions. Both of the above statements are really true for the class of models considered here. The expansion for the skeleton Green functions does not converge uniformly in the temperature, but it converges in a region given by the condition (4.175). The above bounds on the skeleton functions already gave a hint that such behaviour can hold. To really prove this, one needs to control the combinatorics. Such a proof is possible using a technique invented in [55], but it is beyond the scope of this text. On the other hand, the above bounds do not imply that the log 0 is really there. In this and the next section, we motivate why the logarithm actually occurs. In this section, we study a truncation of the right hand side of the differential equation where all terms with number of internal lines i > 3 are left out. This leads to a differential equation whose solution is a resummation of the particle-particle and particle-hole ladders. These graphs are sometimes called the parquet graphs, so we shall call the four-point function obtained by this truncation the parquet four-point function. Under the assumptions we made on the Fermi surface, in particular because of the positive curvature, the particle-particle ladders dominate this flow, and they have the above logarithmic behaviour when the sum of the two ingoing momenta adds up to zero. The terms with i > 3 will be treated in the next section. We show that the solution to the truncation of the RGE where the i = 2 terms are left out in the equation for the four-point function is bounded uniformly in the temperature,

4.5 The Four-Point Function

147

that is, there are no factors tr or (log ) 3 ) r in that case. Thus indeed, the ladder term i = 2 in the RGE is the only cause of high powers of log $. The "overlapping loop" method used to prove this applies very generally and will afterwards be of use for bounding derivatives of the selfenergy as well. 4.5.2 The Parquet Four-Point Function We now discuss the truncation of the flow obtained by keeping only the i = 2 term on the RHS of the RGE for the skeleton four-point function. In Q4 r , m 1 + m2 = 4 + 2i, so j? + m2 — 4) > 2 since the skeleton condition has removed m 1 = 2 and m2 = 2. Let 0 ,4,f,2 be the i = 2 term in this sum; it corresponds to the 'bubble' graph drawn in Fig. 4.1.

Fig. 4.1. The graph corresponding to

Q4,T,2

More explicitly, Q4, r,2 is given by Ch,r,2(t I Pi, • • • , P4) = -1 A4

E

[144

E

ri-Fr2=--r

14,ri (t I Pi, P2 ) Ift ,K) with K = (k,ii, cri) and Ch,r,>3(t

=

f ,r2 (t

( —Pi — P2

I2 1 ) = Q4,r(t

dk .b(K)

I

(K t)

(4.176)

P37 P4)]

— ki2,u2). Let 2 1 ) Ch,r,2(t

El

(4.177)

be the contribution from all terms where at least i > 3 lines connect the two vertices in Q4, r . Although the graph in Fig. 4.1 looks like a bubble graph, the iterative solution to the according differential equation

a at

—14 (t .12) = 'r

2

r 2(t

"

21 )

(4.178)

generates not just the ladder or the RPA graphs, but many more arising from the combinations of the two types. We call the truncated flow given by (4.178) the parquet flow.

148

4. The Fermi Surface Problem

To make the above statement about graphs clearer, we now switch to the more conventional representation in terms of 1P and IT) (we also resum the perturbation series over r and denote F = Er )t'. /4,1. , where /4 0. is the solution to the truncated RGE (4.178)). Since the j index in P = (p, j) fixes whether V) or 17) appears, the four-point function with two incoming fermions and two outgoing ones can be defined by

a,

(4.179) (Pi, cri), (P2, (72), (P3,u3), (No74)) = F(t I (— Pi, ui, 1 ), (—P2, 62, 1 ), (P3,(73, 2 ), (P4, 0-4, 2 ))• The minus signs in front of p1 and p2 mean that the momenta are now counted positive when they are outgoing, so that in f, momentum conservation reads pi + P2 = P3 + P4. The function f in (4.179) fixes F uniquely because of the antisymmetry A 4 F = F. A natural graphical representation of f is drawn in .f(t i

Fig. 4.2.

Fig. 4.2. The vertex corresponding to f In terms of f, (4.178) reads, with af —

at

G=

(pk,uk),

(ti 6 ' ...,6) = °PP (t I 61. • • 16 ) - o ph (t I 6 , 6 , 6 , 6) + o ph (t I

(4.180) 1, 6,6,4.3),

where °pp(tI 6 , • • • , 6) = 12

f dk [1 (Dt(k)Dt(Pi + P2 - k))]

E f(t I 6,6, (p 1 +P2 - k , T2), ( 1c , TO)

(4.181)

Ti ,T2

f (t I (k, TO , (1311 + P2 and Oph(t

I

6,6,6,6) = 24 f

dk

[1 (Dt(k)Dt(P2 — p3 + ic))]

E f(t I (k, To, 6, 6, (j 2 - p3 + k , T2 ) ) Ti ,T2

f (t I 6, (P2 -P3 + k, T2 ), (k, TO, 6)

(4.182)

4.5 The Four-Point Function

149

and Dt is the scaled propagator Dt = Ct , with Ct given by (4.130). There are two particle-hole terms because the function f has to be antisymmetric under exchange of 6 and 6, and under exchange of 6 and 6. The difference of the two 4Pph functions has these properties. (4.180) has the graphical interpretation shown in Fig. 4.3.

)0(

+

-

Fig. 4.3. The graphs in the parquet flow equation

We now discuss more explicitly the particle-particle and particle-hole flow and show how the particle-particle flow leads to the restriction IAI log f3 < const and why the particle-hole flow does not lead to singularities if the Fermi surface has positive curvature. The mixing of the two flows will be discussed after that.

4.5.3 The One Loop Volume Bound -

The right hand side of the parquet equation contains the functions

Let

±k). bi(±) (q) = 8i f dk Dt (k)D t (q

(4.183)

t 13 ±) (q) = — f ds b(8±) (q), o

(4.184)

then

./3 ±) (q) = f dk (Do (k)D0 (q ±k)— Dt (k)D t (q ±k)) = .13 1) ( — q). (4.185) ./3 --) is called the particle-particle bubble, and BP-) the particle-hole bubble. The power counting applied in Sect. 4.4 gives the bound ie) (q)i

2 1Ibtiloo iiptili < 2J 1 B.

(4.186)

This bound is independent of q. A better bound can be obtained as follows. We use that D vanishes for t > log(Ofo to write log(f/f0)

D(k) = f t

de bt, (k),

(4.187)

150

4. The Fermi Surface Problem

to get log(i30) lb(q) I

<2

f

de

f

dk

Iiiti(k) bt(q

(4.188)

± k)I.

t We now use the pointwise bound (4.131) of Lemma 4.6, bound the indicator function II (we ± wo (kol < ft') < 1, and bound the frequency sum with the restriction lwl < Et, by et, (see (4.137)), to get log(Øo) lb(±) (q)I

< 2131,ft-1 f

de e(q, t,

e)

(4.189)

t with

Vi(±) (q,t,e) =

f (27r)' 1 (IE(k)I

Et') ' ( gg ± k) I 5_ Et) .

(4.190)

B

Let R,„ be the neighbourhood of the Fermi surface defined by

= {13 E B :1 (13)1 -0

(4.191)

ft } ,

then W) (ci, t, t') is the intersection volume

W) (q, t, e) = (27r) —d vold (7?,(Eti) n

(7-get ) T q))

(4.192)

of 1, and the translate of R(ft ) by Tg. By the symmetry E( p) = E(p), —

14v) (q, t, t l ) is independent of y E { 1, 1}. -

Lemma 4.10. Let licill B be the distance of q to 0 on the torus B, and let t' > t. Then there is a constant Qo > 0 such that for all q E B, if IlcillB N/ 14±) (q, t, t') < Qo ft , { 1 -s,X otherwise.

(4.193)

The geometric idea in the proof of this lemma is that, for the class of Fermi surfaces considered here, the intersection of the Fermi surface and its translate by a vector q is transversal for almost all q, in which case the intersection volume is bounded by the product of the thicknesses of the neighbourhoods, ftft, < ft, -V. . For q = 0, both neighbourhoods lie on top of each other, and no gain is possible. For q E 28 the intersection is tangential, but the curvature of S provides a gain -N,X. The various cases are drawn in Fig. 4.4. We prove Lemma 4.10 for a spherical Fermi surface in Appendix B.8.1, and also discuss the generalization to nonspherical Fermi surfaces there. We now turn back to the analysis of the four-point function. The volume bound implies the following bound.

4.5 The

Four-Point Function

(a)

151

(c)

Fig. 4.4. The intersections of a shell around the Fermi surface with its translate by a vector q. (a) transversal (13) tangential, with q near to 28 (e) q P.: 0 -

Corollary 4.11. Let B(q) be as in

(4.184). For all qo E R, all q

E

8,

and all t> 0,

W ) (q0,q) < 2BgQ0 (2 + log( iti-)) ik .

(4.194)

Proof. By (4.189) and Lemma 4.10, log(13co)

Ib1±) (q)I < 2B1,Q0 4-1

f

de et,

t

The integral over t' is bounded by

1 141) MI

Et ,

11 L \X

if

Ilcilly -,/

otherwise.

(4.195)

so

..< 213gQ0 { v 1. _ if 11c111 .6 ft otherwise.

(4.196)

To get a bound for fi ±) , we now have to integrate the right hand side of (4.196) over t. For t < log( a-j), the bound for the integra.nd is constant, so the integral gives this constant times a factor log( ic lt), 2). For t . >_ log —i

114IB '

the integral is bounded by a constant times Lc° dt e—t/ 2 <2. Thus the only place where 13(1) (q) can diverge as t3 investigate if that is the case.

•

oa is q = 0. We now

4.5.4 The Particle—Particle Flow

Let us look first at the modification of (4.178) where the 'Pp h terms on the right hand side are dropped, i.e. -85f = IPpp . For an initial four-point function 1(0) that is local in position space, one can solve this equation explicitly. That is, the solution with f (016 , • " with some initial coupling constant

= -- A0CŒ1472ecr3 Œ4 Ao, is

5

(4.197)

152

4. The Fermi Surface Problem

Ao

f 416 , • • • , 6) =

1 + 24A0e-) (pi + p2 )

(4.198)

e.cri 0'2 60.30.4 '

where el (q) is the particle-particle bubble defined in (4.184). The minus sign in the initial condition appears because we wrote the Boltzmann factor as eç and the initial Boltzmann factor as e — Al ) . Thus Ao > 0 corresponds to a repulsive interaction, and Ao < 0 to an attractive one. The 6 symbols make f antisymmetric under exchange of i and 6, and 6 and 4 4. The factor 24 and the extra sign are due to 7 — 1-1,72 ET1T2E 7271 = —2. The solution is so simple because the momentum dependence of f (t16 , ... , ,i) is only via the sum of the ingoing momenta pi + /32 . Thus integrating this truncated equation amounts to a resummation of the particle-particle ladders. For oo, D(k) — > 0, so

t

B (-) (q) = lim B(q) = t-400

f

dk Do(k)D0(q - k).

(4.199)

For T > 0, B( —) (0) is finite, but proportional to log 0: up to a term coming from the ultraviolet cutoff that is bounded uniformly in the temperature, we have, by the symmetry E(—k) = E(k),

B (—) (0) =

h. E

f

dk

1 1 iw — E(k) —iw — E(—k)

(4)EmlE(k)15_c0 fO =

f

dE N(E) 113

_EC)

E wEm

1 2 ±E2

(4.200)

with the density of states N(E) as in (4.77). Because E w -2 converges we can put in a factor ei" and get , by (4.68), 1

1 V"`

7

to EM

w2 +E2

= lim 1 N--. 40 /3

=

•(--'d

wEM

eiEw LiJ2 + E2

1 ( ei" + 40 o E -2E ic,, E wEm 1 ( 1 —2E 1 + OE 1 b OE 2E tan " 2 '

fin, i

-

=

--=

—

icd

—E (4.201)

SO fo

B (—) (0) =

fE

dE

—

0

tanh

q N(E) =

qQ-

f

136 0 2

k il tanhz

NC1). -

(4.202)

4.5 The Four-Point Function

153

As shown in Sect. 4.2.6, N(E) is Ck0 -1 around zero, so a Taylor expansion gives for large 0

1 0E0 B (-) (0) = - N(0) log( — ) + 0(1) 2 2

(4.203)

(if ko > 3 then one can use Taylor expansion to second order; the first order term then cancels because the integrand is odd, and one gets 0(A) instead of 0(1)). N(0) is the density of states at the Fermi surface. For a repulsive interaction (Ao > 0), the function f remains bounded and vanishes for 0 -> co at pi +p2 = 0 ("asymptotic freedom": a running coupling constant defined by

A(t) =

A0

(4.204)

1 + 24A0B -) (0)

decreases to 0 as t -> oc. For an attractive interaction (A0 < 0), A(t) has a singularity at a finite value of t (the BCS case: the effective coupling constant grows as the energy scale is lowered), and f becomes singular if 0 gets large. However, f remains finite and analytic in Ao as long as IA0 IN(0) log(0f0) is small enough (this statement was proved in the last section for the skeleton four-point function, not just the ladder resummation). If one fixes Ao and increases 0, a singularity indicating the onset of superconductivity occurs at a temperature To = 1/00 of order To cc foe-241AIN(0). (4.205) It is natural that c o sets the scale in this equation: the above considerations are valid only when the particle-particle ladders dominate the RG flow. If the Fermi surface has positive curvature, it is possible to choose fo such that the ladders dominate the flow for energies below fo. The role of the curvature for this will become clear when the volume improvement from overlapping loops is shown; it will be discussed further in Sect. 4.8. The coupling constant A0 is not the original coupling constant, but the effective four-point coupling at scale eo. It is a regular, in fact, analytic, function of the original coupling constant A, if I AI is small enough (dependent on co ) but it is of course not true in general that this effective interaction is local, i.e. a constant in momentum space. This can be taken into account in the following way. Let

00-(t1Q)k)i) = f (tI( - k + ?) al)) (k + L, (72)2 +

as), ( -1 +

0'4)) , (4.206)

then

04,

at- (t I Q k , 1) =

12 f dr [4(Dt(-r + ?)Dt(r + ?))]

E 00Ti ,T2

3. 0. 2n.7-2( t

I Q5 kf r)

727134 (t

(4.207) 1).

4. The Fermi Surface Problem

154

Note that the sum of the ingoing momenta Q now plays the role of a parameter in this function. Because Q = 0 is the singular case, we now restrict to Q = O. The flow for 0, has the initial condition 0,0 I 0, k,1) = v,(k,l)

(4.208)

where y is a C2 function obeying t-independent bounds, since it is the initial condition. The particle-particle bubble in (4.207) is independent of the momenta k and 1. Thus derivatives with respect to k and 1 do not act on the propagators Di and therefore do not deteriorate the scaling behaviour. Writing the integration momentum as r = (w,r(p, 0)), we may thus do a Taylor expansion in p without influencing the t-dependence. The terms proportional to p have power counting that is a factor e-t better. Denoting

w(t I 0i,u29 0

(4.209)

cf4,01 ) = Off(t I 0 , 1r( 0 3 0)2 1r( 0 5 01 )),

we thus have, up to terms that are multiplied by an extra factor e - t,

aw

at (t

cr i , (72 , ; cr3 , (74 , Of )

=

(w2 P2))2 f (xi(fT2 12 f dw dpgt (10J(0,0") co 2 p2

(4.210)

"

E w(t I ul,o2,0; 715 7-2; °H )W(t

7-2 2 71 5

0" ; (73, (74,

01 ).

Ti T2

We can now interpret [44] the function w as the kernel of an operator acting on the weighted L2 space L2 ({-1, 1} 2 x Sd-1 ) where the weight is given by the local density of states J(0,0). An explicit construction of an ONB for spaces associated to Fermi surfaces (also called Fermi surface harmonics ) was given by Allen [49]. An expansion in an ONB of eigenfunctions of w, with t-dependent coefficients g, (t), 00 w(t

Ial , cr 2 ,0 ; 0.35 0'4) Of)

=

gk(t)Wk(cr i ,a2 ,0)Wk(a3,a4,00,

(4.211)

k=1

then gives, because of orthogonality, the uncoupled system of equations

agh(t) = 12-)(0)gk(t)2.

at

(4.212)

Thus, in the ladder approximation, the coupling constants associated to the various symmetry channels (which generalize angular momentum channels) flow independently, and all according to the law gk(t) =

gk(0) 1 + 12gk(0).131 -) (0) .

(4.213)

4.5 The Four-Point Function

155

Any coupling that starts out negative (gk (0) < 0) will eventually grow, but one can expect that the one that has largest absolute value will determine the symmetry of the superconducting order parameter. The full flow does not lead to a decoupling of the different angular momentum sectors (even here, we had to do a Taylor expansion and omit some terms to get this decoupling property). Kohn and Luttinger observed, by a second-order calculation [50], that even if the initial interaction is purely repulsive, the effective action at scale co, which determines the initial condition (4.208), must have gk(0) < 0 for some (in general very large) k. Feldman, Kniirrer, Sinclair and Trubowitz [51] have shown by a very careful analysis that although in d = 2 dimensions, the Kohn—Luttinger effect is not there in second order, there is a similar third order effect that drives the full RG flow to an 1 = 1 superconducting state [51]. The analysis of [51, 52] goes far beyond the original proposal [50]. In particular, the RG flow, and not just the Bethe-Salpeter kernel, is studied in [52].

4.5.5 The Particle—Hole Flow The particle-hole flow is obtained by leaving out the pp term on the right hand side of (4.178). Similarly to the particle-particle case, this now resums the particle-hole bubbles

13 +) (q) = f dk (Do (k)D 0 (q + k) — D t (k)D t (q + k)) .

(4.214)

In contrast to the particle particle case, this function is bounded uniformly co. This follows for NI away from in the temperature even in the limit t zero from Corollary 4.11 and for II near zero from the following theorem, which is based on an interpolation formula for Fermi surfaces. Theorem 4.12. Let q E B, qo E .2#Z, and B (+) (q)

=

f ([,214

1 iwn 1E(p) z(cdn+qc)_E(p+q) .

(4.215)

nEZ There is r > 0 and K > 0 such that for all II
(4.216)

1B (+) (01
r is determined by the nearest van Hove singularity (i. e. the nearest point where VE(p) = 0). Let x(s) = (1 + eex) -1 be the Fermi distribution and (50(x) = ds 60(x) = 1, and —axo(x). The function 6,3 is an approximate unity: > 6 as 3 —> co. Let q 0. Using a Taylor expansion, B (+) (q) can be rewritten as

Proof. —

fR

156

4. The Fermi Surface Problem

f ddp 1 (27r) 1 iqo + E(p) — E(p +

B(+) (q) =

1

1

iw E (P)

( /1 3E nEZ

f

•

iWn E( P

))

dd p Xf3(E(P)) — X0 (E(p + q))

(27r)d iq o + E(p) — E(p +

(4.217)

ddp +E( p)-E(p+q) f dt 60(E(p, q, t)) (2 iod igo where

E(p, q, t) = (1 — t)E(p) + tE(p + q).

(4.218)

VE(p, q, t) = VE(p) + tE n (p + Vq) • q

(4.219)

Because with some '0 E [0,1] (here E" denotes the matrix of second partial derivatives of E), and because I VE(p)I > go on the compact set S and the second derivative of E is bounded, there is r > 0 such that for all II < r, VE(p, q, > go/ 2 . Thus the interpolating Fermi surface St = { p : E(p, q, t) = 0} is still a submanifold for all t E [0,1] (it is, in our case, even positively curved). We can therefore introduce a radial coordinate p = E(p, q, t) and a tangential coordinate 0 E Sd-1 in the standard way [45], with a bounded Jacobian and get

B(q) < f dt f dp d0

p, 0)60(p)
(4.220)

If q = 0 and qo = 0, use

)13 w E ( iw_E ) 2 =

delExp(E)=-6,,(E)

and an analogue of (4.220) to see that for

B( +) (0) =

—

(4.221)

—> co,

f dû J(0,0) = —N(0)

(4.222)

is just the density of states at the Fermi energy, which is finite by our assumptions on E. If q = 0, but qo 0 0, a similar calculation shows that • B (+) (q0, 0) = O. Equation (4.217) also implies that at qo = 0 the particle-hole bubble is a Cko function of q for II < r if the dispersion relation E is Ck°.

