Mathematical Methods of Many-Body Quantum Field Theory

Mathematical Methods of Many-Body Quantum Field Theory

CHAPMAN & HALL/CRC Research Notes in Mathematics Series Main Editors H. Brezis, Université de Paris R.G. Douglas, Texas A&M University A. Jeffrey, University of Newcastle upon Tyne (Founding Editor) Editorial Board R. Aris, University of Minnesota G.I. Barenblatt, University of California at Berkeley H. Begehr, Freie Universität Berlin P. Bullen, University of British Columbia R.J. Elliott, University of Alberta R.P. Gilbert, University of Delaware D. Jerison, Massachusetts Institute of Technology B. Lawson, State University of New York at Stony Brook

B. Moodie, University of Alberta L.E. Payne, Cornell University D.B. Pearson, University of Hull G.F. Roach, University of Strathclyde I. Stakgold, University of Delaware W.A. Strauss, Brown University J. van der Hoek, University of Adelaide

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Detlef Lehmann

Mathematical Methods of Many-Body Quantum Field Theory

CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C.

Library of Congress Cataloging-in-Publication Data Lehmann, Detlef. Mathematical methods of many-body quantum field theory / Detlef Lehmann. p. cm. -- (Chapman & Hall/CRC research notes in mathematics series ; 436) ISBN 1-58488-490-8 (alk. paper) 1. Many-body problem. 2. Quantum field theory--Mathematical models. I. Title. II. Series. QC174.17.P7L44 2004 530.14'3--dc22

2004056042

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1

2

6.1

3

7

4

8 5 9

6.2,3

10

Chapter 1 Introduction

The computation of field theoretical correlation functions is a very difficult problem. These functions encode the physical properties of the model under consideration and therefore it is important to know how these functions behave. As it is the case for many mathematical objects which describe some not too idealized systems, also these functions, in most cases, cannot be computed explicitly. Thus the question arises how these functions can be controlled. A quantum many-body system is given by a Hamiltonian H(λ) = H0 + λHint . Here, usually, the kinetic energy part H0 is exactly diagonalizable and Hint describes the particle-particle, the many-body interaction. There are many situations where it makes sense to consider a small coupling λ. In such a situation it is reasonable to start with perturbation theory. That is, one writes down the Taylor series around λ = 0 which is the expansion into Feynman diagrams. Typically, some of these diagrams diverge if the cutoffs of the theory are removed. This does not mean that something is wrong with the model, but merely means first of all that the function which has been expanded is not analytic if the cutoffs are removed. The following example may be instructive. Let Gδ (λ) :=


∞ 0

dx


1 0

1 dk 2√k+λx+δ e−x

(1.1)

where δ > 0 is some cutoff and the coupling λ is small and positive. One may think of δ = T , the temperature, or δ = 1/L, Ld being the volume of the system, and Gδ corresponds to some correlation function. By explicit computation √ √
∞ √ G0 (λ) = lim Gδ (λ) = 0 dx ( 1 + λx− λx) e−x = 1+O(λ)−O( λ) (1.2) δ→0

Thus, the δ → 0 limit is well defined but it is not analytic. This fact has to show up in the Taylor expansion. It reads Gδ (λ) =

n    −1
∞ 2

j j=0

0

dx


1 0

dk

xj e−x 1

2(k+δ)j+ 2

λj + rn+1

(UR)

Apparently, all integrals over k diverge for j ≥ 1 in the limit δ → 0. Now, the whole problem in field theoretic perturbation theory is to find a rearrangement which reorders the expansion (UR) (‘UR’ for ‘unrenormalized’) into a new

1

2 expansion G0 (λ) =

n   
∞ 1 2



0

dx x e−x λ − c

√ λ + Rn+1

(R)

=0

(‘R’ for ‘renormalized’) which, in this √ explicitly solvable example, can be obtained from (1.2) by expanding the 1 + λx term. In (R), all coefficients are finite and, for small λ, the lowest order terms are a good approximation since |Rn+1 | ≤ n! λn+1 , although the whole series in (R), obtained by letting n → ∞, still has radius of convergence zero. That is, the expansion (R) is asymptotic, the lowest order terms give us information about the behavior of the correlation function, but the expansion (UR) is not, its lowest order terms do not give us any information. The problem is of course that in a typical field theoretic situation we do not know the exact answer (1.2) and then it is not clear how to obtain (R) from (UR). Roughly speaking, this book is about the passage from (UR) to (R) for the many-electron system with short-range interaction which serves as a typical quantum many-body system. Thereby we will develop the standard perturbation theory formalism, derive the fermionic and bosonic functional integral representations, consider approximations like BCS theory, estimate Feynman diagrams and set up the renormalization group framework. In the last chapter we discuss a somewhat novel method which is devoted to the resummation of the nonanalytic parts of a field theoretical perturbation series. In the first three chapters (2-4) we present the standard perturbation theory formalism, the expansion into Feynman diagrams. We start in chapter 2 with second quantization. In relativistic quantum mechanics this concept is important to describe the creation and destruction of particles. In nonrelativistic many-body theory this is simply a rewriting of the Hamiltonian, a very useful one of course. The perturbation expansion for exp{−β(H0 + λV )} is presented and Wick’s theorem is proven. In chapter 4 we introduce anticommuting Grassmann variables and derive the Grassmann integral representations for the correlation functions. Grassmann integrals are a very suitable tool to handle the combinatorics and the rearrangement of fermionic perturbation series. In the fifth chapter, we use these formulae to write down the bosonic functional integral representations for the
correlation functions. These are typ
ically of the form F (φ) e−Veff (φ) dφ/ e−Veff (φ) dφ. Here F depends on the particular correlation function under consideration but the effective potential Veff is fixed once the model is fixed. Usually it is given by a quadratic part minus the logarithm of a functional determinant. In particular, we consider the case of an attractive delta-interaction and we give a rigorous proof that the global minimum of the full effective potential in that case is in fact given by the BCS configuration. This is obtained by estimating the functional determinant

Introduction

3

as a whole without any expansions and is thus a completely nonperturbative result. In chapter 6, we discuss BCS theory, the Bardeen-Cooper-Schrieffer theory of superconductivity. Basically the BCS approximation consists of two steps. The interacting part of the full Hamiltonian, which is quartic in the annihilation and creation operators, comes, because of conservation of momentum, with three independent momentum sums. The first step of the approximation consists in putting the total momentum of two incoming electrons equal to zero. The result is a Hamiltonian, which is still quartic in the annihilation and creation operators, but which has only two independent momentum sums. Sometimes this model is called the ‘reduced BCS model’ but one may also call it the ‘quartic BCS model’. The model, which has been solved by Bardeen, Cooper and Schrieffer in 1958 [6] is a quadratic model. It is obtained from the quartic BCS model by substituting the product of two annihilation or creation operators by a number, which is chosen to be the expectation value of these operators with respect to the quadratic Hamiltonian, to be determined selfconsistently. This mean field approximation is the second step of the BCS approximation. In section 6.2 we show that the quartic BCS model is already explicitly solvable, it is not necessary to make the quadratic mean field approximation. This result follows from the observation that in going from three to two independent momentum sums one changes the volume dependence of the model in such a way that in the bosonic functional integral representation the integration variables are forced to take values at the global minimum of the effective potential in the infinite volume limit. That is, the saddle point approximation becomes exact. Even for the quartic BCS model the effective potential is a complicated function of many variables but with the results of chapter 5 we are able to determine the global minimum which results in explicit expressions for the correlation functions. For an s-wave interaction the results coincide with those of the quadratic mean field formalism, but for higher -wave interactions this is no longer necessarily the case. Chapter 7 provides a nice application of the second quantization formalism to the fractional quantum Hall effect. We show that, in a certain long range limit, the interacting many-body Hamiltonian in the lowest Landau level can be exactly diagonalized. However, the long range approximation which is used there has to be considered as unphysical. Nevertheless we think it is worth discussing this approximate model since it has an, in finite volume, explicitly given eigenvalue spectrum which, in the infinite volume limit, most likely has a gap for rational fillings and no gap for irrational fillings. This is interesting since a similar behavior one would like to prove for the original model. Chapters 8 and 9 are devoted to the rigorous control of perturbation theory in the weak coupling case. These are the most technical chapters. Chapter 8 contains bounds on individual Feynman diagrams whereas chapter 9 estimates sums of diagrams. First it is shown that the value of a diagram depends on

4 its subgraph structure. This is basic for an understanding of renormalization. Then it is shown that, for the many-electron system with short range interaction, an n’th order diagram without two- and four-legged subgraphs allows a constn bound which is the best possible case. Roughly speaking, one can expect that a sum of diagrams, where each diagram allows a constn bound, is at least asymptotic. That is, the lowest order terms of such a series would be a good approximation in the weak coupling case and this is all one would like to have. Then it is shown that n’th order diagrams with four-legged subgraphs but without two-legged subgraphs are still finite but they produce n!’s. This is bad since, roughly speaking, a sum of such diagrams cannot expected to be asymptotic. That is, the computation of the lowest order terms of such an expansion does not give any information on the behavior of the whole sum. For that reason diagrams without two- and four-legged subgraphs are called ‘convergent’ diagrams but this does not refer to diagrams with four-legged but without two-legged subgraphs, although the latter ones are also finite. Finally diagrams with two-legged subdiagrams are in general infinite when cutoffs are removed (volume to infinity, temperature to zero). In the ninth chapter we consider the sum of convergent diagrams. As already mentioned, such a sum can be expected to be asymptotic. More precisely, for a bosonic model one can expect an asymptotic series and for a fermionic model, one may even expect a series with a positive radius of convergence. In fact this is what we prove. We choose a fermionic model which has the same power counting as the many-electron system and show that the sum of convergent diagrams has a positive radius of convergence. The same result has been proven for the many-electron system in two dimensions and can be found in the research literature [18]. For those who wonder at this point how objects like the ‘sum of all diagrams without two- and four-legged subgraphs’ are treated technically we shortly remark that these sums are generated inductively by integrating out scales in a fermionic functional integral and then at each step Grassmann monomials with two and four ψ’s are taken out by hand. Diagrams with two-legged subdiagrams have to be renormalized. Conceptually, renormalization is nothing else than a rearrangement of the perturbation series. However, due to technical reasons, it may be implemented in different ways. One way of doing this is by the use of counterterms. In this approach one changes the model under consideration. Instead of a model with kinetic energy, say, ek = k 2 /(2m) − µ, µ the chemical potential, one starts with a model with kinetic energy ek + δe. The counterterm δe depends on the coupling and may also depend on k. Typically, for problems with an infrared singularity, like the many-electron system, where the singularity is on the Fermi surface ek = 0, the counterterm is a finite quantity. It can be chosen in such a way, that the perturbation series for the altered model with kinetic energy e(k) + δe does no longer contain any divergent diagrams. In fact, for the many-electron system with short-range interaction, it can be proven

Introduction

5

[18, 20, 16] that, in two dimensions, the renormalized sum of all diagrams without four-legged subgraphs is analytic for sufficiently small coupling. This is true for the model with kinetic energy ek = k 2 /(2m) − µ which has a round Fermi surface F = {k | ek = 0} but also holds for models with a more general ek which may have an anisotropic Fermi surface. Then, the last and the most complicated step in the perturbative analysis consists in adding in the four-legged diagrams. These diagrams determine the physics of the model. At low temperatures the many-electron system may undergo a phase transition to the superconducting state by the formation of Cooper pairs. Two electrons, with opposite momenta k and −k, with an effective interaction which has an attractive part, may form a bound state. Since at small temperatures only those momenta close to the Fermi surface are relevant, the formation of Cooper pairs can be suppressed, if one substitutes (by hand) the energy momentum relation ek = k 2 /(2m) − µ by a more general expression with an anisotropic Fermi surface. That is, if momentum k is on the Fermi surface, then momentum −k is not on F for almost all k. For such an ek one can prove that four-legged subdiagrams no longer produce any factorials, an n’th order diagram without two-legged but not necessarily without fourlegged subgraphs is bounded by constn . As a result, Feldman, Kn¨orrer and Trubowitz could prove that, in two space dimensions, the renormalized perturbation series for such a model has in fact a small positive radius of convergence and that the momentum distribution a+ kσ akσ  has a jump discontinuity across the Fermi surface of size 1 − δλ where δλ > 0 can be chosen arbitrarily small if the coupling λ is made small. Because of the latter property this theorem is referred to as the Fermi liquid theorem. The complete rigorous proof of this fact is a larger technical enterprise [20]. It is distributed over a series of 10 papers with a total volume of 680 pages. J. Feldman has setup a webpage under www.math.ubc.ca/˜feldman/fl.html where all the relevant material can be found. The introductory paper ‘A Two Dimensional Fermi Liquid, Part 1: Overview’ gives the precise statement of results and illustrates, in several model computations, the main ingredients and difficulties of the proof. As FKT remark in that paper, this theorem is still not the complete story. Since two-legged subdiagrams have been renormalized by the addition of a counterterm, the model has been changed. Because ek has been chosen anisotropic, also the counterterm δek is a nontrivial function of k, not just a constant. Thus, one is led to an invertability problem: For given ek , is there a e˜k such that e˜k + δ˜ ek = ek ? If this question is addressed on a rigorous level, it also becomes very difficult. See [28, 55] for the current status. The articles of [28] and [20] add up to one thousand pages. Another way to get rid of anomalously large or divergent diagrams is to resum them, if this is possible somehow. Typically this leads to integral equations for the correlation functions. The good thing in having integral equations is that the renormalization is done more or less automatically. The

6 correlation functions are obtained from a system of integral equations whose solution can have all kinds of nonanalytic terms (which are responsible for the divergence of the coefficients in the naive perturbation expansion). If one works with counterterms one more or less has to know the answer in advance in order to choose the right counterterms. However, the bad thing with integral equations is that usually it is impossible to get a closed system of equations without making an uncontrolled approximation. If one tries to get an integral equation for a two-point function, one gets an expression with two- and four-point functions. Then, dealing with the four-point function, one obtains an expression with two-, four- and six-point functions and so on. Thus, in order to get a closed system of equations, at some place one is forced to approximate a, say, six-point function by a sum of products of two- and four-point functions. In the last chapter we present a somewhat novel formalism which allows the resummation of two- and four-legged subdiagrams in a systematic and relatively elegant way which leads to integral equations for the correlation functions. Although this method too does not lead to a complete rigorous control of the correlation functions, we hope that the reader feels like the author who found it quite instructive to see renormalization from this point of view.

Chapter 2 Second Quantization

In this chapter we introduce the many-body Hamiltonian for the N -electron system and rewrite it in terms of annihilation and creation operators. This rewriting is called second quantization. We introduce the canonical and the grand canonical ensemble which is the framework in which quantum statistical mechanics has to be formulated. By considering the ideal Fermi gas, we try to motivate that the grand canonical ensemble may be more practical for computations than the canonical ensemble.

2.1

Coordinate and Momentum Space

Consider one electron in d dimensions in a finite box of size [0, L]d. Its kinetic energy is given by h0 =


2 2m ∆

(2.1)

and its Schr¨ odinger equation h0 ϕ = εϕ is solved by plane waves ϕ(x) = eikx . Since we are in a finite box, we have to impose some boundary conditions. Probably the most natural ones are Dirichlet boundary conditions ϕ(x) = 0 on the boundary of [0, L]d but it is more convenient to choose periodic boundary conditions, ϕ(x) = ϕ(x+Lej ) for all 1 ≤ j ≤ d. Hence eikj L must be equal to 1 which gives k = (k1 , ..., kd ) = 2π L (m1 , ..., md ) with mj ∈ Z. Thus, a continuous but bounded coordinate space gives a discrete but unbounded momentum space. Similarly, a discrete but unbounded coordinate space gives a continuous but bounded momentum space and a discrete and bounded coordinate space, with a finite number of points, gives a discrete and bounded momentum space with the same number of points. To write down the Hamiltonian for the many-electron system in second quantized form, we will introduce annihilation and creation operators in coordinate space, ψ(x) and ψ + (x). Strictly speaking, for a continuous coordinate space, these are operator-valued distributions. To keep the formalism simple, we found it convenient to introduce a small lattice spacing 1/M > 0 in coordinate space which makes everything finite dimensional. We then derive suitable expressions for the correlation functions in the next chapters and at

7

8 the very end, the limits lattice spacing to zero and volume to infinity are considered. Thus, let coordinate space be
 1 Γ= x= M (n1 , · · · , nd ) | 0 ≤ ni ≤ M L − 1  1 d Z /(LZ)d (2.2) = M Momentum space is given by
 M := Γ = k = 2π L (m1 , · · · , md ) | 0 ≤ mi ≤ M L − 1  d = 2π /(2πM Z)d L Z

(2.3)

such that 0 ≤ kj ≤ 2πM or −πM ≤ kj ≤ πM since −kj = 2πM − kj . Removing the cutoffs, one gets    2π d  L→∞ 1 1 dd k = (2π)d L → , (2.4) Ld (2π)d m

[−πM,πM]d

m

1 Md



M→∞



→

dd x

(2.5)

[−L/2,L/2]d

n

d

A complete orthonormal system of L2 (Γ) = CN , N = M L, is given by the plane waves ϕk (x) ≡ ϕm (n) =

2π

1 d (ML) 2

ei M L

Pd i=0

mi ni

=

1 d

e2πi

mn N

(2.6)

N2

The unitary matrix of discrete Fourier transform is given by F = (Fmn ) where Fmn = One has

1 d

N2

e−2πi

mn N

(2.7)

⎛

F ∗ = F −1

⎞ | = F¯ = ⎝ · · · ϕk (x) · · · ⎠ |

The discretized version of  fˆ(k) = dd x e−ikx f (x) ,

f (x) =



dd k (2π)d

(2.8)

eikx fˆ(k)

(2.9)

reads in terms of F   L  d2   L  d2 e−ikx f (x) = M Fkx f (x) = M (F f )(k) fˆ(k) = M1d x

f (x) =

1 Ld

 k

x

e

ikx

fˆ(k) =

 M  d2  L k

∗ ˆ Fxk f (k) =

 M  d2 L

(F ∗ fˆ)(x) (2.10)

Second Quantization

9

Derivatives are given by difference operators  n+ei  ∂ n ∂xi f (x) = M f ( M ) − f ( M )  = M1d M M d(δx+ ei ,y − δx,y )f (y) (∆f )(x) =

1 Md

(2.11)

M

y



M 2M d

y

d 

(δx+ ei ,y + δx− ei ,y − 2δx,y )f (y) (2.12) M

M

i=1

which are diagonalized by F , ∂ ∂xj ϕk (x)

which gives

∂ ∗ F 1i ∂x F j

m,m


[F (−∆)F ∗ ]m,m


=

M (e2πi

M→∞ 2π → L mj d 

− 1)ϕk (x)

− 1) δm,m
(2.14)

 2π L

m

2

πmi ML

δm,m


i=1

δm,m
= k2 δk,k


In the following we will write kj for the Fourier transform of writing the exact discretized expressions.

2.2

(2.13)

δm,m
= kj δk,k


i=1

→

mj N

mj N

d    i M 2 2 − 2 cos( 2πm ) = 4M 2 sin2 ML

=

M→∞

= M (e2πi

(2.15) 1 ∂ i ∂xj

instead of

The Many-Electron System

The N -particle Hamiltonian HN : FN → FN is given by 1 HN = − 2m

N  i=1

∆xi +

1 2



V (xi − xj )

(2.16)

i,j=1 i=j

which acts on the antisymmetric N -particle Fock space     FN = FN ∈ L2 (Γ × {↑, ↓})N = (C2|Γ| )N  ∀π ∈ Sn :

(2.17)  FN (xπ1 σπ1 , · · · , xπN σπN ) = signπ FN (x1 σ1 , · · · , xN σN )

Since we assume a small but positive temperature T = 1/β > 0, we have to do quantum statistical mechanics. Conceptually, the most natural setting would be the

10 Canonical Ensemble: An observable has to be represented by some operator AN : FN → FN and measurements correspond to the expectation values T rFN AN e−βHN T rFN e−βHN

AN FN =

(2.18)

Example (The Ideal Fermi Gas): The ideal Fermi gas is given by

1 H0,N = − 2m

N 

∆xi

(2.19)

i=1

We compute the canonical partition function QN := T rFN e−βH0,N

(2.20)

To this end introduce an orthonormal basis of F1 of eigenvectors of −∆ which is given by the plane waves  φkσ (xτ ) := δστ

1 d L2

  eikx  (k, σ) ∈ M × {↑, ↓}

(2.21)

where the set of momenta M is given by (2.3). The scalar product is (φkσ , φk
,σ
, )F1 :=

1 Md



φkσ (xτ )φ¯k
σ
(xτ ) = δσ,σ
δk,k


(2.22)

xτ

and we have −∆φkσ = ε(k)φkσ ,

ε(k) =

d 

M→∞

2M 2 (1 − cos[ki /M ]) → k2 (2.23)

i=1

An orthogonal basis of Fn is given by wedge products or Slater determinants φk1 σ1 ∧ · · ·∧φkn σn (x1 τ1 , · · · , xn τn ) :=

1 n!



signπφk1 σ1 (xπ1 τπ1 ) · · · φkn σn (xπn τπn )

π∈Sn

=

1 n!

det [φki σi (xj τj )]1≤i,j≤n

(2.24)

Second Quantization

11

The orthogonality relation reads   φk1 σ1 ∧ · · · ∧ φkn σn , φk
1 σ1
∧ · · · ∧ φk
n σn
F =

1 M nd



N

φk1 σ1 ∧ · · · ∧ φkn σn (x1 τ1 , · · · , xn τn ) ×

x1 τ1 ···xn τn

φk
1 σ1
∧ · · · ∧ φk
n σn
(x1 τ1 , · · · , xn τn ) =

1 M nd

=

1 M nd

=

1 M nd

=

1 n!

=

1 n!

 x1 τ1 ···xn τn

 x1 τ1 ···xn τn



x1 τ1 ···xn τn

1 n!2



signπ φk1 σ1 (xπ1 τπ1 ) · · · φkn σn (xπn τπn ) ×

π∈Sn

1 n! 1 n!

  det φk
i σi
(xj τj )

  φk1 σ1 (x1 τ1 ) · · · φkn σn (xn τn ) det φk
i σi
(xj τj ) 

signπ φk1 σ1 (x1 τ1 ) · · · φkn σn (xn τn ) ×

π∈Sn



φkπ1 σπ1 (x1 τ1 ) · · · φkπn σπn (xn τn )

det (φki σi , φk
j σj
)F1 

±1 if {k1 σ1 , · · · , kn σn } = {k1 σ1 , · · · , kn σn } 0 else

(2.25)

Thus an orthonormal basis of FN is given by √   N ! φk1 σ1 ∧ · · · ∧ φkN σN  k1 σ1 ≺ · · · ≺ kN σN , (ki , σi ) ∈ M × {↑, ↓} where ≺ is any ordering on M × {↑, ↓}. Another way of writing this is √     N ! (φkσ )nkσ  nkσ ∈ {0, 1}, nkσ = N (2.26) kσ

kσ

Since 1 − 2m

N  i=1

∆xi

   (φkσ )nkσ = nkσ ε(k) (φkσ )nkσ kσ

kσ

(2.27)

kσ

one ends up with QN = T r e−βH0,N =



e−β

P kσ

ε(k)nkσ

{nkσ } P nkσ =N

for the canonical partition function of the ideal Fermi gas.


(2.28)

12  Because of the constraint nkσ = N formula (2.28) cannot be simplified any further. However, if we consider a generating function for the QN ’s which involves a sum over N , we arrive at a more compact expression: Z(z) :=

∞ 

z N QN =

N =0

=

 {nkσ }

∞ 



N =0

{nkσ } P nkσ =N

 nkσ −βε(k) Π ze

kσ

 nkσ

−βε(k) = Π 1 + z e−βε(k) Π ze

kσ

kσ

(2.29)

Thus, from a computational point of view, it is not too practical to have the constraint of a given number of particles, nkσ = N . Therefore, instead of using the canonical ensemble one usually computes in the ∞ Grand Canonical Ensemble: Let F = ⊕∞ N =0 FN , H = ⊕N =0 HN . An ∞ observable has to be represented by some operator A = ⊕N =0 AN : F → F and measurements correspond to the expectation values

A F =

T rF A e−β(H−µN) T rF e−β(H−µN)

(2.30)

where the chemical potential µ has to be determined by the condition N = N , N being the given number of particles and N the number operator, N(1, F1 , F2 , · · · ) := (0, F1 , 2F2 , · · · ). Example (The Ideal Fermi Gas): We compute the chemical potential µ = µ(β, N, L) = µ(β, ρ) for the ideal Fermi gas with density ρ = N/Ld and we calculate the energy for the ideal Fermi gas. To this end introduce the ‘fugacity’ z which is related to µ according to z = eβµ . One has ∞ d N Z(z) z dz d =0 N z QN N = N F = N∞ = z dz = log Z(z) N Z(z) N =0 z QN

 z e−βε(k)  d log 1 + z e−βε(k) = 2 = z dz 1 + z e−βε(k) kσ k  z e−βε(k) d dd k ≈ 2L (2.31) (2π)d 1 + z e−βε(k) where we have used (2.4) in the last line and the integral goes over [−πM, πM ]d . Recalling that z = eβµ and introducing ek := ε(k) − µ we obtain in the zero temperature limit   e−βek β→∞ d dd k N = 2Ld (2π) → 2L d 1 + e−βek

(2.32)

dd k (2π)d

χ(ek < 0)

(2.33)

Second Quantization

13

which determines µ as a function of the density ρ = N/Ld . The expectation value of the energy is obtained from Z(z) according to d d H0 F = − dβ log Z + µN = − dβ 2

=2

 k



   log 1 + e−βek + µ2

k

e−βek ε(k) ≈ 2Ld 1 + e−βek



k d k (2π)d d

e−βek 1 + e−βek



ε(k) χ ε(k) < µ)

(2.34)

and the last approximation holds for large volume and small temperature.


2.3

Annihilation and Creation Operators

2.3.1

Coordinate Space

Let α ∈ {↑, ↓} be a spin index and let δxα (x α ) := δα,α
M d δx,x


(2.35)

For FN ∈ FN the wedge product δxα ∧ FN ∈ FN +1 is defined by (δxα ∧ FN )(x1 α1 , · · · , xN +1 αN +1 ) := 1 N +1

N +1 

(−1)i−1 δxα (xi αi ) FN (x1 α1 , · · · , x i αi , · · · , xN +1 αN +1 )

(2.36)

i=1

Then the creation operator at x is defined by ψ + (xα) : FN → FN +1 , √ ψ + (xα)FN := N + 1 δxα ∧ FN (2.37) and the annihilation operator ψ(xα) : FN +1 → FN is defined by the adjoint of ψ + , ψ(xα) = [ψ + (xα)]∗ . Lemma 2.3.1 (i) The adjoint operator ψ(xα) : FN +1 → FN is given by √ (ψ(xα)FN +1 ) (x1 α1 , · · · , xN αN ) = N + 1 FN +1 (xα, x1 α1 , · · · , xN αN ) (2.38) (ii) The following canonical anticommutation relations hold: {ψ(xα), ψ(yβ)} = {ψ + (xα), ψ + (yβ)} = 0 , {ψ(xα), ψ + (yβ)} = δαβ M d δxy .

(2.39)

14 Proof: (i) We abbreviate ξ = (xα) and η = (yβ). One has   FN +1 , ψ + (ξ)GN FN +1 =  √ F¯N +1 (ξ1 , · · · , ξN +1 ) N + 1 (δξ ∧ GN )(ξ1 , · · · , ξN +1 ) ξ1 ···ξN +1

=

√ 1 N +1



F¯N +1 (ξ1 , · · · , ξN +1 ) ×

ξ1 ···ξN +1 N +1 

(−1)i−1 δξ (ξi )GN (ξ1 , · · · , ξˆi , · · · ξN +1 )

i=1

=

√ 1 N +1

N +1 



i=1

ξ1 ··· ,ξˆi ,···ξN +1

(−1)i−1

F¯N +1 (ξ1 , · · · , ξ, · · · , ξN +1 ) ×

GN (ξ1 , · · · , ξˆi , · · · ξN +1 ) = =

√ 1 N +1

√

N +1 



F¯N +1 (ξ, η1 , · · · , ηN ) GN (η1 , · · · , ηN )

i=1 η1 ···ηN

    N + 1 FN +1 (ξ, ·), GN (·) F = ψ(ξ)FN +1 , GN F . N

N

(2.40)

(ii) We compute {ψ, ψ + }. One has √   ψ(ξ)ψ + (η)FN (ξ1 , · · · , ξN ) = N + 1 (ψ(ξ) δη ∧ FN ) (ξ1 , · · · , ξN ) = (N + 1) (δη ∧ FN ) (ξ, ξ1 , · · · , ξN ) = δη (ξ)FN (ξ1 , · · · , ξN ) +

N 

(2.41)

(−1)(i+1)−1 δη (ξi )FN (ξ, ξ1 , · · · , ξˆi , · · · , ξN )

i=1

Since

√   +   ψ (η)ψ(ξ)FN (ξ1 , · · · , ξN ) = N ψ + (η)FN (ξ, ·) (ξ1 , · · · , ξN ) = N N1

N 

(−1)i−1 δη (ξi )FN (ξ, ξ1 , · · · , ξˆi , · · · , ξN )

(2.42)

i=1

the lemma follows.
In the following theorem we show that the familiar expressions for the Hamiltonian in terms of annihilation and creation operators is just another representation for an N -particle Hamiltonian of quantum mechanics. So although these representations are sometimes referred to as ‘second quantization’, there is conceptually nothing new. We use the notation Γs = Γ × {↑, ↓} (‘s’ for ‘spin’) and write L2 (Γs ) = C|Γs | .

Second Quantization

15

Theorem 2.3.2 (i) Let h = (hxα,yβ ) : L2 (Γs ) → L2 (Γs ) (one particle Hamiltonian) and let H0,N =

N 

hi : F N → F N

(2.43)

i=1

where (hi FN )(x1 α1 , · · · , xN αN ) =  1 h(xi αi , yβ) FN (x1 α1 , · · · , yβ, · · · , xN αN ) Md

(2.44)

yβ

Then



1 M 2d

  ψ + (xα)h(xα, yβ)ψ(yβ)

xy αβ

FN

= H0,N

(2.45)

(ii) Let v : Γ → R and let VN : FN → FN be the multiplication operator (VN FN )(x1 α1 , · · · , xN αN ) =

1 2

N 

v(xi − xj )FN (x1 α1 , · · · , xN αN ) (2.46)

i,j=1 i=j

Then 1 1 2 M 2d

  ψ + (xα)ψ + (yβ) v(x − y) ψ(yβ)ψ(xα)

 xy αβ

FN

= VN

(2.47)

Proof: We abbreviate again ξ = (xα) and η = (yβ). One has   ψ + (ξ)h(ξ, η)ψ(η)FN (ξ1 , · · · , ξN ) = = h(ξ, η) √1N

N 

(−1)i−1 δξ (ξi ) (ψ(η)FN ) (ξ1 , · · · ξˆi · · · , ξN )

i=1

= h(ξ, η)

N 

δξ (ξi )FN (ξ1 , · · · , η, · · · , ξN )

i=1

which gives 1 M 2d



 ψ + (ξ)h(ξ, η)ψ(η)FN (ξ1 , · · · , ξN )

ξ,η

=

N 

1 M 2d

i=1

=

N  i=1



δξ (ξi )h(ξ, η)FN (ξ1 , · · · , η, · · · , ξN )

ξ,η 1 Md

 η

h(ξi , η)FN (ξ1 , · · · , η, · · · , ξN )

(2.48)

16 =

N 

(hi FN )(ξ1 , · · · , ξN )

(2.49)

i=1

This proves part (i). To obtain (ii) observe that because of (2.48) one has N    + δξ (ξi ) FN (ξ1 , · · · , ξN ) ψ (ξ)ψ(ξ)FN (ξ1 , · · · , ξN ) =

(2.50)

i=1

Since + ψ= (xα)ψ + (yβ) v(x − y) ψ(yβ)ψ(xα)

= ψ + (xα)ψ(xα) v(x − y) ψ + (yβ)ψ(yβ) − δxα (yβ) ψ + (xα)v(x − y)ψ(yβ) one gets, using (2.48) again,   ψ + (ξ)ψ + (η)v(x − y) ψ(η)ψ(ξ)FN (ξ1 , · · · , ξN ) =

N 

δξ (ξi ) δη (ξj ) v(x − y) FN (ξ1 , · · · , ξN )

i,j=1

−

N 

δξ (η)δξ (ξi ) v(x − y) FN (ξ1 , · · · , ξN )

i=1

=

N 

δξ (ξi ) δη (ξj ) v(x − y) FN (ξ1 , · · · , ξN )

(2.51)

i,j=1 i=j

+

N 

[δξ (ξi )δη (ξi ) − δξ (η)δξ (ξi )] v(x − y) FN (ξ1 , · · · , ξN )

i=1

Since the terms in the last line cancel part (ii) is proven.


2.3.2

Momentum Space

Recall that the plane waves φkσ (xτ ) = δσ,τ L− 2 eikx are an orthonormal basis of F1 . We define  d  akσ = M1d L 2 φ¯kσ (xτ )ψxτ = M1d e−ikx ψxσ (2.52) d

xτ

⇒

a+ kσ

=

1 Md

 xτ

x

L

d 2

+ φkσ (xτ )ψxτ

=

1 Md

 x

+ eikx ψxσ

(2.53)

The following corollary follows immediately from the properties of ψ and ψ + .

Second Quantization

17

Corollary 2.3.3 (i) One has (akσ FN +1 )(x1 σ1 , · · · , xN σN )  √ e−ikx FN +1 (xσ, x1 σ1 , · · · , xN σN ) = N + 1 M1d

(2.54)

(a+ kσ FN )(x1 σ1 , · · · , xN +1 σN +1 )

(2.55)

x

√ 1 N +1

=

N +1 

(−1)j−1 eikxj FN (x1 σ1 , · · · , x j σj , · · · , xN σN )

j=1

(ii) The following canonical anticommutation relations hold + {akσ , ak
σ
} = {a+ kσ , ak
σ
} = 0 d

{akσ , a+ k
σ
} = δσ,σ L δk,k

(2.56)

Theorem 2.3.4 Let H = ⊕∞ N =0 HN where 1 HN = − 2m

N 



∆xi +

i=1

V (xi − xj ) − µN

i,j=1 i=j

Then there is the representation    + 1 H = L1d e k a+ V (k − p) a+ kσ akσ + L3d kσ aq−k,τ aq−p,τ apσ (2.57) kσ

σ,τ k,p,q

where ek =

d 

1 2m

2M 2 (1 − cos[ki /M ]) − µ

M→∞

→

k2 2m

−µ

(2.58)

i=1

Proof: Let H0,N = (i), we have H0 =

1 2m

N

i=1 (−∆xi



1 Md

x,σ



=

1 Md

=

1 L2d

x,σ

 k,p

=

1 Ld

 k

− µ). According to Theorem 1.1.2, part

+ ψxσ (−∆ − µ)ψxσ 1 Ld

 k

1 e−ikx a+ kσ (−∆ − µ) Ld

L δk,p (ε(p) − d



eipx apσ

p

µ)a+ kσ apσ

(ε(k) − µ)a+ kσ akσ

(2.59)

18 Similarly we obtain for the interacting part I according to the second part of Theorem 1.1.2  + + I = M12d ψxσ ψyτ V (x − y)ψyτ ψxσ x,y σ,τ

=

1 M 2d

 x,y σ,τ

1 L4d



e−ik1 x−ik2 y+ik3 y+ik4 x ×

k1 k2 k3 k4 + a+ k1 σ ak2 τ V (x − y)ak3 τ ak4 σ

(2.60)

 The exponential equals e−i(k1 −k4 )(x−y) e−i(k1 +k2 −k3 −k4 )y . Since M1d y eipy = Ld δp,0 , we arrive at   + Ld δk1 +k2 ,k3 +k4 Vˆ (k1 − k4 )a+ (2.61) I = L14d k 1 σ ak 2 τ ak 3 τ ak 4 σ σ,τ k1 k2 k3 k4

If we choose k1 =: k, k4 =: p and q := k1 + k2 we obtain (2.57).


Chapter 3 Perturbation Theory

In this chapter we derive the perturbation series for the partition function. Since it is given by a trace of an exponential of an operator H0 + λV , we first expand the exponential with respect to λ. The result is a power series with operator valued coefficients. Then we compute the trace of the n’th order coefficient. This can be done with Wick’s theorem which lies at the bottom of any diagrammatic expansion. It states that the expectation value, with respect to exp[−βH0 ], of a product of 2n annihilation and creation operators is given by a sum of terms where each term is a product of expectation values of only two annihilation and creation operators. Depending on whether the model is fermionic or bosonic or complex or scalar, this sum is given by a determinant, a pfaffian, a permanent or simply the sum over all pairings which is the bosonic analog of a pfaffian. For the many-electron system, one obtains an n × n determinant. The final result is summarized in Theorem 3.2.4 below.

3.1

The Perturbation Series for eH0 +λV

The goal of this section is to prove the following Theorem 3.1.1 Let H0 , V ∈ CN ×N and let Hλ = H0 + λV . Then e(t2 −t1 )Hλ = et2 H0 Teλ

R t2 t1

ds V (s) −t1 H0

e

(3.1)

where Teλ

R t2 t1

ds V (s)

:=

∞


λn n!

 [t1 ,t2 ]n

ds1 · · · dsn T [V (s1 ) · · · V (sn )]

(3.2)

n=0

and T [V (s1 ) · · · V (sn )] = V (sπ1 ) · · · V (sπn )

(3.3)

if π is a permutation such that sπ1 ≥ · · · ≥ sπn . Furthermore V (s) = e−sH0 V esH0 .

19

20 For the proof we need the following

Lemma 3.1.2 Let Hλ be as above. Then: d (t2 −t1 )Hλ dλ e

=

 t2 t1

ds e(t2 −s)Hλ V e(s−t1 )Hλ

(3.4)

Proof: We have s=t2 e(t2 −t1 )Hλ − e(t2 −t1 )Hλ
= −e(t2 −s)Hλ e(s−t1 )Hλ
s=t1  (t −s)H (s−t )H
 t d 1 λ λ e 2 = − t12 ds ds e  t2 = − t1 ds e(t2 −s)Hλ (−Hλ + Hλ
)e(s−t1 )Hλ
t = t12 ds e(t2 −s)Hλ (λ − λ )V e(s−t1 )Hλ
(3.5) which gives d (t2 −t1 )Hλ dλ e

e(t2 −t1 )Hλ − e(t2 −t1 )Hλ
= lim λ
→λ λ − λ  t2 = ds e(t2 −s)Hλ V e(s−t1 )Hλ
t1

Proof of Theorem: Since for matrices H0 , V the exponential eH0 +λV is analytic in λ, one has e(t2 −t1 )Hλ =

∞


λn n!



 d n dλ |λ=0

e(t2 −t1 )Hλ

(3.6)

n=0

Thus we have to prove 

 d n (t2 −t1 )Hλ dλ |λ=0 e 

=

et2 H0

ds1 · · · dsn T[V (s1 ) · · · V (sn )] e−t1 H0

(3.7)

[t1 ,t2 ]n

We claim that for arbitrary λ  d n (t −t )H e 2 1 λ = dλ  ds1 · · · dsn T[Vλ (s1 ) · · · Vλ (sn )] e−t1 Hλ et2 Hλ

(3.8)

[t1 ,t2 ]n

where Vλ (s) = e−sHλ V esHλ . Apparently (3.8) for λ = 0 gives (3.7). (3.8) is proven by induction on n. For n = 1, (3.8) reduces to the lemma above.

Perturbation Theory

21

Suppose (3.8) is correct for n − 1. Since T[Vλ (s1 ) · · · Vλ (sn )] is a symmetric function, one has  ds1 · · · dsn T[Vλ (s1 ) · · · Vλ (sn )] [t1 ,t2 ]n  = n! [t1 ,t2 ]n ds1 · · · dsn χ(s1 ≥ s2 ≥ · · · ≥ sn )T[Vλ (s1 ) · · · Vλ (sn )] s s t (3.9) = n! t12 ds1 t11 ds2 · · · t1n−1 dsn V (s1 ) · · · V (sn ) and the induction hypothesis reads 

 d n−1 (t2 −t1 )Hλ e = dλ  t2 s (n − 1)! t1 ds1 · · · t1n−2

dsn−1 e(t2 −s1 )Hλ V e(s1 −s2 )Hλ · · · V e(sn−1 −t1 )Hλ

Thus 

 d n (t2 −t1 )Hλ e dλ

= (n − 1)!

 t2 t1

ds1 · · ·

 sn−2 t1

dsn−1

d dλ



 (3.10)

where, if we put s0 := t2 and sn := t1     d d = dλ e(t2 −s1 )Hλ V e(s1 −s2 )Hλ · · · V e(sn−1 −t1 )Hλ dλ =

=

n−1
i=0 n−1


d (si −si+1 )Hλ e(t2 −s1 )Hλ V · · · dλ e · · · V e(sn−1 −t1 )Hλ

e(t2 −s1 )Hλ V · · ·

 si si+1

ds e(si −s)Hλ V e(s−si+1 )Hλ · · ·

i=0

· · · V e(sn−1 −t1 )Hλ

(3.11)

Substituting this in (3.10), we obtain the following integrals  t2 t1

ds1 · · ·

 si−1 t1

dsi

 si

=

si+1



ds

 si

[t1 ,t2 ]n

t1

dsi+1 · · ·

 sn−2 t1

dsn−1

ds1 · · · dsi ds dsi+1 · · · dsn χ( · · · )

where the characteristic function is χ( · · · ) = χ(s1 ≥ · · · ≥ si−1 ≥ si ≥ s ≥ si+1 ≥ · · · ≥ sn−1 ) which results in (renaming the integration variables)  d n (t −t )H e 2 1 λ = dλ  n(n − 1)! [t1 ,t2 ]n χ(s1 ≥ · · · ≥ sn ) e(t2 −s1 )Hλ V · · · V e(sn −t1 )Hλ


22

3.2

The Perturbation Series for the Partition Function

In this section we prove Theorem 3.2.4 below in which the perturbation series for the partition function is written down. Let H = H(λ) = H0 +λHint : F → F be the Hamiltonian where
+ ψxσ h(xσ, yτ )ψyτ (3.12) H0 = M12d xσ,yτ

Hint =

1 M 2d




xσ,yτ

+ + ψxσ ψyτ V (x − y)ψyτ ψxσ

(3.13)

According to Theorem 3.1.1 of the last section we have T r e−βH(λ) =  ∞
(−λ)n n! n=0

(3.14)

ds1 · · · dsn T r e−βH0 THint (s1 ) · · · Hint (sn )

[0,β]n

where Hint (s) = esH0 Hint e−sH0
+ + = M12d ψxσ (s)ψyτ (s)V (x − y)ψyτ (s)ψxσ (s)

(3.15)

xσ,yτ

(3.16) if we define ψxσ (s) := esH0 ψxσ e−sH0 + + −sH0 (s) := esH0 ψxσ e ψxσ

(3.17) (3.18)

+ (s) is no longer the adjoint of ψxσ (s). In (3.14) Observe that for s = 0 ψxσ the ‘temperature’ ordering operator T acts on the operators Hint (s). Instead of this, we may let T act directly on the annihilation and creation operators ψ(s), ψ + (s). We introduce the following notation: For

¯ ξ = (s, x, σ, b) ∈ [0, β] × Γ × {↑, ↓} × {0, 1} =: Γ

(3.19)

let  ψ(ξ) :=

+ ψxσ (s) if b = 1 ψxσ (s) if b = 0

(3.20)

Then we define T[ψ(ξ1 ) · · · ψ(ξn )] := signπ ψ(ξπ1 ) · · · ψ(ξπn )

(3.21)

Perturbation Theory

23

if π is a permutation such that sπ1 ≥ · · · ≥ sπn and, if sπj = sπ(j+1) , one has bπj ≥ bπ(j+1) . The last requirement is necessary since in (3.15) there are ψ(s) and ψ + (s) for equal values of s and these operators do not anticommute. We also define  h if b = 0 (3.22) h(b) = −hT if b = 1 Lemma 3.2.1 The following relations hold: (i) Let ψ (+) (s) = esH0 ψ (+) e−sH0 . Then
− s h ψxσ (s) = e M d (xσ, yτ ) ψyτ

(3.23)

yτ

+ (s) = ψxσ


yτ

s

+ e M d h (yτ, xσ) ψyτ

(3.24)

(ii) The following anticommutation relations hold + + {ψxσ (s), ψyτ (t)} = {ψxσ (s), ψyτ (t)} = 0 + {ψxσ (s), ψyτ (t)} = M d e

(s−t) h Md

(yτ, xσ)

(3.25) (3.26)

Using the notation (3.22), this can be written more compactly as {ψ(ξ), ψ(ξ  )} = δb,1−b
M d e 1 ∆ − µ, In particular, for h = − 2m + {ψxσ (s), ψyτ (t)} = δσ,τ

1 Ld

−(s−s
) h(b) Md




(xσ, x σ  )

(3.27)

eik(x−y) e(s−t)ek

(3.28)

k

Proof: We have (+) (+) = esH0 (H0 ψxσ − ψxσ H0 )e−sH0

(+) d ds ψxσ (s)

Furthermore H0 ψxσ =




1 M 2d

x
σ
,yτ

= − M12d

ψx+
σ
h(x σ  , yτ )ψyτ ψxσ




x
σ
,yτ

=

− M12d




x
σ
,yτ

ψx+
σ
ψxσ h(x σ  , yτ )ψyτ {ψx+
σ
, ψxσ }h(x σ  , yτ )ψyτ + ψxσ M12d

= − M1d


yτ

(3.29)


x
σ
,yτ

ψx+
σ
h(x σ  , yτ )ψyτ

h(xσ, yτ )ψyτ + ψxσ H0

(3.30)

24 and + = H0 ψxσ

1 M 2d




+ ψx+
σ
h(x σ  , yτ )ψyτ ψxσ

x
σ
,yτ

=

1 M 2d




+ ψx+
σ
h(x σ  , yτ ){ψyτ , ψxσ }

x
σ
,yτ

− =

1 Md


x
σ





1 M 2d

x
σ
,yτ

ψx+
σ
h(x σ  , xσ) + 1 + ψxσ M 2d

=

1 Md


x
σ


+ ψx+
σ
h(x σ  , yτ )ψxσ ψyτ


x
σ
,yτ

ψx+
σ
h(x σ  , xσ)

ψx+
σ
h(x σ  , yτ )ψyτ

+ + ψxσ H0

Thus we get the differential equations
d 1 h(xσ, yτ )ψyτ (s) ds ψxσ (s) = − M d

(3.31)

(3.32)

yτ

d + ds ψxσ (s)

1 Md

=


yτ

+ h(yτ, xσ)ψyτ (s)

(3.33)

which proves part (i) of the lemma. Equations (3.25) of part (ii) are obvious. To obtain (3.26), we use part (i):
s t + + {ψxσ (s), ψx
σ
(t)} = e M d h (yτ, xσ)e− M d h (x σ  , y τ  ){ψyτ , ψy
τ
} yτ,y
τ


= Md




e M d h (yτ, xσ)e− M d h (x σ  , yτ ) s

t

yτ s−t

= M e M d h (x σ  , xσ) d

(3.34)

Suppose now that h is the discrete Laplacian. That is h = [h(xσ, yτ )]xσ,yτ where h(xσ, yτ ) =

d 2 1 2m δσ.τ M M

d


(δx+ ei ,y + δx− ei ,y − 2δx,y ) − µM d δx,y (3.35) M

M

i=1

Let F =

1 d

(ML) 2

(e−ikx )k,x

(3.36)

be the matrix of discrete Fourier transform. Since F hF ∗ = [(ek δk,p )k,p ]

(3.37)

Perturbation Theory

25

we have, since hT = h, (s−t)

+ {ψxσ (s), ψyτ (t)} = M d e M d h (xσ, yτ )

(s−t) ∗ = δσ,τ F ∗ M d e M d F hF F x,y
ik(x−y) (s−t)ek 1 = δσ,τ Ld e e

(3.38)

k

which proves the lemma.
Lemma 3.2.2 (Wick) For some operator A : F → F let A0 :=

T r e−βH0 A T r e−βH0

(3.39)

Recall notation (3.20). Then one has ψ(ξ1 ) · · · ψ(ξn )0 = 0 if n is odd and for even n ψ(ξ1 ) · · · ψ(ξn )0 =

n



(−1)j ψ(ξ1 )ψ(ξj )0 ψ(ξ2 ) · · · ψ(ξ j ) · · · ψ(ξn )0 (3.40)

j=2

where ψ means omission of that factor. Furthermore ψx1 σ1 (s1 , b1 )ψx2 σ2 (s2 , b2 )0 = (3.41)
β [Id + e− M d h(b1 ) ]−1 (x1 σ1 , yτ ){ψyτ (s1 , b1 ), ψx2 σ2 (s2 , b2 )} yτ

if we use the notation (3.22).

Proof: Because of part (i) of the previous lemma we have ψxσ (s + β) =




β

e− M d h (xσ, yτ )ψyτ (s)

(3.42)

yτ

+ (s + β) = ψxσ


yτ

β

T

+ e M d h (xσ, yτ )ψyτ (s)

(3.43)

or in matrix notation, regarding ψ(s) = [ψxσ (s)]xσ∈Γ×{↑,↓} as a column vector, β

ψ(s + β) = e− M d h ψ(s),

β

T

ψ + (s + β) = e M d h ψ + (s)

(3.44)

To write this more compact, we use the definition (3.22) which gives β

ψ(s + β, b) = e− M d hb ψ(s, b)

(3.45)

26 Since T r[e−βH0 ψ(s, b)] = T r[ψ(s, b) e−βH0 ] = T r[e−βH0 eβH0 ψ(s, b) e−βH0 ] β

= T r[e−βH0 ψ(s + β, b)] = e− M d hb T r[e−βH0 ψ(s, b)]

(3.46)

we get β

(Id − e− M d hb )T r[e−βH0 ψ(s, b)] = 0

(3.47)

T r[e−βH0 ψ(s, b)] = 0

(3.48)

which implies

β

since all eigenvalues of Id − e− M d hb , namely 1 − e−β(−1) ek , are nonzero (we  d may assume ek = 0 for all k ∈ 2π L Z ). If we have n operators ψ(ξ1 ) ≡ ψ1 , · · · , ψ(ξn ) ≡ ψn , we write b

ψ1 ψ2 · · · ψn = {ψ1 , ψ2 }ψ3 · · · ψn − ψ2 ψ1 ψ3 · · · ψn = {ψ1 , ψ2 }ψ3 · · · ψn − ψ2 {ψ1 , ψ3 }ψ4 · · · ψn + ψ2 ψ3 ψ1 ψ4 · · · ψn n


j · · · ψn − (−1)n ψ2 · · · ψn ψ1 (3.49) = {ψ1 , ψj }(−1)j ψ2 · · · ψ j=2

which gives ψ1 ψ2 · · · ψn 0 + (−1)n ψ2 · · · ψn ψ1 0 n


j · · · ψn 0 = {ψ1 , ψj }(−1)j ψ2 · · · ψ

(3.50)

j=2

Since ψ2 · · · ψn ψ1 0 ≡ ψ2 · · · ψn ψ(s1 )0 = ψ(s1 + β)ψ2 · · · ψn 0 = e− M d h(b1 ) |1 ψ(s1 )ψ2 · · · ψn 0 β

(3.51)

where the subscript |1 at the exponential means matrix multiplication with respect to x1 σ1 from the ψ1 operator, we get β

[Id + (−1)n e− M d h(b1 ) ]|1 ψ1 ψ2 · · · ψn 0 n


j · · · ψn 0 = {ψ1 , ψj }(−1)j ψ2 · · · ψ

(3.52)

j=2

or ψ1 ψ2 · · · ψn 0 = n
β j

[Id + (−1)n e− M d h(b1 ) ]−1 |1 {ψ1 , ψj }(−1) ψ2 · · · ψj · · · ψn 0 (3.53) j=2

Perturbation Theory

27

In particular, for n = 2, β

ψ1 ψ2 0 = [Id + e− M d h(b1 ) ]−1 |1 {ψ1 , ψ2 }

(3.54)

which proves the lemma.
The following theorem computes the trace on the right hand side of (3.14).

Theorem 3.2.3 (Wick) (i) Let T be the temperature ordering operator defined in 3.21. Then Tψ(ξ1 ) · · · ψ(ξn )0 = n

(−1)j Tψ(ξ1 )ψ(ξj )0 Tψ(ξ2 ) · · · ψ(ξ j ) · · · ψ(ξn )0

(3.55)

j=2

and Tψ(ξ)ψ(ξ  )0 = (3.56)  − (s−sd
) h(b) e M (xσ, x σ  )[χ(s > s ) + χ(b > b )χ(s = s )] δb,1−b
M d β Id + e− M d h(b) (s
−s)
 e− M d h(b )      − (x σ , xσ)[χ(s > s) + χ(b > b)χ(s = s )] β
Id + e− M d h(b )

(+)

(+)

(ii) Let x = (x, x0 ) and ψxσ = ψxσ (x0 ). Then   Tψx+1 σ1 ψy1 τ1 · · · ψx+n σn ψyn τn 0 = det Tψx+i σi ψyj τj  1≤i,j≤n

(3.57)

1 + (iii) Let h = − 2m ∆ − µ and x = (x, x0 ). Then Tψxσ (x0 )ψx
σ
(x0 )0 = δσ,σ
C(x − x, x0 − x0 ) where

C(x) =

1 Ld




eikx e−x0 ek

k 1 d  0 βL

= lim


k,k0



χ(x0 ≤0) 1+eβek

−

−ik0 

χ(x0 >0) 1+e−βek

,

e ei(kx−k0 x0 ) ik , x0 ∈ R 0 −ek

x0 ∈ (−β, β) (3.58) (3.59)

Here the k0 sum ranges over πβ (2Z + 1) such that C as a function of x0 is 2β-periodic and satisfies C(x0 + β, x) = −C(x0 , x).

28 Remarks: 1) In the zero temperature limit (3.58) reads


eikx e−x0 ek χ(x0 ≤ 0)χ(ek < 0) − χ(x0 > 0)χ(ek > 0) (3.60) C(x) = L1d k

2) The prescription in (3.59) is only necessary for x0 = x0 . In that case we only get the part inside the Fermi surface given by χ(ek < 0) in (3.60). Proof: (i) Equation (3.55) is an immediate consequence of (3.40) since if s1 ≥ s2 ≥ · · · ≥ sn then we also have s1 ≥ sj and s2 ≥ · · · s j · · · ≥ sn . The temperature-ordered expectation of two fields is evaluated with (3.41) and (3.27). By definition of the T-operator Tψ(ξ)ψ(ξ  )0 = ψ(ξ)ψ(ξ  )0 [χ(s > s ) + χ(b > b )χ(s = s )] − ψ(ξ  )ψ(ξ)0 [χ(s < s ) + χ(b < b )χ(s = s )] By (3.41) and (3.27), ψ(ξ)ψ(ξ  )0 =




β

[Id + e− M d h(b) ]−1 (xσ, yτ ){ψ(y, τ, s, b), ψ(ξ  )}

yτ

=




β

[Id + e− M d h(b) ]−1 (xσ, yτ )δb,1−b
M d e

−(s−s
) h(b) Md

(yτ, x σ  )

yτ

which proves the first part of the theorem. (ii) We use part (i) and induction. For n = 1, the formula is correct. Suppose it holds for n. Then, expanding the determinant with respect to the first row,   det Tψx+i σi ψyj τj 0 1≤i,j≤n+1 = =

n+1


  (−1)1+j Tψx+1 σ1 ψyj τj 0 det Tψx+i σi ψyk τk 0 1≤i,k≤n+1 i=1, k=j

j=1

=

n+1


(−1)1+j Tψx+1 σ1 ψyj τj 0 Tψx+2 σ2 ψy1 τ1 · · · ψx+j σj ψyj−1 τj−1 ◦

j=1

◦ ψx+j+1 σj+1 ψyj+1 τj+1 · · · ψx+n+1 σn+1 ψyn+1 τn+1 0 =

n+1


(−1)1+j Tψx+1 σ1 ψyj τj 0 (−1)j−1 Tψy1 τ1 ψx+2 σ2 · · · ψyj−1 τj−1 ◦

j=1 + + ◦ ψx+j σj ψ
yj τj ψxj+1 σj+1 ψyj+1 τj+1 · · · ψxn+1 σn+1 ψyn+1 τn+1 0

= Tψx+1 σ1 ψy1 τ1 · · · ψx+n+1 σn+1 ψyn+1 τn+1 0

(3.61)

where we used part (i) in the last line and the induction hypothesis in the second line.

Perturbation Theory

29

(iii) From (3.56) we get + Tψxσ (x0 )ψx
σ
(x0 )0 = δσ,σ
M d

−



e

−(x
0 −x0 ) h Md β

Id + e M d h

e−

(x
0 −x0 ) h Md β

Id + e− M d h

(x σ  , xσ)χ(x0 ≤ x0 )  

(x σ

, xσ)χ(x0

 > x0 )

which gives C(xx0 , x x0 ) =

1 Ld










eik(x−x ) e−ek (x0 −x0 )

k



χ(x0 −x
0 ≤0) 1+eβek

−

χ(x0 −x
0 >0) 1+e−βek



This proves (3.58). The x0 variables in C(x, x ) = C(x − x ) = C(x0 − x0 , x − x ) = Tψx+
σ (x0 )ψxσ (x0 )0 range over x0 , x0 ∈ (0, β). In particular, x0 − x0 ∈ (−β, β). On this interval, C is an antisymmetric function. Namely, let x0 − x0 ∈ (−β, β) such that also x0 − x0 + β ∈ (−β, β). That is, x0 < x0 . Then C(x0 − x0 + β, x − x ) = Tψx+
σ (x0 )ψxσ (x0 + β)0  = −T r e−βH0 ψxσ (x0 + β)ψx+
σ (x0 ) T r e−βH0  = −T r ψxσ (x0 )e−βH0 ψx+
σ (x0 ) T r e−βH0  = −T r e−βH0 ψx+
σ (x0 )ψxσ (x0 ) T r e−βH0 = −Tψx+
σ (x0 )ψxσ (x0 + β)0 = −C(x0 − x0 , x − x )

(3.62)

Since C is only defined on (−β, β), we may expand it into a 2β-periodic Fourier series. This gives us ‘frequencies’ k0 ∈ 2π 2β Z. Because of (3.62), we only get the odd frequencies k0 ∈ 2π (2Z + 1). Equation (3.59) is then equivalent to 2β β −β

eik0 x0 C(x0 , k) dx0 =

2β β C(k0 , k)

= 2C(k0 , k)

where C(x0 , k) = e−x0 ek

χ(x ≤ 0) χ(x > 0)

1 0 0 , C(k0 , k) = − βe −βe k k 1+e 1+e ik0 − ek

This is checked by direct computation.
With the above theorem, we are able to write down the perturbation series for the partition function. If we substitute the trace on the right hand side of (3.14) by the determinant of (3.57), we arrive at the following series:

30 Theorem 3.2.4 Let x = (x0 , x), U (x) = δ(x0 ) V (x) and let C be given by (3.58). Then there is the following power series expansion ∞
T r e−βHλ (−λ)n = n! Dn −βH 0 Tre n=0

(3.63)

where the n’th order coefficient Dn is given by 
  2n n Dn = Π dx0,j M 12dn Π U (x2i−1 − x2i ) det δσi ,σj C(xi , xj ) [0,β]2n j=1

i=1 x1 σ1 ,··· ,x2n σ2n

    and det δσi ,σj C(xi , xj ) = det δσi ,σj C(xi − xj ) 1≤i,j≤2n . If we expand the determinant in Dn and interchange the sum over permutations with the space and temperature sums or integrals, we obtain the expansion into Feynman diagrams. That is,
Dn = signπ Gn (π) (3.64) π∈Sn

where the value of the diagram generated by the permutation π is given by Gn (π) = 

2n

Π dx0,j

[0,β]2n j=1




(3.65) n

1 Π U (x2i−1 M 2dn i=1 x1 σ1 ,··· ,x2n σ2n

2n

− x2i ) Π δσi ,σπi C(xi − xπi ) i=1

The diagrammatic interpretation we postpone to section 4.2.

