Many-Particle Theory,

Many-Particle Theory E K U Gross Universitat Wiirzburg E Runge Harvard University 0 He"Inonen University of Central Fl...

Author: E. K. U. Gross | E. Runge | O. Heinonen

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Many-Particle Theory E K U Gross Universitat Wiirzburg

E Runge Harvard University

0 He"Inonen University of Central Florida

Adam Hilger Brist 01, Philadelphia and New York

English edition @ IOP Publishing I991 Originally published it1 German as Vte!i!eilchenlf~eorie, @ 13 G Teubner, Stuttgast 1986

All rights reserved. Na part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior perrnission of the publisher. Multiple copying is onIy permitted under term of the agreement between the Commit tee of Vice-Chancellors and Principals and the Copyright Licensing Agency.

British Library Cataloguing in Publication Dad a Grass,E.K.U. Many-PartitleTheory I. Title II. Runge, E. 111. Heinonen, 0 . 539 -7 ISBN 0-7503-0 155-4 Librdry of Cong~cssCataleging-in-Pza blication Data Gross, E. K. U. (Eberhard K.U.). 1953Many-particle theory f E.K.U. Gross, E. Runge, 0 . Reinonen. 433 p. cm. Translation of: Vielteilchentheorie Includes bibliographi cal references and index. ISBN 0-7503-0155-4 1. Many-body problem. 2 . Ferrnions, 3. Perturbation (Quantum dynamics) 4, Fermi liquid theory. 5. Ba~tree-Fock approximation. I. Runge, E. 11. Reinonen, 0 . 111. Title QG174.17,P7G7613 1991 530.1 '44-dc20 9 1-25412 CIP

Published under the Adam Hilger irnprint by IOP Publishing Ltd Techno House, Redcliffe Way, Bristol BS1 6NX, England 335 East 45th Street, New York, NY 10017-3483, USA US Editorial Office: 1411 Walnut Street, Philadelphia, PA 19102

Typeset in BTEX by Olle Heinonen. Printed in Great Britain by Bookcraft, Bath.

Preface to the English edit ion

The world around us consists of int efact ing many-particle systems. The aim of theoretical atomic, molecular, condensed matter and nuclear physics is t o describe the rich variety of phenomena of such systems. Modern manyparticle theory is a common denominator of all these, in detail rather different, disciplines of phy~icsand is the subject of this book. Perhaps the most important fundamental concept of many-particle theory is based on the idea that an interacting many-particle system can be described approximately as a system of non-interact ing 'qumi-particles'. However, if we want t o mathematically rigorously demonstrate this intuitive idea, a rather complicated formalism is necessary. Tbis formalism rests on three cornerstonea. (1) The so-calied second quantization, (2) t h e method of Green's functions, and ( 3 ) pert urbation-t heore tical analysis using Feynman diagrams. These three cornerstones presently form the framework within which many experiments are formulated and understood. Not in the lemt because of the overwhelming technological iriqortance that many-particle effects in condensed matter physics have attained in recent years, these cornerstones of many-particle physics must be regarded as part of the standard graduate physics curriculum. Our aim in writing this book is to present the basics of many-particle theory in a closed and easily accessible form. The presentation is primarily aimed .Lit dtudents of theoretical and expetimentd physics, but should also be useful t o mathematicians interested in applications, arid t o theoreticalIp oriented chemists. Only a basic grasp of fundamental quantum mechanics taught in introductory courses is assumed. On this foundation, the material is developed in a systematic and self-contained way. The material in this book consists of well-known ~esults,and great emphasis is placed on performing derivations in detail, without skipping intermediate steps. This makes the material ideal for t h e use in courses, miniAziag the p~eparations necessary. The examples are selected to first of all iflustrate the relations between

the mathematical formalism and physically intuitive concepts. A complete overview of the endless uses of many-particle theory cannot and should not be given within the scope ofthis book. It is, however, our goal that this book will give the reader the necessary fundamental knowledge of many-particle theory to understand the original literature in condensed matter physics, nuclear physics and quantum chemistry. There are many people to wham we owe thanks for the preparation of this book. An understandable present ation of Feynman diagrams is impossible without skilled technical artists. Many thanks to Mrs. Buffo for preparing the original artwork, and to the staff at 10P Publishing for very metic~lausly removing the artwork from the original German manuscript and pasting it into the English manuscript. Many people have helped us with the tedious task of proof-reading the manuscript, and mercilessly pointed out gross or subtle errors, and asked pointed questions about 'what we really mean by' vague and ill-formulated statements. In particular Elisab eth Runge , Mike Johnson, Phil Taylor and Jay Shivamoggi have been extremely helpful, We also thank Leslie McDonald and our editor at IOP Publishing, Lauset Tipping, for helping t o transform a strange concoction of Germah, Swedish and English into sumething we hope passes for English.

E ber hard Gross Erich Runge Olle Heinonen

Contents Preface

v

I Fundamentals and Examples 1 Systems of identical particles

3

2 Second quantization fox fermions

17

3 Second quantization far bosons

31

4 Unitary transformations and field operators

33

5

Example the H a d t o n i a n of translationally invariant systems in second quantization

39

6 Density operators

43

7 The Hartree-Fock approximation

51

8 Restricted Hartree-Fock approximation and the symmetry

dilemma

67

9 Hart reeFo ck for t ranslationally invariant systems

73

10 The homogeneous electron gas in the HartreeFock approximation

79

11 Long-range correlations in the electron-gas: plasmons

89

13 Superconductivity

Green's Functions

I1

14 Pictures 15 The single-particle Green's function 16 The polarization propagat or, the two-part icIe Green's function atld the hierarchy of equations of motion

III

Perturbation Theory

17 Time-independent perturbation theory 18 Time-dependent perturbation theory with adiabatic t urning-on of the interaction

19 Particle and hole operators and Wick's theorem 20 Feynman diagrams

2 1 Diagrammatic calculation of the vacuum amplitude 22 An example: the Gell- Mmn-Brueckner correlation energy of a dense electron gas

23 Diagrammatic calculation of the single-particle Green's function: Dyson's equation 24 Diagrammatic analysis of the Green's function G(k,w ) 25 Self-consistent perturbation theory, an advanced

perspective on HartreeFock theory 26 The quasi-particle concept

27 Diagrammatic calculation of the two-par t icle Green's function and the polarization propagator 28 Effective interaction and dressed vertices, an advanced perspective on plasmons

29 Perturbation theory at finite temperatures

IV Fermi Liquid Theory 30 Introduction

31 Equilibrium properties 32 Transport equation and dolIe~tivemodes 33 Microscopic derivation of the Landau Ferrni liquid

t heary 34 Application to

References h r t her reading Index

the Koado problem

Part I Fundamentals and Examples

Chapter 1 Systems o f identical particles Identical particles are particles which have: the same masses, charges, sizes a n d all other physical properties. In t h e realm of classical mechanics, identical particles are distinguishable, For example, consider a collision of two identical billiard balls - we can follow the trajectory of each individual ball and thus distinguish them. However, in the physics of microscopic particles where rve must use quantum mechanics, the sit uatibn is quite different. We can distinguish identical particles which are very far apart from each other; for example, one electron on the moon can be distinguished from one on the earth. On the other hand, when identical particles interact with one another, as in a scattering experiment, the trajectory concept of classical mechanics cannot be applied because of Heisenberg's uncertainty principle. As a consequence, microscopic particles which interact with each other are completely indistinguishable - they cannot be distinguished by any measurement. The measurable quantities of .a stationary quantum mechanical sys tern are the expectation values of operators that represent the observables of the system. If the system consists of identical particles, these expectation values must not change when the coordinates of two particles are interchanged in the wavefunctioa, (If we could find expectation values which do change when the coordinates of two particles are interchanged,we would have found measurable quantities by which we could distinguish the particles.) Thus, we require that for any possible state Q of the system and for all observables B

for all pairs ( j ,k ) . (We use a caret to denote operators.)

SYSTEMS OF IDENTICAL PARTICLES The coordinates z (r,s ) contain the space and.spin degrees af freedom of the particles. We use the notation

/dr=?Jd3r

and

Jd"r=/dil

Jdr,..

Jitrn-

To a certain extent, equation (1.1) above defines a system of identical particles. B y using this equation, we will derive properties of both the wavefunction and of the operators that describe a system of identical particles. In so ddihg, we need only require t h a t the expectation values remain unchatlged when we interchange t w o particles, since each possible permutation of the particles can be expressed as a sequence of two-particle interchanges. In mathematical terms, each permutation P can be expressed as a product of transpositions Pjk

on a many-particle wavefunction. Applying the permutation operator Pjk twice restores the original wavefunction. Hence, it folIowd that

pjkPjk = Id = 1, So

pJ 1 in any ease. The functions defined in this way .are eigenfunctions of

with eigenvalues

We will give

a Brief proaf

for the case N = 2 . The proof is quite analogous

for arbitrary N .

The fermion wavefunctions dA) above are called Slater deterninonis. Upon inspection, we can deduce the following interesting facts about these functions,

(1) If two particles are the same, vi = v j for some i # j,we have

= 0.

(2) If two columns of the determinant are identical, so that x; = r j for some i # j , we have dA) = 0.

The wavefunction vanishes in both cases; and as a result, the probability of finding such a st ate vanishes as well. This has .two consequences. (1) It is impossible to have two ferrnions in the same state; one state can be occupied by no more than one particle. (2) It is impossibIe t o bring two fermions with the same spin projection to the same point.

-

These two statements comprise the Pavli principle. An essential assumption for this principle is that the parkicks ate independent, i.e., that they do not interact with one another, so t h a t the many-particle states can be characterized by occupied and unoccupied single-particle orbitals. Pauli originally fclrrnulat-ed tbe principle for atoms, in which case the assumption that such systems can be realistically regarded as consisting of independent particles (a'. e., that the electron-electron inte~actionscan be reasonably well described by an effective external potential), is the basis ~ Q Tthe principle. We will justify this assumption in later chapters which discuss the H a r t r e e Fock approximation, It should be emphasized that no such principle exists for bosons. Consequently, we can put an arbitrary number of bosons in one single-particle st ate. is completely determined given the singleThe symmetric state particle states that it contains, e . 4 .

is only determined 2 0 within a In contrast, the antisymmetric state sign by the single-particle states that it contains. For example, we can define two-particle states from d,, and d,, by

We can remove this last ambiguity of the fermion wavefunctions by assigning an arbitrary, but hereafter always the same, enumeration to the v;:

and, according to the same principle, by enumerating t h e single-particle orbitals that belong to these quantum numbers. Given an oMcred N-tuple of indices c = ( ~ 1 , ~ 2 1 .- .J N )

with ci E Af, where hfis the set of natural numbers, and c l < c2 < ... < c~ the $later determinant

\

FUNDAMENTALS AND EXAMPLES

11

is then u n ~ b i g n o u s l ydefined; for by giving the sequence, each product of single-particle orbitals distinguished by increasing indices and consequently correspanding to identical permutations, is defined by a positive sign. Hence, the signs of all other permutations are also determined. As a result, the ex amp1e above b ecornes

1 - -[dl (t1)$4(~2) - 41 ( + 2 ) d 4 ( ~ 1 ) ]

-fi

The value of the determinant is invariant under interchange af all rows and columns so long as the enumeration is retained, We use

to denote symbolically a permat at ion of hr elements, For example

Thus, the Slater determinant can be written in either of t w o ways:

These two possibilities correspond to the row-column interchanges in the determinant. Both exist for the completely symmetrized wavefunct ion @, ($1 as well. We will also characterize the symmetric functions by ordered N tuples of indices, but for syrnrnctric functions the ordering is unimportant. In the remainder crf this chapter will we rove one more important propFm each observable erty of the many-particle wavefunctions and B of a system of identical particles, we have

@(ST

SYSTEMS OF IDENTICAL PARTICLm

12

I = 1). and forL@ = dA) (in the latter case, k" nkd! both for O = We will prove this only for the SIater determinant dA) - the proof for is similar. By definition

Since we have becomes

AS

BP = P$ for

p-lm: = sgn(p-l)

- m;

obse~vablesof the system, this expression

= sgn(P)Qg, we may write this as

Since the value of the integraI remains unchanged if we interchange the integration variables x; i xp(;),we therefore obtain the same vdue for each permutation (altogether N ! times), and thus arrive at the expressioh

With these results, we can prove that the orthonormality af the families of many-partide functions {a,( S )] and follows from the orthonormality of the single-particle orbitaIs { + y ( a ) ) , Explicitly, we must show that

{@IA))

For

d A )equation , (1.7)

with b = c becomes


Since we have assumed that the {&;Iare orthogonal, the product of integrals will be non-zero only for the identity permutation

in which case the result is unity because sgn(Id)=l and because the singleparticle orbitals are normalized. If b # c, there exists at least one index b j , which ia not contained in c. It follows that at Ieast one factor in t h e product of integrals shown above aIways vanishes - therefope the entire expression also vanishes. The proof is similar for symmetric functions dS). If, in this case, a particular single-particle state is multiply occupied, the sum over all petmut at ions contains several non-vanishing terms whose contribution is reduced precisely by a factor h' nk(4 .. We conclude this chapter by proving the extremely important. completeness dheorem:

If the family {qbv(x)) is complete, so too are the

families {a,(4) and (a,( S ) of many-particle functions in the corresponding Hilbert spaces of antisymmetric and syrnmetric many-particle functions, respect iveIy.

-

As an example, consider localized spinless p a r t i c k . The theorem assures that if {&(P)) is complete in then ~ @ ! ~ ' ( r.l ,,r N ) }is complete in

LY',

L $ ~ ) ( R ~ X R ~ X

N times

and

(a,(q( r l , ., .,rN)) is complete in

iV times

+

xz3)

.

SYSTEMS OF IDENTICAL PARTICLES

14

where L2 ( S ' A ) ( ~ 3 x . . . x 7Z3) denote the Hilbert spaces of symrnetric/antisymmetric square-integrable functions on 7Z3 x . . . x 713 with 7Z3 the three-dimensional Euclidean space. To prove the theorem, we begin by showing that a n arbitrary manyparticle wavefunction can be expanded in products of single-part iele functions. We fix the last N 1 coordinates at xjO), i = 2 $ 3 ,. . , , N , and expand the wavefunction with respect to the first one:

-

The expansion coefficients are functions of the fixed coordinates: a,, ,

= dvl

(2 2(O), x3

(O)

9 "" !

XF')

so that Q("*,~~,-.-,"N)' C ( l V 1 ( " P I . . .l w ) $ u l ( x l ) .

If we expand further with respect

to

the next coordinates

and so on, we obtain

If the functions Q have definite symmetries $'jk* = +P with '+' for bosons, and minus for fermions, we can conclude that the expansion coefficients u v ,u2 ~ ,-.-,UN have the same symmetry:

15


On the last line, the indices were simply renamed,

vk

H

v j . If we compare

the last line with the first, it follows from the Iinear independence of the product funttions that

Thus, it follows that the coefficients belonging to permuted index combinations (v;! v ; ! . . . , v ~ Y )= P ( q , ~ 2 ,-.., V N ) differ at the most by a multiplicative sign, so that

with '+' for bosons and sgn(P) far ferrnions. In the last step of the proof, we also replace the multiple sums over dl indices in @(XI,

..

- 1

+N)

=

C

4.1

(21) .

-

d u N ( x ~ ) ~ ....,v uN I

v1 ,.-.,YN

by an (infinite) sum over all ordered combinations c = jul , .. . , w},comec ting a (finite) sum over all permutations P ( v l , .. , ,v N ) of these combinations, with the result

By using the above result

it fallows-finally that

We hare shown then that we c a n expand arbitrary symmetric or mtisymmetric functions in and respectively. The coefficients a, differ from the coefficients f, due ta the normalization of the function @ ( S / A ) .

@LA))

Chapter 2 Second quantization for fermions 1; this chapter, me will define annihilation and creation operators. These operators are mappings between the many-particle Hilbert spaces of different particle numbers:

For example, for Iocalized systems, we can think of X ( N ) as

N times

.

the Hilbert space of square-integrable antisymmetric functions on 7Z3 x . .x

a3.

We proved in the previous chapter that the Slater determinants @c(zl,,.,,sN) form a basis in the N-fermion Hilbert space. We define the action of the annihilation operators tk on this basis by

and if k = c j , by

SECOND Q UIANTIZATION FOR FEIUiIIONS

18

The variables xi are only written out for the sake of clarity - the tk are operators on Hilbert spaces and act on abstract functions. The not ation. introduced here is commonly used and its meaning is clear even though the numbers of the arguments on either side of the equivalence differ. Essentially what happens when tk acts on a Slater determinant is that the orbital with index E is crossed out from the determinant. Pictorially speaking, the particle which occupied this orbital is knnihilated', h o r n the fixed sign it is important t o remember the the Slater determinant is onIy unamb~guouslgdetermined by a given a r d e ~ dN-tuple c = (cl, c ~. ,, ., c N ) , with cl < c~ ... < C N , ci E N . The orbital $ k , which is crossed out by the action of Ek, thus has a 'fixed place'inside the Slater determinant. According to the definition given in equation (Z), the orbital to be crossed out must &st be moved to the top of the determinant, which gives rise to the sign ( - The meaning of this sign-convention will be made clear in the following example. If we assume that k = c j , we can illustrate the action of
.

,,

FUNDAMENTALS A N D EXAMPLES

1

- - x ( ( i k 16 i k ) - ( i k 2 t.= l

and since (ij ( fi [ kt) = (ji I 6

a 1 ki))

Ia), the e~pressionabove becomes

We thus obtain h'oopman's theorem for the ionization energies I,:

The ionization energies that are calculated in this way agree rather well with the experimentally determined values. The same holds far the photoabsorption thresholds for knocking electrons out of deep-lying shells, The suggestion to describe excited rnany-particle states by a wavefunction k ~ i r l ~ presents o) itself. In this wavefunction, a state which is occupied in the Bartree-Fock ground state is replaced by an unoccupied one:

THE ITA RTREE-FOCK APPROXIMATION

= x(il f + i i 1 i) i= 1

+2

,

.-

({ij1 a 1 ij) - {ij 16 I j i ) )

213-1

The excitation energy

becomes

(AE)pk= rk - ~e - ((h.e]&]k!) - ( k f ] $ l f k ) ).

(7.3)

At this point it should be emphasized that the excitation energies calculated in this way are not generally any upper bounds to the exact energies, 'in contrast to the Hartree-Fock ground state energy. h t h e r m o r e , as we have just discussed, the unoccupied orbitals of the ground st ate interact formally with N particles, arid not with N - 1, which would have been correct for excited states. For very l a ~ g systems, e this difference between ( N - 1) particles and N particles does not make much difference. For atomic systems, however, this difference has drastic consequences: consider first a neutral atom. An unoccupied orbital is in this case essentially affected by .the total electrostatic potential of the nucleus and the electrons. This potential vanishes exponentially asymptotically, since the -Z/r-potential of the nucleus and the contributions from the elmtrans cancel asymptotically. A true, excited electron should thus asymptotically feel the potential of the nucleus and of only lV - 1 electtons, which together behave as -1Jr. Since the excited electrons is sa to speak outside the atom, pictorially speaking, its energy is essentially determined by the asymptotic behavior of the potential. It follows that the unoccupied orbitals determihed through the HartreeFock procedure are boond far too weakly, As a result, we cannot expect that the excitation energies for atoms calculated by this procedure should agree particularly well with experimental values.


63

The formula above is also interesting from another point of view: the variational principle, which we evoked in the derivation of the H a r t r e e Fock equations, bnly guarantees that EHFis stationary with respect t o small variations in the single-particle orbitals. Hence, it would be very useful t o find a criterion which would enable us t o state whether the solutions we have found correspond to the minimum energy. A necessary condition for this is certainly E H F ( N< ) EHF(J\J - le 4-lk)

For systems with extended states we can neglect the matrix elements in the potential, so that in this case the criterion is reduced to ft; > ~ p . In other words, the orbital energies of the occupied states must Iie deeper than the unoccupied ones, We will now give a crude estimate to show that the matrix elements of the inte~actionpotentid is, on the average, a factor of (1JN) smaller than the single particle energies. (In examples from solid state theory, N is of t h e order of We consider only cases without external potentials and restrict ourselves to interactions which are homogeneous of degree (- 1) in the position coordinates. An exampIe of such a potential is the Coulomb potential. The virial theorem holds under these assumptions. This theorem states the following: let @ be a solution of a minimization problem of the type

which can be obtained by free variations of the problem, so that is not obtained by just fitting parameters. We then consider the normalized functions O, (rl sl , .. ,r N d N )= 0 1 3 N / 2 ~ ( a r l s 1. .,., &rNs N )

.

which are obtained by a scale transformation of @. These functions satisfy

so that

64

T H E HARTREE-FOCK APPROXTMATION

which is the virid theorem. The Hart ree-Fock solution is no? obtained by free variations according to the Rayleigh-Ritz principle, but by restricting the variations to Slater determinants. However, since the scale transformation maps Slater determinant s back onto Slater determinants, we have again

By using the relation equation (7.4) between the total energy and the single particle energies, we see that

which means that

We introduce average quantities

We have then shown that the difference between the matrix elements is proportional to Z/N. We will close this section with a brief discussion of the so-called Hartree approximation, which is the historic predecessor of the Hartree-Fock approximation, In the Hartree approximation, one makes the ansatr of a product of different single par tick orbit a h for the wavefunction of the system:

In this case one also begins with a model of non-interacting particles, but in the Hartree approximation one does not take into consideration the required antisymmetry of the wavefunction. If one varies the total energy obtained from this product-ansatz under the constraint that the single particle orbitals be normalized, the Hartreeequations are obtained:


65

The ,total energy corresptjnding to a self-consistent solution of this system of equations is N

.

N

The intesaction energies, which are counted twice in the sum over the orbital energies, are subt~actedout, just as in the Hartree-Fack energy (cf. equation

17.411.