4.6 Improved Power Counting

157

In Theorem 4.12, we have left out the ultraviolet cutoff on Do . This is justified because in the integrand for the difference, at least one of the propagators is cut off away from the Fermi surface, hence bounded, and thus the integral converges because f dpiDo (p)1 is finite. In summary, the finiteness of the particle-hole ladders holds under two conditions: there is no van Hove singularity on the Fermi surface, and the Fermi surface has positive curvature. If one of these conditions fails, the particle-hole ladders can have singularities. A relevant example for that is the half-filled Hubbard model, where singularities occur (in d = 2) on the two lines p i = p2 and pi = —p2 (here p = (pi , p2 )). Near to, but not exactly at, half-filling, these singularities are replaced by large values of the function, which may be important in the RG flow, although they do not introduce true instabilities at very low energy scales for small values of the coupling constant.

4.5.6 The Combined Flow An iteration of the full parquet equation (4.178) also generates terms that mix the particle-particle and particle-hole bubbles. However, these terms all contain overlapping loops (defined in Sect. 4.6), and therefore they affect the leading low-energy behaviour, which is given by the particle-particle ladders, only by exponentially small terms. The particle-particle singularity is more generic in that it occurs at any value of the filling factor, whereas the particle-hole singularity occurs only for specific filling values (e.g. half-filling in the Hubbard model).

4.6 Improved Power Counting In the last section, we have seen that the ladder approximation to the flow of the four-point function has a logarithmic growth when the sum of the ingoing momenta equals zero. This by itself does not prove the occurrence of such a singularity in the full four-point function because there may be other contributions that cancel the singularity in the ladders. In this section we shall prove, among other things, that if the ladder term is left out, the fourpoint function is bounded uniformly in the temperature. Thus, although in the full flow, the ladder term and the others are coupled, such cancellations do not occur.

4.6.1 Overlapping Loops We now proceed to discuss the bounds that are at the heart of the analysis of perturbation theory to all orders. They apply to contributions from graphs with overlapping loops. Examples for such graphs are given in Fig. 4.5. The

158

4. The Fermi Surface Problem

first one is a contribution to the four-point function, the second one to the selfenergy. The third one is a graph that occurs when the integral equation corresponding to the parquet RGE is iterated, and when the particle-particle and particle-hole flows mix. All these graphs have at least two loops, and two different loops have one line in common. This line is drawn heavy in the figure.

:CŒ -a-K)Fig. 4.5. Examples for overlapping graphs To motivate the bounds that follow, we discuss the two-legged graph in detail. If associate a vertex function that is identical to one to each of the two vertices, we get the second order contribution to the skeleton selfenergy in the Hubbard model with on-site interaction, given by the integral

È2(q) = f dkidk2 Dt(ki)Dt(k2)1 5 t(q + viki + v2k2)

(4.223)

with vi E { 1, 1} . The property that a linear combination of two integration momenta, like the q + viki + v2k2 in (4.223), appears in the argument of a propagator, is a defining property of graphs with overlapping loops. A purely graphical definition of the property of a graph to be overlapping can be given [45]. If a spanning tree T for a graph G is fixed, the loop generated by a line l i fl T consists of ii and the closed nonselfintersecting path connecting one endpoint of / 1 to the other one. If there are two lines 1 1 ,12 ¢ T, the graph is called overlapping if the loops generated by l i and 1 2 contain a common line of T. Whether a graph is overlapping or not does not depend on which spanning tree one chooses to check this property [45]. Although the graphical definition and interpretation just given will not be necessary for understanding how the bounds work in the RG differential equation, they allow to decide rather easily if there are gains in power counting also when one does the analysis with discrete renormalization steps. One can give a complete characterization of nonoverlapping Feynman graphs with two and four external legs [45]. The power counting of Sect. 4.4 gave —

( .4)±2(q)1

iia a btlioollptiii

130,(8.11)2 E.

(4.224)

For a = 0, this leaves a decay Et oc e—t, so that E2 (q) = f(!,, ds È2 is finite. For a > 1, however, the integral grows with 3 and diverges for zero temperature,

4.6 Improved Power Counting

159

thus indicating that already the first derivative of the selfenergy is divergent at zero temperature, if the momentum is put on the Fermi surface. This is the true behaviour in one dimension, but not for curved Fermi surfaces for d > 2: Proposition 4.13. [45, 46] For the class of models obeying the assumptions in Sect. 4.2.6, there is a constant (independent of temperature) such that t led

()c' ±2(q) 1 <

const f 2

f 1+t 1

if d = 2 if d> 3.

Thus the first derivative of E2 is bounded uniformly in derivative grows at most as a power of log O.

(4.225) and the second

Here and in the following, derivatives with respect to po are understood as a difference operation (f(po + — f (po)). The RGE actually defines the Green functions for (almost all) real values of po, not just the discrete set of Matsubara frequencies, so the effect of such a difference can be bounded in terms of a derivative by Taylor expansion. This changes at most constants, so we shall not write this out explicitly in the proofs. The proof of this proposition will use the volume improvement bound from overlapping loops. It is a generalization of the one-loop volume bound given in Lemma 4.10, but it has much stronger consequences because it is uniform in the external momentum. It applies to all contributions to the right hand side of the RGE with number of internal lines i at least three. Using this bound we shall generalize Proposition 4.13 to the full skeleton selfenergy. Proof. We write k • = ki) for the integration variables, use again (4.187), (4.131), and the inequality 1(1 qo + vi col + v26)21 _< et ) < 1, and do the frequency sums over co l and 6)2 . This gives the bound log(0€0) Oq

±2(q) <

ET1-1,11

log(/3f0)

dti

dt2 V2 (q,

t, ti , t2)

(4.226)

with

V2(q, t, ti, t2) =

dki (27r)d

max v1,v2=±1 lz(ft i )

t

dk2 (27r)d

lz(ft2)

11(IE(q+ viki + v2k2)I < Et) •

(4.227)

and 7Z (6) given in (4.191). The proposition now follows from the Theorem 4.14 in the next section, by integration over t 1 and t2 in (4.226).

160

4. The Fermi Surface Problem

4.6.2 Volume Improvement from Overlapping Loops Theorem 4.14. Under the assumptions stated in Sect. 4.2.6, there is a constant QT such that for all q E B and all t1 ,t2 ,t with t 1 > t and t2 > t, V2(q)t) ti 5 t2)

QT€t1ft2 et {

1+t

if d = 2

(4.228)

The constant QT depends on the minimal curvature of the Fermi surface. The naive bound for V2 (q, t, t 1 , t2) , obtained by dropping the indicator function in (4.227), is V2

(q, t, ti, t2) < const

Et 1 Et 2)

(4.229)

so, up to the factor t in two dimensions, the Lemma implies an improvement by a full power Et. We call Et cc e —t the volume improvement factor. The most important feature of this bound is that the improvement is uniform in the external momentum q. It does therefore not matter if q is near to the Fermi surface (as would be the case in the selfenergy) or if q is any linear combination of momenta, as is the case for the four-point function and the higher m-point functions. Volume improvement from overlapping loops was first proven in [45]. The analogue of Theorem 4.14 proven in [45] gave an improvement factor e-6t with some 6 > 0 under much weaker assumptions on the Fermi surface (it need not be convex, not even connected, and the curvature may vanish on subsets of positive codimension of S, e.g., in d = 3 on a curve on S). On the other hand, the t-behaviour stated in Theorem 4.14 is optimal, that is, one cannot gain more than one power of ft from an overlap of two loops. Theorem 4.14 was first proven in [46] in the form stated in Appendix B.8.2. We prove it for spherical Fermi surfaces in Appendix B.8.2. The proof for the nonspherical case, including surfaces that are not invariant under p —> —p, is in Appendix B of [46]. It does not use Lemma 4.10: in fact, in absence of the symmetry p —> —p, VI+) and V(—) i behave differently: an analogue of Lemma 4.10 holds for 14+) , but only a weaker bound holds for 14—) (see Appendix C of [46]). The existence of a volume improvement by a factor e—t/ 2 , as well as its uniformity in q, can be seen in a very simple way by the following argument

[48] using Lemma 4.10. By definition (4.227), V2

(q, t, t i , t2 ) =

f clici ,Vi(v2) max (q + viki , t, t2). vi,v2=±1 f (27r)a

(4.230)

By Lemma 4.10, VI(±) (P2 t, t2) < Qoft2

Ii / .N

if 1106 otherwise.

(4.231)

4.6 Improved Power Counting

161

Inserting this bound into (4.230), we get V2 (q, t, t i , t2) < Qo Et2(N/

+

vol R(Et i )

vol (R,(E t ,)

n

NR)

(4.232)

with B(q,r) = {13 e B iiP IL < r}. We have vol R.(eti ) < const et l . The set B(-v i q, VEt-)nS is either empty or a piece of the Fermi surface with diameter 2.%/Fri". Thus R(Et 1 ) n B( -v1cl7VEi) is a subset of the cylinder of height 2et 1 above that region, and therefore -

V01(1?(Et 1

n B(—viq,

)

< const et i et(d-1)/2

(4.233)

const ft 1 Et2

(4.234)

Thus, for all d > 2, V2 (CI, t7 tl, t2 ) <

In the following three subsections, we prove stronger ('improved') power counting bounds for the skeleton functions. The proofs in these subsections are taken from [53].

4.6.3 Volume Improvement in the RGE We now use the volume improvement of Theorem 4.14 to bound those contributions to the right hand side of the RGE that have i > 3, We also want to bound derivatives. The derivative applied to Qm, gives

vrnr (t I P) = f

ac

i!X(t I P),

(4.235)

where, after a change of variables ki

Xct,i(t I /2)

=E

a! E EEtokti!a2!

crE{-1,1}i /e{ 1,2}i

f

dk2

Rx13

f

dki a'°./it(ki)

Rx /3

13 ctlImlr1 (t

a

H D(k)

(4.236)

j=2

p(1) K) aCt217n2r2 (t

K, p(2))

with Kj = (ki, g , 1 j), the sum over a running over all triples (c o , ai, a2) with ao + al + a2 = a, and ki = - Eii =2( -1 ) ilki - (PI + - • • +Pytt,-t-Ei)• X„,i depends also on (m i ,ri , m2 , r2 , r). Application of I k gives < 4i

E

iaal imiri (010 la a21m2r2 ( t )10 Yeto,i( t)

(4.237)

with

ya,i (t) =

sup CIEMx5

„,EI-1,1)

f

TT dki IDt (k)I pa i=2

(E viki + Q)I. j=2

(4.238)

162

4. The Fermi Surface Problem

Lemma 4.15. y(t) = o for t > log

, and

Ya,i(t) < (8.I i ) i-1 Ba E i -2- II .

(4.239)

Y(t) < ic(1)--1 (1+t) cr i- l al.

(4.240)

If i > 3,

with K (1) = BI3Ba(8M i-3 QT. Proof. By (4.131), Yo,,i(t) 3, the integral (4.238) for y„,i (t) contains the subintegral S(Eii .4 viki + Q), with

S(q) = f dk2 f dk3IDt(k2)Dt(k3)it(v2k2 + v3k3 + q)I

(4.241)

which we recognize as the function A (see (4.223)). Thus by ProPosition 4.13, IS(q)I < BI,BaQTE t2 Inserting this bound, the remaining i -3 integrations are independent and contribute a factor IlDt(k) Ill i 3 . Thus (4.240) holds. • ' al .

Lemma 4.16. For all i,

IX«,i(t)10

4i (8J1) i-1

al

E aolaila21 1 aal irni ri (t) 10 1 a a2 /m2 r2 (0 10 a

B„Eit -2-la°1 .

(4.242)

For i > 3 IX0,,i(t)10 <

4i (8J1) i-1

E aolaailkt21 laal /rniri (0119 149 (x2 /m2r2 (t)1 0 a

BckoK(I)Eit-1—lacd { (1 +

t)

1

if d = 2 if d > 3.

(4. 24 3 )

In the next sections we use Lemma 4.16 to prove bounds on the non-ladder skeleton functions and on the derivatives of the selfenergy.

4.6.4 Bounds on the Non-Ladder Skeletons

We define the non-ladder skeleton functions as the solutions to the skeleton RGE where the ladder contributions to the four-point function are left out. That is, on the right hand side of the equation for in = 4, m i = 4 and m2 = 4 are left out.

4.6 Improved Power Counting

163

Definition 4.17. The functions A,N7), obtained from the truncation of the skeleton RGE, in which the equation for the four-point function is given by

_a pm (t ) = A4 Q 4 r,>3(t) at 2

(4.244)

4'r \

(i.e., where m 1 = 4 and m2 = 4 are left out in the sum non ladder skeleton functions.

f dk 4,r) are the

-

The next theorem [53] states that the non-ladder skeleton four-point function is bounded and that the non-ladder skeleton selfenergy is C I uniformly in Theorem 4.18. For all r > 1, the non-ladder r th order skeleton selfenergy -uso 12 r (t) converges for t —> oc to a C2 function. There are constants Lr and L., independent of 0, such that

l

1

if lal 1 Lr aa I,Nr ) (t)o < (log Ofo) 2 if la I = 2 and d = 2 log 0E0 if lal = 2 and d > 3.

and

(4.245)

1

if lal =0 log Oeo ifla' = 1

<

(4.246)

if = 2. 0E0H

Thus the removal of the ladder contributions to the four-point function indeed makes Ifri. ) 1 0 uniformly bounded in 0. Theorem 4.18 has an extension to the m > 6-point functions with bounds in the L I norm. The following theorem implies Theorem 4.18. Note that all bounds in this Theorem are independent of O. Theorem 4.19. Let a be a multiindex with la 1 <2. For all m,r, let Imr (0) be C2 in (p,,... ,pm), and assume that there are K° > 0 such that parmr (o)i < K, with K.Z. = 0 for m > 2r + 2, and 41 = Jm 4v , where

> O. Let a) (t) be the Green functions generated by the non-ladder skeleton RGE, as given in Definition 4.17, with initial values Imr (0). Let Km i = 4°1 and for r > 2, let Km ,. = K(°) 1 Mlf ditmri i!(3 2 Ji) i-1 KmiriKM2r2 mr+rn

(4.247)

where Al l = 240B K (1) , with B = maxe, B. Then Km,. = 0 if m> 2r + 2, and for all t > 0 and all m,r 2— rt—Ick:

laa (2(N) mr

< 2K

0—

Et Inr

El-1°,1 1+1 t

2 (j ) äd,2 k 2

if m > 6 if m =4 if m

2.

(4.248)

164

4. The Fermi Surface Problem

Moreover, for

m > 6,

l aa irr) (t)

Hai

(4.249)

< K4r

1 if = 0 (L2±1)2 if ict i 1 f T i p if _ 2,

(4.250)

1

if

lal 1

if

lal = 2

0< K

Mr Et2

for m = 4 l aa4zNr)(t) 1 ° and for

m=2 (L+1‘2

K2r

2) I-+ 2

if la'

=

and d = 2 2 and d = 3.

(4.251)

Proof. Induction in r, with the statement of the Theorem as the inductive hypothesis. r = 1 is trivial because iel) = 0 and because the statement holds for inzi (0). Let r > 2 and the statement hold for all r' < r. The inductive hypothesis applies to both factors 47,N)rk in 0;n - Nr) . For mk = 4, it implies that

1 for all

.-/Zk l o < Kink rk E T:al = Krnkrk EcT lal etaal+

(4.252)

t > O. Recall (4.242) and (4.243), and

la ' Arn -Q.Z.)(t)L = <

'Anz a I

<

c' C2Wr)(t)

o

) (t)

f dei i!IX0(t)1 0 .

(4.253)

Let m > 6. By (4.242) and m1 + m2 = m + 2i, the t-dependent factors in X,i(t) are

et( "41- —2 -Ficti

e t( 2?-2+Icy2D e t(2-i+Ictol)

= et(*1 -2-Flal) .

(4.254)

Since ffz - 2 + lal > ¶ - 2? 1, integrating the RGE gives

actirr) (o)

aairr)(t)lo <

o 2

ds (9°4) o

o

M7?). + m 1- 4 limrEt2—

(4.255)

E

(4.256)

where

vmr = 4B f dkemri i!(32Ji) i-1

aokra1! !a2! Krnirl Km2r2 '

a

For m > 6, m -4 follow.

. Moreover, E

< 310d, so (4.248) and (4.249)

4.6 Improved Power Counting

165

Let m = 4. One of mi and m2 must be at least six because the ladder part is left out in ÎLN,. ) . Thus i = (rri, i + m2 — 4) ? 3. By (4.243), there is an extra small factor et = coe —t , so

la

c, f)(N) (4 ,1 < 1+ t ei -1,1 K(1)14m. "4r" 10 2 t 9

(4.257)

which proves (4.248). (4.250) follows by integration. Let m = 2. The case m1 = m2 = 2 is excluded since this is the skeleton RG. Thus i = (mi + m2 — 2) ? 3, and (4.243) implies

actOr,r)(t)L < ( 1 -:t

) d

'2 f ried K (I) v2

r•

(4.258)

Thus (4.248) holds, and (4.251) follows by integration.

III

Theorem 4.18 follows from Theorem 4.19 by the following argument. Let lal < 1. By (4.251), l5cq-Z ) (01 < K2,,.. By (4.248), <

laag-Ccoi

K2,,f0e-t L2+1 _> 0 as t

oc. Thus the limit t —> oc of 4Nr) (t) exists and is a C I function of (Pi, p2 ). All constants are uniform in /3. The second derivative of Ce.) is 0(t) in d = 2 and 0(1) in d = 3. Since Orr) = 0 for t> log /3e0 , the integral for ir r ) over t runs only up to log /3€o, which gives the stated dependence on log /3€0. A look back at the proof shows that no factorials from powers of t were generated in the recursion for the coefficients (because there are no powers of t that grow with r).

4.6.5 The Derivatives of the Skeleton Selfenergy If the ladder contribution is kept in the RGE, powers of log )3E0 appear in the Green functions. We now prove [53] that if I Al log 13e0 is bounded by a constant, then the derivatives of the selfenergy are bounded independently of 13.

12,,. and 140. converge for t -> There are constants Lm,,. such that for all 0,

Theorem 4.20. The skeleton functions

(9Im ,,.(t)L < tm ,r (logOco)r

00.

(4.259)

for m = 2, lal <2, and m = 4, lal = O. Theorem 4.20 is a consequence of the following, more detailed, theorem.

166

4. The Fermi Surface Problem

Theorem 4.21. Let a be a multiindez with !al < 2. For all m,r, let In„(0) be C2 in (pi , ... ,pm ), and assume that there are Kg). > 0 such that Imr(0)1 < Km, with Ifrç rC12 = 0 for m> 2r + 2, and 44)), = 6„7,4v, where > 0. Let Inzr (t) be the Green functions generated by the skeleton RGE, as given in Definition 4.7, with initial values Im ,.(0). Let Km1 KZ and for r > 2, let Km ,. be given by (4.247). Then for m > 4

(t)1 0 < 2Kmr (log OE° ) and

inzr(t)L

Km ,. (log Seori

(4.260) f7 Hal

(4.261)

For m = 2,

< 2K2, (log 0€0 )r -2 +5,1,24 — lal l aV2r(t) 10 —

(4.262)

and aq2r I

10

K2 r (log Ofo)' 2 Ii

‘ (log 0€0) 1+5d,2

lal < lal = 2.

(4.263)

Proof. The proof is by induction in r, with the statement of the theorem as the inductive hypothesis. It is similar to the proof of Theorem 4.19, with only a few changes. Note that m 1 = 2 and m2 = 2 never appear on the right hand side of the RGE because of the skeleton truncation in Definition 4.7. For m > 4, use (4.242); this gives aa(?Trzr(t)10

9vmr (log /36 0 )r -2

(4.264)

For m > 6, the scale integral is as in the proof of Theorem 4.19. For m = 4, the scale integral is now .1: ds t < log0E0 . This produces the powers of log 0€0

upon iteration. For m > 2, use (4.243); this gives K(net2- l a l (1(1 + t)) 6a.2 instead of Et 2 in (4.264). The theorem now follows by integration over t, recalling that the upper integration limit is at most log /3e0 . • Theorem 4.21 implies Theorem 4.20 as follows. Convergence of the selfenergy follows for lal < 1 as in the proof of Theorem 4.18. For m = 4, the function is bounded uniformly in t. For lal = 2, convergence at > 0 holds because iihr(t) = 0 for t > log 0E0. Corollary 4.22. For R> 1, let Ir) =limt,c„) Eril 12,r (Or . Then for all R > 1, there are constants -yR, independent of 0, such that for all I a I < 2, and all À with 1 Al log 0E0 < const , the skeleton selfenergy satisfies

a ct e) 1 0 < 'YR.