3.3

The Perturbation Series for the Correlation Functions

The perturbation series for the correlation functions can be obtained by differentiating the perturbation series for the partition function. To this end we introduce parameters skσ and λq and the Hamiltonian H = H({skσ , λq })
(ek + skσ ) a+ = L1d kσ akσ + kσ

Hλ = H(skσ = 0, λq = λ)

1 L3d



σ,τ k,p,q

(3.66) λq V (k −

+ p) a+ kσ aq−k,τ aq−p,τ apσ

(3.67)

Perturbation Theory

31

We consider the two point function a+ kσ akσ  =

T r e−βHλ a+ kσ akσ T r e−βHλ

and the four point function

+ Λ(q) = L13d V (k − p)a+ kσ aq−k,τ aq−p,τ apσ 

(3.68)

(3.69)

σ,τ k,p

The expectation on the right hand side of (3.69) is also with respect to e−βHλ , as in (3.68). From these functions we can compute the expectation of the energy according to

Hλ  = L1d ek a+ Λ(q) (3.70) kσ akσ  + q

kσ

Since we defined the creation and annihilation operators such that {akσ , a+ pσ } = a  and Λ(q) are intensive quantities such Ld δk,p we will find that L1d a+ kσ kσ that Hλ  is proportional to the volume as it should be (for constant density and short-range interaction). Theorem 3.3.1 Let Z({skσ , λq }) =

T r e−βH({skσ ,λq }) T r e−βH0

(3.71)

Then 1 a+ kσ akσ  Ld

= − β1 ∂s∂kσ

Λ(q) = − β1 ∂λ∂ q

|s=0

log Z({skσ })

(3.72)

log Z({λq })

(3.73)

|λq =λ

Proof: According to Lemma 3.1.2 one has for some differentiable matrix A = A(s)  1 d A(s) ˙ evA(s) dv e = e(1−v)A(s) A(s) (3.74) ds 0

which gives, since the trace is linear and cyclic,  1 A(s) d ˙ evA(s) ]dv T r e = T r[e(1−v)A(s) A(s) ds 0



1

˙ T r[evA(s) e(1−v)A(s) A(s)]dv

= 0

˙ = T r[eA(s) A(s)]

(3.75)

Using (3.75) with A being the Hamiltonian (3.66) proves the theorem.


Chapter 4 Gaussian Integration and Grassmann Integrals

In this chapter we introduce anticommuting variables and derive the Grassmann integral representations for the partition function and the correlation functions. Grassmann integrals are a suitable algebraic tool for the rearrangement of fermionic perturbation series. We demonstrate this in the first section by considering a model with a quadratic perturbation which is explicitely solvable. In that case the perturbation series can be resumed completely. By doing this in two ways, first a direct computation without anticommuting variables and then a calculation using Grassmann integrals, we hope to be able to convince the reader of the utility of that formalism. In the last two sections Gaussian integrals are discussed. They come into play because they produce exactly the same combinatorics as the outcome of the Wick theorem. That is, the expectation of some monomial with respect to a bosonic or Grassmann, scalar or complex Gaussian measure is given by a determinant or pfaffian or permanent or simply the sum over all pairings. For example, the 2n×2n determinant in (3.63) can be written as the expectation of 4n anticommuting variables or ‘fields’ with respect to a Grassmann Gaussian measure which is independent of n and it turns out that the remaining series can be rewritten as an exponential which leads to a compact expression, the Grassmann integral representation. In chapter 9 we estimate sums of diagrams for a fermionic model and show that the sum of all convergent diagrams has a positive radius of convergence. The proof of that, actually already the precise statement of the theorem, makes heavy use of Grassmann integrals. The reason is that, in order to get sensible estimates, we have to get Feynman diagram-like objects, that is, we have to expand the 2n × 2n determinant in (3.63) to a certain amount, but, since we have to use the sign cancellations in that determinant, we are not allowed to expand too much. To handle the algebra of that expansion it turns out that Grassmann integrals are a suitable tool. The key estimate which makes use of sign cancellations is given in Theorem 4.4.9. This theorem, including Theorem 4.4.8, Lemma 4.4.7 and Definition 4.4.6 will not be used before chapter 9.

33

34

4.1

Why Grassmann Integration? A Motivating Example

To demonstrate the utility of Grassmann integrals, we consider in this section only a quadratic perturbation instead of a quartic one. In that case the partition function is explicitly known and its perturbation series can be completely resumed. First we will do this without using Grassmann integrals which requires about two pages. Then we show the computation with Grassmann integrals which, once the properties of Grassmann integration are known, fits in three lines. We consider the Hamiltonian

λ e k a+ a+ (4.1) Hλ = H0 + λH
:= L1d kσ akσ + Ld kσ akσ kσ

kσ

According to (2.29) its partition function is given by  Z(λ) = T r e−βHλ T r e−βH0 = Π

1+e−β(ek +λ) −βek kσ 1+e

(4.2)

On the other hand, according to chapter 3, the perturbation series for Z(λ) is given by Z(λ) =

∞


(−λ)n n!

 [0,β]n

ds1 · · · dsn
TH
(s1 ) · · · H
(sn )0

n=0

=

∞


(−λ)n n!

β 0

ds1 M1d

β x1 σ·1 · · 0



dsn M1d

 xn σn


Tψx+1 σ1 (s1 )ψx1 σ1 (s1 ) · · ·

n=0

=

∞


· · · ψx+n σn (sn )ψxn σn (sn )0 (−λ)n n!



dξ1 · · · dξn det [C(ξi , ξj )]1≤i,j≤n

(4.3)

n=0

where we abbreviated  β  dξ := 0 ds M1d xσ = limh→∞

1 hM d

 s,x,σ

(4.4)

and in the last equation in (4.4) we wrote the integral as a limit of a Riemannian sum, s = h1 j, 0 ≤ j ≤ βh − 1. Computation without Grassmann Integrals: We expand the determinant in (4.3):
signπ C(ξ1 , ξπ1 ) · · · C(ξn , ξπn ) det [C(ξi , ξj )] = π∈Sn

Gaussian Integration and Grassmann Integrals

35

Every permutation can be decomposed into cycles. A 1-cycle corresponds to a factor C(ξ, ξ), a 2-cycle corresponds to C(ξ, ξ
)C(ξ
, ξ) and an r-cycle corresponds to (eventually relabeling the ξ’s) C(ξ1 , ξ2 )C(ξ2 , ξ3 ) · · · C(ξr−1 , ξr )C(ξr , ξ1 )

(4.5)

If π consists of b1 1-cycles, b2 2-cycles up to bn n-cycles we say that π is of type t(π) = 1b1 2b2 · · · nbn

(4.6)

where in the last equation 1b1 2b2 · · · nbn is not meant as a number but just as an abbreviation for the above statement. Since the total number of permuted elements is n, one has 1b1 + 2b2 + · · · nbn = n

(4.7)

The number of such permutations is given by   {π ∈ Sn | t(π) = 1b1 2b2 · · · nbn } =

n! b1 ! · · · bn ! 1b1 · · · nbn

(4.8)

In the denominator on the right hand side of (4.8) of course the number 1b1 · · · nbn is meant. The sign of such a permutation is given by Pn

signπ = (−1)

r=1 (br −1)

= (−1)n−

P r

br

if t(π) = 1b1 · · · nbn

(4.9)

Now, the basic observation is that, since the ξ-variables are integrated over, every permutation of given type t(π) = 1b1 2b2 · · · nbn gives the same contribution br n  dξ1 · · · dξr C(ξ1 , ξ2 ) · · · C(ξr , ξ1 ) (4.10) Π r=1

Thus, the perturbation series (4.3) equals Z(λ) =

∞


n


(−λ)n n!

n=0




(−1)n−

π∈Sn b1 ,··· ,bn =0 P b r rbr =n t(π)=1 1 ···nbn

n

Π



r=1

=

∞
n=0

=

∞


λn n!

n


n

r

br

×

br dξ1 · · · dξr C(ξ1 , ξ2 ) · · · C(ξr , ξ1 )

  br − dξ1 · · · dξr C(ξ1 , ξ2 ) · · · C(ξr , ξ1 )

n! Π b1 !···bn ! 1b1 ···nbn r=1 =0

b1 ,··· ,bn P r rbr =n

n


n

Π

n=0

P

bP 1 ,··· ,bn =0 r rbr =n

r=1

1 br !

 r br − λr dξ1 · · · dξr C(ξ1 , ξ2 ) · · · C(ξr , ξ1 )

(4.11)

36 Let b = (b1 , b2 , · · · ) ∈ NN . Since lim

N →∞

N


n


n=0

b1 ,··· ,bn =0 P r rbr =n

N


F (b) = lim

N →∞

F (b)

(4.12)

b1 ,··· ,bN =0

we arrive at Z(λ) =

=

∞
∞

1 br !

 br  λ r − 1r T r ( hM d C)

r=1 br =0 ∞

− r1 T r [(

e

λ hM d

C)r ]

= e−

P∞

1 r=1 r T r

[( hMλ d C)r ]

r=1

 λ = eT r log[Id− hM d C ] = det Id −  = Π 1 − ik0λ−ek kσ  −β(ek +λ) = Π 1+e −βe k 1+e

λ C hM d

(4.13) (4.14)

kσ

where the last two lines, (4.13) and (4.14), are shown in the following Lemma 4.1.1 (i) Let C = [C(x − x
)]x,x denote the matrix with elements
 1 C(x − x
) = βL eik(x−x ) ik0 1−ek (4.15) d k

Let F = (Fkx ) be the unitary matrix of discrete Fourier transform in d + 1 dimensional coordinate-temperature space with matrix elements Fkx = √

1 (ML)d hβ

e−ikx

(4.16)

Then 1 (F ∗ CF )k,k hM d

=

δk,k ik0 −ek

(4.17)

(ii) The k0 -product has to be evaluated according to the ε-prescription and gives   −β(ek +λ) e−ik0 ε = 1+e (4.18) Π 1 − ik0λ−ek := lim Π 1 − λ ik 1+e−βek 0 −ek ε0 k0

k0

Proof: (i) We have 1 (F ∗ CF )k,k hM d

=

1 1 (hM d )2 βLd






e−ikx C(x − x
)eik x

x,x

= =

1 1 (hM d )2 βLd 1 hM d


y




e−iky C(y)




y

e−iky C(y) δk,k =





ei(k −k)x

x δk,k ik0 −ek



Gaussian Integration and Grassmann Integrals

37

which proves (i). To obtain (ii), observe that, from (3.59) we have 1 β



1 k0 ik0 −ek

Furthermore 1 β

sym

1 ik0 −ek

k0



1 0 β

:= lim

:=

1 2β

=

1 β

e−ik0  k0 ik0 −ek




1 ik0 −ek

=

+

1 1+eβek

1 −ik0 −ek

(4.19) 

k0


k0

−ek k02 +e2k

= − 21

Thus the product in (4.18) is found to be



 P − ∞  r=1 Π 1 − ik0λ−ek = Π 1 − ik0λ−ek = e k0

k0

−

P∞

λr

r=1 r =e

= Π sym 1 −

k0

= Π

k0 >0

Psym k0

1 (ik0 −ek )r

λ ik0 −ek

k02 +(ek +λ)2 k02 +e2k



λr r

1−e−βek 1+e−βek

P

Psym

e

k0

(4.20)

1 k0 (ik0 −ek )r

λ ik0 −ek

−

P

λ k0 ik0 −ek

β

e− 2 λ

β

e− 2 λ

(4.21)

Since [37] ∞

Π

1+

n=0

λ (2n+1)2 +e2

 =

√ cosh( π e2 +λ) 2 cosh π 2e

(4.22)

the product in (4.21) becomes Π π

k0 ∈ β (2N+1)

k02 +(ek +λ)2 k02 +e2k

β

e− 2 λ =

cosh β 2 (ek +λ) cosh β 2 ek

e− 2 λ = β

1+e−β(ek +λ) 1+e−βek

which proves part (ii).
Computation with Grassmann Integrals: The systematic theory of Grassmann integration is presented in section 4.4. Here we simply use the two basic properties of Grassmann integration, (4.28) and (4.30) below, without proof to reproduce the result (4.14) for the partition function with a quadratic perturbation. Let ψ1 , · · · , ψN , ψ¯1 , · · · , ψ¯N be 2N Grassmann variables. That is, we have the anticommutation relations ψi ψj = −ψj ψi , ψi ψ¯j = −ψ¯j ψi , ψ¯i ψ¯j = −ψ¯j ψ¯i

(4.23)

The reason for writing ψ¯1 , · · · , ψ¯N for the second N Grassmann variables instead of ψN +1 , · · · , ψ2N is the following. For the many-electron system we

38 will have to use Grassmann 2N Gaussian integrals where the quadratic part is not of the general form i,j=1 ψi Aij ψj in which case the result is the pfaffian of the matrix A, but in case the quadratic N term in the exponential is of our N the more specific form i,j=1 ψi+N Bij ψj or i,j=1 ψ¯i Bij ψj in which case one obtains the determinant of B. We introduce the Grassmann algebra G2N = G2N (ψ1 , · · · , ψN , ψ¯1 , · · · , ψ¯N ) 
 = αI,J Π ψi Π ψ¯j  αI,J ∈ C I,J⊂{1,··· ,N }

i∈I

(4.24)

j∈J

The Grassmann integral is a linear functional on G2N .  N ¯ Definition 4.1.2 The Grassmann integral · ΠN i=1 dψi Πj=1 dψj : G2N → C is the linear functional on G2N defined by 
N N αI,J Π ψi Π ψ¯j Π dψi Π dψ¯j := α{1,··· ,N },{1,··· ,N } (4.25) I,J⊂{1,··· ,N }

i∈I

j∈J

i=1

j=1

N ¯ The notation ΠN i=1 dψi Πj=1 dψj is used to fix a sign. For example, on G4   3ψ1 ψ¯1 − 4ψ1 ψ2 ψ¯1 ψ¯2 dψ1 dψ2 dψ¯1 dψ¯2 = −4 (4.26)

but

  3ψ1 ψ¯1 − 4ψ1 ψ2 ψ¯1 ψ¯2 dψ1 dψ¯1 dψ2 dψ¯2 =  + 4 ψ1 ψ¯1 ψ2 ψ¯2 dψ1 dψ¯1 dψ2 dψ¯2 = +4

(4.27)

The two most important properties of Grassmann integrals are summarized in the following Lemma 4.1.3 Let A = (Aij )1≤i,j≤N ∈ CN ×N and for some Grassmann val¯ define the exponential by eF := 2N 1 F n . Then ued function F = F (ψ, ψ) n=0 n! we have a)  PN N ¯ (4.28) e− i,j=1 ψi Ai,j ψj Π (dψi dψ¯i ) = det A i=1

b) Let A be invertible, C = A−1 be the ‘covariance matrix’ and ¯ := dµC (ψ, ψ) Then

1 det A

e−

PN i,j=1

¯i Aij ψj N ψ

Π (dψi dψ¯i )

(4.29)

i=1

 ¯ = det [Ci ,j ] ψi1 ψ¯j1 · · · ψir ψ¯jr dµC (ψ, ψ) k l 1≤k,l≤r

(4.30)

Gaussian Integration and Grassmann Integrals

39

With these preparations we are now ready to apply the formalism of Grassmann integration to the perturbation series (4.3). We introduce a Grassmann ¯ algebra generated by Grassmann variables ψ(ξ), ψ(ξ) which are labeled by ξ = (x0 , x, σ) ∈ [0, β]1/h × [0, L]d1/M × {↑, ↓}

(4.31)

Recall that [0, β]1/h denotes the lattice of βh points in [0, β] with lattice spacing 1/h. The covariance matrix is given by Cξ,ξ = δσ,σ C(x − x
) = δσ,σ

1 βLd






eik(x−x ) ik0 1−ek

(4.32)

k

and the Grassmann Gaussian measure reads ¯ = det C e− dµC (ψ, ψ)

P

xσ,x σ

¯xσ [C −1 ] ψ xσ,x σ ψx σ

Π (dψxσ dψ¯xσ )

(4.33)

xσ

We rewrite the n × n determinant appearing on the right hand side of (4.3) as a Grassmann integral:  det[C(ξi , ξj )]1≤i,j≤n =

¯ 1 ) · · · ψ(ξn )ψ(ξ ¯ n ) dµC (ψ, ψ) ¯ ψ(ξ1 )ψ(ξ

(4.34)

Interchanging the Grassmann integral with the sum of the perturbation series, we get

Z(λ) =


∞

(−λ)n n!

n=0

=


∞ 

1 n!



¯ 1 ) · · · ψ(ξn )ψ(ξ ¯ n ) dµC (ψ, ψ) ¯ dξ1 · · · dξn ψ(ξ1 )ψ(ξ

 n ¯ ¯ λ dξ ψ(ξ)ψ(ξ) dµC (ψ, ψ)

n=0 P

λ ¯ ¯ e hM d xσ ψxσ ψxσ dµC (ψ, ψ)  h i P ¯xσ C −1 − xσ,x σ ψ − λ d δxσ,x σ ψxσ xσ,x σ hM = det C e Π (dψxσ dψ¯xσ ) xσ  λ = det C det C −1 − hM d Id   λ λ 1 − (4.35) = det Id − hM C = Π d ik0 −ek

=

kσ

which coincides with (4.13). In the fifth line we used (4.28) of Lemma 4.1.3.

40

4.2

Grassmann Integral Representations

The perturbation series for the quartic model was given by (3.63). Abbreviating again ξ = (x0 , x, σ) = (x, σ), this reads Z(λ) = =

∞
n=0 ∞


(−λ)n n! (−λ)n n!

 

dξ1 · · · dξ2n U (ξ1 − ξ2 ) · · · U (ξ2n−1 − ξ2n ) det [C(ξi , ξj )]

n=0

=


∞ 

=

n

dξ1 · · · dξ2n Π U (ξ2i−1 − ξ2i ) × 

i=1

¯ 1 ) · · · ψ(ξ2n )ψ(ξ ¯ 2n ) dµC (ψ, ψ) ¯ ψ(ξ1 )ψ(ξ

n  1


¯
¯ ¯ −λ dξdξ ψ(ξ) ψ(ξ)U (ξ − ξ )ψ(ξ ) ψ(ξ ) dµC (ψ, ψ) n!

n=0

e−λ

R

 ¯ ¯ ) dξdξ  ψ(ξ)ψ(ξ)U(ξ−ξ )ψ(ξ  )ψ(ξ

¯ dµC (ψ, ψ)

(4.36)

The diagrammatic interpretation is as follows. To obtain the coefficient of (−λ)n /n!, draw n vertices ψ(ξ1) ξ1

ψ(ξ2) U(ξ1-ξ2)

ψ(ξ1)

ψ(ξ3) ξ3

ξ2

ψ(ξ2)

ψ(ξ4) U(ξ3-ξ4)

ψ(ξ3)

ψ(ξ2n-1) ξ2n-1

• • •

ψ(ξ2n-1)

ξ4

ψ(ξ4)

• • •

ψ(ξ2n) U(ξ2n-1-ξ2n)

ξ2n

ψ(ξ2n)

Each vertex corresponds to the expression  ¯
)ψ(ξ
) ¯ dξdξ
ψ(ξ)ψ(ξ)U (ξ − ξ
)ψ(ξ

(4.37)

¯ with one of Then connect each of the outgoing legs, labelled by the ψ’s, the incoming legs, labelled by the ψ’s, in all possible ways. This produces

Gaussian Integration and Grassmann Integrals

41

(2n)! terms. The following figure shows a third order contribution for the permutations π = (1, 5, 2, 3, 6, 4): π1

π2

π3

π4

π5

π6 5

⇒

1

6 2

3

1

2

3

4

5

4

6

Its value is given by  dξ1 · · · dξ6 U (ξ1 − ξ2 )U (ξ3 − ξ4 )U (ξ5 − ξ6 )× C(ξ1 , ξ1 )C(ξ2 , ξ5 )C(ξ5 , ξ6 )C(ξ6 , ξ4 )C(ξ4 , ξ3 )C(ξ3 , ξ2 ) The correlation functions we are interested in are given by (3.72) and (3.73). Let
H0 ({skσ }) = L1d (ek + skσ )a+ (4.38) kσ akσ kσ

Then T r e−βHλ ({skσ }) T r e−βHλ ({skσ }) T r e−βH0 ({skσ }) = (4.39) −βH 0 Tre T r e−βH0 T r e−βH0 ({skσ })  R    ¯  ¯ ¯ det C det[C −1 ] = e−λ dξdξ ψ(ξ)ψ(ξ)U(ξ−ξ )ψ(ξ )ψ(ξ ) dµCs (ψ, ψ) s

Z({skσ }) =

where Cs (kσ) = ik0 −e1k −skσ and in the last line of (4.39) we used the analog  result of (4.35) for the quadratic perturbation given by L1d kσ skσ a+ kσ akσ . Since ¯ = det Cs e− dµCs (ψ, ψ)

P

xσ,x σ

¯xσ (C −1 ) ψ xσ,x σ ψx σ s

Π (dψxσ dψ¯xσ ) (4.40)

xσ

the determinant of Cs cancels out and we end up with  R    ¯  ¯ Z({skσ }) = e−λ dξdξ ψ(ξ)ψ(ξ)U(ξ−ξ )ψ(ξ )ψ(ξ ) × e−

P

xσ,x σ

¯xσ (C −1 ) ψ xσ,x σ ψx σ s

(4.41)

det C Π (dψxσ dψ¯xσ ) xσ

To proceed we transform to momentum space. Substitution of variables for Grassmann integrals is treated in the following lemma which is also proven in section 4.4

42 Lemma 4.2.1 Let A ∈ C2N ×2N , and let ψ = (ψ1 , · · · , ψN ), ψ¯ = (ψ¯1 , · · · , ψ¯N ), η = (η1 , · · · , ηN ), η¯ = (¯ η1 , · · · , η¯N ) be Grassmann variables. Let G be some Grassmann valued function. Then     N N ¯ det A Π (dψi dψ¯i ) = G(η, η¯) Π (dηi d¯ G A(ψ, ψ) ηi ) (4.42) i=1

i=1

The Fourier transform for Grassmann variables is defined by the following linear transformation:      ψkσ F 0 ψxσ = (4.43) 0 F∗ ψ¯kσ ψ¯xσ where F is the unitary matrix of discrete Fourier transform given by (4.16).

F 0 In particular det 0 F ∗ = 1. Since the bared ψ’s transform with F ∗ = F¯ , one may indeed treat a bared ψ as the complex conjugate of ψ. One obtains the following representations. Theorem 4.2.2 Let
e k a+ Hλ = L1d kσ akσ +

λ L3d

kσ



σ,τ k,p,q

+ V (k − p) a+ kσ aq−k,τ aq−p,τ apσ

(4.44)

and let
a+ kσ akσ  and Λ(q) be given by (3.68) and (3.69). Then there are the following Grassmann integral representations  P P ¯q−k,τ ψpσ ψq−p,τ ¯kσ ψ − λ V (k−p)ψ dµC (4.45) Z(λ) = e (βLd )3 στ kpq
+ 1 1 1 ¯ (4.46) Ld
akσ akσ  = β βLd
ψkσ ψkσ  k0 ∈ π β (2Z+1)

Λ(q) =

1 β




Λ(q0 , q)

(4.47)

q0 ∈ 2π β Z

Λ(q) =

1 (βLd )3





V (k − p)
ψ¯kσ ψ¯q−k,τ ψpσ ψq−p,τ 

(4.48)

σ,τ k,p

where  ¯ =
G(ψ, ψ)

P

P

¯

¯

¯ e− (βLd )3 στ kpq V (k−p)ψkσ ψq−kτ ψpσ ψq−pτ dµC G(ψ, ψ)  − λd 3 Pστ Pkpq V (k−p)ψ¯kσ ψ¯q−kτ ψpσ ψq−pτ e (βL ) dµC λ

(4.49)

and ¯ =Π dµC (ψ, ψ)

kσ

βLd ik0 −ek

e

−

1 βLd

P

¯

kσ (ik0 −ek )ψkσ ψkσ

Π (dψkσ dψ¯kσ )

kσ

(4.50)

Gaussian Integration and Grassmann Integrals Proof: We have




ψxσ = √

1 (ML)d βh

eikx ψkσ , ψ¯xσ = √

1 (ML)d βh

k




43

e−ikx ψ¯kσ

(4.51)

k

The quartic part becomes

1 ψ¯xσ ψxσ U (x − x
)ψ¯x σ ψx σ h2 M 2d σ,σ x,x

=

1 1 h2 M 2d (ML)2d (βh)2












e−i[(k1 −k2 )x+(k3 −k4 )x ] ×

σ,σ k1 k2 k3 k4 x,x

ψ¯k1 σ ψk2 σ U (x − x
)ψ¯k3 σ ψk4 σ =

1 1 h2 M 2d (ML)2d (βh)2







hM d U (k1 − k2 ) hM d βLd ×

σ,σ k1 k2 k3 k4

δk1 +k3 ,k2 +k4 ψ¯k1 σ ψk2 σ ψ¯k3 σ ψk4 σ =

1 1 (hM d )2 (βLd )2







βLd δk1 +k3 ,k2 +k4 ψ¯k1 σ ψk2 σ ψ¯k3 σ ψk4 σ

(4.52)

σ,σ k1 k2 k3 k4

The quadratic term becomes

ψ¯xσ (C −1 )xσ,x σ ψx σ = ψ¯kσ (F C −1 F ∗ )k,k ψk σ xσ,x σ

kσ,k σ




=

0 −ek ψ¯kσ ikhM d ψkσ

(4.53)

kσ

and the normalization factor, the determinant of C, is det C = Π

kσ

hM d ik0 −ek

(4.54)

To arrive at the claimed formulae we make another substitution of variables   d ¯kσ = hMdd η¯kσ η , ψ (4.55) ψkσ = hM kσ d βL βL d

, changes the normalization The determinant of this transformation, Πkσ hM βLd factor in (4.54) to the one in (4.50) and (4.50,4.45) follow. To obtain (4.46), we have according to (3.72) and (4.41) + 1 Ld
akσ akσ 

=

1 ∂ β ∂skσ |s=0

= log

1 ∂ β ∂skσ |s=0



e

−

log Z({skσ })

λ (βLd )3

P

e =

1 β


k0

P στ

−

1 βLd

kpq

P

¯q−k,τ ψpσ ψq−p,τ ¯kσ ψ V (k−p)ψ ¯

kσ (ik0 −ek −skσ )ψkσ ψkσ

×

Π (dψkσ dψ¯kσ )

kσ 1
ψ¯kσ ψkσ  βLd

(4.56)

44 and similarly according to (3.73) Λ(q) = − β1 ∂λ∂ q

log Z({λq })  P P 1 ¯q−k,τ ψpσ ψq−p,τ ¯kσ ψ − λq V (k−p)ψ = − β1 ∂λ∂ q log e (βLd )3 στ kpq dµC |λq =λ


1 = β1 V (k − p)
ψ¯kσ ψ¯q−k,τ ψpσ ψq−p,τ  (4.57) (βLd )3 |λq =λ

q0

στ

kp

which proves the theorem.


4.3

Ordinary Gaussian Integrals

In the next two theorems we summarize basic facts about Gaussian integration. The first theorem treats the real case and the second one the complex case. symmetric and positive definite and let Theorem 4.3.1 Let A ∈ Rn×n be  n C = A−1 . For x, k ∈ Rn let
k, x = j=1 kj xj . Then a) 

e−i k,x e− 2 x,Axdn x = 1

Rn √

b) Let dPC (x) :=

det A n (2π) 2

n

(2π) 2 √ det A

e− 2 k,Ck 1

(4.58)

e− 2 x,Axdn x and let 1


xj1 · · · xjm C :=

Rn

xj1 · · · xjm dPC (x)

(4.59)

Then one has, if C = A−1 = (cij )1≤i,j≤n
xj1 · · · xjm C =

m


cj1 ,jr
xj2 · · · x jr · · · xjm C

(4.60)

r=2

where x jr means omission of that factor. In particular,  pairings σ cjσ1 ,jσ2 · · · cjσ(m−1) ,jσm for m even
xj1 · · · xjm C = 0 for m odd

(4.61)

where the set of all pairings is given by those permutations σ ∈ Sm for which σ(2i − 1) < σ(2i) and σ(2i − 1) < σ(2i + 1) for all i.

Gaussian Integration and Grassmann Integrals

45

Proof: (i) We can diagonalize A by an orthogonal matrix SAS T = D = diag(λ1 , · · · , λn ). Change variables y = Sx, dn y = | det S|dn x = dn x, p = Sk to get   1 −i k,x − 12 x,Ax n e d x = e−i Sk,y e− 2 y,Dy dn y e =

n 

2π λi

e

1 − 2λ p2i i

i=1 n

=

(2π) 2 det D

=

(2π) 2 det A

n

e− 2 p,D 1

e− 2 k,A 1

−1

−1

p

k

(4.62)

(ii) We have 
xj1 · · · xjm C =

xj1 · · · xjm dPC (x)



e−i k,x dPC (x)

=

1 ∂ (−i)m ∂kj1

···

=

1 ∂ (−i)m ∂kj1

· · · ∂k∂j

e− 2 k,Ck

=

1 ∂ (−i)m ∂kj2

· · · ∂k∂j

n 
1 cj1 , k e− 2 k,Ck −

=

m
s=1

=

m


∂ ∂kjm | k=0

1

m |k=0

∂ 1 (−i)m−2 ∂kj2

m |k=0

=1

 ∂ · · · ∂k · · · ∂k∂j j s

cj1 ,js e− 2 k,Ck 1

m |k=0

cj1 ,js
xj2 · · · x js · · · xjm C

(4.63)

s=1

Then (4.61) is obtained from (4.60) by iteration.


and positive definite and let Theorem 4.3.2 Let A ∈ Cn×n be self adjoint n C = A−1 . For z, k ∈ Cn let
k, z = j=1 kj zj (linear in both arguments). For zj = xj + iyj , z¯j = xj − iyj let dzj d¯ zj := dxj dyj . Then a) For k, l ∈ Cn 

e−i k,z e−i l,¯z e− ¯z,Az dn zdn z¯ =

e− k,Cl

(4.64)

e− ¯z,Az dn zdn z¯ and let  zi1 · · · z¯jm dPC (z, z¯)
zi1 · · · zim z¯j1 · · · z¯jm C :=

(4.65)

R2n

b) Let dPC (z, z¯) :=

πn det A

det A πn

R2n

46 Then one has, if C = A−1 = (cij )1≤i,j≤n 


zi1 · · · zim z¯j1 · · · z¯jm C =

m


ci1 ,jr
zi2 · · · zim z¯j1 · · · z¯ ¯jm C jr · · · z

r=1

(4.66) where z jr means omission of that factor. In particular, 

π∈Sm ci1 ,jπ1


zi1 · · · zim z¯j1 · · · z¯jm C =

0

· · · cim ,jπm for m = m
for m = m
(4.67)

Proof: (i) We can diagonalize A with a unitary matrix U = R + iS, U AU ∗ = D = diag(λ1 , · · · , λn ). Change variables ξ = u+iv := U z = (R+iS)(x+iy) = Rx − Sy + i(Sx + Ry). Since det

∂(u, v) = det ∂(x, y)



R −S S R

 = | det(R + iS)|2 = 1

¯ k, q = U l we get with p = U  e

−i k,z −i l,¯ z  − ¯ z ,Az n

e

e

 n

d zd z¯ =

¯

¯

e−i p,ξ−i q,ξ e− ξ,Dξ dn ξdn ξ¯

Now for arbitrary complex numbers a and b and positive λ one has 

e−ia(u+iv)−ib(u−iv) e−λ(u +v ) dudv R2   2 −i(a+b)u −λu2 = e e du e(a−b)v e−λv dv 2

=

π λ

e− 4λ (a+b) 1

2

2

1 + 4λ (a−b)2

=

π λ

e− λ

q

=

ab

(4.68)

which gives 

e−i k,z e−i l,¯z e− ¯z,Az dn zdn z¯ =

n

π λi

e

−

pi qi λi

=

πn det A

e− p,D

−1

πn det A

e− k,Cl

i=1

¯ k, D−1 U l =
k, U −1 D−1 U l =
k, A−1 l. since
p, D−1 q =
U

Gaussian Integration and Grassmann Integrals

47

(ii) We have
zi1 · · · zim z¯j1 · · · z¯jm C = = =

· · · ∂k∂i

∂ ∂lj1

· · · ∂lj∂

1 ∂ (−i)m+m ∂ki1

· · · ∂k∂i

∂ ∂lj1

· · · ∂lj∂

· · · ∂k∂i

∂ ∂lj1

· · · ∂lj∂

(−i)

m+m

1

(−i)m+m

∂ ∂ki2

m

m

m



=



∂ ∂ki1

1

m


∂ 1 (−i)m+m −2 ∂ki2

m

m

m

∂ ∂ ···  ∂lj · · · ∂lj

|k=l=0 |k=l=0

|k=l=0

m

s

s=1

e−i k,z−i l,¯z  dPC (z, z¯)

e− k,Cl n 
− ci1 , l e− k,Cl

|k=l=0

=1

ci1 ,js e− k,Cl



=

m


ci1 ,js
zi2 · · · zim z¯j1 · · · z¯ ¯jm C js · · · z

(4.69)

s=1

(4.67) then follows by iteration of (4.66) and the observation that  1 ∂ ∂ · · ·
zi1 · · · zir C = (−i) e−i k,z dP (z, z¯) r ∂k ∂ki i r |k=0

1

=

4.4

1 ∂ (−i)r ∂ki1

···

∂ ∂kir | k=0

1 = 0.




Theory of Grassmann Integration

In this section we prove some basic facts about Grassmann integration. The first two theorems are the analogs of Theorems 4.3.1 and 4.3.2 for anticommuting variables. The third theorem introduces the ‘R-operator’ which we will use in chapter 9 to prove bounds on the sum of convergent diagrams. The main advantage of this operation is that it produces the diagrams in such a way that sign cancellations can be implemented quite easily. While the first two theorems are standard material [9], the R-operation has been introduced only recently in [22]. For N < ∞ let a1 , · · · aN be anticommuting variables. That is, we have the relations ai aj = −aj ai

∀i, j = 1, · · · , N

The Grassmann algebra generated by a1 , · · · , aN is given by 
GN = GN (a1 , · · · , aN ) = αI Π ai | αI ∈ C I⊂{1,··· ,N }

i∈I

(4.70)

(4.71)

48  Definition 4.4.1 The Grassmann integral · ΠN i=1 dai : GN → C is the linear functional on GN defined by 
N αI Π ai Π dai := α{1,··· ,N } (4.72) i∈I

I⊂{1,··· ,N }

i=1

The notation ΠN i=1 dai is used to fix a sign. For example, on G3   3a1 a2 − 5a1 a2 a3 da1 da2 da3 = −5 but



 3a1 a2 − 5a1 a2 a3 da3 da2 da1 = +5

 a3 a2 a1 da3 da2 da1 = +5

We start with the following Lemma 4.4.2 Leta1 , · · · , aN and b1 , · · · , bN be Grassmann variables and N suppose that bi = j=1 Tij aj where T = (Tij ) ∈ CN ×N . Then db1 · · · dbN = det T da1 · · · daN . That is,  F

N




Tij aj





N



det T Π dai =

i∈{1,··· ,N }

i=1

j=1

 N  F {bi }i∈{1,··· ,N } Π dbi i=1

(4.73) for some polynomial function F . Proof: We have b1 b2 · · · bN =

N


T1,j1 aj1

j1 =1

=




N
j2 =1

T2,j2 aj2 · · ·

N


TN,jN ajN

jN =1

T1,π1 · · · TN,πN aπ1 aπ2 · · · aπN

π∈Sn

=




signπ T1,π1 · · · TN,πN a1 a2 · · · aN

π∈Sn

= det T a1 a2 · · · aN

(4.74)

which proves the lemma.
Before we prove the analogs of Theorems 4.3.1 and 4.3.2 for anticommuting variables, we briefly recall some basic properties of pfaffians since Grassmann Gaussian integrals are in general given by pfaffians.

Gaussian Integration and Grassmann Integrals

49

Lemma 4.4.3 For some complex skew symmetric matrix A = (aij ) ∈ Cn×n , AT = −A, with n = 2m even, let
PfA := signσ aσ1,σ2 · · · aσ(n−1),σn (4.75) pairings σ

=




1 2m m!

signπ aπ1,π2 · · · aπ(n−1),πn

π∈Sn

be the pfaffian of the matrix A. Here the set of all pairings is given by those permutations σ ∈ Sn for which σ(2i − 1) < σ(2i) and σ(2i − 1) < σ(2i + 1) for all i. Then   a) For any B ∈ Cn×n we have Pf B T AB = det B PfA b) (PfA)2 = det A c) If A is invertible, then Pf(A−1 ) = (−1)m /PfA d) Let A ∈ Rn×n , AT = −A. Then there is an S ∈ SO(n) such that 



0 λ1 0 λm , · · · , (4.76) SAS T = diag −λ 0 0 −λ 1 m

Proof: We first prove part (d). Since B := iA is self adjoint, B ∗ = −iAT = iA = B, B and therefore A can be diagonalized and A has pure imaginary eigenvalues. If iλ, λ ∈ R, is an eigenvalue of A with eigenvector z, then also −iλ is an eigenvalue with eigenvalue z¯. If z = x + iy, then Az = Ax + iAy = iλ(x + iy) = −λy + iλx or   0 λy (4.77) A xy = −λ 0 x Since
x, x = λ1
x, Ay = − λ1
Ax, y =
y, y we have, since z and z¯ are orthogonal (observe that
z, z
 is linear in both arguments by definition), 0 =
z, z =
x, x −
y, y + i
x, y + i
y, x = 2i
x, y which results in
x, y = 0. Thus the matrix with rows y1 , x1 , · · · , ym , xm is orthogonal.  (a) We have (B T AB)i1 i2 = j1 ,j2 bj1 i1 aj1 j2 bj2 i2 which gives Pf(B T AB) =

1 2m m!

=

1 2m m!




signπ (B T AB)π1,π2 · · · (B T AB)π(n−1),πn

π∈Sn







signπ bj1 ,π1 · · · bjn ,πn aj1 ,j2 · · · ajn−1 ,jn

j1 ,··· ,jn π∈Sn

=

1 2m m!




εj1 ···jn det B aj1 ,j2 · · · ajn−1 ,jn

j1 ,··· ,jn

= det B PfA

(4.78)

50 (b) Since both sides of the identity are polynomials in aij , it suffices to prove it for real A. Let S ∈ SO(n) be the matrix of part (d) which makes A blockdiagonal. Then 

m m 0 λi = Π λi PfA = det S PfA = Pf(SAS T ) = Π Pf −λ (4.79) i 0 i=1

Thus m

m

i=1

i=1

(PfA)2 = Π λ2i = Π det

0 λi −λi 0



i=1

= det(SAS T ) = det A

(4.80)

(c) Let D be the blockdiagonal matrix on the right hand side of (4.76). Then A = S −1 DS and we have Pf(A−1 ) = Pf(S −1 D−1 S) = Pf(D−1 ) 



0 −1/λ1 0 , · · · , = Pf diag 1/λ 0 1/λm 1 =

(−1)m λ1 ···λm

=

−1/λm 0



(−1)m PfA

(4.81)

which proves the lemma.
Theorem 4.4.4 Let N be even, A ∈ CN ×N be skew symmetric  and invertible N and let C = A−1 . Let ai , bi be Grassmann variables,
a, b = i=1 ai bi and N N d a = Πi=1 dai . Then a)



e−i a,b e− 2 a,Aa dN a = 1

b) Let dµC (a) = PfC e− 2 a,C 1

−1

1 PfC

e− 2 b,Cb 1

(4.82)

a N

d a and let 
aj1 · · · ajm C := aj1 · · · ajm dµC (a)

(4.83)

Then one has, if C = (cij )1≤i,j≤N ,
aj1 · · · ajm C =

m
(−1)r cj1 ,jr
aj2 · · · a jr · · · ajm C

(4.84)

r=2

In particular




aj1 · · · ajm C =

pairings σ

 =

0

signσ cjσ1 ,jσ2 · · · cjσ(m−1) ,jσm for m even for m odd

Pf [(cjr js )1≤r,s≤m ] for m even 0 for m odd

(4.85)

where the set of all pairings is given by those permutations σ ∈ Sm for which σ(2i − 1) < σ(2i) and σ(2i − 1) < σ(2i + 1) for all i.

Gaussian Integration and Grassmann Integrals

51

Proof: a) Since both sides of (4.82) are rational functions of the matrix entries aij (in fact, both sides are polynomials in aij ), it suffices to prove (4.82) for real A. For every real skew symmetric matrix A there is an orthogonal matrix S ∈ SO(N, R) such that 



λN/2 0 λ1 0 , · · · , ≡D (4.86) SAS T = diag −λ 0 −λ 0 1 N/2 Thus, changing variables to c = Sa, d = Sb, dN c = dN a, we get   T 1

a,b − 12 a,Aa N e e d a = e Sa,Sb e− 2 Sa,SAS Sa dN a  1 = e c,d e− 2 c,Dc dN c

(4.87)

Since


N/2

− 21
c, Dc

=

− 21

(c2i−1 , c2i )

0 λi −λi 0



c2i−1 c2i



=−

N/2 i=1

λi c2i−1 c2i (4.88)

i=1

and because of  ed1 c1 +d2 c2 e−λc1 c2 dc1 dc2   1 + d1 c1 + d2 c2 + 12 (d1 c1 + d2 c2 )2 (1 − λc1 c2 ) dc1 dc2 =  = −λ + 12 (d1 c1 d2 c2 + d2 c2 d1 c1 ) dc1 dc2 = −λ − d1 d2 = −λ(1 + λ1 d1 d2 ) = −λ e 1

= −λ e 2 = −λ e we get  e

0 (d1 ,d2 )( −1/λ

0 − 12 (d1 ,d2 )( −λ

1/λ 0 λ 0

e

d a=

)

−1

)



a,b − 12 a,Aa N

“

d1 d2

“

d1 d2

”

(4.89)

e c,de− 2 c,Dc dN c 1

N/2

=

d1 d2 λ

”



−λi e

− 12 (d2i−1 ,d2i )

“

0 λi −λi 0

”−1 “

d2i−1 d2i

”

i=1

= (−1)N/2 PfD e− 2 d,D 1

−1

d

= (−1)N/2 Pf[S T DS] e− 2 b,S 1

= (−1)N/2 PfA e− 2 b,A 1

=

1 Pf(A−1 )

e

− 12 b,A−1 b

−1

−1

D−1 Sb

b

(4.90)

52 which proves part (a). To obtain part (b), observe that ∂ ∂bjm ∂ ∂ ∂bjm−1 ∂bjm ∂ ∂bj1

· · · ∂b∂j

m

e b,a = ajm e b,a = e b,a ajm , e b,a =

∂ ∂bjm−1

e b,a ajm = e b,a ajm−1 ajm

e b,a = e b,a aj1 · · · ajm = aj1 · · · ajm e b,a

which gives
aj1 · · · ajm C



=

∂ ∂bj1

· · · ∂b∂j

=

∂ ∂bj1

· · · ∂b∂j

m |b=0

1

2

(−1)m−1 ∂b∂j 2

= (−1)m−1 ∂b∂j

2

=

m


e− 2 b,Cb 

 1 ∂ 1 N − e− 2 b,Cb |b=0 · · · ∂b∂j c b b r,s=1 rs r s ∂bj1 2 m

 1   − 12 N · · · ∂b∂j cj1 s bs + 12 N crj1 br e− 2 b,Cb |b=0 s=1 r=1 m

  1 N ∂ − s=1 cj1 s bs e− 2 b,Cb |b=0 · · · ∂bj

m |b=0

= (−1)m−1 ∂b∂j =

e b,a dµC (a)

m

  1  ∂ (−1)m−1 (−1)m− ∂b∂j · · · ∂b · · · ∂b∂j −cj1 j e− 2 b,Cb |b=0 j 2

=

=2 m


=

m

1  ∂ (−1) cj1 j ∂b∂j · · · ∂b · · · ∂b∂j e− 2 b,Cb |b=0 j 2

=2 m






m

(−1) cj1 j
aj2 · · · a j · · · ajm C

(4.91)

=2

which proves (4.84). Then (4.85) is obtained by iterating (4.84).
In the next theorem, we specialize the  theorem to the case that the

above 0 B in which case the exponential matrix A has the block structure A = −B T 0 becomes e− a,B¯a if we redefine a := (a1 , · · · , aN/2 ) and a ¯ = (aN/2+1 , · · · , aN ).

Theorem 4.4.5 Let B ∈ CN ×N be invertible, C := B −1 , and let a1 , · · · , aN , a ¯1 , · · · , a ¯N and b1 , · · · , bN , ¯b1 , · · · , ¯bN be anticommuting variables. Let
¯b, a =  N ¯ i=1 bi ai (linear in both arguments). Then a) 

¯

¯

e b,a+ b,¯a e− a,B¯a dN adN a ¯ = det B e− b,C b

(4.92)

Gaussian Integration and Grassmann Integrals e− a,B¯a dN a dN a ¯ and let  ¯im aj1 · · · ajm C := a ¯i1 · · · a ¯im aj1 · · · ajm dµC (a, a ¯)
¯ ai1 · · · a

¯) := b) Let dµC (a, a

53

1 det B

(4.93)

Then
¯ ai1 · · · a ¯im aj1 · · · ajm C = 

m


(−1)m−1+r ci1 jr
¯ ai2 · · · a ¯im aj1 · · · a jr · · · ajm C

(4.94)

r=1

In particular,
¯ ai1 · · · a ¯im aj1 · · · ajm C = 0 if m = m
and
¯ ai1 aj1 · · · a ¯im ajm C = det [(cir js )1≤r,s≤m ]

(4.95)



0 B . Then
a, B¯ a = 12
c, Ac, Proof: Let c = (a, a ¯), d = (¯b, b) and A = −B T 0
¯b, a +
b, a ¯ =
d, c and we get according to Theorem 4.4.4  e

¯ b,a+ b,¯ a − a,B¯ a N

e

 N

d ad a ¯=

e d,c e− 2 c,Ac d2N c 1

= PfA e− 2 d,A 1

−1

d

¯

= det B e− b,C b

since Pf

0 B −B T 0

(4.96)

 = det B and

 ¯  

T −1 b = 2
b, B −1¯b
d, A−1 d = (¯b, b) , B0−1 −B0 b

  T −1 T = C0 −C . which proves (a). To obtain (b), let C˜ = A−1 = B0−1 −B0 0 Then, for 1 ≤ i, j ≤ N (observe that ci is a Grassmann variable and cij is the matrix element of C) C˜i,N +j = −cj,i ,

C˜i,j = 0,

C˜N +i,j = ci,j ,

C˜N +i,N +j = 0

(4.97)

and we get
¯ ai1 · · · a ¯im aj1 · · · ajm C =
cN +i1 · · · cN +im cj1 · · · cjm C˜ 

=

m


(−1)m−1+r C˜N +i1 ,jr
cN +i2 · · · cN +im cj1 · · · c ˜ jr · · · cjm C

r=1 

=

m
r=1

(−1)m−1+r ci1 ,jr
¯ ai2 · · · a ¯im aj1 · · · a jr · · · ajm C

(4.98)

54 Finally (4.95) follows from ¯im ajm C =
¯ ai1 aj1 · · · a

m


ci1 jr
aj1 a ¯i2 aj2 · · · a ¯im ajm C

r=1

=

m


(−1)r−1 ci1 jr
¯ ai2 aj1 · · · a ¯ir ajr−1 a ¯ir+1 ajr+1 · · · a ¯im ajm C

r=1

which is the expansion of the determinant with respect to the first row.
We now introduce the R operation of [22]. To do so, we briefly recall the idea of Wick ordering. A Gaussian integral of some monomial ai1 · · · ai2r is given by the sum of all pairings or contractions: 
signσ ciσ1 ,iσ2 · · · ciσ(2r−1) ,iσ(2r) ai1 · · · ai2r dµC (a) = pairings σ

= Pf [(cik ,i )1≤k, ≤2r ]

(4.99)

 Graphically one may represent every aik by a half line or ‘leg’ and then · dµC means to contract the 2r half lines to r lines in all possible (2r − 1)!! ways. To each line which is obtained by pairing the halflines aik and ai we assign the matrix element cik ,i . Now given two monomials Πi∈I ai and Πj∈J aj , I, J ⊂ {1, · · · , N }, it is useful to have an operation which produces the sum of contractions between ai and ai ‘fields’ and between ai and aj fields but no contractions between aj and aj  fields, i, i
∈ I, j, j
∈ J. This is exactly what Wick ordering : Πj∈J aj : is doing,    (cii )i,i ∈I (cij  )i∈I,j  ∈J (4.100) Π ai : Π aj : dµC (a) = Pf (cji )j∈J,,i ∈I 0 i∈I j∈J Usually (4.100) is accomplished by defining : Πj∈J aj : as a suitable linear combination of monomials of lower degree and matrix elements cjj  , for example : aj aj  : = aj aj  − cjj  . Here we use the approach of [22] which simply doubles the number of Grassmann variables. Definition 4.4.6 Let C ∈ CN ×N be invertible and skew symmetric, C := 

C C C 0 , let a1 , · · · , aN , a1 , · · · , aN be anticommuting variables, ai aj = −aj ai , ai a j = −a j ai , a i a j = −a j a i , and let dµC (a, a ) be the Grassmann Gaussian measure on G2N (a, a ). Then, for some polynomials f and g, we define Wick ordering according to   f (a) : g(a) : dµC (a) := f (a) g(a ) dµC (a, a ) (4.101) GN

G2N

Gaussian Integration and Grassmann Integrals

55

As a consequence, we have the following Lemma 4.4.7 Let f and g be some polynomials. Then    f (a) g(a) dµC (a) = f (a) : g(a + b) :a dµC (b) dµC (a) (4.102) GN

GN

GN

where : · :a means Wick ordering with respect to the a-variables, see (4.103) below. Proof: By definition,   f (a) : g(a + b) :a dµC (b) dµC (a) = GN GN   f (a) g(a + b) dµC (b) dµC (a, a ) G2N

We claim that for every polynomial h = h(a, b)    h(a, a + b) dµC (b) dµC (a, a ) = G2N

(4.103)

GN

GN

GN

h(a, a) dµC (a) (4.104)

To prove this, it suffices to prove (4.104) for the generating function h(a, b) = e a,c+ b,d. Then the left hand side of (4.104) becomes   e a,c+ a +b,d dµC (b) dµC (a, a ) G2N GN  

b,d = e dµC (b) e a,c+ a ,d dµC (a, a ) GN

= e

G2N

− 12 d,Cd

e

− 12


(c,d),( CC C0 )( dc )

= e− 2
(c,d),( C C )( d ) = e− 2 c+d,C(c+d) 1

 = GN

C C

c

e a,c+d dµC (a)

1

(4.105)

which proves (4.104). Then (4.103) follows by the choice h(a, b) = f (a)g(b) .
In Theorem (4.2.2) we found that the Grassmann integral representations   for the correlation functions are of the form f (a) eW (a) dµC (a)/ eW (a) dµC (a). In chapter 9 on renormalization group methods we will generate the sum of all diagrams up to n’th order by an n-fold application of the following

56 Theorem 4.4.8 Let W = W (a) be some polynomial of Grassmann variables, dµC be a Gaussian measure on the Grassmann algebra GN (a) and let Z = let W e (a) dµC (a). For some polynomial f let (Rf )(a) ∈ GN (a) be given by  (Rf )(a) = f (b) : eW (b+a)−W (a) − 1 :b dµC (b) (4.106) = Then we have 1 Z

 N
1 n f (b) : [W (b + a) − W (a)] :b dµC (b) n! n=1

 f (a) eW (a) dµC (a) =  f (a) dµC (a) +

 1 Z

(Rf )(a) eW (a) dµC (a)

(4.107)

Proof: We have with Lemma 4.4.7   W (a) 1 1 dµC (a) = Z f (a) e f (a) : eW (a+b) :a dµC (b) dµC (a) Z  = Z1 f (a) : eW (a+b)−W (b) :a eW (b) dµC (b) dµC (a)  1 f (a) : eW (a+b)−W (b) − 1 :a dµC (a) eW (b) dµC (b) = Z  + Z1 f (a) eW (b) dµC (b) dµC (a)   = Z1 (Rf )(b) eW (b) dµC (b) + f (a) dµC (a) which proves the theorem.
Finally we write down Gram’s inequality which will be the key estimate in the proof of Theorem 9.2.2 where we show that the sum of all convergent diagrams has in fact a small positive radius of convergence instead of being only an asymptotic series. We use the version as it is given in the appendix of [22]. ¯1 ,··· ,a ¯N) be anticomTheorem 4.4.9 Let (b1 , · · · , b2N ) := (a1 , · · · , aN , a muting variables, let C ∈ CN ×N be invertible, let S = dµS (b) = PfS e− 2 b,S 1

−1

b 2N

d

b = det C e− a,C

0 C −C T 0

−1

a ¯ N

and let

d a dN a ¯

(4.108)

Suppose in addition that there is a complex Hilbert space H and elements vi , wi ∈ H, i = 1, · · · , N such that   (4.109) C =
vi , wj H 1≤i,j≤N

Gaussian Integration and Grassmann Integrals

57

and vi H , wj H ≤ Λ

(4.110)

for some positive constant Λ ∈ R. Then, for some subsets I, J ⊂ {1, · · · , 2N }     (4.111)  Π bi dµS (b) ≤ Λ|I| i∈I   √  ( 2 Λ)|I|+|J| if |J| ≤ |I|   (4.112)  Π bi : Π bj : dµS (b) ≤ 0 if |J| > |I| i∈I j∈J Proof: To prove the first inequality, suppose that I = {i1 , · · · , ir }. Then  (4.113) bi1 · · · bir dµS (b) = Pf [(Sik i )1≤k, ≤r ] where

Sik i

⎧ 0 ⎪ ⎪ ⎨ Cik ,i −N = ⎪ −Ci ,ik −N ⎪ ⎩ 0

if if if if

1 ≤ ik , i ≤ N 1 ≤ ik ≤ N and N < i ≤ 2N N < ik ≤ 2N and 1 ≤ i ≤ N N < ik ≤ 2N and N < i ≤ 2N

(4.114)

More concisely, 

 bi1 · · · bir dµS (b) = Pf

0 U −U T 0

 (4.115)

where U = (Uk ) is the ρ := max{k|ik ≤ N } by r − ρ matrix with elements Uk = Cik ,i+ρ −N =
vik , wi+ρ −N H

(4.116)

Since  Pf

0 U −U T 0



 =

0 if ρ = r − ρ ρ(ρ−1) 2 det U if ρ = r − ρ (−1)

(4.117)

we get, for r = 2ρ, using Gram’s inequality for determinants          bi1 · · · bir dµS (b) = det
vik , wi+ρ −N H  ρ

≤ Π vik H wi+ρ −N H ≤ Λ2ρ

(4.118)

k=1

which proves (4.111). To prove (4.112), recall that according to definition 4.4.6  

(4.119) Π bi : Π bj : dµS (b) = Π bi Π bj dµS (b, b ) G2N i∈I

j∈J

G4N i∈I

j∈J

58 where  S =

S S S 0



⎛

0 ⎜ −C T =⎜ ⎝ 0 −C T

⎞ C 0⎟ ⎟ 0⎠ 0

C 0 0 −C T C 0 0 0

(4.120)

This matrix is conjugated by the permutation matrix ⎛ ⎞ Id 0 0 0 ⎜ 0 0 Id 0 ⎟ ⎜ ⎟ ⎝ 0 Id 0 0 ⎠ 0 0 0 Id to 



0 C T −C 0

⎛

0 0 ⎜ 0 0 =⎜ ⎝ −C T −C T −C T 0

C C 0 0

⎞ C 0⎟ ⎟ 0⎠ 0

(4.121)

Also, for 1 ≤ i ≤ 2N define the vectors vi , wi ∈ H ⊕ H by  (vi , 0) for 1 ≤ i ≤ N

vi = (vi , vi ) for N < i ≤ 2N  (0, wi ) for 1 ≤ i ≤ N wi = (wi , 0) for N < i ≤ 2N Then   C =
vi , wj H⊕H 1≤i,j≤2N

(4.122)

and vi H⊕H , wj H⊕H ≤

√

2Λ

(4.123)

for all 1 ≤ i, j ≤ 2N . Thus (4.112) has been reduced to (4.111) for the matrix   0 C T −C 0 the Hilbert space H ⊕ H and the vectors vi , wj .


Chapter 5 Bosonic Functional Integral Representation

In this chapter we apply a Hubbard-Stratonovich transformation to the Grassmann integral representation to obtain a bosonic functional integral representation. The main reason for this is not only that we would like to have integrals with usual commuting variables amenable to standard calculus techniques, but is merely the following one. By writing down the perturbation series for the partition function, evaluating the n’th order coefficients with Wick’s theorem and rewriting the resulting determinant as a Grassmann integral, we arrived at the following representation for the two-point function G(k) =
a+ k ak  R

G(k) =

P

P

¯ ¯ ¯k ψk e−λ ψ¯ψψψ ¯ ψ e− (ip0 −ep )ψψ dψdψ P P ¯¯ R ¯ − (ip0 −ep )ψψ −λ ψ ψψψ ¯ e e dψdψ

(5.1)

If we would have started with a bosonic model, the representation for G(k) would look exactly the same as (5.1) with the exception that the integral in (5.1) would be a usual one, with commuting variables. Regardless whether the variables in (5.1) are commuting or anticommuting, in both cases we can apply a Hubbard-Stratonovich transformation and in both cases one obtains a result which looks more or less as follows: R

G(k) =

R

√

−1 λ φp−p
k,k −Veff (φ) e dφ

[(ip0 −ep )δp,p
+

]

e−Veff (φ) dφ

(5.2)

with an effective potential Veff (φ) = |φ|2 +  log

√ det[(ip0 −ep )δp,p
+ λ φp−p
] det[(ip0 −ep )δp,p
]

(5.3)

Thus, we have to compute an inverse matrix element and to average it with respect to a normalized measure which is determined by the effective potential (5.3). The statistics, bosonic or fermionic, only shows up in the effective potential through a sign, namely  = −1 for fermions and  = +1 for (complex) bosons ( = 1/2 for scalar bosons). Besides this unifying feature the representation (5.2) is a natural starting point for a saddle point analysis. Observe that the effective potential basically behaves as |φ|2 +  log[Id + φ]. Thus, it should be bounded from below and have one or more global minima.

59

60 In fact, in section 5.2 we rigorously prove this for the many-electron system with attractive delta interaction. Therefore, as a first approximation one may try the following R

G(k) =

Fk (φ) e−Veff (φ) dφ R −V (φ) e eff dφ

R

=

Fk (φ) e−[Veff (φ)−Veff (φmin )] dφ R −[V (φ)−V (φ min )] dφ eff eff e

or, if the global minimum φmin is not unique, G(k) ≈ the integral is an average over all global minima.




≈ Fk (φmin ) (5.4)

Fk (φmin )dφmin where

Approximation (5.4) would be exact if there would be an extra parameter in front of the effective potential in the exponents in (5.4) which would tend to infinity. In fact, the effective potential is proportional to the volume, but this
 is due to the number of integration variables, k |φk |2 ∼ βLd dk|φk |2 , and if this is the case, usually fluctuations around the minimum are important. However, as we will see in chapter 6, the BCS approximation has the effect that we can separate off a volume factor in front of the effective potential and the number of integration variables does not depend on the volume. Thus, in the BCS approximation the saddle point approximation becomes exact. In section 6.2 we show that this already holds for the quartic BCS model, it is not necessary to make the quadratic mean field approximation. In the next section, we derive the representation (5.2) for the many-electron system and in section 5.2 we determine the global minimum of the full effective potential with attractive delta interaction and show that it corresponds to the BCS configuration. We also compute the second order Taylor expansion around that minimum. The representation (5.2) will also be the starting point for the analysis in chapter 10.

5.1

The Hubbard-Stratonovich Transformation

In this section we derive bosonic functional integral representations for the quantities we are interested in, for the partition function, the momentum distribution and for our four-point function Λ(q). We specialize to the case of a delta interaction in coordinate space, V (x−y) = λδ(x−y) or V (k−p) = λ in momentum space. A more general interaction is discussed in chapter 6 where we also consider BCS theory with higher -wave terms. There we expand V (k − p) into spherical harmonics on the Fermi surface and include the pand d-wave terms. We shortly recall the Grassmann integral representations from which the bosonic functional integral representations will be derived below. According

Bosonic Functional Integral Representation

61

to Theorem 4.2.2 we had Z(λ) = T r e

−βHλ

/T r e

−βH0

+ 1 Ld
akσ akσ  λ L3d

 k,p

 = =

e

−

λ (βLd )3

P



1 β

kpq

¯k↑ ψ ¯q−k,↓ ψp↑ ψq−p,↓ ψ

dµC

(5.5)

1 ¯ βLd
ψkσ ψkσ 

(5.6)

k0 ∈ π β (2Z+1) +
a+ k↑ aq−k,↓ aq−p,↓ ap↑  =



1 β

Λ(q0 , q)

(5.7)

q0 ∈ 2π β Z

Λ(q) =

λ (βLd )3




ψ¯k↑ ψ¯q−k,↓ ψp↑ ψq−p,↓ 

(5.8)

k,p

where
¯ =
G(ψ, ψ)

P

¯

¯

¯ e− (βLd )3 kpq ψk↑ ψq−k↓ ψp↑ ψq−p↓ dµC G(ψ, ψ)
− λd 3 Pkpq ψ¯k↑ ψ¯q−k↓ ψp↑ ψq−p↓ e (βL ) dµC λ

(5.9)

and ¯ =Π dµC (ψ, ψ)

kσ

βLd ik0 −ek

e

−

1 βLd

P

¯

kσ (ik0 −ek )ψkσ ψkσ

Π dψkσ dψ¯kσ

(5.10)

kσ

The main idea of the Hubbard-Stratonovich transformation is to make the quartic action in the Grassmann integral (5.9) quadratic by the use of the identity (5.11) below and then to integrate out the Grassmann fields.