The effective R a r t ~ e epotential differs from the Hartree-Fock potential in that it does not cont-ain the exchange terms. Consequently, the Hartree potential is a local operator, which significantly facilitates solving equation (7.10) self-consistentlycompared to solving the HartreeFock equations (7.3). Moreover, the Eartree potential is different for each state. This property is not only physically wrong (it contradicts the indistinguishability of identical microscopic particles), but also complicates numerical solutions, since one must solve a different eigenvalue equation for each orbital. As a consequence, one cannot in general choose the Hartree orbitals to be ort hogonal, since they originate from different eigenvalue problems. In contrast to @, , the Hart ree-Fock wavefunct ion satisfies

since it is a Siater determinant. This is precisely the corrtent of the Pauli principle, according to which two ferrnions with equal spin s; = s j cannot be at the: same position rd = ri. Rence, the motions of fermions with equal spin are correlated in the Rartree-Fock model, in contrast to the Rartree model. There is certainly no such restriction for two particles with different spin; their motions are largely uncortelated. A repulsive particle-particle interaction, such as the Coulomb int etaction, should make it unlikely that two particles come close to each other (even when they have opposite spin). On the basis of this fact, and because the B a t t r e e h c k energies are obtained from a variational prilirciple they are too high . The difference between t h e exact ground s t a t e energy and the Hartree-Faek ground s t a t e energy h a s been given the name correlation energy. An important goal of this hook is to develop systematic ways to calculate the correlation energy. However, we will first acquaint ourselves with a n d h e r variation of t h e Hartree-Fock approximation, which is important for practicaI calculations,

Chapter 8 Restricted Hartree-Fock approximation and the symmetry dilemma The Hamiltonian of a non-relativistic atom comrnutes with the operators for the square and z-component of both the total angular momentum and total spin of the system:

As a resuit, the exact solution of the many-pa~ticleSchrodinger equation must be simultaneous eigenfuctionsof all these operators, The Hart reeFock Harniltonian, however, does not commute with any of these operators. Consequently, the solutions of the Hattree-Fack equations do not have proper symmetry. The unrestticted variation of the Hartree-Fock tsnslatz led to singleparticle arbitals of the form

where X& detiote the Pauli spinors, equation (5.3). The simplest way to construct Hattree--k'ack wavefunctions with better symmetry properties is t o restrict the form of the single-particle orbitals from the beginning; .for example p : H T ( ~ ) = p i + ' ( r ) . ~f+o ~ r v = l , ..., N+ UHF

(-1

' P ( N + + ~ ) ( " ='P~++~(')'X-(~) )

where Nt

+N-

forv= lj..*,N-

(a.z)

= N is the total number of particles. The orbitals selected

RESTRICTED HARTREE-FOCK. ,.

68

in this way are eigenfhnctions of the spin projection operators

so that we have

far the correspohding Slater determinants

UHF

If we vary the expectation vhlue (aUKF I H 1 0UHF ) with respect to the single particle orbitals equation (8.2), we obtain a simplified self-consistent procedure. This commonly used, restricted procedure, is often referred to as the hart me-Fock approximat ion. When necessary, the difference between this and other, further restricted procedures can be emphasized by calling i t the 'unrestricted Hartree-Fock' (UHF). This terminology is somewhat misleading - the ansat2 equation (8.2) is in fact already a rest rietion of the general Hartree-Fock orbitals in equation (8.1). Since the UHF-equations are ofsuch practical importance in calculations, we will now derive them for the case crf spin-independent potentials. From equation (7,2)we have the expectation value of t h e energy: UHF

-

N i= 1

2m

If we insert the UHF ansnir, equation (8.2), into

~g~p;(xt)pj(z)

we ob-

FUNDAMENTALS AND EXAMPLES t ain

We define the spin-independent density matrix by

with diagonal elements (r) E

p(il (r, r')

the total density is, in terms df the density matrix,

If we also use

C,~ , ( s ) ~ , , (s) = bg0,,

we finally arrive at

The structure of equation (8.3) shows that only interactions between particles with the same spins contribute t o the exchange energy. In analogy with the detivation of the HartreeFock equations, equation (7.31,we vary equation (8.3) with respect to the functions pj(+I (r) and rp!-)(r), which depend only on position, t a obtain the following coupled equations:

FUNDAMENTALS AND EXAMPLES hole originates from the exchange with the polarization of the medium, i.e., the repulsion betweeen electrons with the same spin. At this point, we should recall equation (8.2), in which we assumed that the single-particle orbitals are eigenfunctions 6f the spin projection operator g., If we further assume that angular momentum is a good quantum number,

we obtain the so-called 'rmtricted Hartree-Fock' (RH F) procedure. The solutions in this approximation usually have better symmetry properties. In our example, we have

and thus

A particulsr variation

the so-called analytic Hart ree-Fock or NartreeFock-Roothaan procedure, One starts with parametrized single-particle orbitals q , ( x ; aCl1. ,. , , (4 a M )with known analytic forms. The optimal parameters are determined by setting the dkrivat ive of the energy expect ation value with respect to the parameters equal to zero, We should now emphasize that the ground state energy calculated with the restricted Flartree-Fock procedure increases as the restrictions on the single-part icle or bit ds becomes stronger. We encounter a fundament a1 problem: if the best possible ground state energy is sought, t h e Hartree-Fock orbitals should bc free to vary; however, the resulting wavefunctions have poor symmetry properties. If the symmetry is improved by restricting the variatian of the orbitals, the ground state energy incre&;es. This fact is occasionally referred t o as the 'symmetry dilemma' of the Hartree-Fock procedure. Projection methods offer a way out by selecting the solution which has rrlinirnal energy and correct symmetry properties from solutions of the free-variation Hartree-Fock procedure [2]. is

I

Chapter 9

Hartree-Fock for t ranslationally invariant systems We will show in this chapter that the self-consistent problem of the BartreeFock approximation can be solved trivially for translationally invariant systems. These are systems of infinite extent, where the external potential is canst ant and the particle-par ticle interaction only depends on the relative distance between the particles. The plane waves discussed in Chapter 5 are solutions of the Hartree-Fock equations (or more precisely, of the UHF equations) for such systems., We will prove this by showing that the matrix elements of the U H F operators (cf. equation (8.4))

" (-1 and the corresponding matrix elernefits (P ] h,,, ( a) for particles with negative spin, are diagonal in the plane wave representation

We first show that the single-particle term is diagonal, and we wiU then show that the direct and the exchange terms are diagonal. With k replacing

74

HARTREE-FOCK FOR TRANSLATIONALLY,,,

the index a,and k' the index

p, we have

since u(r) = constant. To calculate the direct term, we first evaluate the density

We then insert this result in the direct term, which yields

and expand the interaction potential in a Fourier se~ies:

The direct term then becomes

Rere we made me of the assumed translational invariance of the interaction potential, which allows for a simple Fourier series in ( r - r') . We now consider the exchange term. Let C? denote the s u m over all k momenta which are occupied by particles with positive spins, and the k

x=

75


corresponding sum over particles with negative spin. With this not ation, the density matrix becomes

Again, we expand the interaction potential in a Fourier series and obtain

and the proof is complete, The diagonal elements of the Hartree-Fock operators, which are the single-paticle ehetgiest, a r e given by

For t h e total energy

EHF= x(i( i + t1 i)+ i=1

we obtain

((ij 1 G I ij)- (ij 1G 1 ji)) i,f=l

analogously

t ~ h function e r k is called the d i s p e ~ s i o nr e lation. Thjs terminology is borrowed from aptics, where one in general talks about dispersion relations whenever one is concerned with quantities that depend on wavelength.

76

HA RTREE-FOCK FOR TRANSLATIONALLY...

The calculatians show that the plane waves always satisfy the Rartree-Fock equations, completely independantIy of which momentum states are occupied and how many of the occupied stsateshave negative and positive spins. This is vem important, Since the Hartree-Fock equations only guarantee that the energy functional is stationary, the question arises for which occupations and, in particular, for which spin distributions the EartreeFock energy is manimal. A possible distribution can be obtained from the following prescription: put all N+ particles with positive spins in the momentum space sphere lk 1 = 0, . . . , k+ , and correspondingly all N - particles with negative spins in the momentum sphere ]kI = 0, .. . , k- (see f i w e 9.1). If

-

ALLowed states

Figure 9.1 All positive-spin state with k < k+, and all hegathespin states with k < I- are occupied in the JiartreeFock g ~ o u n dstate.

the Fourier transform of the particleparticle interaction depends only on Iq [ and is monotonically decreasing, this prescriy tion is certain ta produce the lowest possible Hmtree-Fock energy for the given numbers N+ and N- , since by occupying the states irl this way we will simultaneously obtain: (I$the smallest possible kinetic energy, which is positive definite in

EHF;

(2) the largest possible magnitude of the exchange energy, which is negative definite in XHF.The magnitude of the exchange energy is made as large as possible since a sphere is certainly the geometrical object for which the sum of the distances between interior points are the smalle s t . Ln this way, the sum &,; wjr-i;, is maximized for monotonically

decreasing functians vq.

1I

FUNDAMENTALS AND EXAMPLES For this occupation af the states we have

and

For the explicit evaluation of these energies, we substitute the sums by integrals according to the prescription

This approximation becomes exact in the thermodpamic limit, i.e., for an infinitely large aystern. The approximation corresponds to the transit ion from Fourier series to Foutier integrals. The volume occupied in moment urn space by each allowed state is precisely a box with sides (27rlL) (see figure 9.1). Thus, each volume element d3k in momentum space contains

This is the origin of the normalization factor of the momentum integral above. The number of particles with positive and negative spins, respectively, are then obtained as states.

and N ,

=

n k! -2 3x2'

The density of the system is %

x.

78

HARTREE-FO CK FOR TRANSLATIONALLY. ..

Far unpolarized systems, i,e,, systems for which N+ = N , = N / 2 , which implies that k+ = k, G k F , we obtain the fundamental relation

The momentum kF of the occnpied state with the highest energy is called the Fermi rnornen-lum. The single-particle energies are obtained as

and the total Hartree-Fock energy becomes

We will now explicitly go to the thermodynamic limit, a.e, we will let the volume R and the particle number iV go to infinity in such a way that the densities (IV+/R) and (N,/i2) remain constant. The momenta

will then remain constant, and it is clear that all energy contributions to EHFdiverge in this limit. This is not surptising in view of the fact that the energy is an extensive quantity. Consequently, we will instead consider t h e HartreeFock energy per particle (or per unit volume). The kinetic energy contribut iona can be evaluated by elementary integration, which yields -the following total contribution t o t h e Hartree-Fock energy per particle:

Chapter 10

The homogeneous electron gas in the Hartree-Fock approximation We will in this chapter discuss the interacting homogeneous electron gas, also known as the free electron gas or the jelliurn model, as an example of a translationally invariant system. This particular sys tern can be thought of as a model for many metals, if we assume that the charge density of the positive ions of the metal is uniformly smeared out over the volume of the system so t h a t the electrons can move practically freely through the material. This drastic assumption can only be expected to bald in some senw when the electrons which arc bound t o the ionic cores form closed shells. The valence electrons are then only weakly bound and are therefore only weakly localized ih the crystal lattite. Hence, the eIectron gas c a n be expected ta give some reasonable results for the alkali metals, where the valence electrons indeed are bound very weakIy. In fact, we can describe the cohesion of alkali metals pather well with the electron gas model in the Rartxee-Fock apptoxirnation. (While the cohesive forces of an ionic crystal, such as NaCl, are the immediate result of the electrostatic attraction between the ions, the cohesion of an alkali metal cannot be unde~stoodso easily; it originates directly from the delocalizat ion of the electrons, i. e., from the fact that binding t valence electrons to 'many' ion cores yields a state of lower energy than the state consisting of separate atoms.) The complete Hamiltonian for the electron gas is

Be.

Here H~ is the electrostatic energy of the background ions:

T H E HOMOGENEOUS ELECTRON GAS IN, ..

80

which is a simple constant. & contains the kinetic energy of the electrons as well as their mutual Coulomb repuIsion:

and Ve-i is the interaction of the electrons with the background ions. Hence, v,-; corresponds t o the external potential 0 in the p~eviousnotation:

We take the system to be a cube of volume R. With t h e imposition of periodic boundary conditions can expand the Coulomb potential i n a Fourier series:

with vq

= J, d3

rd'U'~

z 5IRI

Using equation (10.1), we can write

4re2

-{G~R&

for q

#0

f o q~ = 0-

0 as

The fact that we here obtain a constant potential is a result of the artificially imposed periodicity of the problem; we have t o a certain extent solved the Poisson equation with periodic boundary conditions.

FUNDAMEIVTALS AND EXAMPLES

81

If we use equation (9.4) for the Hartree-Fock singIe-part icle energies, the external potential

As aresult, the single-particle exactly cancels the direct tern (N/C?)U,=~. energies become

The constant energy contribution Hi from the ionic background must be included in the calculation of the total energy. Because of the imposed periodicity, we obtain after the substitution X R - R' the r e s ~ l t

=

If we now use equation (9.5) for the Har-tree-Fock energy per particle, the constant term u and 2 vq,o exactly caneel the contribution from the positive background, so that we obtain in total:

=

We briefly indicate how to caIculate the exchange integrals. We take the z-axis in the integration over k' t o be parallel to k. Thus

Ik - k'12 = (k- k') . (k- k') = k2 + kJ2 - 2kk'

cos 0

where e is the angle between k' and k, With this inserted in the exchange integrals, we obtain

'

$,

THE HOMOGENEOUS ELECTRON GAS IN..,

-

sin 6'dOd$ k-2 + kt' - 2 k k t cos B '

With the variable substitution x E cos 8 , this expression becomes

This intkgral can be evaluated by elementary methods. With the result for the exchange integral we finally obtain for the single-particle energy:

In orddr to calculate the exchange-contribution t o the total energy per particle, we must perform one additional one-dimensional integration over k. This integral is also an elementary integral. The final result is

We obtain a frequently used represent ation of the Hartrec-Fock energy per particle by expressing the Fermi momenta k+ and k- in terms of the density

and the magnetization, or spin-polarization

An unpolarized system then corresponds t o the special case 6 = 0 , whereas for = &1 we have a totally poIarized ('ferromaghetic') system. From 2N - N N- = N - N + , so that [ = + , it follows that

t' we then have - H i t t -iHt'~ Q ~ ) (Po I eiHt Il(x)se $(*')SF

Toget her with the corresponding expression for it > 1 we obtain in all for the Green's function:

iG(s t , x't ')

From this expression, we see that the Green's function depends only on (t - t ') if H is timeindependent. If we use the transformation of the ground state to the Schrodinger picture,

we obtain

156

THE SINGLEPA RTICLE GREEN'S FUNCTION

This form of the Green's function allows for a clear interpret at ion: f o t~> t', we add a particle at space-spin paint x' to the ground state ] Qo(t'))S by the action of $ t ( ~ ' ) ~The . ( N + 1)-partide state created this way then propagates from t' t o t under the influehce of the Hamiltmian H (cf. equation (14.3)) and then fotms the ouei-lap with the (N + l)-particle state d t ( ~ ' I) Bo(t))S. ~ Thus, the Green's function for t > t' is precisely the transition amplitude far the propagation from x' to a: in the time interval ( t - t i ) of a test particle added to the many-particle ground state. Similarly, for 1' > t an ( N - I)-particle state propagates. I n other words: for t' > t the Green" k n c t i o n describes a particle propagation and for t > t' a hole propagation, The interpret ation of a general Green's function G(Xt, At') is completely analogous . Next, we will prove that for system for which the Hamiltonian commutes with the total-momentum operator

the Green's function depends only on the difference (r - r') between the spaee coorditiates:

The translat ionally invariant systems discussed in Chapter 5 are exarnpIes of system of this kind. It is clearly plausible that the Green's function, with the interpret ation given above as a transition amplitude, should o d y depmd on the coordinate difference for stlch systems. The explicit proof is nevertheless somewhat long. For the proof, we will use the following cmlmutator identities, which we will also use later:

[A,Bq = ABC-BCA ABC BAC - BAC - BCA [ A , B q = {A,B)C-B(A,C} [A,BC) = [ A , B ] C - - B [ C , A ]

+

(15.8)

(15.7)

and, similarly

{ A ,B C )

= { A y 3 } C- B[AyC].

(15.8)

With the use of equation (15.6) and with A = $ a ( r ) , B = 1/;$(m), and C = (-iV,) qy( a ) , we readily obtain

GREEN'S FUIVCTIUNS

Thus

This equation represents an operator differential equation for be formally integrated w,ith the result

6,(r). 1t can

This can be proved by inserting

We now return t o the the expression equation (15.5) for the Green's function. To show that it depends only on the coordinate difference, we write the field operators in the Schrodinger picture: iGwp(rt,r't')

Here we insert the completeness elation in Fock space

where yields

I *LN')

denote the eigenfunctions of a system of N particles. This

158

.

THE SINGLE-PARTICLE GREEN'S FUNCTTON

The only non-vanishing contribution comes from the states with (N AZ 1) particles, since the Fock space scalar product between wavefunctions with different particle numbers vanishes. Under the assumption that 2 and P commute, we can choose 1 9, ( N fI)) as eigenfunctions of P:

If we then insert equatioh (15.10) and also choose the coordinate system so that P I Vo) = 0, it finally follows that

This equation proves the statement that the single-particle Green's function only depends on the coordinate difference if [ H , $ = 0 . We can then form the Fourier transform with respect to this coordinate difference:

The inverse transformation is

GREEN'S FUNCTIONS

159

We will now show that for translationally invariant systems, t h e momentum Green's function defined earliet in equation (15.4)is diagonal and that the diagonal elements corre'spond to the simple Fourier transform of the space Greeh's function:

To prove this, we insert the explicit Heisenberg transformation into the definition of the momentum Green's function

i ~ , ~ ( kk't') t, =

I

1

(Qo

(90 90)

I~

1

[ q( ~a ) X C :(~')wJ ,~

and insert the ~ompleteness elation equation (15-11) of the exact system of eigenfunctians of H . This yields

iGaD(kt,kit')

Since ct

k'P

I Qo)

is an e i g e d ~ n e t ~ i oofn P with eigenvalue kt,and furthermore

(*kNfr+')l

can be chosen as oigenfunctions of P with eigenvaIue P,(N'l), the overlap vanishes:

Correspondingly, we have ,

(Bo

I ckD, I ~ 5 ' ~=)(cLaba ) 1 8kN+'))= 0

for k .f P,(NS-1)

F~~rlherrnore, cka ( lo)is ah eigenfunction of P with eigenvalue -k. Hence, the overlap

{@LN-''

vanishes for -k

I cka 1 *(I)

# P, ( N - I), and the corresponding overlap

TAE SINGLE-PARTICLE GREEN'S FUNCTTON

160

vanishes for -k' # P,( N - l ) - Altogether, all the terms for which vanish, which yields

k # k'

Gap(kt,k't') = hkk1~ ~ ~ k(t')k. t , It is then only left to show that these diagonal elements correspond precisely to the above Fourier transform of the spatial Green" function. This foIlows from the construction of the momentum Green's function by a b a i s transformation:

1 G,@ (rt ,r't ') - - C C e i ( k ' r d k ' . r ' ) ~&P (kt,klt')

"

k

k1

Thus, in the continuum limit

we obtain

G , ~(rt, rFtrj=

eik.(r-r')~,o(kt,M') .

In comparison with equation (15.12) and the uniqueness of the Fourier transform, the relatian

iG,p(k, t ,t ' ) = i(3,p(lrt, kt') =

1

P o I *0) ( g o I ~ [ c k(at )HC&

(~'IHII *o)

fallows. The procedures which have led us here to the diagonal momentum Green's function in a translationally invariant s y s t m is typical of the use of symmetries: bne makes an effort t o choo~ca representation in which the Green's function is diagonal. Hence, the Green's function is sometimes in the literature defined in its symmetric form

iG(h, t , t') =

1

P oI go)

POI T [ C A ( ~ ) H C :

%)=

If the Rarniltonian commutes with the total momentum, and is i r ~addition explicitly time-independent, we have for the spatial Green's function Gap(rt,r't') = Gap ((r - r')(t - t'), 0 0)

.

In this case, we can form the four-dimensional Fourier transform

GREEN5S FUNCTIONS

161

A formalism which is manifestly covariant can be constructed with this Green's function. This formalism has obvious special importance for relativist ic r n a n ~ - ~taicle r system (see Chapter 241, The inverse transformation

The single-particle Green's function contains a great deal of information about the system uhder consideration. We wiIl prove that with the help of the Green's function we can calculate

(1) the ground st ate expectation value of any single-particle operator, (2) t h e ground state energy of the system,

(3) the excitation energies of the system. Thus, given a single-particle Green's function of a system, we can obtain the most important physical properties of the system. We will first show property (1). For the sake of simplicity, we will show this property for a single-particle operator which is local in the space-coordinates, but nundiagonaI with respect to the spin-coordinates (thus, the most cwnmon cases are included). Such an operator is in second quantization

arrd the gmund s t a t e expectation value is

If AD,(r) is a differential operator, it will also act on the space-dependence of the field operators. In order for such a case not to complicate the expectation value, we use t1l-e trick of writing the expectation value as

where it is understood that AaCi(r)acts before we perform the limit. We then transform the field operataw to the Heisemberg picture

T H E SLNGLE-PARTICLE GREENS FUNCTION

Thus, the statement follows:

-

-i

J d3r lim Tr ( ~ ( r~f( r rtt,t + ] ). rf+r

In the short-hand notation of .the last part, 'Tr' means the trace over the spin variables, and t+ implies the limit t 1 4 t + . We will now consider a few examples.

(4 The kinetic energy is in first quantization

According ta the formula above, we then have

(ii) The density operator (see Chapter 6) is in first quantization N

p(R) = C6(IL-ri). t=l

From the formula abovc, it follows that the, ground state density of

the system is

Similarly, we ab tain

GREEN'S FUNCTIONS ( z i i ) The kinetic energy density +(EL) =

r ( R ) = (.i(R)) = -2'

zzlb(R - ri) (-2) :

lim

rl+r

and

EL,

(iv) the spin density Z(R) = b ( R - r i ) u i , where i? denotes the vector whose components are the three Pauli spin matrices:

We conclude this discussion by investigating the relation between the singleparticle Green's function and the density matrix discussed in Chapter 6. The general space-spin density matrix is in second quantization (see equation (6.5)): P(% .'I = $t(2')3(4 (x = (r, 8 ) ) Thus, the ground state density matrix is obtained by

=

rim ti-+t

I W o I d t ( x * t ' ) ~ 4 ( x t ) Qo) ~1 P o l *o}

whme we have ornittsd the individual steps in the proof. For Ramiltonians which in addition are explicitly time independent, so that

G(x1,x't') = G ( ~ oxf(t' ,

-1))

the ground state density matrix expresses precisely the Greeh's function at the origin of time: p(z,2') = -iG(xO, ztO+). (15.15) Heme, the Green's function can be interpreted as a time-dependent extension of the density matrix. Tndeed, this time-dependence contains a great d e d of information, since one cannot calculate the exact ground state energy from the single-particle density matrix - for this, the two-part icle density matrix is required. According to the statement (2) above, which we wilI now prove, t h e ground state energy can nevertheless be calculated from the single-particle Green's function. We will carry out the proof for the case of'

T H E SINGLE-PARTICLE GREEN'S FUNCTION

164

a particle-par t icle interaction which is local in space, but non-diagonal with respect t o the apin indices, so that it does have a genuine spin-dependence.