(4.265)

4.7 Renormalization

Subtractions

167

It is expected that 7R < const r, so that the skeleton selfenergy is analytic in A as long as the above temperature-dependent restriction holds. The present treatment of the combinatorics is far from showing anything of the kind, but a proof should be possible by combining the method of [55] with the bounds from overlapping loops [45, 46, 47, 53].

4.7 Renormalization Subtractions We now discuss what happens when the restriction to skeletons is removed, that is, when two-legged insertions are allowed.

4.7.1 Motivation; the Counterterm The two-point function in momentum space, 12,r(t I 1311 P2 (Pk ,Crk ik) of the form 12 7,-(t I Pi, P2) = Eii.12410.2 E2,r(t1( - 1) iPi)

is for Pk = (4.266)

(this holds by antisymmetry and because ./1H/2, r =1-2,r , where 17H projects to the subset pl = —p2). Accordingly, we write the right hand side of the RGE for m = 2 as 1 (4.267) 112 7 r (t = — Q2,r (t 2 with a function Q2 7r that is defined in terms of Q2 7 r in the same way as 1 2,r is determined by Also, in the neighbourhood of the Fermi surface where ,7r(p, 0)), with p and 0 the radial and the flow takes place, p = (co , p) = tangential coordinates around S. The skeleton m-point functions had the simple power counting behaviour ei (m -4); this bound iterated through the RGE in an almost trivial way because on the right hand side of the skeleton RGE, the m = 2 contributions were left out. For m = 2, e ( m-4) = e—t , but we have already seen that the skeleton two-point function is not 0(e —t ), but 0(1). Thus putting in the two-legged functions leads to a failure of the bounds. On the other hand, the derivative gii2,2 = Q2,2 was of order e—t , so one can once again attempt to solve the problem by a change of boundary conditions. The goal will be to choose an initial condition for the flow such that lim 112 r (tI0, ir(0, 0)) = 0

for all 0 E Sd-1 .

t-400 '

with the idea that, if RGE as

Q2, r

(4.268)

remains of order e—t , one can then integrate the

1[2AtI(0,7r(0,0))) =

00

f ds 02 7r(810,7r(0,6))

(4.269)

168

4. The Fermi Surface Problem

which implies that 1[2,7- (t I (0 ) Ir (0 ) 0))) is of order e —t , and hence gives hope that the bounds would not be upset when the skeleton equation is generalized to the full RGE. This strategy works [45, 46, 47], but this is not at all trivial. From a technical point of view, one has to be careful because in order to get from the Fermi surface (0, 7r(0, 0) to general values of p, one has to do Taylor expansion, and taking derivatives potentially generates singularities. From a conceptual point of view, one has to justify the change of boundary conditions. This is an important point, which itself generates further technical complications. One might think of bypassing the above discussion about Taylor expansion 0 as t co for all pEMx B. Indeed, there is by requiring that E2,r (tip) no unique way of imposing the boundary condition, so one has some freedom, However, both from a conceptual and from a technical point of view, there are significant differences between different boundary conditions. The boundary oc is really achieved by an appropriate initial condition condition for t E2,r ( 0 1P) = ]I()r)(P)- This initial condition changes the original model to one ( where the quadratic part of the original action, f dp E(P))0(P) •is changed to

IRA

f

dP /P(P)(iw

E(P) Irr (6) , P))09)*

(4.270)

This change has to be interpreted correctly (and justified, see below). For instance, it makes a difference whether the extra term really depends on w or not. If it does, then the quadratic part of the initial action will in general not correspond to a time-independent Hamiltonian, because the w-dependence of the action is fixed as ico in the functional integral we derived. We shall therefore require that the initial condition is given by a function K that depends only on p, and, to fix K uniquely, we furthermore require that K(p) = K (ir (p, 0)) depends only on the angular variable 0.

4.7.2 Full Amputation The singular behaviour of derivatives makes it necessary to transform from E2 to another function. The best way of getting optimal bounds is to transform to the one-particle irreducible functions. For the purposes of the present discussion, we shall, however, not need to Legendre transform, but just transform from 12 = Er›1 x71 2 , r to an analogue of the full selfenergy and amputate full propagator—s from the m-point functions for m > 4, as follows. Let (t1/9)

/(t1P) = 1+ Ct (P) 112(t1P)

(4.271)

This equation, as all following ones, is to be read as a formal power series in A, i.e. E = E E2(C 11 2)P. Since 112 is 0(A), E is a well-defined formal power series in A. The inverse relation is

4.7 Renormalization Subtractions

12(4)

E(t1p) - 1- Ct(p)E(tIP)

169 (4.272)

•

This transformation is motivated by the relation 112 (CO) = C-1 (G2 —

(4.273)

relating 12 to the full unamputated two-point function G2. In this equation, C gets subtracted because 112 contains only terms of order Ar with r > 1, whereas G2 contains also the free term, and the factors C-1 are there because 12 is an amputated function. The derivative with respect to t gives

È(t) = (1- CtE(t)) 2 012 (t )

— li2 (t)ÛE2 (0) .

(4.274)

The term 12 (t)CtE2(t) is the j = 1 term in the RGE for 12, so we see that this term drops out of the RGE for E. Similarly, we define the fully amputated m-point functions for m > 4 by setting (4.275) A(tilD) = (1- Ct ((-1)ip)E(t1(-1)ip)) 1 and defining

Pm) 11 A(tIPa). a=1

Wrnr(I P1 2 • 2 Pm) =

(4.276)

When inserting this into the RGE, on the internal lines, Dt(p) gets replaced by Dt (P)( 1 Ct(P)E(tin)) -2 , and A (p) gets replaced by tot (p)(1 Ct(p)E(t1p)) -2 , and the external amputations cancel out. The i = 1 term /m et h gets replaced by W m Ct È(t)(1 - C t E(t)) -1 . Thus the RGE for E and the Wm reads -

gEr (t ip) = L? )r ( t ip) Wmr

(4.277)

+ 02,r (tip)

( t i P ) = 411,)r (t1P) aVr(tIP) + mr( t IE)

with

1,2 ),r tIP) (

1 )Ct(( [Wm (t1 2

—1 )3m Pm)È(t1 —1)3m linz) A( t iPm)] r

f dk Wm+2(tIPI .. 2 Pm-2 (k, 1, 2), (—k, 1,1))

1472z)r(tIE)

i3(k)D(k)

Qmr( t 112)

[

riclkg L e ,j q=2

E(tik) (1 - CtE)3(k)jr Dt(kq)

[

(4.278) 15

t(k1)

(1— CtE(t)) 2 (kg ) (1 — CtE(t)) 2 (ki)

wmi (tip (1) ,K)wm2

170

4. The Fermi Surface Problem

0,,,,r

Apart from the replacements on the internal lines, is of the same form as the right hand side of the skeleton RGE. The notation [ - ],. means that the coefficient of Ar of the formal power series is taken.

4.7.3 Bounds for a Truncation In the following, we consider the truncation of the equations (4.277), where for m = 2 and m = 4, the "generalized tadpole" terms 14.V r are left out. The only reason for this is to keep the following proofs and theorems as simple as possible; it will be discussed at the end how one can take these terms into account. Again, we also distinguish between the cases where the ladder contribution i = 2 to Ch,r is removed from (non-ladder) or left in the RGE. As before, the difference in the result is that powers of log Ofo occur when the ladder term is taken into account. The recursion for the constants will be similar to the ones we have done up to now. We therefore do not keep track of these constants any more but use the following shorthand. We say that

Wrnr(tIE) = (9 (fa, [3))

(4.279)

if there is a constant cm ,., independent of 0 and t, such that

1Wmr(t1/2)1 <_ c, f (t, /3).

(4.280)

Let the nonladder equation be the RGE (4.277) with Lo,,. = 0 for m = 2,4 and without the ladder term (i = 2 in iChr . Also, for m > 4 and r > 1 let

WZ- be bounded Ck° functions on (R x B)m satisfying ./THWTT, )r = WZ-, and for p = (co, ir(p, 0)) E Mi x B let

1 7 (p) = (0, r(0, 0)).

(4.281)

The following theorem is an analogue of Theorem 4.8 for the full Green functions. It implies finiteness of the two- and four-point function. An analogue of Theorem 4.9, which implies finiteness of all the m-point functions with m> 6 as well, can be proven similarly, by using the norms I - I,. Theorem 4.23. For all r > 1, there are functions Kr E Cko (B, R) satisfying

. a — x-r (Tr(p,o))= 0 ap

(4.282)

such that the solutions Er and Wmr of the non-ladder equations with the initial condition Er(OIP) = Kr(P) Wmr (O1P) =

W(P)

are Ck° in p and satisfy for all a with lal <1

(4.283)

4.7 Renormalization Subtractions

(1), (2), (3)r (4)r (5)r (6)r (7)r (8) r

171

8E(tip) = 3 a 02,r(tIP) = 0 k( (1 + t)e - t( 2- 1a1)) aa Kr(P) = 0 (1) aaEr (tIH(P)) = 0 ((1 + t)e - t( 2- 10, 1)) act E(tip) = 0 (1) Er (tip) = 0 (e -t + ip - 11(p)i) Er(tIP)Ct(P) = 0 ( 1 ) if p E suPP Dt 3aW4,r (t1P) = 0 ((1 + Oa) aa Wirt ,r(tIP) = 0 (et( -2 +1a1).

Thus, as t co, for all r > 1, the selfenergy E r (t) and the non-ladder fourpoint function W4, r (t) converge to functions that are uniformly bounded in

0 Proof. The proof is by induction on r, with (4.282), (1) r-(8) r , and the hypothesis E(t I — Po , P) = E( tip° 2 P)

(4.284)

wmf(tIR12) = wmr(tIE),

where RP changes all (p)0 to -(pi) 0 , as the inductive hypothesis. We do the case r = 2 explicitly, to show the procedure for construction of Kr before going into the more complicated argument for general r. Note that the right hand side of the equation for Er and W,,,r contains only Er, and W„,,r, with I.' < r, thus the inductive hypothesis applies to all factors in there. Let r = 1. Set Ki (p) = 0 for all p. Then E1 (tip) = 0 for all t > O. Thus (1)1-(6)1 hold trivially. Since Orni = 0 for all m, (7)1 and (8)1 hold by the properties of Wri . Let r = 2. The Er, (t) occurring on the right hand side of (4.277) must have r' = 1, so all of them vanish. Thus E2(tip) is the second order skeleton selfenergy, and hence the results of Sect. 4.6 imply that (1)2 holds for &E2(tip) = a a 022(4)- In the integral for È2 (t1 — po, p), change variables from ko to -kc, for every loop momentum k. Because Dt( -PolP) = Dt(Po113 )1 and because the same holds for the potential f, by the hypotheses stated in Sect. 4.2.6, it follows that

±2(tl

—

po,p) = ±2 (tip° , p)

(4.285)

so (4.284) holds by integration over t. Set log )3f0 K2 (p) — —

f

dt 422 (417 (p)) ,

(4.286)

o

.ti

then K2(p) E R by (4.285), and (4.282) holds for K2 because H(p) = ir(0, 0) = O. Also, (1)2 implies (2)2 because dt 0ckQ22(t) converges for lal < 1. Integration of the RGE gives

g

r

172

4. The Fermi Surface Problem

t E2(t1p) = K2 (p) + f ds Q22 (.9 119).

(4.287)

0

Thus, by definition of K2, 00

E(t1//(p)) = — f ds Q22 (1(p)) t

(4.288)

which, together with (1)2, implies (3) 2 . (4.287), (1) 2 , and (2) 2 imply (4) 2 . Now (5)2 follows by Taylor expansion in p and co, and by (3)2 with a = 1. On supp Dt2 IP — H(P)I < const e - t, and I I Ct I 100 < const et, so (6) 2 holds. In second order, L( 1 ) and L(2 ) are zero because El = 0, so (7)2 and (8)2 follow from the results of Sect. 4.4, and (4.284) follows again by the properties of Di and f). Let r > 3. The main difference to the r = 2 case is that alz ),,- occurs for m > 4, L2 ), r occurs for m> 6, and that 1 — CL(t) is now 1 because the 0(r'), r' < r selfenergy will in general be nonvanishing. Whenever a factor (1— Ct E(t)) -1 appears, we expand it in a formal power series En (ct z(o)lL. As mentioned, since E is 0(A), all terms on the right hand side are of order
a(ct (koz(tiko) =

0 (et) ,

(4.289)

which is a factor et larger than the bound for CL(t) itself. This is the same factor that occurs when a derivative acts on bt (k1). Thus the skeleton proof applies, and (1),. holds. Let 00

Kr (p) = -

f dt 02,r(tIMP))-

(4.290)

o Then (2),. and (4.282) hold by (1),.. The same sign-flipping argument as for r = 2, and the inductive hypothesis (4.284) for r'
4.7 Renormalization Subtractions

8 (Ct (k)E(t110) = 0 (e t( lŒ 1-1) (1+ t))

173

(4.291)

is 0 (1) for a = 0 and 0 (1 = t) for lai = 1 so (7), follows directly from (7)<, and the product rule. Similarly, (8), follows by the skeleton case, which applies to by (4.289), and by (8)
Om,.

f

dk b t (k)D t (k)E(tik)(C t (k)E(tIk)) = 0 (e-t )

which follows from the properties of Dt , from (5)
(4.292) •

If the ladder term i = 2 is included for m = 4, it produces powers of logl3E0 in the same way as it did in the skelton case. Once one allows for such logarithms, one can also bound the second derivative of E. There is an analogue of Theorem (4.23) where (7), is dropped, a factor (log /3E0)' appears everywhere on the RHS, and where the second derivative of the selfenergy is bounded as in Theorem 4.21 and Corollary 4.22. In other words, if I Al log /3e0 < const , the selfenergy satisfies

< ,yR

l aci, E(R)1

o—

(4.293)

for lai < 2, with constants •yR independent of /3, and the four-point function is bounded. The generalized tadpole term L2), is a technical complication for two reasons. First, it leads to one-particle reducible contributions to E (this is why we called E only an analogue of the selfenergy), and second, it leads to graphs that are nonoverlapping. Both these complications can be dealt with the first one by doing a Legendre transform, the second one by a more careful analysis, using techniques of [45]. Instead of going through these proofs, we turn back to the question of what adding K means. The discussion in the next section is based on the results of [45, 46, 47].

4.7.4 The Meaning of K In the last section, we renormalized the perturbation expansion by changing the initial condition by a formal power series CO

K(A,E,p) =

E ArK, (p).

(4.294)

r=--1

Thus, the initial Hamiltonian now has a A-dependent free part

HI; (A) — AN =

E f dk (E(k) + K(A, E, k))4,,.ak,ff •

(4.295)

a-

K is a formal power series in A, and a functional of E. The goal, and the effect, of this was to make the selfenergy vanish on the Fermi surface (see

174

4. The Fermi Surface Problem

(5), of Theorem 4.23: for t oo, Er oc I1) 11(p)1). This means that the Fermi surface S does not change with A because Er vanishes on S for all r; thus S is the Fermi surface of the interacting system. It is this property that removes the infrared divergences, which came from expanding around the propagator with the singularity at the wrong place. The renormalized expansion still amounts to expanding the full propagator (iw — E E1) -1 in terms of the free propagator (iw — E) -1 , but the powers of arising in this expansion are now bounded because E vanishes on S and because its first derivative is bounded. There remains the question how to justify putting in the counterterm function K. In the jellium case, where one has rotational symmetry, K does not depend on k and thus is a correction to p. In the case without rotational symmetry, things are not as simple. In order to have a fixed, that is, Aindependent, starting Hamiltonian, we have to require that E(k) + K(A, E, k) = e(k)

(4.296)

is a fixed dispersion relation (e.g. that of the Hubbard model). In other words, once one does renormalization subtractions, the dispersion relation in the propagator is not the original dispersion relation of the model any more, but it has to be obtained from the latter by solving an implicit function equation. To solve such an equation, we must now regard A as a real parameter, not just a formal variable, to have 45 (E) = E + K(A,E) a real-valued function. With the above bounds, we can do this if we restrict to contributions up to order MX, instead of K. K(R) is a polynomial in A, so it R, i.e., take K(R) = can be made arbitrarily small by restricting to I AI < AR. It is therefore natural to try an iteration to solve (4.296). To do this, one first needs a set of E that (R)(E) = E + K(R)(A,E). This is the is mapped to itself by the map E reason why we were careful about finding bounds for derivatives of K. The above results' imply that for Ic = 2 and I AI log fleo < const , l9a1C(R) I < Thus, for every E satisfying the assumptions of Sect. 4.2.6 and every R > 1, there is an open ball BR around E such that (R) (BR) C BR, and we can iterate the map in this ball. We have not proven the convergence of the iteration; to do this, one needs more detailed bounds, obtained in [46, 47] by an analysis involving a discrete RG flow (such bounds can, with a good deal more work, be given also using the continuous RG flow developed here). Uniqueness of the solution to (4.296) holds under weaker assumptions [45]. Once one can solve (4.296), one has a clear procedure for doing renormalization. Given the interaction, the dispersion relation e of the model, e.g. the tight-binding dispersion relation, and given any R > 1, one solves for E = E(R) = (45 (R) ,) — 1 (e). Then E + K(R) = e by construction of E, and one can put E into the propagator by the Gaussian shift formula

>L

,

PP dpce (0) = det(1 — CeK(R)) dpcE ( 0)e— f dP I7(P)K(R)()0() (4.297) (which follows directly from the definition of the Gaussian measure; see Appendix B). This automatically puts the counterterm K( R) into the action,

4.8

Conclusion

175

and therefore the expansion in A up to order R is now the renormalized expansion. One can interpret the procedure of solving (4.296) as calculating the interaction effects in two steps: first, one determines K( R), which is the part of the selfenergy that changes the Fermi surface. Going from e to E means changing from the Fermi surface of the free system to that of the interacting system. Once this is done, all other effects of the interaction can be calculated at the fixed Fermi surface { p : E(p) = 0}. It should be clear from the description of the solution of (4.296) that with the bounds we have, one can only show that the solution E = 45-1 (e) is C2 in A and in p. Thus a solution of (4.296) does not make sense in a formal power series setting, and even if the power series in A for K converges, the solution of (4.296) cannot be analytic in A (if it were, the infrared divergent unrenormalized expansion would converge, by the identity theorem).

4.8 Conclusion At the end of this long, and partly technical, discussion we review some of the arguments and results, to put them into perspective. The discussion in this chapter consisted of some rigorous bounds for the perturbative Green functions, together with an analysis of truncations of the flow that provide heuristic guidelines of how to understand the bounds. Needless to say, this leaves many open problems, a few of which we shall discuss briefly now.