Lemma 5.1.1 Let a, b ∈ C or commuting elements of some Grassmann algebra. Let φ = u + iv, φ¯ = u − iv and dφ dφ¯ := du dv. Then  2 ¯ ¯ φ eaφ+bφ e−|φ| dφd (5.11) eab = π R2

Proof: We have
2 ¯ eaφ+bφ e−|φ| R2

¯ dφdφ π

=

√1 π

=e


R

(a+b)2 4

e(a+b)u e−u du 2

e

−(a−b)2 4

√1 π




ei(a−b)v e−v dv 2

R

= eab


The lemma above will be applied in the following form:  P P P 2 ¯ ¯ dφ dφ − q aq bq e = ei q (aq φq +bq φq ) e− q |φq | Π qπ q R2N

q

(5.12)

62 where N is the number of momenta q = (q0 , q) and  aq =

λ (βLd )3

 12 

 ψk↑ ψq−k↓ ,

bq =

λ (βLd )3

 12 

k

ψ¯k↑ ψ¯q−k↓ (5.13)

k

The result is √ Theorem 5.1.2 Let ak = ik0 − ek , κ = βLd , g = λ and let dP (φ) be the normalized measure   ig ¯ P √ a δ φ 2 k k,p p−k κ dP ({φq }) = Z1 det ig (5.14) e− q |φq | Π dφq dφ¯q √ φk−p a−k δk,p q κ Then there are the following integral representations   ak δk,p √igκ φ¯p−k  det √ig φ a−k δk,p P 2 dφ dφ¯ κ k−p

e− q |φq | Π q q (5.15) Z(λ) = q π a δ 0 det k k,p 0 a−k δk,p −1   ak δk,p √igκ φ¯p−k 1 ¯ dP (φ) (5.16) ig κ
ψtσ ψtσ  = √ φ a−k δk,p κ k−p tσ,tσ

 |φq |2 dP (φ) − 1

Λ(q) =

(5.17)

Proof: We have 1 κ


ψ¯tσ ψtσ  =

∂ ∂stσ |s=0,λq =λ

Λq = −λ ∂λ∂ q if we define



Z({skσ }, {λq }) =

e− κ3 1

P

Π

kσ

kpq

log Z({skσ }, {λq })

|s=0,λq =λ

log Z({skσ }, {λq })

¯k↑ ψ ¯q−k,↓ ψp↑ ψq−p,↓ λq ψ

κ ik0 −ek

e− κ 1

P

(5.18) (5.19)

×

(5.20)

¯

kσ (ak −skσ )ψkσ ψkσ

Π dψkσ dψ¯kσ

kσ

Using (5.12) and (5.13) we get P

¯

¯

e− κ3 kpq λq ψk↑ ψq−k,↓ ψp↑ ψq−p,↓ =  P  λq 1 P 1 2 ¯ λq 2 k ψk↑ ψq−k↓ +φq κ3 e i q φq κ 3 1

P k

¯k↑ ψ ¯q−k↓ ψ



(5.21) e−

P q

|φq |2

¯q dφq dφ π q

Π

Bosonic Functional Integral Representation

63

which gives (5.22) Z({skσ }, {λq })     gk−p gp−k 1 ψk↑ i √ φk−p ψ−p↓ + ψ¯−p↑ i √ φ¯p−k ψ¯k↓ + = exp − κ κ κ k,p   ¯ ¯ + ψk↑ (ak − sk↑ )δk,p ψp↑ + ψ−k↓ (a−k − s−k↓ )δk,p ψ−p↓ × Π

kσ

κ ak

− Π dψkσ dψ¯kσ e

P q

|φq |2

kσ

¯q dφq dφ π q

Π

The exponent in (5.22) can be written as    gk−p  √ φk−p −(a − s )δ i k k↑ k,p ψ¯p↑ κ 1 ¯ −κ (ψk↑ , ψ−k↓ ) gp−k ¯ −i √ (a−k − s−k↓ )δk,p ψ−p↓ φ κ p−k k,p

Because of dψk↑ dψ¯k↑ dψ−k↓ dψ¯−k↓ = −dψk↑ dψ¯k↑ dψ¯−k↓ dψ−k↓ the Grassmann integral gives    gk−p  κ i √ φ 1 (ak − sk↑ )δk,p k−p κ det = (5.23) gp−k ¯ i √κ φp−k (a−k − s−k↓ )δk,p κ ak kσ  

gk−p  i √ φ (ak − sk↑ )δk,p ak δk,p 0 κ k−p det det gp−k ¯ i √κ φp−k (a−k − s−k↓ )δk,p 0 a−k δk,p which results in Z({skσ }, {λq }) = (5.24)   gk−p i √κ φk−p (ak − sk↑ )δk,p gp−k ¯  det √ i κ φp−k (a−k − s−k↓ )δk,p P 2 ¯ dφ dφ

e− q |φq | Π qπ q q ak δk,p 0 det 0 a−k δk,p Now, using ⎡ ⎡ ⎤ n | | |  d det ⎣ x1 (t) · · · det ⎣ x1 (t) · · · xn (t) ⎦ = dt i=1 | | |

⎤ | | d xi (t) · · · xn (t) ⎦ (5.25) dt

| |

and observing that, if we fix a column labelled by p and letting k label the entries of the column vector,     (ak − sk↑ )δk,p δk,t δk,p d = − (5.26) dst↑ 0 i √gκ φ¯p−k

64 we get by Cramer’s rule 

(ak − sk↑ )δk,p i √gκ φk−p det g i √κ φ¯p−k (a−k − s−k↓ )δk,p   (ak − sk↑ )δk,p i √gκ φk−p det i √gκ φ¯p−k (a−k − s−k↓ )δk,p

 

d dstσ |s=0

=

ak δk,p i √gκ φk−p i √gκ φ¯p−k a−k δk,p

−1 tσ,tσ

which results in (5.16). To obtain (5.17), we substitute variables 

ak δk,p i √1κ φk−p  det i √1 φ¯p−k a−k δk,p κ

Z({λq }) = ak δk,p 0 det 0 a−k δk,p

 e

−

P q

|φq |2 λq

Π q

¯q dφq dφ πλq

(5.27)

to obtain −Λq = λ ∂λ∂ q

=

=

=

=

|λq =λ

log Z({λq }) 

 ak δk,p i √1κ φk−p  det i √1 φ¯p−k a−k δk,p P |φq |2 ¯ ∂ κ q d φq

e− q λq Π dφπλ λ log q ∂λq |λq =λ q ak δk,p 0 det 0 a−k δk,p    P |φ |2 √1 φk−p
 |φq |2 a δ i ¯q k k,p − q λqq dφq dφ κ 1 det − e Πq πλq 2 1 ¯ λq λq i √κ φp−k a−k δk,p   λ
ak δk,p i √1κ φk−p − Pq |φλq |2 ¯q dφq dφ q det √1 ¯ e Πq πλq i κ φp−k a−k δk,p  


ak δk,p i √gκ φk−p − P |φq |2 ¯q dφq dφ 2 q e |φq | − 1 det √g ¯ Πq π i κ φp−k a−k δk,p  
ak δk,p i √gκ φk−p − P |φq |2 ¯q dφq dφ q det √g ¯ e Πq π i κ φp−k a−k δk,p  (|φq |2 − 1) dP (φ)

which proves the theorem.


Bosonic Functional Integral Representation

5.2

65

The Effective Potential

In this section we find the global minimum of the full effective potential for the many-electron system with delta interaction and compute the second order Taylor expansion around it. The results of this section are basically the content of [45]. ¯ two-point function for In the preceding section we showed that the
ψψ the many-electron system has the following representation
G(tσ, φ) e−Veff (φ) Πq dφq dφ¯q 1 ¯
(5.28) κ
ψtσ ψtσ  = e−Veff (φ) Πq dφq dφ¯q where  G(tσ, φ) =

ak δk,p √igκ φ¯p−k ig √ φk−p a−k δk,p κ

−1 (5.29) tσ,tσ

and the effective potential is given by Veff ({φq }) =



 |φq | − log det 2

q

δk,p ig φk−p √ κ a−k

¯p−k ig φ √ κ ak

δk,p

 (5.30)

To obtain (5.30) from (5.14), we divided

the determinant and the normalizaak δk,p 0 tion factor Z by det . In general, Veff is complex. Its real 0 a−k δk,p part is given by    ¯ φ    δk,p √igκ p−k   2 ak Re Veff ({φq }) = |φq | − log det ig φk−p (5.31)    √ δ k,p q κ a−k min be Apparently the evaluation of (5.28) is a complicated problem. Let Veff the global minimum of Veff ({φq }). Since




G(tσ, φ) e−Veff (φ) Πq dφq dφ¯q
= e−Veff (φ) Πq dφq dφ¯q




min G(tσ, φ) e−(Veff (φ)−Veff ) Πq dφq dφ¯q
−(V (φ)−V min ) eff e eff Πq dφq dφ¯q

the relevant contributions to the functional integral should come from configurations {φq } which minimize the real part of the effective potential. Thus as a first approximation one may start with the saddle point approximation which simply evaluates (5.28) by substituting the integrand by its value at the global minimum, or, if this is not unique, averages the integrand over all configurations which minimize the effective potential. In the following theorem

66 we rigorously prove that the global minimum of the real part of the effective potential of the many-electron system with attractive delta interaction is given by  φq = βLd r0 eiθ δq,0 (5.32) where θ is an arbitrary phase and |∆|2 := λr02 is a solution of the BCS gap equation  1 λ =1 (5.33) d βL k2 +e2 +|∆|2 k

0

k

In the next chapter we discuss the BCS model and show that the saddle point approximation with the minimum (5.32) corresponds to the BCS approximation.

Theorem 5.2.1 Let ReV be the real part of the effective potential for the many-electron system with attractive delta interaction given by (5.31). Let ek satisfy ek = e−k and: ∀k ek = ek+q ⇒ q = 0. Then all global minima of ReV are given by  φq = δq,0 βLd r0 eiθ , θ ∈ [0, 2π] arbitrary (5.34) where r0 is a solution of the BCS equation (5.33) or, equivalently, the global minimum of the function  VBCS (ρ) = Veff ({δq,0 βLd ρ eiθ }) (5.35)

√2     β 2 cosh( 2 ek +λρ2 ) d 2 = βL log 1 + k2λρ ρ − log = βLd ρ2 − +e2 cosh β e k

0

k

2

k

k

More specifically, there is the bound ReV ({φq }) ≥





VBCS ( φ ) − mink log

 Π 1−

q=0

+ log

| λd βL

1 βLd

 q

2 ¯ p φp φp+q |

 12 

(5.36)

(|ak |2 +λφ2 )(|ak−q |2 +λφ2 )



 Π

q=0

where φ 2 :=

P

1−

λ |φq |2 |ak −ak−q |2 βLd (|ak |2 +λφ2 )(|ak−q |2 +λφ2 )

 12 

|φq |2 and |ak |2 := k02 + e2k . In particular, ReV ({φq }) ≥ VBCS ( φ )

since the products in (5.36) are less or equal 1.

(5.37)

Bosonic Functional Integral Representation

67

Proof: Suppose first that (5.36) holds. For each q, the round brackets in (5.36) are between 0 and 1 which means that − log(Πq · · · ) is positive. Thus ReV ({φq }) ≥ VBCS ( φ ) ≥ VBCS (r0 ) (5.38)  which proves that φq = δq,0 βLd r0 eiθ are indeed global minima of ReV . On the other hand, if a configuration {φq } is a global minimum, then the logarithms in (5.36) must be zero for all k which in particular means that for all q = 0  φp φ¯p+q = 0 (5.39) p

and for all k and q = 0 |φq |2 |ak − ak−q |2 = |φq |2 [q02 + (ek − ek−q )2 ] = 0 which implies φq = 0 for all q = 0. It remains to prove (5.36). To this end, we write (recall that κ = βLd and Ck = a1k = ik0 1−ek ) ⎡ ⎤   | | ig ¯ ∗ √ Cφ Id κ det ig = det ⎣ bk bk
⎦ √ Cφ Id κ | | where (k, k  fixed, p labels the vector components) ⎛ ⎛ ⎞ ⎜ √ig φ¯p−k
⎜ δk,p ⎟

bk = ⎜

b
= ⎜ ⎟ ⎜ κ a¯k
k ⎝ ⎠ ⎝ ig φk−p √ δk
,p κ ak

bk , | bk |

ek =

⎟ ⎟ ⎟ ⎠

(5.42)

(5.43)

b
k | b
k |

⎤ ⎤ ⎡   | | | | 2

φ

det ⎣ bk bk
⎦ = Π 1 + λ det ⎣ ek ek
⎦ |ak |2 k | | | | ⎡

(5.41)

⎞

If | bk | denotes the euclidean norm of bk , then we have  |φk−p |2 φ2 λ

 2 | bk |2 = 1 + βL d p |ak |2 = 1 + λ |ak |2 = |bk | Therefore one obtains, if ek =

(5.40)

(5.44)

From this the inequality (5.38) already follows since the determinant on the right hand side of (5.44) is less or equal 1. To obtain (5.36), we choose a fixed but arbitrary momentum t and orthogonalize all vectors ek , ek
in the determinant with respect to et . That is, we write ⎡ ⎡ ⎤ ⎤ | | | | | | det ⎣ et ek ek
⎦ = det ⎣ et ek − ( ek , et ) et ek
− ( ek
, et ) et ⎦ (5.45) | | | | | |

68 Finally we apply Hadamard’s inequality, n

| det F | ≤ Π |f j | =

'

n

n 

Π

j=1

j=1

|fij |2

( 12

(5.46)

i=1

if F = (fij )1≤i,j≤n is a complex matrix, to the determinant on the right hand side of (5.45). Since | ek − ( ek , et ) et |2 = 1 − |( ek , et )|2  ⎡ ⎤   | |   det ⎣ ek − ( ek , et ) et e 
− ( e 
, et ) et ⎦ k k     | |

one obtains

 2 Π (1 − |( ek , et )| ) 2

1

≤ | et | Π (1 − |( ek , et )|2 ) 2 k∈Mν k=t

1

(5.47)

k∈Mν

or with (5.41) and (5.44)    Id  |φq |2 − log det ig √ φC  κ

   (5.48) ReV ({φq }) =   q  ⎡ ⎤     | | 2   

φ

2   = κ φ − log 1 + λ − log det ⎣ ek ek
⎦ 2 |ak |   | | k  ⎡ ⎤   | |     = VBCS ( φ ) − log det ⎣ ek ek
⎦   | | ( ' 1 1 ≥ VBCS ( φ ) − log Π (1 − |( ek , et )|2 ) 2 Π (1 − |( ek , et )|2 ) 2 

φ∗ C¯ Id

ig √ κ

k∈Mν k=t

Finally one has ( bk , bt ) =

k∈Mν



ig φt−p ig φk−p √ √ κ ak κ at

p

which gives |( ek , et )|2 =

| λκ

P

¯t−p | φk−p φ

2

p

(|ak |2 +λφ2 ) (|at |2 +λφ2 )

=

| λκ

¯t−k ig φ √ ¯k κ a

+

ig φt−k √ κ at

=

¯t−k+p | φp φ

2

p

(|ak |2 +λφ2 ) (|at |2 +λφ2 )

and ( bk , bt ) =

P

ig ¯ √ φ κ t−k



1 a ¯k

−

1 a ¯t

(5.49)

 (5.50)

which gives |( ek , et )|2 =

2 2 λ κ |φt−k | |at −ak |

(|ak

|2 +λφ2 )

(|at |2 +λφ2 )

(5.51)

Bosonic Functional Integral Representation

69

Substituting (5.49) and (5.51) in (5.48) gives, substituting k → q = t − k ReV ({φq }) ≥

VBCS ( φ ) − log

k=t−q

VBCS ( φ ) − log

=

' '

2 Π (1 − |( ek , et )| )

 2 Π (1 − |( ek , et )| )

1 2

k∈Mν k=t

1 2

(5.52) (

k∈Mν

2  2 Π (1 − |( et−q , et )| ) 2 Π (1 − |( et−q , et )| ) 2 1

q=0

1

(

q=0

Since t was arbitrary, we can take the maximum of the right hand side of (5.52) with respect to t which proves the theorem.


Thus in the saddle point approximation we obtain 1 ¯ κ
ψk↑ ψk↑ 

=

1 2π





2π

dθ 0

ak δk,p igr0 e−iθ δp−k,0 iθ igr0 e δk−p,0 a−k δk,p

a−k −ik0 − ek = = 2 |ak |2 + λr02 k0 + e2k + |∆|2

−1 k↑,k↑

(5.53)

which gives, at zero temperature, 1
a+ k↑ ak↑  Ld

 ik dk0 (−ik0 −ek )e 0 = lim 2 +e2 +|∆|2 2π k 0 k

0   = 12 1 − √ 2ek 2 ek +|∆|

(5.54)

This coincides with the result obtained from the BCS model which we discuss in the next chapter. To compute corrections to the saddle point approximation one Taylor expands the effective potential around the global minimum up to second order. In this approximation the integration measure becomes Gaussian and quantities like Λ(q) =
|φq |2  can be computed. Usually it is not clear to what extent this gives the right answer. In fact, in section 10.4 we will argue that the result for Λ(q) obtained in this way is most likely a wrong one in one (and two) dimensions, while it seems to be the right answer in three space dimensions. In the following theorem the second order Taylor expansion of the effective potential for the many-electron system with attractive delta interaction around the global minimum (5.34) is given. Theorem 5.2.2 Let V be the effective potential (5.30), let κ = βLd and let  √ √ (ρ0 − κ r0 )eiθ0 for q = 0 iθ0 (5.55) ξq = φq − δq,0 κ r0 e = ρq eiθq for q = 0.

70 Then V ({φq }) = Vmin + 2β0 (ρ0 − +

 √ κ r0 )2 + (αq + iγq )ρ2q 1 2



(5.56)

q=0

βq |e−iθ0 φq + eiθ0 φ¯−q |2 + O(ξ 3 )

q=0

where, if Ek2 = k02 + e2k + λr02 , αq =

1λ 2κ

 q2 +(e 0

k

2 k −ek−q ) 2 Ek2 Ek−q

γq = − λκ

> 0 , βq =

λ κ

 k

k

0 ek−q −(k0 −q0 )ek 2 Ek2 Ek−q

λr02 2 Ek2 Ek−q

> 0,

∈R

(5.57)

(5.58)

k

and  Vmin = κ r02 −

1 κ



k log

cosh( β 2

√

cosh

e2k +λr02 )

 .

β 2 ek

(5.59)

Proof: We abbreviate κ = βLd and write  V ({φq }) =

 q

det |φq | − log 2

A

ig ∗ √ φ κ



A¯

A 0 det 0 A¯ ig √ φ κ



where A = C −1 = (δk,p ak )k,p∈Mκ and ak := 1/Ck = ik0 − ek . Then √ V ({φq }) − V ({ κ δq,0 r0 eiθ0 }) =   A √igκ φ∗ det ig √ φ A¯  κ 2 2

ρq − κr0 − log A igr0 e−iθ0 q det A¯ igr0 eiθ0 where igr0 eiθ0 ≡ igr0 eiθ0 Id in the determinant above. Since ⎤ ⎡

a ¯ k δk,p igr0 e−iθ0 δk,p −iθ0 −1 − A igr0 e |ak |2 +λr02 |ak |2 +λr02 ⎦ = ⎣ igr iθ0 δk,p ak δk,p 0e A¯ igr0 eiθ0 − |ak |2 +λr2 |ak |2 +λr02 0

A¯ −igr0 e−iθ0 1 ≡ |a|2 +λr2 0 −igr0 eiθ0 A

(5.60)

(5.61)

(5.62)

Bosonic Functional Integral Representation and because of  

A √igκ φ∗ A igr0 e−iθ0 + = ig √ A¯ igr0 eiθ0 φ A¯ κ  0 ig √ φ − igr0 eiθ0 κ



A ig¯ γ 0 ig ξ ∗ = + ig ξ 0 igγ A¯

ig ∗ √ φ κ

71

− igr0 e−iθ0 0



(5.63)

where γ = r0 eiθ0 and ξ = (ξk−p )k,p is given by (5.55), the quotient of determinants in (5.60) is given by 

  A¯ −ig¯ γ 0 ig ξ ∗ 1 det Id + |a|2 +λr2 0 ig ξ 0 −igγ A    ¯ γ ¯ A ∗ λ |a|2 +λr 2 ξ ig |a|2 +λr 2 ξ 0 0 = det Id + (5.64) γ ∗ ig |a|2 A ξ λ |a|2 +λr 2 ξ +λr 2 0

0

Since ∞  (−1)n+1 log det[Id + B] = T r log[Id + B] = T rB n n n=1

= T r B − 12 T r B 2 + 13 T r B 3 − + · · ·

(5.65)

one obtains to second order in ξ:    ¯ γ ¯ A ∗ λ |a|2 +λr 2 ξ ig |a|2 +λr 2 ξ 0 0 log det Id + γ ∗ ig |a|2 A ξ λ |a|2 +λr 2 ξ +λr02 0   ¯ γ ¯ A ∗ λ |a|2 +λr 2 ξ ig |a|2 +λr 2 ξ 0 0 = Tr γ ∗ ig |a|2 A ξ λ |a|2 +λr 2 ξ +λr02 0 2   ¯ γ ¯ A ∗ λ |a|2 +λr 2 ξ ig |a|2 +λr 2 ξ 1 0 0 − 2Tr + O(ξ 3 ) γ ∗ ig |a|2 A 2 ξ λ 2 ξ 2 +λr |a| +λr 0

(5.66)

0

γ ξ + T r |a|2λγ ξ∗ = T r |a|2λ¯ +λr02 +λr02  λ¯ γ ig A¯ λ¯ γ igA − 12 T r 2 ξ 2 ξ + Tr 2 ξ∗ 2 ξ 2 2 2 |a| + λr0 |a| + λr0 |a| + λr0 |a| + λr02  ¯ + T r |a|2λγ ξ ∗ |a|2λγ ξ ∗ + T r |a|2igA ξ igA ξ ∗ + O(ξ 3 ) +λr 2 +λr 2 +λr 2 |a|2 +λr 2 0

One has 

λ¯ γ |a|2 +λr02

 ξ

= k,p

λ¯ γ |ak |2 +λr02

0

0

 ξk,p ,

λγ |a|2 +λr02

ξ∗

0

 = k,p

λγ |ak |2 +λr02

ξ¯p,k

72 

λ¯ γ |a|2 +λr02

  γ λ¯ γ λ¯ γ ξ |a|2λ¯ ξ = p |ak |2 +λr02 ξk,p |ap |2 +λr02 ξp,k +λr02 k,k 2   λ¯ γ ξk,k λ¯ γ λ¯ γ + ξ ξ = |ak |2 +λr 2 |ak |2 +λr 2 |ap |2 +λr 2 k,p p,k 0



1 κ

=

0

p p=k

2 √ λr0 (ρ0 − κr0 ) |ak |2 +λr02

+

1 κ



(5.67)

0

λ¯ γ λ¯ γ |ak |2 +λr02 |ak−q |2 +λr02

φq φ−q

q=0



λγ |a|2 +λr02

  λγ λγ ∗ ¯ ¯ ξ ∗ |a|2λγ ξ = p |ak |2 +λr02 ξp,k |ap |2 +λr02 ξk,p +λr02 k,k 2   λγ ξ¯k,k λγ λγ + ξ¯ ξ¯ = |ak |2 +λr 2 |ak |2 +λr 2 |ap |2 +λr 2 k,p p,k 0



1 κ

=

0

p p=k

2 √ λr0 (ρ0 − κr0 ) |ak |2 +λr02

+

1 κ



(5.68)

0

λγ λγ |ak |2 +λr02 |ak−q |2 +λr02

φ¯q φ¯−q

q=0

and 



igA |a|2 +λr02



igA |a|2 +λr02



k,p

igak |ak |2 +λr02

ξk,p ,

¯





ξ

=

A ∗ ξ |a|2ig+λr 2 ξ 0

= k,k

¯ igA |a|2 +λr02

igak p |ak |2 +λr02

2 k| = −λ (|ak ||a2 +λr 2 )2 |ξk,k | − λ 2

0

k| = − κλ (|ak ||a2 +λr 2 )2 (ρ0 − 2

0

 p p=k

ξ∗

 = k,p

ig¯ ak |ak |2 +λr02

ξ¯p,k

ig¯ ap ¯ ξk,p |ap |2 +λr 2 ξk,p

(5.69)

0

ak |ak |2 +λr02

√ κr0 )2 −

λ κ



a ¯p ¯ ξk,p |ap |2 +λr 2 ξk,p 0

a ¯ k−q ak |ak |2 +λr02 |ak−q |2 +λr02

ρ2q

q=0



¯ igA |a|2 +λr02

ξ ∗ |a|2igA ξ +λr 2 0

 = k,k



ig¯ ak p |ak |2 +λr02

2 k| = −λ (|ak ||a2 +λr 2 )2 |ξk,k | − λ 2

0

k| = − κλ (|ak ||a2 +λr 2 )2 (ρ0 − 2

0

√

 p p=k

igap ξ¯p,k |ap |2 +λr 2 ξp,k

(5.70)

0

a ¯k |ak |2 +λr02

κr0 )2 −

λ κ

 q=0

ap ξ¯p,k |ap |2 +λr 2 ξp,k 0

ak+q a ¯k |ak |2 +λr02 |ak+q |2 +λr02

ρ2q

Bosonic Functional Integral Representation

73

Therefore (5.66) becomes    ¯ γ ¯ A ∗ λ |a|2 +λr 2 ξ ig |a|2 +λr 2 ξ 0 0 log det Id + γ ∗ ig |a|2 A ξ λ |a|2 +λr 2 ξ +λr 2 0

0

γ ξ + T r |a|2λγ ξ∗ = T r |a|2λ¯ +λr02 +λr02  igA λ¯ γ ig A¯ λ¯ γ − 12 T r 2 ξ 2 ξ + Tr 2 ξ∗ 2 ξ 2 2 2 |a| + λr0 |a| + λr0 |a| + λr0 |a| + λr02  ¯ igA igA λγ ∗ ∗ ∗ + T r |a|2λγ ξ ξ + T r ξ ξ + O(ξ 3 ) 2 2 2 2 +λr0 |a|2 +λr0 |a|2 +λr0 |a|2 +λr0  √κ r (ρ −√κ r ) 0 0 0 = 2 λκ |ak |2 +λr 2 0

k

−

1 2

  λr

√

0 (ρ0 − κr0 ) |ak |2 +λr02

1 κ

k

+

1 κ

  λr

√

0 (ρ0 − κr0 ) |ak |2 +λr02

k

− λκ − λκ



2 +





q=0

2 +



1 κ



q=0

|ak |2 (ρ0 (|ak |2 +λr02 )2

1 κ

λ¯ γ λ¯ γ |ak |2 +λr02 |ak−q |2 +λr02

λγ λγ |ak |2 +λr02 |ak−q |2 +λr02

ak+q |ak+q |2 +λr02

ρ2q

a ¯ k−q ak |ak |2 +λr02 |ak−q |2 +λr02

ρ2q

0

k

q=0

k





−

φ¯q φ¯−q

k

  √ a ¯k λ − κr0 )2 − κ |ak |2 +λr 2



|ak |2 (ρ0 (|ak |2 +λr02 )2

φq φ−q

k

√ κr0 )2 −

k

λ κ

q=0



k

3

+ O(ξ )

(5.71)

Using the BCS equation (5.33),

λ  κ

1 k |ak |2 +λr02

= 1 and abbreviating

Ek2 := |ak |2 + λr02 = k02 + e2k + λr02 this becomes  λr2 −|a |2 √ √ √ k 0 2 κr0 (ρ0 − κr0 ) · 1 − (ρ0 − κr0 )2 λκ E4 k

+



ρ2q

  λ κ

q=0

k

 a ¯ k ak−q 2 Ek2 Ek−q

−

−2iθ 0 φq φ−q q=0 Re e



 λ κ

 k

λr02 2 Ek2 Ek−q



k

 λr2 √ √ √ √ 0 = 2 κr0 (ρ0 − κr0 ) · 1 + (ρ0 − κr0 )2 − (ρ0 − κr0 )2 2 κλ E4 k

+



ρ2q

  λ κ

q=0

 a ¯ k ak−q 2 Ek2 Ek−q

−

k



Re e−2iθ0 φq φ−q

 

λ κ

q=0

k λr02 2 Ek2 Ek−q



k

 λr2 √ 0 = ρ20 − κr02 − (ρ0 − κr0 )2 2 κλ E4

(5.72)

k

+

 q=0

ρ2q

  λ κ

k

k

 a ¯ k ak−q 2 Ek2 Ek−q

−

 q=0

Re e−2iθ0 φq φ−q

  λ κ

k

λr02 2 Ek2 Ek−q



74 √1 κ

Therefore one obtains, recalling that ξk,p =

φk−p − γ δk,p ,

√ V ({φq }) − V ({ κ δq,0 r0 eiθ0 })    ¯ γ ¯ A ∗  λ |a|2 +λr 2 ξ ig |a|2 +λr 2 ξ 2 2 0 0 = ρq − κ r0 − log det Id + γ ∗ ig |a|2 A ξ λ |a|2 +λr 2 ξ +λr02 q 0     λr2   a¯ a √ 2 2 λ 2 λ k k−q 0 = ρq + (ρ0 − κr0 ) 2 κ − ρq κ E4 E2 E2 k

q=0

+



Re e−2iθ0 φq φ−q

  λ κ

q=0

λ κ

 k

a ¯k ak−q 2 Ek2 Ek−q

 q=0 1 2 (1

=

  |a

=

1 2

=

1λ 2κ

2 2 k | +λr0 2 Ek2 Ek−q

λ κ

+

λ κ

k

k ak−q 2 Ek2 Ek−q



2 2 k−q | +λr0 2 Ek2 Ek−q

−

1λ 2κ

k

λ κ



 2¯a

k ak−q 2 Ek2 Ek−q

k

λr02 2 Ek2 Ek−q

 q2 +(e k

1λ 2κ

+

λ κ

 a¯

¯ k−q k ak−q −ak a 2 Ek2 Ek−q

k

ak −¯ ak−q ) k −ak−q )(¯ 2 Ek2 Ek−q

0

−

k

 (a k

1λ 2κ

(5.73)

¯k −ak a ¯ k−q −¯ ak ak−q +ak−q a ¯k−q ka 2 Ek2 Ek−q

+

=



 2¯a

1λ 2κ

 |a

a k

=

λr02 2 Ek2 Ek−q

k−q

k

ρ2q . It is given by

+ 1) −

k

1λ 2κ

q=0

k

Consider the coefficient of 1−

k

k

2 k −ek−q ) 2 Ek2 Ek−q

+

λ κ

 k



λr02 2 Ek2 Ek−q

− i λκ

k λr02 2 Ek2 Ek−q

 Im(¯a

k ak−q ) 2 Ek2 Ek−q

k

− i λκ

k

0 ek−q −(k0 −q0 )ek 2 Ek2 Ek−q

k

= αq + iγq + βq

(5.74)

Inserting (5.74) in (5.73), one gets √ V ({φq }) − V ({ κ δq,0 r0 eiθ0 })   √ 2 ρ2q 1 − = (ρ0 − κr0 ) 2β0 + q=0

+



Re e−2iθ0 φq φ−q

λ κ

  λ κ

q=0

k



 2¯a k

k ak−q 2 Ek2 Ek−q

λr02 2 Ek2 Ek−q



 √ ρ2q (αq + iγq ) = (ρ0 − κr0 )2 2β0 + +

 q=0

q=0

ρ2q βq +

 e−2iθ0 φ

2iθ0 ¯ ¯ φq φ−q q φ−q +e

2 q=0

βq

(5.75)

Bosonic Functional Integral Representation

75

Since βq = β−q , the last two q-sums in (5.75) may be combined to give  e−2iθ0 φ φ +e2iθ0 φ¯ φ¯  ρ2 +ρ2 q −q q −q q −q βq + βq 2 2 q=0

=

1 2

=

1 2



q=0

(φq φ¯q + φ−q φ¯−q + e−2iθ0 φq φ−q + e2iθ0 φ¯q φ¯−q )βq

q=0



(e−iθ0 φq + eiθ0 φ¯−q )(eiθ0 φ¯q + e−iθ0 φ−q )βq

q=0

=

1 2



|e−iθ0 φq + eiθ0 φ¯−q |2 βq

(5.76)

q=0

which proves the theorem
Finally we consider the effective potential in the presence of a small U (1) symmetry breaking external field. That is, we add to our original Hamiltonian the term   + + 1 r ¯ a (5.77) a + ra a −k↓ k↑ k↑ −k↓ Ld k

where r = |r| e ∈ C. In section 6.2 below we show that in that case the effective potential can be transformed to   ak δk,p √igκ φ¯˜p−k det ig ˜ √ φk−p a−k δk,p √ |r| 2  2 2 κ

Ur ({φq }) = u0 + v0 + κ g + |φq | − log a δ 0 q=0 det k k,p 0 a−k δk,p iα

(5.78) where

 φ˜q :=

φq for q =  0 eiα φ0 for q = 0

(5.79)

We obtain the following Corollary 5.2.3 Let Ur be the effective potential (5.78) where φ˜ is given by (5.79). Then: (i) The global minimum of Re Ur ({φq }) is unique and is given by √ φmin = δq,0 κ iy0 (5.80) q where y0 = y0 (|r|) is the unique global minimum of the function VBCS,r : R → R,

√ VBCS,r (y) := Ur u0 = 0, v0 = κ y; φq = 0 for q = 0  2    |r| λy 2 1 −κ log 1 + k2 +e2 = κ y+ g (5.81) 0

k

k

76 (ii) The second order Taylor expansion of Ur around φmin is given by

√ Ur {φq } = Ur,min + 2β0 (v0 − κy0 )2   + (αq + iγq )|φq |2 + 12 βq |e−iα φq − eiα φ¯−q |2 q=0

+

|r| g|y0 |



(5.82)

q=0

u20

 

√ + (v0 − κy0 )2 + |φq |2 + O (φ − φmin )3 q=0

where Ur,min := Ur {φmin q }) and the coefficients αq , βq and γq are given by (5.57, 5.58) of Theorem 5.2.2 but Ek in this case is given by Ek2 = |ak |2 +λy02 = k02 + e2k + λy02 .

Remark: Of course one has lim|r|→0 λy0 (|r|)2 = λr02 = ∆2 where ±r0 is the global minimum of VBCS,r=0 . Proof: (i) As in the proof of Theorem 5.2.1 one shows that    Id  log det ig ¯ ˜ √  Cφ κ

C φ˜∗ Id

ig √ κ



 P  2 λ  q |φq | κ log 1 + |ak |2  ≤  k    2 2 ) =: log 1 + λ(x|ak+y 2 |

(5.83)

k

where we abbreviated x2 :=

1 κ

   u20 + |φq |2 ,

y 2 := κ1 v02

(5.84)

q=0

Thus   

2 √ 2 2 + u + |φ | − log 1+ ReUr ({φq }) ≥ v0 + κ |r| q 0 g q=0

λ(x2 +y 2 ) |ak |2



k

=: κWr (x, y)

(5.85)

where Wr (x, y) = x2 + y +

|r| 2 g

−

1 κ



 log 1 +

λ(x2 +y 2 ) |ak |2

 (5.86)

k

The global minimum of Wr is unique and given by x = 0 and y = y0 where √ y0 is the unique global minimum of (5.81). Since Ur at u0 = 0, v0 = κ y; φq = 0 for q = 0 equals VBCS,r (y), part (i) follows.

Bosonic Functional Integral Representation

77

(ii) Part (ii) is proven in the same way as Theorem 5.2.2. One has Ur ({φq }) − Ur,min 

2 √

2 √ √ = u20 + |φq |2 + v0 + κ |r| − κ y0 + κ |r| g g q=0





− log det = u20 +



  ˜p−k  ak δk,p ak δk,p √igκ φ¯ det ig iα ig ˜ √ e iy0 δk,p √ a δ φ κ κ k−p −k k,p

|φq |2 + (v0 −



a−k δk,p

√

√ √ √ κ y0 )2 + 2(v0 − κ y0 ) κ y0 + κ |r| g

q=0

 − log det Id +

ig −iα √ e (−i)y0 δk,p κ

ak δk,p ig iα √ e iy0 δk,p κ

ig −iα √ e (−i)y0 δk,p κ

−1 ×

a−k δk,p 

0 ig √ ξk−p κ

where in this case ξk−p := φ˜k−p −

√

ig ¯ √ ξ κ p−k



0

κ eiα iy0 δk,p

(5.87)

The expression log det[Id + · · · ] is expanded as in the proof of Theorem 5.2.2. One obtains, if Ek2 := |ak |2 + λy02 ,  √ √ 1 log det[Id + · · · ] = 2 κy0 (v0 − κy0 ) λκ E2 k

+

λ κ

 λy2 (u2 −(v0 −√κy0 )2 ) 0

0

Ek4

k

+

 |ak |2 (u2 +(v0 −√κy0 )2 )

λ κ

0

Ek4

k

+

1λ 2κ

−

1λ 2κ

k

 a

¯ k−q ka 2 Ek2 Ek−q

+

a ¯ k ak+q 2 Ek2 Ek+q



|φq |2

q=0 k

   λy2 e2iα

0 2 Ek2 Ek+q

φ¯q φ¯−q +

λy02 e−2iα φq φ−q 2 Ek2 Ek−q

 (5.88)

q=0 k

Since y0 is a minimum of VBCS,r , one has the BCS equation     2y0 1 − λκ = 0 ⇔ λκ =1− 2 y0 + |r| g E2 E2 k

k

k

|r| g|y0 |

(5.89)

k

Using this, one gets (observe that y0 is negative)

√ |r| 2 Ur ({φq }) = Ur,min + g|y u0 + (v0 − κ y0 )2 0|   λy2 (v −√κy )2   a¯ a 0 0 k k−q λ 0 + + 2 λκ 1 − |φq |2 4 2 2 κ E E E k

k

+

1λ 2κ

   λy2 e2iα

0 2 Ek2 Ek+q

q=0 k

k

q=0

φ¯q φ¯−q +

k−q

k λy02 e−2iα φq φ−q 2 Ek2 Ek−q

 (5.90)

78 Using the BCS equation (5.89) again, one obtains (compare (5.74))  a¯ a |r| k k−q = αq + iγq + βq + g|y 1 − λκ E2 E2 0| k

k

(5.91)

k−q

Substituting this in (5.90) and rearranging as in the proof of Theorem 5.2.2 proves part (ii).


Chapter 6 BCS Theory and Spontaneous Symmetry Breaking

The Hamiltonian for the many-electron system in finite volume [0, L]d is given by H = H0 + Hint where
H0 = L1d e k a+ (6.1) kσ akσ kσ

ek =

k 2m 2

− µ, and Hint =

1 L3d



σ,τ k,p,q

+ V (k − p)a+ kσ aq−k,τ aq−p,τ apσ

(6.2)

As usual, the interacting part may be represented by the following diagram:

p,σ

q-p,τ V(k-p)

Hint = k,σ

q-k,τ

Since there is conservation of momentum, there are three independent momenta. One may consider three natural limiting cases with two independent momenta:

k

p

p

0

k

p

forward

k

p

k

p

exchange

-p k-p

k-p

k

-k

BCS

In chapter 8 where we prove rigorous bounds on Feynman diagrams we show that the most singular contributions to the perturbation series come from the

79

80 q = 0 term in (6.2). This corresponds to the third diagram in the above figure. To retain only this term, the q = 0 term of Hint given by (6.2), is the basic approximation of BCS theory. Furthermore V (k − p) is substituted ˆ−p ˆ = kF k , ˆ ) where k by its value on the Fermi surface, V (k − p) ≈ V (k
k
√ kF = 2mµ, and the momenta k, p are restricted to values close to the Fermi surface, |ek |, |ep | ≤ ωD , where the cutoff ωD is referred to as the Debye frequency. Thus the BCS approximation reads Hint ≈ HBCS

(6.3)

where HBCS =

1 L3d







σ,τ

k,p |ek |,|ep |≤ωD

+ ˆ−p ˆ ) a+ V (k kσ a−k,τ a−p,τ apσ

(6.4)

We first make a comment on the volume factors. In (6.4) we retained only the q = 0 term of (6.2), but we did not cancel a volume factor. It is not obvious that (6.4) is still proportional to the volume. However, in the case of the forward approximation which is defined by putting k = p in (6.2) without canceling a volume factor,

+ V (0)a+ (6.5) Hint ≈ Hforw := L13d kσ ap,τ ap,τ akσ σ,τ k,p

there is an easy argument which shows that (6.5) is still proportional to the volume (for constant density). Namely, on a fixed N -particle space FN the interacting part Hint is a multiplication operator given by Hint|FN =

1 2

N


V (xi − xj )

(6.6)

i,j=1 i=j

Let δy (x) = δ(x − y). Then ϕ(x1 , · · · , xN ) := δy1 ∧ · · · ∧ δyN (x1 , · · · , xN ) is an eigenfunction of Hint with eigenvalue E=

1 2

N


V (yi − yj ) =

i,j=1 i=j

N



1 2Ld

i,j=1 i=j

e−i(yi −yj )q V (q)

(6.7)

q

which is, for V (x) ∈ L1 , proportional to N or to the volume for constant density. One finds that ϕ is also an eigenvector of the forward term, Hforw ϕ = Eforw ϕ where Eforw is obtained from (6.7) by putting q = 0 without canceling a volume factor, Eforw =

1 2Ld

N
i,j=1 i=j

V (q = 0)

(6.8)

BCS Theory and Spontaneous Symmetry Breaking

81

which is also proportional to the volume. Although the model defined by the Hamiltonian HBCS in (6.4) is still quartic in the annihilation and creation operators, it can be solved explicitly, without making a quadratic mean field approximation as it is usually done. The important point is that the approximation ‘putting q = 0 without canceling a volume factor’ has the effect that in the bosonic functional integral representation the volume factors enter the formulae in such a way that in the thermodynamic limit the integration variables are forced to take values at the global minimum of the effective potential. That is, fluctuations around the minimum configuration are suppressed for Ld → ∞. ˆ p ˆ ) in (6.4) can be expanded into spherical harmonics: The interaction V (k−  1 ∞ iϕk −iϕp e + λ20 for d = 2 =−∞ λ|| e 2 ˆ −p ˆ) =  V (k (6.9)  ∞  ˆ ¯ p/kF ) for d = 3 =0 m=− λ Ym (k/kF )Ym (ˆ where k = |k|(cos ϕk , sin ϕk ) for d = 2. We start with the easiest case where only the  = 0 term in (6.9) is retained. That is, we approximate =0 HBCS ≈ HBCS

where =0 HBCS =

λ L3d




+ a+ k↑ a−k,↓ a−p,↓ ap↑

(6.10)

(6.11)

k,p |ek |,|ep |≤ωD

where we put λ = 2λ0 , the factor of 2 coming from spin sums. In the next section we present the standard quadratic mean field formalism which approximates (6.11) by a quadratic mean field Hamiltonian which then can be diagonalized with a Bogoliubov transformation. In section 6.2 we show that already the quartic model defined by (6.11) is explicitly solvable and we discuss symmetry breaking. We find that the correlation functions of the model defined by (6.11) coincide with those of the quadratic mean field model. However, if higher angular momentum terms in (6.9) are taken into account, the quadratic mean field formalism does not necessarily give the right answer. This is discussed in section 6.3.

6.1

The Quadratic Mean Field Model

In this section we present the standard quadratic BCS mean field formalism for a constant (s-wave) interaction as it can be found in many books on manybody theory or superconductivity [57], [31]. We use the formulation of [23].

82 =0 be the quartic BCS Hamiltonian given by (6.11) and (6.1). Let H0 + HBCS + We add and subtract expectation values of products of field operators a+ k↑ a−k↓ , a−p↓ ap↑ which will be defined later selfconsistently. Finally we neglect terms which are quadratic in the differences aa − aa , a+ a+ − a+ a+ which are supposed to be small:
+ =0 HBCS = Lλ3d a+ k↑ a−k↓ a−p↓ ap↑ k,p |ek |,|ep |≤ωD

=

λ L3d


 + + + + + [a+ k↑ a−k↓ − ak↑ a−k,↓ ] + ak↑ a−k↓ ×

k,p |ek |,|ep |≤ωD

≈

λ L3d

λ L3d

 + + + + a+ a [a a − a a ] + a a a a −p↓ p↑ −p↓ p↑ −p↓ p↑ k↑ −k,↓ k↑ −k↓


 + + + a+ k↑ a−k↓ a−p↓ ap↑ + ak↑ a−k↓ a−p↓ ap↑

k,p |ek |,|ep |≤ωD

=

λ Ld

 [a−p↓ ap↑ − a−p↓ ap↑ ] + a−p↓ ap↑


 + + + [a+ k↑ a−k↓ − ak↑ a−k,↓ ]a−p↓ ap↑

k,p |ek |,|ep |≤ωD

=






 + − a+ k↑ a−k↓ a−p↓ ap↑

 + ¯ a−p↓ ap↑ − λLd |∆|2 =: HMF a+ a ∆ + ∆ k↑ −k↓

(6.12)

k |ek |≤ωD

where we put ∆=

1 Ld


p

1 Ld a−p↓ ap↑

(6.13)

and the expectation value has to be taken with respect to HMF . Thus, the quadratic mean field approximation consists in 
 + ¯ a−k↓ ak↑ − λLd |∆|2 (6.14) a+ Hint ≈ HMF = Lλd a ∆ + ∆ k↑ −k,↓ k |ek |≤ωD

where ∆ has to be determined selfconsistently by the BCS equation ∆=

λ L2d


p |ep |≤ωD

T r [a−p↓ ap↑ e−β(H0 +HMF ) ] T r e−β(H0 +HMF )

This mean field model is diagonalized in the following

(6.15)

BCS Theory and Spontaneous Symmetry Breaking  Theorem 6.1.1 Let H = H0 + HMF = k hk where  + hk = L1d ek a+ k↑ ak↑ + a−k↓ a−k↓  + d 2 ¯ − Lλd χ(|ek | ≤ ωD ) a+ k↑ a−k↓ ∆ + ∆ a−k↓ ak↑ − λL |∆|

83

(6.16)

Then one has a) [hk , hk ] = 0 ∀k, k , that is, the hk can be simultaneously diagonalized. b) Let ⎧ ⎪ ⎨0

  for ek > ωD ∆ arctan(−π,0) − ek for |ek | ≤ ωD tk = ⎪ ⎩ π −2 for ek < −ωD 1 2

(6.17)

and Sk :=

1 Ld

 + + ak↑ a−k↓ − a−k↓ ak↑ ,

Uk := etk Sk

Then [Uk , Uk ] = 0 ∀k, k and  + Uk hk Uk+ = L1d Ek a+ k↑ ak↑ + a−k↓ a−k↓ − (Ek − ek )

(6.18)

(6.19)

where Ek :=

 2 ek + |∆|2 for |ek | ≤ ωD |ek | for |ek | > ωD

(6.20)

c) Let U := Πk Uk . Then U HU + =

1 Ld


kσ

and E0 := inf σ(H) = − E1 = E0 .

Ek a+ kσ akσ −



k (Ek


(Ek − ek )

(6.21)

k

− ek ) and E1 ≥ E0 + |∆| ∀E1 ∈ σ(H),

d) The ground state Ω0 = U + 1 of H is given by       + 1 + + cos tk − L1d sin tk a+ 1 (6.22) Ω0 = a a a k↑ −k↓ Ld k↑ −k↓ k |ek |≤ωD

k ek <−ωD

where 1 = (1, 0, 0, · · · ) ∈ ⊕n Fn is the vacuum state. e) The BCS-equation (6.15) reads
λ Ld

k |ek |≤ωD

tanh(βEk /2) 2Ek

=1

(6.23)

84 and for attractive λ > 0 there is the nonanalytic zero temperature solution ∆(T = 0) ≈ 2ωD e

− n1 λ

(6.24)

d

where nd = ωd−1 mkFd−2 /(2π)d , ωd being the surface of the d-dimensional unit sphere. f ) The momentum distribution is given by   tanh(βEk /2) + 1 1 a a = 1 − e kσ k kσ 2 Ek Ld   β→∞ 1 → 2 1 − Eekk

(6.25)

and the anomalous aa , a+ a+ expectations become nonzero and are given by 1 a−k↓ ak↑ Ld

=

∆ 2Ek

tanh(βEk /2)

(6.26)

+ and a+ k↑ a−k↓ = a−k↓ ak↑ .

Proof: a) This follows from [AB, CD] = A{B, C}D − AC{B, D} + {A, C}DB − C{A, D}B (6.27) and k = k ⇒



(+) (+) (+) (+)  ak↑ or a−k↓ , ak ↑ or a−k ↓ = 0

(6.28)

b) Since Sk+ = −Sk we have Uk+ = e−tk Sk = Uk−1 and Uk is unitary. Because of (6.27,6.28) all Sk commute. Furthermore  + Uk hk Uk+ = L1d ek a+ (6.29) k↑ (tk )ak↑ (tk ) + a−k↓ (tk )a−k↓ (tk )  + + λ ¯ a−k↓ (tk )ak↑ (tk ) − Ld χ(|ek | ≤ ωD ) ak↑ (tk )a−k↓ (tk ) ∆ + ∆ where for some operator A A(t) := etSk Ae−tSk

(6.30)

Since d dt A(t)

we have

= etSk [Sk , A]e−tSk

⎞ ak↑ (t) +  ⎟ d ⎜ ⎜ a−k↓ (t) ⎟ = etSk 1d a+ a+ − a−k↓ ak↑ , k↑ −k↓ L dt ⎝ a−k↓ (t) ⎠ + ak↑ (t) ⎛

(6.31) ⎛

⎞ ak↑ ⎜ a+ ⎟ ⎜ −k↓ ⎟ e−tSk (6.32) ⎝ a−k↓ ⎠ a+ k↑

BCS Theory and Spontaneous Symmetry Breaking

85

Using [AB, C] = A{B, C} − {A, C}B the commutator is found to be ⎛ + ⎞ ⎞ ak↑ −a−k↓ +   ⎜ ⎜ ⎟ ⎟ a + 1 ⎜ −k↓ ⎟ = ⎜ ak↑ ⎟ a+ + k↑ a−k↓ − a−k↓ ak↑ , ⎝ a Ld ⎝ ⎠ ⎠ a −k↓ k↑ a+ −a −k↓ k↑ ⎛ ⎞⎛ ⎞ a 0 −1 k↑ ⎜1 0 ⎟ ⎜ a+ ⎟ −k↓ ⎟⎜ ⎟ = ⎜ ⎝ ⎝ ⎠ 0 1 a−k↓ ⎠ −1 0 a+ k↑ ⎛

(6.33)

which gives  

ak↑ (t) a+ −k↓ (t) a−k↓ (t) a+ k↑ (t)



 =



 =

cos t − sin t sin t cos t cos t sin t − sin t cos t

 

ak↑ a+ −k↓ a−k↓ a+ k↑

 (6.34)  (6.35)

Thus, putting χk := χ(|ek | ≤ ωD ), Uk hk Uk+ =    + + 1 a a + a a ek (cos2 tk − sin2 tk ) − 2χk ∆ sin tk cos tk k↑ −k↓ d k↑ −k↓ L    + + L1d a+ −ek 2 sin tk cos tk − χk ∆(cos2 tk − sin2 tk ) k↑ a−k↓ + a−k↓ ak↑ + 2ek sin2 tk + 2χk ∆ sin tk cos tk

(6.36)

Now we choose tk such that the wavy brackets in the third line of (6.36) vanish. That is, ek sin 2tk + χk ∆ cos 2tk = 0

(6.37)

or, for |ek | ≤ ωD , tk =

1 2

  arctan − e∆k

This gives cos 2tk = √

ek , e2k +∆2

sin 2tk = − √

∆ e2k +∆2

(6.38)

and 2ek sin2 tk + 2∆ sin tk cos tk = ek (1 − cos 2tk ) + ∆ sin 2tk = ek − Ek (6.39) which proves part (b). Part (c) is an immediate consequence of (b).

86 d) Since the vacuum state 1 is the ground state of U HU + , the ground state of H is given by Ω0 = U + 1 = Πk e−tk Sk 1. One has

e

−tk Sk

1=

∞


(−tk )n 1 n! Lnd

 n + a+ 1 k↑ a−k↓ − a−k↓ ak↑

n=0

and   + + + a+ a − a a −k↓ k↑ 1 = ak↑ a−k↓ 1 k↑ −k↓ 2  + + 2d a+ a − a a 1 = −a−k↓ ak↑ a+ −k↓ k↑ k↑ −k↓ k↑ a−k↓ 1 = −L 1 which gives Sk2n 1 = (−1)n 1,

+ Sk2n+1 1 = (−1)n L1d a+ k↑ a−k↓ 1

(6.40)

and part (d) follows. e,f ) The momentum distribution is given by  −βH + a+ ak↑ ak↑ ] T r e−βH k↑ ak↑ = T r[e

 + + −βH = T r[U + U e−βH U + U a+ k↑ U U ak↑ ] T r U U e  + + + −βUHU + = T r[e−βUHU U a+ k↑ U U ak↑ U ] T r e  β P + = T r e− Ld kσ Ek akσ akσ (a+ k↑ cos tk − a−k↓ sin tk ) ×  P + − βd kσ Ek akσ akσ L sin t ) T r e (ak↑ cos tk − a+ k −k↓  β P  + + 2 2 T r e− Ld kσ Ek akσ akσ (a+ k↑ ak↑ cos tk + a−k↓ a−k↓ sin tk ) = P + − βd kσ Ek akσ akσ L T r e   β P + T r e− Ld kσ Ek akσ akσ a+ k↑ ak↑ = cos 2tk + Ld sin2 tk β P + T r e− Ld kσ Ek akσ akσ d = cos 2tk Ld 1+e1βEk + L2 (1 − cos 2tk )   = Ld 12 1 − cos 2tk [1 − 1+e2βEk ]   (6.41) = Ld 12 1 − Eekk tanh(βEk /2)

where we used (6.38) in the last line and in the sixth line we used a+ k↑ ak↑ =

BCS Theory and Spontaneous Symmetry Breaking

87

a+ −k↓ a−k↓ . This proves (6.26). Similarly  β a−k↓ ak↑ = T r e− Ld

P

Ek a+ kσ akσ

(a−k↓ cos tk + a+ k↑ sin tk ) ×  + β P (ak↑ cos tk − a+ T r e− Ld kσ Ek akσ akσ −k↓ sin tk )   β P + T r e− Ld kσ Ek akσ akσ a+ k↑ ak↑ 1 = sin 2tk − Ld sin 2tk P − βd Ek a+ akσ 2 kσ kσ L T r e  1 d = L sin 2tk 1+eβEk − 12 = Ld

∆ 2Ek

kσ

tanh(βEk /2)

(6.42)

where we used (6.38) again. The BCS equation (6.15) becomes

∆ =




λ L2d

k |ek |≤ωD

=




λ Ld

k |ek |≤ωD

L→∞



→ λ∆

T r [a−k↓ ak↑ e−β(H0 +HMF ) ] T r e−β(H0 +HMF ) ∆ β→∞ tanh(βEk /2) → 2Ek




λ Ld

k |ek |≤ωD

∆ 2Ek

dd k χ(|e √k |≤ωD ) (2π)d 2 e2 +∆2 k



ω

d−1 = λ∆ 2(2π) d

 ω

d−1 ≈ λ∆ 2(2π) d

χ(|ρ2 /2m − µ| ≤ ωD ) dρ ρd−1  (ρ − kF )2 (ρ + kF )2 /4m2 + ∆2 χ(

dρ ρd−1 r 

ωd−1 d−1 1 ≈ λ∆ 2(2π) d kF ∆

χ(

d−2 ωd−1 kF m (2π)d

log

≈ λ∆

d−2 ωd−1 kF m (2π)d

log 2

const λ

(ρ−kF )2 +∆2 kF

|ρ−kF |≤

k F ∆ 2 m2

ωD ∆

0

= λ∆

which gives ∆ = 2ωD e−

|ρ−kF |≤ωD )

dρ r ∆m2 

ωd−1 d−1 m = λ∆ 2(2π) d kF kF 2

kF m k2 F m2

ωD ∆

+ ! ωD

(ρ−kF

ωD ∆

)

)2 +1

√ dx x2 +1

 ωD ! ( ∆ )2 + 1

∆

as stated in the theorem.


(6.43)

88

6.2

The Quartic BCS Model

In this section we explicitly solve the model defined by the quartic Hamiltonian =0 H = H0 + HBCS
= L1d e k a+ kσ akσ +

λ L3d

kσ




+ a+ k↑ a−k,↓ a−p,↓ ap↑

(6.44)

k,p |ek |,|ep |≤ωD

The correlation functions are identical to those of the quadratic mean field model. In the next section we add higher -wave terms to the electron-electron interaction. While that model is still solvable, since the important point in the BCS approximation is ‘putting q = 0 without canceling a volume factor’ which results in a volume dependence which force the integration variables in the bosonic functional integral representation to take values at the global minimum of the effective potential, the correlation functions of that model are no longer identical to those obtained by applying the quadratic Anderson Brinkmann mean field formalism. This is discussed in the next section. Here we prove

Theorem 6.2.1 Let H be the quartic BCS Hamiltonian given by (6.44) and for some small ‘external field’ r = |r|eiα ∈ C let 
 + a + ra a r¯a+ (6.45) Hr = H + L1d −k↓ k↑ k↑ −k↓ k

 For some operator A let A = T r[e−βH A] T r e−βH and A r the same expression with H replaced by Hr . Then   tanh(βEk /2) + 1 1 lim lim L1d a+ (6.46) kσ akσ r = lim Ld akσ akσ = 2 1 − ek Ek r→0 L→∞

L→∞

and there is symmetry breaking in the sense that lim lim 1d a−k↓ ak↑ r r→0 L→∞ L

=

∆ 2Ek

tanh(βEk /2) = 0

(6.47)

while lim lim 1d a−k↓ ak↑ L→∞ r→0 L

=0

if ∆ = |∆|e−iα and |∆| is a solution of the BCS equation (6.23).