The Rarniltonian is then

For the kine tic energy contributian, we have

The proof runs completely anaIogously t o the proof of equation (15.91, For the commutator of the field operators with the interaction potential

we

obtain with equation (15.6)

-~;w {&bl,@;bKa$Y N O f (2,

~ Y l ? h ~ Z ~ } ]

PD'

-

-

1 -

2

J d 3 y ~ ~ ( y ) v , . . ~ ( r . ~ ) ~ p t ~ ~ ~ a+t (Zr )

ofPP'

PP'

where, with equatioh (15.8), we furthermore have

T H E SINGLEPARTICLE GREEN'S FUNCTION

x (Qo I - $ L ( ~ ' ~ ~ ) H $ L ( Y ~ )(r, HV ~ )~d~ '' @ ' ( f l ) ~ d ' a ' @ I @0). t)~ PP'

We now take the limits r' -+ r and t' 4 t in this equation, then integrate over d3r and sum over a. Rorn the definiti~nof the single-particle Green's function it then fallows that

(15.19) With the help of this equation can the ground state energy

we

eliminate the potential energy from

to obtain the final result Eo =

-i2

/

d'r lirn lim t

s)

[ig at 2m

Tr [ G * (ri ~ ,rtf')].

(15 2 0 )

Fur tr anslationally invariant, explicitly t time-indep endent systems we obtain a particularly simple equation if we use the four-dimensional Fourier transform of the Green's function :

Before we can prove statement (3) above, we will first discuss the Green's function of a system of free, non-interacting fermions, i.e., Ho = T and V G 0. In this case, the Heisenberg picture and the interacting picture are identical, so that we can calculate the Green's function fronr

We transform to creation and annihilation operators of the momentum cigcnfunctions and use the representation equation (14.10) in the interaction picture:

and

GlXi3EN'S FUNCTIONS where

kZ This yields

Bere I ao) is the ground state wavefunction of Ho = T , i.e., a Slaterdeterminant of all plane waves of momentum less than IcF. Consequently, the matrix elements vanish unless the created &ate (kt/3)is identical to the annihilated st ate (ka). This yields

If we now also use

we finally obtain in

the continuurn limit

By definition, it follows that the momentum Green's function

is

For the derivation of the four-dimensional Fourier transform, we will first show that the step function has the fdlawing representation:

THE SINGLE-PARTICLE GREEN'S FUNCTlON

Figure 15.1 htegration paths for .r < 0 and r

The integtand has bution of

a simple pole at w

w

+ ir)

= -iq and the residue leaves a cantrie-%WT

e-iw~

Res-

> 0.

= limn (w w4-17j

+ i q ) (u+ iq) -- e - q 7

If we want to calculate the integral with the help of the theorem of residues, we have t o close the integration path with a serni-circle. To ensure that the integratiiorl over the arc in the limit of an infinitely large radius leaves a vanishing cont~ibution,we must, because of

close the arc iu the lower half-plane for r > O and in the upper half-plane for T < 0. Since there are no singulasities enclosed by the semi-circle, the integral over the closed path vanishes according l o Cauchy 's t heorern- Thus, for T < O the, integral leaves a null result, which we had to show. For T > 0 we obtain, by using the theorem of residues

dw-

=

= -2iri x (sum of residues)

-

- 2 i ~ i ~ - ~ ~ ,

The minus sigh comes from the negative orientation af the integration path, For r > 0 the statement then follows:

GREEN'S FUNCTIONS One can analogously show that

If we insert these into the above expression equation (15.22) for the Green's function, we obtain

With the substitution w

tk

+ w it finally follows that

so that

This function ha5 fur k 2 k F , i . e . , for

and for k

tk

=

k2

k >'Z;fi =t

~a simple ,

pole at

< k ~ i. e,,, for tc, < CF,a simple pole at

It is essenlial to note that we only obtain discrete poles so long as wts are only considering a finite system. h t h u continuum limit, the analytical structure of the Green's function changes, and we obtain a branch cut along the real axis, We will now test thc this formalism by using the Green's function that we have jwt determined together with eqliation (15.21) to caIculate the well-knowh ground state energy of the free non-interacting electron gas:

T H E SINGLE-PARTICLE GREEN'S FUNCTION

Figure 1 5 2 The Iocafr'ons of the poles of the Green%function are just above the real axis for fk < CF,and just below the real axis for r k > E F . Since T > 0 we choose an integration path I? which closes with a counterclockwise semi-circle in the upper half-plane:

The first term ( k > kF) of the integrand does nut leave any contribution, since this tmrn has no singularities in the u p p e ~half-plane. This leaves only the contribution from the second term:

The calculation of the integral over the closed path yields, with the help of the residue theorem

=

lim ( 2 7 4 7-O+

,,-ot

lirn

w-ck+iq

[eiw7(ck

+ u)]

GREEN'S FUNCTIONS Thus, we obtain the familiar expression

for the total energy. We will now for later purposes write the Green's function of a noninteracting Ferrni gas in yet another way. 7b do so, we first consider a system with ( N 1) particles. The ground state energy of this system is obtained by adding a particle a t the Fermi level to the N-particle system, so that

+

We obtain an excited s t a t e of the (N+1)-particle system by adding a particle with momentum K such that K > k~ t o the N-particle system. The energy of this state is then EkN+I) -

C 'k+[~.

Hence the different excitation energies of the ( N + I$-particlesystem are

We obtain the ground state energy af a system with ( N - 1) particles by removing one patticle from the Fermi level

On the other hand, an excited state of the ( N -1)-partielrz system is obtained by removing any particle with momeritum K and encrgy CK < EF fro111 the grotlnd state of the N-particle system. The entire system has then a momentum -K (since the previously fully occupied Ferrni sphere had zero mamen t u rn) and energy

Thus, thc excitation ehergies of the (JV - 1)-particle system are givcn by

We can now write the Green's function of t h e non-interacting N-particle system as a function of the excitation energies of the ( N - 1)- and t h e

THE SINGLE-PARTICLE GREEN'S FUNCTION

172

(N+ l)-particle systems:

ctj (k,w ) =

lim Sap rl+~+

=

lirn 6 4 ql-ko+

w - EF

-(fk-cF)+iv) + O l t - I"F)

w - c F - w k (N+') + il

4w

e l b - k$

I

W - E ~ + ( E ~ - E ~ ) - ~ ~

W F- k)

- r p + wL:-)l

- iq

Thus, the Green" functions has its pales at the exact excitation energies relative to the Fermi level of t h e

( N + l)-particle system:

w

( N - I)-particle system:

w = 6 ~ --

= 6~

+ wk( N + 1 ) - irT

arid of the

WLT-~) +it,,

We will now shew that this statement also holds for the exact Green" function of an interacting system, which proves statement ( 3 ) . Far the sake of simplicity, we will restrict ourselves to explicitly timeindependent, translation ally invariant systems, It then suffices to consider the (diagonal) momentum Green's functions. We will furthermore also restrict ourselves t o spin-independent interactions, so that we have

with

-

I (Qo! Qo)

B(t

- t')

C e-i (E$,N+"-E:"')

(t- t ) )

We hive here p erfurmed the transformation of the Heisenberg operatots to the Schrijdinger picture and inserted the completeness elation equation

GREEN'S FUNCTIONS

173

(15.11). With the representation equations (15.24) and (15.25) of the step function, we obtain

-

iG(k,t t ' )

W - w - E n(*-I)

a

B EL^+') - E:

in the first integral, and in the second integral, this expression becomes

With the substitutiofis w

+ EF

w

We now look a little cldser at the denominators t h a t appear in this expression. In the first term we have

The difference

(N+1) -

wn

(N+l)-E;~+l)

= En

is precisely an excitation energy of the ( N

>0

+ l)-particle system. Hence

is the change of the ground state energy in addifig one particle, which i~ precisely the chemical potential. The second denominator is

The difference (-1)

Wn

&FLN-I) - EhN-Q

30

TrrE SINGLE-PARTICLE GILEEN'S FUIVCTDN

174

is this time an excitation energy of the ( N - 1)-particle system, and

#-

W) 11 = p ( N - l ) Eo is furthermore the chemical potentid. For very large systems, we can set = p, P (N)= -pv-ll -

(However, we have t o be careful in the case of systems for which the ground state energy Ea is discontinuous as a function of the quasi-continuous variable N , so that there are gaps in the ground state energy, such as the band gap in isolators. In such cases, p ( N ) # p(N-') .) Finally, we take into consideratian that the matrix element (9, ( N f l ) I c i l qo)vanishes unless V,( N - 1 ) is a state with tdal momentum P, = k, and similarly the matrix element (!DLN-') I ck 1 !PO) vanishes unless 8 (,N - l ) is a state with total momentum P, = -k. Hence, we can restrict the sum over the intermediate states and the final result is the secalled Lehmann t-epresentation of the Green's function:

Thus, the Green's function of an interacting many-particle system has its poles at t h e exact excitation energies, relative to the chemical potential, af an

and of an

( N + 1)-particle systems: w = p

+ ivm ( N - I)-particle systems: w = p - w w-1) ~ , - ~

We can dso write G(k,w) in the form

and

+ w (,N, +~ O - ilJ

GREEN'S FUNCTIONS

175

The spectral functions defined in this way are by construction real and posit ive:

Because w,'N'l'

> 0, it furthermore follows that

and we obtain the sum rule

1"

dr (A(k, 6 ) 4B(k,t)) = 1.

(15.32)

To prove this last equation we simply integrate:

Similarly, we obtain

and the sum ruIe is proved. For the case of non-interacting Ferrnions it follows from equation (15.27) that

In this case the s u m rule is trivially satisfied. We can now express the time-dependent Green" function using the spectral functions:

-

0

rbbm this it readily follows that

-ti

Q;I

dr ~ ( kE)~-'(~-')("'I ,

. (15.33)

THE SLNGLE-PARTICLE GREEN'S FLTNCTION

176

Hence the Green" function has a discontinuity at the point r = 0:

Equation (15.34) is for practical reasons a very important represent at ion of the momenttun distribution (nk). With the help of this representation, we will prove in Chapter 26 that the function ( n k ) has a discontinuity at E = k ~ i,,e ra Fermi edge, also for interacting systems. We will now conclude this chapter by further investigating the analytical structure of the Green's function. From the general formula

it follows that

Since A and 3 are real (equation (15.30')), we have

(15.36) This last iderrtity, which follows from equation (15.311, shows t h a t Trn Gjk, w ) changes sign at w = p, since both A and B are positive according to equation (15.30). We can also re-write equation (15.36) as

(for r

> 0). If we insert

equation (15.37) in equatio~r(15.351,we ohtain

GREEN'S FUNCTIONS

177

With the subsitutions F, = p + ~ ~ lind the first integral, and E , , in the second integral, we finally obtain the dispersion relation

Ke G(k, U ) = -

1

00

di

ImGCL, E ) W-E

1

= p-t,l$ . (15.38)

Here we see a possible application af the spectral function. Let us assume that we have calculated the single-particle Green's function within same particular approximation (for example the diagrammatic method which we will discuss shortly). In this approximation, the Green's function will in general not have the correct analytic properties; we could for example see that the dispersion relation derived above is not satisfied. In this case, we could for example only use the imaginary part of the approximate Green's function and from this then calculate the spectral function by using equation (15.37) and then finally determine anew the real part of the Green's function from equation (15.35). The Green's function corrected in this way will by construction satisfy the dispersion relation. One can also attempt to introduce further corrections so that, for example, the sum rule equation (15.32) is satisfied. We will close this chapter with a remark on the theory of superconducting systems. The single-partide Green's function is the expectation value ($ttyl) and ($if), respectively. In superconductors, on the other hand expectations values of the type ( t t ) and ' 1 1 the s*called anomalous propagators, are of fundament a1 importance, The order parameter A is for example an expectation value of this sort, The prope~tiesof the single-par ticle Green's function discussed here and in what follows (a~alytic structure, equation of motion and diagrammatic expansion) can easily be: carried over to the anomalous propagators,

,

Chapter 16

The polarization propagator, the two-particle Green's function and the hierarchy of equations of motion As we have seen, the single-particle Green's function contains a large mount of int ercsting physical information of a system. However, there are quantities which cannot be calculated from this Greeh's function. One e ~ a m p l eis the correlation between the observables R and B , which is usually defined by

This quantity gives a measure of the mutual dependence of the fluctuations of A and B on one another. If the deviation of the quantity A from its mean vdue ( A ) is somehow coupled t o the deviation of the quantity B from its meah value { B ) ,then f A will ~ have a non-zero value. If, on the other hand, ~ A = B 0, the observabIes A and 3 are said to be statistically independent. If A and B are local operators in the space-spin representation, so that

then their ground state expectation values are given by

T H E POLARTZATION PROPAGATOR

180 Hence, the correlatirm function is

If we also introduce

the correlation function f A can ~ be written

If we furthermore are only interested in correlations of quantities at different times, it is natural t o dcfine a time-ordered density correlation function in the following way:

For re&ons that will become clear later, I1 is also known as the polarixa-lion prbpagrria~. By using this quantity, we can calculate the ground state expectation value of a local (in space-spin representation), but otherwise arbitrary, two-particle potcntial, (v) =

1 -

2 (Qo

I *u)

J dx J d

x ~X I , )

1dt( x ) i t ( x i )$ ( Z ~ ) + ( X ~ I'PO).

To show how this is done, we use the fermion anti-commutation relations and obtain

GREEN'S FUNCTIONS

If

we

then use the definition of the polarization propagator, we obtain in

total

If we want t o continue t o calculate the correlation between two nan-local observables

we arrive at

Hence, the function IT defined earlier is not sufficient t o calculate such a correlation function - a generdization is necessary. This leads us for timedependent extensions of II to the definition of the two-particle Green's function, which in spacespin representation is

THE POLARIZATION PROPAGATOR

182

Here, the time-ordered product of several Ferrnion creation or annihilation operators is defined as

where the permutations P E S, are chosen such that

t ~ ( l> ) t ~ ( 2 >) - . > t ~ ( , ) .

From this definition of the time-ardered product we obtain the identity

Equation (16,6) is proved by, for example,

For the single-particle Green's function, we needed t o examine only the two cases t > t' and t' > t . For the two-particle Green's function there am 24 distinct cases to examine, corresponding to the possible permutations of the times t l , t 2 , t 3 ,t 4 . Due t o the symmetries given by equation (16.61, this number is fortunately reduced t o six: (1) tl > t l > 13 > t q : i2~(1234)= ( t , b 1 ~ 2 ~t 3 $t4 ) . Because of equation (16.61, this time arderhg includes the cases

> t l > 't3 > t 4 t ] > t 2 > t 4 3 tg tz >tl > t 4 > t3 t2

[as ta the sign, see equation (16.6)J.The basic structure is

G = ($Nt$f) with all possible indices. (2)

13

> t d > 1, > t , : i2G(1234) = ( Y t, L Jf ~ ~ ~ $ ~ $Here ~ ) . the t$ > t 3 > t l > t 2 t3

>t 4

t4

> t3 > t 2

3 t2

> t]. > tl

are included. The basic structure is G = ($t$tyh)).

cases

GREEN'S FUNCTIONS

183

t and all other indexings (3) t i > t g > t g > t d : ~ ( 1 2 3 4 = ) -(d~l$J&$~), with the structure G = ($J$~$J$~). We will only give the three basic structures far the remaining 12 possibilities. Each basic structure includes four cases, just as above.

The polarization propagator, equation (16.2) (without the prefactor ahd the additional term (p(x)) ( p ( x ' ) ) ) is obviously a special ease of case (4). The intetpret ation of the two-particle Green's function is analogous to that of the single-particle Green's function: In case (1) the two-particle Green's function describes the propagation of two additional particles and gives the transition amplitude for finding the particles at (xltl) and (xzt2) if they were added t o the system at ( x 3 t 3 ) and (x4t4). Case (2) similarly describes the propagation of a holcpair, whereas cases (3) - ( 6 ) treat the propagation of particlehole pairs. We will here refrain from a thorau$ discussion of the analytical properties of the two-particle Green's function. Instead, we conclude this chapter by deriving the equation of motion for the single-particle Green's function, which wiIl turn out to contain the t w e p a r t i c l e Green's function. First, we calculate the time derivative of the time-ordered product of two field operators in the Heisenberg representation:

We then use equation (15.1 8) for a potential which is local in space and spin:

T H E POLARIZATION PROPAGATOR This yields

T h e ambiguity of the time-ordered product for operators with the same time argument can in this case be easily overcome. The product q5t(yt)$(yt)@(rt) originates from a single operator, namely d~(xt')/8t,For this reason, the product must remain together in this sequence and regarded as one unit, Hence, the only possible ways of time-ordering are

which furthermore can be written as

Finally, if we use the anti-commutation relation y t ) $ ( x t ) = -$(zt)$(Yt> (which holds only for Heisenberg operators at equal times), it follows that this expression becomes

W e then obtain the equation of motion for the single-particle Green's func-

tion:

(16.7)

In particular, for a non-interacting system we have

In Chis case, the Green's function is a true Green's Functirrn in the mathematical sense, whercas the name is strictly not correct for interacting systems. The derivation of equations (16.7) and (16.8) assnmed a system witlioui an external potenlid - only in this case is the use of equation (15.18) valid. If there is an external ~otentialu(2), the operator o u the left hand side of equations (16.1) and (16.8) must be replaced according to

GREENS FUNCTIONS

185

The equation of motion, equation (16.71, contains the t w ~ p a r t i c l eGreen's function for interacting system. Hence, if we wish to calcuIate the singleparticle Green's function from this equation, we must first have an expression for the twu-particle Greeh's function. This raises the question of an equation of motion for the two-particle Green's function, which will in turn contain a three-particle Green's function, and so on. We can define an n-particle Green's function by

For each n, these functions contain information about higher o r d e ~correlations. Analogous to the previous derivation, we then obtain a n equation of motion far the n-particle Green's function:

which always contains t h e next higher Green's function. In this way, we obtain a sysl-ern of coupled equations for the different Green's function. This system is just as impossible to solve exactly as it is t o calculate the exact many-part icle wavefunction. However, the advantage of the Green's function lies precisely in this partitioning of the full problem into a hierarchy of simple equations. The exact wavefunction of an interacting many-particle system is an enormously complicated structure which contains information on arbitrarily high correlations. Even if this function were given t o us, it would be practically impossible to calculate physically relevant information, such a the ground state energy, for macroscopic systems (i.e., with particles). Far that reason, it is more favorable to work with quantities such as the Greert's function, which indeed contain much less information, but are more intimately related to measurable quantities. If the hierarchy is broken at any point 'by assuming an approximate Greens?unction, all Green's functions of lower order c a n in principle be calculated from the equation of mation equation (16.10).

Part I11

Perturbat ion Theory

Chapter 17 perturbation theory We begin by separating the complete Hamiltonian

into an 'unperturbed' part Ho and a 'perturbation' v , which in most of the problems discussed here will represent the part icle-particle interactions. We assume that the eigenvalue problem associated with Ho has been soIved. We separate out a coupling constant from the perturbing potential v =gT and interpret the eigenvalues and wavefunetions of the complete problem as functions of this coupling constant. The basic idea of perturbation theory is simply to expand each quantity in a power series in g . One expands about g = 0, i e . , in some sense about the unperturbed problem. Jig is small, so that the potentid really is a weak perturbation, one can hope that only a few t e r m in the series will be adequate. However, it is often necessary t o sum either the entire series, or at least the dominant terms of the series, to infinity. In this chapter, we will briefly review the traditional methods of stationary perturbation theory. Our goal is t o obtaiil an expression for the ground state of the complete problem

First of dl

190

TIME-INDEPENDENT PERTURBATION THEORY

From this we obtain the general equation

We denote the projection

ope~atoronto the ground state of the unper-

turbed problem as p~

I @o)(@o 1

and the projection operator on .the remainder of the Hilbert space

This operator commutes with

(17,2) as

&:

Thus, for any number E , Q commutes with the operator ( E - Ho), and we can write

Hence

If we define

equation (17.4) can be written as

If we iterate this equation, we obtain

PERTURBATION THEORY

191

We have found

a general equation from which we can in principle calculate the solution of the complete problem from the ground state of the unperturb ed problem. If we insert equation (17.7) into equation (17. I), we obtain for the energy shift

We obtain the Rayleigh-Schrodinger version of stationary perturbation theory by setting E = Wo. This yields

Next, we separate out from equation (17.10) all terms of different orders in the coupling constant g. We do so explicitly for the first term of the energy shift. The first terms of the series above are

TIME-INDEPENDENT PERTURBATION THEORY

192

If we arrange the contributions according to their orders in g , the result is

where

Initially, the t e r m in the Rayleigh-Schrodinger series run in parallel to their orders in g. However, this pattern changes at the n = 2 'term, at which point arbitrarily high orders in g are included. We observe that the terms in the Rayleigb-Schr~dingerseries, with n 3, are of at least f o u ~ t horder in g . Hence, the third-order correction A E ( ~is) obtained from the n = 2 term above, with AE replaced by A E ( ~=) (Go I v ] Qo);

>

As the order in g intteases, it becomes correspondingly difficult to sort out the terms for each order. The Brillouin- Wigner version of stationary perturbation theory possibly provides a more systematic approach. In it, we set E = EQ in the general formulas, equations (17.7) and (17.8). This yields

As in the Rayleigh-SchrSdinger series, the first term of the Brillouin-Wigner series

for the energy shift corresponds to the first-order term in g:

However, the second term of the Brillouin-Wigner series

PERTURBATIONTHEORY already contains terms of arbitrariIy high orders in g:

When we separate the powers of g , Rayleigh-Schrijdinger expansion:

we obtain the same result as

in the

The expressions for each order of the energy shift, derived from the BriloflinWigrler series, are identical t o those from t h e Rayleigh-Scbrbdinger series because the expansion in powers of g is unique. However, sepat-ating t h e powers of g from the BrilIouin-Wigner series is at least as difficult as in the Rayleigh-Schr Sdhger series . In conclusidn, whereas b d h the Rayleigh-Schriidinger and the BrillouinWigner methods of stationary perturbation theory formally give exact expansions of the complete problem, separating the orders in the perturbing potential (i. e.., the powers of g) bccornes rather tedious. This limits thcir usefulness to the firat few of orders in g. If we want to sum the series in g to infinity, it wilt be necessary t o have a perturbation expansion that will directly give us the individual o r d e ~ in s g.

Chapter 18 Time-dependent perturbat ion theory with adiabatic t urning-on of the interact ion The series derived in Chapter 14 for the time evolution operator in the interaction picture ( c f . equations (14.19) and (14.20))is a direct expansion in orders of the potential:

In this chapter, we are going to construct a fictitious time-dependent problem such the wavefunction of the completec problem c a n be obtained by applying the time e v ~ I u t i o noperator to the unperturbed wavefunction

The series for U given in equation (18.1) then gives a direct method for expansion in orders of the potentiali We will now discuss in detail how and under w h a t general cdrlditions such an expansion c a n be constructed. We define an explicitly time-dep endent Harrril tonian in the Schrijdinger picture by H , ( l ) s G Ho+e-'ltl v with E > 0. This Hahltanian has the foliowing properties:

Thus, we will slowly turh on the poteritial at t = =too to the unper'turbed problem Ha, and at t = 0 the potential is completely turned on. The time-

ADIABATIC TURNING-ON dependent SchrGdinger equation

at t = A m goes over to the asymptotic equation

The required initial conditions for equation (18.2) at t = -m is that the interacting ground state evolves from the non-interacting one:

The equation of motion is in the interaction picture

"1

a

@ c ( t ) ) ~= e - ' l ' l ~ ( t ) 1~8 ,( t ) )I .