4.8.1 Summary We have analyzed perturbation theory in the coupling constant A to all orders for the -skeleton Green functions, and also for the ones with properly subtracted two-legged insertions (i.e. with the appropriate boundary condition). The results show that in d > 2 spatial dimensions, and for Fermi surfaces that have strictly positive curvature, for all R > 1, the order < R renormalized selfenergy E(R) = limt_H„,E,R 1 Er (t)Ar exists and satisfies

'YR

(4.298)

for all lal < 2, and the order < R renormalized four-point function ir) = limt,,„ E rR_ I 14,7.(t)Ar exists and satisfies i 0 11 < Yi1:1)

(4.299)

with ,yR and YR independent of fl, provided the condition IA I log Ofo < const

(4.300)

176

4. The Fermi Surface Problem

holds. The higher Green functions are bounded in an L' norm. The detailed bounds also imply that if a single term in the RGE for the four-point function, the ladder term (i = 2), is left out, then the statements for the four-point function still hold, but without the restriction in (4.300), i.e. for Pti < const, uniformly in the temperature. This conclusion comes from the overlapping loop bounds; it applies to rather general Fermi surfaces because it holds whenever any kind of volume improvement is there (see [45] for more precise statements). One can get a heuristic insight in the behaviour of the full solution by the opposite truncation of the RGE where only the ladder term i = 2 is kept in the four-point function. The solution of the RG differential equation then describes a flow of a coupling constant that reproduces the familiar BCS formulas, e.g. the form of the critical temperature (4.205). Renormalization was done by introducing a counterterm function K that, in general, is momentum dependent. The justification for this requires the solution of a functional equation that relates the free and the interacting Fermi surface . This is a mathematically nontrivial problem even in finiteorder perturbation theory, and we have given only a hint at its solution

[46, 47]. We have not given any nonperturbative bounds here. Doing so is possible, at least in d = 2 spatial dimensions [55]. The nonperturbative analysis of these systems is the subject of ongoing research [56, 61] and goes beyond the scope of this introductory text. On the whole, it is expected that the above perturbative analysis is confirmed in the nonperturbative setting (but a proof of this is difficult). Also, some technical parts of it, in particular the overlapping loop method, are important for nonperturbative studies as well. The continuous RG method also works nonperturbatively for fermionic quantum field theory if the RGE is integrated [57, 63]; the continuous method may work without explicit integration as well [54, 53], but whether it does is at the moment an open problem. 4.8.2 A Fermi Liquid Criterion The results lead us to formulate a criterion for Fermi liquid behaviour at positive temperatures [53 ] . This is necessary because Fermi liquid (or non-Fermi liquid) behaviour is observed only above the critical temperature for superconductivity. Because of the Kohn-Luttinger effect, one may expect that every system satisfying the assumptions of Sect. 4.2.6 has such a superconducting transition (if the reflection symmetry is broken, the Cooper instability is suppressed and Fermi liquid behaviour can occur at zero temperature, as shown in [56]). It is not completely straightforward to formulate such a criterion that involves only equilibrium observables and that can distinguish between Fermi liquids and other states of many fermion systems (such as the Luttinger liquid behaviour that occurs in d = 1 [59, 601) because at positive temperatures, there is no discontinuity in the occupation number density which could serve

4.8 Conclusion

177

as a distinction (at T = 0 there isn't either, because the energy gap in the superconducting state removes the discontinuity). A criterion for Fermi liquid behaviour at positive temperature is as follows. 1. The renormalized perturbation expansion in A for the full Green functions and the counterterm function K converges in a region R. = {(A, 0) : I Al log 0 < const }. 2. The selfenergy is C2 in p, and its second derivative is uniformly bounded on compact subsets of R. 3. The counterterm function K is Ck° in p, with ko > d (where d is the spatial dimension), and there is a unique solution to (4.296). A slightly different, but equivalent, definition of Fermi liquid behaviour at positive temperature (referring to the skeleton functions instead of the full Green functions) was given in [53]. We conjecture that the above conditions can be verified in d = 2 dimensions for the jellium band relation and the Hubbard band relation, in the latter case for a filling that is sufficiently far away from half-filling, by a combination of existing techniques [45, 46, 47, 55]. The jellium model is much easier than the Hubbard model because in that case K is a constant, hence Condition 3 is trivially fulfilled. Condition 3 was proven in [45, 46, 47 1 in perturbation theory by a rather detailed singularity analysis [46] and an extension of the overlapping loop method [47]. Such an analysis is also possible using the RGDE, but not included here. Condition 3 is needed to invert the map i and to have certain summability properties of the propagator. While a proof that the above version of Fermi liquid behaviour occurs is, even for the jellium model, beyond what can be explained here, it is easy to explain why it is sensible to call systems fulfilling Conditions 1-3 Fermi liquids, and in particular, why the above definition allows to distinguish between Fermi and Luttinger liquids. This distinction is in the selfenergy. By Condition 2, one can do a first-order Taylor expansion in p 11(p), to get (writing 11(p) = (0, P(p))) —

EV)

= Po (ao E) (0 ,P (P)) + (P — P(P)) • VE( 0, P (P)) + E(p)

(4.30 1 )

from which one obtains a finite wave function renormalization

Z(p) = 1 + i(a0 E)(0,P(p))

(4.302)

and a finite correction to the Fermi velocity VE(p), and the Taylor remainder È(p) vanishes quadratically in the distance of the momentum p to its projection 11(p) on the Fermi surface. This property distinguishes Fermi liquids from other possible states of the many-fermion system, such as Luttinger liquids: In one dimension (where `Luttinger liquid behaviour' has been proven [58, 59, 601), already the first derivative of the selfenergy does not obey Condition 2. The geometric reason for this is that in one dimension, the Fermi "surface" consists only of two

178

4. The Fermi Surface Problem

points. There is no transversal direction and hence, of course, no curvature, and all the volume improvement effects are absent. Higher-dimensional systems with a flat Fermi surface also have singularities in the first derivative of the selfenergy, again because of absence of any volume improvement. Note that this distinction between the behaviours of the selfenergy in one and more than one dimension can only be made if )3 is allowed to vary; at fixed 0, the requirement that something is bounded independently of is trivial. This is the reason why a whole range of values of fi and À is included in the above definition. As discussed, the restriction (4.300) comes from the four-point function, in particular the ladder terms analyzed in Sect. 4.5.4. Thus this restriction is there to avoid the BCS instability: the temperature is above the critical temperature for superconductivity. Under the above definition, one can justify using renormalized perturbation theory: given a model with dispersion relation e, interaction 0, and coupling constant A, and at temperature T such that I Al logfl is small enough, solve the inversion equation (4.296) for E and change to E, whose zero set is the interacting Fermi surface, as described in Sect. 4.6. Because À is fixed, E is now fixed. By condition 1, and because À is fixed with log0 small enough, the renormalized expansion in À converges.

4.8.3 How the Curvature Sets a Scale Finally, we revisit the heuristic discussion about the validity of the ladder flow near to half-filling in the Hubbard model in the light of the above volume and power counting bounds. It was mentioned that an important role in the perturbative justification of the ladder approximation by the overlapping loop bound was played by the energy scale at which the flow is started, the 'initial' energy scale co. The reason for this is that the volume improvement is curvature dependent, and will lead to effects only below a certain scale, as follows. For the band relation E(p) = —tE cos ki — p, the curvature of the Fermi surface vanishes for p = O (half-filling). For small p, the size ro of the neighbourhood will therefore depend on p and vanish with p. The initial energy scale c o for the flow must be chosen < ro . The analysis of Sect. 4.6 showed that the i > 3 terms have an extra scaling factor QTE t , where the constant QT is proportional to 1/tc. Since tc p, we now see that the chemical potential sets a scale co , below which the ladders dominate, in a natural way: if cap > 1 is large, then the "improvement factor" QTct > 1, so there is no improvement at all, and thus no justification for restricting to the ladder flow. If co is chosen such that co /p < 1, the improvement holds for all t > 0, and then the ladders dominate the flow. The flow of the coupling constants in the various angular momentum sectors is then essentially independent, and one that started out negative will eventually diverge.

4.8 Conclusion

179

Thus the geometrical picture given by the volume bounds is simply that when Et /p > 1, the neighbourhood is so thick that curvature effects do not lead to any improvement; the scaling behaviour is similar to that of a onedimensional system. Only below scale p, the curvature becomes visible and its effects single out the ladders in the flow of the four-point function (see Fig. 4.6).

Fig. 4.6. A shell around the Fermi surface intersecting its translate, for the tightbinding dispersion relation near half-filling p ::---, O. The energy scale of the shell is above the scale set by p, therefore the shell is so thick that curvature effects do not improve power counting

There are a number of other effects that become important near to halffilling. The effective one-dimensionality suggested by the above considerations also implies that the selfenergy behaves rather differently above and below scale p, so that in particular a careful analysis of the wave-function renormalization is important for scales above p. It should be noted that above scale p, the parquet flow is also not justified as the leading contribution by the overlapping loop bounds. Finding out which approximation is tractable and gives the correct behaviour requires different arguments.

A. Appendix to Chapters 1-3

A.1 A Topology on the Ring of Formal Power Series In this appendix, we recall briefly some basic facts about formal power series. The ring of formal power series is the set F = 010 of sequences S = ( sk)k>o with addition defined in the obvious way and multiplication as k

(ssi)k =Est sik _i .

(A.1)

1= 0

F is a domain of integrity with unit element (1, 0, 0, ...) and the map 1: C F, a E-+ (a, 0, 0, ...) is an injection. The rule for the product is remembered easiest if one associates a formal expression S(z) = k>0 ZkS k to a formal variable z, to the sequence S. The lower degree of S E F is defined as w(S) = min{ k > 0 : sk 0 0} for S 0 0, and w(0) = oo. Then w(S + S') > ininlw(S), w(S 1 )}, and we can introduce a metric on F by the definition cl(S, S) = 0 and for S 0 S',

E

,

d(S,S') = 2 -w (s-s' ) .

(A.2)

With this, F becomes a complete metric space. We call a family (Sa ) OEE N of formal power series summable if the limit

n

lim

n—).co

E SI,

(A.3)

v=1

exists in F. Then a sequence (S„,),, E N is summable if and only if w(S) —i oc' as a oc. In particular, the substitution of one formal power series in another, S(S'(z)), is a well-defined formal power series if and only if cd(S') > 1.

A.2 Fourier Transformation Recall that

. N, so M =

t-- E 2N. The dual lattice to

r = eZ d I LZ d = { x E EZd

L

:

—

—< 2

xk

<

L —

2

Vk E {1, ... , 4}

(A.4)

182

A. Appendix to Chapters 1-3

is 2Ir 2Ir 27r d r ir F* = --z-Ey 7-Zu• = {p E —Z : —e- < pk < -. L ..I

..I

Vk E {1, . . , d}}.

(A.5)

Both lattices have the same number of points

Ill

=

If* I

= Md = ( e )d•

(A.6)

d

We define p • x =

E PkXk,

x=1

f dp.fr (p) = L — d

E .fr (p),

per'.

r*

(Al)

and

(A.8)

Sr* (P, q) = L d8pq.

The Fourier transform of f : F —i C is

f : F*

C, given by

ir dx e —i" f (x).

(A.9)

1(x) = Lf dp eiP'x I (p).

(A.10)

The inverse relation is

The Fourier sums have the periodicity properties f (p +

r n) = f (p)

Vn E Zd

(A.11)

Vn E Zd .

(A.12)

and f (x ± Ln) = f (x)

We denote the scalar product for functions çh ,

V; E Cr* by

E

()r* = f dPikP)V;(P) = L—d ç.4) 11- 4) • F* pEr*

(A.13)

As mentioned, one can view the transition from F to F* as a change of basis to the basis given by F(x) = ei", so that F(q) = 8r* (p, q). The orthogonality of this basis, (Pp , 1-4)r* = 81-* (p, q), implies

= (0,0)r

(A.14)

so that, with this choice of scalar products, the Fourier transform is indeed a unitary map. Moreover,

A.2 Fourier Transformation

Sr (x, 0) fr. fr dx eiP'x = So L d = Sp* (p, 0)

ifx,--,

dpe ip, = sx0E -d =

and

183

if p E

The matrix representation of a linear operator is in the Fourier basis

(A04) = f dp Â(p, q)q3(q) I-.

(A.17)

with

Â(p, q) = (Fp , AFq ) = f dx f dy e -i"A(x,y)ei ". r r The inverse formula is A(x , y) = fr dp

fr. dq eiPx Â(p, q) e -in.

(A.18)

(A.19)

We have chosen our conventions such that the limiting cases E —Y 0 and L —Y 00 can be discussed separately without further rescalings. They are (a) the thermodynamic limit L —> oc:

r = f(p) = f (x) = E d E e ipx

ad , F* = Rd / 27 Zd =: B

Ed

E e

_ipx f(x)

E

xEr ddp •ipx f (p) e fB (27)d

= (5 (p)

(A.20)

for p E F*

xEr ipx I ddp e js (270d

=

8x0 - E —d .

(b) the continuum limit E —y 0:

r ,„ _ 27r zd

F = Rd / LZ d ,

AP)

L

= f dd x e -iPx f (x)

1(x) = L-d

E

eiPx f(p)

pEr* L—d

E

EiPx

= (5(x)

pEr*

f dx e'Px = (5po L d if

(A.21)

184

A. Appendix to Chapters 1-3 Note that under our conditions, the limit e -> 0 at fixed L is the limit of a sequence EN =F. LA-ffor N -). 00.

A.3 Properties of the Boson Propagator In this Appendix, we prove Lemma 1.10. Write p = (pi, k), k E Rd-1 .

1 +p2 = 1 ± k2 + p

=

(pi

+ iti(k))(Pi - iti(k))

(A.22)

with p(k) = VI -- c 2 . Since t> 0, and the integrand is 0(1k1 -2 ) as lki -> oo, we can evaluate the integral using the residue theorem, closing in the upper half plane. This gives dd-ik e -tp,(k) Cd(t) =

=

f (27r)d-1 2p(k) 1 is ci-21 2(2r)d -1 i

f

00

dn ()d— 2 e -til(p)

0

r r

P (P)

(A.23)

This shows that Cd(t) is finite for all t> 0 and that (i) holds. The constant Isd-21 = 2ir -1 d- f(q1 ). We call the constant in front of the integral in (A.23) Ad.

Since P

p2 u > 1+ p(p) = 1 + f du 2, 0 (1+u2 ) 1 /2 2(1+ p2 ) 1 /

we obtain

1 + Sp2 if 0 1

with S = -. Therefore, for t > 1 1 e -2 f dp pd Ad(

0d (t)

0

11(P)

f oci e + dp p d-2 1

P(P)

)

1 —t6p / -t( Ad e dp pd-2 e-t6P2 + 1 CC) dp pd-2 e ) Sp ' o t 1 00 Ad e-t (t - d-/dXX(1-2 e -6X2 ± t-d 7. f dx sd- ie- 'x) 0 t 00 + 1 f °° dx X d— 1 e -6x ) A de-t t- = dx xd-2 e-5x 2

fo

d-1 (

<

f 13

const - e -t t- 41 ; 1

(t > 1 was used to replace ftc° dx by fr dx). For 0 < t < 1, write p = f, dp = -Ids, to get

3 i

(A.26)

A.4 Wick Reordering for Bosons

Cd(t) =

Adt —(d-1) 1. 0°ds sd-2e—tt")

so

td-2 0d(t) = Ad

fo CI°

dss"

185 (A.27)

e —sr-L(1)

(A.28)

4(!)

(we used tp(1) = tV1 + (1) 2 =N/.F t2 = sV1 + (1)2 = skt(t9)).

For d > 3, simply use p(!) > 1 to get

td-2 d(t) <

Ad

00

f dS S c1-3e-8

(A.29)

so Cd(t) < Adr(d 2)t— (' -2 ). Since p(!) —> 1 as t —> 0, this becomes exact O. as t I)N 2 1, pdp = d s , For d = 2, write s = 41(p) = 41+p2)1/2 , so p = then

Ô2 ( t) =

A2 f ecids 2s t e-8

t2 s

Jt

ds e —8

J(f)2 —1

1

00

A2 2 f

1

Vs2 t2

log t.

(A.30)

A.4 Wick Reordering for Bosons It is convenient to derive the component form of the RG in position space and take the Fourier transform afterwards. The quadratic term on the right hand side of the RG equation is r7t(ri) r71(r2)

E y (7.1 , m, r2, m2) E m1=1 m2=1

Q1 ° 0) = 7? A

(

(A.31)

with r2, m2) =

where x (1) = xmi+m2),

(,( 1 )1 rIAAo /(2)1 f rbr f7AAo m2 r2 —m1 r1

- - - ,xini), x (2) =

P(1, 0) = f dxdy OAA O (x, y) (here OAA0 =k-4 ACAA0 ) and

VL-')V"/

(A.32)

,xmi-Em2), x =

,y(0)

(A.33)

A. Appendix to Chapters 1-3

186

w x,y (0)

= - -x 6 0( )

\

so(y ) QcOA

0(xki))) (fl ki=1

"CPA

( M2 H TT txxini-Ek2)) • k2=1 (A.34)

Let

E ak Sp (x, xk),

a (1)

m2

a (1)

aml+k Sr(ac,xm l +k),

k=1

a )

and a = a( 1 ) a( 2). With this,

[

6

840) Qc OA

(A.35)

k=1

0(xki)) =

aa (50(xk

oc (e(a(1) mr)] °A

a=0

u m '

kl1 =1 aak 4(x)

k-=.1

a=0

a

[mi

(A.36)

aak

Similarly, n1.2

QCOA

(II 0(X 0) = k=1

Il rn2

a

k=1 aam i-Ek

a=0

(A.37)

Since a( 1) is independent of a mi +1, , ami +m2 , the derivatives with respect to a (2) can be commuted through so that [

P(X-) =

arni+m2

'J ai • • • aami+m2

Z (a, 0)]

(A.38) a=0

with

= (a (1) , 6AA0 a( 2 )) e (a,O)r- 4 (c. (1) , coA a(2) )r-i(c,(1) , CoA a (2) )r . (A.39)

Wick ordering of e ( '' 0) r, the symmetry of CAA () and CoA, and Ônno = - 4/1, give

Z(a, 0) = (a (1) , 6A0 a (2) ) ee(", (c' C" = (a (1) ,

(- 60A)

a (2) )

e (, a( 1) coA

CoA a (2 )) r )

2)r

[2: (: 9 (:: 0

)2),)

Q

ocoA (er )

(A.40)

Let m 1 + m2 = p. By Lemma 2.6 att

[aai aaz(ct ,

a

,0 =[z(— ai • • aid acx a=o

(A.41)

A.4 Wick Reordering for Bosons

187

By definition of a( 1 ) and

a

a

A

mi

a

EE

COA 811 ( 2 ) ) =

a

,

,,ctk COA(Xk i , Xk 2 )

(A.42)

aak2

ki=i k2=rni+i ° 1

Every derivative can act only once on a l ...ap, without giving zero because the a's are all different. So

,_,

a

a

)

(A.43)

exp ( u oii5-c--17)-ac 7iTj al • • . alL

=

a ,oaak1 Co A(Xki,Xk2) , -ff (— ) ai ... am n

E

ua k2

LCM1 xM2 (ki ,k2)EL

where M1 = { 1, ... , mi} and M2 = 117/1 +1, • • • , pj. L = 0 contributes to the sum, but gives the t-independent result a l ... ao , so the t-derivative removes this term in (A.40). Let r i (k1 ,k2) = ki and ir2 (ki , k2 ) = k2 . For a term given by L $ 0 to be nonzero, ri IL and 7r2 IL must be injective, because otherwise a derivative would act twice. Thus the sum can be restricted to the set

L= { LCM 1 X M2 :L(1,1, and irk IL injective for k = 1,2}

(A.44)

If L E L, ri (L) = r2 (L). Thus

min{mi,m2}

r=

U i=1

U

r(Bi, B2)

(A.45)

Bicmi 1 1311=1B21=i

with

L(B i , B2 ) =

IL E L: 71- 1 (L) = B 1 and r2(L) = B2}.

(A.46)

Let

B1 =

{N I • • • ,bi} ,

B2 = {bi+1, • • • ,b2},

1 < b i < . . . bi <

171,1,

M1 ± 1 < bi+1 < ... b2i < M1 + M2,

(A.47)

then for any L E .031, B2 ), there is a unique permutation it E Si such that

L= {(bk,b i+ ,(k )) : k E { 1 , ...,0}.