(6.48)

BCS Theory and Spontaneous Symmetry Breaking

89

Proof: The Grassmann integral representation of Z = T r e−βHr /T r e−βH0 , H0 given by (6.1) (not to be confused with Hr=0 = H0 + HBCS ) is given by Z = Z(skσ = 0, rk = r) where (recall that κ := βLd )  P λ ¯ ¯ Z({skσ , rk }) = e− κ3 k,p,q0 ψk0 ,k,↑ ψq0 −k0 ,−k,↓ ψp0 ,p,↑ ψq0 −p0 ,−p,↓ × (6.49) 1

eκ Π

kσ

P k

(r¯k ψ¯k↑ ψ¯−k↓ +rk ψ−k↓ ψk↑ ) ×

κ ik0 −ek

e− κ 1

Here k = (k0 , k), k0 , p0 ∈ πβ (2Z + 1), Stratonovich transformation, we obtain  det · · · Z({skσ , rk }) = det ·

P

¯

kσ (ik0 −ek −skσ )ψkσ ψkσ

Π dψkσ dψ¯kσ

kσ

q0 ∈ 2π β Z. By making a Hubbardas in Theorem 5.1.2 ! ! e−

P

q0

|φq0 |2

¯q dφq0 dφ 0 π q0

Π

(6.50)

where the quotient of determinants is given by ⎡   ⎤ ig ¯ √ δ (ak − sk↑ )δk,p φ − r ¯ δ p −k k p ,k p,k 0 0 0 0 κ ⎦  det ⎣  ig √ φk −p + rk δk ,p δ (a − s )δ k,p −k −k↓ k,p 0 0 0 0 κ & ' = (6.51) a δ 0 det k k,p 0 a−k δk,p ( ) ig ¯ √ (ak0 ,k − sk0 ,k,↑ )δk0 ,p0 − r¯k0 ,k δp0 ,k0 φ κ p0 −k0 det ig √ φk −p + rk ,k δk ,p (a−k ,−k − s−k ,−k,↓ )δk ,p  0 0 0 0 0 0 0 0 0 κ & ' a δ 0 k det k0 ,k k0 ,p0 0 a−k0 ,−k δk0 ,p0 Observe that in the first line of (6.51) the matrices are labelled by k, p = (k0 , k), (p0 , p) whereas in the second line of (6.51) the matrices are only labelled by k0 , p0 and there is a product over spatial momenta k since the matrices in the first line of (6.51) are diagonal in the spatial momenta k, p. Since we are dealing with the BCS approximation which is obtained from the full model by retaining only the q = 0 term of the quartic part, the integration variables are no longer φq,q0 but only φq=0,q0 ≡ φq0 . As in Theorem 5.1.2, the correlation functions are obtained from (6.50) by differentiating with respect to stσ or rt , t = (t0 , t): )−1  ( ig ¯ √ a δ − r ¯ δ φ k ,t k ,p p −k p ,k 0 0 0 0 0 0 0 1 ¯ β dPr ({φq0 }) ig κ ψtσ ψtσ = √ φ + rδk0 ,p0 a−k0 ,−t δk0 ,p0 β k0 −p0 t0 σ,t0 σ

 ( 1 κ

ψt↑ ψ−t↓ =

ig √ β

ak0 ,t δk0 ,p0 φk0 −p0 + rδk0 ,p0

ig √ β

φ¯p0 −k0 − r¯δp0 ,k0 a−k0 ,−t δk0 ,p0

)−1

(6.52) dPr ({φq0 })

t0 ↓,t0 ↑

(6.53)

90 where d e−L Vr ({φq0 }) Πq0 dφq0 dφ¯q0 dPr ({φq0 }) = * −Ld V ({φ }) r q0 e Πq0 dφq0 dφ¯q0

(6.54)

with an effective potential Vr ({φq0 }) =
q0

(

1
|φq0 | − d log L 2

k

det

(6.55) ) ¯ ak0 ,k δk0 ,p0 φp0 −k0 − r¯δp0 ,k0 ig √ φ + rδk0 ,p0 a−k0 ,−k δk0 ,p0 β k0 −p0 & ' a δ 0 det k0 ,k k0 ,p0 0 a−k0 ,−k δk0 ,p0 ig √ β

* dd k  If we substitute the Riemannian sum L1d k in (6.55) by an integral (2π) d, then the only place where the volume shows up in the above formulae is the prefactor in the exponent of (6.54). Thus, in the infinite volume limit we can evaluate the integrals by evaluating the integrands at the global minimum of the effective potential, or, if this is not unique, by averaging the integrands over all minimum configurations. To this end observe that the external field r√only appears in conjunction with the φ0 variable through the combination φ0 / β − ir/g. Thus, by substitution of variables (φ0 = u0 + iv0 and r = |r| eiα ) 

 f

R2



R2

φ0 √ β

 2 2 ¯0 φ − i gr , √ + i gr¯ e−[u0 +v0 ] du0 dv0 = β

(6.56)

  √ |r| 2 2 ¯0 φ φ0 −iα √ e−[u0 +(v0 + β g ) ] du0 dv0 f eiα √ , e β β

we obtain  (

)−1 ¯ ak0 ,t δk0 ,p0 √igβ φ˜p0 −k0 dQr ({φq0 }) ig ˜ √ φ a δ β k0 −p0 −k0 ,−t k0 ,p0 t0 σ,t0 σ )−1  ( ig ¯ ˜p −k √ δ a φ k ,t k ,p 0 0 0 0 0 1 β dQr ({φq0 }) ig ˜ κ ψt↑ ψ−t↓ = √ φ a δ k −p −k ,−t k ,p 0 0 0 0 0 β 1 κ

ψ¯tσ ψtσ =

(6.57)

(6.58)

t0 ↓,t0 ↑

where d e−L Ur ({φq0 }) Πq dφq0 dφ¯q0 dQr ({φq0 }) = * −Ld U ({φ }) 0 r q0 e Πq0 dφq0 dφ¯q0

(6.59)

BCS Theory and Spontaneous Symmetry Breaking

91

with an effective potential   Ur ({φq0 }) = u20 + v0 + β

|r| g

2 +




|φq0 |2

(6.60)

q0 =0

) ak0 ,k δk0 ,p0 √igβ φ¯˜p0 −k0 det ig ˜ √ φk −p a−k ,−k δk ,p 0 0 0 0 0 1
& β ' log − d L a δ 0 k det k0 ,k k0 ,p0 0 a−k0 ,−k δk0 ,p0 (

and  φ˜q0 :=

φq0 for q0 = 0 eiα φ0 for q0 = 0

(6.61)

The effective potential (6.60) is discussed in Corollary 5.2.3. More precisely, ˜r the effective potential considered in Corollary 5.2.3, then if we denote by U  ˜r according to Ur ({φq0 }) = U ˜r {φq0 ,q = 0 for q = Ur of (6.60) is related to U ˜r fulfills φq0 ,q = 0 0, φq0 ,0 ≡ φq0 } . In particular, if the global minimum of U ˜r is also the global if q = 0, which is case, then the global minimum of U minimum of Ur . Thus we have  β iy0 (6.62) φmin q0 = δq0 ,0 where lim λy02 = |∆|2

(6.63)

r→0

and |∆| the solution of the BCS equation. Hence, in the infinite volume limit, & 1 d L→∞ βL

lim

ψ¯tσ ψtσ = =

a−t |at |2 +λy02

& 1 d L→∞ βL

lim

ψt↑ ψ−t↓ = =

ak0 ,t δk0 ,p0 ge−iα y0 δp0 ,k0 −geiα y0 δk0 ,p0 a−k0 ,−t δk0 ,p0 =

ge y0 |at |2 +λy02

=

t0 σ,t0 σ

−it0 −et t20 +e2t +λy02

ak0 ,t δk0 ,p0 ge−iα y0 δp0 ,k0 −geiα y0 δk0 ,p0 a−k0 ,−t δk0 ,p0 iα

'−1

iα

ge y0 t20 +e2t +λy02

(6.64) '−1 t0 ↓,t0 ↑

(6.65)

Performing the sums over the t0 variables and using (6.63) one obtains (6.46) and (6.47). (6.48) follows from symmetry.


92

6.3

BCS with Higher -Wave Interaction

We now consider the BCS model in two or three dimensions with an electronelectron interaction of the form (6.9). To have a uniform notation for d = 2 or d = 3 we write ˆ−p ˆ) = V (k

J


ˆ l (ˆ λl y¯l (k)y p)

(6.66)

l=0

where we assume ˆ = (−1)l yl (k) ˆ yl (−k)

(6.67)

Thus our Hamiltonian is l≤J H = H0 + HBCS

=

1 Ld




e k a+ kσ akσ +

kσ

(6.68) 1 L3d




J





+ ˆ l (ˆ λl y¯l (k)y p)a+ kσ a−k,τ a−p,τ apσ

k,p l=0 |ek |,|ep |≤ωD

στ

In the easiest case only even l terms contribute to (6.68). In that case only a↑ a↓ pairs give a nonzero contribution since
k

ˆ a+ a+ y¯l (k) k↑ −k↑ =


k

= −

ˆ  ) a+  a+ y¯l (−k −k ↑ k ↑


ˆ  ) a+ a+  (−1)l y¯l (k k ↑ −k ↑ k

l even

= −


k

ˆ a+ a+ y¯l (k) k↑ −k↑

(6.69)

Thus, for even l, the Hamiltonian becomes H=

1 Ld


kσ

e k a+ kσ akσ +

1 L3d




J


+ ˆ l (ˆ 2λl y¯l (k)y p)a+ k↑ a−k,↓ a−p,↓ ap↑

k,p l=0 |ek |,|ep |≤ωD l even

(6.70) In the next theorem we compute the a+ a correlation function for the model defined by (6.70) in two and three dimensions. This model, although being quartic in the annihilation and creation operators and with a nontrivial interaction, is still exactly solvable. The important point, again, is that, compared to the full model, we put ‘q = 0 without canceling a volume factor’ which has the effect that, in the infinite volume limit, in the bosonic functional integral

BCS Theory and Spontaneous Symmetry Breaking

93

representation the integration variables are forced to take values at the global minimum of the effective potential which can be computed. Then we compare our result to that one of the standard Anderson Brinkman mean field formalism [3, 5]. While in two dimensions the results are consistent, which has been shown by A. Sch¨ utte in his Diploma thesis [58], the quadratic mean field formalism in general gives the wrong answer in three dimensions, at least in the case where there is no explicit SO(3) symmetry-breaking term present. We have the following Theorem 6.3.1 a) Let H be given by (6.70) with 2λl substituted by λl . Then
+ 1 1 1 a a = lim ψ¯kσ ψkσ e−ik0  (6.71) kσ d kσ β L βLd  0

k0 ∈ π β (2Z+1)

where * 1 ψ¯kσ ψkσ βLd

−ik0 −ek ¯ k Φk k02 +e2k +Φ

=

*

e−βL

d

J

VBCS ({φl })

Π dul dvl

l=0 l even

(6.72)

J

e−βLdVBCS ({φl }) Π dul dvl l=0 l even

and (φl = ul + ivl , φ¯l = ul − ivl , gl = J


Φk :=

√

λl )

¯ k := Φ

gl φl yl (k),

l=0 l even

J


gl φ¯l y¯l (k)

(6.73)

l=0 l even

and the BCS effective potential is given by VBCS ({φl }) =

J


 |φl |2 −

dd k 2 (2π)d β

& log

cosh( β 2

√

¯ k Φk ) e2k +Φ

cosh

l=0 l even

β 2 ek

' (6.74)

¯ k is not necessarily the complex conjugate of Φk if some of the Observe that Φ λl are negative. b) Let d = 2 and suppose λl0 > 0 is attractive and λl0 > λl for all l = l0 . Then lim 1 d ψ¯kσ ψkσ L→∞ βL

ik0 +ek = − k2 +e 2 +|∆ l 0

k

0

where |∆l0 |2 is a solution of the BCS equation
tanh( β √e2 +|∆l |2 ) λl0 0 √2 2 k =1 2 Ld k |ek |≤ωD

2

ek +|∆l0 |

|2

(6.75)

(6.76)

94 c) Let d = 3 and suppose that the electron-electron interaction in (6.66) is given by a single, even  term, 


ˆ−p ˆ ) = λ V (k

ˆ m (ˆ Y¯m (k)Y p)

(6.77)

m=−

Then, if eRk = ek for all R ∈ SO(3), one has  ik0 +ek 1 ¯ lim βLd ψkσ ψkσ β,L = k2 +e2 +λ ρ2 |Σm α0 L→∞

S2

0

k

m Ym (p)|

0

2

dΩ(p) 4π

(6.78)

where ρ0 ≥ 0 and α0 ∈ C2+1 , Σm |α0m |2 = 1, are values at the global minimum (which is degenerate) of & '2  √2 ˆ 2) cosh( β ek +λ ρ2 |Σm αm Ym (k)| 2 dd k 1 2 W (ρ, α) = ρ − log (6.79) (2π)d β cosh β e 2

M

k

In particular, the momentum distribution is given by    √2 ek +|∆(p)|2 ) tanh( β dΩ(p) + 1 1 2 √ lim Ld akσ akσ β,L = 2 2 1 − ek 4π 2 L→∞

ek +|∆(p)|

S2

(6.80)

1

and has SO(3) symmetry. Here ∆(p) = λ2 ρ0 Σm α0m Ym (p). Proof: a) The Grassmann integral representation of Z = T r e−βH /T r e−βH0 , H given by (6.70), is given by Z = Z(skσ = 0) where (recall that κ := βLd )  
1
¯ ˆ l (ˆ Z({skσ }) = exp − 3 λl y¯l (k)y p) × ψk0 ,k,↑ ψ¯q0 −k0 ,−k,↓ κ k,p,q0 l even + ψp0 ,p,↑ ψq0 −p0 ,−p,↓ × P

¯ ψ −1 (ik −e −s )ψ Π ik κ−ek e κ kσ 0 k kσ kσ kσ Π dψkσ dψ¯kσ kσ 0 kσ  


√ 1 ˆ ψ¯k ,k,↑ ψ¯q −k ,−k,↓ × = exp − 3 λl y¯l (k) 0 0 0 κ l even q0 k0 ,k + √
λl yl (ˆ p)ψp0 ,p,↑ ψq0 −p0 ,−p,↓ × p0 ,p

Π

kσ

κ ik0 −ek

e− κ 1

P

¯

kσ (ik0 −ek −skσ )ψkσ ψkσ

Π dψkσ dψ¯kσ (6.81)

kσ

Here k = (k0 , k), k0 , p0 ∈ πβ (2Z + 1), q0 ∈ 2π β Z. By making the HubbardStratonovich transformation  P P P 2 ¯ ¯ dφ dφ − l,q al,q0 bl,q0 0 e = ei l,q0 (al,q0 φl,q0 +bl,q0 φl,q0 ) e− l,q0 |φl,q0 | Π l,q0π l,q0 l,q0

BCS Theory and Spontaneous Symmetry Breaking

95

with al,q0 := bl,q0 :=

 λ 12  l κ3

p)ψp0 ,p,↑ ψq0 −p0 ,−p,↓ p0 ,p yl (ˆ

κ3

¯l (k)ψk0 ,k,↑ ψq0 −k0 ,−k,↓ k0 ,k y

 λ 12  l

,

¯

ˆ ¯

we arrive at  Z({skσ }) =

det det

! P 2 ··· ! e− l,q0 |φl,q0 | Π l,q0 ·

¯l,q dφl,q0 dφ 0 π

(6.82)

where the quotient of determinants is given by (

)  igl ¯ ˆ p,k √ (ak − sk↑ )δk,p ¯l (k)δ l κ φl,p0 −k0 y det  igl ˆ √ (a−k − s−k↓ )δk,p l κ φl,k0 −p0 y (k)δk,p & ' = (6.83) ak δk,p 0 det 0 a−k δk,p ( ) ˆ ¯ √i Φ (ak0 ,k − sk0 ,k,↑ )δk0 ,p0 (k) κ p0 −k0 det ˆ √i Φk −p (k)  (a−k0 ,−k − s−k0 ,−k,↓ )δk0 ,p0 0 0 κ & ' a δ 0 k ,k k ,p 0 0 0 k det 0 a−k0 ,−k δk0 ,p0 and we abbreviated ˆ := Φq0 (k)




ˆ , gl φl,q0 yl (k)

ˆ := ¯ q0 (k) Φ

l




ˆ gl φ¯l,q0 y¯l (k)

(6.84)

l

Observe that in the first line of (6.83) the matrices are labelled by k, p = (k0 , k), (p0 , p) whereas in the second line of (6.83) the matrices are only labelled by k0 , p0 and there is a product over spatial momenta k since the matrices in the first line of (6.83) are diagonal in the spatial momenta k, p. As in Theorem 5.1.2, the correlation functions are obtained from (6.82) by differentiating with respect to stσ :  ( 1 κ

ψ¯tσ ψtσ =

¯ p0 −k0 (t) ak0 ,t δk0 ,p0 √iβ Φ √i Φk −p (t) a−k ,−t δk ,p 0 0 0 0 0 β

where, making the substitution of variables

)−1 dP ({φl,q0 }) (6.85) t0 σ,t0 σ

√1 φl,q 0 Ld

→ φl,q0 ,

d e−L V ({φl,q0 }) Πl,q dφl,q0 dφ¯l,q0 dP ({φl,q0 }) = * −Ld V ({φ }) 0 l,q0 e Πl,q0 dφl,q0 dφ¯l,q0

(6.86)

96 with an effective potential (

¯ p0 −k0 (k) ak0 ,k δk0 ,p0 √iβ Φ det √i Φ (k) a−k0 ,−k δk0 ,p0
1
β k0 −p0 & ' |φl,q0 |2 − d log V ({φl,q0 }) = L ak0 ,k δk0 ,p0 0 l,q0 k det 0 a−k0 ,−k δk0 ,p0

)

(6.87) * dd k  If we substitute the Riemannian sum L1d k in (6.87) by an integral (2π) d, then the only place where the volume shows up in the above formulae is the prefactor in the exponent of (6.86). Thus, in the infinite volume limit we can evaluate the integral by evaluating the integrand at the global minimum of the effective potential, or, if this is not unique, by averaging the integrand over all minimum configurations. Since the global minimum of (6.87) is at Φq0 (k) = 0 for q0 = 0 (compare the proof of Theorem 6.2.1) we arrive at (Φq0 =0 (k) ≡ Φ(k))  ( 1 κ

ψ¯t↑ ψt↑ =  =

¯ ak0 ,t δk0 ,p0 √iβ Φ(t)δ k0 ,p0 i √ Φ(t)δk ,p a−k ,−t δk ,p 0 0 0 0 0 β

|at

|2

)−1 dP ({φl }) t0 ↑,t0 ↑

a−t dP ({φl }) ¯ + Φ(t)Φ(t)

(6.88)

where d e−βL V ({φl }) Πl dφl dφ¯l dP ({φl }) = * −Ld V ({φ }) l e Πl dφl dφ¯l

(6.89)

with an effective potential V ({φl }) =


l

|φl |2 −

1 βLd




 log

¯ |ak |2 +Φ(k)Φ(k) |ak |2

 (6.90)

k0 ,k

which coincides with (6.72). b) This part is due to A. Sch¨ utte [58]. We have to prove that the global minimum of the real part of the effective potential (6.74) is given by φl = 0 for l = l0 . To this end we first prove that for e ∈ R and z, w ∈ C , ,2   , , z + iw) ¯ , ≤ cosh2 e2 + |z|2 ,cosh e2 + (z + iw)(¯ and , ,2   , , z + iw) ¯ , = cosh2 e2 + |z|2 ,cosh e2 + (z + iw)(¯

⇔

w=0

(6.91)

(6.92)

BCS Theory and Spontaneous Symmetry Breaking

97

√

a2 + b2 , a + ib = r eiϕ one has   √ - 1+cos ϕ √ √  1−cos ϕ a + ib = r cos ϕ2 + i sin ϕ2 = r ± i 2 2 r−a = r+a 2 ±i 2

Namely, for real a and b, r =

and | cosh(a + ib)|2 = cosh2 a − sin2 b which gives ,2 ,  , , z + iw) ¯ , = ,cosh e2 + (z + iw)(¯ 2 2 2 cosh2 R2 + e +|z|2 −|w| − sin2 R 2 −

e2 +|z|2 −|w|2 2

where R=

 (e2 + |z|2 − |w|2 )2 + 4(Re w¯ z )2

Since e2 +|z|2 −|w|2 2

≤ e2 + |z|2  ⇔ (e2 + |z|2 − |w|2 )2 + 4(Re w¯ z )2 ≤ e2 + |z|2 + |w|2 ⇔ (e2 + |z|2 − |w|2 )2 + 4(Re w¯ z )2 ≤ (e2 + |z|2 + |w|2 )2 R 2

+

⇔

(Re w¯ z )2 ≤ (e2 + |z|2 )|w|2

and because of (Re w¯ z )2 ≤ |w|2 |z|2 (6.91) and (6.92) follow. Now suppose that {λl } = {λ } ∪ {λm } where the λ are attractive couplings, λ > 0, and the λm are repulsive couplings, λm ≤ 0. Then ¯ k = (zk + iwk )(¯ Φk Φ zk + iw ¯k )  √  √ ¯k are where zk =  λ φ y (k), wk = m −λm φm ym (k) and z¯k and w the complex conjugate of zk and wk . Thus, using (6.91) and (6.92) , ,  
, cosh( β2 √e2k +Φ¯ k Φk ) ,2  2 d2 k 1 , , min Re VBCS ({φl }) = min |φl | − (2π)2 β log , , cosh β e φl

φl

= min φ




 |φ |2 −



l d2 k 1 (2π)2 β

2

, , , cosh( β2 √e2k +|zk |2 ) ,2  , , log , , cosh β 2 ek

Using polar coordinates k = k(cos α, sin α), φl = ρl eiϕl , we have
 g g ρ ρ ei[ϕ −ϕ +(− )α] |zk |2 = ,

k

(6.93)

98 , , , cosh( β2 √e2k +x) ,2 , is a concave , |zk | = Since W (x) := log , cosh β e and 0 , 2 k function, we can use Jensen’s inequality to obtain * 2π



dα 2π

2

d2 k 1 (2π)2 β



1 β

2  λ ρ .

, ,  , cosh( β2 √e2k +|zk |2 ) ,2 , log ,, ≤ , cosh β e 2

k

dk k 1 2π β

 dk k 1 2π β

=

, , cosh( β √e2 +R 2π k 2 0 log ,, cosh β e 2

dα 2 2π |zk | )

k

, , , cosh( β2 √e2k +P λ ρ2 ) ,2 , , log , , cosh β 2 ek

,2 , , ,

and  
2 min Re VBCS ({φl }) = min ρ − ρ

φl

dk k 1 2π β



, , , cosh( β2 √e2k +P λ ρ2 ) ,2  , , log , , cosh β 2 ek

By assumption, we had λl0  ≡ λ0 > λ for  all  = 0 . Thus, if there is some ρ > 0 for  = 0 , we have  λ ρ2 < λ0  ρ2 and, since W (x) is strictly monotone increasing,  dk k 1 2π β

, , , cosh( β2 √e2k +P λ ρ2 ) ,2 * , < , log , , cosh β 2 ek

dk k 1 2π β

, , , cosh( β2 √e2k +λ0 P ρ2 ) ,2 , , log , , cosh β 2 ek

Hence, we arrive at   min Re VBCS ({φl }) = min ρ20 − ρ0

φl

dk k 1 2π β

, , , cosh( β qe2 +λ ρ2 ) ,2  k 2 0 0 , , (6.94) log , , cosh β , , 2 ek

which proves part (b). c) We have to compute the infinite volume limit of * a−p −κV (φ)  Πm=− dum dvm R4+2 ap a−p +|Φp |2 e 1 ¯ * κ ψpσ ψpσ = −  e−κV (φ) Πm=− dum dvm R4+2 where V (φ) =


m=−

&

 |φm | − 2

M

dd k 2 (2π)d β

log

cosh( β 2

√

e2k +|Φk |2 )

cosh

β 2 ek

and (vm → −vm ) 1

Φk = λ2




ˆ φ¯m Ym (k)

m=−

Let U (R) be the unitary representation of SO(3) given by
ˆ = ˆ Ym (Rk) U (R)mm Ym (k) m

'

(6.95)

BCS Theory and Spontaneous Symmetry Breaking  ˆ =: (U Φ)k . and let m (U φ)m Ym (k)  U (R)Φ k = ΦR−1 k and  
2 V U (R)φ = |[U (R)φ]m | − m

=




 |φm |2 − M

m

=




M

 |φm |2 −

M

m

99

Then for all R ∈ SO(3) one has

dd k 2 (2π)d β

&

dd k 2 (2π)d β

log

dd k 2 (2π)d β

log

& log

cosh( β 2

cosh( β 2

cosh( β 2

e2k +|[UΦ]k |2 )

cosh

β 2 ek

√

e2k +|ΦR−1 k |2 )

cosh

&

√

β 2 ek

√

e2k +|Φk |2 )

cosh

'

'

'

β 2 ek

= V (φ)

(6.96)

 Let S 4+1 = {φ ∈ C2+1 | m |φm |2 = 1}. Since U (R) leaves S 4+1 invariant, S 4+1 can be written as the union of disjoint orbits, S 4+1 = ∪[α]∈O [α] where [α] = {U (R)α|R ∈ SO(3)} is the orbit of α ∈ S 4+1 under the action of U (R) and O is the set of all orbits. If one chooses a fixed representative α in each orbit [α], that is, if one chooses a fixed section σ : O → S 4+1 , [α] → σ[α] with [σ[α] ] = [α], every φ ∈ C2+1 can be uniquely written as ρ = φ ≥ 0, α =

φ = ρ U (R)σ[α] ,

φ
φ
,

[α] ∈ O and R ∈ SO(3)/I[α]

σ where I[α] = I[α] = {S ∈ SO(3) | U (S)σ[α] = σ[α] } is the isotropy subgroup of σ[α] . Let

* R4+2

Πm dum dvm f (φ) =

* R+

Dρ

* O

D[α]

* [α]

 DR f ρ U (R)σ[α]

be the integral in (6.95) over R4+2 in the new coordinates. That is, for example, Dρ = ρ4+1 dρ. In the new coordinates ,
 ,
,2  ,,2  , , , ˆ , U (R)σ[α] Ym (ˆ p), = λ ρ2 , σ ¯[α],m Ym R−1 p |Φp |2 = λ ρ2 , m

m

m

such that −a−p ap a−p +|Φp |2

=

−a−p ˆ )|2 ap a−p +λ ρ2 |Σm σ ¯[α],m Ym (R−1 p

≡ f (ρ, [α], R−1 p)

100 Since V (φ) = V (ρ, [α]) is independent of R, one obtains * 1 ¯ κ ψpσ ψpσ

=



−a−p ap a−p +|Φp |2

e−κV (φ) Π dum dvm m=−

(6.97)  e−κV (φ) Π dum dvm m=− * * * Dρ O D[α] [α] DR f (ρ, [α], R−1 p) e−κV (ρ,[α]) R+ * * * = −κV (ρ,[α]) R+ Dρ O D[α] [α] DR e R * * DRf (ρ,[α],R−1 p) −κV (ρ,[α]) [α] R e R+ Dρ O D[α] vol([α]) DR [α] * * = Dρ O D[α] vol([α]) e−κV (ρ,[α]) R+ *

It is plausible to assume that at the global minimum of V (ρ, [α]) ρ is uniquely determined, say ρ0 . Let Omin ⊂ O be the set of all orbits at which V (ρ0 , [α]) takes its global minimum. Then in the infinite volume limit (6.97) becomes R

lim 1 ψ¯pσ ψpσ κ→∞ κ

=

Omin

R

DR f (ρ,[α],R−1 p)

R D[α] vol([α]) [α] [α] DR R D[α] vol([α]) O

(6.98)

min

Consider the quotient of integrals in the numerator of (6.98). Since  ,  ,2  ˆ , f (ρ, [α], R−1 p) = f ρ2 ,Σm σ ¯[α],m Ym R−1 p  ,   ,2  = f ρ2 ,Σm U (S)σ[α] m Ym R−1 p ,  , ,2   ˆ , = f (ρ, [α], (RS)−1 p) ¯[α],m Ym (RS)−1 p = f ρ2 ,Σm σ for all S ∈ I[α] , one has, since [α]  SO(3)/I[α] * * * −1 −1 p) I[α] DS p) SO(3)/I[α] DR f (ρ, [α], R [α] DR f (ρ, [α], R * * * = DR DR I[α] DS [α] SO(3)/I[α] * * −1 p) SO(3)/I[α] DR I[α] DS f (ρ, [α], (RS) * * = DR DS SO(3)/I[α] I[α] * −1 p) SO(3) DR f (ρ, [α], R * = SO(3) DR * * −1 p) S 2 dΩ(t) SO(3)t→p DR f (ρ, [α], R * * = S 2 dΩ(t) SO(3)t→p DR * * S 2 dΩ(t) f (ρ, [α], t) SO(3)t→p DR * * = (6.99) S 2 dΩ(t) SO(3)t→p DR that DR has the where SO(3)t→p = {R ∈ SO(3) | Rt = p}. If one assumes * usual invariance properties of the Haar measure, then SO(3)t→p DR does not

BCS Theory and Spontaneous Symmetry Breaking

101

depend on t such that it cancels out in (6.99). Then (6.98) gives R

*

dΩ(t) f (ρ,[α],t)

D[α] vol([α]) S2 R 2 dΩ(t) Omin S * lim κ1 ψ¯pσ ψpσ = κ→∞ D[α] vol([α]) Omin

(6.100)

Now, since the effective potential, which is constant on Omin , may be written as   , ,2  dΩ(t) 2, V (ρ, [α]) = ¯[α],m Ym (t), 4π G ρ Σm σ S2

with G(X) = ρ − 2

also

* S2

*

&

dk k2 2π 2

log

cosh( β 2

dΩ(t) f (ρ, [α], t) * = S 2 dΩ(t)

√

e2k +λ X)

cosh

β 2 ek

' , it is plausible to assume that

 S2

ip0 +ep dΩ(t) 4π p20 +e2p +λ ρ2 |Σm σ ¯[α],m Ym (t)|2

is constant on Omin . In that case also the integrals over Omin in (6.100) cancel out and the theorem is proven.


We now compare the results of the above theorem to the predictions of the quadratic Anderson-Brinkman Balian-Werthamer [3, 5] mean field theory. This formalism gives  √2 ∗    tanh( β ek +∆k ∆k ) + 1 1 2 √ lim Ld akσ akσ = 2 1 − ek (6.101) 2 ∗ ek +∆k ∆k

L→∞

σσ

where the 2 × 2 matrix ∆k , ∆Tk = −∆−k , is a solution of the gap equation  √2 ∗ ek +∆k ∆k ) tanh( β dd k 2 √ ∆p = U (p − k) ∆ (6.102) k d 2 ∗ (2π) Mω

2

ek +∆k ∆k

Consider first the case d = 2. The electron-electron interaction in (6.102) is given by  U (p − k) = l λl eil(ϕp −ϕk ) (6.103) Usually it is argued that the interaction flows to an effective interaction which is dominated by a single attractive angular momentum sector λl0 > 0. In other words, one approximates U (p − k) ≈ λl0 eil0 (ϕp −ϕk )

(6.104)

where l0 is chosen as in the theorem above. The proof of part (b) of the theorem gives a rigorous justification for that, the global minimum of the effective potential for an interaction of the form (6.103) is identical to the global minimum of the effective potential for the interaction (6.104) if λl0 > λl

102 for all l = l0 . Once U (k − p) is approximated by (6.104), one can solve the equation (6.102). In two dimensions there are the unitary isotropic solutions   cos lϕk sin lϕk ∆(k) = d (6.105) sin lϕk − cos lϕk for odd l and

 ∆(k) = d

0 eilϕk −eilϕk 0

 (6.106)

for even l which gives ∆(k)+ ∆(k) = |d|2 Id such that (6.101) is consistent with (6.75). In three dimensions, for an interaction (6.77), it has been proven [21] that for all  ≥ 2 (6.102) does not have unitary isotropic (∆∗k ∆k = const Id) solutions. That is, the gap in (6.102) is angle dependent but part (c) of the above theorem states that a+ kσ akσ has SO(3) symmetry. For SO(3) symmetric ek also the effective potential has SO(3) symmetry which means that also the global minimum has SO(3) symmetry. Since in the infinite volume limit the integration variables are forced to take values at the global minimum, the integral over the sphere in (6.80) is the averaging over all global minima. However, it may very well be that in the physically relevant case there is SO(3) symmetry breaking. That is, instead of the Hamiltonian H of (6.70) one should consider a Hamiltonian H + HB where HB is an SO(3)-symmetry breaking term which vanishes if the external parameter B (say, a magnetic field) goes to zero. Then one has to compute the correlations lim lim 1d a+ kσ akσ B→0 L→∞ L

(6.107)

which are likely to have no SO(3) symmetry. However, the question to what extent the results of the quadratic mean field formalism for this model relate to the exact result for the quartic Hamiltonian H + HB (in the limit B → 0) needs some further investigation. Several authors [10, 12, 7, 36] have investigated the relation between the reduced quartic BCS Hamiltonian

+ HBCS = H0 + L13d U (k − p) a+ (6.108) kσ a−kτ apσ a−pτ σ,τ ∈{↑,↓} k,p

and the quadratic mean field Hamiltonian 

+ HMF = H0 + L13d U (k − p) a+ kσ a−kτ apσ a−pτ σ,τ ∈{↑,↓} k,p

(6.109)

 + + + + a+ kσ a−kτ apσ a−pτ − akσ a−kτ apσ a−pτ

BCS Theory and Spontaneous Symmetry Breaking

103

where the numbers apσ a−pτ are to be determined according to the relation apσ a−pτ = T r e−βHMF apσ a−pτ T r e−βHMF . The idea is that H  = HBCS − HMF  

+ + + = L13d U (k − p) a+ kσ a−kτ − akσ a−kτ × σ,τ ∈{↑,↓} k,p

(6.110)

  apσ a−pτ − apσ a−pτ

is only a small perturbation which should vanish in the infinite volume limit. It is argued that, in the infinite volume limit, the correlation functions of the models (6.108) and (6.110) should coincide. More precisely, it is claimed that 1 d L→∞ L

lim

−βHBCS

log TTrree−βHMF

(6.111)

vanishes. To this end it is argued that each order of perturbation theory, with  respect to H  , of T r e−β(HMF +H ) /T r e−βHMF is finite as the volume goes to infinity. The Haag paper argues that spatial averages of field operators * like L1d [0,L]d dd xψ↑+ (x)ψ↓+ (x) may be substituted by numbers in the infinite volume limit, but there is no rigorous control of the error. However, in part (c) of Theorem 6.3.1 we have shown for the model (6.70), that the correlation functions of the quartic model do not necessarily have to coincide with those of the quadratic mean field model. Thus, in view of Theorem 6.3.1, it is questionable whether the above reasoning is actually correct.

Chapter 7 The Many-Electron System in a Magnetic Field

This chapter provides a nice application of the formalism of second quantization to the fractional quantum Hall effect. Throughout this chapter we will stay in the operator formalism and make no use of functional integrals. Whereas in the previous chapters we worked in the grand canonical ensemble, the fractional quantum Hall effect is usually discussed in the canonical ensemble which is mainly due to the huge degeneracy of the noninteracting system. For filling factors less than one, a suitable approximation is to consider only the contributions in the lowest Landau level. This projection of the many-body Hamiltonian onto the lowest Landau level can be done very easily with the help of annihilation and creation operators. The model which will be discussed in this chapter contains a certain long range approximation. This approximation makes the model explicitly solvable but, as it is argued below, this approximation also has to be considered as unphysical. Nevertheless, we think it is worthwhile considering this model because, besides providing a nice illustration of the formalism, it has an, in finite volume, explicitly given eigenvalue spectrum, which, in the infinite volume limit, most likely has a gap for rational fillings and no one for irrational filling factors. This is interesting since a similar behavior one would like to prove for the unapproximated fractional quantum Hall Hamiltonian.

7.1

Solution of the Single Body Problem

In this section we solve the single body problem for one electron in a constant magnetic field in two dimensions. We consider a disc geometry and a rectangular geometry. The many-body problem is considered in the next section in a finite volume rectangular geometry.

105

106

7.1.1

Disk Geometry

We start by considering one electron in two dimensions in a not necessarily constant but radial symmetric magnetic field
= (0, 0, B(r)) B Let H(r) =

1 r2


r 0

(7.1)

B(s) s ds

(7.2)

then the vector potential A(x, y) = H(r) (−y, x) = rH(r) eϕ

(7.3)

rotA = (0, 0, B(r))

(7.4)

satisfies

and the Hamiltonian is given by  2 H =
i ∇ − eA   ∂ =
2 −∆ + 2ih(r) ∂ϕ + r2 h(r)2 ≡
2 K

(7.5)

if we define h(r) =
e H(r)

(7.6)

∂ K = −∆ + 2ih(r) ∂ϕ + r2 h(r)2

(7.7)

and

As for the harmonic oscillator, we can solve the eigenvalue problem by introducing annihilation and creation operators. To this end we introduce complex variables

∂ ∂z

=

such that ∂ ∂z z

=

1 2



1 2



∂x ∂x

z = x + iy,  ∂ ∂ ∂x − i ∂y ,

− i2 ∂y ∂y

 =1=

z¯ = x − iy   ∂ 1 ∂ ∂ ∂ z¯ = 2 ∂x + i ∂y

∂ ¯, ∂ z¯ z

∂ ¯ ∂z z

=0=

∂ ∂ z¯ z

Now define the following operators a = a(x, y, h) and b = b(x, y, h): √ √ ∂ a = 2 ∂∂z¯ + √12 h z, a+ = − 2 ∂z + √12 h z¯ √ √ ∂ b+ = 2 ∂∂z¯ − √12 h z b = − 2 ∂z − √12 h z¯, Note that b(h) = a+ (−h) and b+ (h) = a(−h). There is the following

(7.8) (7.9)

The Many-Electron System in a Magnetic Field

107

Lemma 7.1.1 a) There are the following commutators: [a, a+ ] =
e B(r) = [b, b+ ]

(7.10)

and all other commutators [a(+) , b(+) ] = 0. b) The Hamiltonian is given by H =
2 K where K = a+ a + aa+ = 2a+ a +
e B(r) Proof: We have   ∂ [a, a+ ] = ∂∂z¯ , h¯ = z − hz, ∂z

∂(h¯ z) ∂ z¯

+

∂(hz) ∂z

(7.11)

∂h = h + z¯ ∂h ∂ z¯ + h + z ∂z

Since h = h(r) depends only on r, one obtains with ρ = r2 = z z¯ ∂(z z¯) dh dh z ∂h ¯ ∂h ∂z = z ∂z dρ = ρ dρ = z ∂ z¯

which results in

Using H(r) =

1 r2

(7.12)

    e dH [a, a+ ] = 2 h + ρ dh dρ = 2
H + ρ dρ

(7.13)


r

B(s) s ds, one finds 
r 2 H + ρ dH − r14 0 B(s)s ds + dρ = H + r 0

1 = = B(r)r 2r

1 dr r 2 B(r)r d(r 2 )



B(r) 2

(7.14)

Thus one ends up with [a, a+ ] =
e B(r) = [b, b+ ] Furthermore [b+ , a+ ] = [a(−h), a+ (h)] = =









∂ ∂ z + hz, ∂z ∂ z¯ , h¯ ∂(h¯ z) ∂(hz) ¯ ∂h ∂ z¯ − ∂z = z ∂ z¯

− z ∂h ∂z = 0

because of (7.12). This proves part (a). To obtain part (b), observe that  2  ∂ ∂ 1 ∂ ∂2 = + (7.15) = 14 ∆ 2 2 ∂z ∂ z¯ 4 ∂x ∂y and ∂ ∂ϕ

= = =

∂y ∂ ∂x ∂ ∂ ∂ ∂ϕ ∂x + ∂ϕ ∂y = −y ∂x + x ∂y   1 ∂ − 2i (z − z¯) ∂z + ∂∂z¯ + 12 (z +  ∂  i z ∂z − z¯ ∂∂z¯

z¯) 1i



∂ ∂ z¯

−

∂ ∂z

 (7.16)

108 which gives ∂ + r2 h(r)2 K = −∆ + 2ih(r) ∂ϕ   ∂ ∂ ∂ = −4 ∂z ¯ ∂∂z¯ + h2 z z¯ ∂ z¯ − 2h z ∂z − z

(7.17)

Since ∂ ∂ a+ a = −2 ∂z ∂ z¯ −

= =

∂ z ∂∂z¯ + 12 h2 z z¯ ∂z hz + h¯   ∂ ∂ ∂ −2 ∂z ¯ ∂∂z¯ + 12 h2 z z¯ − ∂(hz) ∂ z¯ − h z ∂z − z ∂z 1 e B(r) 2K −
2

the lemma follows.
We now focus on the case of a constant magnetic field B(r) = B = const

(7.18)

eB 2


(7.19)

In that case h(r) = where

=

B =

1 22B


eB

(7.20)

denotes the magnetic length. We change variables w=

√1 z 2 B

(7.21)

∂ c+ = B a+ = − ∂w + 12 w ¯

(7.22)

− 12 w

(7.23)

and introduce the rescaled operators c = B a =

∂ ∂w ¯

+ 12 w,

∂ − 12 w, ¯ d = B b = − ∂w

d+ = B b+ =

∂ ∂w ¯

which fulfill the commutation relations [c, c+ ] = [d, d+ ] = 1,

[c(+) , d(+) ] = 0

Then the Hamiltonian becomes 2 
 +  1 1 =
eB 2m i ∇ − eA m c c+ 2

(7.24)

(7.25)

and the orthonormalized eigenfunctions are given by ¯ = φnm (w, w)

√ 1 n!m!

n

m

(c+ ) (d+ ) φ00

(7.26)

The Many-Electron System in a Magnetic Field

109

where 2

e−

1

φ00 (w, w) ¯ =

(2π2B )1/2

ww ¯ 2

=

1 (2π2B )1/2

e

− |z|2 4


(7.27)

B

The eigenfunctions can be expressed in terms of the generalized Laguerre polynomials Lα n . One obtains φnm (w, w) ¯ = =

1 (2π2B )1/2 1 (2π2B )1/2

 n!  12

wm−n Lm−n (ww) ¯ e− n

m!

 n!  12  m!

√z 2 B

m−n

Lm−n n

ww ¯ 2



|z|2 22B



2

e

− |z|2 4


B

(7.28)

We summarize some properties of the eigenfunctions in the following lemma. By a slight abuse of notation, we write φnm (z) = φnm (z, z¯) in the lemma z z¯ . below instead of φnm √2 , √2 B

B

Lemma 7.1.2 Let φnm (z) be given by the second line of (7.28). Then (i) φnm (z) = (−1)n−m φmn (z)

(7.29)

(ii) φˆnm (k = k1 + ik2 ) =


R2

dx dx e−i(k1 x+k2 y) φnm (x + iy)

= 4π2B (−1)n φnm (−2i2B k)

(7.30)

(iii) ∞ 

φn1 m (z1 )φn2 m (z2 ) =

m=0

1 (2π2B )1/2

φn1 n2 (z1 − z2 ) e

i

Im(z1 z ¯2 ) 2
2 B

(7.31)

Proof: Part (i) is obtained by using the relation L−l n (x) =

(n−l)! l l n! (−x) Ln−l (x)

for l = n − m. To obtain the second part, one can use the integral ∞  2  y2 n ax2 y l l e− 2a , a > 0 xl+1 Lln (ax2 ) e− 2 Jl (xy) dx = (−1) y L l+1 n a a 0

110 where Jl is the l’th Bessel function. One may look in [43] for the details. The third part can be proven by using a generating function for the Laguerre polynomials, ∞ 

tk Lm−k (x)Lk−n (y) = tn (1 + t)m−n e−tx Lm−n n n k

 1+t t

 (tx + y) ,

|t| < 1

k=0

but probably it is more instructive to give a proof in terms of annihilation and creation operators. To this end we reintroduce the rescaled variables √ w = z/( 2B . Then we have to show ∞ 

φn1 m (w1 )φn2 m (w2 ) =

m=0

1 (2π2B )1/2

φn1 n2 (w1 − w2 ) eiIm(w1 w¯2 )

Since φ¯jm = (−1)j−m φmj we have to compute (c+ 1 denotes the creation operator with respect to the w1 variable) (−1)j j (c+ )n (d+ 2) (n!j!)1/2 1

∞ 

+ m (−d+ 1 c2 ) φ00 (w1 )φ00 (w2 ) m!

m=0

=

j

(−1) j (c+ )n (d+ 2) (n!j!)1/2 1

∞ 

¯ 2 )m (w1 w φ00 (w1 )φ00 (w2 ) m!

m=0

=

(−1)j j w1 w ¯2 1 (c+ )n (d+ 2) e 2π2B (n!j!)1/2 1

=

(−1)j 1 j 12 (w1 w ¯ 2 −w ¯ 1 w2 ) (c+ )n (d+ φ00 (w1 2) e (2π2B )1/2 (n!j!)1/2 1

e− 2 (w1 w¯1 +w2 w¯2 ) 1

− w2 )

Now, because of A

e Be

−A

=

∞ 

1 k!

  A, A, · · · [A, B] · · ·

k=0

and

 1 + w1 ∂ = (w w ¯ − w ¯ w ), d (w w ¯ − w ¯ w ), 1 2 1 2 1 2 1 2 2 2 2 ∂w ¯2 = − 2

1 one gets

¯ 2 −w ¯1 w2 ) 2 (w1 w e− 2 (w1 w¯2 −w¯1 w2 ) d+ = d+ 2e 2 − 1

1

=

∂ ∂w ¯2

1

−

and j 2 (w1 w ¯ 2 −w ¯1 w2 ) j e− 2 (w1 w¯2 −w¯1 w2 ) (d+ = (−d+ w1 −w2 ) 2) e 1

1



¯2 − w ¯1 w2 ), d+ 2 2 (w1 w + 1 1 2 w2 + 2 w1 = −dw1 −w2

The Many-Electron System in a Magnetic Field

111

Similarly one finds n 2 (w1 w ¯ 2 −w ¯1 w2 ) n = (c+ e− 2 (w1 w¯2 −w¯1 w2 ) (c+ w1 −w2 ) 1) e 1

1

and we end up with 1 + 1 1 n + j e 2 (w1 w¯2 −w¯1 w2 ) (n!j!) 1/2 (cw1 −w2 ) (dw1 −w2 ) φ00 (w1 (2π2B )1/2

=

1 (2π2B )1/2

− w2 )

eiIm(w1 w¯2 ) φnj (w1 − w2 )

which proves the third part of the lemma.


7.1.2

Rectangular Geometry

In this section we solve the eigenvalue problem for a single electron in a constant magnetic field in two dimensions for a rectangular geometry. We start with the half infinite case. That is, coordinate space is given by [0, Lx ] × R. The following gauge is suitable A(x, y) = (−By, 0, 0)

(7.32)

Then H=




i∇

2 + eA

 ∂ =
2 −∆ + 2i eB
y ∂x +

e2 B 2 2
2 y



≡
2 K

(7.33)

where, recalling that B = {
/(eB)}1/2 denotes the magnetic length, K = −∆ +

2i y∂ 2B ∂x

+

1 2 y 4B

(7.34)

We make the ansatz ψ(x, y) = eikx ϕ(y)

(7.35)

2π Imposing periodic boundary conditions on [0, Lx ] gives k = L m with m ∈ Z. x The eigenvalue problem Hψ = εψ is equivalent to   2 2ky ∂2 2 + y4 ϕ(y) = ε ϕ(y)
2 − ∂y 2 + k − 2 B B   ∂2 1 2 2 ⇔
− ∂(y−2 k)2 + 4 (y − B k)2 ϕ(y) = ε ϕ(y) B

B

This is the eigenvalue equation for the harmonic oscillator shifted by 2B k. Thus, if hn denotes the normalized Hermite function, then the normalized eigenfunctions of H read   (7.36) ψnk (x, y) = √L1  eikx hn (y − 2B k)/B x B

112 with eigenvalues 1/(2m)Hψnk = εn ψnk , εn =
eB m (n + 1/2). As in the last section, the eigenvalues have infinite degeneracy. This is due to the fact that, as in the last section, we computed in infinite volume. If one turns to finite volume, the degeneracy is reduced to a finite value. Namely, the degeneracy is equal to the number of flux quanta flowing through the sample. A flux quantum is given by
= 4, 14 · 10−11 T cm2 (7.37) e and there is flux quantization which means that the magnetic flux through a given sample has to be an integer multiple of φ0 . Thus, for finite volume [0, Lx ] × [0, Ly ] the number of flux quanta is equal to φ0 = 2π

M :=

Lx Ly ! BLx Ly Φ = = ∈N φ0 2π
/e 2π2B

(7.38)

This is the case we consider next. So let coordinate space be [0, Lx ] × [0, Ly ]. The following boundary conditions are suitable (‘magnetic boundary conditions’): 2

ψ(x, y + Ly ) = eixLy /B ψ(x, y)

ψ(x + Lx , y) = ψ(x, y) ,

(7.39)

The finite volume eigenvalue problem can be solved by periodizing the states (7.36). This in turn can be obtained by a suitable superposition. To this end observe that in the y-direction the wavefunctions (7.36) are centered at 2π 2π m by K := L M , then the center yk+K yk = 2B k. Now, if we shift k = L x x 2π is shifted by 2B L M = 2B L2πx x eigenfunctions:

Lx Ly 2π2B

= Ly . As a result, one finds the following

Lemma 7.1.3 Let H be given by (7.33) on finite volume [0, Lx ] × [0, Ly ] with the Landau gauge (7.34) and with magnetic boundary conditions (7.37). Then a complete orthonormal set of eigenfunctions of H is given by ψn,k (x, y) =

√ 1 L x B

∞ 

ei(k+jK)x hn,k (y − jLy )

(7.40)

j=−∞

n = 0, 1, 2, ... where K :=

2π Lx M

=

Ly 2B

k=

2π Lx m,

m = 1, 2, ..., M

  y2 and hn,k (y) = hn (y−2B k)/B , hn (y) = cn Hn (y) e− 2

the normalized Hermite function, cn = π − 4 (2n n!)− 2 . The eigenvalues are given by   1 1 (7.41) εn =
eB 2m Hψn,k = εn ψn,k , m n+ 2 1

1

For the details of the computation one may look, for example, in [42].

The Many-Electron System in a Magnetic Field

7.2

113

Diagonalization of the Fractional Quantum Hall Hamiltonian in a Long Range Limit

Having discussed the single body problem in the previous two sections, we now turn to the N -body problem. We consider the many-electron system in two dimensions in a finite, rectangular volume [0, Lx ] × [0, Ly ] in a constant
= (0, 0, B). The noninteracting Hamiltonian is magnetic field B H0,N =

N  


i ∇i

2 − eA(xi )

(7.42)

i=1

with A given by the Landau gauge (7.32). Apparently the eigenstates of (7.42) are given by wedge products of single body eigenfunctions (7.40), Ψn1 k1 ,··· ,nN kN = ψn1 k1 ∧ · · · ∧ ψnN kN

(7.43)

with eigenvalues En1 ···nN = εn1 +· · ·+εnN . Recall that N denotes the number of electrons and M (see (7.38) and (7.40)) is the degeneracy per Landau level which is equal to the number of flux quanta flowing through the sample. The quotient ν=

N M

(7.44)

is the filling factor of the system. For integer filling, there is a unique ground state, namely the wedge product of the N = νM single body states of the lowest ν ∈ N Landau levels. This state is separated by a gap of size
eB m from the other states. The existence of this gap leads to the integer quantum Hall effect. If the filling is not an integer, the ground state of the noninteracting N -body system is highly degenerate. Suppose that ν = 1/3. Then M = 3N 2π m with m, the ground states are given by Ψ0m1 ,··· ,0mN and, identifying k = L x where m1 , ..., mN ∈ {0, 1, ...M − 1} and m1 < · · · < mN . Apparently, there are   3 N    √ (3N )! M 3 √ 3 (7.45) = 3N = (2N )! N ! ≈ 22 N N 4πN √ such choices. Here we used Stirling’s formula, n! ≈ 2πn (n/e)n to evaluate the factorials. Since 33 /22 = 6 43 , this is an extremely large number for macroscopic values of N like 1011 or 1012 which is the number of conduction electrons in Ga-As samples where the fractional quantum Hall effect is observed. Now, if the electron-electron interaction is turned on, some of these states are energetically more favorable and others less, but, due to the huge degeneracy of

114 11

about (6.75)(10 ) , one would expect a continuum of energies. However, the discovery of the fractional quantum Hall effect in 1982 demonstrated that this is not true. For certain rational values of ν with odd denominators, mainly n , p, n ∈ N, the interacting Hamiltonian ν = 2pn±1 HN =

N  


i ∇i

 2 − eA(xi ) + V (xi − xj )

i=1

(7.46)

i,j=1 i
=j

with V being the Coulomb interaction, should have a gap. Since then, a lot of work has been done on the system (7.46) (see [34, 39, 13] for an overview). Numerical data for small system size are available which give evidence for a gap. However, so far it has not been possible to give a rigorous mathematical proof that the Hamiltonian (7.46) has a gap for certain rational values of the filling factor. In this section we consider the Hamiltonian (7.46) in a certain approximation in which it becomes explicitly solvable. We will argue below that this approximation has to be considered as unphysical. However it is interesting since it gives a model with an, in finite volume, explicitly given energy spectrum which, in the infinite volume limit, most likely has a gap for ν ∈ Q and no gap for ν ∈ / Q. For that reason we think it is worth discussing that model. We consider the complete spin polarized case and neglect the Zeemann energy. We take a Gaussian as the electron-electron interaction, V (x, y) = λ e−

x2 +y2 2r2

(7.47)

Opposite to Coulomb, this has no singularity at small distances. We assume that it is long range in the sense that r >> B , B being the magnetic length. This length is of the order 10−8 m for typical FQH magnetic fields which are about B ≈ 10T . The long range condition is used to make the approximation (see (7.68, 7.70) below)



2 B

 2

ds ds hn (s) hn (s ) e− 2r2 (s−s )

≈




ds ds hn (s) hn (s )

(7.48)

s2

where hn (s) = cn Hn (s) e− 2 denotes the normalized Hermite function. With this approximation, the Hamiltonian PLL HN PLL , PLL being the projection onto the lowest Landau level, can be explicitly diagonalized. There is the following Theorem 7.2.1 Let HN be the Hamiltonian (7.46) in finite volume [0, Lx ] × [0, Ly ] with magnetic boundary conditions (7.39), let A(x, y) = (−By, 0, 0) and let the interaction be Gaussian with long range, V (x, y) = λ e−

x2 +y2 2r2

,

r >> B

(7.49)

The Many-Electron System in a Magnetic Field

115

LL be the projection onto the lowest Landau level, where Let PLL : FN → FN LL FN is the antisymmetric N -particle Fock space and FN is the antisymmetric Fock space spanned by the eigenfunctions of the lowest Landau level. Then, with the approximation (7.48), the Hamiltonian HN,LL = PLL HN PLL becomes exactly diagonalizable. Let M be the number of flux quanta flowing through [0, Lx ] × [0, Ly ] such that ν = N/M is the filling factor. Then the eigenstates and eigenvalues are labelled by N -tuples (n1 , · · · , nN ), n1 < · · · < nN and ni ∈ {1, 2, · · · , M } for all i,

HN,LL Ψn1 ···nN = (ε0 N + En1 ···nN )Ψn1 ···nN

(7.50)

where ε0 =
eB/(2m) and En1 ···nN =

N 

W (ni − nj ) ,

W (n) = λ

i,j=1 i
=j



2

e− 2r2 (Lx M −jLx ) 1

n

(7.51)

j∈Z

and the normalized eigenstates are given by Ψn1 ···nN = φn1 ∧ · · · ∧ φnN where φn (x, y) = √π

−1 4

B L y

∞ 

−

1 2
2 B

2 n Lx −sLx ) (x− M

ei(x− M Lx −sLx )y/B n

2

(7.52)

s=−∞ 1

=

e

− √1 √π 4 M B L x

∞ 

e

−

1 2
2 B

n r (y− Mr Ly )2 i(x− M Lx ) M Ly /2B e

(7.53)

r=−∞

Before we start with the proof we make some comments. The approximation (7.48) looks quite innocent. However, by reviewing the computations in the proof one finds that it is actually equivalent to the approximation 2 2 2 2 2 V (x, y) = λ e−(x +y )/(2r ) ≈ λ e−x /(2r ) . That is, if we write in (7.49) 2 2 V (x, y) = λ e−x /(2r ) then the above Theorem is an exact statement. Recall that the single body eigenfunctions are localized in the y-direction and are given by plane waves in the x-direction. Since the eigenstates are given by wedge products, we can explicitely compute the expectation value of the energy. One may speculate that for fillings ν = 1/q the ground statesare la belled by the N -tuples (n1 , · · · , nN ) = j, j +q, j +2q, · · · , j +(N −1)q which have a q-fold degeneracy, 1 ≤ j ≤ q. A q-fold degeneracy for fillings ν = p/q, p, q without common divisor, follows already from general symmetry considerations (see [42] or [61]). In particular, for ν = 1/3, one may speculate that there are three ground states labelled by (3, 6, 9, ..., 3N ), (2, 5, 8, ..., 3N − 1) and (1, 4, 7, ..., 3N −2). Below Lemma 7.2.3 we compute the expectation value of the energy of these states with respect to the Coulomb interaction. It is about as twice as big as the energy of the Laughlin wavefunction. This is not too surprising since a wedge product vanishes only linearly in the differences xi − xj whereas the Laughlin wavefunction, given by P √ 2 1 wi = zi /( 2B ) (7.54) ψ(w1 , ...wN ) = Π (wi − wj )3 e− 2 j |wj | , i<j

116 vanishes like |xi − xj |3 which results in a lower contribution of the Coulomb energy 1/|xi − xj |. Thus we have to conclude that the approximation (7.48), implemented in (7.68) and (7.70) below, has to be considered as unphysical. Nevertheless, the energy spectrum (7.51) seems to have the property that it has a gap for rational fillings in the infinite volume limit. That is, we expect   > 0 if ν ∈ Q  (7.55) ∆(ν) := lim E1 (N, M ) − E0 (N, M ) N,M →∞ = 0 if ν ∈ /Q N/M =ν E0 being the lowest and E1 the second lowest eigenvalue in finite volume. This is interesting since a similar behavior one would like to prove for the original Hamiltonian (7.46). However, it seems that the energy (7.51) does not distinguish between even and odd denominators q. That is, it looks like the approximate model does not select the observed fractional quantum Hall fillings (see [34, 39, 13] for an overview). This should be due to the unphysical nature of the approximation (7.48).

Proof of Theorem 7.2.1: We proceed in three steps: Projection onto the lowest Landau level using fermionic annihilation and creation operators, implementation of the approximation (7.48) and finally diagonalization. (i) Projection onto the Lowest Landau Level To project HN onto the lowest Landau level, we rewrite HN in terms of fermionic annihilation and creation operators   2 d2 x ψ + (x)
i ∇ − eA(x) ψ(x) HN =   (7.56) + d2 x d2 x ψ + (x)ψ + (x  )V (x − x  )ψ(x  )ψ(x)  FN

where FN is the antisymmetric N -particle Fock space. We consider the complete spin polarized case in which only one spin direction (say ψ = ψ↑ ) contributes and we neglect the Zeeman energy. Let ψ and ψ + be the fermionic annihilation and creation operators in coordinate space. In order to avoid confusion with the single body eigenfunctions ψn,k (7.40), we denote the latter ones as ϕn,k . Introducing an,k , a+ n,k according to ψ(x) = an,k =






ϕn,k (x)an,k ,

n,k

d2 x ϕ¯n,k (x)ψ(x) ,

a+ n,k =






ϕ¯n,k (x)a+ n,k

(7.57)

d2 x ϕn,k (x)ψ + (x)

(7.58)

ψ + (x) =

n,k

the an,k obey the canonical anticommutation relations {an,k , a+ n ,k } = δn,n δk,k

(7.59)

The Many-Electron System in a Magnetic Field

117

and (7.56) becomes, if H = ⊕N HN , H = Hkin + Hint

(7.60)

where Hkin =



ε n a+ n,k an,k

(7.61)

n,k

The interacting part becomes 
Hint = d2 x d2 x ψ + (x)ψ + (x  )ϕ¯n,k (x) nk|V |n k  ϕn ,k (x  )ψ(x  )ψ(x) n,k n ,k

=





(n1 l1 ; n2 l2 ; nk) nk|V |n k  ×

n,k n1 ,··· ,n4 n ,k l1 ,··· ,l4

+ (n k  ; n3 l3 ; n4 l4 ) a+ n1 ,l1 an3 ,l3 an4 ,l4 an2 ,l2

(7.62)

where we used the notation

nk|V |n k  := d2 x d2 x ϕn,k (x) V (x − x  ) ϕ¯n ,k (x  )
(n1 l1 ; n2 l2 ; nk) := d2 x ϕ¯n1 ,l1 (x) ϕn2 ,l2 (x) ϕ¯n,k (x)

(7.63) (7.64)

Now we consider systems with fillings ν =

N < 1 M

(7.65)

and restrict the electrons to the lowest Landau level. Since the kinetic energy is constant, we consider only the interacting part, HLL := PLL Hint PLL =

  n,k n ,k

(7.66)

+ (0l1 ; 0l2 ; nk) nk|V |n k  (n k  ; 0l3 ; 0l4 ) a+ l1 a l3 a l4 a l2

l1 ,··· ,l4

where we abbreviated al := a0,l ,

+ a+ l := a0,l

(7.67)

(ii) The Approximation The matrix element nk|V |n k  is computed in part (a) of Lemma 7.2.2 below. For a gaussian interaction (7.47) the exact result is (7.68) n, k|V |n , k  =
2 √

 2 r2 2 B 2πB δk,k λ r [e− 2 k ]M ds ds hn (s) hn (s ) e− 2r2 (s−s )

118 where, if k = 2πm/Lx , 2

[e

− r2 k2

]M :=

∞ 

2

e

− r2 (k−jK)2

j=−∞

=

∞ 

2

2 r 2π e− 2 [ Lx (m−jM)]

(7.69)

j=−∞

is an M -periodic function (as a function of m). For a long range interaction r >> B , we may approximate this by √

r2 2 n, k|V |n , k  ≈ 2πB δk,k λ r [e− 2 k ]M ds hn (s) ds hn (s )

=: δk,k vk ds hn (s) ds hn (s ) (7.70) Then HLL becomes HLL =   n,n k

=

(0l1 ; 0l2 ; nk) vk




hn (s)ds




+ hn (s )ds (n k; 0l3 ; 0l4 ) a+ l1 a l3 a l4 a l2

l1 ,··· ,l4

 

+ (0l1 ; 0l2 ; 1y k) vk (1y k; 0l3 ; 0l4 ) a+ l1 a l3 a l4 a l2

(7.71)

k l1 ,··· ,l4

Here we used that ∞ 

hn (y)




hn (s)ds = 1

(7.72)

n=0

which is a consequence of ∞ 

ϕ¯n,k (x, y)




∞ n=0

hn (y) hn (s) = δ(y − s). Thus

hn (s)ds

n=0

=

=

√ 1 L x B

√ 1 L x B

∞  j=−∞ ∞ 

e−i(k+jK)x

∞ 

 
hn (s)ds hn (y − yk − jLy )/B

n=0

e−i(k+jK)x

(7.73)

j=−∞

and (7.71) follows if we define (0l1 ; 0l2 ; 1y k) :=

dxdy ϕ¯0,l1 (x, y) ϕ0,l2 (x, y) √L1 

∞ 

x B

e−i(k+jK)x

j=−∞

(7.74) These matrix elements are computed in part (b) of Lemma 7.2.2 below and the result is M √ 1 (0l1 ; 0l2 ; 1y k) = δm,m [e− 2 −m1 L  x B


2 B 4

(l1 −l2 )2

]M

(7.75)

The Many-Electron System in a Magnetic Field

119

2π M m, lj = L2πx mj and δm = 1 iff m1 = m2 mod M . In the if k = L 1 ,m2 x 2π M following we write, by a slight abuse of notation, also δl,l  if l = Lx m. Then the Hamiltonian (7.71) becomes

HLL =

1 L x B

 


2 B 4

M δk,l [e− 2 −l1

(l1 −l2 )2

M ]M vk δk,l × 3 −l4

k l1 ,··· ,l4

=

[e−



1 Lx


2 B 4

(l3 −l4 )2

+ + ]M a+ l1 a l3 a l4 a l2

+ δlM wl2 −l1 a+ l1 a l3 a l4 a l2 2 −l1 ,l3 −l4

(7.76)

l1 ,··· ,l4

where the interaction is given by wk :=

√

2π λ r [e−

r2 2

k2

]M [e−


2 B 4

k2 2 ]M

(7.77)

(iii) Diagonalization Apparently (7.76) looks like a usual one-dimensional many-body Hamiltonian in momentum space. Thus, since the kinetic energy is constant, we can easily diagonalize it by taking the discrete Fourier transform. For 1 ≤ n ≤ M let ψn :=

√1 M

M 

nm

e2πi M am ,

ψn+ =

M 

√1 M

m=1

e−2πi M a+ m nm

(7.78)

m=1

or am =

√1 M

M 

e−2πi M ψn , nm

a+ m =

√1 M

n=1

M 

nm

e2πi M ψn+

(7.79)

n=1

Substituting this in (7.76), we get HLL =



ψn+ ψn+ W (n − n ) ψn ψn

(7.80)

n,n

with an interaction W (n) =

1 Lx

M 

nm

e2πi M wm

(7.81)

m=1

where wm ≡ wk is given by (7.77), k = 2πm/Lx . The N -particle eigenstates of (7.80) are labelled by N -tuples (n1 , · · · , nN ) where 1 ≤ nj ≤ M and

120 n1 < n2 < · · · < nN and are given by Ψn1 ···nN = ψn+1 ψn+2 · · · ψn+N |1  2πi + 1 = M N/2 e− M (n1 j1 +···nN jN ) a+ j1 · · · ajN |1 j1 ···jN

=



1 M N/2

e− M

2πi

(n1 j1 +···nN jN )

ϕ0j1 ∧ · · · ∧ ϕ0jN

j1 ···jN

= φn1 ∧ · · · ∧ φnN

(7.82)

if we define φn (x, y) :=

M 

√1 M

nj

e−2πi M ϕ0j (x, y)

(7.83)

j=1

The energy eigenvalues are HLL Ψn1 ···nN = En1 ···nN Ψn1 ···nN

(7.84)

where N 

En1 ···nN =

W (ni − nj )

(7.85)

i,j=1 i
=j

The Fourier sums in (7.81) and (7.83) can be performed with the Poisson summation formula. This is done in part (c) and (d) of Lemma 7.2.2. If we √ r2 2 approximate wk ≈ 2π λ r [e− 2 k ]M , since by assumption r >> B , we find for this wk W (n) = λ



e− 2r2 (Lx M −jLx ) 1

j∈Z

=λ



e

n

2

“ ”2 − 2r12 2B 2πn Ly −jLx

(7.86)

j∈Z

Thus the theorem is proven.