(18.4)

Since the right-hand side vanishes for t + rtoo

the state vector in the interaction picture becomcs time-independent in this limit: [ Q,(t 1 = constant.

For t + -m we obtain from equation (18,3) the initial condition lirn

t-dm

(

I

I

- lirn eiHot * J t ) ) S = iHote-iWotGo)

I - ,+km

= I @o>.

(18.5) The forrrial mlutiori of t h e equation of motion, equation (18.41,with the initial condition equation (18.5), is

The potential has reached its full strength at the time t = 0. The question then arises in which sense the state

is related to the exact ground state j ao).(We will from now on drop the index i at t = 0, since at this time all pictures are identical). Claarly, I Q,(0)) dcpeulds on the magnitude of c, f . e . , how fast the potential is turned on- If the potential is turned on sufficiently slowly ('adiabatically'), we can hope

PERTURBATION THEORY

197

that at each paint in time, the ground state has adjusted to the potential strength at that time, and that we obtain the exact ground state from

1 q0)= E-+O lim ( Q,(O)). This question is not answered trivially, since we explicitly used E. > O when we established the initial condition equation (18.5). The question if and under what conditions such a limit gives sensible results is answered by the Gell-M ann-Low theorem [13]:

(1) If the quantity I () = lim,,o

a

exists to a11 orders

in per-

turbation theory ji.e., if in the perturbation expansion

exists for each n), then I () is an exact eipenthe limit lirn,,a I function H . (The theorem does not guarantee that this state is the ground st ate!)

(2) The limit lim,,o

as

F +0

I i P c Q O ) ) does not

exists; in fact

under the conditions in I.

The infinite phase that appears in (2) will obviously cancel with the denominator in (1). If we pause for a while to consider the Gell-Mann-Low 'theorem, we realize that it is a fairly weak theorem, We would rather have a confirmation that the limit c -t 0 indeed exists, or at least know the conditions under which it does exist. how eve^, the theorem does guarantee the desired final result, e q u a t i o ~(18.71, ~ if we only can calculate the limit t + 0 using perturbation theory. This guarantee is of course also very valuable. We now proceed with the proof of the statements af the theorem. We have

(no- woj 1

QE(0))

= ( H o - w ~ ) 1 1 , ( 0 , - ~ )(Po): l

= [ H ~ , u c ( o , - ~1 )@o) J

PERT URBATIOIV THEORY

It follows that

Each factor ~ ( tcontains ) one factor g; thus

and

We then finally arrive at

*

As-

ADIABATIC TURNING-ON

202

1.t should be emphasized that neither the numerato~nor the denominator exists in the limit E 4 0;only for the ratio of them wilt the infinite phases cancel out simultaneously~ We use ( cf equation (14.16))

together with the property

to obtain

It remains to be shown that 00

OD

dt, e-t(lt'i+.b.+l'vl)~[~(tl)l ., . ~ ( t , ) j O ( t ) l ]

To show this, we split up the Y-hmensional integration in distinct pieces, the boundaries of which form the surfaces t ; = t , In each piece, there are a certAin number of variables with t ; > t aad a number with t i < t . The situation it1 two dimensions, for example, is depicted in figure 18.1. For each piece, we introduce the not atioris for eachti withti > t : and for each ti with-ti < i :

T ~ . ~ . T ~ UI

...u~-~.

The number n depends on the particular piece. Within a single piece, it holds that

and the contribution of this piece

to

the: integral above is

The entire>integral is obtained by summing aver the 2u pieces. It is not necessary to considet each piece individually since most of the integrals give

Figure 18.1 Integration sreas for u = 2,

the same value: only t h e number n plays a role. For fixed n , which of the original 1; are transformed to T and which are transformed t o u is completely irrelevant, since they are only dummy integration variables. Given v and n , there are

possibilites t o select n times t; such that 1; > t . All corresponding pieces give the same ccsntribution. Hence, it is necessary only to sum over ail possible values of rl from n = O to n = Y:

The desired result follows:

ADIABATIC TURNING-ON

204

X.-'(I~~I+..-+~~~~)T = Uc(-, t)Otf )rUclt,

[ v ( o l ) r . . .~(r,)~]

-4,

which concludes the proof of equation (18.9) Another interesting conclusion we can make is that

(*o

--

1 T [ O ( t ) ~ O ( t ' /) ~ ] lirn ,-0

1

{Go

I st I @o)

-03

v=o

xT [ V ( ~ I ) I. - . v ( t u ) ~ O ( t ) ~ ~ 1( @dm t')~] Just as in the preceding proof, for ($0

t > d'

(18.10)

we can write:

I T [ o ( ~ ) H o ( ~ ' ) H I]

The identity for the numerator can then be proved by spitting up the calculation of the v-dimensional t-integrals in single pieces with boundary surfaces -ti = t a n d t i = f' and then calculate t h e i ~contributions by a similar summat ion. With the identity equation (18.10) we finally obtain the following import ant represent ation of the single-par ticle Green's function:

iGap(rt , r't')

This is -the form of the Green's function that we will use in the diztgrammatic, analysis. Corresponding equations can be obtained for the momentumGreen's function or any other choice af representation.

Chapter 19

Particle and hole operators and Wick's theorem Most of the work in the calculation of the energy shift and the Green's function according to equations (18.8) and (18,11) is the evaluatian of the matrix elements relative t o ( ao)of time-ordered products of the potential in the interaction representation. If this potential is represented in second quantization, a time-clrdered product of creation and annihilation operators in the interaction representation is obtained. Wick's theorem [14] provides a method to calculate the I @o)-expettationvaIue of such time-ordered products. In this chapter W B will prove Wick's theorem. and hole operators. We assume that the eigenWe first define value problem given by N

is solved by the single-particle orbitals \

the otbl* occupied. Furthermore, let ct

with single-partick energies up to rF are a ~ denote ~ the i creation and annihilation operators of the single-particle orbit$L,p,. We then dsfirne the following operators for pa.rticles and holes, respectively. For particles:

Tn the ground

state,

creation operator annihilation opera'tor

a; E ci

creatian apwator

bj

anriihilation operator

bj ~

and for holes:

+- I L Cj

+

c

3

-

fort. < E F .

WICK'S THEOREM

206

Clearly, the annihilation of a real particle with energy less than 6~ cortesponds t o the creation of a hole with the same single-partide: energy. We obtain in the interaction picture

Since

{ ct; , c k }

= hik we have furthermore { ati , ak] = bik

and

{ btj , bi] = b j l .

Moreover, any possible particle operator anticommutes with any possible hole operator, since they act on different states. The operators defined in this way satisfy a;

1 ad) = 0

and

bj I G o ) = O

(I9-4)

i.e., the ground state of the Ifo-problem contains neither particles not holes. Thus, we will call I Go) the vacuum state (relative to the particlehole represent ation). We can now express the operatar Ha in t e r m of the particle and hole operators:

where nH and mp arethe number operators for holes and particles, respectively. From this pepresentat ion of Ho we see that thc energy of an arbitrary eigenstate of Ho is obtained as the ground state energy plus the sum of the particle energies minus the sum of the hole energies. Ncxt, we exprcss the field operators in terms of particle and hole operatoss.

PERTURBAT7UN THEORY

207

We can apparently separate $(c) into a hole-creation part

$+(XI

and a particle-annihilation part @- (x). Similarly, we can separate $ J ~ ( x )into a. particle-creation part and a hole-annihilation part:

The rninus-index always denotes the annihilation part (for both particles and holes), and the plus-index denotes the creation part. From equation (19.4), we have the important equation

The separation of creation and annihilation parts is invariant under ttansformations to other pictures. For example, we have

$+ ( ~ t )=~

with

pi(x)b!(t)~,

The normal o r d e r N of a product of particle and hole operators Is defined by rearranging the product in such a way that all annihilation operators are to the right, multiplied with the sign of the permutations necessary t o achieve this. In other words, t h e norrnaI-ordered product is obtained by bringi~lgall annihilation operators to the right, and in doing so, treating a11 operators as if they anticommute. For examp15

_------

_._---

N c i ( t 1 )qv;)c:(13)1

for

Ei, € k

CF,

fj

< EF.

The time dependence of t h e operators in the i~teractionpicture does not matter for the normal order. It is very important to recognize that the definition of a normal-ordered product is unique. Indeed, one could kommute oper-ators within the product of creation operators or within the product of annihilation operators and, for example, write

208

WICK'S THEOREM

for the example above. However, all expressions obtained in this way are identical, since the creation operators anticommute (and similarly, the annihilation operators anticommute). We now generalize our definition to include linear combinat ions

P

where A and B are products of particle and hole operators and a, are

complex numbers. It follows that

or, in general

As an example, we calculate

In the last line, we have used the ~r_e~riotts~epresentation of the field operators in terms of particle aihhCie operators; the third term, for exanlple, is

Next, we define the so-caljed pairing of two arbitrary particle or hole operators A and B as

PERTURBATION THEORY

209

From the tule we established earlier for the normal order of linear combinations of operators we readily obtain the generalization of the pairing to linear combinat ions:

It s h o ~ l dbe emphasized, that the pairing is not defined for the cme when both A and 3 themselves are products of partide and hole operators. Let us consider some examples:

and, analogously

b;(f)bf

( 1 ~ ,= ) e'% ('-''I6

ZJ' -.

u For all other possible combinations of pairing vanishes, since the,pairing

and hole operators, the for two anticommu ting z

operators:

k L . ,

{A,B)==O

AB=O. LI

xi

The statement hold9 also for linear combinations A = ajAi, B = Cj,#jBj,if all components anticommute, ( i . e . , {&,B ~ =} 0):

210

WICK'S T E O R E M

Most pairings are also zero. However, if they do not vanish, the result is always a c-number, At this point w e can begin to see the purpose of the definition: with it, we can very simply calculate the vacuum expectation value of two linear cornbinat ims A , B of particle and hole operators:

Since the vacuum expectation value of a normal ordered product always vanishes, this expression is

Thus, in the end we only have t o cdculate the pairings t o determine the vacuum expectation value. We will now calculate another pairing as an exercise:

and, analogously

All dther 14 pairings which can be formed from the operators and (ybf)* vanish, since the operators either anticommute or are already normal-

PERTURBATION THEORY

u

u

From this we obtain the pairings for the field operator themselves:

and, analogously,

Finally, we also define the normal order products of operators which cmtain pairings:

WICK'S THEOREM

212

Here q is the number of commutations necessary to bring the paired operators to the left of the product, i e . , in this example the number of pairwise commutations needed to get from ABCL) . . . X Y Z t o ADCYBE . . . XZ. Wkk's theorem for normal products states that

+

,

. . all other terms with one pairing

+ . . all other terms with two pairings ,

+ all completely paired terms (they appear only for even n ) . For a proof, we first show the following lemma:

-.I

In the case that '2.-k.-. an annihilation operator, all p a i r i n g ATI3 vanish, L.

l...

u

and the lemma is trivialIy satisfied, If B is a creation operator, all pairin@ A,B where A, also is a creation operator vanish. Hence, it suffiecs to prove

u

the lernma for the case where all A; are annihilation operators. A11 cases with an additional creation operator A; can then readily be canstrueled by multiplying the equation without the creation operator A, on t h e left by A j . The normal order is then satisfied and additional pairings do not arise. Thtzs, for annihilation operators AI .. . A , it remains to be shown that

(The slashed operator 4, means that this operator is omitted.) The prefactor (-- I ) ' + ~ comcs from (T - 1) commutations of A, and (n - 1) commutalions of B , We prove the stahement by induction. For n = I, we have from the definition of the pairing

PERTURBATION THEORY

213

We assume that the statement is true for n . By multiplication from the left by an annihilation operator Ao, we have

where we have used AoB

r -BAa

+AoB. This is the statement for (n+ 1) u

and the proof of the lemma is complete. With the help of this lemma we will now prove Wiek's theorem in a similar way by induction. For n = 1,2 the statement is trivially satisfied by t h e definition of the pairing. Assume that the statement is true for n , By multiplying from the right by an operator A,+1 we obtain

Bythelemma,weobtainfor t \\

and, .for the second term

first tern

WICK'S THEOREM

+

The underlined expression is the first term of Wick's theorem -for (a 1). The two expression marked by underbraces yield all t e r m with one pairing, and so on. Repeated application of the lemma wi1I eventually lead t o all terms for Wick's theorem for ( n 4- 1). Due to the linearity of normal-ordered products (and of t h e pairings), Wick's theorem also holds for linear combinations of particle and hole nperatars, and cbnsequently also for the field operators themselves. We have now come to the last crucial step: the calculation of vatuum expect at ion values of time-ordered products. For two linear combinations

of particle and'hole operators A i l B j , we define the so-called contraction by

From the properties of nbrmal-ordered and time-ordered praduc t s it follows that

----_ -. -

-.

With the help of the definition of tirne-ordered products we can readily

n

calculate the contraction A(t B(t'):

-Bjt')A(t)

i f t > i!' if t' r t

PERTURBATION THEORY

Thus, contractions correspond t o two different pairings, depending on the time-order. In particular, contractions are always c-numbers, so that

From the corresponding property of pairings, it follows that

W

n n

N ( A B C D E . . . XYZ) = (-1)gAD BY N ( C E ... XZ). With this result, we can now state Witk's theorem for time-ordered products:

+. . . all other terms with one contraction +N(AI A2 .n . . A,) + ...

+ - . . alI other terms with two contractions

+ all completely contracted terms

>. '
.. . and we have indicated sums over all term with one pairmg, two pairings and so on,Since the times are ordered, the pairings are precisely equal t o the contractions:

and the proof is complete. We conclude this chapter by calculating a few additional contract ions. First, 'we consider the creation and annihilation operators of the Hrproblem:

for

fi,rk

> .EF :

aj(t)dL

for t

> t'

- ok( t )aj ( t ) = O

for t'

>t

b+3 ( t ) b k ( t )= O

for t

> 1'

u

-

t r

u

-

-for y , r k S E F : b tj ( t ) b k ( t r ) =

I

-bk(tt)b:(t)

fortl>t

u =

{

0

otherwise :

~ k e - i c j ~ t - t ' ~ cj

>f

-bike-",

_S C F , t'

,(t-tl)

tj

~

i> , t1

>t

otherwise

This expression is reminiscent of the n~omentumGreen's function for transIationally invariant systems discussed earlier ( cf. equation (15.23):

(kt,kY) = dp,61,e

-itk(t-t')

- $l)$(k - k F )

[ ~ ( f

- #(t' - t ) 6 ( k p - k)] .

n In fact, the contraction c j ( t ) c l ( t ' )is in general the frr:c Hb-propagator:

Green's function far the

PERTURBATION THEORY so for H = Ho we have

n i d O ) ( j t kt') ,

= (@a T [ ~ ~ ( t ) ~ t I (/ tQ'O ) ~=] ~

t

~ ( t ) ~ c ~ ( t ' ) ~ .

(l9.11) For the remaining contractions that can be formed from the Ho creation and annihilation operators, we have

and

from the general proof above. The last equations follow direttly from the earlier calculated pairings. In an analogous manner, we obtain fot the field operators:

Thus, all contrac-tions t h a t can be formed from field operators or from thc creation and annihilation operators of the Ho problem either vanish or correspond to free Ho-propagators,

Chapter 20 k'eynrnan diagrams We have now arrived at the diagrammatic calculation of the energy shift AE and t h e singleparticle Green's function. By using the procedure of 'adiabatic turning-on', we derived the foIlowing perturbation series in the coupling constant g (cf. equations (18.1), (18.8) and (18.11)) for the energy shift m d the Green's function:

a

AE = Iim i-ln(Qo ~ 4 0 dt

1 U E ( t , - o o ) f ao>

and

If we now use the second-quantized representation of the potentia1, either through the Ho-creation and annihilation operators

or through the field operators

FEYNMAN DIAGRAMS we will obtain terms of the form

and

respectively. Apart from the matrix elements (ij 1 v I lit) in the Horepresentation, or v ( x , x t ) in the space-spin representation, which we as usual assume known, we only have to calculate vacuum expectation values of t he-ordered products of creation and anhihilation operators. Accarding to Wick's theorem, the result is the sum overoa11 completely contracted combinations of these creation and anhihilation operators. Depending on each separate contraction, the terms either identically vanish or consist of a product of free propagators ~ ( ' 1 . Hence, all that appears in the perturbation expansion are matrix elements of the potential and free propagators do). The general form of the nth order term in the perturbation series is then , .

n factors

2n factors for AE (2n 1) Iactota for G

+

The product of the matrix elements of the potential is the same in all terms of nth order; hence, we only have to find all possible products of free propagators with the corresponding arguments. A diagrammatic method for the investigation of the terms that arise uses t h e famous Feynrnan diagrams. In this method, each occurring term is assigned a diagrammatic repre~edtzttion through a unique translation recipe. The translation recipe is usually summarized in a rlumber af rules, the so-called Fey~lmanrnles: (1) We imagine a time-axis with time increming from below to above.

(2) The Green's function G(')(x~,x'~'),which in any represen~alionA describe the 'free' propagation, i,e., the Ho-propagation of particles (for t > t'), and holes (for 1 < t'), respectively, from state A to state A',

PERTURBATION THEORY

22 1

is reprcsmted by a continuous line from (At) to (A'f'). According t o the propagation from A' t o A, we draw an arrow, which points from the second t o the first argument, on the line. The endpoints of the line must be ordered according to the imagined timeaxis (see figure 20,I). The length, curvature or tilt of the lines do not matter; the

for t * t t

for + ' > f

for ti= P

Figure 20 -1 T h e pr-upagator is represented by a line, with the direetion of pr~pagationindicated by armws. only important thing is the order of the endpints with respect to the imagined time-axis. Thus, a link with the arrow pointing up describes the propagation of a particle, and a line with the arrow pointing down describes the propagation of a hole. We will later discuss how to calculale and interpret the propagator for t = t f . Furthermore, it should be remarked that we cannot use these rules to represent the four-dimensional Fourier transform G(k, w ) , since this one does not contain any time argument, We will treat the diagrammatic rcpresentation of this Green's function in Chapter 24.

Thc simplest example that we know is the spatial Green's function ~ ( ' 1 (rt,r't'),or, with spin-indices labeled, Gap(rt ,r't') = b a D ~ ( ' ) ( rr't'). t, These Green's functions are depicted diagrammatically in figure 20.2. Another example is the momentum Green's ianction. This one is diago-

Fi ute 20.2 Diagrammatic represent at ions of the Green 's f m c tions G (rt,r't') and Ga8(rt,r't').

61

FEYNMAN DIAGRAMS

222

na1 for Ho = Ckto k 2 / ( 2 m )t c k , o ~: ~Gap(kt, ,o k't') = 6ap6kkf~(0)(k, t - t') ( s e e figwe 20.3). akf lo l Gab

Ikt,ktt'l= b u R 6 k , k ~ ~ 1 0 1 i k , t - t ' ~ Pk' f '

1' +Ika Or

Figure 20.3 Diagrammatic rep~esentiltionof themomentum Green's function (0) G,,(kt, kit') = 6,p6k,kt~(o)(k,l - t'). (3) The matrix elements of the interaction are represented by a wiggly line with the endpoint6 labeled according t o figure 20.4, The point

Figure 20.4 Diagrammatic representation of the intere~cti~h matrix element ( i j I v ( jt). The connection points between the interaction Line and the propagator u e called internal vertices.

where the interaction lines are connected ta propagators are called internal vertices. They are start or endpoints far each ~ ( ~ I - l i which ne is the result of a contraction of four operators belonging to this matrix element: (20.1) (ij I v I kt)cl(t)cf(t)cr(t)ck(t). The direction of the arrow on the particle lines relative t o the imagined timeaxis is not fixed. For example, the lines of the ~ ( ' ) - ~ r o ~ a ~ a t o r in the matrix element equation (20.1) can be represented as either of the diagrams in figure 20.5, The crucial point is that at each internal vertex, there is one line with the arrow pointing toward the interaction line, and one line wit b the arrow pointing away, The indices of the two arrows that point away correspond to the creation operators c;t and ct.; the indices of the arrows that point in to the vertex corresporld to the annihilation operators cb and y.

Within the framework of the non-relativistic theory, which we are considering here, the Coulomb interaction between electrons is instantaneous; hence, the interaction lines run horizontally. (Other intetactions, such as

PERTURBATION THEORY

Figure 20.5 Possible diagrammatic represeent ations of the propagator lines belonging to t h e matrix element in equation (20.1). elec tron-p honon interactions, are retarded even in non-relativistic theories . Hence, interaction lines for such interactions begin and end at different times.) This means that the left and right endpoints of these (as later 'k "-. starting points or endpoints of the Green's function lines) belong to the e*' time-point of the imagined vertical time axis. This time-point is common b all four operators which are connected by this matrix element. In relativistic theories, in which the Coulomb interaction has a finite velocity of propagation, the requirement that the interaction l i n e are horizontal will be dropped (see Chapter 24). A potential which is local in the space-representation and diagonal with respect to spin-coordinates, i. e,, v ( z ,x') = v ( s s , rs'), will be represented as shown in figure 20.6. If t h e pcrtential is local in space, but non-diagonal i n the

Figure 20,6 Fur potentials which are locaf i n space, only one index is needed at each internal vertex. spin coordinates, the diagrarnmatical represefitation is as depicted in figure 20.7. For the t ranslationally invariant interaction discussed in Chapter 5, the momentum represent at ion is particularly simple:

In this case, the matrix element depends only on

index, q. T h e creation and annihilation operators that belong to the incoming and outgoing particle Ohe

F E Y N M A N DIAGRAMS

Figure 20.7 The representation of a potential which is local in space, but non-diagonal in spin. ,---4&s

a vertex telI us how to label these lines (see figure 20.8). Apparently, conservation of momentum holds at the internal vertices. Thus, the matrix elements of the interaction can be interpreted as a propagator for a particle with momentum. q. at

Figure 20 -8 I n teraetian line for a translationally invariant pat en tiaI in momentum rcprescntation.

With the three above rules, we will now draw a few Feynman diag k a r s . We begin with the representation of the so-called vacuum ampliZude (Qo 1 UEI Qe}, which is necessary for the calculation of the energy shift AE, and which also appears in the denominator of the single-particle Green's function. The quantity has acquired this name, since it is the probability amplitude for the transition from the vacuum state I Go) back t o I iPO). In the vacuum amplitude, all the contractions that appear have indices which are also found on the interaction matrix elements, i. e., all do)-lines begin and end on the endpoints of interaction lines. In first order, i , e . , one interaction line, there are only two different diagrams. We draw them in figure 20.9 Tor the case of a potential which is local in space, but noh-diagonal in spin-indices. In contrast t o the vacuum arnplitudr, dl lines in the diagrammatical reprcsentatioa of the numerato~of the single-particle Green's function do not start and end a n an dpoints of interact ion lines. Furt,hermore, t the two operators &(r,t) and $o(r'tl), which correspond t o the argunients of the exact Grccn's function, define two additional endpoints. In first order, we obtain the six diagrams shown in figure 20,10, Since only the starting

PERTURBATIONTHEORY

Figure 20.9 The two first-order vatu urn amplitude diagrams.