(A.48)

Thus the sum over L splits into a sum over i > 1, a sum over sequences b = (b i , . . . , b2i ) with 1 < bi < b2 < ... < bi < mi < bi+1 < . . . < b2i < mi + m2, and a sum over permutations it E Si, and

(A.49)

A. Appendix to Chapters 1-3

188

exp

a

:0)

,c0A acta(2) )

(A.5 0 )

min{mi,m2}

E. H D(xb k ,xb i+ , ( ,, ) ) H(i, b, r)

E i=1

rESi

b k=1

with H(i,b,7r) =

( k=1

a

8

=

The derivatives give (calling

(A.51)

-

8abk aabi+w(k)

, IL

H(i,b,r) = II ak

(A.52)

k=1

The remaining derivatives acting in (A.41) come from the Wick ordered exponential in (A.40). They now act on H. By Lemma 2.6, they give

I

[(2c0A (e (445) )

H k=1

43 / C04

kgg

a=.0

(mi+m2

QCO A

H k=1

106

(A.53)

ocxo)

The final step is to rename the integration variables to rewrite the Wick ordered product in the form in which it appears in the result. Before that, we use the permutational symmetry of the G,7,r (t I x(k)) to permute its arguments such that x l , , xb i appear as the first i entries. Then the summand does not depend on b and it any more, so the sum over b and r, with the constraint (A.49), gives

E cE c

rESi 131

mi,B2

(M1) (TT12) il

(A.54)

M2

1B11=IB21=i

and y becomes min{m i ,m2 }

Y(r 1 ,m 1 ,r2 ,m2 )

f

axm

i=i

Yri ,m i ,r2 ,m2 ,i(xi, • • • , Xm)

with

f2C0A

(H

0( X 0)

k=i

(A.55)

A.5 Yr1,m1,r2,rn2,i (X)

=

189

The Lower Bound for the Sunset Graph

H COA(Vk,Wk)

f dvi ... dvi dwi ... dwi (— N 8

k ---71 Gin n

, vi, xi) Grn All,,,° 2 (w 1 ,

, wi , x2 ) (A.56)

with x i = (xi, , xm l -i) and x2 = (xm i -i+1, • • • , xm ). The derivative with respect to t and an additional reordering give another factor i. Let 1 = i — 1, then (m1) (m2)

i

i

= m in2 n (m i

(A.57)

1) (m2

1 which is the combinatorial factor defined in (3.34). The constraint 2i + m m 1 + m2 implies the constraint in (3.33). Performing the Fourier transform and renaming m1 = m', m2 = m" etc. gives the RG equation in momentum space. In the graphical interpretation, most permutations ir id occurring in the sum give rise to nonplanar graphs. If one restricts the sum to planar graphs, only ir = id and the shifts r(i) = i+ k mod i, k E {1, , i 1}, remain. Thus only i of the i! permutations give rise to planar graphs. The combinatorial factor corresponding to a `planarized' theory obtained by leaving out the 1! is therefore .1- 711 1 m 2 1 = m1 M2

(m1 —

1

1) (m2 — 1)

(A.58)

A.5 The Lower Bound for the Sunset Graph In this appendix, we prove (3.104). Since K' (x) = 0 unless x E (1,4), aAcAA 0 (q)

= —K

a2 2 q2 2) A2 A3 q2 + mo2

vanishes unless A2
/Alio (0) = f d 3P CoA(p) f d 3 q CoA(q —

(A.60)

aAcAA 0 ( q) •

The integrand is nonnegative, so /AA,, (0) > J(A) where

J(A) = f d 3p A (' ., 4)

2

P2 + 1mo

f A( IA,

d3 q CoA (q - p)

aAcAA0

(q)

(A.61)

IX)

with

A(a,b) = { p E R3 : a < II < b}. Since Ki(x) < 0 for all x E (1,4), there is K0 > 0 such that minf I K i (x) I

E 4) 2 , (751 ) 2 ll Ko•

(A.62)

(A.63)

190

A. Appendix to Chapters 1-3

Thus, for A > m and q E A(1A, aAcAA0(q)

(1.4) 2 2K0 K0 1 > A3 (0) 2 + mo 2> 3 A3.

Moreover, in the support of the integrand for

J(A),

— III > A— = A,

1g — A

5 lq

A

iPi < 4 A +

(A.65)

3A 2

(A.66)

> 0,

so, with L0 = miniK(x) : (i) 2 < x < COA

(A.64)

1 = (q — p) 2 + m 2 K( m 2

)

1,0 (A.67) A2'

4°

3 2+ mo 2 (0)

Inserting these bounds, we get A

AA)

>f

d3p L 0 K 0 p2 mo2 A2 3A3

4

f

d3q = k 1

f IrciP

A2 p2 + m02

tA)

A(4„1-)

9

(A.68)

8

with

= (-423r-)2KoL o 4) 3 — ( -9§- ) 3 ).

(A.69)

In the last integral, we use m 7 2 < A2 < (8p) 2 , i.e. , mo2 + p2 <65p2 , to get A

A 4

p2 dp

A I dP 520 .

>1

p2 + mo2 fig

A 8

Thus the lower bound in (3.104) holds with Qi = 5k2o •

(A.70)

B. Appendix to Chapter 4

BA Fermionic Fock Space We just give a brief summary here; for details, see [64]. If 7-t is the oneparticle Hilbert space (assumed as separable), the configuration space for an n-particle system is lin =1-1 0 ... 0 7-1 (tensor products are defined, e.g., in [65 ] ). The Fock space .F over 74 is defined as

00

T = 0 1-1 12

(B.1)

n=0

with 7-1 ° = C. elements of .F are sequences V) = (0(0) , 0(1) , ...) with 0°) E C, 0 1 ) E 71, etc. The infinite direct sum is a Hilbert space with the natural inner product (B.2) ((p I oF = 70°) + E((p(n) i on))-k.• n>i The set Y0 of finite vectors, which have only a finite number of 1/,(11 ) nonzero, is dense in T. Let P± be the linear operator defined by

1

P±(fi 0 • • • 0 fn) = 7 E (±1)hr ( 1 ) 0 • • • ® fw(n) n

(B.3)

• 71-Esn

where (+1) 7 = 1 and (-1) 7 = E(R)is the sign of the permutation r. The P± are projectors. The Bose Fock space is I.+ = RE T, the Fermi Fock space is .F_ = P__ T. For f E 7-1, the creation and annihilation operators a* (f) and a(f) are defined on .F0 as the linear operators satisfying a(f)0 (°) = 0, a* (f)(°) = f, and a(f)(fi 0 • • • 0 fn) =

AFL

(f

I

h)7.1 f2 0 • • • 0 fn

a*Cf)(flo•••ofn) = Vn+ifofio•••ofn.

(B.4)

The map f P,4 a* (f) is linear and the map f I-4 a(f) is antilinear. An elementary calculation shows that on 7tn, Ila(f )11 < AFL Ilf II and Ila*(f )1I

192

B. Appendix to Chapter 4

r\ri7— F 1 11f II. The bosonic and fermionic creation and annihilation operators are defined by

a(:) (f) = P±a( *) (f)P±.

(B.5)

From now on we focus on the fermionic case. The fermionic operators obey the canonical anticommutation relations (CAR) {a _ ( f), a_ (g)}

= { al (f), a!_(g)} = 0 {a_(f),a(g)} = 1 (f I g)7.1.

(B.6)

A consequence of the CAR is that the fermionic creation and annihilation operators are bounded,

11a(f)II= Ila*(f)11= If II -k ,

(B.7)

and thus extend uniquely to bounded operators defined on T and that, if (fcx)crEA is an ONB of 7-t, a ONB of T is given by the vectors 9 = (1, 0, 0, .. .) and HaEs a(f)Q, where S runs over all finite subsets of A. The physical interpretation is that fl is the vacuum, the state without any particles, and that the n-particle state n

H a(f)fl = -VW!P_(f al ®... 0

fa n )

(B.8)

kr=1

has n fermions occupying the states fai , ... , fan . The CAR imply in particular that a* (f) 2 = 0, i.e. that no state can be occupied twice. In the text, we only consider the Fermionic Fock space, so we drop the subscript and write a(f) = a_(f).

B.2 Calculus on Grassmann Algebras Let N be a finite set, without loss of generality Ar = {1, ... , N} (this is no loss of generality because the Grassmann Gaussian measure introduced below changes at most by a sign if the elements of Ar are permuted). The formal algebraic objects 01 , ... ON are called Grassma,nn variables if they obey

:= lkilAi + tAilPi = 0

(B.9)

for all i, j E A. The Grassmann algebra C { i, - - -, ON} generated by 0 1 , . ON consists of polynomials

g = b(g) + E E ci i ,...,in (g)Oi l .. . 0in n>i ii,...,inEA

(B.10)

where b,ci1 ,...,in (g) E C, with addition and multiplication defined as usual for polynomials and by taking account of the anticommutation rules for the generators 01 , ... , ON, namely, for any permutation ir,

B.2 Calculus on Grassmann Algebras • • '. Ikr(n) =

E(7)1141 " . Viin .

193

(B.11)

The coefficients c 1 ,...,j (g) in (B.10) are not uniquely fixed by g because the set of all monomials in the generators is not linearly independent by (B.11). However, the set B = {1} U { lPi, ...tpin I n > 1,i 1 , • • • , in E A, ii < i2 < . • . < in}

(B.12)

is a basis of the vector space C{0 1 , . .. , OAT}, so any element g can be written as

g

=

b(g)

+E

E

sii ,...,in (g)0i1 • • • iPin

n>1 ii<...
= b(g) +

E silk!

(B.13)

icA and the map A -4 C2N , g 1-+ (b, s) is a bijection. In other words, we may assume that the coefficients are totally antisymmetric, ci,r ,,,,...,i,r(n) (g) --, E(7)c i1 ,..., in (g)

(B.14)

because any other part would cancel by (B.11). There are plenty of ways to see that A = C{01 , ... , ON}, as rather sketchily introduced above, is a well-defined mathematical object, indeed, a C-algebra. In fact, if (fa ) a is an ONB of a Hilbert space 1-t, then the fermionic creation operators a* (fa) form a Grassmann algebra, and the same holds for the annihilation operators (if 1-1 has infinite dimension, this Grassmann algebra is infinite-dimensional). An explicit matrix representation p(A) C M(2 N , C) is generated by

_

P ( N) = (73 (0 ... (0 u3 0 u..1 0 (73 0 ... 0 0-3,

(B.15)

where a l , u2, 0 3 are the Pauli matrices and a+ = al +iu2 appears as the k-th factor of the product (B.15). The nilpotency Vi E A (B.16) implied by the anticommutativity (B.9) has many important consequences: the degree of any element g E A, defined as the maximum value of n for which a monomial with n generators occurs in (B.10), cannot exceed N. Moreover, viewed as a ring, C{0 1 , ... , ON} is certainly not a domain of integrity since every element without constant term b is a divisor of zero. In the framework of C*-algebras, the situation may be summarized by the statement that the spectrum of every g E A is the set consisting only of the point b(g). If b(g) 0 0, the inverse of g can be calculated as a geometric series which terminates after a finite number of terms because of nilpotency.

194

B. Appendix to Chapter 4

Functions on a Grassmann algebra are definable whenever they have some formal or even convergent power series expansion. Let f be a formal power series in some indeterminate z,

f

= E f kZ

3

(B.17)

k>0

then for all g E A with b(g) = 0, f(g) can be defined as

f(g)

Efkgic

(B.18)

k>0

since the sum on the right side involves only a finite number of terms. In the case b(g) 0, f (g) can be defined similarly by Taylor expansion around z = b(g) if the first N derivatives of f at this point exist. In analogy to the bosonic, that is, usual, differential and integral calculus, derivatives and integrals with respect to the Grassmann variables are given by special linear mappings. The derivative with respect to Ok, acting to the right, is the map

a

A

(B.19)

E(-1) , 16k,ifoil•••ipir_loil + 3.•• .1)b_n

(B.20)

490k

:

A

-4

determined by its action on elements of B,

490k

• • • Oin

1=1

and the requirement that it be linear. Similarly, the derivative with respect to IN, acting to the left, is the linear map 4--

a

alPk 473— IPT:1 Win

We

A A

E(_ir —i6k,i

0.21 • • •

(B.21)

For all i, k E A

(B.22) tNi 490k

aOk a'çbi

themselves generate another Grassmann algebra, and that the 4-a same holds for the —. (90 We have seen already when introducing Fock space that changes of basis in the one-particle space 1-1 correspond to linear transformations of the annihilation operators. Since the algebra of operators exists independently of the basis chosen in 1-1, such transformations do not change the Grassmann algebra generated by them. Similarly, the Grassmann algebra generated by so that the

195

B.2 Calculus on Grassmann Algebras

V = A(01 , ... opN)T is the same as that generated by = ON, • • • if A E GL(N, C). In particular, if 01, • • • ,I,bN and 771, • • • ,riN are Grassmann variables, then /Pi +7,1, ••• ON +riN generate the same Grassmann algebra as /Pi , • • • , IPN and 711,...,7w (This is a change of generators (0) 71 ) T 1- A(0) n) T (1 1) \ with A = 0 1) )•

Under a linear change of generators, = (01, • • •

with A

E

,ON) T 1-+ V = A(tPi, • • • ,ON) T ,

(B.23)

GL(N,C), the derivatives transform as 0 a = (A-1Y — IM' ,Mt

that is, for all a

E

(B.24)

A

a ,a ath

N

1--a.

= k=1

a'çbk

(B.25)

Contrary to the bosonic case, the integral with respect to the variable is defined to be identical to the derivative,

a

f do k g = —g 8114

Ok

Vg E A.

Multiple integrals are denoted by

f

d, .. . dOir, =

f dOi l f d 2 ... f (17,bir, ,

the most important case being the integral with respect to all generators (B.28)

which maps A-4 C. Lemma B.1. For all a E C,

f

diP1 F(01 — anl ) 02, • • . , N) — f di F(IPI, 02, • • • 2 ON) •

Proof. In every polynomial of the Grassmann algebra, such as F, the vari-

able 0 1 can appear only linearly by unchanged by addition of aiii .

nilpotency.

Thus the coefficient of

Vii

is Ill

From now on we assume that we have an even number N = 2L of generators which we label as /Pi, • • • '0/, and 6, • • • ,'Ç-b-L• Corollary B.2. Let A,B E MN(C). Then L

f 11

k=1

dtPkdOk f

L

OP + All, V) + BO

(B.30) k=1

196

B. Appendix to Chapter 4

B.3 Grassmann Gaussian Integrals For Q E ML(C) let (7,b

=i,j=1 E

I op)

(B.31)

and let DODO = d1 d1 • • • chbLd'IPL•

Lemma B.3.

f DODO

(B.32)

e - O-P-1 Qt1') = det Q.

Proof. Expanding the exponential, we have

I = f DI-PDO e-(;-61 Q1P) =

(1)P Eœ — f Dopo (7i p=0

PI

op)P.

(B.33)

(IT) I OOP is a monomial of degree p in and p in IT). Thus only p = L can contribute to the integral, and

f(o goLD,,T)Do

E 11 Qikim f DtpDo II t

Pikik .

(B.34)

k=1

k=1

The anticommutation rules for the Grassmann variables give

H 1,-/).0p20. =

L

L

k=1

k=1

( - 1) 1(.+1)12

k=1

=

H

-1) (L +1) /2Eil , , iL

Oik

, ...,3L

H

k=1

111Pk k=1

(B.35)

HI,T4Ok• k=1

Since

f chAr d0 r 7.1)r,1.1)r

15

1

L!

the Grassmann integral gives a factor (-1) L , and

E

Qikik

il,•...iL k=1

H Q w(k)(7(k) D 7r,crESL, 2 ME(47) k=1 -d

E E(p) H pESL,

(B.36)

E

Qk,p(k) .T! Le 1 = k=1 71. ES L

det Q. 1111

B.3 Grassmann Gaussian Integrals

197

Definition B.4. Let Q E GL(N,C). The Grassmann Gaussian measure with covariance C = Q-1 is the linear functional f i--+ (f)Q on the Grossmann algebra given by

,

L

1

TT (1

(f ) c2 = det Q f : LI

1-P dtPic e

f (IP 5 0).

(B.37)

k

We also write this as

(f)Q =

f

clitic(0)0f(0)0)-

(B.38)

If f depends on no other Grassmann variables, (f)c2 E C. But f may also depend on other Grassmann variables I?, in which case (f)Q is an element of the 'remaining' subalgebra generated by the 9. Remark B.5. Since f 1.-+ (f)Q is a linear map on finite-dimensional spaces, it is continuous in f. If f depends differentiably on a parameter A, derivatives with respect to A can be taken under the integral sign. Lemma B.6.

f clize (

5 0)e( 4 11,1))+0,T177) = e (771c77) .

(B.39)

Proof. Completing the square and using Corollary B.2, we get

f =

DOD/pe - (17'1Q0 )+01,,b)+(17, 177)

e Olen) f D,1-pDoe -(1:6-c T filc2(7,b-07))

(B.40)

= e (41c77) f NDIPe - OPIQO = eN cn) det Q.

• Lemma B.7. If fN -' (f)l-fil lfri , then = L_,I,Jco,...,L1401.,

(n cj =

E

(p 1 j (f)(_1)1 / 1(1 / 1 -1 ) 1 2 det Fj/ (C)

(B.41)

I ,J C{1,...,L }

1 1 1=1.1 1

where I'Ll(C) is the III x I/I matrix with elements (r.,(c)),,, = c i,, (and I = fi l , ... y ia J = 1:715 • • • f iLl, with i l < ... < iL and il < ... < iL). Remark B.8. The çoij(f) may be Grassmann-algebra valued, as long as they are independent of i/, and tp .

198

B. Appendix to Chapter 4

Remark B.9. Setting j = 0 can be done by putting Ilk = akxk, ak E R and Xk Grassmann variables, and letting ak -4 O. Since all anticommutators of Grassmann generators vanish, the ijk form a Grassmann algebra. Proof. By linearity of ( • )Q, it suffices to calculate (17/0.1 )Q . By expansion of IJI because only monomials the exponential we see that (i i tirl )c2 = 0 if I/I of the same degree L in and have a nonzero integral. Let I/I = I JI = m.

We write

in V-' 'V

Ti *e oltp)+(,--pi,) [H nt (— ,...,„i k )

=

(B.42)

77=o Exchanging the differentiation with respect to q at ii = 0 and the integral with respect to 0 is allowed since all operations are continuous and they commute. By Lemma B.6, k=1

H

(0 I 1P" c2 {

o 77.9, )

k=1

TT a e771)] eqjk

k=1 k=1

rm

77=0 L

M

HH

(— _

1=1 r=1

=

m

L

E

17=0

in

H(

II cirri

[k=1

ri,..•,r,n=i i=1

in

a7) .,.

in

=

II 7int] 77=0

1 =1

in

M

E H ci,„ [H ( ani ) H n E k=1 1.1 1
h

E6

irES,n

(7)

E

77=0

,rm (-1)m(m+1) /2 Citr1) H (5im

1
(-1) m(m+1) / 2+m

.r,r(01

k=1

det rji(C).

(B.43)

• Corollary B.10. If C depends continuously or differentiably on a parameter t, then for all f E A, f dpc(1 1- ),IP)f et-li,i,b) depends continuously or differen-

tiably on t. Proof. f dpc (tp ,

f (11), 0) is a polynomial in C, hence smooth in C.

1111

B.4 Gram's Inequality; Bounds for Gaussian Integrals Definition B.11. Let q > O. For f=Efi x rpi , with fij E C, let

Ilf II, =

E ihAqi (1/1 + 14 .

(B.44)

B.4 Gram's Inequality; Bounds for Gaussian Integrals

199

Lemma B.12. For all q > 0, II • Il q is a norm on the Grassmann algebra oh}, and for all f and g E A, A = C{th,

Ilfg11,

(B.45)

11fIl q 11g11, •

Proof. It is obvious that II• 11 g is a norm on A. The product inequality holds because

E ih jgic

i lfg li q =

(B.46)

InK =0 J

and

E

IlfIl q 11911 g =

(B.47)

I ,J,K,L

•

Lemma B.13. (Gram's inequality) Let 1-1 be a separable Hilbert space, and for f = (f11 • • • fn) E lin and g = g71) E na let

(B.48)

(f, g) = det (((fi I gi))1
Then

I

f,

012 < (f5 f) (615 g

I ( f,f)1 = Idet (((fi I fi))i,i)I S

(B.49)

)

11 lifill2

(B.50)

i=1

and

1-k=1111.611 ligkil •

1(.f 01

(B.51)

fi fn and Proof. Let .T be the Fock space over 1-t, and let F = G = NrniP_gi 0 ... 0 gn (where P_ is the antisymmetrization projection). Then (F I G).F.

= =

fi 0 • • • 0 in 1 n!