Lemma 7.2.2 a) For the matrix element in (7.63) one has nk|V |n k  = (7.87)
2 √

 2 r2 2 B 2πB δk,k λ r [e− 2 k ]M ds ds hn (s) hn (s ) e− 2r2 (s−s )

The Many-Electron System in a Magnetic Field

121

where, if k = 2πm/Lx , [e−

r2 2

k2

∞ 

]M :=

e−

r2 2

(k−jK)2

∞ 

=

j=−∞

2

r 2π e− 2 [ Lx (m−jM)]

2

(7.88)

j=−∞

is an M -periodic function (as a function of m). b) The matrix elements (7.74) are given by M √ 1 (0, l1 ; 0, l2 ; 1y , k) = δm,m [e− 2 −m1 L 


2 B 4

(l1 −l2 )2

x B

if k =

2π Lx m, lj

=

2π L x mj

]M

(7.89)

M and δm = 1 iff m1 = m2 mod M . 1 ,m2

c) For m ∈ Z let vm = [e−

r2 2

k2

]M =



e−

r2 2

2π 2 (L ) (m−jM)2 x

(7.90)

j∈Z

and let V (n) =

1 Lx

M

nm

e2πi M vm . Then  − 1 L n −jL 2 1 x) e 2r2 ( x M V (n) = √2π r m=1

(7.91)

j∈Z

d) Let k = 2πm/Lx and let ϕ0,m ≡ ϕ0,k be the single-body eigenfunction (7.40). Then M 

√1 M

e−2πi M ϕ0,m (x, y) nm

m=1

= √π

−1 4

B L y

∞ 

−

1 2
2 B

2 n Lx −sLx ) (x− M

e

i

(x− n Lx −sLx )y M
2 B

(7.92)

s=−∞ 1

=

e

− √1 √π 4 M B L x

∞ 

e

−

1 2
2 B

n r (y− Mr Ly )2 i(x− M Lx ) M Ly /2B e

(7.93)

r=−∞

Proof: a) We have
n, k|V |n , k  = d2 x d2 x ϕn,k (x) V (x − x ) ϕ¯n ,k (x ) =

1 L x B






dxdx dydy  ei(k−jK)x−i(k −j



K)x

(7.94)

×

j,j 

hn,k (y − jLy ) hn ,k (y  − j  Ly ) V (x − x ) =

1 L x B








dxdx dydy  ei(k−jK)(x−x ) ei(k−jK−k +j



K)x

×

j,j 

    (x−x )2 (y−y )2 hn (y − yk − jLy )/B hn (y  − yk − j  Ly )/B λ e− 2r2 e− 2r2

122 The x -integral gives Lx δm−jM,m −j  M = Lx δm,m δj,j  if k = 2πm/Lx, k  = 2πm /Lx , 0 ≤ m, m ≤ M − 1. Thus we get n, k|V |n , k  = 
  x2
λ  dx ei(k−jK)x e− 2r2 dydy  hn (y − yk − jLy )/B × B δk,k j

  (y−y )2 hn (y  − yk − jLy )/B e− 2r2 =

 r2
2 √  2 2
B 2πB λr δk,k e− 2 (k−jK) dsds hn (s) hn (s ) e− 2r2 (s−s ) j


2 √
 2 B r2 2 = 2πB λr δk,k [e− 2 k ]M dsds hn (s) hn (s ) e− 2r2 (s−s )

(7.95)

and part (a) follows. b) One has (0, l1 ; 0, l2 ; 1y , k) =  √ 1 3 L x B

(7.96) dxdy e−i(l1 +j1 K)x ei(l2 +j2 K)x e−i(k+jK)x h0,l1 (y) h0,l2 (y)

j1 ,j2 ,j

The plane waves combine to

 2π exp i L (m + j M − m − j M − m − jM )x 2 2 1 1 x

(7.97)

and the x-integral gives a volume factor Lx times a Kroenecker delta which is one iff m2 + j2 M − m1 − j1 M − m − jM = 0 or m = m2 − m1 ∧ j = j2 − j1

if m2 ≥ m1

m = m2 − m1 + M ∧ j = j2 − j1 − 1

if m2 < m1

(7.98)

Thus (7.96) becomes (0, l1 ; 0, l2 ; 1y , k) = M δm,m 2 −m1

√ 1 3 √1 Lx B

M √1 = δm,m 2 −m1

B

3

√1 Lx

 j1 ,j2

0

 j1 ,j2

Ly

0

  dy h0 (y − yl1 − j1 Ly )/B ×   h0 (y − yl2 − j2 Ly )/B

Ly

  dy h0 (y − yl1 − j1 Ly )/B ×   h0 (y − yl2 − j1 Ly + (j1 − j2 )Ly )/B

The Many-Electron System in a Magnetic Field  Ly   M 1 √1 √ = δm,m dy h0 (y − yl1 − j1 Ly )/B × 2 −m1 3 Lx

B

M √1 = δm,m 2 −m1

B

0

j1 ,j

3



√1 Lx



M √ 1 = δm,m 2 −m1 L 

x B

−∞

j

e

∞

− 12 4
B

123

  h0 (y − yl2 − j1 Ly + jLy )/B     dy h0 (y − yl1 )/B h0 (y − yl2 + jLy )/B

(yl1 −yl2 +jLy )2

j



M √ 1 = δm,m 2 −m1 L 

x B

e−


2 B 4

(l1 −l2 +jK)2

j M √ 1 = δm,m [e− 2 −m1 L 


2 B 4

(l1 −l2 )2

x B

]M

(7.99)

M  where δm,m  equals one iff m = m mod M and equals zero otherwise.

c) It is [e−

r2 2

k2



]M =

e

2 M2 L2 x

− r2

m −2πj ) (2π M

2

(7.100)

j∈Z

We use the following formula which is obtained from the Poisson summation theorem   t 2  2 1 t e− 2t (x−2πj) = 2π e− 2 j eijx (7.101) j∈Z

j∈Z

with m x = 2π M ,

t=

L2x r2 M 2

(7.102)

Then vm =

Lx √1 2π r M



L2 x

e− 2r2 M 2 j e−2πi M 1

2

jm

(7.103)

j∈Z

and V (n) becomes V (n) =

1 Lx

M 

nm

e2πi M

Lx √1 2π r M

m=1

=

1 √1 2π r M

1 √1 2π r M

L2 x

e− 2r2 M 2 j e−2πi M 1

2

jm

j∈Z



e

L2 − 2r12 Mx2

j2

e

L2 − 2r12 Mx2

2



M 

e2πi

(n−j)m M

m=1

j∈Z

=



j

M M δn,j

j∈Z

=

√1 2π r

 s∈Z

L2 x

e− 2r2 M 2 (n−sM) 1

2

(7.104)

124 which proves part (c). d) According to (7.40) we have

√1 M

M 

e−2πi M ϕ0,m (x, y) = nm

m=1 √1 M

M 

e−2πi M

nm

m=1

=

∞ 

√ 1 L x B

2π

ei Lx (m+jM)x h0,k (y − jLy )

j=−∞ ∞ 

−1

√π 4 MLx B

M 

2π

n

2π

n

2π

ei Lx (x− M Lx )m ei Lx jMx e

−

1 2
2 B

(y−2B L2πx m−jLy )2

j=−∞ m=1

=

M ∞  

−1

√π 4 MLx B

ei Lx (x− M Lx )m e

m=1 j=−∞

√1 2π

1

=

− √π 4 √1 MLx B 2π



dq e−

q2 2

e

iq
y

i

Ly
2 B

dq e−

M 

B

jx

q2 2

× “

e

iq

y
B

L

2π −B L m−j
y x

”

B

ei Lx (x− M Lx −qB )m × 2π

n

m=1 ∞ 

e

“ ”L i
x −q
y j B

B

(7.105)

j=−∞ 1

=

− √π 4 √1 MLx B 2π



dq e−

q2 2

e

iq
y

M 

B

ei Lx (x− M Lx −qB )m × 2π

n

m=1

  L 2π δ ( xB − q) By − 2πr

∞  r=−∞

The delta function forces q to take values q=

x B

B − 2πr L y

which gives x−

n M Lx

2

n − qB = − M Lx + 2πr LBy =

r−n M

Lx

Therefore the m-sum in (7.105) becomes M  m=1

ei Lx (x− M Lx −qB )m = 2π

n

M  m=1

e2πi

(r−n)m M

M = M δr,n

(7.106)

The Many-Electron System in a Magnetic Field

125

and we get √1 M

M 

e−2πi M ϕ0,m (x, y) = nm

m=1 ∞ 

√ √ −1 4 B π√ M 2π L L x B y

= √π

∞ 

−1 4

B L y

= √π

−1 4

B L y

s=−∞ ∞ 

e e

e

−

1 2
2 B

„ «2 “ ” 2π
2
x−r LyB i
x −2πr LBy
y

e

B

B

M δr,n

r=−∞ „ «2 “ ” 2π
2
− 12 x−(n+sM) LyB i
x −2π(n+sM) LBy
y 2


e

B

−

1 2
2 B

2 n Lx −sLx ) (x− M

e

i

B

(x− n Lx −sLx )y M
2 B

B

(7.107)

s=−∞

This proves (7.92). (7.93) is obtained directly from (7.40) by putting r = m + jM .
Since the eigenstates of the approximate model are given by pure wedge products, we cannot expect that their energies, for a Coulomb interaction, are close to those of the Laughlin or Jain wavefunctions. The reason is that a wedge product only vanishes linearly in xi − xj while the Laughlin wavefunction vanishes like (xi − xj )3 if xi goes to xj . This gives a lower contribution to the Coulomb repulsion 1/|xi − xj |. In general it is not possible to make an exact analytical computation of the expectation value of the energy if the wavefunction is not given by a pure wedge product, like the Laughlin or composite fermion wavefunction. One has to rely on numerical and analytical approximations or exact numerical results for small system size. For a pure wedge product, the exact result can be written down and it looks as follows. Lemma 7.2.3 a) Let ψ(x1 , ..., xN ) be a normalized antisymmetric wavefunc
N tion and let W (x1 , ..., xN ) = i,j=1 V (xi − xj ). Let A = d2 x be the sample i<j

size, let nν = N/A = ν/(2π2B ) be the density
and let ρ be the density of the constant N -particle
wavefunction such that dx1 · · · dxN ρ = 1, that is, ρ = 1/AN . Let W = d2N x W (|ψ|2 − ρ). Then 
W  1 1 dx1 dx2 V (x1 − x2 ) g(x1 , x2 ) − 1 (7.108) N = 2 nν A where g(x1 , x2 ) =

N (N −1) n2ν




dx3 · · · dxN |ψ(x1 , ..., xN )|2

(7.109)

b) Suppose that ψ(x1 , ..., xN ) = φm1 ∧ · · · ∧ φmN (x1 , ..., xN ) = √1N ! det [φmi (xj )1≤i,j≤N ]

(7.110)

126 Then 1 n2ν

g(x1 , x2 ) =

  P (x1 , x1 )P (x2 , x2 ) − |P (x1 , x2 )|2

(7.111)

where P is the kernel of the projector onto the space spanned by {φm1 , ..., φmN }. N That is, P (x, x ) = j=1 φmj (x)φ¯mj (x ). Proof: a) One has W =

d2N x

N 

  V (xi − xj ) |ψ(x1 , ..., xn )|2 − ρ

i,j=1 i<j

=

N (N −1) 2

=

1 2





dx1 dx2 V (x1 − x2 )



dx1 dx2 V (x1 − x2 ) × 

= N NA−1 12 A1



N (N − 1)

  dx3 · · · dxN |ψ(x1 , ..., xn )|2 − ρ

dx3 · · · dxN |ψ(x1 , ..., xn )|2 −

N (N −1) A2



dx1 dx2 V (x1 − x2 ) ×   N (N −1) dx3 · · · dxN |ψ(x1 , ..., xn )|2 − 1 n2 ν

where we used nν = N/A ≈ (N − 1)/A. This proves part (a). b) Let ψ = φm1 ∧ · · · ∧ φmN . Then dx3 · · · dxN |ψ(x1 , ..., xN )|2  = N1 ! επ εσ dx3 · · · dxN φmπ1 (x1 )φ¯mσ1 (x1 ) · · · φmπN (xN )φ¯mσN (xN ) π,σ∈SN

=

1 N!

   |φmπ1 (x1 )|2 |φmπ2 (x2 )|2 − φmπ1 (x1 )φ¯mπ1 (x2 )φmπ2 (x2 )φ¯mπ2 (x1 )

π∈SN

=

1 N (N −1)

N    |φmi (x1 )|2 |φmj (x2 )|2 − φmi (x1 )φ¯mi (x2 )φmj (x2 )φ¯mj (x1 ) i,j=1 i
=j

=

1 N (N −1)

N 

{· · · }

i,j=1

=

1 N (N −1)

  Pm (x1 , x1 )Pm (x2 , x2 ) − Pm (x1 , x2 )Pm (x2 , x1 )

where Pm (x1 , x2 ) =

N j=1

φmj (x1 )φ¯mj (x2 ).


(7.112)

The Many-Electron System in a Magnetic Field

127

Now consider the eigenvalues En1 ...nN given by (7.51). For −M/2 ≤ n ≤ M/2, the dominant contribution from the periodizing j-sum for W (n) is L2 n 2 the j = 0 term which is exp{− 2rx2 ( M ) }. This is small if n = ni − nj is large. Thus, it seems that those configurations (n1 , ..., nN ) have low energy for which ni − nj in average is large. Hence, one may speculate that, for ν = 1/3, the minimizing configurations are (3, 6, 9, ..., 3N ), (2, 5, 8, ..., 3N − 1) and (1, 4, 7, ..., 3N − 2) and the lowest excited states should be obtained from these states by just changing one ni or a whole group of neighboring ni ’s by one each. The projection P{3k} onto the space spanned by φ3 , φ6 , ..., φ3N can be explicitly computed. In the infinite volume limit, one simply obtains P{3k} = 13 Pν=1 where Pν=1 is the projection onto the whole lowest Landau level, spanned by all the ϕn ’s. Thus, the energy per particle U{3k} for the  2 2 wavefunction φ3 ∧ φ6 ∧ · · · ∧ φ3N is 13 Uν=1 = − 31 π8 eB ≈ −0, 21 eB which is much bigger than the energy of the Laughlin wavefunction which is about 2 Uν=1/3 = −0, 42 eB .

Chapter 8 Feynman Diagrams

8.1

The Typical Behavior of Field Theoretical Perturbation Series

In chapters 3 and 4 we wrote down the perturbation series for the partition function and for some correlation functions. We found that the coefficients of λn were given by a sum (d + 1)n-dimensional integrals if the space dimension is d. Typically, some of these integrals diverge if the cutoffs of the theory are removed. This does not mean that something is wrong with the model, but merely means first of all that the function which has been expanded is not analytic if the cutoffs are removed. To this end we consider a small example. Let
∞
1 1 e−x (8.1) Gδ (λ) := 0 dx 0 dk √k+λx+δ where δ > 0 is some cutoff and the coupling λ is positive. One may think of δ = T , the temperature, or δ = 1/L if Ld is the volume of the system. By explicit computation, using Lebesgue’s theorem of dominated convergence to interchange the limit with the integrals, √ √
∞ G0 (λ) = lim Gδ (λ) = 0 dx 2( 1 + λx − λx) e−x δ→0 √ = 2 + O(λ) − O( λ) (8.2) Thus, the δ → 0 limit is well defined but it is not analytic. This fact has to show up in the Taylor expansion. It reads Gδ (λ) =

n    −1
∞ 2

0

j j=0

dx


1 0

dk

xj e−x 1

(k+δ)j+ 2

λj + rn+1

(8.3)

Apparently, all integrals over k diverge for j ≥ 1 in the limit δ → 0. Now, very roughly speaking, renormalization is the passage from the expansion (8.3) to the expansion G0 (λ) =

n   
∞ 1 2



0

dx 2x e−x λ − c

√ λ + Rn+1

(8.4)

=0

129

130 √ where the last one is obtained from (8.2) by expanding the 1 + λx term. One would √ say ‘the diverging integrals have been resumed to the nonanalytic term c λ’. In the final expansion (8.4) all coefficients are finite and, for small λ, the lowest order terms are a good approximation since (θλ ∈ [0, λ])  
 1    ∞ 1 2 xn+1 λn+1 e−x  |Rn+1 | = 2 0 dx n+1 n+ 1 2 (1+θλ x)  1 
∞  2 dx xn+1 e−x λn+1 ≤ 2 n+1 0 √  n n n+1 1 (2n)! n+1 n→∞ = 22n ∼ 2 e λ (8.5) n! λ Here we used the Lagrange representation√of the n + 1’st Taylor remainder in the first line and Stirling’s formula, n! ∼ 2πn(n/e)n , in the last line. An estimate of the form (8.5) is typical for renormalized field theoretic perturbation series. The lowest order terms are a good approximation for weak coupling, but the renormalized expansion is only asymptotic, the radius of convergence of the whole series is zero. The approximation becomes more accurate if n approaches 1/λ, but then quickly diverges if n > e/λ. Or, for fixed n, the n lowest order terms are a good approximation as long as λ < 1/n. In this small example we went from ‘the unrenormalized’ or ‘naive’ perturbation expansion (8.3) to the ‘renormalized’ perturbation expansion (8.4) by going through the exact answer (8.2). Of course, for the models we are interested in, we do not know the exact answer. Thus, for weak coupling, the whole problem is to find this rearrangement which transforms a naive perturbation expansion into a renormalized expansion which is (at least) asymptotic. In the next section we prove a combinatorial formula which rewrites the perturbation series in terms of n’th order diagrams whose connected components are at least of order m, which, for m = n, results in a proof of the linked cluster theorem. This reordering has nothing to do with the rearrangements considered above, it simply states that the logarithm of the partition function is still given by a sum of diagrams which is not obvious. In section 8.3 we start with estimates on Feynman diagrams. That section is basic for an understanding of renormalization, since it identifies the divergent contributions in a sum of diagrams.

Feynman Diagrams

8.2

131

Connected Diagrams and the Linked Cluster Theorem

The perturbation series for the partition function reads Z(λ) =

∞ 

(−λ)n n!




dξ1 · · · dξ2n U (ξ1 − ξ2 ) × · · ·

(8.6)

n=0

· · · × U (ξ2n−1 − ξ2n ) det [C(ξi , ξj )]1≤i,j≤2n If we expand the 2n × 2n determinant and interchange the sum over permutations with the ξ-integrals, we obtain the expansion into Feynman diagrams: Z(λ) =

∞ 

(−λ)n n!

n=0



signπ G(π)

(8.7)

π∈S2n

where the graph or the value of the graph defined by the permutation π is given by  n (8.8) G(π) = dξ1 · · · dξ2n Π U (ξ2i−1 − ξ2i ) C(ξ1 , ξπ1 ) · · · C(ξ2n , ξπ2n ) i=1

In general the above integral factorizes into several connected components. The number of U ’s in each component defines the order of that component. The goal of this subsection is to prove the linked cluster theorem which states that the logarithm of the partition function is given by the sum of all connected diagrams. A standard proof of this fact can be found in many books on field theory or statistical mechanics [40, 56]. In the following we give a slightly more general proof which reorders the perturbation series in terms of n’th order diagrams whose connected components are at least of order m where 1 ≤ m ≤ n is an arbitrary given number. See (8.17) below. Linked Cluster Theorem: The logarithm of the partition function is given by the sum of all connected diagrams, log Z(λ) =

∞  n=0

We use the following

(−λ)n n!

 π∈S2n G(π) connected

signπ G(π)

(8.9)

132 Lemma 8.2.1 Let {wn }n∈N be a sequence with wi wj = wj wi ∀i, j ∈ N and let a be given with awi = wi a ∀i ∈ N (for example wi , a ∈ C or even elements of a Grassmann algebra). For fixed m ∈ N define the sequence {vn }n∈N by (n = mk + l, 0 ≤ l ≤ m − 1, k ∈ N)  aj wmk−mj+l vmk+l = . (−1)j (mk + l)! j=0 j! (mk − mj + l)! k

(8.10)

Then the wn ’s can be computed from the vn ’s by  aj vmk−mj+l wmk+l = (mk + l)! j=0 j! (mk − mj + l)!

(8.11)

∞ ∞   wn vn a = e . n! n! n=0 n=0

(8.12)

k

and one has

Proof: One has k  vmk−mj+l aj j! (mk − mj + l)! j=0

=

r=i+j

=

k−j k  wm(k−j)−mi+l aj  ai (−1)i j! i! (m(k − j) − mi + l)! j=0 i=0

r=0

=

wmk−mr+l   r  (−1)i r! (mk − mr + l)! i=0 i

k  ar

r

k  ar r=0

wmk−mr+l wmk+l δr,0 = r! (mk − mr + l)! (mk + l)!

which proves the first formula. The second formula is obtained as follows ∞ ∞ m−1 ∞ m−1 k    wmk+l   vmk−mj+l wn aj = = n! (mk + l)! j! (mk − mj + l)! n=0 j=0 k=0 l=0

=

k=0 l=0

∞ ∞ m−1  aj   vmk−mj+l j! (mk − mj + l)! j=0 k=j l=0

= ea

∞  vn n! n=0

which proves the lemma.


r=k−j

=

∞ ∞ m−1  aj   vmr+l j! r=0 (mr + l)! j=0 l=0

(8.13)

Feynman Diagrams

133

of n’th order diagrams whose connected Now we define the sum Det(m) n components are at least of order m inductively by (1) Det(1) n = Detn (C, U )  n := dξ1 · · · dξ2n Π U (ξ2i−1 − ξ2i ) det [C(ξi , ξj )]1≤i,j≤2n

(8.14)

i=1

and for m ≥ 1, n = mk + l , 0 ≤ l ≤ m − 1 1 j (m) (m+1) k  Detmk−mj+l Detmk+l Det(m) m = (−1)j m! (mk + l)! j=0 j! (mk − mj + l)!

(8.15)

Then the fact that the logarithm gives only the connected diagrams may be formulated as follows. Theorem 8.2.2 The logarithm of the partition function is given by ∞  λn log Z(λ) = Det(n) n (C, U ) n! n=1

(8.16)

and Det(n) n (C, U ) is the sum of all connected n’th order diagrams. Proof: We claim that for arbitrary m Z(λ) =



∞ m  λn λs (s) Det(m+1) Det exp n s n! s! n=0 s=1

(8.17)

For m = 0, (8.17) is obviously correct. Suppose (8.17) is true for m− 1. Then, because of (m+1)

λn

Detmk+l Det(m+1) n = λmk+l n! (mk + l)!  λm  (m) k (m) j  Detmk−mj+l j m! Detm mk−mj+l λ (−1) = j! (mk − mj + l)! j=0

(8.18)

and the lemma, one obtains Z(λ) =

m−1

∞   λs λn (s) Det(m) Det exp n s n! s! n=0 s=1

m−1

∞   λs m λn (m+1) λm! Det(m) (s) m Detn Dets = e exp n! s! n=0 s=1 =



∞ m  λn λs (s) Det(m+1) Det exp n s n! s! n=0 s=1

(8.19)

134 Furthermore we claim that for a given m ∈ N (m+1)

Det1

(m+1)

= Det2

= · · · = Det(m+1) =0 m

(8.20)

Namely, for n = mk + l < m one has k = 0, n = l such that  Det(m+1) n = (−1)j n! j=0 0



 (m) j 1 m! Detm j!

Det(m) Det(m) n n = n! n!

(8.21)

and it follows Det(m+1) = Det(m) = · · · = Det(n+1) n n n

(8.22)

But, for n = m = 1m + 0, by definition  Det(m+1) m = (−1)j m! j=0 1



Det(m) m = − m!

 (m) j 1 m! Detm j! 

 (m) 1 1 m! Detm 1!

(m)

Detm−mj (m − mj)! = 0

(8.23)

which proves (8.20). Thus one obtains  

∞ m  λn λs (m+1) (s) Z(λ) = 1 + Detn Dets exp n! s! n=m+1 s=1

(8.24)

By taking the limit m → ∞ one gets log Z(λ) =

∞  λs s=1

s!

Det(s) s

(8.25)

which proves (8.16). It remains to prove that Det(n) n is the sum of all connected n’th order diagrams. To this end we make the following definitions. Let π ∈ S2n be given. We say that π is of type t(π) = 1b1 2b2 · · · nbn

(8.26)

iff Gπ consists of precisely b1 first order connected components, b2 second order connected components, · · · , bn n’th order connected components, where Gπ is the graph produced by the permutation π. Observe that, contrary to section (1.3.1), the br are not the number of r-cycles of the permutation π, but, as defined above, the number of r’th order connected components of the diagram (8.8) given by the permutation π. Let (b ,··· ,bn )

S2n1

= {π ∈ S2n | t(π) = 1b1 2b2 · · · nbn }

(8.27)

Feynman Diagrams

135

Then S2n is the disjoint union

S2n =

(b ,··· ,bn )

S2n1

(8.28)

(0,··· ,0,bm ,··· ,bn )

(8.29)

0≤b1 ,··· ,bn ≤n 1b1 +···+nbn =n

Let

(m)

S2n =

S2n

0≤bm ,··· ,bn ≤n mbm +···+nbn =n

and (n)

(0,··· ,0,1)

c S2n = S2n = S2n

(8.30)

One has (b ,··· ,bn )

|S2n1

|=

(1!)b1

n! c bn |S c |b1 · · · |S2n | , · · · (n!)bn b1 ! · · · bn ! 2

(8.31)

in particular (0,··· ,0,bm ,··· ,bn )

|S2n

|=

n! (0,··· ,0,bm+1 ,··· ,bn−mbm ) |S c |bm |S2n−2mbm | (n − mbm )! (m!)bm bm ! 2m (8.32)

We now prove by induction on m that  Det(m) n

dξ1 dξ2 · · · dξ2n

=

n 

U (ξ2i−1 − ξ2i ) det(m) [C(ξj , ξk )j,k=1,··· ,2n ]

i=1

(8.33) where det(m) [C(ξj , ξk )j,k=1,··· ,2n ] =



C(ξ1 , ξπ1 ) · · · C(ξ2n , ξπ(2n) ) (8.34)

(m)

π∈S2n

It follows from (8.33) that Det(n) n is the sum of all connected diagrams. Obviously (8.33) is correct for m = 1. Suppose (8.33) is true for m. Let (m+1)

 Det n

=

 dξ1 dξ2 · · · dξ2n

n 

U (ξ2i−1 − ξ2i ) det(m+1) [C(ξj , ξk )j,k=1,··· ,2n ]

i=1

(8.35)

136 Then, for n = mk + l with k ∈ N and 0 ≤ l ≤ m − 1 2n   

Det(m) = n

i=1

(m) π∈S2n

  2n

bm ,bm+1 ,··· ,bn =0 mbm +···+nbn =n

(0,··· ,0,bm ,bm+1 ,··· ,bn ) π∈S2n

bm =0

c π∈S2m

i=1





dξi

m 

dξi · · ·

i=1

U (ξ2i−1 − ξ2i )



bm C(ξr , ξπr ) ×

2m  r=1

i=1

2(n−mbm )

(m+1) π∈S2n−2mbm

bm =0

C(ξr , ξπr )

n! × (n − mbm )! (m!)bm bm !

   2m

k 

2n  r=1

i=1



k 

=

U (ξ2i−1 − ξ2i )

n 

=

=

n 

dξi

dξi

i=1

n−mb m



2(n−mbm )

U (ξ2i−1 − ξ2i )

C(ξr , ξπr )

r=1

i=1

bm  (m+1) n! (m)  Det Det m n−mbm b m (n − mbm )! (m!) bm !

(8.36)

or (m) k  Detmk+l = (mk + l)!



(m) 1 m! Detm

bm

bm !

bm =0

(m+1)

 Det mk−mbm +l (mk − mbm + l)!

(8.37)

(m+1)

Then, by the lemma and the definition of Detmk+l , (m+1)

k  mk+l  Det = (−1)bm (mk + l)! bm =0



(m) 1 m! Detm

bm !

bm

(m)

(m+1)

Detmk−mbm +l Detmk+l = (mk − mbm + l)! (mk + l)!

which proves (8.33).


8.3 8.3.1

Estimates on Feynman Diagrams Elementary Bounds

In this section we identify the large contributions which are typically contained in a sum of Feynman diagrams. These large contributions have the

Feynman Diagrams

137

effect that the lowest order terms of the naive perturbation expansion are not a good approximation. The elimination of these large contributions is called renormalization. We find that the size of a graph is determined by its subgraph structure. For the many-electron system with short range interaction, that is, for V (x) ∈ L1 , the dangerous subgraphs are the two- and four-legged ones. Indeed, in Theorem 8.3.4 below we show that an n’th order diagram without two- and four-legged subgraphs is bounded by constn which is basically the best case which can happen. A sum of diagrams where each diagram is bounded by constn can be expected to be asymptotic. That is, the lowest order terms in this expansion would be indeed a good approximation if the coupling is not too big. However usually there are certain subgraphs which produce anomalously large contributions which prevent the lowest order terms in the perturbation series from being a good approximation. For the many-electron system with short range interaction these are two- and four-legged subdiagrams. Fourlegged subgraphs produce factorials, that is, an n’th order bound of the form constn n!, the constant being independent of the cutoffs. Diagrams which contain two-legged subgraphs in general diverge if the cutoffs are removed. The goal of this section is to prove these assertions. We start with a lemma which sets up all the graph theoretical notation and gives the basic bound in coordinate space. The diagrams in Lemma 8.3.1 consist of generalized vertices or subgraphs which are represented by some functions I2qv (x1 , · · · , x2qv ) and lines to which are assigned propagators C(x − x ). A picture may be helpful. In figure 8.1 below, G = G(x1 , x2 ) and there is an integral over the remaining variables x3 , · · · , x10 .

C(x6-x3)

I(x1,x3,x4,x5) x1

5

3 4

C(x5-x9)

C(x7-x4)

I(x9,x10) 9

10

Figure 8.1

I(x2,x6,x7,x8) 6 7

8

C(x10-x8)

x2

138 Lemma 8.3.1 (Coordinate Space Bound) Let I2q = I2q (x1 , · · · , x2q ), xi ∈ Rd , be a generalized vertex (or subgraph) with 2q legs obeying

  2q I2q ∅ := sup sup dxj |I2q (x1 , · · · , x2q )| < ∞ xi

i

(8.38)

j=1 j=i

Let G be a connected graph built up from vertices I2qv , v ∈ VG , the set of all vertices of G, by pairing some of their legs. Two paired legs are by definition a line  ∈ LG , the set of all lines of G. To each line, assign a propagator v v C (xvi , xi ) = C (xvi − xi ). Suppose 2q legs remain unpaired. The value of   G is by definition G(x1 , · · · , x2q ) =

  v∈VG

2qv 

dxvi

i=1 xv int. i



I2qv (xv1 , · · · , xv2qv )

v∈VG



v

C (xvi , xi ) 

∈LG

(8.39) v v where x1 , · · · , x2q ∈ ∪v∈VG ∪2q i=1 {xi } are by definition the variables of the

v

v v unpaired legs and xvi , xi ∈ ∪v∈VG ∪2q i=1 {xi } are the variables of the legs  connected by the line . Let f1 , · · · , f2q be some test functions. Fix s = |S| legs of G where S ⊂ {1, · · · , 2q}. These s legs, which will be integrated against test functions, are by definition the external legs of G, and the other legs, which are integrated over Rd , are called internal. For S = ∅, define the norm       G S := (8.40) dxk |fk (xk )| dxi |G(x1 , · · · , x2p )|

i∈S c

k∈S

Then there are the following bounds: a) G ∅ ≤



C L1

∈T



C ∞



I2qv ∅

(8.41)

v∈V

∈L\T

where T is a spanning tree for G which is a collection of lines which connects all vertices of G such that no loops are formed. b) G S ≤

 ∈T¯

C L1

 ∈L\T¯

C ∞



I2qv Sv

(8.42)

v∈V

w where now T¯ = i=1 Ti is a union of w trees which spans G and w is the number of vertices to which at least one external leg is hooked which, by definition, is the number of external vertices. Each Ti contains precisely one external vertex. Finally, Sv is the set of external legs at I2qv .

Feynman Diagrams

139

c) Let G be a vacuum diagram, that is, a diagram without unpaired legs (q = 0). Then    |G| ≤ Ld C L1 C ∞ I2qv ∅ (8.43) ∈T

∈L\T

v∈V

where |G| is the usual modulus of G and Ld =




dx 1.

Proof: a) Choose a spanning tree T for G, that is, choose a set of lines T ⊂ L = LG which connects all vertices such that no loops are formed. For all lines not in T take the L∞ -norm. Let xi ∈ {x1 , · · · , x2q } be the variable where the supremum is taken over. We get G ∅ ≤

  v∈VG

≤

  v∈VG

2qv 

dxvi

i=1 xv i =xi

2qv  i=1 xv i =xi



|I2qv (xv1 , · · · , xv2qv )|

v∈VG

dxvi





v

|C (xvi − xi )| 

∈LG

|I2qv (xv1 , · · · , xv2qv )|

v∈VG



v

|C (xvi − xi )| × 

∈T



C ∞

∈LG \T

The vertex to which the variable xi belongs we define as the root of the tree. To perform the integrations, we start at the extremities of the tree, that is, at those vertices I2qv which are not the root and which are connected to the tree only by one line. To be specific, choose one of these vertices, I2qv1 (xv11 , · · · , xv2q1v ). Let xvr1 be the variable which belongs to the tree. This 1 variable also shows up in the propagator for the corresponding line , C (xvr1 −  xvr ) where v  is a vertex necessarily different from v since  is on the tree. Now we bound as follows    dxv11 · · · dxv2q1v |I2qv1 (xv11 , · · · , xv2q1v )||C (xvr1 − xvr )| |I2qv ({xvj })| 1 1  2q  v1   dxvi 1 |I2qv1 (xv11 , · · · , xv2q1v )||C (xvr1 − xvr )| |I2qv ({xvj })| = dxvr1 Π i=1  ≤

i=r

  dxvr1 sup v

xr 1

  dxvi 1 |I2qv1 (xv11 , · · · , xv2q1v )| |C (xvr1 − xvr )| × Π i=1

2qv1 i=r

  |I2qv ({xvj })|  2qv1 
  v v v = sup Π dxi 1 |I2qv1 (x11 , · · · , x2q1v )| dxvr1 |C (xvr1 − xvr )| |I2qv ({xvj })| v

xr 1

i=1 i=r



≤ I2qv1 ∅ C L1 |I2qv ({xvj })|

(8.44)

140 Now we repeat this step until we have reached the root of the tree. To obtain an estimate in this way we refer in the following to as ‘we apply the tree identity’. For each line on the tree we get the L1 norm, lines not on the tree give the L∞ norm and each vertex is bounded by the · ∅ norm which results in (8.41). b) Choose w trees T1 , · · · , Tw with the properties stated in the lemma. For each line not in T¯ take the L∞ -norm. Then for each Ti apply the tree identity with the external vertex as root. Suppose this vertex is I2qv (y1 , · · · , y2qv ) and y1 , · · · , ypv are the external variables. Then, instead of I2qv ∅ as in case (a) one ends up with pv  

2qv     dyk fjk (yk ) dyi |I2qv (y1 , · · · , y2qv )|

(8.45)

i=pv +1

k=1

which by definition is I2qv Sv . c) Here we proceed as in (a), however we can choose an arbitrary vertex I2qv to be the root of the tree, since we do not have to take a supremum over some xi . We apply the tree identity, and the integrations at the last vertex, the root I2qv , are bounded by   2qv 2qv v v v v v v v Π dxj |I2qv (x1 , · · · , x2qv )| ≤ dxi sup Π dxj |I2qv (x1 , · · · , x2qv )| xv i

j=1

j=1 j=i

≤ Ld I2qv ∅ which proves the lemma


For an interacting many body system the basic vertex is  ¯  )ψ(ξ  ) ¯ (ξ − ξ  )ψ(ξ dξdξ  ψ(ξ)ψ(ξ)U

(8.46)

which corresponds to the diagram ψ(ξ) ξ ψ(ξ)

ψ(ξ’) U(ξ-ξ’)

ξ’ ψ(ξ’)

In order to represent this by some generalized vertex I4 (ξ1 , ξ2 , ξ3 , ξ4 ) which corresponds to the diagram

Feynman Diagrams ψ(ξ3)

141

ψ(ξ4)

I4(ξ1,ξ2,ξ3,ξ4)

ψ(ξ1)

ψ(ξ2)

we rewrite (8.46) as  ¯ 3 )ψ(ξ1 )ψ(ξ ¯ 4 )ψ(ξ2 )I4 (ξ1 , ξ2 , ξ3 , ξ4 ) dξ1 dξ2 dξ3 dξ4 ψ(ξ

(8.47)

which coincides with (8.46) if we choose I4 (ξ1 , ξ2 , ξ3 , ξ4 ) = δ(ξ3 − ξ1 )δ(ξ4 − ξ2 )U (ξ1 − ξ2 )

(8.48)

I4 ∅ = U L1 (Rd+1 ) = V L1 (Rd ) < ∞

(8.49)

Then

if we assume a short range potential. The next lemma is the momentum space version of Lemma 8.3.1. We use the same letters for the Fourier transformed quantities, hats will be omitted. For translation invariant I4 (ξ1 , ξ2 , ξ3 , ξ4 ) = I4 (ξ1 + ξ  , ξ2 + ξ  , ξ3 + ξ  , ξ4 + ξ  ) we have  ¯ 4 )ψ(ξ2 )I4 (ξ1 , ξ2 , ξ3 , ξ4 ) = ¯ 3 )ψ(ξ1 )ψ(ξ dξ1 dξ2 dξ3 dξ4 ψ(ξ  d-k1 d-k2 d-k3 d-k4 (2π)d δ(k1 + k2 − k3 − k4 ) I4 (k1 , k2 , k3 , k4 )ψ¯k ψk ψ¯k ψk 3

1

4

2

dd k where we abbreviated d-k := (2π) d . For example,  dx1 dx2 dx3 dx4 ei(k1 x1 +k2 x2 −k3 x3 −k4 x4 ) δ(x3 − x1 )δ(x4 − x2 )V (x1 − x2 )  = dx1 dx2 ei(k1 −k3 )x1 +i(k2 −k4 )x2 V (x1 − x2 )  = dx1 dx2 ei(k1 −k3 )(x1 −x2 ) V (x1 − x2 ) ei(k1 +k2 −k3 −k4 )x2

= (2π)d δ(k1 + k2 − k3 − k4 ) V (k1 − k3 )

(8.50)

The value of the graph G defined in the above Lemma 8.3.1 reads in momentum space (2π)d δ(k1 + · · · + k2q ) G(k1 , · · · , k2q ) = (8.51)     v v dk C (k ) (2π)d δ(k1v + · · · + k2q )I2qv (k1v , · · · , k2q ) v v ∈LG

∈LG

v∈VG

142 where kiv ∈ ∪∈L {k } is the momentum ±k if  is the line to which the i’th leg of I2qv is paired. Observe that G(k1 , · · · , k2q ) is the value of the Fourier transformed diagram after the conservation of momentum delta function has been removed. However, for a vacuum diagram, we do not explicitly remove the factor δ(0) = Ld . In (8.51) we have |LG | =: |L| integrals and |VG | =: |V | constraints. The |V | delta functions produce one overall delta function which is explicitly written in (8.51) and |V | − 1 constraints on the |L| integration variables. Let T be a tree for G, that is, a collection of lines which connect all the vertices such that no loops are formed. Since |T | = |V | − 1, we can use the momenta k for lines  ∈ L \ T which are not on the tree as independent integration variables. We get     d-k C (K ) I2qv (Kv1 , · · · , Kv2qv ) (8.52) G(k1 , · · · , k2q ) = ∈L\T

∈L

v∈V

Here the sum of all momenta K flowing through the line  is defined as follows. Each line  not on the tree defines a unique loop which contains only lines on the tree with the exception of  itself. To each line in this loop assign the loop momentum k . The sum of the external variables k1 + · · · + k2q vanishes, thus we may write G(k1 , · · · , k2q ) = G(k1 , · · · , k2q−1 , −k1 − · · · − k2q−1 )

(8.53)

There are 2q −1 unique paths γi on G, containing only lines on the tree, which connect the i’th leg, to which the external momentum ki is assigned, to the 2q’th leg, to which the momentum k2q = −k1 − · · · − k2q−1 is assigned. To each line on γi assign the momentum ki . Then K is the sum of all assigned momenta. In particular, K = k for all  ∈ L \ T . The Ki appearing in I2qv (K1 , · · · , K2qv ) are the sum of all momenta flowing through the leg of I2qv labelled by i .

Lemma 8.3.2 (Momentum Space Bound) Define the following norms   I2q (k1 , · · · , k2q ) I2q ∞ = sup (8.54) k1 ,··· ,k2q ∈Rd k1 +···+k2q =0

and for S ⊂ {1, · · · , 2q}, S = ∅,     d-ks δ(k1 + · · · + k2q ) I2q (k1 , · · · , k2q ) I S = sup ks ∈Rd s∈S /

(8.55)

s∈S

Recall the notation of Lemma 8.3.1. Then there are the following bounds

Feynman Diagrams

143

a) G ∞ ≤





C ∞

∈T

C 1



I2qv ∞

(8.56)

v∈V

∈L\T

b) G S ≤



C ∞

∈T¯



C 1

∈L\T¯



I2qv Sv

v∈V Sv =∅



I2qv ∞ (8.57)

v∈V Sv =∅

c) Let G be a vacuum diagram, that is, G has no unpaired legs (q = 0). Then    |G| ≤ Ld C ∞ C 1 I2qv ∅ (8.58) ∈T

v∈V

∈L\T

where Ld is the volume of the system. Remark: Observe that the · S -norms in coordinate space depend on the choice of testfunctions whereas in momentum space the · S -norms are defined independent of testfunctions. Proof: a) We have   G(k1 , · · · , k2q ) ≤          C (K ) C (K ) I2qv (K1 , · · · , K2qv ) d-k ∈L\T

≤ =



C ∞

∈T

v∈V





C ∞

∈T

=

∈T





∈T



I2qv ∞  I2qv ∞

v∈V

C ∞



v∈V

v∈V

∈L\T



d-k

∈L\T



∈L\T

d-k

∈L\T

I2qv ∞



   C (K )    C (k ) ∈L\T

C 1

(8.59)

∈L\T

b) Let ki1 , · · · , ki|S| be the external momenta of G with respect to S . Observe that exactly |Svi | , i = 1, · · · , w , of these momenta are external to I2qvi and |Sv1 | + · · · + |Svw | = |S|

(8.60)

For each j = 1, · · · , w , let kj∗ , be one of the momenta external to Ivj . That is, kj∗ = kij∗ for some 1 ≤ j ∗ ≤ |S| . Let k˜1 , · · · , k˜w be the complementary external momenta. Here, w = |S| − w . We have  ∗    ∗ ˜ k1 , · · · , kw (8.61) , k1 , · · · , k˜w = ki1 , · · · , ki|S|

144 By definition, G S = sup

 

ks ∈Rd s∈S /

= sup ks ∈Rd s∈S /

d-k˜j

j=1



(8.62)

s∈S

  w

= sup

  d-ks δ(k1 + · · · + k2q ) G(k1 , · · · , k2q )

w 

  d-kj∗ δ(k1 + · · · + k2q ) G(k1 , · · · , k2q )

j=1 

w 

d-k0

ks ∈Rd s∈S /

d-k˜j

j=1

w 

∗ d-kj∗ δ(k0 + k1∗ + · · · + kw )×

j=1

  δ(k1 + · · · + k2q ) G(k1 , · · · , k2q )

Observe that   δ(k1 + · · · + k2q ) G(k1 , · · · , k2q ) ≤    d-k |C (k )| |I2qv (k1 , · · · , k2qv )| δ(k1 + · · · + k2qv )

(8.63)

v∈V

∈LG

Thus,  G S ≤ sup



dk0

ks ∈Rd s∈S /

w 

d-k˜j H(k0 , k˜1 , · · · , k˜w , ks ; s ∈ / S)

(8.64)

j=1

where / S) = H(k0 , k˜1 , · · · , k˜w , ks ; s ∈   w  ∗ d-kj∗ d-k |C (k )| δ(k0 + k1∗ + · · · + kw ) j=1

(8.65)

∈LG

×



|I2qv (k1 , · · · , k2qv )| δ(k1 + · · · + k2qv )

v∈V

Let G∗ be the graph obtained from G and the vertex I∗ = δ(k0 + k1∗ + ∗ ) with w + 1 legs by joining the leg of Ivj with momentum kj∗ · · · + kw to one leg of I∗ other than the leg labeled by k0 . Let Ti∗ be the tree obtained from Ti by adjoining the line that connects the external vertex at the end of Ti to I∗ . Observe that T ∗ = T1∗ ∪ · · · ∪ Tw∗ is a spanning tree for G∗ . For each  ∈ LG∗ \ T ∗ there is a momentum cycle obtained by attaching  to the unique path in T ∗ that joins the ends of  . Let p∗ be the momentum flowing around that cycle. The external legs of G∗ are labeled by the momenta k˜1 , · · · , k˜w , ks ; s ∈ / S , and k0 . Connect each of the external / S , to k0 by the unique path in T ∗ that joins them legs k˜1 , · · · , k˜w , ks ; s ∈ and let the external momenta flow along these paths. We have   / S) = δ · · · G∗ (k0 , k˜1 , · · · , k˜w , ks ; s ∈ / S) (8.66) H(k0 , k˜1 , · · · , k˜w , ks ; s ∈

Feynman Diagrams

145

where      ks δ · · · = δ k0 + k˜1 + · · · + k˜w +

(8.67)

s∈S /

and G∗ (k0 , k˜1 , · · · , k˜w , ks ; s ∈ / S) =     d-p∗ |C (K∗ )| |I2qv (K∗1 , · · · , K∗qv )| ∈LG∗ \T ∗

v∈V

∈LG

Here, K∗ is the sum of all momenta flowing through  and K∗i appearing in I2qv is the sum of all momenta flowing through the leg of I2qv labelled by i . Observe that, by construction, G∗ (k0 , k˜1, · · · , k˜w , ks ; s ∈ / S) is  independent of k0 . Now split LG = T1 ∪ · · · ∪ Tw ∪ LG \ (T1 ∪ · · · ∪ Tw ) . Then 

∗



G =



=







=



d-p∗ w 

, K∗2qv )| 

|C (K∗ )|

∈T1 ∪···∪Tw

∈LG∗ ) \T ∗

|C (p∗ )| ×

∈L\(T1 ∪···∪Tw )

|I2qv (K∗1 , · · ·

v∈V



, K∗qv )| 

|C (K∗ )|

∈T1 ∪···∪Tw

∈LG∗ \T ∗

|C (K∗ )| ×

∈L\(T1 ∪···∪Tw )

|I2qv (K∗1 , · · ·

v∈V



d-p∗



|C (K∗ )|

∈T1 ∪···∪Tw

∈LG∗ \T ∗





d-p∗

|C (p∗ )| ×

∈L\(T1 ∪···∪Tw )

|I2qvi (K∗1 , · · · , K∗2qv )|



i

i=1

|I2qv (K∗1 , · · · , K∗2qv )|

v=vi i=1,··· ,w

Thus we get  G S ≤ sup



dk0

ks ∈Rd s∈S /

w 

   d-k˜j δ k0 + k˜1 + · · · + k˜w + ks × s∈S /

j=1 ∗

= sup ks ∈Rd s∈S /

  w j=1

G (k0 , k˜1 , · · · , k˜w , ks ; s ∈ / S) d-k˜j G∗ (k˜1 , · · · , k˜w , ks ; s ∈ / S)

146   w

≤ sup ks ∈Rd s∈S /



d-k˜j



w 

≤

C ∞

ks ∈Rd s∈S /

I2qv ∞

v=vi i=1,··· ,w

  w

× sup

|I2qv (K∗1 , · · · , K∗2qv )|

v=vi i=1,··· ,w



∈T1 ∪···∪Tw

|C (p∗ )|

∈L\(T1 ∪···∪Tw )



|I2qvi (K∗1 , · · · , K∗2qvi )|

i=1





|C (K∗ )|

∈T1 ∪···∪Tw

∈LG∗ \T ∗

j=1



d-p∗



d-k˜j

 ∈LG∗ \T ∗

j=1 w 



d-p∗

|C (p∗ )| ×

∈L\(T1 ∪···∪Tw )

|I2qvi (K∗1 , · · · , K∗2qvi )|

(8.68)

i=1

Exchanging integrals, we obtain   w

d-k˜j



 ∈LG∗ \T ∗

j=1

=

|I2qvi (K∗1 , · · · , K∗2qvi )|

i=1





d-p∗

∈LG∗ \T ∗



 ∈LG∗ \T ∗

|C (p∗ )| ×

∈L\(T1 ∪···∪Tw )

  w

=

|C (p∗ )| ×

∈L\(T1 ∪···∪Tw )

w 





d-p∗

j=1

d-p∗

d-k˜j

w 

|I2qvi (K∗1 , · · · , K∗2qv )| i

i=1



|C (p∗ )| ×

∈L\(T1 ∪···∪Tw )

w  



d-k˜j |I2qvi (K∗1 , · · · , K∗2qv )| i

i=1

(8.69)

˜ ∈Sv k j i j=1,··· ,w

For each i = 1, · · · , w , exactly one of the arguments, K∗1 , · · · , K∗2qv , say for i convenience the first, is the momentum flowing through the single line that connects I2qvi to I∗ . It is the sum of all external momenta flowing into I2qvi and some loop momenta. Furthermore, exactly |Svi | − 1 of the arguments K∗1 , · · · , K∗qv appearing in I2qvi are equal to external momenta in the set i {k˜1 , · · · , k˜w } . By construction, no other momenta flow through these legs. For convenience, suppose that K∗2 = k˜2 , · · · , K∗|Sv | = k˜|Svi | . The remaining i arguments on the list K∗1 , · · · , K∗qv are sums of loop momenta only. Recall i

Feynman Diagrams

147

that K∗1 = −K∗2 − · · · − K∗2qv . Thus, i   d-k˜j |I2qvi (K∗1 , · · · , K∗2qv )| = i

˜ ∈Sv k j i j=1,··· ,w



dp



  d-k˜j δ p + K∗2 + · · · + K∗2qvi |I2qvi (p, K∗2 , · · · , K∗2qvi )|

˜ ∈Sv k j i j=1,··· ,w

 =

 d-k˜j δ p + k˜2 + · · · + k˜|Svi | + K∗|Sv



dp

|+1 i

 + · · · + K∗2qvi ×

˜ ∈Sv k j i j=1,··· ,w

|I2qvi (p, k˜2 , · · · , K∗2qv )| i

 ≤

K∗ |S

sup vi |+1

,··· ,K∗ 2qv

 d-k˜j δ p + k˜2 + · · · + k˜|Svi | +



dp

˜ ∈Sv k j i j=1,··· ,w

i

+ K∗|Sv

|+1 i

+ · · · + K∗2qv

 i

|I2qvi (p, k˜2 , · · · , K∗2qv )| i

= I2qvi Svi

(8.70)

Combining (8.69) and (8.70), we arrive at   w

d-k˜j

j=1



 ∈LG∗ \T ∗

≤



d-p∗

w 

∈L\(T1 ∪···∪Tw )



I2qvi Svi

=



|I2qvi (K∗1 , · · · , K∗2qvi )| 

d-p∗



I2qvi Svi

w  i=1

∈LG∗ \T ∗

i=1 w 

|C (p∗ )|

|C (p∗ )|

∈L\(T1 ∪···∪Tw )

C 1

(8.71)

∈L\(T1 ∪···∪Tw )

i=1

Finally, G S ≤



C ∞

∈T1 ∪···∪Tw

 v=vi i=1,··· ,w

I2qv ∞

w  i=1

I2qvi Svi



C 1

∈L\(T1 ∪···∪Tw )

which proves the lemma.


8.3.2

Single Scale Bounds

The Lemmata 8.3.1 and 8.3.2 estimate a diagram in terms of the L1 - and 1 L - norms of its propagators. An L∞ bound of the type, say, 1+x 2 ≤ 1 is ∞

148 of course very crude since the information is lost that there is decay for large x. In order to get sharp bounds, one introduces a scale decomposition of the covariance which isolates the singularity and puts it at a certain scale. To this end let M be some constant bigger than one and write C(k) = =

=:

χ(|ik0 −ek |≤1) 1 k |>1) + χ(|ikik00−e ik0 −ek = ik0 −ek −ek ∞  χ(M −j−1 <|ik0 −ek |≤M −j ) k |>1) + χ(|ikik00−e ik0 −ek −ek j=0 ∞  j UV

C (k) + C

(k)

(8.72)

j=0

For each C j (k) we have the momentum space bounds C j ∞ ≤ M j ,

C j 1 ≤ c M −j

(8.73)

 since vol{(k0 , k) | k02 + e2k ≤ M −j } ≤ c M −2j . Here c is some j independent constant. Then the strategy to bound a diagram is the following one. Substitute each covariance C of the diagram in (8.51) by its scale decomposition. Interchange the scale-sums with the momentum space integrals. Then one has to bound a diagram where each propagator has a fixed scale. The L1 and L∞ -bounds on these propagators are now sharp bounds. Thus we apply Lemma 8.3.2. The bounds of Lemma 8.3.2 depend on the choice of a tree for the diagram. Propagators on the tree are bounded by their L∞ -norm which is large whereas propagators not on the tree are bounded by their L1 -norm which is small. Thus the tree should be chosen in such a way such that propagators with small scales are on the tree and those with large scales are not on the tree. This is the basic idea of the Gallavotti Nicolo tree expansion [33] which has been applied to the diagrams of the many-electron system in [29],[8]. An application to QED can be found in [17]. Instead of completely multiplying out all scales one can also apply an inductive treatment which is more in the spirit of renormalization group ideas which are discussed in the next section. Let C

≤j

(k) :=

j 

C i (k)

(8.74)

i=0

such that C ≤j+1(k) = C ≤j (k) + C j+1 (k)

(8.75)

We consider a diagram up to a fixed scale j, that is, each covariance is given by C ≤j . Then we see how the bounds change if we go from scale j to scale j +1 by using (8.75). If the diagram has L lines, application of (8.75) produces 2L terms. Each term can be considered as a diagram, consisting of subdiagrams

Feynman Diagrams

149

which have propagators C ≤j , and lines which carry propagators of scale j + 1. Then we apply Lemma 8.3.2 where the generalized vertices I2q are given by the subdiagrams of scale ≤ j which we can bound by a suitable induction hypothesis and all propagators are given by C j+1 . The next lemma specifies Lemmata 8.3.1, 8.3.2 for the case that all propagators have the same scale and satisfy the bounds (8.73) and in Theorem 8.3.4 diagrams are bounded using the induction ∞ outlined above. In the following we will consider only the infrared part j=0 C j of the covariance which contains the physically relevant region around the singularity at the Fermi surface ek = 0 and k0 = 0 and neglect the ultraviolet part C UV (k) in (8.72). The proof that all n’th order diagrams with C UV (k) as propagator are bounded by constn can be found in [29]. Lemma 8.3.3 Let G be a graph as in Lemma 8.3.1 or 8.3.2 and C be some covariance. a) Coordinate Space: Suppose that each C satisfies the estimates C (x) ∞ ≤ c M −j ,

C (x) 1 ≤ c M αj

(8.76)

Then there are the bounds   G ∅ ≤ c|LG | I2qv ∅ M −(qv −1−α)j M (q−1−α)j v∈VG

G S ≤ c|LG |

(8.77)

  I2qv ∅ M −(qv −1−α)j ×



v∈VG,int.





I2qv Sv M −(qv −

|Sv | 2 )j

(8.78) 

M (q−

|S| 2 )j

v∈VG,ext.

and if G has no unpaired legs   I2qv ∅ M −(qv −1−α)j M −(1+α)j |G| ≤ βLd c|LG |

(8.79)

v∈VG

b) Momentum Space: Suppose that each C (k) satisfies the estimates C (k) 1 ≤ c M −j

(8.80)

Then there are the bounds   I2qv ∞ M −(qv −2)j M (q−2)j G ∞ ≤ c|LG |

(8.81)

C (k) ∞ ≤ c M j

v∈VG

G S ≤ c|LG |

  I2qv ∞ M −(qv −2)j ×



v∈VG,int.



v∈VG,ext.



I2qv Sv M −(qv −

|Sv | 2 )j

(8.82) 

M (q−

|S| 2 )j

150 and if G has no unpaired legs   I2qv ∞ M −(qv −2)j M −2j |G| ≤ βLd c|LG |

(8.83)

v∈VG

Proof: a) Apply Lemma 8.3.1. Choose a spanning tree T for G. Suppose G is made of n = v 1 vertices. Then  1  |T | = n − 1, |L| = 2qv − 2q (8.84) 2 v∈V   qv − q − n + 1 = (qv − 1) − q + 1 (8.85) |L \ T | = v∈V

v∈V

Thus  n−1  P G ∅ ≤ c M αj c M −j v∈V = c|L| M −j ( = c|L| M −j



qv −q−n+1

I2qv ∅

v∈V

P v∈V



qv −(1+α)n+1+α−q)

I2qv ∅

v∈V

P v∈V

(qv −1−α)

M j(q−1−α)



I2qv ∅

v∈V

= c|L|

  I2qv ∅ M −(qv −1−α)j M (q−1−α)j

(8.86)

v∈V

which proves (8.77). To wobtain (8.78), we use (8.42). Let w be |Vext |. Choose a union of trees T¯ = i=1 Ti such that each Ti contains one external vertex and T¯ spans G. Then |T¯ | = n − 1 − (w − 1),   |L \ T¯ | = qv − q − (n − 1 − (w − 1)) = qv − p − n + w v∈V

v∈V

Recall that |Sv | is the number of external legs of I2qv and |S| the number of external legs of G, so v∈Vext |Sv | = |S|. Thus one gets n−w  P  c M −j v∈V G S ≤ c M αj = c|L| M −j = c|L|

I2qv Sv

v∈V

P v∈V

(qv −1−α)

M −j((1+α)w−q)

   I2qv ∅ M −(qv −1−α)j × v∈Vint



qv −q−n+w



I2qv Sv

v∈V

   I2qv Sv M −j(qv −1−α) M −(1+α)j M qj

v∈Vext

= c|L|

Feynman Diagrams    I2qv ∅ M −(qv −1−α)j × v∈Vint

 

 P |S| |S| I2qv Sv M −j ( v∈Vext qv − 2 ) M j(q− 2 )

v∈Vext

= c|L|

  v∈Vint

I2qv ∅ M −(qv −1−α)j

 

151



I2qv Sv M −j(qv −

× |Sv | 2 )



M j(q−

|S| 2 )

(8.87)

v∈Vext

The bounds of part (b) are obtained in the same way by using Lemma 8.3.2. The L1 - and L∞ -norms are interchanged on the tree and not on the tree but also the covariance bounds for the L1 - and L∞ -norm are interchanged (now α = 1) which results in the same graph bounds.