Figure 20.10 The six possible first-ordcr diagrams in the numerator of the Green 's frrnction,

FEYNMAN DIAGRAMS

Figure 20,11 The last ofthe diagrams in figure 20.10 can, for example, be drawn in either of these two ways. point and endpoint are important for the do)-line, the last two diagrams can for example be drawn as shown in figure 2 0 , l l . A11 Green" functions diagrams drawn here are for particle p~opagationIt > t'). The carrespanding diagrams for holepropagation are obtained by exchanging the indices and by changing the direction of a11 arrows. In all depicted diagrams for the vacuum amplitude, do)-functions with two equal time-arguments appear. To determihe these pa~ticularones, we once again return to the originaI formula for the first-order vacuum m p l i tude;

where we use Wick's theorem for normal products to calculate the matrix element M :

Only pairings of the form

lit$ appear, since the sequence

of creation and

U

annihilation operators is already determined by the representation of the potentid ~ ( t in) second ~ quantization. Furthermore, we have

PERTURBATION THEORY

227

Since the Green's functions with equal time-arguments, d m in higher order, always arise from pairings within a ~ ( t we ) can ~ ~formulate the fourth Feynman rule :

(4) Green's functions with equd time-arguments of the form shown in figure 20.12 shall be interpreted as G(')(x~,~ ' t).+

Figure 20.12 Equal-time Green's functions.

In all, we obtain for the first order vacuum amplitude:

The two expressions clearly correspond to the two diagrams depicted in figure 20.9. At this point, it is clear that t o evaluate the diagrams, we need the following additional rule: (5) All indices and coordinates attached to an internal vertex should be summed and integrated over, respectively. The adiabatic switching factor e-'ltl shall be added to the time-integrations.

We must now determine the overall sign and the prefactor. The rule which determines the sign is known as the 'lodp theorem': (6) The sign of a term of arbitraty order is (-I)(, where 1 is the number of closed loops formed by ~(')-lines. For example, the third-order term in figure 20.13 has three loops which result in a negative sign.

FEYNMAN DIAGRAMS

Figure 20,13 A third-order diagram with three loops.

To prove this theorem, we consider an arbitrary fully contracted term which contributes t o the vacuum amplitude:

Such a term consists of distinct groups of four operators of the form cfctcc. We can say with certainty that if the first and last operator in such a fourgroup belong to the same loop, then they bebng to the same internal vertex and are consequently connected diagrammatically (see Fig 20.14). The same

Figure 20.14 U; for example, the indices k and rt belong to the same loop, they must be connected diagrammatically.

holds for thee two rniddlt operators of each four-group. 'la indicate this fact, we have connected the corresponding operators in the expression above with vertex 'bows'. By rearranging, we obtain

PERTURBATION THEORY

229

The sign of this expression is obtained from the fact that in this rearrange ment, two operators per four-group have been commuted, A single loop apparently consists of a group of operators which are connected by a closed chain of alternating contraction brackets and vertex bows within a contraction. We now separate the general term above in single factors, each of which corresponds to a single Ioop:

By this teat~angementwe also obtain the overall sign of the expression, since by factorizing a particular loop we can determine if each pair cte belongs to the loop or not. By rearranging, each pair is thus shifted as a unit, which gives the sign. Far example

It then only remains t o show that each loop-factor can be written in the form

n n n - cct cc+ cct

n

. . .CCt .

(20.2)

n Since cct = i d 0 ) , the contribution from the cornplele expression is then ( G ) ( i d u ) ) . ..,where t is the total number of loops, as we stated. To show equation (20.21, we must factor s u t the single contractions:

FEYNMAN DIAGRAMS

-

t t - c t c . . . c a . . .c t c ~ . .C . t CC,C~C,.

(20,3)

n If the remaining term to the left only contains the contraction c,cf (which is the case if c, = c p ) , we have precisely equation (20.2). If there are other contractions, we bring cp to the right, with the result

As far as the left factor is concerned, we are now at t h e beginning of the expressions equation (20.3) or equation (20.41, and can factor out the contraction that belongs to c ~ as, shown in equation (20.3)-(20.31, and so on. In this way, we go from vertex to vertex in a closed chain, so long as we come back to the starting point: this is done in the last step in equation

n

(20.31, where the the eontraction c g c i is moved through in the remaining term on the Ieft, which closes the circle, and the proof of the loop theorem for the vacuum amplitude is complete. We now consider a completely contracted contribution t o the numerator of' the single-particle Green's function:

t t

t et c c , . . ct ct ccc,c, t .

C C CCC

Apart from the groups of four operators, this term also contains two additional operators, which represent the external fixed points z and y , A do)-line either belongs to a loop, ar it is a part of the continuous chain which connects the t w o external points z and y. Therefore, if we factorize the loops as we did with the vacuum amplitude, we also obtain a cantribution from t h e continuous line:

Each loop gives a factor of (-11, as above. Hence, it remains to be shown that the part, coming from the line does not contribute a minus sign. To d o so, we factor out the single contractions:

The first one of these two factors gives rise to a minus sign, just as a loop, so the term will have an overall positive sign:

It now only remains t o determine the p~efattorsof the 'translation recipe. First, each term of nth order has a prefactor ( - i J n/ ( r 1 ! 2 ~ )In . the vacuum amplitude we have furthermore (2n) contractions of the form i~('). Thus, we obtain in all a prefactor ( i ) 2 " ( - i ) n / ( n ! 2 " )= ( i ) * / ( n ! 2 * ) .In the nurnerator of the Green's function we have in the nth order (2n+ 1) contractions of the form i~('). Finally, since the expansion for the Green's function really is an expansion for hG, the diagrams for the Green's function acquire an additional factor (-z'). Thus, the prefactor is in all (-i)(i)"+' (-i)"/(n!2") = ( i ) n / ( n ! 2 n )The . recipe is then complete with the following additional rule: (73 The prefactor of each term of nth order is

This holds both for the vacuum amplitude and the numerator of the single-part icle Green's function. As an exercise, we will now show all second-order diagrams which cdntribute to the vacuum wrlplitu Je. To construct these diagrams, we first draw the two interaction lines at the times t l and d 2 and find a11 ways to connect them with ~('1-lines. We obtain the diagrams shown in figure 20.15. The question arises as to how many distinct diagram there are of a certain order. The answer is quite simple: assurne that we have drawn the n interaction lines sf the vacuum amplitude. We then have to place the 2n arrows at the vertices of thc diagram, There are 2n possible ways of placing the first arrow. On the other hand, there are only (2n - 1) possible ways of placing the second arrow, correspondiag to (271 - 1) contractions, and so on. This gives (2n)!different dit~gramsfor the vacuum amplilude. For the numerator of the Green's function, them are the additional two external points. Correspondingly, there are ( 2 n + I)! different diagrams for particle propagation ( I > t ' ) and hole-propagation ( t < t i ) . Moreover, the contributions from visually different diagrams which give the same contributions ta

FEYNMAN DTAGRA MS

Figure 20,15 AII possible second-order vacuum amplitude diagra~ru~

PERTURBATION THEORY

233

the perturbation expansion are included in t h e prefactors. This is clear for the first two and the last two diagrams of the second row in figure 20.15, These diagrams can be obtained from one another by permuting the internal vertices on a single interaction line. Their cant ribu t ions are identical, since we sum over all indices. We will discuaa this kind of Uegeneracy' in detail in the next chapter. We wiI1 now translate two examples of the Feynman diagrams depicted above t o mathematical language by using the Fe'ynman rules. The first example is the diagram shown in figure 20.16. According t a the rules, this

Figure

Second-order vacu urn amplitude diagram.

diagram gives a contrib~tion

Since, up to this point, we have assumed that t i > t 2 , the expression we obtain is strictly only valid in the domain of integration, where f l > t 2 , We obtain a family of equivalent diagrams if we assume that iz > t l . We will furthe? consider this kind of degeneracy in the next chapter by performing the time integrations.

As a second example, we translate the two disconnected diagrams shown in figure 20.17 by using the F e y n ~ n mrules.

Figure 20.17 These two disconnected first-order diagrams are included in the set of all second-order vacuum amplitude diagrams.

x ( j k 1 v 1 rnn)(pq 1 v 1 rs)~(')(mt~,jlt)~(')(nt~, kt:) ~ ( O ) ( s t zp, l : ) ~ ( D ) ( r t 2 ,

8

.

1

dtl e-'tti i ( j k 1 u 1 r n n ) ~ ( O ) ( r nj it ~: ), ~ ( O l ( n t kt:) ~

This example illustrates a very important fact: the contribution from a disconnected diagram is factorized into parts which belong to connected diagrams of lower order. If the interaction consetves mamcfitum, most of the depicted diagrams vanish. For example, in the diagram shown in fi re 20.18, we must have q = 0. IIence, both the vertical and diagonal G O)-lines have momentum k, which cannot happen, since the vertical one is a particle line (k > kF) and the diagonal one a haIe-line ( k .< kF) , Consequently, this diagram must vanish. This may be shown explicitly by using the occurring 6-functions. We have lerzrril that 6utlr in the vacuum amplitude and the numerator of the single-particle Green's function, ihcrc arc connected and disconnected diagrams. 1n the next chapter, we will show that we only need to consider the connected diagrams: in the case of the Green's function, the disconnected diagrams are canceled by the denominator; in the vacuum amplitude, the disconnected diagrams can be summed analyticalIy. At that point, we will make a discourse to calculate thc correlation energy of a dense electron gas (Chapter 22). After that, we will continue the cdculation of the singleparticle Green's fur~ction.

ff-'

Figure 20 .I8 A second-order diagram in moment urn representation.

Chapter 21 Diagrammatic calculation of the vacuum amplitude In this chapter, we will sum all disconnected diagkams which contribute to the perturbation expansion of the vacuum amplitude. The goal is to show that only connected diagrams appear in the remaining expression of the vacuum amplitude. The firial result of this summation is the smcalled Linked-Cluster Theorem by Goldsto~le[15], which states that

The natation (@o 1 U 1 @O)L means that only connected diagrams are included in the sum. For example, the expansion for the case of interactions which conserve momentum is shown in figure 21.1. Acca~dingto the

Figure 21.1 Diagram for the perturbation expansion of the vacul~rrrarnplit ude for moment urn conserving in terattions.

linked-cluster theorem, only t h e connected diagrams shown in figlire 21.2 t.ont4rit~~ite. As a corollary of Goldstone's Iheoreln, we obtain a very simple

CALCULATION OF THE VACUUM AMPLITUDE

F i g u ~ e21.2 The connected diagram for the vacuum amplitude in figure 21.1,

formula for the energy shift:

To prove the theorem, we first consider two arbitrary diagrams, -y(") and y('), of order n and ii.These diagram may be connected or disconnected. In the expansion of the vacuum amplitude, there will aIso be a disconnected diagram l?(".+;) of order (n E), which is constructed from the first t w o di agr amsr

+

From the fact discussed earlier, that disconnected diagrams may be factorized into contributions from their sub-diagram, and from Feynman rule ( 7 ) for the prefactors, we ob Lain the folIowing important equation:

With the help of this equation, we can factorize an arbitrary disconnected diagram which contributes to (ao1 U 1 QO) in a prodnct of connected diagrarm will1 givetl prefactors. There are in general also many other diagrams, which give the s a m e contribution t o the perturbation expansion. These are the hagrams which consist of the same sub-diagrams and which all can be factorized into thc same product, equation (21.3). As an example, we considcr all diagrams of fifth order, which can be constructed from the three first-order sub-diagrams shown in figure 2 1-3. All such fifth-order diagrams are shown in figure 21.4. These diag~amsare obtained from one another by permuting two single inte~aetionlines, if., the indices of the interaction lines are interchanged, Howcvcr , i t is clear that all 5! possible petmutations of the interaction lincs will not lead t o distinct diagrams, For exan~ple,permuting the interaction lines within a particular connected sub-diagram will

PERTURBATION THEORY

Figure 21.3 Three connected diagrams. The first one, the ather two ones, which are the same diagram g2(2) .

gF),

is distinct from

Figure 21.4 All fifth-order diagrams that can bu constructed from the diagrams in figure 2 1.3.

240

CA L CULATlOiV OF THE VACUUM AMPLITUDE

not lead to a new diagram. If we want t o calculate the number of distinct diagrams which can be constructed from the three sub-diagrams, we must divide 5 ! by the number of permutations of the sub-diagrams, which is two times 2!. Fu~thermore,we note that permuting identical sub-diagrams does not lead t o distinct diagrams. Since in this example we have two identical sub-diagram, we must d s o divide by another factor of 2!. This leaves 5!/(2!2!2!$= 15 distinct diagrams, which is in agreement with figure 21.4. The contribution of all diagrams in figure 21.3 to the perturbation expansion is obtained by applying equation (21,3) twice:

After this example, we will now prove the theorem. We start by emmerating all connected diagrams of t h e perturbation series and denoting them

where the superscript denotes the order. An arbitrary diagram af the perturbationseriescanbeconstructedby tldiagramsg~')~ k2 diagramsg2In2 ) , k3 diagrams

gp', and so on. By repeated use of equation (21.31, we obtain

the following contribution from this diagram:

As we have seen, there is in general a whole series of different diagrams, which all give the same contribution t o the perturbatioa expansion, All these degenerate diagrams can be obtained by suitablc pcrnmtxtions of interaction lincs. If wc want to determine the number of these different diagrams, we must divide the number ( k l n l + k 2 n 2 + . , ,)! of all perrrlt~tationsby thc number of permutations which lead t o dcgencrate diagrams. First of all, idedical

PERTURBATION THEORY

241

diagrams are obtained if interact ion lines within connected sub-diagrams are permuted. In each sub-diagram there are (n;!)such permutations passible, so in total we must divide by (nl!)lD1. . . ( n ; ! ) k i. . . Secondly, we also obtain identical diagrams when entire identical sub- diagrams are permuted of sub-diagrams, there are (k;!) such with one another. In each class permut atims, so that we must divide by another factor of (kl! k a !. . , ki! . , .). In all, we obtain

glnr',

different diagrams, which contain kl x gl, k p x g 2 , . . ., and which all give the same contribution , equation (21.41, to the perturbation expansion. If we sum over all these contributions, we obtain -

C

all distinct diagrams which cbntain

1

kl!.. . ki! . - -

kl X g 1 , ~ 2 X ~ 2 , . - .

All diagrams of the vacuum amplitude are then obtained by summing over a11 kl,k 2 ,. . . from zero to infinity:

This concludes bhc proof of the linked-cluster theor em. We also nccd to examine the connected diagrams. In these, too, we can factor out degenerate ones,just as we hare seen earlier. For example, two of the diagrams shown in figure 21.5 leave identical contributions, s h c c they can be obtained from one another by interchanging the internal vertices of

Figure 21.5 Two connected diagrams, gf) and $), of second order. The diagrams givc identical contributions to the vacuum amplitude.

an interaction line. Since t hc matrix elements always satisfy the symmetry relation

( j k ] v I nin} = (kj I 'lr ( a m )

CALCULATION OF THE VACUUM AMPLITUDE

Figure 21.6 Third-order ring-diagrams. T h e ones depicted here are mymmetric under reAec tion i n a vertical line through the middle of the diagrams.

and since all indices will be summed over, the contributions are identical. Such diagrams ate usually called topologically eqaivalen t . At this point, the question arises of how many topologrcally equivalent diagrams there are to a certain diagram of order n. The answer is simple. If we fix the lowest interaction line, there are two permutations possible for each of the ternaining In - 1) interaction lines, so there are in total 2n-1 different topologically equivalent diagtams. If the diagrams are symmetric under reflection in a vertical line through the middle of the diagram, we do not obtain any new diagrams by intetchanging the vertices of the lowest interaction line. In this case the family then contains 2n-1 topologically equivalent di~grams.If the diagrams are asymmetric under reflection in such a lihe, N e obtain twice a many, 2. e ,,Zn different diagrams in a family of topologically equivalent ones. The secaIled ringdiagrams of third order shown in figure 21.6 are examples of asymmetric diagrams.

All diagrams that we have considered so far correspond to a fixed order of the times t l , t 2 , .. . , t,, as we have mentioned, with which the individual interaction lines are labeled. For each of the (nl) other orders of the time arguments, we obtain a diagram which looks identical and which differs from the first only with respect to the order af the times. This form of degeneracy can be taken ihto consideration quite simply: if we consider the equation

2

is i

i

PERT IJRBATION THEORY proven in Chapter 14

it is sufficient to consider the family of diagrams with t1 > t z > ... > t,. The only thing that we have t o do now, is to explicitly perform the timeintegration. We first investigate an example of low order, the firstorder connected diagram g!) of figure 21.7 ( a ) :

and @) g,Ill,

Figure 21.7 The first-order connected diagrams (a)

J

=

gd

jkmn

t

dtl e-'ltl1(jk

1 v 1 mn)

-03

x G ( O ) (mtl, j t f ) d 0 ) (ntl,kt+).

If we insert the free Green's functions (cf. equations (19.10) and (19.11)) i ~ ( ' ) ( j tkt') , = 6jke-'c~(t-"l [ ( ~ (-t t t ) 8 ( 5 - r F ) - B(tf - t)B(cF

and use tl t

> t l 9 ... > t,,

we have

so t h a t we can write

In contrast to OUT earlier derivation, we will now perform the time integration at the operator level, For the last factor, we obtain

If we add the next factor

It,-* ecfn-1

v(t,-

to

this, we obtain

It,-'

dt, ectnv(t,)r

I ao)

The ( n- 1)-fold repetition of this procedure finally leads

to

PERTURBATION THEORY

x

The Iimit

E

4

ltn-' dt,

eltnv ( t ,II

1 @O)L

0 can then be performed with the result

If we now insert the completeness relation of the anpertu~bedeigenfunctions

between the potential operator vs and the factars

a cornparison with equation (21.13) shows that the energies Wj belong to the int ermediatc st at es. If we compare the result with the Rayleigh-Schrodinger

formtlla

the restriction to connected diagrams has the same effect as the projection operator Q = 1- ] aO)(a01, namely, to prevent I QO) from appearing as an intermediate state, which would result in a zero denominator.

Chapter 22

An example: the correlation energy of a dense electron gas We will in this chapter use the methods discussed in the previous chapters in an important example: the calculation of the cor'f.elationenergy of an electron gas in the limit of high density, according t o Gell-Mann and Brutsckner [16], The interacting electron gas is, according t o equation (10.10), described by the Hamiltonian

It is important that the the contribution from q = 0 is explicitly removed from the sum over q, since this Fourie~component of thc interaction potential is canceled out by t h e energy contribution from the urliform positive background charge density. Thrt matrix elements of the interaction have the form shown in figure 22.1, which is 47re2

((k + q ) ~(kt , - q)d 1 v 1 kc,kro3 = Rq2 ' This diagram can be interpreted as a momentum-transfer process: the first electron with momentum k' loses a momentum q , which is transferred t o a

ELECTRON GAS CORRELATION ENERGY

Figure 22.1 Interaction diagram for the electron gas.

second electron by the interaction. From this, it follows immediately that the diagrams which contain the part shown in figure 22.2 vanish, since they require q = 0, and this term was omitted from the sum, In consequence,

Figure 22.2 Any diagram, which contains the s ~ c a l l e dtadpole diagram, vanishes.

the direct k r m in the Hart.rce-Fnck energy, which was discussed earlier, vanishes:

(AE),Ill - 0.

(22.1)

In first-order p~rturbatiorltheory, only the exchangc terms represented in figure 22.3 remain. The energy contribution from these is

&%

Figure 22.3 The exchange diagrams of' the electron gas.

PERTURBATION THEORY

(_)I+"

-

dne2

2l

w

km

symmetric diagram qf

' kF), 2.e.) their mrnnenta cannot be equal. Therefore, the diagrams vanish, However, diagrams which contain the factors shown in figure 22.6 can appear. We have already encountered such diagrams as first-order contributions to the single-particle Green" function. In a connected diagram of second order for the vacuum amplitude of the electron gas, the bottom and top p a t s can only look like figure 22.7. When

EL ECTROIV GAS CORRELATION ENERGY

Figure 22.6 Diagrams with these factors may give non-vanishing contributions.

Figure 22.7 Connected second-order diagrams for the vac u urn amplitude can only have these bottom and fop parts. these two are taken into account, of the 24 diagrams of second order, which were constructed in Chapter 20, only the four connected diagrams shown in figure 22.8 remain, and of these, only two are topologically distinct. We now label these according t o figure 22 "9, taking into accouht the momentum conservation. We eliminate k; , ki and q by considering the unperturbed single-particle Green's functions labeled by the numbers 1 through 4:

2

, k ~ 6-o;ol ~ 6 6k2,k:-q7

3

Ski ,k:+qf6o1u:

:

:

; k '

ki=ki-q qt=k; - k z = k l - k ~ - q J. k h = k l - q ' = k r -kl+k2+qrk2+q 3 kh = kE+ q consistent with 3 3

la)

~ = L =Iu ; 2=u;=(T

(.b I

Ir )

(dl

Figure 22.8 The only connected second-order diagrams far the vacuum an?plitude. The diagrams (a) and @), and (c) and (d) are topologically equivalen t .

PERTURBATION THEORY

Figure 22.9 Labeling of the diagrams in figure 22.8.

Thus, we obtain indexing shown in figure 22.10. We also obtain

This is a 'safe method', where the momentum conservation at each vertex

Figure 22.10 Momentum conservation simplifies the indexing of figure 22.9. is taken into consideration. This method should be used for complicated diagrams and, in particular, in doubtful cases where it is not immediately clear if momentum conservation can be satisfied. In the case of simpler diagrams, the correct labeling can be found by inspection; see, for example, figure 22.11. In this figure


Figure 22.11 in a diagram such as this one, it is straightforward to Iabel the ]in es.

W7e will now as an exercise label the ring diagram of third a ~ d e rin figurc 22.1 2. A ring diagram is a diagram which is an uninterrupted ScqUehCe of interactim lines and particle hole 'bubbles'. This example illustrates an important property of ring diagrams: thc same momentum transfer q appears in all interactioh lines.

Figure 22.12 A ring diagram only contains interaction lines and particlehole 'bubbles'.

Let us now calculate the energy contribution of the two topologically equivalent diagrams figure 22.8 (c) and 22.8 ( d ) , according to squation (21.11):

ELECTRON Gd S CORRELATION ENERGY

+

+

where (kl (,lkzj < 1 < lkl ql, \kg q(. The resulting expression can be integrated analytically [17],with the result

= 0.0484i'V Ry.