E

E(r)e(u)(f7T(i) 0 • •

fir(n)

I 9,(1) 0 • • • 0 ger(n))

,crE S n

1 TT.

E 1 =— E n! =

P-91 • .. 0 9n)

E(r)E(7)

k=1

E (ir-i

0 0.) H(fk I

I gOE(k)) N.-1 0 , (10 )

k=-1

7r,aES,

E or) 11 7TESn

11(f,o)

(fk

I gir(k))

k=1

= det ((fk I gi ))

(B.52)

200

B. Appendix to Chapter 4

Thus (B.49) follows from the Cauchy-Schwarz inequality for the inner product on T. Consider now the quadratic form on Cn defined by 2

(B .53)

u 1-+ Q(u) =

Obviously, Q(u) > 0 for all u E Cn. Since Q(u) = Ei,j u,i (fi I fi )ui , we see that the hermitian matrix 0 given by 0ii = (fi I fi) satisfies 0 > 0, so all eigenvalues of 0 are nonnegative, anddet 0 = (F I F).F > O. If fi , ... , fn are linearly dependent, then, by (B.53), Q(u) = 0 for a nonzero u, so 0 has a zero eigenvalue. Then det 0 = 0 and (B.50) holds trivially. If fi , ... fn are linearly independent, Q(u) > 0 for all u 0, and the Gaussian integral

n f TT ......., du (det 0) -1 = 11

dft-ie _ ,1. Q(u)

27r

(B.54)

converges (here for u = a + ib, dudft = dadb). Since fi 0, we have P = (fi I fl) > 0 2 and (B.55) Q(u) = (p lui l 2 + ( du l + a g i ) + 0- (ii) where Q.-- (ii) -- E 1,i > 2 %O&M and a = a(E) = Enj=20 1 .7AL.7'- Thus (

Q(u)

4 + (ft)

= (Plul — <

12 _ (u ). (dUl - 2 1 + Q 4/

(B.56)

By Fubini's theorem we can take the u rintegral first in (B.54) and use (B.56) to get duid% _,..,„-,) f -1 > f H ( d e t 0 Û ) 2ir e 2

0, i=2

duldfil e -i(plu1-,-'12

V'

2ir

(B.57)

The n i -integral gives 1 ; the integral over ii then gives the inverse determinant of the matrix where trie first row and column of 0 are removed. Thus

I

(B.58) Ilf111 2 det (((fi I fj))2
Theorem B.14. Let C = Q-1 have the representation

Cii = (ai I bi),

(B.59)

where al, .. . , an and bl,...,bn are vectors in a separable Hilbert space 1-1, and for all i E {1, .. . , n}, let Ilaill2 < q and iibiii 2 < q. Then

I(f)QI

11f11, •

(B.60)

B.5 Gra.ssmann Integrals for Fock Space Traces

Proof. Let

'.

f

By Lemma B.7 and (B.59), det((a I bi))i er JE J

E

(f)Q =

201

.

(B.61)

1/1=1.1 1

By Lemma B.13,

1-11

E IhJIMaill11b1 11

i(f)c2i

il

< E

(B.62)

Dropping the constraint that I/I = IJI from the sum, we obtain the bound by 11f11,. Remark B.15. This is not the only determinant bound one could use. There is also Hadamard's bound n

I det AI <

11 i=1

(

n

1/2

E lAii i2 i=„.

(B.63)

or the much cruder bound I det AI < —H7 1.1(E73 .1 IA • • I)

B.5 Grassmann Integrals for Fock Space Traces In this appendix, we derive the Grassmann integral representation for the Fock space traces. The presentation follows [66] (except for the convergence proof, which was not given in [66]). Recall that a basis of F is given by FicEA ae* f2. Let A be the Grassmann algebra generated by (1k))c Er, and let p be the linear map p : A, defined by p(OA) = = HcEA 7RO. Since p maps a basis to a basis, it is an isomorphism of and A. If y E has the representation y = E A VA0,6 5 then p(v) = EA vb = v(z-p- ). The action of the creation and annihilation operators is

=

p(div) = //)(6v(I7j),

a

p(ay) = av_,(e) vo,b)•

(B.64)

Since

I

A')

=

= [(-40 3 VAi l 1,==o 5

(B.65)

where i =[LE but with the product taken in the opposite order, the inner product on .F has the representation

202

B. Appendix to Chapter 4 (.1)

1w

)

= [1-) (-Lotp ) w

(

j)] 1,0=0 -

(B.66)

-

where 00) = EA Any linear operator Q E £(.F, .T) can be written as a polynomial in the creation and annihilation operators. The normal ordered form Q (a* , a) of such a Q is the form where all creation operators are moved to the left of all annihilation operators. Specifically, we define

Q(a*, a) =

E

Qxy

H a'c' IT an

(B.67)

C EX nEY

X,YCr

where the , on the product means that it is taken in the opposite order. The action of a linear operator Q with normal ordered form Q(a*, a) is transferred to the Grassmann algebra generated by the as

p(Q (a* , a)v) = A

i) v(0).

(tp, a

(B.68)

B.5.1 Delta Functions and Integral Kernels We now derive an integral representation for the action of the operator A on an element y in the Grassmann algebra, as given in (B.68). For two Grassmann variables ,t-p and i-ki, the `Grassmann delta function' is simply given by

6 (0 — 0' ) = (0' — 0)

(B.69)

because for f (i) = a + b,

f do' (01 — 0) f (0 1) = f

().

By an expansion of the exponential and the nilpotency of the Grassmann variables, one sees that

(5 (V; —

1) = f

do

(B.71)

One should keep in mind, however, that the thus introduced 'delta function' is an odd element of the Grassmann algebra, so that one has to take care of the order in products. To avoid this, one combines the formulas to

f

ciO'

f do J; - ') 1P f (01 ) =

(B.72)

Now everything is reexpressed in bilinears that commute. The generalization of this formula to finitely many Grassmann variables is as follows.

B.5 Grassmann Integrals for Fock Space Traces

Lemma B.16. Let f

f (

()

E .4r(0).

IIEr dOi(x)(10(x))

203

Then

eE.Er((z)-01(x))0(x)

PO'-= f(0 ). )

(B. 73)

x

Proof, f is of the form f(i) = EAcr fstr»d . By linearity of the integral,and because bilinears commute, the left hand side of (B.73) is

h f II (dol(x)do(x)e(0(x)-1(x))0(x)) E Acr

(11)1 ) ,6 ,

(B.74)

xEr

and it does not matter in which order the product over z E r is taken. Consider the contribution from a fixed A, and reorder the product over z E r such that the integrals with z E A are taken first. For z E A,

f lit-pi(x)(hp(x) eo(x) -

7 -1 (x))cx)

(0°

= f d0x) (V(z) - OW) (0 ° = f di '(x) ( -0(x)) (01) A

(B.75)

= 0(x) f dV(x) (V) ° = '0(x) ak-b-t?(x) W) ° . Because ri)(x)--2— av(x) is bilinear, it commutes with all factors /- '(x'), with z' 0

z, in the product (,-//)°. Thus its effect is just to replace 17)' (x) by (z). After this replacement has been done for all z E A, the result is f('), which does not depend on

i-pt any more, so the remaining integrals leave it unchanged. Ill

Corollary B.17. Let Dr0.13r0 = Ei,Irl

fi

d0(x)4(x). Then

xEr

f DrI- DrIP e(''.

f (' -b-) =

(B.76)

Let y and w be in Tr. Then

(y I w) =

f DrODFIP e-(04) Op) w(0).

(B.77)

Let A(c* ,c) be the normal ordered form of the operator A. Then

(Ay)(1 1- )) = f Dr5eDrX e -( "'x) AN j ) X) v(i- )

(B.78)

24(0,x) = A(), x) e (?-P- 'x)

(B.79)

where is the integral kernel of A.

204

B. Appendix to Chapter 4

Proof. The extra power of E-1 in Drt-p-DrIP is there because (0,g) is defined as e EIP(s)q (x). By (B.76) f

\A ,f5f() = f DrIpDrIp e(x -1,-6 =0) 0A g t-p) ,

(B.80)

-z )

so (B.77) holds by (B.66). Because in the normal ordered form of A, all derivatives are on the right of all factors of , (B.80) implies (B.78). Note • that (B.78) holds only with the normal ordered form of A.

rp,

In (B.78), the action of A is expressed as a Grassmann integral. The integral kernel of the product AB is the convolution

(AB

)

(0) 0) = f Dri-Drxe

6x) A(0,X) 13- MO).

(B.81)

B.5.2 The Formula for the Trace The trace of an operator A on Tr can be evaluated in the basis Tr A =

0 j

E (0,d I A OA).

as

(B.82)

Acr

Lemma B.18. If A(c* ,c) is the normal ordered form of A, Tr A =

f DrODrIP A ( ,b0P)e(04) .

Proof. By (B.77) and (B.78), Tr A = E Acr

(B.83)

j with

ao, = f DrgDrODri-Drx e-0 '9)-(5-c'x) 60 A(,) XA •

(B.84)

The only nonvanishing contributions to the integral are those whose degree in j and 6) is the same, and similarly for x. Let 14 = d. Then only terms of A of total degree 2d contribute to aA. These terms have even degree, so in the integrand. Because we may replace 0 3 A(, x) k-A by Ã(Ü, x) û 03 k° = (-(x)61(s)), the sum over .A gives

IT

xEA

die' II (—k(x)0(x)) E dCr xEd = 11 e

= II ( , _ Er(X)0(X))

0 (x ) =

z Er

e -(R ' 8) .

(B.85)

xEr Renaming the integration variables as 't- = 0-) V) = xf A (B.83) because a shift in the integration variable gives

= 0, X = i",

we get

B.5 Grassmann Integrals for Fock Space Traces

f

205

DrADrA

= f DrADrA

f

=

=

e -(

+11j »(11) '''b) (B.86)

DriDrA e -( k A)

The trace of a product of n factors is n-1

n-1

Tr II Ak k=1

I Dri- k-DIAk = f Drk0DrXne ("° ' Xn) If k=1 n-1

(Ak H k=1

(kk- 1 , Xk)e-('7h-

5-0c-1 AO

•

(B.87)

Renaming the integration variables as 77k = k- k and qk = Xk-i-i , we get n-1 Tr H ilk

n-1

= f H (DrqkDripc

Ak(77k,11k))

k=0

k=1 n-2

eXP (E( 71k — 771k÷1 7 710+ ( 71-1 + 1)0) 71n-1)) •

(B.88)

k=0

The form of the exponent shows that the positive sign in the exponent of the integral (B.83) for the trace leads to antiperiodic boundary conditions. Thus

we define (B.89) rin = —7 10 ) 71-1 = —71n-1 whenever these variables appear in the above sums. The exponent then becomes 11n =

n-1

n-1

771k) 71k — Ilk-1). E (fik — fik+1 ) 71k) = E( k=0 k=0

(B.90)

B.5.3 The Time Continuum Limit

We have now made all preparations for deriving the Grassmann integral representation for traces of the form C(A) = Tr e—A , for A any linear operator on .7'. Since all linear maps between finite-dimensional spaces are continuous, - -„(0A), where ((/3A) is an entire function of 0, and (CM) = lim n-4 oc ,(0i1) =

.-71

Tr ((1 - ,,,ii)n)

.

(B.91)

B. Appendix to Chapter 4

206

We shall use this to rewrite of the partition function; rewriting the expectation values of observables is similar. Let A(a*, a) be the normal ordered form of A. Then, by (B.88), n-1

n-1

n (f3

.'

E oh mi. -77k - i) n-1

II (i. - ...i A(ii-k oik )) .

A) = f II DrqkDrincek=i k=0

(B.92)

k=0

Note that no extra reordering needs to be done when one uses (B.91). If one uses the Lie product formula, Lemma 1.1, one has ta calculate the normal ordered form (e -A) (a*, a) of eA , which is not the same as e—A(at 'a) . This is not difficult for simple Hamiltonians, but can get arbitrarily complicated for general A. We would now like to show that one can reexponentiate by writing

1 - .A(qk, Ilk) :-._- e4'1(71k,77k) with 00 n

B

(

n) -

,

E

1)=2

2—p

P.i

(

1 Di _ n iunknk, 11k)

(B.93)

— -

(B.94)

pAcq,q»P

and showing that any integral containing a remainder factor B vanishes in the limit n .- oo. The remainders are 0( 3 7 -), but the measure also depends on n and therefore the proof is not trivial; it makes essential use of the Gram estimate. We define the discrete Euclidean time variable rk = and denote T = Irk : k E {O,... , n - 1}}. The lattice spacing in time direction is er = Tk -f- i - Tk --= 2'. We rename the Grassmann generators as ti)(7-k) =-- 71k and Vi ( rk) = flk, all define (497- 0)(7-k) = -i-17 (0(7-k) - 0(71_ 1 )). With this, and the notation Dni-p- Dnti) = IL-(1) Drt-f)(7-k)Dr1,'(rk),

--g + to

er E[(0(7),(0,-lib)(7)) — A( 11;0110( 7 )))

4--71 (13A) = f D n 'tPD n IP

with

e T ET

(B.95)

Dn

n-1 Dn =

II

(1+ 3 7 - R(1-k(rk))

(B.96)

11'(rk)))

k=0

and R(rp- ,0) = - eci- A( 0,0) B(1)). Theorem B.19. Let A(a* ,a) be the normal ordered form of A,

and

6,(n (/3 A) =

f DD n IP e

-ET

Then ( (fi A) = Tr e -flA = lim (n (f3A). n-400

(B.97)

B.5 Grassmann Integrals for Fock Space Traces

207

The proof of Theorem B.19 constitutes the remainder of this section. We first prove that the Gram bound applies to the covariance 8,-1 ; this also holds for the more general case where a quadratic part of A is included - y)aay ,,, and that in the measure. Assume that A(a*, a) = Ex,y:8(x ,cr the Fourier transform of E is E(p). Then

E (cr) , (-0,--Ft)cr»

= f Dnl-ppno e

r

det(E, (-Or + E))

ET

Dn

f dye OP,

Dn

(B.98)

where C = (E r (-Or + E)) -1 . By Fourier transformation, we see that the eigenvalues of -Or + E are -iED+ E(p), labelled by (w, p, a) E Mn x r, where i6.) =E r(1 - e- k"-). Thus

det(E r (-0,_ + E)) =

(1 - e7-E(p)-

.

(B.99)

(.0,p,Œ)Emn xr

iwe, = Now WE,- = (2/ + 1), so, as l varies, ezi of - 1. Thus

ll

(X —

6 = Xn +

(21+1) runs over all roots (B.100)

1,

woin and therefore

(1+ ( 1_ fl E(p)) n ) .

det(E r (-(9,_ + E))=

(B.101)

(p,a)ErIn the limit n oc, this gives the partition function Ha + e-SE(P)) of the Fermi gas. The covariance C is the Fourier sum

=

1

o wENIn

PEAS

eiP(x- xi ) 50yr , • + E(p)

(B.102)

Lemma B.20. For all w G Mn , WE,- E Fir,r). Let E E R. For all n >

2014

f . sin WE, 0-E=1 Er

Per I

> i, then cos WE T

!PI

1(.0E71

<

(B.103)

IwErl <

(21+1). Since / E

Proof. For w E Mn , WEr = Write

If

if

, -721 -11, WE E [ - r,r).

+ -I- (1 - cosujer) - E. 6,

(B.104)

O, so 1 — cos we, > 1, and thus

i; . lirf) - El ? --..(1 - coswe r ) - .E >lii. - E ?

(B.105)

208

B. Appendix to Chapter 4

sinz is decreasing on [0,7r/2], so inf [0,7r/2] x

The function x Per < 112 ,

sin WET

- El >

tation as a scalar product IlaC11 2

=

2

w

ir

WE T

Lemma B.21. Let = (7-,x,o-) and .'

= a. 7r Thus for

= (7-1 , x', a-'). then C has a represen-

e) = (aCI bV) with

dEP—

(B.107)

n > max{4, 20 imEali IE(P)I}

(B.108)

11bVII 2

EL - c.00tr,

= 731

pEA*

If then

ei l (1 + logn).

iiacii2 = Ike

(B.109)

Proof. Let 7-1 = L2 (Mn x A* x f0, 270), with the scalar product 27r 1 de a(w, p, 0)b(w, p, 0). L—d E (a 1 b) =

f—

E

LeEm,‘

pEA* 0

27r

(B.110)

Choose at (w, p, 0) = erdIi3 — E(p)1- 112 e—i9a

(cv ,p,O)

=

• e •ILJT —ipx —

E(P)i

-1/2

—iea'

- E(p)

(B.111)

then C(Ce) = (ac 1 be) and (B.107) hold. On the right hand side of (B.107), sum over w first, and split the sum into those cd.) for which lwe r l < 7r/2 and we have the rest. By Lemma B.20 and by L- d Ep 1 =

<

L—d

E A pE /i*

E -1

(1 —

wEMn

ir 21w1

1

Iwt r i
(2

1+ wEmn

n/2-1 v-,

11 1=-10121 +

)

(B.112)

+ log n).

• The proof of the following lemma contains both the essential argument about convergence as n -4 oc and a hint at the complications that arise in the full proof.

B.5 Gra.ssmann Integrals for Fock Space Traces

209

Lemma B.22. For all (5 > 0 and all A E n-1

n -1-6 -, f diLe (1 P

as n

(e

E A(t-P( 7k)Aberk))

k.0

(B.113)

0

- 1)

oc.

co, we may Proof. Since we want to prove something about the limit n restrict to those n that satisfy (B.108). By Theorem B.14 and Lemma B.21, it suffices to show that, for qn = Er-1 (1 + log n), n-1

X n = exp (n —1-6E A(Ork), 001))) - 1 k=0

0 qn

oc. By the binomial theorem, applied in the form

as n

H eak - 1 = H(1+ (eh - 1)) - 1 = E 11 (eŒh - 1),

(B.115)

SOO kES

and the product inequality (B.45),

E sc{0,....n-1 so0

}

H kES

(B.116)

1

e

A((rk)4(7k))

qn

•

A(1 l-i(rk), 0(7-k)) is an element of the Grassmann algebra generated by the 11'1 generators (IN, /-,bc)c - Er. The maximal degree of such an element is 211'1. Thus, by definition of the norm, liA(' P- (11), 1,b(rk))II < aqinri with a constant a that depends on r but not on n. Expansion of the exponential and the product inequality (B.45) imply

H

e1 -5A(1J,(7-0,(7-0) q —1 11„

kEs

1 1) 11 "/ exP ( n-1-651

SO

n-1

Xn <

E s=1

(e n

This vanishes for n

n - 1 (en -l_s aisig eir 8

'

) n

oc because n -6 qinri

-1=

ean

qn

-

I.

0 in that limit.

Note that for (5 = 0, this is not sufficient to prove convergence. This is the reason why one has to use some analyticity property to prove Theorem B.19. The deeper reason for the nonconvergence of the norm estimate for (5 = 0 is

210

B. Appendix to Chapter 4

that the bound by log n overestimates the actual behaviour of C. It is nontrivial to prove better estimates because the sum over w is only conditionally convergent in the limit n -- oc. We now turn to the proof of Theorem B.19. Note that E(p) = 0 is allowed in Lemma B.21, so the lemma applies to C = Theorem B.23. Let

=

Ten

f

n-1

—

n-1 Ale II

diLC (CP 2 0 (II e n —

k=0

(B.119)

( 1 — .1:140)

k=0

where Ak is an element of the Grassmann algebra generated by et-P(7-k ,), 11)(1-k , V cEr . Thus IlAkii q < akel . Assume that there is an a > 0 such that

for all n, maxk< n — 1 for all p > 2,

00

I ak I

Rn ,p .— 0

a . Let 7-Z n,p be defined by Ten =

E

p=2

OP Ten,p . Then

as n .— oc.

Proof. By Theorem B.14, it suffices to show that iiRntpli gn - Oasn -400 for every fixed p, where qn = Er-d (1 + log n). Let Bk =

1 E rz' p>2

(-mot) n2_p __EBk,pop 1 P.

(B.120)

p>2

then

(B.121)

1 bpq111 iiBk,pli q < -1 with bp = aP/p!. Let Ar = {0, ... , n - 1}. Rn

=

E S U T=A( T#0

H (1 — t Ak)

H

kES

kET

Bk

=EE S u T=.1■1"

H(4,Ak) ri E Bk,r,„3". kET rk >2 ucskE,

(B.122)

TOO

Collecting powers of 0, we have AhL II (- n )

)

R71,1) =

E1

kEU

11

D. k,rk -I-)

(B.123)

kET

where E' denotes a sum over SUT =IV, with T 00, over U c S, and over sequences (rk)kET, rk > 2 for all k E T, such that

Erk +

in =P.

(B.124)

kET

We now take the norm and use the product inequality, and the estimates for iiAkiiq and 11Bk, r,",. With the notation Py = I r 1, we then have

B.5 Grassmann Integrals for Fock Space Traces E,

11Rn,plign <

H y±lz,,_ n-1-riiBk,rhii gn n

kEU

kET

7 ,y> WI Tr qn b E/f1 kn- agn ) n 7.1. 2 rk '

G -

211

(B.125)

!CET -

By (B.124), lUi< p, so arl (lagn l iU1 < qn " (-n.

(B.126)

Inserting this, we can resum the rk and U and get, with b = Er>2 b r <

I IR'n,P I lqn

<

E E (a\ lul

qn 11)

S LJT=AT

ucs

n)

TOO

E (1+ a\

qn 7P

1'

g7;0 7n 17, 1

n) Su T.--.1V TOO

(1

n qn " (0- X n r - 1)

where

(B.127)

n -y

xn = (1 -1- 17 ) -1

(B.128)

For x >0, log(1 + x) < x and ex —1 < xex, nso

(1 + xn )n _1 < en l°g(1 +xn ) —1 < enxn —1 < nxnenxn Since nxn < const n with

0
(B.129)

8> 0,

((1 + xn )n — 1) < const qn 7Pn -6

0

as n # oc, so Ten,p vanishes in that limit.