8.3.3

Multiscale Bounds

Theorem 8.3.4 Let G be a 2q-legged diagram made from vertices I2qv as in Lemma 8.3.1. Suppose that each line  ∈ LG , the set of all lines of G, carries ∞ the covariance C = j=0 C j where C j satisfies the momentum space bounds (8.73), C j 1 ≤ cM M −j

C j ∞ ≤ cM M j ,

a) Suppose that q ≥ 3, qv ≥ 3 for all vertices and that G has no two- and four-legged subgraphs (which is formalized by (8.101) below). Then one has  |L | I2qv Sv , (8.88) G {1,··· ,2q} ≤ aM G v∈VG

where aM = cM

∞

M−3 . j

j=0

b) Suppose that q ≥ 2, qv ≥ 2 for all vertices and that G has no two-legged subgraphs (which is formalized by (8.115) below). Then one has |L |

G {1,··· ,2q} ≤ rM,|LG | cM G



I2qv Sv ,

(8.89)

v∈VG

where rM,|LG | = 1 +

∞  j=1

j

|L |

j |LG | M − 2 (1 − M − 2 ) ≤ constM G |LG |! (8.90) 1

152 Proof: Let C ≤j =

j i=0

C i and

(2π)d δ(k1 + · · · + k2q ) G ≤j (k1 , · · · , k2q ) =     ≤j v v dk C (k ) (2π)d δ(k1v + · · · + k2q )I2qv (k1v , · · · , k2q ) v v ∈LG

v∈VG

∈LG

We write C ≤j+1 = C ≤j + C j+1 and multiply out: (2π)d δ(k1 + · · · + k2q ) G ≤j+1(k1 , · · · , k2q )      ≤j C + C j+1 (k ) × = dk ∈LG

=

   A⊂LG

=

∈LG

v v (2π)d δ(k1v + · · · + k2q )I2qv (k1v , · · · , k2q ) v v

v∈VG

dk



C ≤j (k )

∈LG \A

∈LG

   A⊂LG A=∅





∈A

v (2π) δ(· · · )I2qv (k1v , · · · , k2q ) v



∈LG \A

∈LG



C j+1 (k ) ×

d

v∈VG

dk



C ≤j (k )



C j+1 (k ) ×

∈A

v (2π)d δ(· · · )I2qv (k1v , · · · , k2q ) v

v∈VG

+ (2π) δ(k1 + · · · + k2q ) G ≤j (k1 , · · · , k2q ) d

(8.91)

The lines in A have scale j + 1 propagators while the lines in LG \ A have C ≤j as propagators. Generalized vertices which are connected by ≤ j lines we consider as a single subdiagram such that, for given A ⊂ LG , every term in (8.91) can be considered as a diagram consisting of certain subgraphs H2qw and only scale j + 1 lines. See the following figure 8.2 for some examples. Thus, let WG,A be the set of connected components which consist of vertices I2qv which are connected by lines in LG \ A. Each connected component is labelled by w ∈ WG,A and is itself a connected amputated diagram with, say, 2qw unpaired legs. These legs may be external legs of G or come from scale j + 1 lines. We write the connected components as (2π)d δ(k1w + · · · + w w )H2qw (k1w , · · · , k2q ) such that we obtain k2q w w (2π)d δ(k1 + · · · + k2q ) G ≤j+1(k1 , · · · , k2q )      w = dk C j+1 (k ) (2π)d δ(· · · )H2qw (k1w , · · · , k2q ) w A⊂LG A=∅

∈A

∈A

w∈WG,A

+ (2π)d δ(k1 + · · · + k2q ) G ≤j (k1 , · · · , k2q )

(8.92)

Feynman Diagrams

153

j I4

I4

j+1

H6

j+1

⇒

j+1

j+1 j+1

I2

j+1 H2=I2

j+1 I4

I4

j+1

j

j+1 j+1

⇒

j+1

H4

I2

H4=I4 j+1

Figure 8.2 w where the delta function δ(· · · ) = δ(k1w + · · · + k2q ). The above expression w can be considered as the value of a diagram with generalized vertices H2qw and lines  ∈ A. Let TG,A ⊂ A be a tree for that diagram. Then we can eliminate all the momentum-conserving delta functions by a choice of loop momenta:

G ≤j+1(k1 , · · · , k2q ) =     dk C j+1 (K ) A⊂LG A=∅

∈A

∈A\TG,A



w H2qw (Kw 1 , · · · , K2qw )

w∈WG,A

+ G ≤j (k1 , · · · , k2q )

(8.93)

where, as in (8.52), K is the sum of momenta flowing through the line . Then we get with the momentum space bound of Lemma 8.3.3 G ≤j+1 − G ≤j ∞ ≤



w∈WG,A

A⊂LG A=∅

G ≤j+1 − G ≤j S ≤

   M −(j+1)(qw −2) H2qw ∞ M (j+1)(q−2)

c|A|



 

c|A|

A⊂LG A=∅



(8.94)

M −(j+1)(qw −2) H2qw ∞ ×

w∈WG,A w int.

   |Sw | |S| M −(j+1)(qw − 2 ) H2qw Sw M (j+1)(q− 2 )

w∈WG,A w ext.

(8.95)

154 Part a) We verify the following bounds by induction on j: For each connected ≤j

amputated 2q-legged diagram G2q with q ≥ 3 one has ≤j

G2q ∞ ≤ c|LG | s(j)|LG | M −j 3 M j(q−2) q



I2qv ∞

(8.96)

v∈VG

and for all S = ∅, {1, · · · , 2q} : ≤j

j

G2q S ≤ c|LG | s(j)|LG | M − 3 (q− 

≤j

G2q {1,··· ,2q} ≤ c|LG | s(j)|LG |

|S| 2 )

M j(q−

|S| 1 2 −3)



I2qv Sv (8.97)

v∈VG

I2qv Sv

(8.98)

v∈VG

j j where s(j) = i=0 M − 3 . Then part (a) is a consequence of (8.98). For j = 0 one has C = C 0 for all lines and one obtains with Lemma 8.3.3: G ∞ ≤ c|LG |

  I2qv ∞ M −(qv −2)0 M (q−2)0 v∈VG

=c

|LG |



s(0)|LG |

I2qv ∞

v∈VG

G S ≤ c|LG |



  I2qv ∞ M −(qv −2)0 ×

v∈VG,int.





I2qv Sv M −(qv −

|Sv | 2 )0



M (q−

|S| 2 )0

v∈VG,ext.

= c|LG | s(0)|LG |



I2qv Sv

(8.99)

v∈VG

Suppose (8.96) through (8.98) are correct for j. Then, to verify (8.96) for j + 1, observe that by (8.93) and the induction hypothesis (8.96) ≤j+1

G2q ≤



c|A|

A⊂LG A=∅

≤



A⊂LG A=∅

≤j+1

∞ ≤ G2q

≤j

≤j

− G2q ∞ + G2q ∞

   ≤j M −(j+1)(qw −2) H2qw ∞ M (j+1)(q−2) + G2q ∞ w∈WG,A

c|A|

  qw M −(j+1)(qw −2) cLw s(j)Lw M −j 3 M j(qw −2) ×

w∈WG,A

 v∈Vw

 I2qv ∞ M (j+1)(q−2) +

Feynman Diagrams  q I2qv ∞ + c|LG | s(j)|LG | M −j 3 M j(q−2) 

=

 

c|LG | s(j)LG \A

v∈VG

M −(qw −2) M −j

qw 3

 

w∈WG,A

A⊂LG A=∅

155

I2qv ∞ M (j+1)(q−2)

v∈VG

q 3

+ c|LG | s(j)|LG | M −j M j(q−2)



I2qv ∞

(8.100)

v∈VG

Now the assumption that G has no two- and four-legged subgraphs means qw ≥ 3

(8.101)

for all possible connected components H2qw in (8.93). Thus we have  

M −(qw −2) M −j

qw 3



≤

w∈WG,A



M −(j+1)

qw 3

|A|+q  j+1 = M− 3

w∈WG,A

and (8.100) is bounded by  |A|+q  j+1 c|LG | s(j)|LG \A| M − 3 I2qv ∞ M (j+1)(q−2)



v∈VG

A⊂LG A=∞ q

q

+ c|LG | s(j)|LG | M −j 3 M (j+1)(q−2) M − 3



I2qv ∞

v∈VG q

= c|LG | M −(j+1) 3

|LG | 



|LG | k

  k  j+1 s(j)|LG |−k M − 3 I2qv ∞ M (j+1)(q−2)

k=1

+c

|LG |

|LG |

s(j)

M

−(j+1) q3

M

(j+1)(q−2)



v∈VG

I2qv ∞

v∈VG q

= c|LG | M −(j+1) 3

|LG | 



|LG | k

  k  j+1 s(j)|LG |−k M − 3 I2qv ∞ M (j+1)(q−2) v∈VG

k=0

q

= c|LG | M −(j+1) 3 =c

|LG |

M

−(j+1) q3

 |LG |  j+1 I2qv ∞ M (j+1)(q−2) s(j) + M − 3 |LG |

s(j + 1)



v∈VG

I2qv ∞ M (j+1)(q−2)

(8.102)

v∈VG

which verifies (8.96). To verify (8.97) and (8.98) for scale j + 1, observe that

156 by (8.95) and the induction hypothesis (8.97,8.98) ≤j+1

G2q

≤j+1

≤j

S ≤ G2q

≤ M (j+1)(q−



|S| 2 )

≤j

− G2q S + G2q S  

c|A|

A⊂LG A=∅

 M −(j+1)(qw −2) H2qw ∞ ×

w∈WG,A w int.

   |Sw | M −(j+1)(qw − 2 ) H2qw Sw + G≤j 2q S w∈WG,A w ext.

≤ M (j+1)(q−



|S| 2 )

 

c|A|

A⊂LG A=∅

M j(qw −2)



j

|Sw | 2 )

I2qv ∞



=M

 

×

  |Sw | M −(j+1)(qw − 2 ) c|Lw | s(j)Lw ×

w∈WG,A w ext.

M j(qw −



|Sw | 1 2 −3)

I2qv Sv

v∈Vw (j+1)(q− |S| 2 )

qw 3

w∈WG,A w int.

v∈Vw

M − 3 (qw −

M −(j+1)(qw −2) c|Lw | s(j)Lw M −j



c|LG |

A⊂LG A=∅

M −(qw −

 

s(j)|LG \A|

|Sw | 2 )

j

|Sw | 2 )

+ G≤j 2q S

M −(qw −2) M −j

w∈WG,A w int.

M − 3 (qw −



M −j 3 1

 

qw 3



×

I2qv Sv

v∈VG

w∈WG,A w ext.

+ G≤j 2q S

(8.103)

Now, for S = ∅ but S = {1, · · · , 2q} may be allowed, one has  

M −(qw −2) M −j

w∈WG,A w int.

≤ M

− j+1 3

qw 3



 

M −(qw −

w∈WG,A w ext.

P

w∈WG,A w int.

M

− 23

qw

P

M

− j+1 3

w∈WG,A w ext.

− 23

|Sw | 2 )

P w∈WG,A w ext.

(qw − |S2w | )

P

= M−

|S| j+1 3 (|A|+q− 2 )

≤ M−

|S| j+1 3 (|A|+q− 2 )

M−3×2 M−3

= M−

|S| j+1 3 (|A|+q− 2 )

M−

M

2

j+1 3

w∈WG,A w ext.

1

j

M − 3 (qw −

M

|Sw | 2 )

(qw − |S2w | )

M −j 3 1



×

− j3 |Vext. |

(qw − |S2w | )

j

M − 3 |Vext. |

j

(8.104)

Feynman Diagrams

157

Therefore it follows from (8.103) for S = ∅ ≤j+1

G2q

S ≤ c|LG | M −

|S| j+1 3 (q− 2 )



M (j+1)(q−



|S| 1 2 −3)

I2qv Sv ×

v∈VG

s(j)|LG \A| M −

j+1 3 |A|

≤j

+ G2q S

(8.105)

A⊂LG A=∅

Now suppose first that S = {1, · · · , 2q}. By the induction hypothesis (8.97) one has  |S| |S| j 1 ≤j I2qv Sv G2q S ≤ c|LG | s(j)|LG | M − 3 (q− 2 ) M j(q− 2 − 3 ) v∈VG

= c

|LG |

|LG |

s(j)

M

− 23 (q− |S| 2 )

≤ c|LG | s(j)|LG | M −

1 3

M M

|S| j+1 3 (q− 2 )

|S| − j+1 3 (q− 2 )

M (j+1)(q−

M

1 (j+1)(q− |S| 2 −3)



|S| 1 2 −3)



I2qv Sv

v∈VG

I2qv Sv

(8.106)

v∈VG

Substituting this in (8.105), one obtains ≤j+1

G2q

S ≤ c|LG | M − 

|S| j+1 3 (q− 2 )

I2qv Sv

v∈VG

|S|

M (j+1)(q− 2 − 3 ) ×  j+1 s(j)|LG \A| M − 3 |A| 1

A⊂LG A=∅

+ c|LG | s(j)|LG | M −

|S| j+1 3 (q− 2 )

M (j+1)(q−

|S| 1 2 −3)



I2qv Sv

v∈VG

= c|LG | M −  v∈VG

= c|LG | M −

|S| j+1 3 (q− 2 )

I2qv Sv

|S|

M (j+1)(q− 2 − 3 ) ×  j+1 s(j)|LG \A| M − 3 |A| 1

A⊂LG |S| j+1 3 (q− 2 )

M (j+1)(q−



|S| 1 2 −3)

I2qv Sv s(j + 1)|LG |

v∈VG

which verifies (8.97) for j + 1. Now let S = {1, · · · , 2q}. Then by the induction hypothesis (8.98) one has ≤j

G2q {1,··· ,2q} ≤ c|LG | s(j)|LG |

 v∈VG

I2qv Sv

(8.107)

and (8.105) becomes ≤j+1

G2q

{1,··· ,2q} ≤ c|LG | M −(j+1) 3 1



I2qv Sv

v∈VG

+ c|LG | s(j)|LG |

 v∈VG

 A⊂LG A=∅

I2qv Sv

s(j)|LG \A| M −

j+1 3 |A|

158

≤ c|LG |



I2qv Sv

v∈VG



s(j)|LG \A| M −

j+1 3 |A|

A⊂LG



= c|LG | s(j + 1)|LG |

I2qv Sv

(8.108)

v∈VG

which verifies the induction hypothesis (8.98) for j + 1. Part b) We verify the following bounds by induction on j: For each connected ≤j

amputated 2q-legged diagram G2q with q ≥ 2 one has 

|L |

≤j

G2q ∞ ≤ cM G j |LG | M j(q−2)

I2qv ∞

(8.109)

v∈VG

and for all S = ∅, {1, · · · , 2q}: |L |

≤j

G2q S ≤ cM G j |LG | M j(q−



|S| 1 2 −2)

I2qv Sv

(8.110)

v∈VG |L |

≤j

G2q {1,··· ,2q} ≤ rM (j) cM G



I2qv Sv

(8.111)

v∈VG

where rM (j) = 1 +

j−1 

i|LG | (M − 2 − M − i

i+1 2

j

) + j |LG | M − 2

(8.112)

i=1

Then part (b) is a consequence of (8.111). For j = 0 one has C = C 0 for all lines and one obtains with Lemma 8.3.3: |L |

G ∞ ≤ cM G



  |L | I2qv ∞ M −(qv −2)0 M (q−2)0 = cM G I2qv ∞

v∈VG |L |

G S ≤ cM G



v∈VG,int.

=



|L | cM G v∈VG

  I2qv ∞ M −(qv −2)0 × 

v∈VG

  |Sv | |S| I2qv Sv M −(qv − 2 )0 M (q− 2 )0

v∈VG,ext.

I2qv Sv

(8.113)

Suppose (8.109) through (8.111) are correct for j. Then, to verify (8.109) for

Feynman Diagrams

159

j + 1, observe that by (8.93) and the induction hypothesis (8.109) ≤j+1

G2q ≤

≤j+1

∞ ≤ G2q



w∈WG,A

A⊂LG A=∅

≤

  |L | M −(j+1)(qw −2) cM w j |Lw | M j(qw −2) ×

|A|

cM

A⊂LG A=∅

≤j

   ≤j M −(j+1)(qw −2) H2qw ∞ M (j+1)(q−2) + G2q ∞

c|A|



≤j

− G2q ∞ + G2q ∞

w∈WG,A

 I2qv ∞ M (j+1)(q−2)



v∈Vw



|L |

+ cM G j |LG | M j(q−2)

I2qv ∞

v∈VG

=



    M −(qw −2) I2qv ∞ M (j+1)(q−2)

|L |

cM G j |LG \A|

w∈WG,A

A⊂LG A=∅

v∈VG



|L |

+ cM G j |LG | M j(q−2)

I2qv ∞

(8.114)

v∈VG

Now the assumption that G has no two-legged subgraphs means qw ≥ 2

(8.115)

for all possible connected components H2qw in (8.93). Thus we have 

M −(qw −2) ≤ 1

w∈WG,A

and (8.114) is bounded by 

|L |

cM G j |LG \A|

A⊂LG A=∞



I2qv ∞ M (j+1)(q−2)

v∈VG |L |

+ cM G j |LG | M (j+1)(q−2)



I2qv ∞

v∈VG

=

|L | cM G

|LG | 



k=1

+

|LG | k



j |LG |−k

 v∈VG

|L | cM G j |LG | M (j+1)(q−2)

I2qv ∞ M (j+1)(q−2)  v∈VG

I2qv ∞

160 |L |

= cM G

|LG | 



|LG | k



k=0

=

|L | cM G (j

=

|L | cM G (j

|LG |

+ 1)



j |LG |−k 

I2qv ∞ M (j+1)(q−2)

v∈VG

I2qv ∞ M (j+1)(q−2)

v∈VG



|LG |

+ 1)

I2qv ∞ M (j+1)(q−2)

(8.116)

v∈VG

which verifies (8.109). To verify (8.110, 8.111) for scale j + 1, observe that by (8.95) and the induction hypothesis (8.110, 8.111) ≤j+1

G2q

≤j+1

S ≤ G2q

≤ M (j+1)(q−

|S| 2 )

≤j

≤j

− G2q S + G2q S



|A|

cM

A⊂LG A=∅

 

 M −(j+1)(qw −2) H2qw ∞ ×

w∈WG,A w int.

   |Sw | M −(j+1)(qw − 2 ) H2qw Sw + G≤j 2q S w∈WG,A w ext.

≤ M (j+1)(q−

|S| 2 )



|A|

cM

A⊂LG A=∅

M j(qw −2)



w∈WG,A w int.

I2qv ∞

|Sw | 1 2 −2)



|L |

M −(j+1)(qw −2) cM w j |Lw | ×

 



=

|Sw | 2 )

|L |

cM w j |Lw | ×

 I2qv Sv + G≤j 2q S

v∈Vw |S| |L | M (j+1)(q− 2 ) cM G

M −(j+1)(qw −

w∈WG,A w ext.

v∈Vw

M j(qw −

 





j |LG \A|

A⊂LG A=∅

M −(qw −2) ×

w∈WG,A w int.

    |Sw | 1 I2qv Sv + G≤j M −(qw − 2 ) M −j 2 2q S

(8.117)

v∈VG

w∈WG,A w ext.

Now, for S = ∅ but S = {1, · · · , 2q} may be allowed, one has  w∈WG,A w int.

M −(qw −2)

   |Sw | 1 M −(qw − 2 ) M −j 2 w∈WG,A w ext.

≤ 1 · M − 2 M −j 2 = M − 1

1

j+1 2

(8.118)

since there is at least one external vertex and at least one internal line because

Feynman Diagrams

161

of A = ∅. Therefore it follows from (8.117) for S = ∅ ≤j+1

G2q

|L |

S ≤ cM G M (j+1)(q−

|S| 1 2 −2)





I2qv Sv

≤j

j |LG \A| + G2q S

A⊂LG A=∅

v∈VG

(8.119) Now suppose first that S = {1, · · · , 2q}. By the induction hypothesis (8.110) one has  |S| 1 ≤j |L | I2qv Sv G2q S ≤ cM G j |LG | M j(q− 2 − 2 ) v∈VG

≤

|S| 1 |L | cM G j |LG | M (j+1)(q− 2 − 2 )



I2qv Sv

(8.120)

v∈VG

Substituting this in (8.119), one obtains ≤j+1

G2q

|L |

S ≤ cM G M (j+1)(q−

|S| 1 2 −2)



I2qv Sv

v∈VG |L |

|L |

|S| 1 2 −2)

|S| 1 2 −2)



|L |

|S| 1 2 −2)

I2qv Sv

v∈VG



I2qv Sv

v∈VG

= cM G M (j+1)(q−

j |LG \A|

A⊂LG A=∅

+ cM G j |LG | M (j+1)(q− = cM G M (j+1)(q−





j |LG \A|

A⊂LG



I2qv Sv (j + 1)|LG |

(8.121)

v∈VG

which verifies (8.110) for j + 1. Now let S = {1, · · · , 2q}. By the induction hypothesis (8.111) one has |L |

≤j

G2q {1,··· ,2q} ≤ rM (j) cM G

 v∈VG

I2qv Sv

(8.122)

where rM (j) = 1 +

j−1 

i|LG | (M − 2 − M − i

i+1 2

j

) + j |LG | M − 2

i=1

and (8.119) becomes ≤j+1

G2q

|L |

{1,··· ,2q} ≤ cM G M −(j+1) 2 1



I2qv Sv

v∈VG |L |

+ rM (j) cM G

 v∈VG

 A⊂LG A=∅

I2qv Sv

j |LG \A|

(8.123)

162    1 |L | I2qv Sv = cM G M −(j+1) 2 (j + 1)|LG | − j |LG | 

|L |

+ rM (j) cM G

v∈VG

I2qv Sv

v∈VG |L |



= rM (j + 1) cM G M −(j+1) 2 1

I2qv Sv

(8.124)

v∈VG

since   1 rM (j) + M −(j+1) 2 (j + 1)|LG | − j |LG | = 1+

j−1   i=1

i|LG | M − 2 − i|LG | M − i

i+1 2



+ j |LG | M − 2 j

  1 + (j + 1)|LG | − j |LG | M −(j+1) 2

= rM (j + 1)

(8.125)

which verifies the induction hypothesis (8.111) for j + 1.


8.4

Ladder Diagrams

In this section we explicitly compute n’th order ladder diagrams and show that they produce factorials. Consider the following diagram:

s+q/2

t+q/2 s

k1

k2

...

kn

t

-s+q/2

-t+q/2

Its value is given by  n Λn (s, t, q) := Π

i=1

dd+1 ki (2π)d+1

n

Π C(q/2 + ki )C(q/2 − ki ) ×

i=1

n−1

V (s − k1 ) Π V (ki − ki+1 ) V (kn − t) i=1

(8.126)

Feynman Diagrams

163

Specializing to the case of a delta function interaction in coordinate space or a constant in momentum space, V (k) = λ, one obtains Λn (s, t, q) = λn+1 Λn (q) where  n n dd+1 k Λn (q) = Π (2π)d+1i Π C(q/2 + ki )C(q/2 − ki ) i=1 i=1 
d+1 n n d k = C(q/2 + k)C(q/2 − k) = {Λ(q)} (8.127) d+1 (2π) and Λ(q) =




dd+1 k (2π)d+1

C(q/2 + k)C(q/2 − k)

(8.128)

is the value of the particle-particle bubble,

Λ(q) =

q→

k

→q

The covariance is given by C(k) = C(k0 , k) = 1/(ik0 − ek ). For small q the value of Λ(q) is computed in the following Lemma 8.4.1 Let C(k) = 1/(ik0 − k2 + 1), let d = 3 and for q = (q0 , q), |q| < 1, let  ∞  dk0 d3 k Λ(q) := (8.129) 2π (2π)3 C(q/2 + k)C(q/2 − k) −∞

|k|≤2

Then for small q Λ(q) = − 8π1 2 log[q02 + 4|q|2 ] + O(1)

(8.130)

Proof: We first compute the k0 -integral. By the residue theorem  ∞ 1 dk0 1 2π i(k + q /2) − e i(−k + q /2) − e−k+q/2 0 0 0 0 k+q/2 −∞  ∞ 1 dk0 1 = 2π k + q /2 + ie k − q /2 − ie−k+q/2 0 0 0 k+q/2 0 −∞

χ(ek+q/2 > 0)χ(e−k+q/2 > 0) 2πi χ(ek+q/2 < 0)χ(e−k+q/2 < 0) = + 2π −q0 − iek+q/2 − ie−k+q/2 q0 + iek+q/2 + ie−k+q/2 =

χ(ek+q/2 > 0)χ(ek−q/2 > 0) χ(ek+q/2 < 0)χ(ek−q/2 < 0) + iq0 + |ek+q/2 | + |ek−q/2 | −iq0 + |ek+q/2 | + |ek−q/2 |

(8.131)

164 To compute the integral over the spatial momenta, we consider the cases ek+q/2 < 0, ek−q/2 < 0 and ek+q/2 > 0, ek−q/2 > 0 separately. Case (i): ek+q/2 < 0, ek−q/2 < 0. Let p = q/2 and let k := |k|, p := |p|, and cos θ = kp/(kp). Then k 2 + 2kp cos θ + p2 < 1, k 2 − 2kp cos θ + p2 < 1 ⇔ k 2 < 1 − p2 , 2kp| cos θ| < 1 − p2 − k 2

(8.132)

then the last inequality in (8.132) gives no If 2kp < 1 − p2 − k 2 or k < 1 − p, restriction on θ. For 1 − p ≤ k < 1 − p2 one gets | cos θ| <

1 − p2 − k 2 2kp

(8.133)

Thus we have 

d3 k χ(ek+q/2 < 0)χ(ek−q/2 < 0) 3 iq0 + |ek+q/2 | + |ek−q/2 | |k|<2 (2π)  2  π χ(ek+q/2 < 0)χ(ek−q/2 < 0) 1 2 dk k dθ sin θ = 4π 2 0 iq0 − 2(k 2 + p2 − 1) 0  1−p  π 1 1 2 + = dk k dθ sin θ 2 2 4π iq0 − 2(k + p2 − 1) 0 0   2 2  √1−p2  π χ | cos θ| < 1−p2kp−k

dk k 2 dθ sin θ iq0 − 2(k 2 + p2 − 1) 0 1−p  1−p 2 1 + dk k 2 = 4π 2 0 iq0 − 2(k 2 + p2 − 1)

 √1−p2 2 2 1−p2 −k2 dk k (8.134) 2kp iq0 − 2(k 2 + p2 − 1) 1−p  Now we substitute x = 1 − p2 − k 2 , k = 1 − p2 − x = kx to obtain  d3 k χ(ek+q/2 < 0)χ(ek−q/2 < 0) 3 iq0 + |ek+q/2 | + |ek−q/2 | |k|<2 (2π)  1−p 1 2 = 2 dk k 2 + 2 4π iq0 − 2(k + p2 − 1) 0

 √1−p2 2 1 − p2 − k 2 dk k 2 + 2kp iq0 − 2(k 2 + p2 − 1) 1−p =

1 4π 2



1−p2

dx kx 2p(1−p)

1 + iq0 + 2x



2p(1−p)

dx 0

1 x 2p iq0 + 2x



Feynman Diagrams

165

 1 kx  log[iq + 2x] + O(1) −  0 4π 2 2 x=2p(1−p) 

 2p(1−p) 1 iq0 dx + 1− 4p iq0 + 2x 0 2p(1−p)

iq0 1−p 1  log[iq0 + 4p(1 − p)] + O(1) − log[iq0 + 2x] = 2 − 4π 2 8p 0

1 1 iq0 = 2 − log[iq0 + 4p(1 − p)] + O(1) − log[1 + 4p(1 − p)/(iq0 )] 4π 2 8p =

=−

1 log[iq0 + 4p(1 − p)] + O(1) 8π 2

(8.135)

Case (ii): ek+q/2 > 0, ek−q/2 > 0. The computation is similar. As above, let p = q/2, k = |k|, p = |p| and cos θ = kp/(kp). Then k 2 + 2kp cos θ + p2 > 1, k >1−p , k 2 > 1 − p2 , 2

2

k 2 > 1 − p2 ,

k 2 − 2kp cos θ + p2 > 1

⇔

±2kp cos θ > 1 − p − k ⇔ 2 2 ∓2kp cos θ < k − (1 − p ) ⇔ 2

2

2kp |cos θ| < k 2 − (1 − p2 )

(8.136)

If 2kp < k 2 − (1 − p2) or k > 1 + p (recall that  |q| < 1), then the last inequality in (8.136) gives no restriction on θ. For 1 − p2 ≤ k < 1 + p one gets | cos θ| < Thus we have 

k 2 − (1 − p2 ) 2kp

(8.137)

d3 k χ(ek+q/2 > 0)χ(ek−q/2 > 0) 3 −iq + |e 0 k+q/2 | + |ek−q/2 | |k|<2 (2π)  2  π 1 χ(ek+p > 0)χ(ek−p > 0) = 2 dk k 2 dθ sin θ 4π 0 −iq0 + 2(k 2 + p2 − 1) 0   k2 −(1−p2 )  1+p  π χ | cos θ| < 2kp 1 + = 2 √ dk k 2 dθ sin θ 2 + p2 − 1) 4π −iq + 2(k 2 0 0 1−p

 π  2 1 dk k 2 dθ sin θ −iq0 + 2(k 2 + p2 − 1) 1+p 0  1+p 2 2 1 2 ) + dk k 2 k −(1−p = 2 √ 2kp 2 + p2 − 1) 4π −iq + 2(k 0 1−p2

 2 2 2 dk k (8.138) −iq0 + 2(k 2 + p2 − 1) 1+p

166  Now we substitute x = k 2 − (1 − p2 ), k = x + 1 − p2 = kx to obtain  d3 k χ(ek+q/2 > 0)χ(ek−q/2 > 0) 3 −iq + |e 0 k+q/2 | + |ek−q/2 | |k|<2 (2π)  1+p 2 k 2 − (1 − p2 ) 1 + = 2 √ dk k 2 4π 2kp −iq0 + 2(k 2 + p2 − 1) 1−p2

 2 2 2 dk k −iq0 + 2(k 2 + p2 − 1) 1+p 

 3+p2 x 1 1 + dx dx kx 2p −iq0 + 2x −iq0 + 2x 0 2p(1+p)  2p(1+p)  1 1 iq0 = 2 dx 1+ 4π 4p −iq 0 + 2x 0

 kx  log[−iq0 + 2x] + O(1) − 2 x=2p(1+p) 2p(1+p) iq0 1  log[−iq0 + 2x] = 2 + O(1) 4π 8p 0

1+p − log[−iq0 + 4p(1 + p)] + O(1) 2

1 iq0 1 = 2 log[1 − 4p(1 + p)/(iq0 )] − log[−iq0 + 4p(1 + p)] + O(1) 4π 8p 2

1 = 2 4π

=−

2p(1+p)

1 log[−iq0 + 4p(1 + p)] + O(1) 8π 2

(8.139)

Combining (8.131), (8.135) and (8.139) we arrive at Λ(q) = −

 1  log[iq + 4p(1 − p)] + log[−iq + 4p(1 + p)] + O(1) 0 0 8π 2

Since p = |q|/2 > 0 we have arg[±iq0 + 4p(1 ∓ p)] ∈ (−π/2, π/2) and therefore log[iq0 + 4p(1 − p)] + log[−iq0 + 4p(1 + p)]   = 12 log[q02 + 16p2 (1 − p)2 ] + log[q02 + 16p2 (1 + p)2 ] + iO(1) = log[q02 + 4|q|2 ] + O(1)

(8.140)

which proves the lemma.
Thus the leading order behavior of Λn (q) for small q is given by  n Λn (q) = {Λ(q)}n ∼ constn log[1/(q02 + 4|q|2 )]

(8.141)

Feynman Diagrams

167

If Λn (q) is part of a larger diagram, there may be an integral over q which then leads to an n!. Namely, if ω3 denotes the surface area of S 3 ,    n ω3 n 1 3 2 2 2 dq0 d q log[1/(q0 + 4|q| )] = dρ ρ3 (log[1/ρ])n 8 0 q02 +4|q|2 ≤1  ∞ ρ=e−x ω3 n 2 = dx e−4x xn 8 0 =

constn n!

(8.142)

Chapter 9 Renormalization Group Methods

In the last section we proved that, for the many-electron system with a short range potential (that is, V (x) ∈ L1 ), an n’th order diagram Gn allows the following bounds. If Gn has no two- and four-legged subgraphs, it is bounded by constn (measured in a suitable norm), if Gn has no two-legged but may have some four-legged subgraphs it is bounded by n! constn (and it was shown that the factorial is really there by computing n’th order ladder diagrams with dispersion relation ek = k2 /2m − µ), and if Gn contains two-legged subdiagrams it is in general divergent. The large contributions from two- and four-legged subdiagrams have to be eliminated by some kind of renormalization procedure. After that, one is left with a sum of diagrams where each n’th order diagram allows a constn bound. The perturbation series for the logarithm of the partition function and similarly those series for the correlation functions are of the form (8.16) log Z(λ) =

∞
n=1




λn n!

signπ Gn (π)

(9.1)

π∈S2n Gn (π) connected

where each permutation π ∈ S2n generates a certain diagram. The sign signπ is present in a fermionic model like the many-electron system but would be absent in a bosonic model. The condition of connectedness and similar conditions like ‘Gn (π) does not contain two-legged subgraphs’ do not significantly change the number of diagrams. That is, the number of diagrams which contribute to (9.1) or to a quantity like F (λ) =

∞
n=1




λn n!

signπ Gn (π) =:

∞


gn λn

(9.2)

n=1

π∈S2n Gn (π) connected, without 2−,4−legged subgraphs

is of the order (2n)! or, ignoring a factor constn , of the order (n!)2 , even in the case that diagrams which contain two- and four-legged subgraphs are removed. If we ignore the sign in (9.2) we would get a bound |F (λ)| ≤

∞
n=0

|λ|n n!

constn (n!)2 =

∞


n! (const|λ|)n

(9.3)

n=0

169

170 and the series on the right hand side of (9.3) has radius of convergence zero. Now there are two possibilities. This also holds for the original series (9.2) or the sign in (9.2) improves the situation. In the first case, which unavoidably would be the case for a bosonic model, there are again two possibilities. Either the series is asymptotic, that is, there is the bound n  
  gj λj  ≤ n! (const|λ|)n (9.4) F (λ) − j=1

or it is not. If one is interested only in small coupling, which is the case we restrict to, an asymptotic series would already be sufficient in order to get information on the correlation functions since (9.4) implies that the lowest order terms are a good approximation if the coupling is small. For the many-electron system with short range interaction the sign in (9.2) indeed improves the situation and one obtains a small positive radius of convergence (at least in two space dimensions) for the series in (9.2). One obtains the bound n  
  gj λj  ≤ (const|λ|)n (9.5) F (λ) − j=1

Concerning the degree of information we can get about the correlation functions or partition function we are interested in (here the quantity F (λ)), there is practically no difference whether we have (9.4) or (9.5). In both cases we can say that the lowest order terms are a good approximation if the coupling is small and not more. We compute the lowest order terms for, say, n = 1, 2 and then it does not make a difference whether the error is 3! (const|λ|)3 or just (const|λ|)3 . However, from a technical point of view, it is much easier to rigorously prove a bound for a series with a small positive radius of convergence, as in (9.5), which eventually may hold for the sum of convergent diagrams for a fermionic model, than to prove a bound for an expansion which is only asymptotic, as in (9.4), which typically holds for the sum of convergent diagrams for a bosonic model. The goal of this section is to rigorously prove a bound of the form (9.5) on the sum of convergent diagrams for a typical fermionic model. By ‘convergent diagrams’ we mean diagrams which allow a constn -bound. In particular, for the many-electron system, this excludes for example ladder diagrams which, although being finite, behave like n! constn . The reason is the following. There are two sources of factorials which may influence the convergence properties of the perturbation series. The number of diagrams which is of order (n!)2 , which, including the prefactor of 1/n! in (9.2), may produce (if the sign is absent) an n! and there may be an n! due to the values of certain n’th order diagrams, like the ladder diagrams. Now, roughly one can expect the following: • A sum of convergent diagrams, that is, a sum of diagrams where each diagram allows a constn bound, is at least asymptotic. That is, its

Renormalization Group Methods

171

lowest order terms are a good approximation if the coupling is small, regardless whether the model is fermionic or bosonic. • A sum of finite diagrams which contains diagrams which behave like n! constn is usually not asymptotic. That is, its lowest order contributions are not a good approximation. Those diagrams which behave like n! constn have to be resummed, for example by the use of integral equations as described in the next chapter. For the many-electron system in two space dimensions a bound of the form (9.5) on the sum of convergent diagrams has been rigorously proven in [18, 16]. The restriction to two space dimensions comes in because in the proof one has to use conservation of momentum for momenta which are close to the Fermi surface k2 /2m − µ = 0 which is the place of the singularity of the free propagator. Then in two dimensions one gets more restrictive conditions than in three dimensions [24]. If the singularity of the covariance would be at a single point, say at k = 0, these technicalities due to the implementation of conservation of momentum are absent and the proof of (9.5) becomes more transparent. Therefore in this section we choose a model with covariance C(k) =

1 d

|k| 2

, k ∈ Rd

(9.6)

and, as before, a short range interaction V ∈ L1 . This C has the same power counting as the propagator of the many-electron system which means that the bounds on Feynman diagrams in Lemma 8.3.3 and Theorem 8.3.4 for d both propagators are the same (with M substituted by M 2 in case of (9.6)). However, as mentioned above, the proof of (9.5), that is, the proof of the Theorems 9.2.2 and 9.3.1 below will become more transparent.

9.1

Integrating Out Scales

In this subsection we set up the scale by scale formalism which allows one to compute the sum of all diagrams at scale j from the sum at scale j − 1. We start with the following

Lemma 9.1.1 Let C, C 1 , C 2 ∈ CN ×N be invertible complex matrices and 1 , dµC 2 be the corresponding Grassmann Gaussian C = C 1 + C 2 . Let dµC , dµCP ¯ −1 − N ¯ ¯ i,j=1 ψi Cij ψj measures, dµC = det C e ΠN i=1 (dψi dψi ). Let P (ψ, ψ) be some

172 polynomial of Grassmann variables. Then   ¯ = ¯ dµC (ψ, ψ) P (ψ 1 + ψ 2 , ψ¯1 + ψ¯2 ) dµC 1 (ψ 1 , ψ¯1 ) dµC 2 (ψ 2 , ψ¯2 ) P (ψ, ψ) (9.7)

Proof: Since thePintegral is linear and every monomial can be obtained from N ¯ the function e−i j=1 (¯ηj ψj +ηj ψj ) by differentiation with respect to the Grass¯ ¯ = e−i¯η,ψ−iη,ψ , mann variables η and η¯, it suffices to prove (9.7) for P (ψ, ψ) N ¯ η , ψ := j=1 η¯j ψj . We have 

¯

e−i¯η,ψ−iη,ψ dµC = e−i¯η,Cη = e−i¯η,C η e−i¯η,C η  1 2 ¯1 ¯2 = e−i¯η,ψ −iη,ψ  dµC 1 e−i¯η ,ψ −iη,ψ  dµC 2  1 2 ¯1 ¯2 = e−i¯η,ψ +ψ −iη,ψ +ψ  dµC 1 dµC 2 (9.8) 1

2

which proves the lemma.
Now let C

≤j

=

j


Ci

(9.9)

i=0

be a scale decomposition of some covariance C and let dµ ≤j := dµ

C ≤j

= det[C

≤j

]e

fi fl ≤j −1 ¯ − ψ,(C ) ψ

¯ Π(dψdψ)

(9.10)

be the Grassmann Gaussian measure with covariance C ≤j . We consider a general interaction of the form ¯ =λ λV(ψ, ψ)

m 
q=1

2q

¯ 2 ) · · · ψ(ξ2q−1 )ψ(ξ ¯ 2q ) Π dξi V2q (ξ1 , · · · ξ2q ) ψ(ξ1 )ψ(ξ

i=1

(9.11) which we assume to be short range. That is, m
q=1

V2q ∅ < ∞

(9.12)

Renormalization Group Methods

173

The connected amputated correlation functions up to scale j are generated ¯ by the functional (from now on we suppress the ψ-dependence, that is, for ¯ brevity we write F (ψ) instead of F (ψ, ψ))  ≤j ≤j 1 (9.13) eλV(ψ+ψ ) dµ ≤j (ψ ≤j ) W (ψ) := log ≤j Z  ≤j (9.14) Z ≤j := eλV(ψ ) dµ ≤j (ψ ≤j ) and the sum of all 2q-legged, connected amputated diagrams up to scale j is ≤j

given by the coefficient W2q in the expansion W

≤j

(ψ) =

∞ 


2q

≤j ¯ 2 ) · · · ψ(ξ2q−1 )ψ(ξ ¯ 2q ) (9.15) Π dξi W2q (ξ1 , · · · ξ2q ) ψ(ξ1 )ψ(ξ

i=1

q=1

W ≤j can be computed inductively from W ≤j−1 in the following way. Lemma 9.1.2 For some function F let F (ψ; η) := F (ψ) − F (η)

(9.16)

Let dµj := dµC j be the Gaussian measure with covariance C j . Define the quantities V j inductively by  P j−1 i j j 1 V (ψ) := log Yj (9.17) e( i=0 V +λV )(ψ+ψ ;ψ) dµj (ψ j )  P j−1 i j e( i=0 V +λV )(ψ ) dµj (ψ j ) (9.18) Yj = where, for j = 0,

−1 i=0

V i := 0 and λV is given by (9.11). Then

W ≤j =

j


V i + λV = W ≤j−1 + V j

(9.19)

i=0

Proof: Induction on j. For j = 0 W ≤0(ψ) = log Z 1

≤0

 eλV(ψ+ψ



= log Y10

eλV(ψ+ψ

0

≤0

;ψ)

)

dµ ≤0

dµ0 + λV(ψ)

= V 0 (ψ) + λV(ψ) since

 Z ≤0 =

0

eλV(ψ ) dµ0 = Y0

(9.20)

(9.21)

174 Furthermore W ≤0(0) = V 0 (0) = 0. Suppose (9.19) holds for j and V i (0) = 0 for all 0 ≤ i ≤ j. Then, using Lemma 9.1.1 in the second line,  ≤j+1 ≤j+1 ) W (ψ) = log eλV(ψ+ψ dµ ≤j+1 − log Z ≤j+1     λV(ψ+ψ j+1 +ψ ≤j ) = log exp log e dµ ≤j dµj+1 − log Z ≤j+1  ≤j j+1 = log eW (ψ+ψ ) dµj+1 + log Z ≤j − log Z ≤j+1  ≤j j+1 ≤j = log eW (ψ+ψ ;ψ) dµj+1 + W ≤j (ψ) + log Z≤j+1 Z    P j i j+1 ≤j V (ψ+ψ ;ψ) 1 Z ≤j i=0 = log Yj+1 dµj+1 + W (ψ) + log Yj+1 ≤j+1 e Z   ≤j = V j+1 (ψ) + W ≤j (ψ) + log Yj+1 Z≤j+1 (9.22) Z

and, since V i (0) for 0 ≤ i ≤ j, 1 V j+1 (0) = log Yj+1

 e

Pj i=0

V i (ψ j+1 )

dµj+1 = log 1 = 0

Since also by definition W ≤j+1 (0) = W ≤j (0) = 0 the constant log Yj+1 in (9.22) must vanish and the lemma is proven


9.2

(9.23) Z ≤j Z



≤j+1

A Single Scale Bound on the Sum of All Diagrams

We consider the model with generating functional  1 W(η) = log Z eλV(η+ψ) dµC (ψ)  Z = eλV(ψ) dµC (ψ) and covariance C(ξ, ξ  ) = δσ,σ C(x − x ) where  dd k C(x) = eikx χ(|k|≤1) d (2π)d |k| 2

(9.24)

(9.25)

The interaction is given by (9.11) which we assume to be short range, that is, V2q ∅ < ∞ for all 1 ≤ q ≤ m. The norm  · ∅ is defined in (8.38). The strategy to controll W2q,n , the sum of all n’th order connected amputated 2q-legged diagrams, is basically the same as those of the last section. We

Renormalization Group Methods

175

introduce a scale decomposition of the covariance and then we see how the bounds change if we go from scale j to scale j +1. Thus, if ρ ∈ C0∞ , 0 ≤ ρ ≤ 1, ρ(x) = 1 for x ≤ 1 and ρ(x) = 0 for x ≥ 2 is some ultraviolet cutoff, let C(k) :=

ρ(|k|) d

|k| 2

=

=

∞
ρ(M j |k|)−ρ(M j+1 |k|) d

j=0 ∞


|k| 2 f (M j |k|) d

j=0

|k| 2

=:

∞


C j (k)

(9.26)

j=0

1 ≤ x ≤ 2 which implies where f (x) := ρ(x) − ρ(M x) has support in M  1 supp C j ⊂ k ∈ Rd | M (9.27) M −j ≤ |k| ≤ 2M −j

Lemma 9.2.1 Let C j (k) = f (M d|k|) be given by (9.26) and let C j (x) = |k| 2

dd k ikx j e C (k). Then there are the following bounds: (2π)d j

a) Momentum space: d

C j (k)∞ ≤ cM M 2 j ,

C j (k)1 ≤ cM M − 2 j d

(9.28)

b) Coordinate space: C j (x)∞ ≤ cM M − 2 j , d

d

C j (x)1 ≤ cM M 2 j

(9.29)

where the constant cM = c(M, d, f ∞ , · · · , f (2d) ∞ ) is independent of j. Proof: The momentum space bounds are an immediate consequence of (9.27). d Furthermore C j (x)∞ ≤ (2π)− 2 C j (k)1 . To obtain the L1 bound on C j (x), observe that  d   j  d k  |(M −2j x2 )N C j (x)| =  (2π) (−M −2j ∆)N eikx f (M d|k|)  d |k| 2   d j   d k ikx (−M −2j ∆)N f (M d|k|)  =  (2π) d e |k| 2  N

dd k  ∂ 2 f (M j |k|)  d−1 ∂ ≤ (M −2j )N (2π)  d  ∂|k|2 + |k| ∂|k| d |k| 2

≤ constN sup{f ∞, · · · , f (2N ) ∞ } |k|≤2M −j dd k 1 d |k| 2

≤ constN sup{f ∞, · · · , f (2N ) ∞ } M

−d 2j

(9.30)

since each derivative either acts on f (M j |k|), producing an M j by the chain d rule, or acts on |k|− 2 , producing an additional factor of 1/|k| which can be

176 estimated against an M j on the support of the integrand. Thus, including the supremum over the derivatives of f into the constant, |[1 + (M −2j x2 )d ]C j (x)| ≤ const M − 2 j d

or M−2j |C (x)| ≤ const 1 + (M −j |x|)2d d

j

(9.31)

from which the L1 bound on C j (x) follows by integration.
Let  W ≤j (ψ) := log =

Z

≤j

∞ 
q=1

eλV(ψ+ψ

1

≤j

)

dµ ≤j (ψ ≤j )

(9.32)

2q

≤j ¯ 2 ) · · · ψ(ξ2q−1 )ψ(ξ ¯ 2q ) Π dξi W2q (ξ1 , · · · ξ2q ) ψ(ξ1 )ψ(ξ

i=1

≤j

be as in (9.13), then W2q is given by the sum of all connected amputated j 2q-legged diagrams up to scale j, that is, with covariance C ≤j = i=0 . We ≤j

≤j

want to control all n’th order contributions W2q,n which are given by W2q = ∞ ≤j n n=1 λ W2q,n . According to Lemma 9.1.2 we have, for all n ≥ 2, ≤j

W2q,n =

j


i V2q,n

(9.33)

i=0

where the V i ’s are given by (9.17). The goal of this subsection is to prove the i . following bounds on the V2q,n

j Theorem 9.2.2 Let V2q,n be given by

 V j (ψ) = log Y1j =

∞
∞
q=1 n=1

e

Pj−1 i=0

 λn



V i +λV (ψ+ψ j ;ψ)

dµj (ψ j )

(9.34)

2q

j ¯ 2q ) Π dξl V2q,n (ξ1 , · · · , ξ2q ) ψ(ξ1 ) · · · ψ(ξ

l=1

where C j is given by(9.26) and let the interaction λV be given by (9.11). n j j Define V2q S, ≤n := =1 |λ| V2q, S where the norms  · ∅ and  · S are

Renormalization Group Methods

177

defined by (8.38) and (8.40). Then there are the following bounds: d

j ∅, ≤n ≤ M 2 j(q−2) V2q

sup

1≤r≤n P qv ≥1, qv ≤nm

 P P q −q 211 v qv cM v v × r  

(9.35)

≤j−1

M − 2 j(qv −2) W2qv ∅, ≤n d



v=1

and for S ⊂ {1, · · · , 2q}, S = ∅ d

j S, ≤n ≤ M 2 j(q− V2q

|S| 2 )

sup

1≤r+s≤n, s≥1 P qv ≥1, qv ≤nm |Sv |<2qv

 P P q −q 211 v qv cM v v ×

r  

(9.36)

 d ≤j−1 M − 2 j(qv −2) W2qv ∅, ≤n ×

v=1 r+s 

  |Sv | ≤j−1 −d j(q − ) v 2 M 2 W2qv Sv , ≤n

v=r+1

where cM is the constant of Lemma 9.2.1.

Proof: To isolate all contributions up to n’th order, we apply n times the R-operation of Theorem 4.4.8. This operation has been introduced in [22] and is an improvement of the integration by parts formula used in [18]. Since the scale j is kept fixed in this proof, we use, only in this proof, the following notation to shorten the formulae a bit. W := W ≤j−1 = V := V

j−1 i=0

V i + λV

j

(9.37) (9.38)

dµ(ψ) := dµj (ψj )

(9.39)

The fields which are integrated over are now called ψ and the external fields, in the statement of the theorem denoted as ψ, will now be denoted as η. In this new notation, the functional integral to be controlled reads  V(η) = log

1 Y

eW(η+ψ;η) dµ(ψ)

(9.40)

and Y such that V(0) = 0. Furthermore we let ξ = (x, σ, b) where b ∈ {0, 1} ¯ for b = 1, or an unbarred one, indicates whether the field is a barred one, ψ, ψ, for b = 0. Recall that W(η + ψ; η) = W(η + ψ) − W(η). We start the

178 expansion as follows:  (9.41) eW(η+ψ;η) dµ(ψ) V(η) = log Y1   1 d dt log Y1 eW(tη+ψ;tη) dµ(ψ) = dt 0   1 d + ψ; tη) eW(tη+ψ;tη) dµ(ψ) dt W(tη

= eW(tη+ψ;tη) dµ(ψ) 0   1 ∞
2q
d |I c | = dξ W (ξ) dt ( dt t ) Π η(ξi ) × Π l 2q c q=1

l=1

0

I⊂{1,··· ,2q} I=∅

i∈I

Πi∈I ψ(ξi ) eW(tη+ψ;tη) dµ(ψ) eW(tη+ψ;tη) dµ(ψ)

where I c := {1, · · · , 2q} \ I denotes the complement of I. For a fixed choice of q and I, we may represent the term in (9.41) graphically as in figure 9.1. ψ η ψ

ψ ψ

W8 ψ

ψ

η

Figure 9.1 Since the bare interaction (9.11) is by assumption at most 2m-legged, m ≥ 2 some fixed chosen number, a 2q-legged contribution is at least of order max{1, q/m}. Since we are interested only in contributions up to n’th order, we can cut off the q-sum at nm. To the functional integral in the last line of (9.41) we apply the R-operation of Theorem 4.4.8. In order to do so we write Wtη (ψ) := W(ψ + tη) − W(tη) such that with Theorem 4.4.8 we obtain

Πi∈I ψ(ξi ) eWtη (ψ) dµ(ψ)

= (9.42) eWtη (ψ) dµ(ψ)

 R (Πi∈I ψ(ξi )) (ψ) eWtη (ψ) dµ(ψ)

Π ψ(ξi ) dµ(ψ) + eWtη (ψ) dµ(ψ) i∈I

Renormalization Group Methods

179

where  ∞  
1 ˜ ˜ − Wtη (ψ)] ˜ k : dµ(ψ) (9.43) R Π ψ(ξi ) (ψ) = Π ψ(ξi ) : [Wtη (ψ + ψ) k! i∈I

k=1 ∞


=

i∈I

 k Π ψ(ξi ) : [W(ψ + ψ˜ + tη) − W(ψ˜ + tη)] : dµ(ψ)

1 k!

i∈I

k=1

Since we are interested only in contributions up to n’th order, we could cut off the k-sum at n. Letting η˜ := ψ˜ + tη, we get   ˜ = R Π ψ(ξi ) (ψ) (9.44) i∈I ∞


k=1






∞


1 k!

dξ 1 · · · dξ k W2q1 (ξ 1 ) · · · W2qk (ξ k ) ×

q1 ,··· ,qk =1

···

I1 ⊂{1,··· ,2q1 } I1 =∅






k

Ik ⊂{1,··· ,2qk } Ik =∅

Π Π c η˜(ξi )

k

Π ψ(ξi ) : Π Π ψ(ξi ) : dµ(ψ)

r=1 i∈Ir

r=1 i∈Ir

i∈I

Substituting this in (9.42), we obtain

 Πi∈I ψ(ξi ) eWtη (ψ) dµ(ψ)

W (ψ) = Π ψ(ξi ) dµ(ψ) + e tη dµ(ψ) i∈I  ∞ ∞

1 dξ 1 · · · dξ k W2q1 (ξ 1 ) · · · W2qk (ξ k ) × k! q1 ,··· ,qk =1

k=1




I1 ⊂{1,··· ,2q1 } I1 =∅




···



Ik ⊂{1,··· ,2qk } Ik =∅

(9.45)

k

Π ψ(ξi ) : Π Π ψ(ξi ) : dµ(ψ) × r=1 i∈Ir

i∈I

Πkr=1 Πi∈Irc (ψ + tη)(ξi ) eW(ψ+tη;tη) dµ(ψ)

eW(ψ+tη;tη) dµ(ψ)

The first term in (9.45) we can define as the k = 0 term of the k-sum in (9.45). By multiplying out the (ψ + tη) brackets in the last line of (9.45) we obtain

Πi∈I ψ(ξi ) eWtη (ψ) dµ(ψ)

W (ψ) = (9.46) e tη dµ(ψ)  ∞ ∞

1 dξ 1 · · · dξ k W2q1 (ξ 1 ) · · · W2qk (ξ k ) × k! k=0

q1 ,··· ,qk =1




I1 ⊂{1,··· ,2q1 } I1 =∅


J1 ⊂I1c

···




···

Ik ⊂{1,··· ,2qk } Ik =∅


Jk ⊂Ikc

k



k

i∈I

Π Π c tη(ξi )

r=1 i∈Jr

Π ψ(ξi ) : Π Π ψ(ξi ) : dµ(ψ) ×

r=1 i∈Ir

Πkr=1 Π i∈Jr ψ(ξi ) eW(ψ+tη;tη) dµ(ψ) eW(ψ+tη;tη) dµ(ψ)

180 Graphically we may represent this as follows. The functional integral on the left hand side of (9.46) comes with a generalized vertex W2q , see (9.41) and figure 9.1. |I c | legs of W2q are external (η-fields) and are no longer integrated over. |I| legs of W2q are internal (ψ-fields) and are integrated over, that is, they have to be contracted. The value of k in the first sum of (9.46) is the number of new vertices produced by one step of the R-operation to which the |I| legs of W2q may contract. In figure 9.2, k = 3. If k = 0, then all these fields have to contract among themselves. q1 , · · · , qk in the second sum of (9.46) fix the number of legs of the new vertices. I1 , · · · , Ik specify which of those legs have to contract to the |I| legs of W2q . These legs are not allowed to contract among themselves. Observe that, because of the restriction Ir = ∅, there is at least one contraction between a new vertex and W2q . This contraction will be made explicit in Lemma 9.2.3, part (a), below and also has been made explicit in figure 9.2.

η η

(1)

ψI

ψI

ψI

(1)

W(0) 8

W2

η

ψI

η

W4

(1)

W4

ψJ

ψI ψI

ψJ

Figure 9.2 Finally, J1 , · · · , Jk specify which legs of the new vertices which have not been contracted to the W2q fields will be external (those in Jrc ) or will produce new vertices in a second step of the R-operation. The improvement of this expansion step compared to the integration by part formula used in [18], Lemma II.4, is the following. The only place where ‘dangerous’ factorials due to the number of diagrams (or number of contractions) may arise is the integral  k (9.47) Π ψ(ξi ) : Π Π ψ(ξi ) : dµ(ψ) i∈I

r=1 i∈Ir

In [18], this corresponds to the ‘primed integral’ in formula (II.4). Now the point is that to the integral in (9.47) we can apply Gram’s inequality of Theorem 4.4.9 which immediately eliminates potential factorials and gives a bound constnumber of fields times the right power counting factor whereas in [18] we could not use Gram’s inequality because the determinant which is

Renormalization Group Methods

181

given by the ‘primed integral’ has a less ordered structure. To produce all contributions which are made of at most n generalized vertices W2qv , we repeat the R-operation n-times. This produces a big sum of the form nm

n
q=1

I

nm


n






k1 =1 q11 ···q11 =1 I11 ···I 11 J11 ···J 11 k

n 


···

k

k

nm








k

k

k


   K q, I, {k , q , I , J } (η)

nm


···

k2 =0 q12 ···q22 =1 I12 ···I 22 J12 ···J 22

(9.48)

n =1 n kn =0 q1n ···qk I1 ···Iknn n

The purpose of the square brackets above is only to group sums together which belong to the same step of R-operation. In the n’th application of R we have J1n = · · · = Jknn = ∅ such that the corresponding functional integral in the last line of (9.46) is equal to 1 since otherwise this would be a contribution with at least n + 1 generalized vertices W2qv . Figure 9.3 shows the term in figure 9.2 after a second step of R-operation.