(22-5)

Here, c(3) is the Riernann zeta-function

The important conclusion is that this energy contribution does not depend on kF. Hence, it does not depend on the 'average atomic radius>r, introduced emlief (cJ. equation (10.6)):

We now calculate the other sccond-order energy contribution (the diagJarns figure 22.8 ( a) and j b ) ) :

By going to the continuum limit and by making the substitulions

we obtain, analogously with ( A E ) ~ ~ )

(

~

(2) ~ )

3N (

1

d39 =

Q)

~

/

T

/

~

~

~

l

/

~Ry. ~

(22.7) ~ 2

+

Here, the integrals are performed over the region k1 , k2 < k F ; )kl + ql ,(kZ ql > kF. This term, too, is independent of r,. We will now proceed to

~

.

(

~

PEITTURBATION THEORY

263

show that the resulting inkegral diverges. Thus, we will prove the statement made in Chapter 10, that second-order perturbation theory divetges for the electron gas. We consider the integtal for q -+ 0 and introduce the notation

The integration region lkal < 1 < Ik; 1 - qxi < ki < 1 for small q , since

+ ql of the k-integrations becomes

In the integral I ( ~= ) Jd3t,

1

d3k2

1 9.(kl

+ k2 + 9) '

we take the z-axis parallel t o q , which leads to

and, finally, by sing the estimate above that tcrms af order q 2 , we arrive at

k; = I + OCq) and neglecting

It remains t o perform the p-integration:

This integral diverges at the lower limit. At the upper limit of the integration, k. e,, for q 4 w, our estimate does not hold. However, a glance at the term q4 in the denominator of the original integral shows that no problem arises at the upper limit.

264


Let us look at what happens in t hird-order perturbation theory. Consider the third-order ring diagram shown in figure 22.13, with

Figure 22,13 A third-orderring diagram.

The energy contribution is

Ry now, it is clear how t o proceed from here: we go ovcr to the cor~tinuurn

PERTURBATION THEORY

265

limit, scale the wavevectors by k F , change the signs of

kl and q

to

obtain

with the usual integration limits

We cafi establish the following important facts:

(1) The total expression is proportional t o r,, due to the factor of I l k F (cf. equation (22.6)).

(2) It is easy to establish that I ( q ) goes linearly to zero a5. q 0, by osing the scheme above. This means that the third-order ring diagram diverges strongIy at the lower limit: 4

In complete analogy, we obtain

The r,-dependence is easy

an arbitrary energy contribution, irrespectively whether or not it is a ting diagram. We have already established that all contributions of second order arc independent of r,. For each additional order, we obtain

- a matrix element ( - a factor

- a sum

t o deternine for

/ v I ) = 4 r i e 2 / ( ~ k 2 )i ,. e . , a factor of k ~ ~ ;

(A +. . .+ A)

-

k2 in the denominator, i,e., another factor of

xk,which aftcr the replacement

by kr, kTLew k0ld/kF, gives a factor of k:,.

--t

d3k and scaling

266


Thus, an additional factor of k ~ l or , rather a factor r $ l , appears for each order: (AE~!:,; r : - 2 . (22.12) Our goal is to calculate the energy for high densities, i. e., for r s -+0. If the individual energy terms ( A E ) ( ~did ) not diverge, this would be straightforward (provided the perturbation theory wouId converge); we would simply take the con'stant second-order term and eventually also include a higherorder correction. However, we must consider the divergence in detail: since the total correlation energy must be finite, we can assume that by summing up all terms, the divergence will be removed, Thus, the task will be to obtain a sensible Tirnit value of such a series of divergent terms, It is clear what happens physically: the electrons screen themselves, so that the long range of the interaction, it., the q -+0-cantribution,is eliminated. The ring diagrams have the strongest divergences in each order. It is then natural t o sum these dominant contributions in each order:

-

This procedure seems plausible. However, we must be aware that there are divergent diagrams other than the ~ i n gdiagrams, for example the third-order diagram shown in figure 22-14, Because of the two identical rnornehtum transfers, this diagram is as divergent as the second-order ring diagram, and it is not at all obvious that this diagram can be omitted, but that the diagram figure 22.10 ( a ) must be included.

Figure 22.14 A third-order ring diagram which is not maxirnalIy divergent,

In tdal, the perturbation series can be written in the following way:

PERTURBATION TETECFRI' where l i n ~ ~I,,,(q) ,~ = constant. We will now argue why it is necessary only ta consider the ring diagrams. The reason why the divergences appear is the long range of the Coulomb potential. However, we may expact that an effective screening-effect appears in the electron gas, so that the Fourier cornponefits are only important t o a minimal value kc. Provided kc is small, this results in

and so on. If we recall from our discussion of plasmon theory, that the effective screening comes into effect at

the expressions that we have already calculated are propart ion-a1 t o In(r,), l / r , , and l/rz, respectively. If we insert these into the individual energy contributions, we obtain, together with the explicit r,-dependences: ( A E )(2, ,,, (fi) (AE),,,

(AE)~$

-

In r ,

eonstant

(22.13)

constant

All other contributions go t o zero at least as ( r , In r,)

as r , + 0. Thcrcfore,

i t should be reasonable to put the correlation cncrgy in the fo1Jowing form for high densities:

We will now start with the summation of all ring diagrams. We first make the following represent ation of the cantrihr~tionsfrom the ring diagrams plausible:

with

268

ELEC'JRON GAS CORRELATION ENERGY

and

The statement holds for n = 2:

(cf. equation (22.711, and for n = 3 :

To evaluate this expression, we partition the integration regions in six pieces:

tljhlit) co&sporrds to the zero-order polarization rn-o~iigatwin(*) (q,t) , which will

be discussed in detail in Chapter 27.

PERTURBATION THEORY

269

By making the substitutions t l 4 -tl, t2 -+ - t 2 , we see that the contributions I) and 41, the contributions 2) and 51, as well as the contributions 3) and 6 ) are pairwise identical. Furthermore, we can calculate that, with the abbreviation DiE (q . p; q 2 / 2 ) , we have

+

contribution I) =

contribution 2)

dtz e-ti

=

,-t2D2 , ( - t l a t z ) D 3

- t l a e + t 2 ~e-(fl a -t2 p l d new = old old new new - iO1d (tYW - 2 t l - t , + t , €1 + t z - 1 1

If we also interchange pl

u pa, SO

that we inte~changeD l

that

-

Qg,

we see

contribution 1) = contribution 2).

A completely analogous calculations shows that contribution 3 ) = contribution 1).

Thus, the dontributions from the six different integrations are equal, and we obtain for J3 (with the notation of the calculation for cantrihtltio~~ 2)):

=

J/Xi is or hole-propagation (fg > t A ) ,there are three possibilities with the twoparticle Green's function: particle-pair propagation, hole-pair propagation and part icle-hole propagation. These may be pepresented as shown in figure 27.1. Again, Feynrnan rule (1) with the imagined time-axis is used when the

Figure 27.1 Possible diagrarnlnatic represcn tat ions for particle-pair propagation. In 0, t~ > tg > tc > tu, in (b) t g > 1~ > tC > b ,in (c) ! A > ta t ~ . t~ > t~ > t~ > t ~ a n, d i n (d) t g

cnd-points are drawn. We also agree to put arrows pointing away from the endpoint if the endpoint corresponds to a creation operator, whereas the arrow points toward the endpoint if it corresponds to an annihilation operator. Accordingly, we obtain the diagrams in figure 2 7 . 2 ( a ) and ( b ) fur holc-pair propagation and particlehole pray itgation, respect ivcly. As an exampie, we consider the first few terms in the numerator of the particle-pair propagatcjr . The corresponding diagratris are shown in figutc 27.3. The corresponding diagrams for the hole-pair propagator are obtained sirnply by reversing the directions of all arrows, as shown in figure 27,4. On the other harrd, new diagrams appear in the particle-hole propagator, for c x a ~ p l ethe diagrams in figurr: 27.5.

PERTURBATION THEORY

Figure 27.2 In la),a diagram for hole-pair propagation for t ~ , t > o t A , flg is shown. There are three more such diagrams, corresponding to three other possible t ime-orderings for hole-pair propagation. In (b), a diagram far particlehole propagation corresponding to t ~ , t >c t g , t D is drawn. There are three more such diagrams. In addition, there are four mom diagrams each for particle-hole propagation for the time-orcferings t B , t~ > t ~t ,~ , tg , t ~and , tg ,tD > t ~t ~, respectively. , t~~t~

Figure 27.3 The first few diagrams from the numeratdr of the particle-pair propagat or.

T WQ-PARTICLE GREEN'S FUNCTIONS

Figure 27.4 The diagrams of the numerator of the hole-pair propagator are ab t ained from the diagrams in figure 27.3 simply. by changing the directions of all arrows.

Figure 27.5 A few particle-hole diagtarns.

The next step is the linked-duster thcorem, which says, just as far the single-par ticle Green's function, that the disconnected sub-diagrams of the vacuum amplitude arc canceled with the denominator:

It should he emphasized that only diagrams which contain pieces that are not connected to any external point are omitted. Thus, diagrams such %T those shown in figure 27.6 are 'linked clusters', in this serrse, ahd arc to be included,

Figure 27.6 These l o w t ~ t n r d e rdiagrams are to be treated as connected diagrams.

PERTURBATION THEORY

325

Bzhm we start t o evaluate the perturbation se~ies,we must add a suppled m a t to the Feynrnan rules for the two-particle Green's function: The translation of G(')(A,B , C,D ) , represented by the diagrams in figure 27.7(a), is surely not

since we can read directly from the definition of the tweparticle Greenjs function that

- G @ ) ( A ,B , C , D ) = G @ ) ( BA, , C ,D) (see

figure 27.7 ( 8 ) ) which is

i.e., t h e same expression which we alteady attempted to write as

Figure 27.7 The two diagram (a) contribute to G(')(A, B, C,lJ),whereas the two diagrams in @) con tribute to G(') ( B ,A , C,D).

+ G ( ~ ) ( A , B ,D). C , This uncertainty of sign can easily be elsrified if we go back t o the Wick-expansion:

The minus-sign in the last line of this equation must then replace the plussign in equation (27.1). {The factors iZ on both sides of the equation cancel out.) Thus, the paradox is resolved. We scc that we must add a Feylramn rule concerning thc overall sign of the diagram. This rule must, for example, correctly give the sign of the diagram in figure 27.8. We state Fey~nman rule (6)A:

TWU-PA RTICL E GREEN'S FUNCTIONS

Figure 27.8 If the referehence diagram (a) has a plus-sign, then the two diagrams @) and (c) must have a minus sign. Two diagrams, which differ by the interchange of two incoming outgoing fermion lines, differ by a minus sign.

OY

Diagram which contribute to G(A,B, C ,13) and which have fermion lines running from D to A (and thus also from C to B ) , then have a plus sign (in addition t o all other pre-factors according t o the other Feynman rules). The notation 'rule (6)A' is to emphasize the close relationship with the loop-theorem, which we will now treat briefly. The d i a g r m in figure 27.9 cah be interpreted as the contributions to the single-partitle Green's func-

Figure 27.9 These two diagrams of the two-particle Green's functions can be interpreted as sihgle-partick Green 's function diagrams by Closihg the propagator lines at the dashed lines. tion G ( A ,D)shown in figure 27.10. These two diagrams differ by a rnixi~is sign, according to the loop-theorem (rule ( 6 ) ) . However, they can also be

Figure 27 -10 Single-part icle Green 's function diagrams corrcspon ding to the twepart icie Green 's function tijagrams in figure 27.9.

PERTURBATION THEORY interpreted as contributions t o

As s ~ hthey , acquire a relative minus sign, according to the complementary rule (6)A. (see figure 27.11).

F i g r e 27.1I The endpoints B and C af the diagrams in figure 27..S can be closed with a single-particle Green's fumtion to yield diagrams for the Green's function G(A,D ) .

One particular possible way of summing up the perturbation series is again by 'dressing the skeletons' by successively inserting self-energy insertions. We will now show that the approximation t o the two-particle Green's function consisting of the sum of t h e t w o lowest-order dressed skeletons

corresponds precisely t o the Kartree-Fock approximation. With the rnachinery that we have at our disposal at this point, the proof is quite simple. We use the relation discussed earlier hetween the irreducible self-energy and the two-particle Green's function:

Tfthis relation is inserted in Dyson's equation, we obtain an ihtegral equation which corresponds to the differentid equation of motion, equation (16.7):

TWO-PARTICLE GREEN'S FUNCTIONS

328

with the diagrammatical representation t

'"

P rf

t'f

Irf

In the interpretation of the diagrammatic represehtation in equation (27.41, it should be borne in mind that each interaction line in a diagram of the single-particle Green's function leaves a factor of a', according to Feynman rule (7) (disregarding additional degeneracy factors, which are in any case omitted in non-indexed diagrams, i, e . , sums over degenerate diagrams). Consequently, the single interac tioa line in equation (27.4) rnus t also contribute a factor of i . Furthermore, a factor (-1) must be added in the translation. This factor takes into account that for each diagram of the two-particle Green's function, which acquires a plus-sign according t o the rule (6)A, becomes a loop, which must have a factor of (-1)' by connecting the two legs on the right-hand side. No loop, which acquires a minus sign according to ~ u l c(6)A and compensates the minus sign, is contained in the other diagrams. The two following illusttations will demonstrate this process for the lowest-order diagrams. These st aternents are easily verified using the formulation of tule (6)A. This rule demands a plus sign precisely whcn the two legs on the right-hand side within the two-particle Green's function are connected with fermion lines; by connecting these legs a loop thus emerges, Tf we insert the approximation equation (27.2) in equation [27.4), we obtain figure 27.12, This is precisely Dyson's equation shown in figure 27.13. This approxinlation for M corresponds precisely to the Hartree-Fock approximation, as we showed in Chapter 25.

Figure 27.12 appro xi ma ti ox^ for the dresscd Green 's function.

Figure 27.13 The diagrams i n figure 27.12 are Dyson's equation (a) with the self-energy M approximated by (b).

We will not here further pursue the diagrammatic analysis af the general two-par ticle Green's function, Instead, we consider a special case, which in ptactice is extremely important, namely, the already earlier introduced polaization propagator. It is defined as

where

Apparently, we have

whorc the correct order of the field nperators is ensured by the infinitesimally larger times t+ > t >and( i f ) +> 2'. The polarizrtt,inn propagator describes, a s thc name implies, the syreading of a particle-hale pair. We will now see which diagrams contribute to it. In zeroth order, we obtain for thc perturbation expansion of t h e first term with the help of Wick's theorem ( x stands for ( r t ) ) :


=

[ i ~ ( ' ) ( x , x ) ][ i (01~(s', b)]- [ido) (I',a)] [i~(')

(x,o')]

These two terms have the diagrammatical representation

With the help of the relation which we proved in Chapter 15 between density and the single-partide Green's function, we have

We have derived the diagrams (27.6) by hand, i.e., with Wick's theorem, to emphasize an important point which can easily lead to trarlslation errors. The two-particle Green's function considered always has t w o (fixed) equal arguments and, accordingly, two legs with identical indices. These legs are tcrminatcd at the same points ih the diagrams. ITence, to zeroth order, we obtain this special two-particle Green's function horn the diagram:

and

PERTURBATION THEORY

33 1

So long as no interaction Tines end on the external endpoints, these are not verfkces; in particular, there are then no loops which have been closed in the sense of the loop theorlem. Instead, the rule (6)A must be used for the diagrams of this special two-par title Green's function. Consequently, the upper graph, (27.7))acquires a minus sign (note that the endpoints AC and D-B are connected), in agreement with the result equation (27.6) obtained by hand. The following point of view, which is easily ve~ifiedfor the diagrams above and also for the general diagranu below, is equivalent. The two endpoints with the same index close one or two loops, and, in particular, precisely one loop if and only if rule (6)A requires a plus sign for the cortesponding two-par t i c k Green's function diagram. Bctueen interaction lines, the diagrams fm iII yield, according to the loop theorem, the factor

-in.

We how consider an additional example: the diagram

acquires a minus sign as a diagram of the tweparticIe Green's function, according t o rule @)A. As a sub-diagram between t w o interaction lines, it acquires a plus sign, according to the loop themem. It is clear that the entire expansion of GCx, x ' , x , z') can be divided into two types of diagrams, those in which the line starting at x is connected in any way (including by interaction lines) t o tbe line starting at x' (type II), and those for which this is not the case (type I). It is also clear, that line starting at x in the diagrams of type I must come back to x. Hence, all these diagrams acquire a minus sigh according to rule (6)A. Thus, if we interpret all t l ~ parts e which are attached to x and s', respectively, z u diagrams of the full single-particle, Green's frlhctimIs G(x,x) and G(x', X I ) , we must insert by hand an overall minus sign in front of the corresponding product, as illustrated in figure 27.16, Thc first term in figure 27.14

cancels out according to the definition of in,so that we obtain


Lit x,x(x,xll

I:+? "W + P " + 1 @ +..I [0 8 +

...

+

+

-.

-*

+

-

+

[Cilm[c?fl +

I

Figure 27.14 The two-particle Green's function G ( x ,x F , x , x ' ) can be cxpanded as a sum of typc 1 ahti type 11 diagrams.

in(+,z') =

G

the sum of all type 11 diagrams in G ( r , xi,x ,x ' ) . (27.10)

Hereafter, we denote iII(x, x') by

The cantributiuns to the polarization propagator are then in zeroth ordcr

we obtain the d i a g r a m in figure 27.15, and in second order have, for example the diagrams in figure 27.16, Jnst as we did with the

in first order, we

333

PERTURBATION THEORY

self-energy, we now define the sscalisd i r d u cdble p lam'zoiion inse d i m s as those diagrams which cannot b e separated into polarization insertions of lower order by cutting a single interaction line. Of the diagrams of second order depicted in figure 27.16,for example, the first three are reducible, whereas the rest are irreducible.

Figure 27.15 Fi~st-orderdiagrams fur the polarization propagator.

Figure 27.16 Somc second-order diagrams for the polarization propagator.

Tf we define

iQ(z$cl)

the sum of all irreducible polarization insertions

the polarization propagator can be represented as shown in figure 27.18. This is Dyson's equation for the polarbatioa propagator. When we translate the diagrams into mathematical expressions, we note that by cutting the interactioh lines for all terms except for the second loop they become identically indexed legs of a two-particle Green's function. The question of the sign associated with this is easily answered if we, z~=qdiscussed in detail above, consider -ill and correspondinglgr -iQ as insertions in an interaction

TWO-PARTICL E GREEN'S FUNCTIONS

Figure 27.17 Dyson's equation for the polarization propagator, line. The number of loops is not changed if we cut the interaction line, so that rule (6)A does not come into effect from this point of view. We can then write

=

[-@(I,

x')]

+'i

/ / d4y

d4y' [ - i Q ( x . Y)] u(gIP') [ - i n ( y t l x')]

We cancel out the factor (--a') and obtain

For translationally invariatl t systems with instantaneous spin-independent interact ions, this equation becomes particularly simple:

with the mlution Q(4,w)

"(qlW'

= 1-~ ( ~ ) Q ( q , w ) *

The use of the polarization propagator has an important application in the theory of linear response. To see this, we consider an interacting system, which ffor t _< 0 is in the ground state I q o )of the Barniltonian H = Ho+ V. At t = 0 we apply a perturbation

We want the lincar rwponsc for an arbitrary observable

of the system, i.e., the part of the deviation of the expectation value

(27.16) I A I v )-)(@o 1 A 1 go} that is linear in the perturbation Hext. Here, I 9 ( t ) )is the solution of the G(A)(t) = ( * ( t )

full time-dependent Schr6dinger equation

To calculate the linear response, we go over t o the interaction picture relative to ( H + HeXt ( t ) ) ,i.e,, t o the Heisenberg picture relative to H (thus, we use the subscript H ) . In this picture, the wavefunction is

and it satisfies the equation of motion (cf, (14.12))

with The solution up to terms linear in HEXt of equation (27.18) is

so that, together with equation

(27.17), we obtain

Fronl this, it follows that

+O(HZX,).

Thc linear response eq~iat~iol~ (27,161 for the observable A is thcn:

336

TWO-PARTICLE GREENS FUNCTIONS

We then insert equations (27.141,(27:15$and the definition af the densityfluctuation operator, equation (27.5), t o obtain

If we introduce a retarded polarization propagator (analogous to the retarder Green's function, cf. equation (26.1))

!

iIIR(rt,r't') = O(t - t') ('YO [art)H 7 F ( r ' t ' ) ~ I] @0) (90 1 q o ) we

have

In particular, we obtain for the linear response of the density:

propagator is identical to the so-called Hence, the retarded 'density-density response function', i . e . , the function which describes the linear responsc of the density to a perturbation which couples to the density (cf. (27.14))- For initially hrrmogeneous systems (systems which are homogeneous for t 5 O), we obtain by Fourier transforming

A direct diagrammatic analysis of IIR is not possible, sinec Witk's theorem can only be applied to titne-ardered products. Fortunately, one can,as with the retarded Green's function (see equation (26.41, as well as equations (26.5) and (26.3)), by using the corresponding Lehtnann representation, derive a relation between Il and TIH. We have

We close this chapter by stating the response function for an, in, practicer par tieularly import ant example of a resp onsp: fiin c t ion: the s@called Lindhard function. 'Phis is the response function of a hornogerreous system ef nun-interacting particles. It is obtained from the palarizatibn propagator with u ( x , x ' ) = 0, i-e., from the wroth-order diagram

PERTURBATION THEORY

~ I I ( ' ) ( ~w, )

The evaluation of this four-dimensional integral is possible, b u t tedious, using elementary methods (see, for example, Fetter and Walecka [22]. The results are, with the abbreviations

and

o

I -

1

[

-

-

)

2

for u > $ + q fur

]

irbk --

9

for

I2 I -

u

v> 1:

Equation (28.12) is a screened Coulomb potential. This is in accordance with the physical picture that we have already made: t h e bare elcctron

PERTURBATION THEORY

343

repels othcr electrons because of the Coulomb repulsion and the exchange interaction, which reduces the probability of finding other electrons in its immediate vicinity, Thus, the electron drags a 'hoIe cloud' with it and hence becomes a quasi-electron with an altered dispersion relation. This can of course be calculated from the co~respondingRPA for the self-energy M . Due t o the screening, the full Coulomb interaction between the bare electrons becomes an effective interaction between qu&i-elec trons, which here is obtained approximately as a Yukilw+potential. (For a close^ loook at the idea that the effective interaction is an interaction between quasiparticles, see Falicov [23].)

Equation (28,lO) implies that the dielectric function dRPA)[0, u)vanishes as w approaches the plasma frequency up. Consequently, it follows from equation (28.6) that the response function II(RPA)of the electron gas in this case becomes infinite in this limit. This means that an infinitesimal perturbation will result in a finite change in the ele~trondensity. This concept is connected with an eigenrnode of the system, in this case a plasma oscillation. At this point, we have the uppor tunity to elegantly calculate the dispersion relation w ( q ) of the plasmons. We set

This yields, for srnalI p,

[ +5

lim w ( ~ zi) w~P2 I 5-0

3(k~$)'*~]

We close this chaptm with a discussion of yet another way of partially summing the perturbation expansion: the dressing of vertices. We discussed in Chapter 25 how the line can bc dressed by successive insertions of self-energy insert ions. The corresponding approximations for the self-energy, for example t hc one shown in figure 28.6, then yields an equation to be salved self-consistently. We c a n use this idea to simultaneously dress the interaction lines with polarization insertions; for example, from figure 28.6, we obtain figure 28.7. Note that dressing the interaction line in the direct term would he i ~ l c ~ r t c csince t, it would lead l o double-counting. Since the effective interaction in general is time-dependent, this results in a time-dependent equation to be solved self-consistently.