(B.130)

1111

We now only have to put all the pieces together to have a proof of Theorem B.19. Recall that ((OA) is an entire function of 0. The 1-n (0,4) are also entire functions of for all n. We know that 4--n (pA) ( (/3A) as n oc. It follows by expanding in 0, writing MA) =

E (2,01 )

-'71(0A) =

p>0

E

(B.131)

p>0

that for all p> 0 Cri,p (p as n oc. Now Cn (/3A) = Cn (/3 A) 7Zn , and'all summands are entire functions of for any n. Let Cri (f3 A) = Ep>0 (11 ,01 ) • By Theorem B.23, lim (n,p liM (B.132) ,

n-400

n-400

so the formal power series Ep>o zpOP with zp= lim n-400

(Tim converges and

represents the function ((OA). Thus Theorem B.19 holds.

212

B. Appendix to Chapter 4

B.5.4 Nambu Formalism It is useful for deriving the component form of the RGE to rename the Grassmann variables such that the distinction between //) and b is in another index. This is a variant of the usual `I•Tambu formalism'; see, e.g., [57]. Let

1= A x {-1, 1} x {1,2 } ,

(B.133)

and denote X = , a, i) E F. For x E A and a E {-1, 1 } , the fields are defined as 0(s, a,1) = 1,6,(z) and '(x, a, 2) = 0,(x). The antiperiodicity condition reads 0(x + Oer,a,i) = -0(x , a, i) with er the unit vector in r-direction. The Grassmann algebra generated by the (0(X))x er is denoted by A 1- [0]. Given another set of Grassmann variables (n(X)),(Er, the Grassmann algebra generated by the and n is denoted by AF[0,11]. Furthermore, denote fr, dX F(X) = ELI EirE{-1,1 } IA dx F(x, a, i) and a, 0, , ,i')) = 6ii' 45crai 6A (X, X I ) ) and define a bilinear form on '4r[/, ii] by

(0) 11)r = f dX

opo npo = - (71 ,0)

.

(B.134)

(tp, AtP) r, where, for X = (x, a-, i)

Then, for any operator a, (i), alp) = and X' = (x' , ct-' , i'),

0

A(X, X') = / a(x , a, x' , a') -a(x' , a', z, a)

if i = e if i = 1 and e = 2 if i = 2 and e = I.

(B.135)

In other words, when written as a matrix in the index i, A takes the form

( _O a a) T

(B.136)

with (aT)(x , a, , a') = a(x' , , x , a-) denoting the transpose of a. A is invertible if and only if a is invertible, and A(X, X') = -A(X 1 , X). With this, ZA n = ArA det a 2A where 2A = f diLOP)e -As4 ( 0) , where C = A-1 , and dit cis the linear functional (`Grassmann Gaussian measure') defined by (Luc (iP) = (det a) i DrIP e''10)". The constant N'A det a drops out of all correlation functions and can therefore be omitted. The 'measure' dpc is normalized, f dpc(0) = 1, and its characteristic function is ,

f dtic (0) e (77,20r = e i(n, c n)r

(B.137)

All moments of duc can be obtained by differentiating (B.137) with respect to n and setting i = 0.

B.5 Grassma,nn Integrals for Pock Space Traces

213

B.5.5 Matsubara Frequencies

To implement the antiperiodic boundary conditions in time direction, it is convenient to define the fields 0 on the torus T2 = ET Z/2/3Z and to require antiperiodicity with respect to translations by 0, (4.57). We have T2 = T U (T + 0) with T = { r E T2 : T = Er k, k E { t, ... , i7 z 1}}. The dual to T2 is T;' = i7 Z/2nT Z = {co = k : k E {-nr ,...,nr - 1}}. The Fourier transform on '1'2 is n - -

f(W)

-

-

1 = ET E enf(r), f(r) = — 7 e i " 'A LLY). T

20 1---d ET2

ET2

(B.138)

W

If fer -0) = -AT), then f(co) = 0 if ui2, is even. In that case, with j(co) = 1 f(r) = —A

E

e1"i(4.0)

(B.139)

,- cool"

with Mn, given by (4.63). The orthogonality relations are

f

dT e i("±wm)T

eh' = — 1 (STo - (57-0).

= 08mn,

Er

(B.140)

T

Lemma B.24. Let 0 > 0, E0 > 0, and a,- > 20(E0+ Em.), where Emax is defined in (4.74), and let 6) be defined as in (4.65). Then for all k E B and all co E Mn,, Ii rj— E(k)I < E0 implies PI < riEo and IE(k)I < 2E0 , and , E ,T3 cd EM n

1(10• — E(k)1

E0)

E0 1(IE001 2E0) .

(B.141)

,

Proof. By (4.65), 1m c2 = ,-(1 - cos(we r )) and Re iD = -, sin(wE T ). The condition Ii rt) - E(k)I < Eo Implies IRe EDI < E0 and 1 1 n 4.-D r+ E(k)I < E0 . Thus 11m .r31 < E0 + Emax . Since 1 cos x > 37 -x2 , -

2 (E0 ± Em.)ET > ET Ilm ï.-4:31= 1 - cos(wE r ) > ;17 (wET ) 2 ,

(B.142)

SO

iwEri

I- 0(E° + En.)(2nT ) -1 ) 1 /2 < ;.

(B.143)

Since -il sh is decreasing on [0, S], E0 > 'Re rA31 ? iwii

sin(wEr) 2 I ? - Icol. 7r wEr

(B.144)

So 'col < frrEo . Since 1- cosx < lx2 ,

1 1m 6)1

<

* P ET)2

(B.145)

214

B. Appendix to Chapter 4

Thus 11m 1.4) E(k)1 < E0 implies 1E(k)1 <2E0 , hence 71(10 - E(k)i < Eo) <E0 ) 11(1E(k)1 <2E0 ) . The summation over Mn, gives < taz,

E

2

E

71 Owl < E0 ) =

wEmn,

ii (12n+ 11 <

< E0. (B.146)

n.-

Note that the last inequality holds in particular if 2P- <1, because then the sum is empty, hence the result zero, because 2n + 1 is always odd. • We end this appendix with a brief proof of (4.68), which concerns the limit n, oo. At temperature T = > 0, the poles of the Fermi distribution f p(E) = (1+ e)' in the complex E-plane are at E = icon , co =-- (2n+1) E M. For n7- -4 oc, the free propagator at positive temperature T > 0 is C(r, E(p)), where

5

E) =

-e-TE (1 — f p(E)) e _ T E f 0(E)

if T if T

>0 < 0.

(B.147)

Note that this obeys the antiperiodicity

C (r + 0, E) = -C(rE)

(B.148)

so that C is periodic with period 20. By (B.148) and because eiwn 0 = -1, the Fourier coefficients of C are O

2 en (E) = — f (IT C 20 -0 The Fourier series

converges for T

ei wnr

1 1 0 icon -E •

1 e -iWnT Cer ± 0, = 1---d icon - E 8 nEZ

OZ. Thus (4.68) follows by taking the limit T t 0 of (B.150).

B.6 Feynman Graph Expansions It is a natural question if there are any changes in the proofs of the connectedness theorems in Chap. 2 if we start from a fermionic theory instead of a bosonic one. In this section, we briefly discuss how the proofs given for the bosonic theory extend to fermions. We shall see that all essential formulas carry over unchanged, only that the variables are now Grossmann variables, and that the few formulas that do get changed get only trivial changes. In particular, we show that the proof that only connected graphs contribute to the effective action is unchanged even though the Grossmann algebra is noncommutative. In a nutshell, the reason for this is that the polymer activities

B.6 Feynman Graph Expansions

215

are elements of the even subalgebra, which is commutative, so that all results of Sect. 2.4 apply. The main problem in generalizing the previous representations to the fermionic case is that the fermionic algebra is noncommutative whereas we have assumed, e.g. in the polymer expansions, that the activities are in a commutative algebra over C. We now show that one can arrange everything such that only elements of the even subalgebra A2 [0, n1 occur, so that the assumption that everything commutes still holds. The only place where the fermion signs appear is at the very end, when the derivatives are taken and evaluated at 7./ = O. The definition of the fermionic effective action,

e a eff (0, c,

-

A v) .

f dttc,(0) e-w(d)+) ,

(B.151)

is formally identical to the one for bosonic fields, only that now the Grassmann integral is implied in (B.151). Similarly, (2.117) carries over, only that the v(mk) are now antisymmetric with respect to permutations, because no other partwould contribute, by the permutation antisymmetry of the Wick ordered polynomials. Thus we have CO

Geff(p,c,v)= EG(eifi(p,c,v)

Ai-

(B.152)

r=1

with Ge ff OA

C, V) = ET ( V,

...,

V, C, I,b)

(B.153)

and (see (2.121))

(B.154) P

f 11 d, 11 vrk)(xo

E Mi,...Mp

bEBO

k=1

a [ ll — E(q, C, CIAO] • am bE Bo n=0

In the product over derivatives, the order is from small to large k, and for every k, ik from 1 to mk, and

f

= L aA1 aP aA log ZFI

(B.155)

...

with ZF

=

f

P

CliiCi (0)

II (I. ±

(B.156)

k=1

(the A's are still ordinary complex variables, of course). Because of the simple algebraic nature of the Grassmann integral, there is not even any convergence question for this integral.

216

B. Appendix to Chapter 4

At this point we see that all factors are bilinear in the fermions, so all activities are in a commuting algebra. Expanding as in (2.126) and doing the Gaussian integrals, we get exactly the same formula as (2.127), only with ia replaced by ri. Thus all activitis commute because they are in the even subalgebra, and Theorem 2.17 applies, hence (2.128)42.139) hold, with ia replaced by ri. Thus the logarithm removes all disconnected graphs, exactly as in the bosonic case. The only notable difference to the bosonic case arises when one now does the derivatives. While Lemma 2.13 would carry over to fermions unchanged if the coefficients in an expansion in ri were C-valued, the extra 0-variables imply that the coefficients are actually in the Grassmann algebra (and in particular, they may have odd degree). The correct replacement of Lemma 2.13 is Lemma B.25. Let A and B be elements of the Grassmann algebra C{n},

that is, A=

E ani ,

B=

E bin/

(B.157)

where ai,bi E C, and let C(x) be the formal power series 00

C(x)

. E cdtr.

(B.158)

r=0

Then

a an

al, an

al, C((—,1,0) = [A( n) B(—)

[A(—) B(7 1) c((71,0))1

ail

71=0

]

(B.159)

where 21, an- is the derivative with respect to li, acting to the left. Moreover

a

a =-- [B(— [A(- ) A(71)]

071 ) B(7ù] 71=0

ail

.

(B.160)

71=0

Proof. The proof is an easy exercise in Grossmann algebra and is left to the • reader. The derivation of the graph rules is now easy. The place where the fermion signs arise is in the derivatives. Commuting them properly gives the usual fermionic sign factors. We leave these details to the reader. We finally note that for the case of QED-type or Yukawa-type or phonon theories, where one has boson and fermion fields with interaction term 0,-*, our results imply that all symmetry factors are equal to one.

B.7 The Thermodynamic Limit in Perturbation Theory

217

B.7 The Thermodynamic Limit in Perturbation Theory The propagator for the many-fermion system without cutoff is given in (4.64). Thus, for k = (w,k) E A* let 1

D(k) = i6) —

(B.161)

E(k) Xi ( €t 2 1 k") — E(k)12)

where i-.4) is defined in (4.65), xi is given in (4.71), and define C(k) similarly, with xi replaced by x2 = 1 — xi . Then C(k) + D(k) = (0 E(k)) -1 is independent of t. The functions Ct and Dt define operators lit (K,IC) and 6-`t (IC, IC) on the functions on F* in the same way as 6 defines C by (4.62). Denote iit = 4Dt, and let 11(A) = 1 if the event A is true and 0 otherwise. —

Proposition B.26. Dt is a C' function of t that vanishes identically t > log ".. If t
if

supp Dt C {k E A* : liri - E(101 < Et} supp Dt C

{lc E A* : .et < liE3

-

(B.162)

E(k)I < Et}.

Moreover

I

lit (k)1
Pt (01

1

11(i - E(k)1

Et) ,

f dk 1.1i t (k)l< 41A

and

A*

Et)

2$ 1 (Ii6)

-

E(k)I

Et) ,

f dk ID t (k)I
(B.163)

(B.164)

-

A8

where V1 is the constant in (4.73). Proof. xi(s) = 0 if s > 1, so D(k) 0 0 implies lic-D — E(k)I < et < 1. 0 only for By Lemma B.24, this implies lwl < 12r- ft. Since PI > Dt

5,

t < log qa. Since iir.7.)—E(k)1 > IRe Cji > L the stated properties of Di follow. The t-derivative gives I Dt (k)1 = 2€T 2 go' — E(k) 1 lxi (€T2 )0 - E(k)I 2 ) i . Since xli (x) = 0 for s 41 (1,1), bt(k) 0 0 implies IQ < liiD E(k)I < Et which implies (B.163) and (B.162). (B.164) follows from these inequalities by • Lemma B.24, by (4.73), and by Dt = 1i°g('3 "12) ds 1)8 . —

The bounds (B.164) are crude because the restriction IE(k)I < et was replaced by IE(k)I <2 when (4.73) was applied. Better bounds would require a more detailed analysis of the behaviour as a function of L. The proof of existence of the thermodynamic limit will proceed inductively in r. It will be an application of the dominated convergence theorem to the integrated RGE

218

B. Appendix to Chapter 4

imr (tI 12) = im,(0 1E) + - Am f ds Qmr(s I 12). 2

(B.165)

o

To this end, it is necessary to make the integration region independent of n, and L. Although /mr (t), given by (B.165), appears evaluated at 13 rm on the RHS of (B.165) only, the integral defines the(P1,•m)E tderivative of a function defined on fm, where /-1 = MxBx {1, -1} x {1,2} with B = G* the first Brillouin zone of the infinite lattice, and M = {(.4.)n = -173r-(2n + 1) : n E Z} the set of Matsubara frequencies in the limit nr Fm : fm -4 C, let I Fm 10 = sup IFm Per*

(B.166)

oc. For a bounded function I

(B.167)

m

Up to now, the dependence of /mr (t) on (n r , L) was not denoted explicitly. We now put it in a superscript and write ar' L) . Let lim = (72 1-,L)-400

limL_4c0

Lemma B.27. Let (44;'L) (0)),,,,,,nr ,i, be a family of bounded functions such that IS L) (0) = 0 if m > 2r + 2, lim(n1.,L)-400 ar' L) (0) = /,„ (0) exists and is a bounded function on and 1,ç,12rr'L) (0) < KZ.. Let (41; 114 (t))m,r,n,,L be the solution to (B.165). Then, for all m and r, the function 14.1- ' 14 (t) = 0 if m> 2r + 2, and _a i(n,,L)/.‘ kt) = 0 Ot mr

for all t > log 2'

(B.168)

there are bounded functions Imr (t) : Pc:pm -4 C such that urn (nr ,L)--*oo

(t)

= imr (t).

(B.169)

-

Let Pmr be the polynomials defined recursively as

Pmf(x) = 6 1-r limr (0)1 ± EcT 1

f dkniri i!

(x)Pm2r2(x)

(B.170)

(in particular, the coefficients of Pm,. are independent of fl), then for all

nr ,L,O,t

1

4Infr'L)(t)10 < (€0,3)r -i Pmr

(LIVi log g;°) .

(B.171)

B.7 The Thermodynamic Limit in Perturbation Theory

219

Proof. Induction in r, with the statement of the Lemma as the inductive hypothesis. Let r = 1. Since ri > 1 and r2 > 1, the right hand side of the (t, = jr(nnir equation is zero, so " o .1-j) for all t. Thus the statement (

,

follows from the hypotheses on Let r > 2, and the statement hold for all r' < r. Let m > 2r + 2. Then ,L) (0) r 1 , m2, r2, i) contributes to the right hand = O. The five-tuple side only if r i + r2 = r, m 1 + m2 = m + 2i, i > 1, and by the inductive hypothesis, only if m i < 2r i + 2 and m2 < 2r2 + 2. Thus 4"4 (t) can be nonzero only if m = m i + m2 - 2i < m i +m2-2 < 2r i +2 + 2r2 + 2-2 =-- 2r+ 2. The integral appearing on the right hand side of (B.165) is a Riemann sum approximation to an integral over the (t, n„, L)-independent region s E [0, oc), kj E M x B. By the inductive hypothesis, the factors 4„r,:,L) have a limit (B.169) satisfying (B.171). Since for all t, .bt is a bounded C2 function, the same holds for the propagators (boundedness holds because f3 < oc). Thus the integrand converges pointwise, and it suffices to show that it is bounded by an integrable function to get (B.169) and (B.171) by an application of the dominated convergence theorem. Let a -= 41)1 log and let g be the function on [0, oc) x (M x B) 1-1 given by

2 (s < log 2?) f foe - s 1.2-1 Pmiri(a)07.1-1 Pm2r2(a)0

g(s,li) =

11 Ùf ds,

i i! (B.172)

8

< leoe-8 3 ) 1 (IE(kil < 260 e-83)

j=2 0

The integrand on the right hand side of (B.165) is bounded by g by Proposition B.26, Lemma B.24, and the inductive hypothesis (B.171). Because g vanishes identically for s > log 2?, it is integrable. By (B.163) and (B.164),

f

ds f

g(k)dk 2 .. . dki <

1

f diçmr i

ai-1 Prniri (c)13m2r2 (a). (B.173)

Thus (B.169) holds, and (B.171) holds with Pm, given by (B.170) (the factor 61- r comes from the assumption that 0€0 > 6). Since the right hand side of (B.165) vanishes for t > log (B.168) holds. • As discussed, 47:1;'1') (0) is not the initial interaction because at t = 0, the (nr ,L) propagator Co 0 O. The /m r (0) are obtained from the original interaction by Wick ordering with respect to C and integrating over all fields with covariance Co. The existence of the limit (n„, L) -4 oc of the 47:fr'L) (0) can be shown by a similar inductive proof (it is not trivial because in the limit n, oc, the absolute value of the propagator is not summable). An alternative proof is in Appendix D of [46].

B. Appendix to Chapter 4

220

B.8 Volume Improvement Bounds In this Appendix, we prove the one-loop and two-loop volume bounds for the case E(p) = p2 1, that is, for the spherical Fermi surface. -

B.8.1 The One Loop Volume Bound -

Let

(19

T(E, q) = f (270d 11 (1E(7r(0, /9) + q)I e)

(B.174)

sd-I

Lemma B.28. Let E(p) = p2 - 1, q E Rd and q = 14. Then for all d> 2

and all

E<

1/4, if q < 2E if 2E
1 q T(E, q) < /2d 7i:7 2d e (d-1)/2

(B.175)

-

-

o

Proof. For E(p) = p2 1, p and 0 are simply polar coordinates, so E(7t (p, 0) =-- p + q(q + 2V1 + p cos(p), where (i9 is the angle between 7r(p,0) and q. For d = 2, we can choose coordinates such that ço = 0; for d > 3, we write d9 = 'sin tiol d-2 dtpdco, with co varying over S'_2 /Z2 , and (19 E [0, 27r]. The integrand is independent of co and f dco/(27r)d -1 < 1, so -

-

-

2w

dtP

f— 27r 'sin WI

T(E, ci)

d

—2 (1q+ 2 cos 491

(B.176)

If q < e, use T(E, q) < 1. Let 2E < q < 1. The indicator function vanishes unless 1(-q - :) < cosy) < 1.( - q + if ). Thus Icoscp1 < 3/4, and therefore 2 sin p> 1. Thus the Jacobian of a change of variables to -y = q + 2 cos cp is bounded by one, and hence T(E '

cl)

Let q > 1. We change variables from 2vV1 - v2 , to get

T(E,

<

2d-2

Ir

(B.177)

071

cp

to y = cos

and use 1sin 01 <

1

f dv i v id-2( 1 _ v 2.) civ_

(14v 2 + q

-

21 < E) .