(1)

W2

(0)

W8

(1)

(2)

W4

W6

(1)

W4

(2)

W4

Figure 9.3 The contributions K look as follows:   K q, I, {k , q , I , J } (η)  n n n k = Π k1 ! dξ Π dξ W2q (ξ) Π Π W2qr (ξ r ) × =1

(9.49)

=1 r=1

=1

 n k    

1  d n Int I, I 1 (ξ) Π Int J −1 , I (ξ) 0 dt dt Π c tη(ξi ) Π Π =2

i∈I

Π tη(ξr,i )

=1 r=1 i∈Jr c

182 

  Π c dξr,i η(ξr,i ) K q, I, {k , q , I , J } (ξi , ξr,i )

k

Π c dξi η(ξi ) Π

=:

r=1 i∈Jr

i∈I

where the last equation defines the kernels K. The integrals Int in the second line of (9.49) are given by    −1  k k , I (ξ) := Int J Π Π ψ(ξr,i ) : Π Π ψ(ξr,i ) : dµC (ψ) (9.50) r=1 i∈Jr

r=1 i∈Ir

 l Let K = ∞ with respect to the l=1 λ Kl be the power series expansion of K ∞ coupling λ (which enters (9.49) only through W2q = l=1 λl W2q,l ) and let |K(ξ)| ≤n :=

n


|λ|l |Kl (ξ)|

(9.51)

l=1

Then we have, suppressing the argument q, I, {k , q , I , J } of K at the moment, ˜ ≤n ≤ (9.52) |K(ξ)|     1  n n n k Π k1 ! dξ Π dξ |W2q (ξ)| ≤n Π Π |W2qr (ξ r )| ≤n Int I, I (ξ) × =1

=1 r=1

=1

   n k   1  d  −1 , I (ξ) 0 dt dt Π Int J Π c tδ(ξi − ξ˜i ) Π Π n

If we abbreviate the sum in (9.48) by all 2p-legged contributions up to n’th

Π tδ(ξr,i − ξ˜r,i )

=1 r=1 i∈Jr c

i∈I

=2



q,I,{k ,q ,I  ,J  } , then a first n order, l=1 λl V2p,l , reads as

|V2p (ξ)| ≤n ≤
q,I,{k ,q ,I  ,J  }

bound on follows: (9.53)



n

n

k



c



χ Σ k ≤ n − 1, |I c | + Σ Σ |Jr | = 2p |K(ξ)| ≤n =1 r=1

=1

with the bound (9.52) on |K(ξ)| ≤n. Here we introduced  1 if A is true χ(A) = 0 if A is false

(9.54)

to enforce a 2p-legged contribution and to restrict the number of generalized vertices to at most n. By integrating (9.53) over the ξ-variables, we obtain the analog bounds for the  · S, ≤n-norms, S = ∅ or S ⊂ {1, · · · , 2p}. V2p (ξ)S, ≤n ≤




  χ · · · K(ξ)S, ≤n

(9.55)

q,I,{k ,q ,I  ,J  }

Although K is not a single diagram, but, if we would expand the integrals Int in (9.52), is given by a sum of diagrams, we can obtain a bound on KS in

Renormalization Group Methods

183

the same way as we obtained the bounds in Lemma 8.3.1 for single diagrams. In that lemma first we had to choose a tree for the diagram and then, in (8.41, 8.42) one has to take the L∞ -norm for propagators not on the tree and the L1 -norm for propagators on the tree. A natural choice of a tree for K is the following one. We choose the vertex W2q which was produced at the very beginning in (9.41) as the root. Every vertex W2qr which is produced at the ’th step of R-operation is connected by at least one line to some vertex W2qs−1 which is the meaning of Ir = ∅. For each W2qr we choose exactly one of these lines to be on the tree for K. The point is that this tree would be a tree for all the diagrams which can be obtained from (9.52) by expanding the integrals Int, but now we can use the sign cancellations by applying a determinant bound, that is, Gram’s inequality, to the integrals  Int J −1 , I (ξ) in (9.52). This is done in part (b) of Lemma 9.2.3 below. First however, we must make explicit the  propagators for lines on the tree which are also contained in Int J −1 , I (ξ) (the factors in the wavy brackets in (9.56) below) since they are needed to absorb some of the coordinate space ξ-integrals (the remaining ξ-integrals are absorbed by the generalized vertices W2qr ). This is done in part (a) of the following Lemma 9.2.3 a) Let J, I1 , · · · , Ik ⊂ N be some finite sets, let Ir = ∅ for 1 ≤ r ≤ k and let ir1 := min Ir . Recall the definition (9.50) for the Integrals Int. Then     Int J, ∪kr=1 Ir (ξ) ≤ (9.56)   
 k   k r Π |C(ξir1 , ξjr )| Int J \ {j1 , · · · , jk }, ∪r=1 (Ir \ {i1 })  j1 ,··· ,jk ∈J jr =js

r=1

b) Let J, I ⊂ N be some finite sets and let  dd k ik(x−x ) f (M j |k|) C(x, x ) = C j (x − x ) = e d (2π)d |k| 2 be the covariance of Lemma 9.2.1. Then |I|+|J|   √ d sup Int(J, I)(ξ) ≤ 2c M − 4 j

(9.57)

ξ

Proof: a) We have, using the notation of Definition 4.4.6,    k k (9.58) Int J, ∪r=1 Ir (ξ) = Π ψ(ξj ) : Π rΠ ψ(ξir ): dµC r=1 j∈J i ∈Ir  k  = Π ψ(ξj ) Π rΠ ψ (ξir ): dµC  r=1 i ∈Ir j∈J  k  = ± ψ  (ξi11 ) · · · ψ  (ξir1 ) Π ψ(ξj ) Π Π r ψ (ξir ): dµC  r j∈J

r=1 i ∈Ir \{i1 }

184 =


j1 ∈J

 ψ  (ξi21 ) · · · ψ  (ξir1 ) ×

(±)C(ξi11 , ξj1 )

k

Π

j∈J\{j1 }




=

ψ(ξj ) Π

Π

r=1 ir ∈Ir \{ir1 }

ψ  (ξir ): dµC 

(±)C(ξi11 , ξj1 ) · · · C(ξik1 , ξjk ) ×

j1 ,··· ,jk ∈J jr =js



ψ(ξj )

Π

j∈J\{j1 ,··· ,jk }

k

Π

Π

r=1 ir ∈Ir \{ir1 }

ψ  (ξir ): dµC 

which proves part (a) by taking absolute value. Part (b) is an immediate consequence of (4.112) of Theorem 4.4.9: We have  1  C(x, x ) = (9.59) dd k fx (k) gx (k) (2π)d   d d where fx (k) = eikx f (M j |k|)/|k| 4 , gx (k) = e−ikx f (M j |k|)/|k| 4 and H = L2 (Rd ) with norms   √ j d fx L2 , gx L2 ≤ dd k f (M d|k|) = C j (k)L1 ≤ c M − 4 j (9.60) |k| 2

Application of (4.112) gives (9.57).
We now continue with the proof of Theorem 9.2.2. The combination of (9.56) and (9.57) gives the following bound on the integrals Int(J −1 , I ). Recall that I = (I1 , · · · , Ik  ). Let i r,1 := min Ir . Then     (9.61) Int(J −1 , I )(ξ) ≤ −1    k |J |+|I |−2k  √
d j 2c M − 4 j Π |C (ξir,1 , ξjr )| j1 ,··· ,jk ∈J −1

r=1

Now that we have made explicit the propagators on the spanning tree for K we can estimate as in Lemma 8.3.1 to obtain the following bound. We substitute (9.61) in (9.52) and get (k 0 := 1, q10 := q, J 0 = I 0 := I)  1 Pn Pk  k n  c 1 d |I c | K∅, ≤n ≤ Π k ! Π W2qr ∅, ≤n dt ( dt t ) t =1 r=1 |Jr | × r=1 =0 0  √ |J −1 |+|I  |−2k 
n k j −d j 4 2c M Π Π C (x)L1 =1

−1 j1−1 ,··· ,j −1  ∈J



n

≤ |I | Π c

=1

k

|J −1 | k

r=1





n

Π W2qr ∅, ≤n c M

d 2j

Pn=1 k

=0

 2c M

−d 2j

Pn=1

„

×

|J −1 |+|I  | −k 2

«

(9.62)

Renormalization Group Methods

185

The sum in (9.53) or more explicitly, the sum in (9.48) we bound as follows:


· · · ≤ 2|Ir | sup · · · ≤ 22qr sup · · · 



Jr ⊂Ir

Jr ⊂Ir






· · · ≤ 22qr

sup Ir ⊂{1,··· ,2qr }

Ir ⊂{1,··· ,2qr } ∞


··· ≤

qr =1




∞
qr =1

··· ≤

k

1  2qr



≤2

Pk−1 r=1



(9.65)

qr

sup  k

k |

(9.64)

qr

k

−1

···

sup 2qr · · · = sup 2qr · · ·


 |J −1 | 

≤ 2|J

(9.63)

Jr ⊂Ir

sup 

1

k

|J −1 | k

2qr−1

sup 

1 |J −1 | k

 ···

 ···

k

1 |J −1 | k

 ···

(9.66)

Thus, combining (9.62), (9.55) and the above estimates we arrive at  V2p ∅, ≤n ≤

n

k

n



k

χ(· · · ) Π Π 27qr |I c | Π Π W2qr ∅, ≤n ×

sup

=0 r=1

q,I,{k ,q ,I  ,J  }

=0 r=1

Pn Pn  d  d  k  =1 c M 2 j =1 2c M − 2 j

 ≤

sup q,I,{k ,q ,I  ,J  }

n

k



n

„

|J −1 |+|I  | −k 2

«

k

χ(· · · ) Π Π 29qr Π Π W2qr ∅, ≤n × =0 r=1

=0 r=1

Pn Pn  d  d  k  =1 2cM 2 j =1 2cM − 2 j

„

|J −1 |+|I  | −k 2

«

(9.67) The factors of M ± 2 j are treated exactly in the same way as in the proof of Lemma 8.3.3. To this end observe that a fixed choice of q, I, {k , q , I , J } fixes the number of generalized vertices and the number of ψ’s which have to be paired. In other words, all diagrams which contribute to a given K(q, I, {k , q , I , J }) have the same number of generalized vertices W2qr ≡ W2qv , 1 ≤ v ≤ ρ, ρ = ρ(q, I, {k , q , I , J }) the number of generalized vertices, and the same number of lines, d

L=

n
|J −1 | + |I | =1

2

ρ n k 
1 

= 2qr − 2p = qv − p 2 r=1 v=1 

=0

(9.68)

186 although the individual positions of the lines are not fixed, only the lines on the tree are fixed. The number of lines on the tree is given by T =

n


k = ρ − 1

(9.69)

=1

Thus (9.67) can be written as  P P ρ V2p ∅, ≤n ≤ sup 211 v qv c v qv −p Π W2qv ∅, ≤n ×

(9.70)

v=1

1≤ρ≤n, qv ≥1 P v qv ≤nm

 d j ρ−1  − d j Pv qv −p−ρ+1 M2 M 2



In the same way we obtain for S = ∅, if w counts the number of external vertices,  P P ρ sup (9.71) 211 v qv c v qv −p Π W2qv ∅, ≤n × V2p S, ≤n ≤ v=1

1≤ρ+w≤n, w≥1 P qv ≥1, v qv ≤nm |Sv |<2qv

ρ+w

Π

v=ρ+1

 d ρ−w  − d j Pv qv −p−ρ+w M 2 W2qv Sv , ≤n M 2 j



The rearrangement of the factors of M ± 2 j in the above inequalities into the form of (9.35) and (9.36) is then exactly done as in the proof of Lemma 8.3.3. Thus we have reached the end of proof of Theorem 9.2.2.
d

9.3

A Multiscale Bound on the Sum of Convergent Diagrams

We saw already in section 8.3 that all diagrams without two- and fourlegged subdiagrams were finite. In this subsection we show that the sum of all those diagrams is also finite and has in fact a small positive radius of convergence. This is a nontrivial result since the number of diagrams is of order (2n)! ∼ constn (n!)2 but there is only a factor of 1/n! which comes from the expansion of the exponential. Thus, if we would expand down to all diagrams, or more precisely, if we would expand  the integrals Int in (9.52), we would be left with a series of the form n n! λn which has zero radius of convergence. However, in the last subsection we generated all n’th order diagrams in the following way. First we chose a tree. For an n’th order diagram, there are about n! ways of doing this. Then we treated the sum

Renormalization Group Methods

187

of all diagrams obtained by pairing all legs which are not on the tree as a single term K. Thus K corresponds to a sum of ∼ n! diagrams. However, by using the sign cancellations between these diagrams, that is, by applying Lemma 9.2.3 which results from Gram’s inequality, we were able to eliminate the factorial coming from pairing all fields (or legs or halflines) which were not on the tree. In this subsection we use Theorem 9.2.2 to give an inductive proof that the sum of all diagrams without two- and four-legged subdiagrams has positive radius of convergence. As in the last section, we consider the simplified model d with propagator 1/|k| 2 . The proof for the many-electron system is given in [18] (for d = 2). First we give a precise mathematical definition of ‘sum of all diagrams without two and four legged subdiagrams’. The sum of all connected 2qlegged diagrams W2q is given by  W(η) = log Z1 eλV(ψ+η) dµC (ψ) =

∞ 


2q

η (ξ2 ) · · · η(ξ2q−1 )¯ η (ξ2q )(9.72) Π dξi W2q (ξ1 , · · · , ξ2q ) η(ξ1 )¯

i=1

q=1

According to Lemma 9.1.2, W2q can be inductively computed: ∞


W =

V j + λV

(9.73)

j=0

where

 V j (ψ) = log Y1j

Pj−1

e(

i=0

V i +λV )(ψ+ψ j ;ψ)

dµC j (ψ j )

(9.74)

 i where, for j = 0, −1 i=0 V := 0 and λV is given by (9.11). Let Q2,4 be the projection operator acting on Grassmann monomials which takes out twoand four-legged contributions. That is, Q2,4

∞ 
q=1

:=

∞ 
q=3

2q

¯ 2 ) · · · ψ(ξ2q−1 )ψ(ξ ¯ 2q ) Π dξi W2q (ξ1 , · · · , ξ2q ) ψ(ξ1 )ψ(ξ

i=1 2q

¯ 2 ) · · · ψ(ξ2q−1 )ψ(ξ ¯ 2q ) (9.75) Π dξi W2q (ξ1 , · · · , ξ2q ) ψ(ξ1 )ψ(ξ

i=1

The sum of diagrams W 2,4 which is estimated in Theorem 9.3.1 below is inductively defined as W 2,4 =

∞
j=0

V 2,4,j + λV

(9.76)

188 where

 V 2,4,j (ψ) := Q2,4 log Y 12,4

Pj−1

e(

i=0

V 2,4,i +λV )(ψ+ψ j ;ψ)

dµC j (ψ j ) (9.77)

j

d

the covariance given by (9.27), C j (k) = f (M j |k|)/|k| 2 , and the interaction λV given by (9.11). This is actually a little bit more than the sum of diagrams without two- and four-legged subgraphs, namely: Consider a diagram.  For each line, substitute the covariance by its scale decomposition C = j C j , interchange the scale sum with the coordinate or momentum space integrals which come with the diagram. This gives a scale sum of labelled diagrams. Then W 2,4 is the sum of all labelled diagrams, the sum going over all diagrams and over all scales, which do not contain any two- and four-legged subdiagrams where the highest scale of the subdiagram is smaller than the lowest scale of its (two or four) external legs. 2,4 (ξ) be the sum of all 2q-legged connected diagrams Theorem 9.3.1 Let W2q without two- and four-legged subgraphs defined inductively in (9.76) and (9.77) above. The interaction given mby (9.11) is assumed to be short range and at most 2m-legged, V ∅ := p=1 V2p ∅ < ∞. Let f1 , · · · , f2q be some test  4 (2q+1)  functions with fi 1,∞ := fi 1 + fi ∞ < ∞. Let M := 3 · 211 d , let cM be the constant of Lemma 9.2.1 and let   1 (9.78) |λ| ≤ min 4·222m1 V 2 , 1 c2m ∅

M

Then there is the bound  2q 2q q 2,4 Π dξi fi (ξi ) |W2q (ξ1 , · · · , ξ2q )| ≤ 3|λ| 2m Π fi 1,∞ i=1

(9.79)

i=1

∞ 2,4,j + λV Proof: According to (9.76) and (9.77) we have W 2,4 = j=0 V where  P j−1 2,4,i +λV )(ψ+ψ j ;ψ) V 2,4,j (ψ) := Q2,4 log Y 12,4 dµC j (ψ j ) (9.80) e( i=0 V j

We now prove the following bounds by induction on j. For λ satisfying (9.78), for all q ≥ 3 and for S ⊂ {1, · · · , 2q}, S = ∅, q

d

2,4,j V2q ∅ ≤ |λ| 2m M 2 j(q−2) 2,4,j V2q S ≤ |λ|

q 2m

M

d 2j

(

1 q− |S| 2 −4

(9.81) ) Π f  i 1,∞

(9.82)

i∈S

We start with (9.81). For j = 0, every vertex is given by some λV2qv , 1 ≤ qv ≤ ∞ j=0 j=0 j=0 be the n’th order contribution to V2q = n=max{1,q/m} λn V2q,n . m. Let V2q,n

Renormalization Group Methods

189

Since the interaction is at most 2m-legged, a 2q-legged contribution must be at least of order (2q)/(2m) = q/m. From the proof of Theorem 9.2.2 we get

j=0 ∅ V2q,n

≤M

d 2 0(q−2)

sup P

qv ≥1,

qv ≤nm

 P n   P  11 v qv −d 0(qv −2) v qv −q 2 cM V2qv ∅ M 2 v=1

 n ≤ 211m cm M V ∅

(9.83)

which gives, for, say,  |λ| ≤

1 1 , 2 211m cm M V ∅




j=0 V2q ∅ ≤

(9.84)

 11m m n q 2 cM V ∅ |λ|n ≤ |λ| 2m

(9.85)

n=max{1,q/m}

and proves (9.81) for j = 0. Suppose (9.81) holds for 0 ≤ i ≤ j. From (9.35) and (9.77) we have d

2,4,j+1 V2q ∅, ≤n ≤ M 2 (j+1)(q−2) × (9.86)  P r   P  2,4, ≤j 11 v qv −d (j+1)(qv −2) v qv −q 2 M cM W2qv ∅, ≤n sup 2 1≤r≤n P qv ≥3, qv ≤nm

v=1

By the induction hypothesis (9.81), 2,4, ≤j

W2qv

∅, ≤n

≤ ≤

2,4, ≤j

W2qv j


∅

2,4,i V2q ∅ + χ(qv ≤ m) |λ|V2qv ∅ v

i=0

≤

j


qv

|λ| 2m M 2 i(qv −2) + χ(qv ≤ m) |λ| V ∅ d

i=0 qv ≥3

≤

M

d 2

≥2

qv

M

d 2 j(qv −2)

|λ| 2m 1−

1

d M − 2 (qv −2) qv

qv

≤

2 M 2 j(qv −2) |λ| 2m + |λ| 2m

≤

3 M 2 j(qv −2) |λ| 2m

d d

qv

+ χ(qv ≤ m) |λ| 2

(9.87)

190 Substituting this in (9.86), the supremum becomes sup 1≤r≤n P qv ≥3, qv ≤nm

= |λ|

q 2m

 P r   P  qv d d q −q M − 2 (j+1)(qv −2) 3 M 2 j(qv −2) |λ| 2m 211 v qv cM v v v=1

sup 1≤r≤n P qv ≥3, qv ≤nm

≤ |λ|

q 2m

sup 1≤r≤n P qv ≥3, qv ≤nm

= |λ|

q 2m

 P P P q −q 211 v qv cM v v |λ| v

q qv 2m − 2m

r  

3M

−d 2 (qv −2)



v=1

 P P P q −q 211 v qv cM v v |λ| v

q qv 2m − 2m

r  

3

qv

M

−d 2

qv 3



v=1



sup

3·2 M 11

−d 6

Pv qv  Pv qv −q  1 2m |λ| cM

1≤r≤n P qv ≥3, qv ≤nm q

≤ |λ| 2m

(9.88)

if we first choose d

M 6 ≥ 3 · 211

(9.89)

and then let |λ| ≤

1 c2m M

(9.90)

This proves (9.81). We now turn to (9.82). For j = 0, we obtain as in (9.83)  n n j=0 S ≤ 211m cm V2q,n Π V Sv M v=1 n  ≤ 211m cm Π fi 1,∞ M V ∅

(9.91)

i∈S

which proves (9.82) for j = 0 if we choose λ as in (9.84). Suppose now that (9.82) holds for 0 ≤ i ≤ j. From (9.36) and (9.77) we get d

2,4,j+1 S, ≤n ≤ M 2 (j+1)(q− V2q

r  

sup

1≤r+s≤n, s≥1 P qv ≥3, qv ≤nm |Sv |<2qv

 P P q −q 211 v qv cM v v ×

2,4, ≤j

M − 2 (j+1)(qv −2) W2qv d

v=1 r+s  

M

v=r+1

|S| 2 )

|Sv | −d 2 (j+1)(qv − 2 )

 ∅, ≤n ×

2,4, ≤j W2qv Sv , ≤n

 (9.92)

Renormalization Group Methods

191

By the induction hypothesis (9.82) we obtain, for |Sv | < 2qv , 2,4, ≤j

W2qv

≤

2,4, ≤j

Sv , ≤n ≤ W2qv j


 Sv

2,4,i V2q Sv + χ(qv ≤ m) |λ|V2qv Sv v

i=0

≤


j

qv

|λ| 2m M 2 i(qv − d

|Sv | 1 2 −4)

 1

+ χ(qv ≤ m) |λ| 2

Π fi 1,∞

i∈Sv

i=0

≤

 |Sv | d 1 M 2 j(qv − 2 − 4 )

qv

|λ| 2m

qv



Π fi 1,∞ |Sv | d 1 i∈Sv 1 − M − 2 (qv − 2 − 4 )   qv |Sv | qv d 1 |λ| 2m j(q − − ) v 2 4 ≤ M2 + |λ| 2m Π fi 1,∞ d i∈Sv 1 − M−8 qv

|Sv | 1 2 −4)

≤ 3|λ| 2m M 2 j(qv − d

+ |λ| 2m

Π fi 1,∞

(9.93)

i∈Sv

d

if we choose M 8 ≥ 2. Substituting (9.93) and (9.87) into (9.92), we obtain 2,4,j+1 V2q S, ≤n

≤M

|S| d 2 (j+1)(q− 2 )

sup

1≤r+s≤n, s≥1 P qv ≥3, qv ≤nm |Sv |<2qv

 P P q −q 211 v qv cM v v ×

r    qv d 3M − 2 (qv −2) |λ| 2m × v=1 r+s  

3M

|Sv | −d 2 (qv − 2 )

M

−d 8j

|λ|

qv 2m

v=r+1 d

≤ M 2 (j+1)(q−

|S| 2 )

sup

Π fi 1,∞



i∈Sv

q

M − 8 (j+1) |λ| 2m ×  P r   Pv qv −q   1 d 11 v qv 2m |λ| cM 3M − 6 qv × 2 d

1≤r+s≤n, s≥1 P qv ≥3, qv ≤nm |Sv |<2qv

v=1 r+s 

 3M

|Sv | −d 2 (qv − 2 )

M

d 8



Π fi 1,∞

(9.94)

i∈S

v=r+1

where we used the fact that there is at least one external vertex, s ≥ 1, to d make the factor M − 8 (j+1) explicit. Since |Sv | < 2qv , we have r+s  v=r+1

r+s    Pr+s ! |Sv | |Sv | d d d M − 2 (qv − 2 ) M 8 ≤ M − 4 (qv − 2 ) ≤ M −ε v=r+1 qv (9.95) v=r+1

192 if r+s 
qv −

d 4

|Sv | 2



≥ε

v=r+1

qv

v=r+1

r+s


⇔

r+s


qv −

|S| 2

≥

v=r+1

qv

v=r+1 r+s


⇔

r+s


4ε d

qv ≥

v=r+1

|S| 1 4ε 2 1− d

(9.96)

Again, since |Sv | < 2qv , we have r+s


2qv ≥ |S| + 1

(9.97)

v=r+1

Thus, (9.96) is satisfied if we can choose ε such that 1 |S| |S| + 1 ≥ 2 2 1 − 4ε d 1 d ⇔ ε≤ 4 |S| + 1 Thus, choosing ε =

d 1 4 |S|+1

≤

d

2,4,j+1 V2q S, ≤n ≤ M 2 (j+1)(q−

d 8

(9.98)

we obtain from (9.94)

|S| 1 2 −4)

q

|λ| 2m Π fi 1,∞ × i∈S

 Pv qv −q   Pv qv  1 11 −ε 2m 3·2 M |λ| cM

sup 1≤r+s≤n, s≥1 P qv ≥3, qv ≤nm |Sv |<2qv d

≤ M 2 (j+1)(q−

|S| 1 2 −4)

q

|λ| 2m Π fi 1,∞

(9.99)

i∈S

if we first choose M sufficiently large such that  1  4 (|S|+1)  M ≥ 3 · 211 ε = 3 · 211 d

(9.100)

and then choose λ sufficiently small such that (9.90) holds. This proves (9.82) provided that |λ| ≤ min



1 1 2 , 2m 4·222m c2m M V ∅ cM

 (9.101)

Renormalization Group Methods

193

In particular, from (9.82) we infer 2,4 W2q {1,··· ,2q} ≤

∞


V 2,4,j {1,··· ,2q} + χ(q ≤ m)|λ|V2q {1,··· ,2q}

j=0

 ≤

|λ|

q 2m

∞


M

−d 8j

j=0 q

+ |λ|

q 2m



2q

Π fi 1,∞

i=1

2q

≤ 3|λ| 2m Π fi 1,∞ i=1

which proves the theorem.


9.4

Elimination of Divergent Diagrams

In this section we show that one possibility to get rid of two-legged subdiagrams is the addition of a suitable chosen counterterm. The corresponding lemma reads as follows. Lemma 9.4.1 Let V be a quartic interaction,

¯ 1 )ψ(ξ ¯ 2 )ψ(ξ3 )ψ(ξ4 ) ¯ = dξ1 dξ2 dξ3 dξ4 V (ξ1 , ..., ξ4 )ψ(ξ V(ψ, ψ)

(9.102)

and let Q be a quadratic counterterm,

¯ = dξ1 dξ2 Q(ξ1 , ξ2 )ψ(ξ ¯ 1 )ψ(ξ2 ) Q(ψ, ψ) (9.103) ∞ where Q(ξ1 , ξ2 ) = n=1 Qn (ξ1 , ξ2 )λn . Consider the generating functional

(9.104) G(η) = log eλV(ψ+η)+Q(ψ+η) dµC (ψ) = W(η) + λV(η) + Q(η) where, using the notation V(ψ; η) := V(ψ) − V(η),  W(η) = log eλV(ψ+η;η)+Q(ψ+η;η) dµC (ψ) =

∞ 


(9.105)

η (ξ1 ) · · · η(ξ2q ) dξ1 · · · dξ2q W2q (ξ1 , ..., ξ2q )¯

q=0

and dµC is the Grassmann Gaussian measure for some covariance C. Then: a) Let W2,n = W2,n (ξ1 , ξ2 ) be the n’th order contribution to the expansion ∞ W2 = n=1 W2,n λn . Then, as a function of the Qj ’s, W2,n depends only on Q1 , ..., Qn−1 ,   W2,n = W2,n Q1 , ..., Qn−1 (9.106)

194 b) Let

be some operation on the W2,n ’s, for example = id or W2 = dξ W2 (ξ, 0). Let r = id− be the renormalization operator. Determine the Qn ’s inductively by the equation   Qn = − W2,n Q1 , ..., Qn−1

(9.107)

Then G2 = rW2 and W2q is given by the sum of all 2q-legged connected amputated diagrams where each two-legged subdiagram has to be evaluated with the r operator. In particular, for = id one has G2 = 0, the full two-point function of the changed model (by the addition of the counterterm Q) coincides with the free propagator, and G2q is the sum of all 2q-legged, connected amputated diagrams without two-legged subdiagrams.

Proof: a) W2 is given by the sum of all two-legged connected amputated diagrams made from four-legged vertices V coming from the interaction and two-legged vertices Q coming from the counterterm. To obtain an n’th order diagram, there may be k four-legged vertices and one Qn−k or n − k Q1 ’s for example. If there is a Qn , there cannot be anything else. Since all diagrams contributing to W have at least one contraction, this diagram would be a loop with no external legs, thus it does not contribute to W2,n . This proves part (a). b) Consider a diagram Γ2q,r of some order r contributing to W2q,r . It is made from four-legged vertices V and two-legged vertices Q. Suppose this diagram has a two-legged subdiagram T = Tn which is of order n. We write Γ2q,r = S2q,r−n ∪ Tn to distinguish between the two-legged subdiagram and the rest of the diagram. Now consider the sum of all diagrams which are obtained by combining S2q,r−n with an arbitrary n’th order two-legged diagram. Apparently, this sum is given by S2q,r−n ∪ W2,n + S2q,r−n ∪ Qn = S2q,r−n ∪ (W2,n + Qn ) = S2q,r−n ∪ rW2,n . Thus, part (b) follows.


Thus, we can eliminate two-legged subdiagrams by the addition of a suitable chosen counterterm. Typically, for infrared problems the counterterms are finite. Using the techniques of this chapter (plus a sectorization of the Fermi surface which is needed to implement conservation of momentum) one can show [18, 20, 16] that for the many-electron system with short-range interaction the renormalized sum of all diagrams without four-legged subgraphs is still well behaved, that is, its lowest order terms are a good approximation. This holds for ek = k2 /(2m)− µ as well as for an ek with an anisotropic Fermi surface. The theorem reads as follows:

Renormalization Group Methods

195

Theorem 9.4.2 Let d = 2, V (x) ∈L1 and C(k) = 1/(ik ∞ 0 − ek ). Then there ∞ is a finite counterterm δe(λ, k) = n=1 δen (k) λn , n=1 δen ∞ |λ|n < ∞, such that the perturbation theory of the model inductively defined by

λ W 0 (ψ) = − βL vk−p ψ¯k↑ ψ¯q−k↓ ψp↑ ψq−p↓ + δe(k, λ)ψ¯kσ ψkσ (9.108) d k,p,q

W j+1 (ψ ≥j+1 ) = (Id − P4 ) log



k,σ

eW

j

(ψ

≥j+1

+ψ j )

dµC j (ψ j )

(9.109)

has positive radius of convergence independent of volume and temperature. The projection P4 is defined by ∞  ¯ ¯ P4 (9.110) k1 ,...,k2q W2q (k1 , ..., k2q )ψk1 ψk2 · · · ψk2q−1 ψk2q := q=2  ¯ ¯ k1 ,...,k4 W4 (k1 , ..., k4 )ψk1 ψk2 ψk3 ψk4 Proof: The proof can be found in the research literature [18, 20, 16].


9.5

The Feldman-Kn¨ orrer-Trubowitz Fermi Liquid Construction

In the preceding sections we saw that the sum of all diagrams without two- and four-legged subgraphs is analytic. Two-legged subgraphs have to be renormalized which can be done by the addition of a counterterm. By doing this one obtains the theorem above, the renormalized sum of all diagrams without four-legged subgraphs is also analytic for sufficiently small coupling, in two dimensions and for short-range interaction. This is true for the model with kinetic energy ek = k2 /(2m) − µ which has a round Fermi surface F = {k | ek = 0} but also holds for models with a more general ek which may have an anisotropic Fermi surface. Thus, the last and the most complicated step in the perturbative analysis consists of adding in the four-legged diagrams. These diagrams determine the physics of the model. In section 8.4 we saw that ladder diagrams, for ek = k2 /(2m) − µ, have a logarithmic singularity at zero transfer momentum which may lead to an n!. This singularity is a consequence of the possibility of forming Cooper pairs: two electrons, with opposite momenta k and −k, with an effective interaction which has an attractive part, may form a bound state. Since at low temperatures only those momenta close to the Fermi surface are relevant, the formation of Cooper pairs can be suppressed, if one substitutes (by hand) the energy momentum relation ek = k2 /(2m)−µ by a more general expression with an anisotropic Fermi surface. That is, if momentum k is on the Fermi surface, then momentum −k is not on F for almost all k, see the following figure:

196

F ∼

,

F ∩ (−F ) ∼

Figure 9.4: An allowed Fermi surface for the FKT-Fermi liquid theorem For such an ek one can prove that four-legged subdiagrams no longer produce any factorials, an n’th order diagram without two-legged but not necessarily without four-legged subgraphs is bounded by constn . As a result, Feldman, Kn¨ orrer and Trubowitz could prove that, in two space dimensions, the renormalized perturbation series for such a model has in fact a small positive radius of convergence and that the momentum distribution a+ kσ akσ  has a jump discontinuity across the Fermi surface of size 1 − δλ where δλ > 0 can be chosen arbitrarily small if the coupling λ is made small. The complete rigorous proof of this fact is a larger technical enterprise [20]. It is distributed over a series of ten papers with a total volume of 680 pages. Joel Feldman has set up a webpage www.math.ubc.ca/˜feldman/fl.html where all the relevant material can be found. The introductory paper ‘A Two Dimensional Fermi Liquid, Part 1: Overview’ gives the precise statement of results and illustrates, in several model computations, the main ingredients and difficulties of the proof. As FKT remark in that paper, this theorem is still not the complete story. Since two-legged subdiagrams have been renormalized by the addition of a counterterm, the model has been changed. Because ek has been chosen anisotropic, also the counterterm δe(k) is a nontrivial function of k, not just a constant. Thus, one is led to an invertability problem: for given e(k), is there a e˜(k) such that e˜(k) + δ˜ e(k) = e(k) ? This is a very difficult problem if it is treated on a rigorous level [28, 55]. In fact, if we would choose = id in (9.107) (which we only could do in the Grassmann integral formulation of the model, since in that case the counterterm would depend on the d + 1-dimensional momenta (k0 , k), as the quadratic action in the Grassmann integral, but ek in the Hamiltonian only depends on the d-dimensional momentum k), then r = 0 and two-legged subgraphs were eliminated completely. In that case the  −1 interacting two-point function would be exactly given by ik0 − ek if the quadratic part of the action is chosen to be ik0 − ek + counterterm(k0 , k), the counterterm being the sum of all two-legged diagrams without two-legged subdiagrams. Thus in that case the computation of the two-point function would be completely equivalent to the solution of an invertibility problem.

Chapter 10 Resummation of Perturbation Series

10.1

Starting Point and Typical Examples

In the preceding chapters we wrote down the perturbation series for the partition function and for various correlation functions of the many-electron system, and we identified the large contributions of the perturbation series. These contributions have the effect that, as usual, the lowest order terms of the naive perturbation expansion are not a good approximation. We found that, for a short range interaction, n’th order diagrams without two- and four-legged subgraphs allow a constn bound, diagrams with four- but without two-legged subgraphs are also finite but they may produce a factorial, that is, there are diagrams with four-legged subgraphs which behave like n! constn . And finally diagrams with two-legged subdiagrams are in general infinite if the cutoffs are removed. Typically one can expect the following: If one has a sum of diagrams where each diagram allows a constn bound, then this sum is asymptotic. That is, its lowest order contributions are a good approximation if the coupling is small. If the model is fermionic, the situation may even be better and, due to sign cancellations, the series may have a positive radius of convergence which is independent of the cutoffs. If one has a sum of diagrams where all diagrams are finite but certain diagrams behave like n! constn , one can no longer expect that the lowest order terms are a good approximation. And, of course, the same conclusion holds if the series contains divergent contributions. Basically there are two strategies to eliminate the anomalously large contributions of the perturbation series, the use of counterterms and the use of integral equations. In the first case one changes the model and the computation of the correlation functions of the original model requires the solution of an invertability problem which, depending on the model, may turn out to be very difficult [28, 55]. In the second case one resums in particular the divergent contributions of the perturbation series, if this is possible somehow, which leads to integral equations for the correlation functions. Conceptually, this is nothing else than a rearrangement of the perturbation series, like in the example of section 8.1 (which, of course, is an oversimplification since in that case one does not have to solve any integral equation). The good thing in having integral equations is that the renormalization is

197

198 done more or less automatically. The correlation functions are obtained from a system of integral equations whose solution can have all kinds of nonanalytic terms (which are responsible for the divergence of the coefficients in the naive perturbation expansion). If one works with counterterms one more or less has to know the answer in advance in order to choose the right counterterms. However, the bad thing with integral equations is that usually it is impossible to get a closed system of equations without making an uncontrolled approximation. If one tries to get an integral equation for a two-point function, one gets an expression with two- and four-point functions. Then, dealing with the four-point function, one obtains an expression with two-, four- and six-point functions and so on. Thus, in order to get a closed system of equations, at some place one is forced to approximate a, say, six-point function by a sum of products of two- and four-point functions. In this chapter, which is basically the content of [46], we present a somewhat novel approach which also results in integral equations for the correlation functions. It applies to models where a two point function can be written as
(10.1) S(x, y) = [P + Q]−1 x,y dµ(Q) . Here P is some operator diagonal in momentum space, typically determined by the unperturbed Hamiltonian, and Q is diagonal in coordinate space. The functional integral is taken with respect to some probability measure dµ(Q) and goes over the matrix elements of Q. [ · ]−1 x,y denotes the x, y-entry of the matrix [P + Q]−1 . The starting point is always a model in finite volume and with positive lattice spacing in which case the operator P + Q and the functional integral in (10.1) becomes huge- but finite-dimensional. In the end we take the infinite volume limit and, if wanted, the continuum limit. Models of this type are for example the many-electron system, the ϕ4 model and QED. In Theorem 5.1.2 we showed, using a Hubbard-Stratonovich transformation, that the two-point function of the many-electron system with delta interaction can be written in momentum space as
1 βLd


ψ¯kσ ψkσ  =

⎡ ⎣

ap δp,p
√ig

βLd

φp−p


√ig

βLd

φ¯p
−p

a−p δp,p


⎤−1 ⎦

dP ({φq }) (10.2)

kσ,kσ

where ap = ip0 − (p2 /2m − µ) and dP ({φq }) is the normalized measure ⎡ ⎤ √ig φ¯p
−p ap δp,p
P 2 d 1 βL ⎦ e− q |φq | Π dφq dφ¯q (10.3) dP ({φq }) = det ⎣ ig √ φp−p
a−p δp,p
Z q d βL

For the ϕ4 -model, one can also make a Hubbard-Stratonovich transformation to make the action quadratic in the ϕ’s and then integrate out the ϕ’s. One

Resummation of Perturbation Series ends up with (see section 10.5 below)
 ap δp,p
−
ϕk ϕ−k  =

√ig Ld

γp−p


−1 k,k

199

dP ({γq })

(10.4)

d where now ap = 4M 2 i=1 sin2 (pi /2M ) + m2 , 1/M being the lattice spacing in coordinate space, and dP ({γq }) is the normalized measure (γ−q = γ¯q ) dP ({γq }) =

1 Z

 det ap δp,p
−

√ig Ld

γp−p


− 12

e−

P q

|γq |2

γq (10.5) Πq dγq d¯

Thus for both models one has to integrate an inverse matrix element. The only difference is that in the bosonic model, the functional determinant in the integration measure has a negative exponent, here -1/2 since we consider the real scalar ϕ4 -model, and for the fermionic model the exponent is positive, +1. An example where the integrand is still an inverse matrix element but where the integration measure is simply Gaussian without a functional determinant is given by the averaged Green’s function of the Anderson model. In momentum space, it reads (see section 10.3 below)
 −1 2 dv d¯ q vq ap δp,p
+ √λ d vp−p

G(k) =
G(k; E, λ, ε) = Πq e−|vq | π L d k,k

RL

d where, for unit lattice spacing in coordinate space, ap = 4 i=1 sin2 (pi /2) − E − iε. The rigorous control of
G(k) for small disorder λ and energies inside the spectrum of the unperturbed Laplacian, E ∈ [0, 4d], in which case ap has a root if ε → 0, is still an open problem [2, 41, 48, 52, 60]. It is expected that limε0 limL→∞
G(k) = 1/(ak − σk ) where Imσ = O(λ2 ). In all three cases mentioned above one has to invert a matrix and perform a functional integral. Now the main idea of our method is to invert the above matrices with the following formula: Let B = (Bkp )k,p∈M ∈ CN ×N , M some index set, |M| = N and let

(10.6) G(k) := B −1 kk Then one has G(k) =

(10.7) 1

Bkk +

N  r=2

(−1)r+1



Bkp2 Gk (p2 )Bp2 p3 · · · Bpr−1 pr Gkp2 ···pr−1 (pr )Bpr k

p2 ···pr ∈M\{k} pi =pj

−1 where Gk1 ···kj (p) = (Bst )s,t∈M\{k1 ···kj } pp is the p, p entry of the inverse of the matrix which is obtained from B by deleting the rows and columns

200 labelled by k1 , · · · , kj . For the Anderson model, the matrix B is of the form B = self adjoint + iε Id, which, for ε = 0, has the property that all submatrices (Bst )s,t∈M\{k1 ···kj } are invertible. There is also a formula for the off-diagonal inverse matrix elements. It reads −1

= −G(k)Bkp Gk (p) + (10.8) B kp N  r=3

(−1)r+1



G(k)Bkt3 Gk (t3 )Bt3 t4 · · · Btr p Gkt3 ···tr (p)

t3 ···tr ∈M\{k,p} ti =tj

These formulae also hold in the case where the matrix B has a block structure Bkp = (Bkσ,pτ ) where, say, σ, τ ∈ {↑, ↓} are some spin variables. In that case the Bkp are small matrices, the Gk1 ···kj (p) are matrices of the same size and the 1/· in (10.7) means inversion of matrices. In the next section we prove the inversion formulae (10.7), (10.8) and in the subsequent sections we apply them to the above mentioned models.

10.2 10.2.1

Computing Inverse Matrix Elements An Inversion Formula

In this section we prove the inversion formulae (10.7), (10.8). We start with the following Lemma 10.2.1 Let B ∈ Ck×n , C ∈ Cn×k and let Idk denote the identity in Ck×k . Then: (i) Idk − BC invertible

⇔

Idn − CB invertible.

(ii) If the left or the right hand side of (i) fullfilled, then C 1 Idn −CB C. Proof: Let

⎞ − b1 − ⎟ ⎜ .. B=⎝ ⎠, .

− bk − ⎛

1 Idk −BC

=

⎞ | | C = ⎝ c1 · · · ck ⎠ | | ⎛

column where the bj are n-component row vectors and the cj are n-component  vectors. Let x ∈ Kern(Id − CB). Then x = CB x = j λj cj if we define  λj := ( bj , x). Let λ = (λj )1≤j≤k . Then [(Id − BC) λ]i = λi − j ( bi , cj )λj =

Resummation of Perturbation Series

201

  ( bi , x) − j ( bi , cj )λj = 0 since x = j λj cj , thus λ ∈ Kern(Id − BC). On the  λj cj ∈ Kern(Id − CB) other hand, if some λ ∈ Kern(Id − BC), then x := which proves (i). Part (ii) then follows from C = 1 Idn −CB C(Idk − BC)


j 1 Idn −CB

(Idn − CB)C =

The inversion formula (10.7) is obtained by iterative application of the next lemma, which states the Feshbach formula for finite dimensional matrices. For a more general version one may look in [4], Theorem 2.1.





A B C D k×(n−k)

Lemma 10.2.2 Let h = are invertible and B ∈ C

h invertible

∈ Cn×n where A ∈ Ck×k , D ∈ C(n−k)×(n−k)

, C ∈ C(n−k)×k . Then A − BD−1 C invertible

⇔

−1

⇔

D − CA

(10.9)

B invertible

and if one of the conditions in (10.9) is fullfilled, one has h−1 = where E= F = −EBD

−1

1 A−BD−1 C −1

= −A

,

BH ,

H=

1 D−CA−1 B −1

G = −HCA

, = −D



E F G H



(10.10) −1

CE . (10.11)

Proof: We have, using Lemma 10.2.1 in the second line, A − BD−1 C inv. ⇔ Idk − A−1 BD−1 C inv. ⇔ Idn−k − D−1 CA−1 B inv. ⇔ D − CA−1 B inv. Furthermore, again by Lemma 10.2.1, D−1 C

1 Id−A−1 BD−1 C

=

1 Id−D−1 CA−1 B

D−1 C =

1 D−CA−1 B

C = HC

A−1 B

1 Id−D−1 CA−1 B

=

1 Id−A−1 BD−1 C

A−1 B =

1 A−BD−1 C

B = EB

and

which proves the last equalities in (10.11), HCA−1 = D−1 CE and EBD−1 = A−1 BH. Using these equations and the definition of E, F, G and H one computes         A B E F F A B Id 0 = E C D G H G H C D = 0 Id

202 It remains to show that the invertibility of  h implies   the invertibility of A −  Id 0 −1 −1 ¯ C= BD C. To this end let P = 0 , P = Id such that A − BD −1 ¯ ¯ ¯ ¯ P hP − P hP (P hP ) P hP . Then (A − BD−1 C)P h−1 P = P hP h−1 P − P hP¯ (P¯ hP¯ )−1 P¯ hP h−1 P = P h(1 − P¯ )h−1 P − P hP¯ (P¯ hP¯ )−1 P¯ h(1 − P¯ )h−1 P = P − P hP¯ h−1 P + P hP¯ h−1 P = P and similarly P h−1 P (A − BD−1 C) = P which proves the invertibility of A − BD−1 C


Theorem 10.2.3 Let B ∈ CnN ×nN be given by B = (Bkp )k,p∈M , M some index set, |M| = N , and Bkp = (Bkσ,pτ )σ,τ ∈I ∈ Cn×n where I is another index set, |I| = n. Suppose that B and, for any N ⊂ M, the submatrix (Bkp )k,p∈N is invertible. For k ∈ M let

G(k) := B −1 kk ∈ Cn×n

(10.12)



GN (k) := {(Bst )s,t∈M\N }−1 kk ∈ Cn×n

(10.13)

and, if N ⊂ M, k ∈ / N,

Then one has the following: (i) The on-diagonal block matrices of B −1 are given by G(k) =

(10.14) 1

Bkk −

N   (−1)r Bkp2 Gk (p2 )Bp2 p3 · · · Bpr−1 pr Gkp2 ···pr−1 (pr )Bpr k r=2

p2 ···pr ∈M\{k} pi =pj

where 1/ · is inversion of n × n matrices. (ii) Let k, p ∈ M, k = p. Then the off-diagonal block matrices of B −1 can be expressed in terms of the GN (s) and the Bst , [B −1 ]kp = −G(k)Bkp Gk (p) −

N  r=3

(−1)r



(10.15)

G(k)Bkt3 Gk (t3 )Bt3 t4 · · · Btr p Gkt3 ···tr (p)

t3 ···tr ∈M\{k,p} ti =tj

Resummation of Perturbation Series

203

Proof: Let k be fixed and let p, p ∈ M \ {k} below label columns and rows. By Lemma 10.2.2 we have ⎛ ⎜ ⎜ ⎝

⎛ ⎞−1 ⎞ E −F − Bkk − Bkp − ⎜ ⎟ ⎜ ⎟ ⎟ ⎟ = ⎜ | ⎟ = ⎜ | ⎟ ⎝ ⎠ ⎝ ⎠ ⎠

∗ Bp p G H Bp k | | ⎞

G(k)

⎛

where

G(k) = E =

Bkk −

 p,p
=k

=

Bkk −



1

Bkp {(Bp
p )p
,p∈M\{k} }−1 pp
Bp
k

Bkp Gk (p)Bpk −

 p,p
=k p=p


p=k

(10.16)

1



Bkp {(Bp
p )p
,p∈M\{k} }−1 pp
Bp
k

and 



Bkt {(Bp
p )p
,p∈M\{k} }−1 tp Fkp = B −1 kp = −G(k) t=k

= −G(k)Bkp Gk (p) − G(k)



(10.17)



Bkt {(Bp
p )p
,p∈M\{k} }−1 tp

t=k,p

Apply Lemma 10.2.2 now to the matrix {(Bp
p )p
,p∈M\{k} }−1 and proceed by induction to obtain after  steps

G(k) =

(10.18) 1

Bkk −



(−1)r

r=2



Bkp2 Gk (p2 )Bp2 p3 · · · Bpr−1 pr Gkp2 ···pr−1 (pr )Bpr k − R +1

p2 ···pr ∈M\{k} pi =pj

Fkp = −G(k)Bkp Gk (p) −

 r=3

(−1)r



(10.19) ˜ +1 G(k)Bkt3 Gk (t3 )Bt3 t4 · · · Btr p Gkt3 ···tr (p) − R

t3 ···tr ∈M\{k,p} ti =tj

204 where R +1 = (−1)



Bkp2 Gk (p2 ) · · · Bp
−1 p
×

p2 ···p
+1 ∈M\{k} pi =pj

(10.20)

{(Bp
p )p
,p∈M\{kp2 ···p
} }−1 p


p
+1

˜ +1 = (−1) R



G(k)Bkt3 · · · Gkt3 ···t
−1 (t )Bt
t
+1 ×

t3 ···t
+1 ∈M\{k,p} ti =tj

Bp
+1 k (10.21)



{(Bp
p )p
,p∈M\{kt3 ···t
} }−1 t


+1 p

˜ N +1 = 0 the theorem follows.
Since RN +1 = R

10.2.2

Field Theoretical Motivation of the Inversion Formula

In many field theoretical models one has to deal with determinants of the form det[Id + B]. The standard approximation formula is det[Id + B] = eT r log[Id+B] ≈ e

Pn r=1

(−1)r+1 r

T rB r

(10.22)

with n = 1 or n = 2. For large B, this approximation is pretty bad. The 2 1 reason is that the approximation, say, det[Id + B] ≈ eT rB− 2 T rB does not keep track of the fact that in the expansion of det[Id + B] with respect to B every matrix element of B can only come with a power of 1, but in the 2 1 expansion of eT rB− 2 T rB matrix elements of B with arbitrary high powers are produced. This can be seen already in the one-dimensional situation. In 1 2 det[Id + x] = 1 + x the x comes with power 1, but in elog[1+x] ≈ ex− 2 x , the x comes with arbitrary high powers. However, one can eliminate these higher powers by introducing anticommuting variables, since they have the property ¯ n = 0 for all n ≥ 2. that (ψψ) Consider first the one-dimensional case. Introduce the  ¯Grassmann algebra ¯ ¯ and eψψ ¯ = 1 as well ¯ We have eψψ = 1 + ψψ dψdψ with generators ψ and ψ.  ¯ ¯ ψψ ψψ ¯ = 1. Thus, abbreviating dµ := e dψdψ ¯ ¯ e dψdψ we may write as ψψ

¯ x] ¯ x)dµ = elog[1+ψψ 1 + x = (1 + ψψ dµ (10.23) Now, if we apply (10.22) for, say, n = 2 to the right hand side of (10.23), we end up with


2 1 ¯ ¯ ¯ ¯ elog[1+ψψ x] dµ ≈ eψψ x− 2 (ψψ x) dµ = eψψ x dµ = 1 + x (10.24)

Resummation of Perturbation Series

205

which is the exact result. The following lemma is the generalization of (10.23) to N -dimensional matrices.

Lemma 10.2.4 Let B = (Bkp )k,p∈M ∈ CN ×N , N = |M|. Then a)

det[Id + λB] =

N 



λn

n=0

detk1 ···kn B

(10.25)

{k1 ,··· ,kn }⊂M

where detk1 ···kn B := det[(Bkp )k,p∈{k1 ,··· ,kn } ].
b)

det[Id + B] =

where dµ := e gral.

P k∈M

¯k ψk ψ



det Id + (ψ¯k ψk Bkp )k,p∈M dµ

(10.26)

Πk∈M dψ¯k dψk and the integral is a Grassmann inte-

Proof: a) Let bp = (Bkp )k∈M denote the column vectors of B and let {e1 , · · · , eN } be the standard basis of CN . Then one has 

n

d dλ |λ=0

det[Id + λB] =

 p1 ···pn pi =pj

=

 p1 ···pn pi =pj

= n!

⎤ | | | | det ⎣ e1 bp1 bpn eN ⎦ | | | | ⎡

detp1 ···pn B 

detp1 ···pn B

{p1 ···pn }⊂M

b) By part (a)

det[Id + B] =

N 



detk1 ···kn B

n=0 {k1 ,··· ,kn }⊂M

In the last sum over k1 , · · · , kn one only gets a contribution if all the k’s are distinct, because of the determinant detk1 ···kn B. Since 


ψ¯k1 ψk1 · · · ψ¯kn ψkn dµ =

1 if all k are distinct 0 otherwise

206 one may write

det[Id + B] =


 N



ψ¯k1 ψk1 · · · ψ¯kn ψkn detk1 ···kn B dµ

n=0 {k1 ,··· ,kn }

=


 N


=





detk1 ···kn (ψ¯k ψk Bkp ) dµ

n=0 {k1 ,··· ,kn }



det Id + (ψ¯k ψk Bkp ) dµ

which proves the lemma.


The utility of the above lemma lies in the fact that application of (10.22) to (10.26) gives a better result than a direct application to det[Id + B]. The reason is that in the expansion

det[Id + λB] = eT r log[Id+λB] = e

∞ r=1

(−1)r+1 T rB r r

∞ 

b2 (−1)2b1 +3b2 +··· b [λT rB] 1 λ2 T rB 2 · · · b b 1 2 b1 ! 1 b2 ! 2 · · · b1 ,b2 ,···=0  b1 ∞ n   (−1)b1 +···+bn  n = (−λ) Bkk × ··· b1 !1b1 · · · bn !nbn b ,··· ,bn =0 n=0

=

(10.27)

k

1 1b1 +···+nbn =n

···×

 

bn Bk1 k2 Bk2 k3 · · · Bkn k1

k1 ···kn

all terms in the sums  b1  bn  · · · · · ≡ k1 ···k1 k k1 ···kn 1

2 2 3 b1 k1 ···k2b2 k1 ···

n knb n

for which at least two kji ’s are equal cancel out. The approximation, say, 1

2

det[Id + B] ≈ eT rB− 2 T rB (10.28)  b1   b2 ∞ n b1 +b2   (−1) (−λ)n B B B = kk k1 k2 k2 k1 b1 !1b1 b2 !2b2 b ,b =0 n=0 1 2 1b1 +2b2 =n

k

k1 ,k2

Resummation of Perturbation Series

207

does not keep track of that information, but the approximation
2 1 ¯ ¯ det[Id + B] ≈ eT r(ψk ψk Bkp )− 2 T r[(ψk ψk Bkp ) ] dµ =


 ∞

(−λ)n

n=0

n 

b1 ,b2 =0 1b1 +2b2 =n

(10.29)

 b1 (−1)b1 +b2  ¯ ψ ψ B × k k kk b1 !1b1 b2 !2b2 k



b2 ψ¯k1 ψk1 ψ¯k2 ψk2 Bk1 k2 Bk2 k1

dµ

k1 ,k2

does. We consider the lowest order approximations. For n = 1, application of (10.22) to (10.26) gives
P  ¯ det[Id + B] ≈ e k Bkk ψk ψk dµ = (1 + Bkk ) (10.30) k

For n = 2 we have (for, say, even N )
P P 1 ¯ ¯ ¯ det[Id + B] ≈ e k Bkk ψk ψk − 2 k,p Bkp Bpk ψk ψk ψp ψp dµ
P P 1 ¯ ¯ ¯ = e− 2 k,p Bkp Bpk ψk ψk ψp ψp e k (1+Bkk )ψk ψk Π dψ¯k dψk

(10.31)

k



N/2

=

(−1)n 1 n! 2n

n=0



n

Π Bki pi Bpi ki ×

k1 ,··· ,kn p1 ,··· ,pn




i=1

n

Π ψ¯ki ψki ψ¯pi ψpi e

P

¯

k (1+Bkk )ψk ψk

i=1





N/2

=

(−1)n


k

n

Π Bki pi Bpi ki ×

{(k1 ,p1 ),··· ,(kn ,pn )} disjoint ordered pairs

n=0

i=1

n

Π ψ¯ki ψki ψ¯pi ψpi e

P

¯

k (1+Bkk )ψk ψk

i=1





N/2

=

(−1)n

= =

N/2





n=0

π∈SN , π consists of n 2−cycles and N −2n 1−cycles



π∈SN π consists only of 1− and 2−cycles

Π dψ¯k dψk k

n

Π Bki pi Bpi ki

{(k1 ,p1 ),··· ,(kn ,pn )} disjoint ordered pairs

n=0

Π dψ¯k dψk

i=1

Π

q∈M\{k1 ,··· ,pn }

(1 + Bqq )

signπ (Id + B)1π1 (Id + B)2π2 · · · (Id + B)N πN

signπ (Id + B)1π1 (Id + B)2π2 · · · (Id + B)N πN

208 to which we refer in the following as two-cycle or two-loop approximation,  det[Id + B] ≈ signπ (Id + B)1π1 (Id + B)2π2 · · · (Id + B)N πN π∈SN π consists only of 1− and 2−cycles

(10.32) Now consider inverse matrix elements. By Cramer’s rule one has [Id + B]−1 tt =

dett [Id + B] det[Id + B]

(10.33)

if we define (opposite to the definition in (10.25)) detI,J B := det[(Bkp )k∈I c ,p∈J c ] for I, J ⊂ M, |I| = |J|, I c = M \ I and detI B := detI,I B. In the following we give an alternative proof of (10.7), (10.8) by using the representation (10.26) and the tool of Grassmann integration. One has  PN −1 (−1)r+1 P ¯ ¯ k1 ···kr ∈Mt Bk1 k2 ···Bkr k1 (ψψ)k1 ···(ψψ)kr r e r=1 dµt dett [Id + B] =  PN (−1)r+1 P ¯ ¯ det[Id + B] k1 ···kr ∈M Bk1 k2 ···Bkr k1 (ψψ)k1 ···(ψψ)kr e r=1 r dµ Zt (10.34) ≡ Z P

¯ ψ ψ where MI := M \ I and dµI := e k∈MI k k Πk∈MI dψ¯k dψk . Separating the ¯ t variables in the denominator, we obtain (ψψ)
 P N −1 (−1)r+1 P ¯ ¯ ¯ k1 ···kr ∈Mt Bk1 k2 ···Bkr k1 (ψψ)k1 ···(ψψ)kr r Z= e r=1 eBtt (ψψ)t ×  PN r+1 P ¯ ¯ ¯ ¯ k2 ···kr ∈Mt Btk2 ···Bkr t (ψψ)t (ψψ)k2 ···(ψψ)kr r=2 (−1) e eψt ψt dψ¯t dψt dµt

 N  = 1 + Btt + (−1)r+1 r=2



 ¯ ¯ Btk2 · · · Bkr t
(ψψ)k2 · · · (ψψ)kr t Zt

k2 ···kr ∈Mt

(10.35) where ¯ k t = ¯ k · · · (ψψ)
(ψψ) r
2 PN −1 (−1)s+1 1 ¯ k e s=1 ¯ s (ψψ)k2 · · · (ψψ) r Zt

P p1 ···ps ∈Mt

¯ p ···(ψψ) ¯ p Bp1 p2 ···Bps p1 (ψψ) s 1

dµt

Resummation of Perturbation Series

209

Since ¯ k · · · (ψψ) ¯ k t
(ψψ) 2 r
PN −r (−1)s+1 P ¯ ¯ 1 p1 ···ps ∈Mtk ···k Bp1 p2 ···Bps p1 (ψψ)p1 ···(ψψ)ps s s 2 e s=1 = dµtk2 ···kr Zt Ztk2 ···kr Ztk2 Ztk2 k3 Ztk2 ···kr ≡ = ··· Zt Zt Ztk2 Ztk2 ···kr−1 =
ψ¯k ψk t
ψ¯k ψk tk · · ·
ψ¯k ψk tk ···k (10.36) 2

2

3

3

2

r

r

2

r−1

one arrives at   ψ¯t ψt =

(10.37) 1

1 + Btt +

N 

(−1)r+1



Btk2 · · · Bkr t
ψ¯k2 ψk2 t · · ·
ψ¯kr ψkr tk2 ···kr−1

k2 ···kr ∈Mt ki =kj

r=2

which coincides with (10.7). The Off-Diagonal Elements: Let s = t. One has [Id + B]−1 st =

dets,t [Id + B] det[Id + B]

and one may write dets,t [Id + B] = det[(Id + B)(s,t) ] where (s,t)

(Id + B)kp

⎧ (Id + B)kp = δkp + Bkp if k = s ∧ p = t ⎪ ⎪ ⎨ 0 if (k = s ∧ p = t) ∨ = (k = s ∧ p = t) ⎪ ⎪ ⎩ 1 if k = s ∧ p = t

˜ (s,t) ∈ C N ×N according to To apply (10.26), we define B ˜ (s,t) (Id + B)(s,t) = Id + B That is, since s = t by assumption, ⎧ ⎨ Bkp if k = s ∧ p = t ˜ (s,t) = −δkp if (k = s ∧ p = t) ∨ (k = s ∧ p = t) B k,p ⎩ 1 if k = s ∧ p = t Then [Id +

B]−1 st

˜ (s,t) ] det[Id + B = = det[Id + B]



¯ B ˜ (s,t) ]dµ det[Id + ψψ  ¯ det[Id + ψψB]dµ

210 and one has  [Id +

B]−1 st

=

e 

PN r=1

e

(−1)r+1 r

¯ B ˜ (s,t) )r ] T r[(ψψ

PN

(−1)r+1 r=1 r

dµ

r] ¯ T r[(ψψB) dµ

Z (s,t) Z

≡

The numerator becomes Z (s,t) =
P ¯ k +P N ˜ (s,t) (ψψ) r=2 e k∈M Bkk

(10.38) (−1)r+1 r

P k1

˜ (s,t) ˜ (s,t) ¯ ¯ ···kr ∈M Bk k ···Bkr k (ψψ)k1 ···(ψψ)kr 1 2

e

¯

×

Π dψ¯k dψk

k∈M (ψψ)k

k


=

P

1

P

e

k∈Mst

¯ k +P N Bkk (ψψ) r=2

(−1)r+1 r

P k1

˜ (s,t) ˜ (s,t) ¯ ¯ ···kr ∈M Bk k ···Bkr k (ψψ)k1 ···(ψψ)kr 1 2

P

e

1

¯

k∈Mst (ψψ)k

×

Π dψ¯k dψk k

Consider the sum



˜ (s,t) · · · B ˜ (s,t) (ψψ) ¯ k1 · · · (ψψ) ¯ kr B k1 k2 kr k1

k1 ···kr ∈M

which is part of the exponent in (10.38). To get a nonzero contribution, all kj have to be different. Furthermore, if, say, k1 = s then k2 has to be equal to t and one gets a contribution  ¯ s (ψψ) ¯ t (ψψ) ¯ k · · · (ψψ) ¯ k ˜ (s,t) · · · B ˜ (s,t) (ψψ) 1·B 3 r tk3 kr s k3 ···kr ∈Mst



=

¯ s (ψψ) ¯ t (ψψ) ¯ k · · · (ψψ) ¯ k Btk3 · · · Bkr s (ψψ) 3 r

k3 ···kr ∈Mst

If k2 = s then k3 has to be equal to t and one gets the same contribution. Thus  ¯ k · · · (ψψ) ¯ k ˜ (s,t) · · · B ˜ (s,t) (ψψ) B 1 r k1 k2 kr k1 k1 ···kr ∈M



=r

¯ s (ψψ) ¯ t (ψψ) ¯ k · · · (ψψ) ¯ k Btk3 · · · Bkr s (ψψ) 3 r

k3 ···kr ∈Mst

and one obtains Z (s,t) = (10.39)
 P r+1 P P (−1) N ¯ ¯ ¯ k1 ···kr ∈Mst Bk1 k2 ···Bkr k1 (ψψ)k1 ···(ψψ)kr e k∈Mst Bkk (ψψ)k + r=2 r ×
P N r+1 P ¯ ¯ ¯ ¯ k3 ···kr ∈Mst Btk3 ···Bkr s (ψψ)s (ψψ)t (ψψ)k3 ···(ψψ)kr e r=2 (−1) dψ¯s dψs ×  P ¯ ¯ dψt dψt e k∈Mst (ψψ)k Π dψ¯k dψk k∈Mst

Resummation of Perturbation Series


Since

211

¯ ¯ eAst (ψψ)s (ψψ)t dψ¯s dψs dψ¯t dψt = Ast

one arrives at Z (s,t) =

N 

(−1)r+1

=

¯ k · · · (ψψ) ¯ k st Zst Btk3 · · · Bkr s
(ψψ) 3 r

k3 ···kr ∈Mst

r=2 N 



(−1)r+1



Btk3 · · · Bkr s Zstk3 ···kr

(10.40)

k3 ···kr ∈Mst

r=2

such that [Id + B]−1 st =

N  Z (s,t)  = (−1)r+1 Z r=2

Btk3 · · · Bkr s

k3 ···kr ∈Mst

=

N  r=2

(−1)r+1



Zstk3 ···kr Z

Btk3 · · · Bkr s
ψ¯s ψs 
ψ¯t ψt s ×

k3 ···kr ∈Mst


ψ¯k3 ψk3 st · · ·
ψ¯kr ψkr stk3 ···kr−1 This is identical to formula (10.8).