EFFECTIVE INTERACTION

Figure 28-6 Self-consisteht Hattree-Fock approximation fur the self-energy M.

Figure 28.71n the Hartree-Eock appfoxlmation for M , the interaction line of the exchange term can also be dressed, in additioa to the Green's functions.

It is easy to erroneously believe that the approximation for M depicted in figure 28.7 yields the exact self-energy if all possible self-energy insertions and all passibIe polarization insertions on the interaction line are included, However, the diagrams in figure 28.8, for example, are not obtained in this way. Such diagrams can also by summed formally. We first define the so-

lo1

Ibl

(cl

Figure 28.8 Diagrams (a), @i) and (c) canhot be obtaihed only by dressing interaction lines and Green 's fun cfions.

called vertex insertion as a diagram with two ~ ( O I - l e ~and s on interaction leg. Such diagrams can clearly be placed at a normal vertex and in this way ' d m s the vertex'. Examples are shown in figure 28.9- It is clear that, for example, the diagram figure 28.8(a) can be obtained by the following replacement shown in figure 28 .lo. W e now define an irreducible vertex insertion as a vertex insertion which has no self-energy insertions on the incoming and outgoing ~ ( ~ I - l i nand e s no polarization insertion on the incoming and ootgoing interaction Iines, Correspondingly (writ ten for translationally invariant systems)

PERTURBATION THEORY

345

Figure 28.9 Examples of diagrams which can be included in the dressed vertex.

Figure 28.10 The diagram figure 28.8(a) can be constructed by a vertexinsertion into a first-order Green's func t i m diagram. irreducible vertex function

A(k, q)l the surn of all irreducible vertex insertions with the corresponding indices an the external legs

(see figure 28.11). Since each vertex insertion has two ~ ( O I - l e ~and s one k+q

Figure 28.1 1 The irreducible vertex function A(k,q) . interaction leg, we can someLimes combinr all vertex insertions which differ only in the self-energy and polarization insertions, respectively, on the external lines by dressing the Green's functions and the interaction lines:

surn over all ve-rtex inser tians =

w

EFFECTIVE INTERACTION

346

In particulas, we have the exact equations

and

Even though these two equations look unappealingly asymmetric, it is clear from their derivations that the vertex function A can only be at one end; otherwise, some diagrams will be counted twice (see figures 28,12 and 28.13).

tn 1

t bl

tcl

(dl

Figure 28.12 The diagram (a) is included in the diagram (b), but also in djagrarn (c) in the form shawn in (dl.

F i g u ~ e28.13 The diagram (a) must not be jncludcd in the direct term (b), since vertex diagrams are included, as shown i n (c).

Of course, the irred~lciblevertex diagrams can also be reduced t o a few skeleton diagrams; examples are shown in figure 28.14. Again, self-consisterlt approximation can be derived through approximations such as the Ohe shown in figure 28.15.We realize what an enormous number of diagrams we obtain if we on the right-hand side insert only the lowest-order approximations for u , ~G , ~ 1 1 dA,

PERTURBATION THEORY

Figure 28.24Examples of skeleton diagrams of the irreducible diagram.

Figure 28.15 Self-consistent approximation fm the irreducible vertex

In many problems, t h e so-called ladder diagrams (see figure 28.16) give thc dominant contribution. These diagrams can then be summed.

Figure 28.16 A vcrtex ladder diagram.

We have how learnt several different ways af pariial snmmation. Which way is best depends on the particular problem, especially on the nature of the interaction.

Chapter 29

Perturbation theory at finite temperatures W e will in this final chapter of Part TI1 briefly indicate h m the formalism discussed up t o this point has t o be generalized in order t o be applied to temperature-dependent systems. The starting point is the so-called ensemble avemge t o calculate equilibrium expectation values of observables of the system under consideration. Let ( !Pi} be the exact states of the system, with energies Ei

I

=&

1%)

and particle numbers Ni Let P; be the probability of finding the system in the state 1 9 ; ) .The expectation value of an arbitrary operator A is then obtained from the ensemble average ( A ) Cpi(*i1 A 1%)

If the system under consideration can exchange energy in the form of heat as well as particles with its environment, one uses the grand canonical ensemble. In this case, the probability P; depehds on the temperature T and the chemical potential p of the system, and is obtained from the fallowing well-known expression;

where k~ is Boltzmann's constant. We now define the so-called Bltreh density operator as

FINITE TEMPERATURES

350

The index G stands for 'grand canonical'. We have

hence

with the trace taken over any complete basis. The denominator has a special name; it is called the grand canonical partition function:

W e denote ensemble average of the non-interacting system with the Hamiltonian H = Hg by the subscript 0:

As an example, we calculate the number of particles with momentum k, i - e . , the momentum distribution function, for a system of non-interacting free fermions. The eigenfunctions of H D are Slater determinants I @;) of plane waves with

=

n(k) I a;) n; (k) 0 if the momentum stale k is unoccupied in 1 a;} 1 if thr: momentum state k i s occupied in ( a$). =

i

We calculate the individual terms in this exprasion:

PERTURBATION THEORY

35 1

The sum over all configurations I a;) can, by an arbitrary but fued renumeration of the momentum, be written as

We then obtain fbr the denominator of equation (29.5)

1

5

1

C C

. * . e-

p ( E y ~ - P ) ~ i ( l ) e - ~ { ~ ~ (2)) -I .~. ~ ~ ,

The numerator of equation (29.5) becomes

We combine these two expressions to arrive at

This is the well-known Fermi distribution function. We will hereafter abbreviate it as (~b(k]) 0 fk.

=

352

FINITE TEMPERATURES

The decisive observation for the pertu~batianexpansion at finite temperatures, which we will arrive at later, is that PC; satisfies a Schrodinger-type of equation, namely the BIach equation

If we make the substitutions

the asual SchrGdinger equation. This similarity makes the construction of the formalism possible in parallel with the zero-temperature theory, First, kre define a 'Reisenberg picture' by we obtain

The two operators transformed in this way are in geaeral not unitary. They are, however, each others' inverse, so that the transformation of a product is the product of transformations. The fact that the transformation operators in general are not unitary has as a consequence that a transformed creation

in general is no2 the adjoint of the transformed annihilation opetator

satisfies the It is important to keep in mind that (li)tlHand not canrsnical anti-commutation relation with the annihilation operator $ H . We now define the temperature-dependent single-particle Green" function for fetmions through the ensemble-average :

The aperator T is defined as before; it ordcrLsthe field operators according t o increasing T and multiplies with the sign of the hCCessaTy permutation. In this definition, as also in the Blorh-equation, no factor of i appears. From the Green's function defined in this way3 we can calculate (1) the (ensemble) expec tatirrrl value of atly single-particle operator, and

PEBTURBATION THEORY

353

(2) the internal energy E ( T , p ) = ( H ) of the system, and thus also the expectation value of the two-particle interaction.

An interaction picture can also be introduced by sepa~atingHo - pN from H-PN: Ir'H-p3V=(Ho-pN)+V=Ke+V and defining

(29.10)

-~e Arorage-h>os O ( T )=

The Heisenberg picture and the interaction picture are related by

where

= e+KO

~ i ( rq l ,)

TI

,-A"(T~-~),-KO~

(29.12)

The operator U defined in equation (29.12) is not in general unitary, but we have as before

One can easiiy show that U satisfies the following equation of mation (wc will not perform the details of this calculation; it runs entirely parallel with the discussions in Chapters 14 and 15):

with the formal sohtion

The representst ion given by equatiorl (29.14) is the basis for t h e perturbation ~xpansionfor temperatures T > 0. For example, from this equation wc ohtain for the partition function

The trace can be elegantly evaluated in the complete basis of eigenfunct ions of the unperturbed system with the result

Thus, the part ition function ZG corresponds to the vacuum amplitude of the T = &formalism; just as the vactlum amplitude, ZG stands in the denominator of the Green's function, and gives rise to disconnected diagrams which cancel out with the disconnected parts of the diagrams in the numerator. A direct evaluation of the numerator of the Green's function yields

We note that the integration intervals here, as we11 as in equation (29.16)) are

Eo,Plb Ih contrast to

the usual (temperature-independent) Green's function, we

have t o determine not a matrix element of 'tirnel-ordered products in each term, but, as we see in thc representation equation (29.16) of the dehorninator, we have t o determine a trace, 1.e-, an infinite sum of such matrix elements. Wick's thmrem in the form that we know it will not help us in the evaluation of these rrlatrix elements, since t h e normal-order products only vanish when they act on the ground state ( aO); in the trace, aII states I Qi) appear, We can, however, find a generalization of Wick's theorem, which will help us, The idea is t o interpret ZG and the numerator and demomi-

nat or of the of the Green's function individually as chscmblc-averages with respect to 0) = e x p [ - P ( H O / I N ) ] . 1.0 do this, we expand 1/2g1and

pL

obtain

-

PERTURBATION THEORY

355

In Chapter 19, the normal-order and contraction af operators were introduced to evaluate the matrix elements. T h e contractions became c-numbers; in pa~ticularwe had that the expectation value of norrnd-ordered products vanishes ( h e ~ writtern e fm translationally invariant systems'):

We>use this property as a model far defining the contraction in the finitetemperature theory:

These

definitions are extended lit~early.For exampIe, we set

Furthermore, we h&ve

These ensemble averages arc obtained, as was the example of the momentum distribution function, djrectly from the definition equation (29.4) of ( }0 as weighted averages of expectation values relative to Slater dete~minants. As in Chapter 14, we find that

We can then c$lcula.te the contraction defined in equation (29.18'):

F I r n TEMPERATURES

and from equations (29.21) and (29.22) it finally follows that

This is the temperature-dependent analog af the momentum Green's function equation (15.23). Correspondingl~t,we have for the space-dependent Green's function

We callnot pr0ve.a Wick's theorem as an operator identity for temperaturedependent systems with these results, but we have nevertheless the following statement for thc cnsemble average [24, 251:

(T

[. .. ck(,) . -

. .I

c:, (.) . )o

=

surn of all completely contracted terms.

This statement is saficient to construct a diagrammatic pertrlrhation expansion with the same diagram and almost the same translation rules as before. Only t hc prcfactors change, sihce no factors d i appear. We find an interesting difference in the evaluation af the diagrams: For temperatures T > 0 each state 1 k) is occupied with a certain probability fk and unoccupied with a certain probability (1 - fk). A principal division in particle and hole states is no longer possible, since each state t o a certain extent is both. For this reason diagrams which vanish at zero ternperatu~es(in t rarrslatio~lallyinvariant sys terns) give non-zcro contributions; see, for example, figure 29.1. Such diagrams are frequently called an omalozls diagrams. All questions that were discussed in the zero-temperature formalism cart also be investigated for systems at finite temperatures. Additional interesting applications include

PERTURBATION THEORY

Figure 29.1 The following diagrams do not wntribute to the ground state energy, but they do contribute to the grand canonical potential at aon-zero ternperat ures. - the summation of

the ring-diagtam contribution to ZG = ZG(T,p).

One obtains an (approximately valid) equation of state

for a dense electron gas; - the derivation of the Hartwe-Fock equations for systems at finite temperaturea through the approximation

of transition temperatures, such as the temperature of the phase transition into a superconducting state;

- calculation

- the general qestion of finite-t emperature earrec tions to the zero-

tempe~atureresults.

We will not here further prlrsrle the possible uses of the finite-temperature formalism, but only address one remarkable point, which puts the usual perturbation theory for T = 0 in a somewhat different light. In many cases, the contributions of the anomalous diagrams do not go t o zero, as one would expect, in the limit T --+ 0. The question then arises of which result gives the correct result for the ground s t a t e energy of the system: the earlier zerotemperature result or the limit T + 0 of a finite-temperature calculation, The answer is relatively simple. The original zero-temperature theory is rrlarred hy a particular uncertainty. In the proof of the Gell-Mann-Low theorem, we could not show t h a t the adiabatically generated state is the

FINITE TEMPEXGATURES

Figure 29.2 The ground state energy at zcro temperature may not evolve adiabatically. ta the ground state of the interacting system.

ground state of the full Hamiltonian; it was only guaranteed, that it is an eigenstate. Hence, if the spectrum of eigenvalues is exhibited as a function of the coupling constant, one cannot expect that the ground state of the full problem is obtained by turning on the interact ions adiabatically; instead, one may end up in an excited state (see figure 29.2). With the help of examples (see far example [26])one can show that in such cases the T 4 0 limit of the temperat ure-dep endent theory gives a st ate with lower energy. Thns, the temperature-dependent theory is really the correct one, since it does not involve any adiabatic turning-on. The integrals that appear in this theory have the structure of exp(-. . .), ie., they are, in conttast ta the earlier integrals with the stkucture exp(-i . . .), cdnvergent without any additional convergence factors. IIowever, in quite a few cases, for example in systems with spherical Fermi surfaces, the results from the two different theories do not differ.

Part IV

Fermi Liquid Theory

Chapter 30 Introduction The ground st ate of a homogeneous, translationally invariant system of fermions has a well-defined Fermi sbrface a t chemical potential p . It is useful t o define a Fermi temperature TF by

This temperature sets a scale for the excitations in the system. For a homogeneous non-interac tihg systern of density n in three dimensions, the Ferrni temperature is given by

where EP is the Fermi energy, rF = k $ / ( 2 r n ) . Some physical systems consist of fermions at temperatures much lower than the Femi temperature. Such systems are said t o be degenerate. One example of such a system is 3 ~ ate temperatures well below its condensation temperature, btlt well above the temperature at which 3 ~ becomes e a superfluid. Another example is given by the conduction electrons in a metal at room temperature. These form a degenerate fermion system because the Fermi temperature is very high, of the order of l o 4 K. The physical proper ties of a degenerate non-int eract ing Fermi system ate usually treated in graduat e-level textbooks in st atistical mechanics, These properties are essentially determined by the statistics of the particles - the Pauli exclusion principle prohibits occupation of any single-particle state by more than one particle. In rcealdegenerate fermioh systems, there are very strong par t i c k interactions, frequently of the order of k B T ~Therefore, . one may expect that the properties af a real degenerate fermion system will differ considerably from

362

INTRODUCTION

those of an ideal non-interacting degene~ateferrnion system. It is, however, a remarkable fact that the properties of seal degenerate systems; are qualitatively very similar to those of non-interacting ones. The reason f o this ~ is that in both interacting and non-interacting systems, the physical properties are determined by low-lying excitations, and at low temperatures there are only a few such excitations present. This is due to the severe restrictions imposed by the Pauli exclusion principle on the available phase-space for low-lying excitations. Hence, there is a strong correspondence between the nature of the low-lying excitations in interacting and non-interacting degenerate fermion systems, In the interacting system, however, many properties of low-lying single-par ticle excitations, such as mass, are typically m e r e n t from those of the non-interacting system. This effect, which is due t o the strength of the interactions,is called renormalization. There is also a residual interaction between these excitations in an interacting system, but this ~ e s i d u dinteraction only quantitatively modifies the values of macroscopic properties even though it may in fact be very strong. Degenerate l'errni systems are we11 understood in t e r m of a theory put forward by Landau in 1950 [27]. This theory applies to no~rnalFermi liquids. These are interacting translationally invariant isotropic ferrnian systems, such as liquid 3 ~ ine the temperature range discussed above. The theory may also be applied with minor modifications to the conduction electrons in a metal. The madifxcations account for the fact that the conduction electrons in a metal do not constitute an isotropic translationally invariant system, due to the crystal lattice of discrete ions, The theory i s a semiphenomcndogical one, in the sense that all properties arc determined in terrrls of a few parameters. These parameters can be determined from a few experiments and then uscd to make new independent predictions. This is the course followed by Landau whcn he introduced She theory and shortly thereafter used it to predict thc existence of zero sound in 3 ~ e The . theoretical hasis for thc Landau Fermi liquid theoty was rigorously demonstrated by Luttinger and Nozikres in 1962 [28, 291, when they verified the theory using infinite order perturbation theory. In other words, the theory is valid in any system where perturbation theory converges. The microscopic derivation by Luttinger and Nozikres also showed how the parameters in the theory can be calculated directly from first p~inciples.Such caZcuIations have been performcd for, for example, simple metals. One of the most extensive calculations to this date of Fermi liquid parameters for various metals was perforrncd by Rice in 1965 f30j . Through the years since the introduction of Landau's Fermi liquid theory, the fundamehtal concepts of the theory, such as the existence of a FerrPli surface and quasi-particles, have become cornerstones in condensed matter physics for the understanding of Fermi systems, Recent developments in condensed matter physics have again shown the importance of these concepts. One such development is the so-called Klondo effect, in which a minimum in

FERMI LIQUID THEORY

363,

the resistivity of met a I s with a smdI concentration of magnetic impurities is observed at low temperatures. A complete theore tical understanding of the Kondo effect did not emerge until after the develapment of renormalization group theory. Nevertheless, the low temperature behavior of the Kando effect could be understood by the application of Fermi liquid theory 1311. We will discuss the application of Fermi liquid theory to the Kondo problem in a later chapter. Another more recent area is the physics of high-T, superconductors. The materials that exhibit high-T, aupe~conductivityall contain planes of copper oxide, and these planes are separated by, for example, rare earth atarns. At the present, the consensus has emerged that all the interesting physics happens in the essentially twedimensional layers of copper oxide. The model that is frequently used to describe this system is the secalled Hubbard Hamiltonian. Not even the normal prope~tiesof this Hamiltonian are cornpletely understood at the present time. However, there haw been some very interesting developments in the theory of the normal properties of these materials, which may be a first steps towards a complete theory of high-T, super conductivity. In these developments, attempts are made t o invest igate to what extent the normal state af the system can be described as a Fcrmi liquid. It is argued by some researchers that this state of the Hubbard Hamiltonian is a somewhat exotic variation of a Ferrni liquid, termed a marginal Ferrni liquid. The essential point far our illustrative purposes here, however, is that it is the concepts from the Landau Ferrni liquid thcory that play a major role in these developments [32].

Chapter 31 Equilibrium properties We consider first a tr anslatiozlaIlg invariant system af N non-interacting fermions in a unit volume. For a translationally invariant system, momentum k and spin o = ff are good quantum numbers. The many-body wavefunction of the system is a Slat er-de terminant of single-par tide states yk,@with energies r i0) = k 2 / 2 m , where rn is the mass of the particles. Any eigenstate of the system may then be completely described by a distribution function nk,, , which gives the occupancies of the single-particle states. In the ground state, this distribution function is nk( 0 ), given by

and the ground state energy of this N-particle system is

+

Let us add one particle to the system. The ground state of this ( N 11)particle system is obtained by placing the additional particle in a momentum state of momentum lkl = kF. The difference between the ground s t a t e energy Eo(*+') and Eo( N )is

where p is the chemical potential af thesystem. Since the additional pastick was added on the Fermi surface, we have

366

EQ UILIBRIUM PROPERTIES

The value of p depends on the temperature, but for degenerate systems the deviation of p from EF is of order ( T / T ~ ) which ', we will ignore. The elementary excitations of the non-intetacting system are particles and holes. These excitations are obtained by adding a particle with k > kF or removing a particle with k < k ~ The . thermodynamic properties of the system are determined by these elementary excitations, so it is convenient to have s measure of the departure of a state of the system from its ground state. Such a measure is given by the quantity 6nk,, , defined by

This quantity measures the difference in the occupation numbers of the single-particle states from their values in the ground state and thus describes the excited states of the system. For example, ank,, = 6 k , k t 6 , , 1 with k > kF corresponds t o the excitation of a particle, and 6nt,, = -6k,kt60,s, with k < kF corresponds t o the excitation of a hole. The internal energy of an excitation is then readily expressed in terms of braka and is

It is clear that at nun-zero but low enough temperatures, such that kBT < p , only particles and holes near the Fcmi energy will be excited thermally, so Snk,, is of order unity only in a narrow region with thickness of order k ~ T / l near r the Fermi surface and zero otherwise. It is then convenient to measure the energies relative ta the chemical potential, which can he done by using the free errePgy F. This is the appropriate thermodynamic potential of a system iri contact with a particle reservoir, so that the number of particles is allawed t o fluctuate, but with the restriction that the average number of particles be fixed at N . This free energy is defined as

and the free energy of an excitation is ,thus

We now turn to a real, interacting system. We assume that the SYStern is isotropic and translationally invariant. 'I'hesc assumptions are made for convenience - the theory can be formulated for anisotropic and nontranslationally invariant systems (see [33, 341). As a consequence of the first assumption, the Fcrmi surface of the interacting system is spherical for reasons of symmet~y. By a theo~emdue t o Luttinger [35], the Fermi surface of the interacting system must enclose the same volume as a carresponding non-interacting system with tJhe same par tide density so that

FERMI LIQUID THEORY the values of kF and p do not change as the interactions are turned on. As a consequence of the second assumption, momentum k and spin a are good quatltum numbers and can be used to label excited states of the intetacting system, precisely as was the case with the hon-interacting system. This enables us to generate quasi-particle states of the interacting system from those of the non-intefacting system. The mechanism that allows us to do so is the adiabatic generation of the interacting eigenstates $J~,from the nan-interacting ones. We imagine that the coupling constant for the particle-particlec interactim can be tuned continuously from zero t o its final, real strength. We then assume that there is a o n e - b o n e correspondence between the excited states of the non-interacting system and those of the interacting system in the following way. We start with a particular quasi-particle state n t , o of the non-interacting system. We then slowly (in a sense which will be made mare precise later) turn on the interactions, In this way we will then generate a quasi-particle state of the interacting system. For example, a non-ifitera6 tihg s t ate with a single-particle excitation nk,, = n p ) bklr,boTol will in this way generate a single quasi-particle excitation of momentum k' and spin a' of the interacting state. In this way, we have a one-to-one correspondence between the excited states of non-interacting and of the interacting systems, and from a state of the noninteracting system containing 6nk,, quasi-pa~ticleswe generate a st at e of the interacting system containing dnk,, quasi-particles. These are precisely the quasi-particles discussed in Chapter 26.

+

For this adiabatic generation of the excited states to be valid, we must assume that all eigenstat es of elcmentary excitetions of the interacting sys tern can hc reached in this way, This holds for the ground state of the intctacting system, provided the requirenlcnts of the Gell-Mann-Low theorem in Chapter 18 are satisfied and if, for example, the Fermi surface is spherical. An example of when all eigenstates of the interacting system may not be generated adiabatically is provided by a system with attractive interactions. In this case t h e interacting ground state can be a superfluid liquid. Also, the adiabatic generation of excited states can only be valid for low-lying excitations near the Fermi-susface. The reason for this is that the adiabatic procedure must obviously take place in a time shorter than the life-time of the excitations - otherwise we would not end up in a well-dcfined excited state. Rut we have already seen that the life-time of an excited s t a k of mmenturn k goes as [(k - k F ) / k F ] - 2 ,so if the momentum of the excitation in the non-interacting is too far from the Fermi surface, we may not be ablc t o generate the emresponding interacting excited state adiabatically. This limits the validity of the Fermi liquid theory t o excitations in the immediate vicinity of the Fermi surface.