(B.178)

-1

If q> 2 + e, 14v2 + q 21 = 4v 2 + q 2 > E and the integrand is identically zero. If lq 21 < 2E, then 14y2 + q 21 < e implies 4v2 <3e and therefore -

-

-

-

B.8 Volume Improvement Bounds

2 d-2 T(e,

f

ir

2d-1 (d-1)/2 < < dv IvI d-2 (1 v2 ) e d-l

221

(B.179)

Finally, if 6 = 2 - q> 2E, then 14v2 + q 21 < E implies 12.-Vri. < 21v1 < 1-Vrr-E. For Isl < x 11 < V2- x, so -

21y 1

-

<

(B.180)

E.

and therefore

2d-2 f 1 T(E, q) <

<

dv v d-2 (1 - v2 ) c1;3

2_-

7r

1 (I 2 1 v I — (si

NfilE)

o

2d-1 if

dv

(B.181)

7 j v F--1--)2 11 (I 2 1 v 1 - 6 I 5o

Because q > 1, 6 < 1, and thus the support of the integrand does not contain y = 1. Therefore T(e, q) < 2d*. • Lemma B.28 implies Lemma 4.10 by Taylor expansion: using the coordinates p and 0, we have €t,

W) ( q, t, ti) =

dp

de

f

(2)d 7 J(P, 0)

(1E(r(P, 0) + EI)1 <

Et)

. (B.182)

d —1

By (4.75) and because et, < Et,

IE(7r(p, 0) + q)

—

2 E(7r(0,0) + q)I < — ipi < Et,

(B.183)

< 2IJ1 0 26 t,

(B.184)

go

SO

(q, t,

(( i

- ' cl)ft ) •

Lemma 4.10 now follows from Lemma B.28 by the observation that for q > Eth < NrEt. Generalizing Lemma B.28 to the nonspherical case and taking into account that B is a torus (so there may be more intersections of the two regions) is not difficult in principle, but it requires some care. The difference to the above proof is that the function q + 2 cos ça is replaced by a 21--periodic function f that, by strict convexity, is nonconstant. The second derivative at the critical points of f is proportional to the curvature of the Fermi surface at the corresponding point r(0, 0((,o)) on S.

222

B. Appendix to Chapter 4

B.8.2 The Two-Loop Volume Bound Theorem B.29. Let /co > 2, denote 7r0(0) = 7r(0,0), and let

W(E) =

f (101 f (102 1 (IE(voro(01) + viro( 02) + c1)1

sup

gEB viE{±1}5d-I

5d

-

E)

•

1

(B.185) There is a constant Qv > 1 such that for all 0 <

MO

11+ I logEl

QV E 1

E

< fo

if d =-- 2 if d > 3.

(B.186)

The constant Qv depends on the curvature of the Fermi surface. Theorem B.29 is Theorem 1.2 of [461. We prove it for the spherical Fermi surface, using Lemma B.28. The proof of Theorem B.29 for the nonspherical case is in Appendix B of [44

Proof. We have

f

W(E) = sup pER d

(10 (27r)d

T(E, P - wo OM.

(B.187)

sd-1

If p denotes the absolute value of p and y the angle between p and 7r0, we have (4,0 ) = IP - r0(0)1 = \Ai - p)2 + 2p(1 - cosy) and, by Lemma B.28, W(E) = supp>0 W(E,p) where 2w

VV(E,P) =

and

,r(E., ) _

I

f 4 I sin wid- 27-(E,

op»

(B.188)

0

1 e

k(m)E

2d e I Nr2=-7 2' 8.(d-1) / 2 0

if <2E if 2E < < 1/M if 1/M < < 2 - 1/M if -1/M < - 2 < -2E if -2E < - 2 < E otherwise.

(B.189)

Here we take M > 4; Lemma B.28 implies that K(M) = max{M,2d /-VM-}. Note that by definition of t, the bound for W follows immediately for the contribution from those y for which (‘,0) is between -17 and 2 - k. The contributions that require more detailed bounds are the ones from the (p for which ((,c) r-z..-1 0 and (ço),c..-1 2. We split the proof in four cases:

1. 0
ki

B.8 Volume Improvement Bounds 2. 1 — 3. 1 +

223

 3+ -k, YV(E,p)= 0). In Case 1, we have ((,o) > 1 - p> ky , and (ço) <1 + p < 2 Thus, by (B.189), VV(E,p) < K(M)E. Note that if p > 1 + then (c)o) > p - 1 > and thus ((,o) Re, 0 can occur only in Case 2. In Case 2, ((,c)) 2 also occurs, and the proof in Cases 3 and 4 is similar (in fact easier). Thus we shall do only Case 2 in detail. We consider the contributions from the various regions given in the definition of separately by inserting the corresponding indicator function in the integral. The integral with 11 E (-14-, 2 is bounded by K(M)E as in Case 1. Let

k,

r

b))

Op)

ci

f

=

dçoisin (pl d-2

6

J1(2€ < (ço) < -L A, )

(ti9 )

(B.190)

. Since p > 3/4, this The restriction (cicr) < implies 2p(1 - cosy) < means that cp is near to zero. Changing variables to y = sin , we have OP) = X(v) = (( 1 - /3) 2 + 4Pv2 ) 1/2 2.Nrpv, and 1

cl < 2d

f

dv vd -2 (1 — v2 )

0 c1-2

For d > 3, we use 7 ; 7- _<

6 1(2E X(V) 11M X(V)

715-- 11d-3 < 1, to get C1 <

(B.191)

2de • For d = 2, we use

-17

implies y < , so (1 - y 2 ) -1 /2 < 2, to make a change of that x(y) < variables from y to w = x(y), with Jacobian bounded by 2 \rf), so

i/m 1605

f

dw -E- <32€ Ilog(2ME)1.

The contribution from those ço that have

((,o)

(B.192)

< 2E,

1 C2 = 2d

f dy yd-2 (1 - y2 ) 41 71((1- p)2 4PV2 < 4E2)

(B.193)

o is nonzero only if p - 1 is of order E and if 4py2 <462, so e C2 < 2d+1

dV V d-2 =

2d+1

d -1

e ) d-1

(B.194)

224

B. Appendix to Chapter 4

We now turn to the contribution from those ço with ((,()) 2. Since ip – 11 < + 2p(1 –cos') 4 can hold only for ii. Thus we shift ço ki/ Ir and introduce v = sin (p'/2. With q(v) = – p) 2 + 4p(1 – v 2 ), ÇOI = we have to bound

1 C3 =2d

f

(2E < 2 –

dV Vd-2 (1- V 2 )

O

5_ ) /2 - ij

(B.195)

< 2E) 2 d

(B.196)

and

1

=

2d f dv Vd-2 - V2 ) -r- (-E < 2 – d—3

d—1

•

O

Now 2– =

4••7)

<(4 – n 2) = 2p(v2 – a(p))

(B.197)

with a(p) = (p – 1)(p ± 3 )/ 4p. By a similar argument, we have 2 – i > 142Ep (v2 – a(p)), and hence

4(v2 — a).

–1 (0 – < 2 – 4

(B.198)

Since lai < -h., the restriction on v imposed by the indicator function implies (1 _ v 2)-1/2 <4. Thus

1 C3 < 2"E f dv Vd-2 CI

1

'1 i ) (E2
vv 2 - a

(B.199)

and

f dv vd-2 (1v 2 — al <

< 4d+1

(B.200)

0 We first bound C3. If a > 0, we change variables to u = v 2 – a. This gives C3 < 2d+2E/3 with 4/M =

f

du —

'NF U

d—3

(a + u) 2

(B.201)

.

6/2

For d > 3, (a + u) d ; 3 < 1 and thus

/3 < 2. For d =-- 2,

4/M

4/M

du

f ,‘Fit:\f_E 11 a _< 6/2

(du u =

6/2

log — 8 . ME

(B.202)

B.8 Volume Improvement Bounds

225

If a < 0, let a = lai. With u = y2 , C3 < 2 61±2EJ73, where

1

1

d-3

=

f du u -r-

< 2 -

0711- a

O

For d > 3, we use that uP < 1 and \/uwe have to bound 1 j3 du 1 AFL

u+

4

1- > NFL to get

(B.203) J3

< 2. For d = 2,

1

(B.204)

ii

with v = max{0,1 - a}. Integration by parts gives

j3 21

1

ILATu +

+ I

du ,u 43/2" 4

(B.205)

v

Because a > 0, -Va(u + a) -1 /2 < 1, so the boundary term is at most one. The same argument implies that the remaining integral is bounded by 1

1-1-cr

f du _ f du < log 2 + 21a1 fu±a f u E

(B.206)

e/2

V

Finally, we bound C4. If a <0, v2 - a > v2 , so

< 4d+IEclia

dy V d-2

<

const Ed-1 .

(B.207)

O

If 0 < a < 16E, then ly2 - al <8e implies v2 < 24E, and C4 < const E follows in the same way as in (B.207). If a > 16E, then we use that y > 0 to get 8E? 1(v - -1)(y +

2 JIv -

2 4vIv - -Val,

(B.208)

that is, y must be in the interval iv - Val <2,Ç. Then the integral over y is < 2Vi, and C4 < const E. • Theorem 4.14 follows from Theorem B.29 by the same Taylor expansion in the p integration variables that was used in Sect. B.8.1 to derive Lemma 4.10 from Lemma B.28.

References

1. P.A.M. Dirac, The Principles of Quantum Mechanics, Second Edition, Oxford, Clarendon Press, 1947 2. R. P. Feynman, A. P. Hibbs, Quantum Mechanics and Path Integrals, McGraw Hill, 1965 3. B. Simon, Functional Integration and Quantum Physics, Academic Press, 1979 4. M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol. II, Academic Press, 1975 5. G. Roepstorff, Funktionalintegrale in der Quantenphysik, Vieweg, Braunschweig, 1992 6. J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, Second Edition, Springer, 1987 7. V. Rivasseau, From Perturbative to Constructive Renormalization, Princeton University Press, 1991 8. S. Weinberg, The Quantum Theory of Fields, Cambridge University Press, 1995 9. Pierre J. van Baal, A Course in Field Theory, http://rulgm4.1eidenuniv.nlivan baal/FTcourse.html 10. M. Aizenman, Comm. Math. Phys. 86 (1982) 1 11. J. Fralich, Nuci. Phys. B 200 (1982), 281 12. R. Fernandez, J. PriShlich, A. Sokal, Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory, Springer, 1992 13. M. Liischer, P. Weisz, Nucl. Phys. B290 (1987) 25 14. K. Wilson, J. Kogut, Phys. Reports 12 (1974) 75 15. N.D. Merrain, H. Wagner, Phys. Rev. Lett. 17 (1966) 1133, R. Dobrushin, S.L. Shlosman, Comm. Math. Phys. 42 (1975) 31 16. D. Maison, H. Reeh, Comm. Math. Phys. 24 (1971) 67, S. Coleman, Comm Math. Phys. 31 (1973) 259 17. R.B. Griffiths, Phys. Rev. 136 437, Jour. Math. Phys. 8 (1967) 478,484, Comm. Math. Phys. 6 (1967) 121 18. D. Ruelle, Statistical Mechanics, W.A. Benjamin, 1969 19. R.B. Israel, Convexity in the Theory of Lattice Gases, Princeton University Press, 1978 20. E. Seiler, Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics, Springer Lecture Notes in Physics 159, 1982 21. B. Simon, The Statistical Mechanics of Lattice Gases, Princeton University Press, 1993 22. A.C.D. van Enter, R. Fernàndez, A.D. Sokal, J. Stat. Phys. 72 (1993) 879 23. F. Wegner, in Phase Transitions and Critical Phenomena, vol. 6, edited by C. Domb and M. Green, Academic Press, 1976 24. L. R. Foulds, Graph Theory Applications, Springer, 1992 25. C. Borgs, J. Imbrie, Commun. Math. Phys. 145 (1992)235 26. G. Gallavotti, Rev. Mod. Phys. 57 (1985) 471

228

References

27. G. Gallavotti, F. Nicolb, Comm. Math. Phys. 100 (1985) 545; 101 (1985) 247 28. J. Feldman, T. Hurd, L. Rosen, J. Wright, QED: A Proof of Renormalizability, Springer Lecture Notes in Physics 312, (1988) 29. J. Polchinski, Nucl. Phys. B 231 (1984), 269 30. C. Wieczerkowski, Comm. Math. Phys. 120 (1988), 149 31. G. Keller, C. Kopper, M. Salmhofer, Helv. Phys. Acta, 65 (1992) 32 32. G. Keller, C. Kopper, Comm. Math. Phys. 148 (1992) 445, Phys. Lett. B273 (1991) 323 33. G. Keller, Comm. Math. Phys. 161 (1994) 311 34. C. Kopper, Renormierungstheorie mit Flufigleichungen, Aachen: Shaker, 1998 35. K. Gawedzki, A. Kupiainen, Comm. Math. Phys. 77 (1980) 33, Comm. Math. Phys. 99 (1985) 197 36. J. Feldman, J. Magnen, V. Rivasseau, R. Sênkr, Comm. Math. Phys. 98 (1985) 273 Comm. Math. Phys. 100 (1985) 23, Comm. Math. Phys. 109 (1987) 437 37. G. Mack, A. Pordt, Comm. Math. Phys. 97 (1985) 267 38. D. C. Brydges, H.-T. Yau, Comm. Math. Phys. 129(1990) 351 39. D. C. Brydges, J. Dimock, T. Hurd, Comm. Math. Phys. 172 (1995) 143, mp_arc 96-538, mp arc 96-681, and references therein 40. A. Abdesselam, V. Rivasseau, Rev. Math. Phys. 9 (1997) 123 41. T. Kennedy, E. Lieb, Physica 138A (1986) 320 42. J. Hubbard, Proc. R. Soc. London A 243 (1957) 336 43. N. W. Ashcroft, N. D. Mermin, Solid State Physics, Sunders College Publishing, 1976 Harcourt Brace College Publishers, New York 44. J. Feldman, E. Trubowitz, Helv. Phys. Acta 63 (1990) 157, ibid. 64 (1991) 213 45. J. Feldman, M. Salmhofer and E. Trubowitz, J. Stat. Phys. 84 (1996) 1209 46. J. Feldman, M. Salmhofer, and E. Trubowitz, Comm. Pure Appl. Math. LI (1998) 1133 47. J. Feldman, M. Salmhofer, and E. Trubowitz, Regularity of the Moving Fermi Surface: The Full Selfenergy, to appear in Comm. Pure Appl. Math. 48. M. Salmhofer, Rev. Math. Phys. 10 (1998) 553 49. P.B. Allen, Physical Review B 13 (1976) 1416 50. W. Kohn and J. M. Luttinger, Phys. Rev. Lett. 15 (1965) 524 51. J. Feldman, H. Knikrer, R. Sinclair, E. Trubowitz, Helv.Phys.Acta 70(1997)154 52. R. Sinclair, Computational Evidence for Superconductivity in a Repulsive Model, ETH Thesis No. 11660 (1996) 53. M. Salmhofer, Comm. Math. Phys. 194 (1998) 249 54. M. Salmhofer, Phys. Lett. 408 B (1997) 245 55. J.Feldman, J.Magnen, V.Rivasseau, E.Trubowitz, Helv.Phys.Acta 65(1992)679 56. J. Feldman, H. Karrer, D. Lehmann, E. Trubowitz, in Constructive Physics, V. Rivasseau (ed.), Springer Lecture Notes in Physics, 1995 57. D. C. Brydges and J. Wright, J. Stat. Phys. 51 (1988) 435 58. G. Benfatto, G. Gallavotti, J. Stat. Phys. 59 (1990) 541 59. G. Benfatto, G. Gallavotti, A. Procacci, B. Scoppola, Comm. Math. Phys. 160 (1994) 93 60. F. Bonetto, V. Mastropietro, Comm. Math. Phys. 172 (1995) 57 61. J. Feldman, H. Karrer, E. Trubowitz, Comm. Math. Phys. 195 (1998) 465 62. A. Abdesselam and V. Rivasseau, to appear 63. M. Disertori and V. Rivasseau, to appear 64. 0. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics II, Springer 1981 65. Reed, M., Simon, B., Methods of Modern Mathematical Physics, vol. 1, Academic Press, New York 1980 66. U. Wolff, Nucl. Phys. B225 (1983) 391-408

Index

acoustical phonons 128 annihilation operator 115 band functions 114 bare coupling constant 105 beta function 111 Bloch walls 45 Boltzmann factor 152 boundary conditions 19 Brownian motion 5 canonical anticommutation relations 115, 192 change of boundary conditions 88, 93 characteristic function 8 charge density 116 chemical potential 118 cluster expansions 40 compatibility relation 44 component form of the RG equation 69 conditional Wiener measure 5 connected amputated Green functions 41 connected correlation functions 16 connected Feynman graph 54 continuum limit 12, 23, 183 convergence of the renormalized Green functions 82 Cooper instability 176 correlation functions 15 Coulomb interaction 128 counterterm function 174 covariance 8 creation operator 115 critical phenomena 12 critical temperature 176, 178 crystallization 113 cutoff removal 49 density of states 127 dominated convergence theorem 62

effective action 47 Euclidean invariance 75 Euclidean time 123 even subalgebra of a Grassmann algebra 130, 214 external leg 37 factorial growth 79 Fermi distribution 121 Fermi gas 119 Fermi liquid behaviour 113, 176 Fermi surface 121 Fermi surface harmonics 154 fermions 57, 113 ferromagnetic 18 Feynman graphs 27 Feynman's path integral formula 4 Feynman-Kac formula 5 filling factor 123 Fock space 115 formal power series 26 four-legged subgraphs 103 free energy density 40 Gaussian fields 9 Gaussian measure 5, 8 generalized tadpole graphs 99 generating functional for the connected amputated Green functions 48 geometry of the Fermi surface 113 Gram's inequality 199 grand canonical ensemble 118 graph 42 Grassmann algebra 192 Grassmann bilinear form 124 Grassmann delta function 202 Grassmann Gaussian measure 124 Grassmann integrals 123 Green functions 15 half-filled Hubbard model 129

230

Index

hard lines 55 heat equation 68 heat kernel 3 Hermite polynomials 32 high-temperature expansion 44 hopping amplitude 117 hopping parameters 114 Hubbard model 114 imaginary time 6 improved power counting 161 infrared problem 113 initial condition 90 integrating out fluctuations 48 interacting Fermi surface 176, 178 internal lines 37 interpolating Fermi surface 156 intersection volume 150 invariance 12 inverse temperature 118 irrelevant terms 106 Ising model 17 jellium model 129 Kallen—Lehmann representation 24 Kohn—Luttinger effect 155, 176

labelled vertex 37 ladder approximation 109 Landau pole 26, 112 lattice Laplacian 13 lattice regularization 12 lattice vibrations 114 Lie product formula 206 local density of states 127 local interaction 20 loop integrations 87 Lorentz invariance 20 low-temperature expansion 45 lower degree 181 Luttinger liquid behaviour 113, 176 magnetic susceptibility 18 marginal terms 106 mass counterterm 88 Matsubara frequencies 125 multigraph 42 multiindex 91 Nambu formalism 212

nearest-neighbour hopping 116 neighbourhood of the Fermi surface 150

non-ladder skeleton functions 163 number density 120 number operator 117 one-loop volume bound 220 overlapping loops 157

parquet four-point function 146 parquet graphs 146 particle-hole bubbles 155 particle-hole flow 155 particle-hole ladders 157 particle-particle ladders 153 partition 43 path integral 1 Pauli principle 121 Pauli-Villars cutoff 24 permutational symmetry 73 perturbative renormalizability 67 perturbatively non-renormalizable 103 phonon interactions 114 planar graphs 189 polymer activities 214 polymer system 53

positive temperature as a physical cutoff 122 propagator 22 random walk 16 relevant terms 106 renormalization 23 renormalization condition 90 renormalization group equation 63 renormalized coupling constant 105 renormalized perturbation theory 25 resummations of perturbation theory 113 ring of formal power series 181 rotation invariance 65, 101 RPA graphs 147

scale parameter 63 Screening 128 self—contraction 35 selfenergy 113 semigroup of effective actions 63 skeleton functions 140 soft lines 55 source terms 11, 47 spin rotation invariance 116

substitution of formal power series 181 summability of a family of formal power series 181 superconductivity 113, 153

Index

symmetry factors 54 thermodynamic limit 12, 114, 183 tight-binding approximation 114 topological Feynman graphs 39 tree—level—estimate 87 Trotter product formula 2 two—legged insertions 86 two-loop volume bound 222 ultraviolet cutoff 12 ultraviolet divergences 85 vertex functions 49 vertex set of a Feynman graph 37 volume improvement factor 160

wave function renormalization 90, 179 Wick lines 55 Wick ordered monomials 30 Wick ordering 23, 28 Wilson's effective action 41

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