10.3

The Averaged Green Function of the Anderson Model

Let coordinate space be a lattice with unit lattice spacing and finite volume [0, L]d with periodic boundary conditions: Γ = {x = (n1 , · · · , nd ) | 0 ≤ ni ≤ L − 1} = Zd /(LZ)d

(10.41)

Momentum space is given by $ % M := Γ = k = 2π L (m1 , · · · , md ) | 0 ≤ mi ≤ L − 1 d  /(2πZ)d = 2π L Z

(10.42)

We consider the averaged Green function of the Anderson model given by
−1
G(x, x ) := [−∆ − z + λV ]x,x
dP (V ) (10.43) where the random potential is Gaussian, dP (V ) = Π e− x∈Γ

2 Vx 2

dVx √ . 2π

(10.44)

212 Here z = E + iε and ∆ is the discrete Laplacian for unit lattice spacing, [−∆ − z + λV ]x,x
= −

d 

(10.45)

(δx
,x+ei + δx
,x−ei − 2δx
,x ) − z δx,x
+ λVx δx,x


i=1

By taking the Fourier transform, one obtains 
eik(x −x)
G(k)
G(x, x ) = L1d 



G(k) =

(10.46)

k∈M

ap δp,p
+

RL d

√λ vp−p
Ld

−1



p,p
∈M k,k

dP (v)

(10.47)

where Ld = |Γ| = |M|, dP (v) is given by (10.50) or (10.51) below, depending on whether Ld is even or odd, and ak = 4

d 

sin2

ki

2

− E − iε

(10.48)

i=1

The rigorous control of
G(k) for small disorder λ and energies inside the spectrum of the unperturbed Laplacian, E ∈ [0, 4d], in which case ak has a root if ε → 0, is still an open problem [2, 41, 48, 52, 60]. It is expected that limε0 limL→∞
G(k) = 1/(ak − σk ) where Imσ = O(λ2 ). The integration variables vq in (10.47) are given by the discrete Fourier transform of the Vx . In particular, observe that, if F denotes the unitary matrix of discrete Fourier transform, the variables  e−iqx Vx ≡ √1 d Vˆq (10.49) vq ≡ (F V )q = √1 d L

L

x∈Γ

would not have a limit if Vx would be deterministic and cutoffs are removed, since the Vˆq are the quantities which have a limit in that case. But since the Vx are integration variables, we choose a unitary transform to keep the integration measure invariant. Observe also that vq is complex, vq = uq + iwq . Since Vx is real, u−q = uq and w−q = −wq . In order to transform dP (V ) to momentum space, we have to choose a set M+ ⊂ M such that either q ∈ M+ or −q ∈ M+ . If L is odd, the only momentum with q = −q or wq = 0 is q = 0. In that case dP (V ) becomes dP (v) = e−

u2 0 2

−(u2 +w 2 ) duq dwq du0 √ Π e q q π 2π q∈M+

(10.50)

For even L we get dP (v) = e− 2 (u0 +uq0 ) 1

where q0 = 2π L m.

2πm L

2

2

du0 duq0 2π

Π+e

q∈M

−(u2q +wq2 ) duq dwq π

is the unique nonzero momentum for which

2π L m

(10.51) = 2π(1, · · · , 1)−

Resummation of Perturbation Series

10.3.1

213

Two-Loop Approximation

Now we apply the inversion formula (10.7) to the inverse matrix element in (10.47),
 −1
G(k) = dP (v) ap δp,p
+ √λ d vp−p
L

RL d

k,k

We start with the ‘two-loop approximation’, which we define by retaining only the r = 2 term in the denominator of the right hand side of (10.7), G(k) ≈

Bkk −

1



p∈M\{k}

Bkp Gk (p)Bpk

(10.52)

Thus, let  G(k) := ap δp,p
+

−1 √λ vp−p
Ld k,k

= G(k; v, z)

(10.53)

In the infinite volume limit the spacing 2π/L of the momenta becomes arbitrarily small. Hence, in computing an inverse matrix element, it should not matter whether a single column and row labelled by some momentum t is absent or not. In other words, in the infinite volume limit one should have Gt (p) = G(p) for L → ∞

(10.54)

and similarly Gt1 ···tj (p) = G(p) as long as j is independent of the volume. We remark however that if the matrix has a block structure, say B = (Bkσ,pτ ) with σ, τ ∈ {↑, ↓} some spin variables, this structure has to be respected. That is, for a given momentum k all rows and columns labelled by k ↑, k ↓ have to be eliminated, since otherwise (10.54) may not be true. Thus the two-loop approximation gives G(k) =

ak +

√λ v0 Ld

−

For large L, we can disregard the according to

λ2 Ld

1 

√λ v0 Ld

G(k) =:

p=k

vk−p G(p) vp−k

(10.55)

term. Introducing σk = σk (v, z)

1 , ak − σk

(10.56)

we get σk =

λ2 Ld

 |vk−p |2 ≈ ap − σp

p=k

λ2 Ld

 |vk−p |2 ap − σp p

(10.57)

214 and arrive at


1


G(k) = λ2 Ld

ak −

 ap −

p

dP (v)

|vk−p |2  |vp−t |2 λ2

(10.58)

t at − λ2 Σ··· d

Ld

L

Now consider the infinite volume limit   L → ∞. By the central limit theorem of probability √1 d q |vq |2 −
|vq |2  is, as a sum of independent random variL √ ables, normally distributed. Note that only a prefactor of 1/ Ld is required for that property. In particular, if F is some bounded function  independent of L, sums which come with a prefactor of 1/Ld like L1d q cq |vq |2 can be substituted by their expectation value,
      ck |vk |2 dP (v) = F lim L1d ck
|vk |2  (10.59) F L1d lim L→∞

L→∞

k

k

Therefore, in the two-loop approximation, one obtains in the infinite volume limit 1


G(k) = ak −

λ2 Ld

 p

ap −

1 ak −
σk 

(10.60)

dd p
|vk−p |2  (2π)d ap −
σp 

(10.61)


|vk−p |2  λ2  |vp−t |2 

=:

t at − λ2 Σ··· d

Ld

L

where the quantity
σk  satisfies the integral equation
σk  =

λ2 Ld


|vk−p |2  ap −
σp  p

L→∞

→


λ2 [0,2π]d

For a Gaussian distribution
|vq |2  = 1 for all q such that
σk  =
σ becomes independent of k. Thus we end up with
G(k) =

4

d i=1

2

sin

1 ki

2

− E − iε −
σ

(10.62)

where
σ is a solution of

σ = λ2 [0,2π]d

dd p 1

 (2π)d 4 di=1 sin2 pi − E − iε −
σ 2

(10.63)

This equation is well known and one deduces from it that it generates a small imaginary part Im σ = O(λ2 ) for ε  0 if the energy E is within the spectrum of −∆, E ∈ (0, 4d).

Resummation of Perturbation Series

10.3.2

215

Higher Orders

We now add the higher loop terms (the terms for r > 2 in the denominator of (10.7)) to our discussion and give an interpretation in terms of Feynman diagrams. For the Anderson model, Feynman graphs may be obtained by brutally expanding (C = (1/ap δp,p
), V = ( √λ d vp−p
) in the next line) L

∞ 

−1 ak δk,p + √λ d vk−p k,k dP (v) ∼ (C[V C]r )kk dP (v) L

=

∞   (−λ)r √

Ld

r=0

r

p2 ···pr


1 ak ap2 ···apr ak

r=0

vk−p2 vp2 −p3 · · · vpr −k dP (v)

(10.64)

For a given r, this may be represented as in figure 10.1 (ck := 1/ak ). vk-p2

vp2-p3

vp3-p4

vpr-1-pr

vpr-k

k

k ck

cp2

cp3

cp4

cpr-1

cpr

ck

Figure 10.1 The integral over the v gives a sum of (r − 1)!! terms where each term is a product of r/2 Kroenecker-delta’s, the terms for odd r vanish. If this is substituted in (10.64), the number of independent momentum sums is cut down to r/2 and each of the (r − 1)!! terms may be represented by a diagram as in figure 10.2.

+

+ Figure 10.2

+

+

···

Here, the value of the, say, third diagram is given by λ4  1 L2d

p1 ,p2 ak ak+p1 ak+p1 +p2 ak+p2 ak

For short:
G(k) = sum of all two-legged diagrams.

(10.65)

216 Since the value of a diagram depends on its subgraph structure, one distinguishes, in the easiest case, two types of diagrams: diagrams with or without two-legged subdiagrams. Those diagrams with two-legged subgraphs usually produce anomalously large contributions. They are divided further into the one-particle irreducible ones and the reducible ones. Thereby a diagram is called one-particle reducible if it can be made disconnected by cutting one solid or ‘particle’ line (no squiggle), like the first diagram in figure 10.3.

irreducible, no 2−legged subgraphs

reducible

irreducible, with 2−legged subgraphs

Figure 10.3

The reason for introducing reducible and irreducible diagrams is that the reducible ones can be easily resummed by writing down the Schwinger-Dyson equation which states that if the self energy Σk is defined through
G(k) =

1 ak − Σk (G0 )

(10.66)

then Σk (G0 ) is the sum of all amputated (no 1/ak ’s at the ends) one-particle irreducible diagrams. The lowest order contributions are drawn in figure 10.4. Observe that the second diagram of figure 10.2 is now absent, but the fourth diagram, which contains a two-legged subgraph, is still present in Σk (G0 ).

k

k

+

k

k

+

k

k

+

···

Figure 10.4

Here we wrote Σk (G0 ) to indicate that the factors (‘propagators’) assigned to the solid lines of the diagrams contributing to Σk are given by the free two-point function G0 (p) = a1p . The diagrams contributing to Σk (G0 ) still

Resummation of Perturbation Series

217

contain anomalously large contributions, namely irreducible diagrams which contain two-legged subdiagrams. In the following we show, using the inversion formula (10.7) including all higher loop terms, that all graphs with two-legged subgraphs can be eliminated or resummed by writing down the following integral equation for
G:
G(k) =

1 ak − σk (
G)

(10.67)

where σk (
G) is the sum of all amputated two-legged diagrams which do not contain any two-legged subdiagrams, but now with propa1 instead of G0 = a1k gators
G(k) = ak −σ k which may be formalized as in (10.76) below. The lowest order contributions to σk (
G) are drawn in figure 10.5. 〈 G 〉(k) k

k

k

k

+

+

⇒

σk (
G) =

k

k

k

k

+

···

Figure 10.5

Observe that the third diagram of figure 10.4, that one with the two-legged subdiagram, has now disappeared. Thus the advantage of (10.67) compared to (10.66) is that the series for σk (
G) can be expected to be asymptotic, that is, its lowest order contributions are a good approximation if the coupling is small, but, usually, the series for Σk (G0 ) is not asymptotic. Therefore renormalization in this case is nothing else that a rearrangement of the perturbation series, Σk (G0 ) = σk (
G),

(10.68)

the lowest order terms to Σk (G0 ) are not a good approximation but the lowest order terms of σk (
G) are. Equations of the type (10.67) can be found in the literature [47, 15], but usually they are derived on a more heuristic level without using the inversion

218 formula (10.7). In section 10.6 we discuss more closely how our method is related to the integral equations which can be found in the literature. We now show (10.67) for the Anderson model. For fixed v one has G(k, v) =

1 ak − σk (v)

(10.69)

where L  d

σk (v) =

r=2

(−1)r





√λ Ld

p2 ···pr pi =pj , pi =k

r

Gk (p2 ) · · · Gkp2 ···pr−1 (pr ) vk−p2 · · · vpr −k (10.70)

We cut off the r-sum in (10.70) at some arbitrary but fixed order  < Ld where  is chosen to be independent of the volume. Furthermore we substitute Gkp2 ···pj (p) by G(p) since we are interested in the infinite volume limit L → ∞. Thus & ' 1
G(k) = (10.71)  ak − r=2 σkr (v) where σkr (v) = (−1)r



√λ Ld

r 

G(p2 ) · · · G(pr ) vk−p2 · · · vpr −k

Consider first two strings srk1 , srk2 where  r srk (v) = √λ d r crkp2 ···pr vk−p2 · · · vpr −k L

(10.72)

p2 ···pr pi =pj , pi =k

(10.73)

p2 ···pr pi =pj , pi =k

and the crkp2 ···pr are some numbers. Then
srk1 srk2  =
srk1 
srk2  + O(1/Ld ) which means that in the infinite volume limit
srk1 srk2  =
srk1 
srk2 

(10.74)

This holds because all pairings which connect the two strings have an extra volume factor 1/Ld. Namely, if the two strings are disconnected, there are √ (r1 +r2 ) giving (r1 + r2 )/2 Rie(r1 + r2 )/2 loops and a volume factor of 1/ Ld mannian sums. If the two strings are connected, there are only (r1 + r2 − 2)/2 loops, leaving an extra factor of 1/Ld. This is visualized in the following figure 10.6 for two strings with r = 2 and r = 4 squiggles. The disconnected contributions have exactly three loops, the connected contributions have exactly two loops, the prefactor is 1/L3d.

Resummation of Perturbation Series

219


=

+

+

···

d

= O(1/L )

= O(1) Figure 10.6

By the same argument one has in the infinite volume limit
(srk1 )n1 · · · (srkm )nm  =
srk1 n1 · · ·
srkm nm

(10.75)

which results in 1


G(k) = ak −



r

(−λ) √ r Ld



(10.76)


G(p2 ) · · ·
G(pr )
vk−p2 · · · vpr −k 

p2 ···pr pi =pj , pi =k

r=2

Now, the condition p2 , · · · , pr = k and pi = pj in (10.76) means exactly that two-legged subgraphs are forbidden. Namely, for a two-legged subdiagram like the one in the third diagram in figure 10.4, the incoming and outgoing momenta pi , pj , to which are assigned propagators
G(pi ),
G(pj ), must be equal which is forbidden by the condition pi = pj in (10.76). Figure 10.7 may be helpful.

⇒ pi = pj, pi

not allowed

pj

Figure 10.7a

⇒ pi = pj, not allowed pi

pj

Figure 10.7b However, we cannot take the limit  → ∞ in (10.76) since the series in the denominator of (10.76) is only an asymptotic one. To see this a bit more clearly suppose for the moment that there were no restrictions on the

220 momentum sums. Then, if V = ( √λ d vk−p )k,p and
G = (
G(k) δk,p )k,p , L



√λ Ld

r 

 
G(p2 ) · · ·
G(pr )
vk−p2 · · · vpr −k  = (V [
GV ]r−1 )kk

p2 ···pr

(10.77) and for  → ∞ we would get
G(k) =

1 ak −
(V

GV Id+ GV )kk 

=

1 ak −
(V

1

G−1 +V

V )kk 

(10.78)

That is, the factorials produced by the number of diagrams in the denominator of (10.76) are basically the same as those in the expansion
R

x2 z+λx

e−

x2 2

√dx 2π

=



λ2r z 2r+1

(2r + 1)!! + R +1 (λ)

(10.79)

r=0

where the remainder satisfies the bound |R +1 (λ)| ≤ ! constz λ . We close this section with two further remarks. So far the computations were done in momentum space. One may wonder what one gets if the inversion formula (10.7) is applied to [−∆ + z + λV ]−1 in coordinate space. Whereas a geometric series expansion of [−∆ + z + λV ]−1 gives a representation in terms of the simple random walk [59], application of (10.7) results in a representation in terms of the self-avoiding walk:

 det (−∆ + z + λV )y,y
∈Γ\γ −1 (10.80) [−∆ + z + λV ]0,x = det [(−∆ + z + λV )y,y
∈Γ ] γ:0→x γ self−avoiding

where Γ is the lattice in coordinate space. Namely, if |x| > 1, the inversion formula (10.8) for the off-diagonal elements gives [−∆ + λV ]−1 0,x = L  d

(−1)r+1

r=3 L  d

=

r=3



G(0)G0 (x3 ) · · · G0x3 ···xr (x) (−∆)0x3 · · · (−∆)xr x

x3 ···xr ∈Γ\{0,x} xi =xj



det (−∆ + λV )y,y
∈Γ\{0} × ··· det [(−∆ + λV )y,y
∈Γ ] =x∈Γ

x2 =0,x3 ,··· ,xr ,xr+1 |xi −xi+1 |=1 ∀i=2···r

which coincides with (10.80).



det (−∆ + λV )y,y
∈Γ\{0,x3 ···xr ,x}

···× det (−∆ + λV )y,y
∈Γ\{0,x3 ···xr }

Resummation of Perturbation Series

221

Finally we remark that, while the argument following (10.73) leads to a factorization property for on-diagonal elements in momentum space,
G(k) G(p) equals
G(k)
G(p), there is no such property for products of off-diagonal elements which appear in a quantity like ' −1  −1  & 1 λ λ ak δk,p + √ d vk−p a ¯k δk,p + √ d v¯k−p (10.81) Λ(q) = Ld L

k,p

L

k,p

k−q,p−q

(2  ( ( . (Each off-diagonal which is the Fourier transform of ([−∆ + z + λV ]−1 x,y √ inverse matrix element is proportional to 1/ Ld , therefore the prefactor of 1/Ld in (10.81) is correct.)

10.4

10.4.1

The Many-Electron System with Attractive Delta Interaction The Integral Equations in Two-Loop Approximation

The goal in this section is to apply the inversion formula (10.7) to the inverse matrix element in (10.2) and to compute the two point function
ψ¯kσ ψkσ  for the many-electron system with delta interaction, or more precisely, to derive a closed set of integral equations for it. We briefly recall the setup. We consider the many-electron system in the grand canonical ensemble in finite volume [0, L]d and at some small but positive temperature T = 1/β > 0 with attractive delta interaction (λ > 0) given by the Hamiltonian H = H0 − λHint  2 k = L1d ( 2m − µ)a+ kσ akσ − kσ

λ L3d

 kpq

+ a+ k↑ aq−k↓ aq−p↓ ap↑

(10.82)

Our normalization conventions concerning the volume factors are such that the d canonical anticommutation relations read {akσ , a+ pτ } = L δk,p δσ,τ . The mo 2π d $  2π d ( (|ek | ≤ mentum sums range over some subset of Z , say M = k ∈ L L Z % 2 1 , ek = k /2m − µ, and q ∈ {k − p |k, p ∈ M}. We are interested in the momentum distribution ) −βH + akσ akσ ] T r e−βH (10.83)
a+ kσ akσ  = T r[e and in the expectation value of the energy 
Hint  = Λ(q) q

(10.84)

222 where Λ(q) =

λ L3d

 k,p

) + −βH T r[e−βH a+ k↑ aq−k↓ aq−p↓ ap↑ ] T r e

(10.85)

By writing down the perturbation series for the partition function, rewriting it as a Grassmann integral
P λ ¯ ψ ¯ ψ ψ ψ T r e−β(H0 −λHint ) ¯ = e (βLd )3 kpq k↑ q−k↓ q−p↓ p↑ dµC (ψ, ψ) (10.86) T r e−βH0 dµC = Π

kσ

βLd ik0 −ek

e

−

1 βLd

P

¯

kσ (ik0 −ek )ψkσ ψkσ

Π dψkσ dψ¯kσ

kσ

performing a Hubbard-Stratonovich transformation (φq = uq + ivq , dφq dφ¯q := duq dvq )
P P P 2 ¯ ¯ dφ dφ e− q aq bq = ei q (aq φq +bq φq ) e− q |φq | Π qπ q (10.87) q

with 1

aq =

λ2 3 (βLd ) 2



ψ¯k↑ ψ¯q−k↓ , bq =

k

1

λ2 3 (βLd ) 2



ψp↑ ψq−p↓

(10.88)

p

and then integrating out the ψ, ψ¯ variables, one arrives at the following representation which is the starting point for our analysis:  + 1 1 1
a+
ψkk ψkk0 σ  (10.89) kσ akσ  = βLd β Ld 0σ k0 ∈ π β (2Z+1) 1

where, abbreviating k = (k, k0 ), κ = βLd , ak = ik0 − ek , g = λ 2 , +−1
* ak δk,p √igκ φ¯p−k 1 ¯ dP (φ) ig κ
ψtσ ψtσ  = √ φ a−k δk,p κ k−p

(10.90)

tσ,tσ

and dP (φ) is the normalized measure * + ig ¯ P √ a δ φ 2 k k,p p−k κ e− q |φq | Π dφq dφ¯q dP (φ) = Z1 det ig √ φk−p a−k δk,p q κ Furthermore Λ(q) =



1 β

q0 ∈ 2π β

where Λ(q) =

λ (βLd )3



Λ(q, q0 )

(10.91)

(10.92)

Z


ψ¯k↑ ψ¯q−k↓ ψq−p↓ ψp↑ 

k,p

=
|φq |2  − 1

(10.93)

Resummation of Perturbation Series

223

and the expectation in the last line is integration with respect to dP (φ). The expectation on the ψ variables in
ψ¯kσ ψkσ  is Grassmann integration, P  λ ¯k↑ ψ ¯q−k↓ ψq−p↓ ψp↑ ψ 1 ¯ ¯ 3 k,p,q
ψkσ ψkσ  = Z ψkσ ψkσ e κ dµC , but these representations are not used in the following. The matrix and the integral in (10.90) become finite dimensional if we choose some cutoff on the k0 variables which is removed in the end. The set M for the spatial momenta is already finite since we have chosen a fixed UV-cuttoff |ek | = |k2 /2m − µ| ≤ 1 which will not be removed in the end since we are interested in the infrared properties at k2 /2m = µ. Our goal is to apply the inversion formula to the inverse matrix element in (10.90). Instead of writing the matrix in terms of four N × N blocks (ak δk,p )k,p , (φ¯p−k )k,p , (φk−p )k,p and (a−k δk,p )k,p where N is the number of the d + 1-dimensional momenta k, p, we interchange rows and columns to rewrite it in terms of N blocks of size 2 × 2 (the matrix U in the next line interchanges the rows and columns): * U

+ ak δk,p √igκ φ¯p−k U −1 = B = (Bkp )k,p ig √ φ a−k δk,p κ k−p

where the 2 × 2 blocks Bkp are given by , Bkk =

ak √igκ φ¯0 ig √ φ a−k κ 0

-

 , Bkp =

ig √ κ

0 φk−p

φ¯p−k 0

 if k = p

(10.94)

We want to compute the 2 × 2 matrix

G(k) =

G(k) dP (φ)

(10.95)

where G(k) = [B −1 ]kk

(10.96)

As for the Anderson model, we start again with the two-loop approximation which retains only the r = 2 term in the denominator of (10.7). The result will be equation (10.101) below where the quantities
σk  and
|φ0 |2  appearing in (10.101) have to satisfy the equations (10.102) and (10.105) which have to be solved in conjunction with (10.110). The solution to these equations is discussed below (10.111).

224 We first derive (10.101). In the two loop approximation,  −1  G(k) ≈ Bkk − Bkp Gk (p) Bpk , = , =:

p=k

ak √igκ φ¯0 ig √ φ0 a−k κ ak ig √ φ κ 0

ig √ κ

φ¯0

a ¯k

+ -

λ κ



φ¯p−k φk−p

p=k



 Gk (p)

−1 + Σ(k)

φ¯k−p

−1

φp−k (10.97)

where, substituting again Gk (p) by G(p) in the infinite volume limit, −1   ,  ig ¯  √ ¯ φ a p 0 φ φ¯k−p p−k κ λ Σ(k) = κ + Σ(p) ig √ φk−p φp−k φ a ¯p κ 0 p=k

(10.98) Anticipating the fact that the off-diagonal elements of
Σ(k) will be zero (for ‘zero external field’), we make the ansatz   σk Σ(k) = (10.99) σ ¯k and obtain   σk = σ ¯k λ κ

 p=k

1 2 (ap +σp )(¯ ap +¯ σp )+ λ κ |φ0 |

(10.100) , (ak + σk )|φp−k |2 − √igκ φ0 φ¯k−p φ¯p−k − √igκ φ¯0 φk−p φp−k (¯ ak + σ ¯k )|φk−p |2

As for the Anderson model, we perform the functional integral by substituting the quantities |φq |2 by their expectation values
|φq |2 . Apparently this is less obvious in this case since dP (φ) is no longer Gaussian and the |φq |2 are no longer identically, independently distributed. We will comment on this after (10.122) below and in section 10.6 by reinterpreting this procedure as a resummation of diagrams. For now, we simply continue in this way. Then , ig ¯ √ a ¯ +
¯ σ  −
φ  k k 0 κ (10.101)
G(k) = |a + σ |21+ λ |φ |2  0 k k − √igκ
φ0  ak +
σk  κ where the quantity
σk  has to satisfy the equation  σp  a ¯ p + ¯
σk  = λκ
|φp−k |2  |a + σ |2 + λ |φ |2  p

p=k

p

κ

0

(10.102)

Resummation of Perturbation Series

225

Since dP (φ) is not Gaussian, we do not know the expectations
|φq |2 . However, by partial integration, we obtain 
ig 2 √
|φq |  = 1 + κ (10.103) φq [B −1 (φ)]p↑,p+q↓ dP (φ) p

Namely,




P

2 φq φ¯q det [{Bkp (φ)}k,p ] e− q |φq | dφq dφ¯q
 P  2 1 ∂ = 1+ Z det [{B (φ)} ] e− q |φq | dφq dφ¯q φq ∂φ kp k,p q ⎤ ⎡ | | |
P  2 ∂Bkσ,pτ Bkσ,p

τ

⎦ e− q |φq | dφq dφ¯q = 1 + Z1 det ⎣ Bkσ,p
τ
∂φ φq q p,τ | | |


|φq |2  =

1 Z



Since ∂ ∂φq Bkp

we have ⎡

|

det ⎣ Bkσ,p
τ
|

|

=

ig √ κ

00 10

 δk−p,q

⎧ ⎤ if τ =↓ ⎨0 . Bkσ,p

τ

⎦ det [{Bkp }k,p ] = ⎩ √ig [B −1 ] p↑,p+q↓ if τ =↑ | κ |

∂Bkσ,pτ ∂φq

|

which results in (10.103). The inverse matrix element in (10.103) we compute again with (10.7,10.8) in the two-loop approximation. Consider first the case q = 0. Then one gets 

|φ0 |2  = 1 + √igκ φ0 G(p)↑↓ dP (φ) p

= 1+

ig √ κ


p

= 1+

λ κ


p

, φ0 |a +σ |21+ λ |φ |2 p p 0 κ

φ0

¯0 φ 2 |ap +σp |2 + λ κ |φ0 |

a ¯p + σ ¯p − √igκ φ¯0 − √igκ φ0 ap + σp

dP (φ)

dP (φ) ↑↓

(10.104)

Performing the functional integral by substitution of expectation values gives 
|φ0 |2  |a + σ |21+ λ |φ |2 
|φ0 |2  = 1 + λκ p

p

p

κ

0

or
|φ0 |2  = 1−

λ κ

 p

1 1 2 |ap + σp |2 + λ κ |φ0 | 

(10.105)

226 Before we discuss (10.105), we write down the equation for q = 0. In that case we use (10.8) to compute [B −1 (φ)]p↑,p+q↓ in the two-loop approximation. We obtain [B −1 (φ)]p↑,p+q↓ ≈ − [G(p)Bp,p+q G(p + q)]↑↓ = − |a

1

p +σp |



=

2 + λ |φ |2 0 κ

ig 1 √ 2 κ |ap+q +σp+q |2 + λ κ |φ0 |

×

ig −√ [(¯ a+σ ¯ )p+q φ¯0 φ−q + (¯ a+σ ¯ )p φ0 φ¯q ] (¯ a+σ ¯ )p (a + σ)p+q φ¯q − λ φ¯2 φ κ 0 −q κ ig 2φ ¯q ¯ ¯ √ (a + σ)p (¯ a+σ ¯ )p+q φ−q − λ φ − [(a + σ) φ + (a + σ) φ p φ0 φ−q ] p+q 0 q κ 0 κ

¯q − λ φ ¯2 (¯ a+¯ σ) (a+σ) φ κ 0 φ−q − √igκ |a +σ |2 + λp|φ |2 p+q ( p p κ 0 )(|ap+q +σp+q |2 + λκ |φ0 |2 )

 ↑↓

(10.106)

which gives
|φq |  = 1 + 2

λ κ




φq

p

= 1+

λ κ

 p

¯q − λ φ ¯2 (¯ a+¯ σ)p (a+σ)p+q φ κ 0 φ−q

(|ap +σp |2 + λκ |φ0 |2 )(|ap+q +σp+q |2 + λκ |φ0 |2 )

¯2 σp )(ap+q + σp+q ) |φq |2 − λ (¯ ap + ¯ κ φ0 φq φ−q  2 + λ |φ |2  2 + λ |φ |2  |a + σ | |a + σ | ( p p )( p+q p+q ) 0 0 κ κ

dP (φ) (10.107)

The expectation
φ¯20 φq φ−q  can be computed again by partial integration:
P 2 2 1 ¯
φ0 φq φ−q  = Z φ¯20 φq φ−q det [{Bkp (φ)}k,p ] e− q |φq | dφq dφ¯q
  P 2 = Z1 φ¯20 φq ∂ φ¯∂−q det [{Bkp (φ)}k,p ] e− q |φq | dφq dφ¯q ⎤ ⎡ | | |
 P 2 ∂B ⎦ e− q |φq | dφq dφ¯q = Z1 det ⎣ Bkσ,p
τ
∂ φkσ,pτ φ¯20 φq ¯−q Bkσ,p

τ

p,τ | | | The above determinant is multiplied and divided by det [{Bkp }k,p ] to give ⎧ ⎡ ⎤ | | | if τ =↑ ⎨0 . ∂B kσ,pτ ⎣ ⎦





det [{Bkp }k,p ] = det Bkσ,p τ ∂ φ¯−q Bkσ,p τ ⎩ √ig [B −1 ] p↓,p+q↑ if τ =↓ | | | κ Computing the inverse matrix element again in the two-loop approximation (10.106), we arrive at / (ap +σp )(¯ap+q +σp+q )φ¯2 φq φ−q − λ φ¯2 φ2 φq φ¯q 0 0 κ 0 0
φ¯20 φq φ−q  = λκ (|ap +σp |2 + λκ |φ0 |2 )(|ap+q +σp+q |2 + λκ |φ0 |2 ) p

Abbreviating gp =

ap + σp  2 , |ap + σp |2 + λ κ |φ0 | 

√λ

fp =

2 κ |φ0 |  2 |ap + σp |2 + λ κ |φ0 | 

(10.108)

Resummation of Perturbation Series this gives λ ¯2 κ
φ0 φq φ−q 

=

λ κ



gp g¯p+q κλ
φ¯20 φq φ−q  −

p

λ κ



227

fp fp+q κλ
|φ0 |2 
|φq |2 

p

or λ ¯2 κ
φ0 φq φ−q 

=

− λκ



fp fp+q κλ
|φ0 |2 
|φq |2   1 − λκ p gp g¯p+q p

Substituting this in (10.107), we finally arrive at  1 − λκ p gp g¯p+q 2
|φq |  = ( (   (1 − λ  gp g¯p+q (2 − λ  fp fp+q 2 p p κ κ

(10.109)

(10.110)

where gp , fp are given by (10.108). Observe that, since dP (φ) is complex, also
|φq |2  is in general complex. Only after summation over the q0 variables we obtain necessarily a real quantity which is given by (10.85) and (10.92).

10.4.2

Discussion

We now discuss the solutions to (10.105) and (10.110). We assume that the solution
σk  of (10.102) is sufficiently small such that the BCS equation λ κ



1 |ap + σp |2 +|∆|2

=1

(10.111)

p

has a nonzero solution ∆ = 0 (in particular this excludes large corrections like
σp  ∼ pα 0 , α ≤ 1/2, which one may expect in the case of Luttinger liquid behavior, for d = 1 one should make a separate analysis), and make the ansatz λ
|φ0 |2  = βLd |∆|2 + η

(10.112)

where η is independent of the volume. Then λ κ

 p

=

λ κ

1 2 |ap + σp |2 + λ κ |φ0 | 

 p

= 1−

1 |ap + σp |2 +|∆|2

= −

λ κ

λ κ





1 η p |ap + σp |2 +|∆|2 + κ

η/κ (|ap + σp |2 +|∆|2 )2

  + O ( κη )2

p

  + O ( κη )2 λ 

c∆ κη

(10.113)

where we put c∆ = κ p (|ap + σp |1 2 +|∆|2 )2 and used the BCS equation (10.111) in the last line. Equation (10.105) becomes κ |∆|2 + η =

c∆

η κ

λ λ  =κ  + O(1) c∆ η + O ( κη )2

(10.114)

228 and has a solution η = λ/(c∆ |∆|2 ). Now consider
|φq |2  for small but nonzero q. In the limit q → 0 the denominator in (10.110) vanishes, or more precisely, is of order O(1/κ) since 1−

λ κ



gp g¯p −

p

1−

λ κ

 p

λ κ



fp fp =

p 1 2 |ap + σp |2 + λ κ |φ0 | 

= O(1/κ)

(10.115)

because of (10.113). If we assume the second derivatives of
σk  to be integrable (which should be the case for d = 3 and
|φq |2  ∼ 1/q 2 by virtue of (10.102)), then, since the denominator in (10.110) is an even function of q, the small q behavior of
|φq |2  is 1/q 2 . This agrees with the common expectations [25, 14, 11]. Usually the behavior of
|φq |2  is inferred from the second order Taylor expansion of the effective potential Veff ({φq }) =



* |φq | − log det

δk,p

2

ig φk−p √ κ a−k

q

¯p−k ig φ √ κ ak

+

δk,p

(10.116)

around its global minimum (see section 5.2) = φmin q

1 βLd

|∆| √ λ

δq,0 eiθ0

(10.117)

where the phase θ0 of φ0 is arbitrary. If one expands Veff up to second order in 2  1 1 d |∆| √ ρ eiθ0 for q = 0 − βL |∆| iθ0 0 d λ (10.118) ξq = φq − δq,0 βL √λ e = for q = 0 ρq eiθq one obtains (see section 5.2) Veff ({φq }) = Vmin + 2β0 (ρ0 − + 12



1 βLd

|∆| 2 √ ) λ

+



(αq + iγq ) ρ2q

q=0

βq |e

−iθ0

φq + e

iθ0

φ¯−q | + O(ξ 3 ) 2

(10.119)

q=0

where for small q one has αq , γq ∼ q 2 . Hence, if Veff is substituted by the right hand side of (10.119) one obtains
|φq |2  ∼ 1/q 2 . For d = 3, this seems to be the right answer, but in lower dimensions one would expect an integrable singularity due to (10.102) and (10.84), (10.85) and (10.92). In particular, we think it would be a very interesting problem to solve the integral equations (10.102), (10.105) and (10.110) for d = 1 and to check the result for Luttinger liquid behavior. A good warm-up exercise

Resummation of Perturbation Series

229

would be to consider the 0 + 1 dimensional problem, that is, we only have the k0 , p0 , q0 -variables. In that case the ‘bare BCS equation’ 

λ β

p0 ∈ π β (2Z+1)

1 p20 +|∆|2

=1

(10.120)

still has a nonzero solution ∆ for sufficiently small T = 1/β and the question would be whether the correction
σp0  is sufficiently big to destroy the gap. That is, does the ‘renormalized BCS equation’ 

λ β

p0 ∈ π β (2Z+1)

1 |p0 + σp0 |2 +|∆|2

=1

(10.121)


σp0  being the solution to (10.102), (10.105) and (10.110), still have a nonzero solution? We remark that, if the gap vanishes (for arbitrary dimension), then also the singularity of
|φq |2  disappears. Namely, if the gap equation has no 1 1 2 solution, that is, if κ p |ap + σ 2 < ∞, then
|φ0 |  given by (10.105) is no p | longer macroscopic (for sufficiently small coupling) and λκ
|φ0 |2  vanishes in the infinite volume limit. And the denominator in (10.110) becomes for q → 0 1−

λ κ



1 |ap + σp |2

p

which would be nonzero (for sufficiently small coupling). Finally we argue why it is reasonable to substitute |φ0 |2 by its expectation value while performing the functional integral. We may write the effective potential (10.116) as Veff ({φq }) = V1 (φ0 ) + V2 ({φq })

(10.122)

where V1 (φ0 ) = |φ0 |2 −  =κ



 log 1 +

k |φ0 |2 κ

−

1 κ



2 λ |φ0 | κ k02 +e2k



|φ |2

log 1 +

k



0 λ κ k02 +e2k



≡ κ VBCS



|φ0 | √ κ

 (10.123)

and V2 ({φq }) =

⎡,  δk,p |φq |2 − log det ⎣ √ig φ0 q=0

δ κ a−k k,p

¯0 ig φ √ δ κ ak k,p

δk,p

-−1 ,

δk,p ig φk−p √ κ a−k

(10.124) -⎤ ⎦

¯ ig φ p−k √ κ ak

δk,p

230 If we ignore the φ0 -dependence of V2 , then the φ0 -integral   1 F κ |φ0 |2 e−V1 (φ0 ) dφ0 dφ¯0  = (10.125) e−V1 (φ0 ) dφ0 dφ¯0   2  −κV (ρ) BCS   F ρ e ρ dρ κ→∞  → F (ρ2min ) = F κ1
|φ0 |2  −κV (ρ) BCS e ρ dρ (10.126) simply puts |φ0 |2 at the global minimum of the (BCS) effective potential.

10.5 10.5.1

Application to Bosonic Models The ϕ4 -Model

In this section we choose the ϕ4 -model as a typical bosonic model to demonstrate our resummation method. As usual, we start in finite volume [0, L]d on a lattice with lattice spacing 1/M . The two point function is given by S(x, y) =
ϕx ϕy  (10.127) P  g2 1 P 4 2 1 − 2 x ϕx Md e− M 2d x,y (−∆+m )x,y ϕx ϕy Πx dϕx N d ϕx ϕy e := R  P P 2 4 2 − g2 1d − 12d x ϕx x,y (−∆+m )x,y ϕx ϕy M M e d e Πx dϕx N R where d    (−∆ + m2 )x,y = M d −M 2 (δx,y−ei /M + δx,y+ei /M − 2δx,y ) + m2 δx,y i=1

(10.128) 

First we have to bring this into the form [P + Q]−1 x,y dµ, P diagonal in momentum space, Q diagonal in coordinate space. This is done again by making a Hubbard-Stratonovich transformation which in this case reads
P P 2 P 2 1 dux − 12 ax x = ei x ax ux e− 2 x ux Π √ (10.129) e 2π x

with ax =

√ g ϕ2 Md x

(10.130)

Resummation of Perturbation Series

231

The result is Gaussian in the ϕx -variables and the integral over these variables gives
 −1 ig 1 2 √ (−∆ + m ) − u δ dP (u) (10.131) S(x, y) = x,y x x,y d M 2d M

RN d

x,y

where dP (u) =

1 Z

 det

1 (−∆ M 2d

+ m2 )x,y −

√ig ux δx,y Md

− 12

e− 2 1

P x

u2x

Π dux x

Since we have bosons, the determinant comes with a power of −1/2 which is the only difference compared to a fermionic system. In momentum space this reads  eik(x−y)
G(k) (10.132) S(x − y) = L1d k

where (γq = vq + iwq , γ−q = γ¯q , dγq d¯ γq := dvq dwq )
 −1 ak δk,p − √ig d γk−p dP (γ)
G(k) = L

RN d

(10.133)

kk

and dP (γ) =

1 Z

 det ak δk,p −

√ig γk−p Ld

− 12

e− 2 v0 dv0 1

2

Π+e

−|γq |2

dγq d¯ γq

q∈M

and M+ again is a set such that either q ∈ M+ or −q ∈ M+ . Furthermore ak = 4M 2

d 

sin2



ki 2M



+ m2

(10.134)

i=1

Equation (10.133) is our starting point. We apply (10.7) to the inverse matrix element in (10.133). In the two loop approximation one obtains (γ0 = v0 ∈ R)  ak δk,p −

−1 √ig γk−p d L kk

≈ =:

ak −

igv0 √ Ld

+

g2 Ld

1  p=k

Gk (p)|γk−p |2

1 ak + σk

(10.135)

where σk = − √ig d v0 + L

g2 Ld

 p=k

|γk−p |2 igv0 ap − √ + σp d L

(10.136)

232 which results in
G(k) =

1 ak +
σk 

(10.137)

where
σk  has to satisfy the equation
σk  = − √ig d
v0  + L


|γk−p |2  ap +
σp 

g2 Ld

(10.138)

p=k

=

g2 2Ld




G(p) +

p


|γk−p |2  = ap +
σp 

g2 Ld

p=k

where the last line is due to
 1
v0  = Z v0 det ak δk,p −
3

g2 Ld

√ig γk−p Ld

− 12

e− 2 v0 dv0 1


|γk−p |2  + ap +
σp 

1 2

p=k

2

e−|γq | dγq d¯ γq 2

Π

q∈M+

− 12 4 1 2  2 e− 2 v0 dv0 Π e−|γq | dγq d¯ ak δk,p − √ig d γk−p γq L + q∈M
 −1  = − 21 (− √ig d ) ak δk,p − √ig d γk−p dP (γ) (10.139) =

1 Z

∂ ∂v0 det

L

L

pp

p

As for the many-electron system, we can derive an equation for
|γq |2  by partial integration:
− 12 v02  2 e− 2 dv0 Π e−|γq | dγq d¯ γq γq γ¯q det ak δk,p − √ig d γk−p
|γq |2  = Z1 L q 
− 12  v02  2 1 = 1 + Z1 γq γq ∂γ∂ q det ak δk,p − √ig d γk−p e− 2 dv0 Π e− 2 |γq | dγq d¯ L q   ∂
√ig γk−p det a δ − k k,p d ∂γq L  dP (γ)  = 1 − 12 γq ig det ak δk,p − √ d γk−p L
 −1  −ig √ √ig γk−p a δ − dP (γ) (10.140) γ = 1 − 12 q k k,p d d L

L

p,p+q

p

Computing the inverse matrix element in (10.140) again with (10.8) in the two-loop approximation, one arrives at  g2 1
|γq |2  = 1 −
|γq |2  2L d (ap + σp )(ap+q + σp+q ) p

or
|γq |2  =

1+

g2 2

 [0,2πM]d

1 dd p 1 (2π)d (ap + σp )(ap+q + σp+q )

(10.141)

Resummation of Perturbation Series which has to be solved in conjunction with  dd p
σk  = g 2 [0,2πM]d (2π) d

233

|γk−p |2 + 12 ap + σp 

(10.142)

Introducing the rescaled quantities
σk  = M 2 s Mp ,


|γq |2  = λ Mq ,

ak = M 2 ε k , ε k =

d 

M

sin2

ki 2

+

m2 M2

i=1

(10.143) (10.141) and (10.142) read sk = M d−4 g 2 λq =

1+



1 dd p λk−p + 2 [0,2π]d (2π)d εp +sp

2

M d−4 g2

(10.144)

1



(10.145)

dd p 1 [0,2π]d (2π)d (εp +sp )(εp+q +sp+q )

Unfortunately we cannot check this result with the rigorously proven triviality theorem since
σk  and
|γq |2  only give information on the two-point function  g2  4 2 S(x, y), (10.127), and on M d x
ϕ(x)  = q Λ(q) where Λ(q) =
|γq |  − 1. However, the triviality theorem [32, 30] makes a statement on the connected four-point function S4,c (x1 , x2 , x3 , x4 ) at noncoinciding arguments, namely that this function vanishes in the continuum limit in dimension d > 4.

10.5.2

Higher Orders

We now include the higher loop terms of (10.7), (10.8) and give an interpretation in terms of diagrams. The exact equations for
G(k) and
|γq |2  are
 −1 0 / 1 ak δk,p − √ig d γk−p (10.146) dP (γ) = ak +σ
G(k) = k L

N  

kk

d

σk = − √ig d v0 + L

√ig Ld

r=2

r 

Gk (p2 ) · · · Gkp2 ···pr−1 (pr ) ×

p2 ···pr =k pi =pj

× γk−p2 γp2 −p3 · · · γpr −k and
|γq |2  = 1 +

ig √ 2 Ld




 γq ak δk,p −

p N   d

p→p2

= 1+

1 2

r=2

√ig Ld

r 

/

−1 √ig γk−p Ld p,p+q

dP (γ)

(10.147)

G(p2 )Gp2 (p3 ) × · · · × Gp2 ···pr−1 (pr ) ×

p2 ···pr =p2 +q pi =pj

× γp2 −p3 · · · γpr−1 −pr γpr −p2 −q γp2 +q−p2

0

234 For r > 2, we obtain terms
γk1 · · · γkr  whose connected contributions are, in terms of the electron or ϕ4 -lines, are at least six-legged. Since for the manyelectron system and for the ϕ4 -model [54] (for d = 4) the relevant diagrams are two- and four-legged, one may start with an approximation which ignores the connected r-loop contributions for r > 2. This is obtained by writing     γk1 · · · γkn ≈ γk1 · · · γkn 2

(10.148)

where (the index ‘2’ for ‘retaining only two-loop contributions’) 

  γk1 · · · γk2n 2 :=


γkσ1 γkσ2  · · ·
γkσ(2n−1) γkσ2n 

pairings σ




γk1 · · · γk2n dP2 (γ)

=

(10.149)

if we define

dP2 (γ) := e

−

P

|γq |2 q |γq |2

γq dγq d¯

(10.150)

Π π |γq |2  q

Substituting dP by dP2 in (10.147) and (10.148), we obtain a model which differs from the original model only by irrelevant contributions and for which we are able to write down a closed set of equations for the two-legged particle correlation function
G(k) and the two-legged interaction correlation function
|γq |2  by resumming all two-legged (particle and squiggle) subdiagrams. The exact equations of this model are
G(k) =


 ak δk,p −


|γq |2  = 1 +

ig √ 2 Ld

−1 √ig γk−p d L kk




dP2 (γ)

 γq ak δk,p −

p

(10.151)

−1 √ig γk−p Ld p,p+q

dP2 (γ)

(10.152)

and the resummation of the two-legged particle and squiggle subdiagrams is obtained by applying the inversion formula (10.7) and (10.8) to the inverse matrix elements in (10.151) and (10.152). A discussion similar to those of section 10.3 gives the following closed set of equations for the quantities
G(k) and
|γq |2 :
G(k) =

1 , ak +
σk 


|γq |2  =

1 1 +
πq 

(10.153)

Resummation of Perturbation Series

235

where
σk  =

g2 2Ld




G(p) +

p


πq  = − 12

  r=2

  r=2

√ig Ld

r−1 r  s=3

√ig Ld

r 


G(p2 ) · · ·
G(pr ) ×

p2 ···pr =k pi =pj

(10.154) ×
γk−p2 γp2 −p3 · · · γpr −k 2   δq,ps+1 −ps
G(p2 ) · · ·
G(pr ) × p2 ···pr =p2 +q pi =pj

5ps −ps+1 · · · γpr−1 −pr γpr −p2 −q 2 ×
G(p2 + q)
γp2 −p3 · · · γ

 (10.155)

In the last line we used that γq in (10.148) cannot contract to γp2 −p3 or to γpr −p2 −q . If the expectations of the γ-fields on the right hand side of (10.154) and (10.155) are computed according to (10.149), one obtains the expansion into diagrams. The graphs contributing to
σk  have exactly one string of particle lines, each line having
G as propagator, and no particle loops (up to the tadpole diagram). Each squiggle corresponds to a factor
|γ|2 . The diagrams contributing to
π have exactly one particle loop, the propagators being again the interacting two-point functions,
G for the particle lines and
|γ|2  for the squiggles. In both cases there are no two-legged subdiagrams. 1 resums ladder or bubble However, although the equation
|γq |2  = 1+ π q diagrams and more general four-legged particle subdiagrams if the terms for r ≥ 4 in (10.155) are taken into account, the right hand side of (10.154) and (10.155) still contains diagrams with four-legged particle subdiagrams. Thus, the resummation of four-legged particle subdiagrams is only partially through the complete resummation of two-legged squiggle diagrams. Also observe that, in going from (10.151), (10.152) to (10.153) to (10.155), we cut off the r-sum at some fixed order  independent of the volume since we can only expect that the expansions are asymptotic ones, compare the discussion in section 10.3.

10.5.3

The ϕ2 ψ 2 -Model

This problem was suggested to us by A. Sokal. One has two real scalar bosonic fields ϕ and ψ or ϕ1 and  ϕ2 (since ψ ‘looks fermionic’) with free action −∆ + m2i and coupling λ x ϕ21 (x)ϕ22 (x). One expects that there is mass generation in the limit m1 = m2 → 0. In the following we present a computation using (10.7) in two-loop approximation which shows this behavior. Let coordinate space be a lattice of finite volume with periodic boundary conditions, lattice spacing 1 and volume [0, L]d : Γ = {x = (n1 , · · · , nd ) | 0 ≤ ni ≤ L − 1} = Zd /(LZ)d Momentum space is given by $ %  2π d M := Γ = k = 2π /(2πZ)d L (m1 , · · · , md ) | 0 ≤ mi ≤ L − 1 = L Z

236 We consider the two-point functions (i ∈ {1, 2})
P 2 2 1 Si (x, y) = Z ϕi,x ϕi,y e−λ x∈Γ ϕ1,x ϕ2,x × R2Ld

e−

P

P2

ϕi,x (−∆+m2i )ϕi,x

x∈Γ

i=1

(10.156) Π dϕ1,x dϕ2,x

x∈Γ

where




e−λ

Z= R

P x∈Γ

ϕ21,x ϕ22,x −

e

P

P2

x∈Γ

i=1

ϕi,x (−∆+m2i )ϕi,x

2Ld

Π dϕ1,x dϕ2,x

x∈Γ

(10.157) Consider, say, S1 . We may integrate out the ϕ2 field to obtain


− 1 ϕ1,x ϕ1,y det −∆ + m22 + λϕ21,x δx,x
2 × S1 (x, y) = Z1 RL d

e−

P x

ϕ1,x (−∆+m21 )ϕ1,x

Π dϕ1,x x

(10.158)

On the other hand, integrating out the ϕ1 field gives the following representation: P 2 2 

e− x ϕ1,x (−∆+m1 +λϕ2,x )ϕ1,x 1 ϕ1,x ϕ1,y dϕ S1 (x, y) = Z Π 1,x × −1 RL d RL d det [−∆ + m21 + λϕ22 ] 2 x P

− 1 2 det −∆ + m21 + λϕ22 2 e− x ϕ2,x (−∆+m2 )ϕ2,x Π dϕ2,x x


−1 = Z1 (10.159) −∆ + m21 + λϕ22 x,y dP (ϕ2 ) RL d

where dP (ϕ2 ) =

1 Z

P

− 1 2 det −∆ + m21 + λϕ22 2 e− x ϕ2,x (−∆+m2 )ϕ2,x Π dϕ2,x x



and dP (ϕ2 ) = 1 (the Z factor above differs from its definition in (10.157) by some factors of 2π, for notational simplicity we have chosen the same symbol). By taking the Fourier transform (and dropping the ‘hats’ and the index 2 on the Fourier transformed ϕ2 field)  eik(x−y)
G1 (k) (10.160) S1 (x − y) = L1d k

where (ϕq = vq + iwq , ϕ−q = ϕ¯q , dϕq dϕ¯q := dvq dwq )
 −1 6 2) a1,k δk,p + √λ d (ϕ dP ({ϕq })
G1 (k) = k−p L k,k RL d


−1  a1,k δk,p + Lλd t ϕk−t ϕt−p k,k dP ({ϕq }) = RL d

(10.161)

Resummation of Perturbation Series

237

To keep track of the volume factors recall that ϕq is obtained from ϕx by applying the unitary matrix F = ( √1 d e−ikx )k,x of discrete Fourier transform, L that is   ϕq = √1 d e−iqx ϕx , ϕx = √1 d eiqx ϕq L

L

x

q

 2

 F ϕx δx,x
F ∗ k,p =



1 Ld

e−i(k−p)x ϕ2x =

√1 Ld

6 2) (ϕ k−p

x

and ϕ2x =

1 Ld



ei(k+p)x ϕk ϕp =

√1 Ld



eiqx

√1 Ld

q

k,p



ϕk ϕq−k

k

which gives 6 2) (ϕ k−p =

√1 Ld



ϕt ϕk−p−t =

√1 Ld

t



ϕt−p ϕk−t

(10.162)

t

Furthermore dP ({ϕq }) =

1 Z

det a1,k δk,p + ×

Π

q∈M+

− 12 λ  × t ϕk−t ϕt−p Ld −(q2 +m22 )|ϕq |2 e dϕq dϕ¯q

(10.163)

and a1,k = 4

d 

sin2

ki

2

+ m21

(10.164)

i=1

Equation (10.161) is the starting point for the application of the inversion formula (10.7). We have to compute
G1 (k) = G1 (k) dP (ϕ) where  G1 (k) = a1,k δk,p +

λ Ld



ϕk−t ϕt−p

t

−1

(10.165)

k,k

In the two-loop approximation one obtains G1 (k) =

1 a1,k + σ1,k

(10.166)

where, approximating G1,k (p) by G1 (p) in the infinite volume limit,     λ λ σ1,k = Lλd ϕk−t ϕt−k − ϕ ϕ G (p) ϕp−t ϕt−k k−s s−p 1 d d L L p p=k

t

=

λ Ld

 t

|ϕt |2 −

2

λ L2d



p,s,t p=k

s

ϕk−s ϕs−p ϕp−t ϕt−k G1 (p)

t

(10.167)

238 Thus
G1 (k) =

1 a1,k +
σ1,k 

(10.168)

where
σ1,k  =

λ Ld




|ϕt |2  −

λ2 L2d

t




ϕk−s ϕs−p ϕp−t ϕt−k 
G1 (p)

(10.169)

p,s,t p=k

To obtain a closed set of integral equations, we ignore connected three and higher loops and approximate
ϕk−s ϕs−p ϕp−t ϕt−k  ≈
ϕk−s ϕs−p 
ϕp−t ϕt−k  +
ϕk−s ϕp−t 
ϕs−p ϕt−k  +
ϕk−s ϕt−k 
ϕs−p ϕp−t  = δk,p
|ϕk−s |2 
|ϕt−k |2  + δt−p,k−s
|ϕk−s |2 
|ϕs−p |2  + δs,t
|ϕk−s |2 
|ϕs−p |2  = δt−p,k−s
|ϕk−s |2 
|ϕs−p |2  + δs,t
|ϕk−s |2 
|ϕs−p |2  (10.170) where the last line is due to the constraint k = p in (10.169) which comes out of the inversion formula (10.7). Thus
σ1,k  has to satisfy the equation
σ1,k  =

λ Ld



2


|ϕt |2  − 2 Lλ2d

t


|ϕk−s |2 
|ϕs−p |2  a1,p +
σ1,p  p,s

(10.171)

p=k

To close the system of equations, one needs another equation for
|ϕ2,q |2 . This can be done again by partial integration as we did in the last section for the ϕ4 -model. However, for the specific model at hand, one has, by virtue of (10.158) with the indices 1 and 2 interchanged,
|ϕ2,q |2  =
G2 (q)

(10.172)

Thus one ends up with
G1 (k) =

1 a1,k +
σ1,k 

(10.173)

where
σ1,k  =

λ Ld

 t

2


G2 (t) − 2 Lλ2d




G2 (k − s)
G2 (s − p)
G1 (p) (10.174)

p,s p=k

In particular, for m1 = m2 = m one has
G1  =
G2  ≡ G, G(k) =

k2

1 + m2 + σk

(10.175)

Resummation of Perturbation Series

239

and σk has to satisfy the equation σk = λ



dd p [−π,π]d (2π)d

G(p) − 2λ2



dd p dd q [−π,π]2d (2π)d (2π)d

G(p) G(q) G(k − p − q) (10.176)

To see how the solution looks like one may ignore the λ2 -term in which case σk = σ becomes of k. In the limit m2 ↓ 0 one obtains, if we d independent 2 substitute 4 i=1 sin [ki /2] by k 2 for simplicity, σ=λ



dd p 1 [−π,π]d (2π)d p2 +σ

(10.177)

which gives ⎧ ⎨ O(λ)   σ = O λ log[1/λ] 2 ⎩ O(λ 3 )

10.6

if d ≥ 3 if d = 2 if d = 1 .

(10.178)

General Structure of the Integral Equations

In the general case, without making the approximation (10.148), we expect the following picture for a generic quartic field theoretical model. Let G and G0 be the interacting and free particle Green function (one solid line goes in, one solid line goes out), and let D and D0 be the interacting and free interaction Green function (one wavy line goes in, one wavy line goes out). Then we expect the following closed set of integral equations for G and D: G=

G−1 0

1 , + σ(G, D)

D=

D0−1

1 + π(G, D)

(10.179)

where σ and π are the sum of all two-legged diagrams without two-legged (particle and wavy line) subdiagrams with propagators G and D (instead of G0 , D0 ). Thus (10.179) simply eliminates all two-legged insertions by substituting them by the full propagators. For the Anderson model D = D0 = 1 and (10.179) reduces to (10.67) and (10.76). A variant of equations (10.179) has been derived on a more heuristic level in [15] and [47]. Their integral equation (for example equation (40) of [47]) reads G=

G−1 0

1 +σ ˜ (G, D0 )

(10.180)

240 where σ ˜ is the sum of all two-legged diagrams without two-legged particle insertions, with propagators G and D0 . Thus this equation does not resum two-legged interaction subgraphs (one wavy line goes in, one wavy line goes out). However resummation of these diagrams corresponds to a partial resummation of four-legged particle subgraphs (for example the second equation in (10.183) below resums bubble diagrams), and is necessary in order to get the right behavior, in particular for the many-electron system in the BCS case. Another popular way of eliminating two-legged subdiagrams (instead of using integral equations) is the use of counterterms. The underlying combinatorial identity is the following one. Let
¯ = S(ψ, ψ)

¯ dk ψ¯k G−1 0 (k)ψk + Sint (ψ, ψ)

(10.181)

be some action of a field theoretical model and let T (k) = T (G0 )(k) be the sum of all amputated two-legged particle diagrams without two-legged par¯ ticle subdiagrams, evaluated with the bare propagator G0 . Let δS(ψ, ψ) = ¯ dk ψk T (k)ψk . Consider the model with action S − δS. Then a p-point function of that model is given by the sum of all p-legged diagrams which do not contain any two-legged particle subdiagrams, evaluated with the bare propagator G0 . In particular, by construction, the two-point function of that model is exactly given by G0 . Now, since the quadratic part of the model under consideration (given by the action S − δS) should be given by the bare Green function G−1 0 and the interacting Green function is G, one is led to the equation G−1 − T (G) = G−1 0 which coincides with (10.180). Since the quantities σ and π in (10.179) are not explicitly given but merely are given by a sum of diagrams, we have to make an approximation in order to get a concrete set of integral equations which we can deal with. That is, we substitute σ and π by their lowest order contributions which leads to the system 1 , dp D(p)G(k − p) 1  D(q) = D0 (q)−1 + dp G(p)G(p + q) G(k) =

G0 (k)−1 +



(10.182) (10.183)

This corresponds to the use of (10.7) and (10.8) retaining only the r = 2 term. Thus we assume that the expansions for σ and π are asymptotic. Roughly one may expect this if each diagram contributing to σ and π allows a constn bound (no n! and of course no divergent contributions). The equations (10.182) and (10.183) can be found in the literature. Usually they are derived from the Schwinger-Dyson equations which is the following nonclosed set of two equations for the three unknown functions G, D and Γ, Γ being the vertex

Resummation of Perturbation Series

241

function (see, for example, [1]):
dp G(p) D(k − p) Γ(p, k − p) G(k)

G(k) = G0 (k) + G0 (k)

(10.184)


D(q) = D0 (q) + D0 (q)

dp G(p) G(p + q) Γ(p + q, −q) D(q)

The function Γ(p, q) corresponds to an off-diagonal inverse matrix element as it shows up for example in (10.103). Then application of (10.8) transforms (10.184) into (10.179). One may say that although the equations (10.182) and (10.183) are known, usually they are not really taken seriously. For our opinion this is due to two reasons. First of all these equations, being highly nonlinear, are not easy to solve. In particular, for models involving condensation phenomena like superconductivity or Bose-Einstein condensation, it seems to be appropriate to write them down in finite volume since some quantities may become macroscopic. And second, since they are usually derived from (10.184) by putting Γ equal to 1 (or actually −1, by the choice of signs in (10.184)), one may feel pretty suspicious about the validity of that approximation. The equations (10.179) tell us that this is a good approximation if the expansions for σ and π are asymptotic. A rigorous proof of that, if it is true, is of course a very difficult problem.


    

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