The free energy of the interacting systern can be expremed in terms of nk,s in particular., it can be expressed i n the deviation 6nk,, from the

EQUILIBRIUM PROPERTIES

368

ground s'tate quasi-particle distribution function nk(0):

If the deviation is small, in a sense which will be made more precise shortly, we can expand the free energy in a power series in fink,, . To second order in the deviation, we obtain

In the first summation an the right-hand side of equation (31.11, € I , - p is the free energy of a single quasi-particle of momentum k and spin o. In the presence of other quasi-particles the interactions bet ween quasi-particles change this free energy t o

It fbllclws immediately from the adiabatic generation that only om qu'asiparticle can occupy each avaiIahle quasi-particle state. By applying statistical mechanics to a gas of quasi-particles, we can then calculate the probability f (c) that there is a quasi-particle with e n e r a r in the system. This probability is

1 f ("

)=

ep(c-')

+1

where = I / k B T , As T + 0, this e x p r ~ s i u ntends to a step function at t = p . At finite temperatures, the energy c will depend on the temperature since thermally excited quasi-particks will affect the cnergy E through equation ( 3 1.21, and f (r) becornes a very complicated function. Note that, as a consequence of the fact that each quasi-particle state can be occupied by only one quasi-particle, the quasi-particles are distributed according to the Fermi-Dirac distribution function, and we can treat them as fermions. It is not at all trivial t o actually prove that the quasi-particles are fermions and this may not even be the case. However, for our purposes here, it suffices to know that we can treat them as ferrnions. The quasi-part,icles are close t o the Fermi surface because of the restriction on k - k F . We may then expand t he energies rk about the Fermi surface. To lowest order in ( k - kF), the expansion can be written

where the efich've

mass m* of a quasi-particle

is defined as

FERMI LIQUID THEORY

369

with the gradient evaluated at the Fermi surface. This definition is the same as equation (26.18) for m**. We will h e ~ efollow convention and use the notation m*. The free energy given in equation (31.1$ has the form of the free ehergy of a gas of interacting pa~ticles,the quasi-particles, where the particle interaction is given by the funk tion fk,,k,,f, usually called the Landau f -function. The expansion of the free energy is an expansion in the ratio of the number ank,, of these particles t o the total number N af particles in the system. Thus, the expansion is valid for Ct,, lbnk,oJ< N . This condition holds for temperatures T much less thah the chemical potential of the system. Note that there is no particdar restriction on the strength of t h e quasi-particle interactions. For systems invariant under space inversion and time-reversal [which is the case in the absence of magnetic fieIds) the Landau f-function must satisfy

Furthermore, we will always take Ikl = kF since the excitations are very close to the Fermi surface, The f-function is then a function only of the relative orientations of k and ktand of 27 and u'. Thus, the spin-component depends only on if a and cr' arc parallel or anti-parallel. It is convenient to separate TT and f:, these two independent spin-components fky into spin-symmetric and anti-symmetric parts:

Be-cause sf the isotropy, the spin-symmetric and anti-symmetric parts only dcpend on the angle E between k and kt,so we can expand these components in a series of Tkgendre polynomials

Usually, the coeficiellts f;(") arc expressed in a dimensionless forrrl F;(') by rilultiplying them by the density-of-states at the Ferrni s ~ ~ r f a c~e ,( 0(see ) belnw), so that

The coefficients F,$") then measure the interaction strength relative t o t h e kinetic eneTgy of the quasi- pa^ ticles and completely determine the interaction between thero. The grcat usefulness of the theory lies in the fact that

370

EQUILIBRTUM PROPERTIES

in practical applications, only a few of the parameters F$') and rn* need t o be included. They can then be determined from a few experiments, and new independent p~edictianscan then be made using these values of the parameters. As we have argued before, the physical properties at low energies depend crucially on the number of states available for low-energy excitations. This number is given by the density-of-states at the Fermi surface, ~(0).Let us calculate v(Oj for a Fermi liquid. We know how t o convert a sum over quasi-particle states k,a to an integral:

where g ( k ) is some function of Ikl only. (Remember that the system has unit volume.) Instead of summing over particle states (k,u),we can sum over energy levels. To do so,we introduce the density-of-states v ( E )which , is the number of states pet unit volurne and unit energj. Thus,

rn*dc/<w,

,

= k 2 / ( 2 m * ) , so that d k = m*dc/k By changing integration variable, we obtain

For quasi-particles,

=

With r p = k 2 / ( 2 r n * ) , we obtain for the density-of-states at thc Fcrmi surface:

We will now calculate the specific heat of a Fermi liquid. The specific heat is given by the increase in internal energy due to an infinitesimal increase ST in temperatu~e: h i 3 = C,bT,

To lowest order in ( T J T F )t,h e increase in internal

energy is given by

increasing the temperature leads to an increase in the number of quasiparticles, bnkIp. There is also an increase in t h e internal energy due. ta

FERMI LIQUID THEORY

371

the quasi-particle interactions, but this term is of order ( T / T ~ and ) ~can be ignored here. Hence, the specific heat is that of a non-interactiag gas sf quasi-particles. The increase in internal energy is then given by the the number of quasi-particles within an energy kBT of the Fermi shrface, This number is given by the product of the numbet of the density-of-states at the Fermi surface, v(O), and ~ B T The , energy of excitation is of the order of kBT, which gives the standard result that Cw u - Y ( o ) ~ ~ TAn . exact calculation is st raightfo~wardand completely analogous to the calculation of C;, for a free elctron gas. The result is

The only change in C, in the interacting Ferrni liquid from that of the non-interacting Fermi liquid is due to the change in the density-of-states at the Fermi surface. With the density-of-st ates given by equation (31.6), we obtain

The effective mass rn* can thus be obtained directly from measurements of the heat capacity of the Fermi liquid. Next, we turn to the propagation of ordinary sound in a Fermi liquid, for example the propagation of sound in liquid 3 ~ c These . sound waves are due to vibrations of the quasi-particles, where the restoring force is provided by frequent collisions between them. The speed of sound u s is obtained from thermodynamics and is given by

where P is the pressure. Using thermodynamic Maxwell relations, it is easy t o show

that

dP 3~ =Nan

a~v

so that

a~ of ap using We will evaluate equation (31.8) by calculating the inverse 7 Y Fermi liquid theory. We begin by considering the change dN in N due ta a change dp in the chemical potential, If the chemical potential increases an amount djt, tllc Fcrrrii wavcr~urr~ber irlcrc&es a11 amcnlnt

Quasiparticles on the new Fermi surface must have an energy r ( p which satisfies

4~ + d

+ db)

4 - ((PI= d ~ .

This increase in quasi-particle energy t o r(p f dp) has two contributions. The first one comes from the fact that k~ increases due t o the increase in N , and this contribution is v F d k F . As a result of increasing the Fermi surface, new quasi-particles are added at k~ dkF, The second contribution comes from the interactions involving these new quasi-particles and is Ck,e, fko,kta,6nktgl. Collecting these two terms we obtain

+

Since the quasi-particles are oh the Ferrni surface, we can write

This cqutttion thcn gives t h e change in kF if the chemical potential increases. The increase dN in thc nr~mberof particles within the expanded Ferrni surface is just

so that

a;E

In an isotropic systenl, the Ferrrii gnrface is spherical and so a k is independent of direction. We perform the integral aver k' in equation (31.9) over t h e Ferrni surface:

-

-

ak; J~~.,~.wb(k' - kr)Kkr2 cas E d k ' d ~ PC 8~ 23 ( 2 ~ ) ~ u'

k3,

FERMI LIQUID THEORY

Here, we used the fact that k also is on the Fermi surface, and in the last line we used equation ( 3 1 3 ) . The result of this integration in equation (31.9) yields a kk -~ 1 V dp l+F:' With this result inserted in equation ('31 , l o ) we obtain

Integrating over k and summing over a,we obtain

If we insert the expression equation (31,ti) for v(O), and also use the relation between the density N (remember that the system has unit volume) and kp, N = 1;;/(3x2), we finally obtain

The interactions thus enter into the speed of sound in two pIaces. First, the density-of-states at the Fermi surface, v(0), is modified by the effective mass mY,and second, the interactions enter in t h e parameter F i . For example, if the quasi-particle interaction is repblsive, the speed of sound increases due to thc positive F i . The parameter mv can be obtained ftom specific heat measurements, so measurement of the speed of sound will yield Ft . Far example, for liquid

EQUILIBRIUM PROPERTIES

374

3 ~ ate temperatures where the Fermi liquid theory applies, F( = 10.8 at a pressure of 0.28 atm. < -1, the system is unstable, because density AuctuaNote that if tions will build up exponentially, Finally, we will calculate the Pauli spin susceptibility x p of a Fermi liquid. In the pr'esence of a uniform external magnetic field H, the quaiparticles acquire an additional energy -gfi3uH, where g is the Land4 factor, which we take to be 2, p~ is the Bohr magneton and a = f$. In equilibrium the system must have a single chemical potential p . As a consequence, the quasi-particles with spin - (which have their energies increased) must have their Ferrni surface reduced by an amount SkF, whereas the Fermi surface of the spin ++quasi-partides must increase an amount 6kF. This will modify the distribution nk,,of the quasi-particles. The change 6nk,, in the quasi-particle distributian function is 6nk,, = + 1 S n k , = -1

for a = + $ and 1

for a = -7 and

kF< k

kF

< kF+6kp

- 6kF

(31.12)

5 k 5 kF

In turn, these changes in the quasi-pa~ticle distribution will modify the quasi-particle energies through the quasi-particle interactions. The relation between hk,, and the change #fk, of the quasi-particles is

We look for a solution for Stkr, of the form

With this ansaiz, the change 6 k in~ the Fermi surface Is

Using equation (31.12), w e can w r i t e equation (31 -13) as

where k and k' are on thc Permi surface kp and dr' is tlie element of solid angle. If we now insert equation (3 1.15) and also use eqv ation ( 3 1.5), we ol>tair~

FERML LIQUID TI-IEORY

Inserting the a n s a h equation (31.14) in equation (31.17) yields for q

The magnetization M of the system is

'I%e Pauli spin susctlptihility ~p is then

The eRec t of the interactions is again a renormalization of thc rcsponse of the system, here by a factor (1 Ft1-l (apart from the change in the density-of-statm at the Fermi surface). Moreover, we note that if F t < -1 the susceptibility becomes infinite, which signals an inst ability of the spin system; for example, the system may bccome ferromagnetic.

+

Chapter 32 Transport equation and collective modes We will now turn our attention to the c s e where the system is nan-uniform. This would be the case, for example, if an external field which varies in space is applied ar if there are internal long-range fluctuations. We restrict ourselves to the case where the system is only weakly non-uniform. The idea is then to imagine that the system is divided into srrlall sub-units, each of which is in local equilibrium. In each sub-unit, we can then define a quasi-particle distribution function nk,,(r, t ) , where r is the position vector of the sub-unit. Of course, for this to makc scnsc, t,hc spatial variation af t h e system must be on a length scale much larger than this size of each sub-nnit. We are only interested in linear resparlse and write DL,*(r,l )

= nf)

+ bny,, (I-,l ) .

We carr simplify further hy Fourier transforming, Since we are only interested in linear response, we can then study the response of each Fourier component q separately, and write

It is tempting to interpret the quantity 6nk*, (r,t ) as the probability of finding a quasi-particle (k,a) at position r at time f . However, this interpretation is riot quite correct. Indeed, attempts to interpret the qomi-particle distribution function in terms of localized wave-packets of quasi-particles will lead to unexpected effects (see [36]). We shall find later that dnk,,(q,u) can bc intcrprctcd as the probability umplatzsde of finding a quasi-particle-quasihole pair of total momentum q. This will become apparent when we derive the Landau theory from infinite-order perturb ation theory. We shall f hen also find that the precise limits of validity on t h e thcory arc

TRANSPORT EQUATION

These conditions then say what one would intuitively expect - that the wavelengths of the disturbances of the system must be much larger than the Ferrni wavelength, and that their energies must be much smaller than the Fermi energv. These conditions will ensure that only quasi-particles in the immediate vicinity af the Fermi surface will be involved in the excitations. With these preliminaries behind us, we can then proceed to expand the total free energy of the system. This expansion now has the form

Again, we will restrict ourselves to translat ionally invariant systems, Tn this case, the quasi-particle energy does not depend on P and is simply c k , and the interaction function fk,,y,,(r,rl) depends on r and r' only through (r-r'). Moreover, the interaction function represents the short-rangeforces between the quasi-particles,If we are considering neutral systems, this is obviously true. It is not clear if we are considering charged systems, such as the conduction electrons in a solid. However, in such systems, the trick is to scparate out the long-range part of the Coulomb interaction between the quasiparticles, which leads to a term in the energy, which is treated separately . This part is the direct Hartree intetaction (not shown here). The rernainder of the Coulomb intmaction are then short-ranged exchange-correlation interactions, which are described by an interaction function fk,kt,r. So in either case, f k - r is short-ranged, with a range of thc nrder of an atomic length. We can then replace this by fkr,kr,,d(r - r') inside the integral. With all these simplifications, the total free energy is then

Frorri this

wc S C : that ~ a

quasi-particle (k,a) at a point r has an energy

This energy now rarics both with momentum k and with respect to position r, so Vkckp(r) and Vrrk, (r) are both non-zero. Thc first of these gradients,

FER MI LIQUID THEORY

379

Vktka(3'; is just the local velocity of the qu-i-particle. The spatial gradient, however, represents a diffusion force, which tends to push the quasi-particle t o a region of low energy, Landau derived the transport equation for quasi-par ticles by treating t h e m as independent particles described by a classical Hamiltmian ck, (P). The time-development of the distribution functim is then given by the explicit time-derivative plus the Poisson bracket of the distribution function with the classical HamiItonian:

where a = x,y, s denotes Cartesian components. In this expression, howevet, we have ignored the rate of change of nk,, (r, t ) due to collisions between the tpasi-particles. R y a procedure which is well-known in the theary of the Boltxrhann equation (see, for example [37]), we can insert such a term by hand, so equation (32.2) becomes

-

(32.3) Here I ( n ) is a collisicrn integral, which is a functional of the quasi-particle distribution function. The collision integral rncamres the rate of change of nk,g(r,t ) due ta collisions. Quasi-par ticle collisions are not important for freq~lenciesw larger than the typical quasi-particle collision frequency v and I ( n ) can be droppcd in these oases. The co1li~;ionfrequency is proportional to the square of the I I I I ~ I I ) ~of~ quasi-particles, which is proportional to ( T / T ~ ) for ' low temperatures.

The quasi-particle distribution nk,, is meaningless other than on the Fcrmi surface. Therefore, we want the transport equation in terms of the physically sensible quantity 6nk,,(r, t ) . The assumption that the distribution function deviates very Iittle from its homogeneous equilibrium valuc n f ) allows us to cast thc transport equation in term3 of the equilibrium value of the distribution function plus corrections linear in the deviation from eq~iilibriurn.This gives an equation linear in $nk,,(r, t ) , The result is

TRANSPORT EQUATION

Out first application of the transport equation (3Z4) is to calculate the particle current density jk in a state containing a quasi-particle 6nkg,. Since the total number of particles is conserved, the continuity equation

must hold, with J the total (particle) current density. We can obtain this equation by summing the transport equation (32.4) over k and a, The result is

(32.6)

The right side of equation (32-6) vanishcs since the collisiorls conserve thc number of particles. Cornparing equation (32 -6)with the continuity equation (32.5), we see that

is the current density carried by a system containing a quasi-particle (k,cr). Similar expressions can be obtained for the energy current Q and the momentum current density napby multiplying equation (32.4)by ~k or k and integrating while making use of energy or momentum co~lservation. The equation (32.7) enables us to derive a relation between the effective mass m' and the f -film c t ion for translat ionally invariant systems. Consider a non-interacting system in a statc containing one quasi-particle with momentnm k. The current carried by this s t a t e is thcn

Now t l ~ r non thc interactions adiabatically, This takes the non-interacting state t o an interacting state containing one quasi-particle with lnomcntum k. Since the system is translationally i nvafian t , the momentum is conserved. The current carried in the interacting state is then d s n $. Cornlr~ringthis

FERM1 LIQUID THEORY with equation (32.7)we obtain

We may, without loss of generality, take k parallel to 2. It is then simple to carry out the integrations in equation (32.9):

-

x k'b

'

an,,101 fkr,kht

Uk',z

=

z

f k m , ~ l O ~ 6 ( fk r/1)vkttz ktd

This result inserted in equation (32.9) together with

vk

= k/m* yield

the

relation

Equation (32 .lo) give important additional information about the quasiparticle interactions. If the effective mass m* is known, for example from specific heat measurements, the parameter F[ can be calculated. The first application of the transpart equation (32.4) was made by Landau when he predicted the existence of a new col2ective mode in liquid 3 ~ e . This mode was detected a few years after Landau's prediction, The mode is a collective oscillatian of the quasi-particle gas where the restoring force comes from the interaction fk,,kt,t between the quasi-particles, and has a dispersion similar to ordinary sound waves in that the energy of the mode is proportional t o the wavelengthof the mode. Therefore, the mode was named 'zero sound'. In an ordinary sound wave, 01, the other hand, the restoring force is by frequent adiabatic collision between the quasi-particles, and these collisions restate local equilibrium. Hence, for the ordinary sound waves to propagate, the frequency w of the mode must be much less than the callision frequency u. In contrast, the collision frequency must be much less than w for the zeto-sound modes, so that collisions do not affect the ascillation. Since the nolliaion frequency goes as ( T / T at ~ low ) ~temperatures, this is always the case fox sufficiently low temperatures.

To analyze the transport equation for collective modes, we first set i(q.r-wt)

Snk,m(r7t ) = bnk,ce

With this form of 6nkqo(r,t ) , the transport equation (32.41 becomes

We simplify this equation by introducing polar and azimuthal angles (0, p) relative ta q and by introducing a dimensionless quantity s defined by

The collective mode is an oscillation of the Ferrni surface. Hence, we ihtroduce the displacement u(ku) of the Fermi surface at the point k on the Fermi surface; in t e r m of u(krr), we have 5nk, = 6 ( f k - p)vFu(ka),s o the transport equation then becomes

where dyt is the elemtsnt of solid ahgle, 5 is the angle between k and k', and F is the reduced interaction function

Equation (311.12) is a homogeneous linear integraI equation for the displacement u(ku) of the Fermi surface due t o the collective modes. The equation is an eigeuvalue equation, and it can be shown that the eigenvalue 8 may only take discrete values. It follows that the energy w = s q v ~of a mode is linear in q. This is a consequence of the short rangc of thc quasipart ide interact ions. For long-range int eraclions, such as the Coulomb interaction, the frequency is 'lifted' and is hon-zero at infinite wavelengths, The reason for this is that a local variation in the quasi-Particle dcnsity, due t o the zero-sotlnd nlode, will 'res111tin a local accunmlation of, charge arid macroscopic electric fields. This is the case for the plasma oscillations of an electron gas, which a t e precisely thc zero-sound mode of the interacting electron gas, discussed in Chapter 11. The frequency of plasma oscillations becomes u p = 4sne2/rn* as q -P 0, Eqrraticlrl (32,123 has many different solutions. Distinction is usually made be tween two main categories, those modes for which thr: different spin orientations oscillate i r l phase, and those for which the spin orientations are

FERMI LIQUID THEORY

383

out, of phase. The latter are called spin waves. The zerusound mode is the simplest in-phase mode. In this case, only the symmetric interactions F 3 matter. We make a further simplification by assuming that only the constant term F , need be included. With these simplifications, equation (32.12) becomes

The right hand side of equation (32.13) does not depehd on the angles ( 8 , p), must be of the form so the solution for u(B, cos 8

2110,

$40: a - cos 8

'

Inserting this in (32.13) and integrating over solid angle we obtain the dispersion elation. The result is

-sl n -s- l+= l- 4 2 s-l

1

I;b"

From equation (32.14) we see that the Fermi surface has an egg-shaped deformation for a zero-sound mode, The deformat ion becomes smaller and smaller as F; vanishes. In this so-called weak-coupling limit we have s = 1 with v = u ~ to, be contrasted with the sound velocity v, = uF/&. On the other hand, as the coupling Ft grows, the strong-coupling limit of s is s = We also see from equation (32.15) that the solutions for s are purely imaginary for < -1 - if the interactions are attractive and

dm.

strong enough, the system is unstable. The zeru-sound modes ih liquid IIe3 wcrc discovered by Abel, Anderson and Wheatley in 1966 [38]. The technique used in the experiment was ta monitor the absorption of sound waves at a fixed temperature, which means that the collision frequency v is fixed. At high frequencies w > v, zerwsonnd waves can propagate, and the damping due to collisions is proportional ta x2. As the frequency w is reduced, however, there will eventually be a crassover to normal (first) sound, for which the darnpihg goes as w 2 / ~ ' (a fact not proven here), By detecting the croswover in the darnpifig at fixed ten~peraturefrom being constant t o varying as w Z the existmice of the zero-sound mode could be established. Finally, there ate, as wc havc said, many other different modes possible, It is a good exercise to solve the transport equation for transverse waves, for which u ( # , p )rn e i p . These mode can be supported if t h e tern1 F1 cos Be" Pis included, Similarly, there tcre the spin (antisymmetric) waves where u(B, p, cr) r u(B,p)u.

Chapter 33 Microscopic derivation of the Landau Ferrni liquid theory In this chapter, we will derive the Landau Ferrni Iiquid theory using ordinary perturbation theory to infinite order. We will thus verify the theory for any system in which the perturbation theory converges. Any such system is then a normal Ferrni liquid, but there may of course exist normal Ferrni liquids for which the Landau themy holds, but for which perturbation theory does not converge. Moreover, we will exclude systems with attractive interactions between the particles. Attractive interactions may cause a phase-transition tD occur, such as the superfluid transition in 3 ~ or e the superconducting transition in certain metals, ih which case the s t r u c t ~ ~of r e the ground state changes and the system is not normal, In this case, ordinary pefturbation theory, in which we expand about the nan-interacting ground state, does not converge. It is possible, however, to construct ather perturbation theories for such systems but we will not further discuss that here, We will also restrict the derivation to systems with short-range interactions. It is fairly straightforward t o extend the derivatiori to long-range interactions, such as the Coulomb interaction, h u t a bit tedious, For a derivation which includes lung-range interactions, see Nozikres 1341. Finally, we will also assume that the a y s t e ~ nis isotropic and translationally invariant. These assumptions are not rcstrictive, but simplify the notation. Specificakly, we will verify the following hypothesis and properties of the Landau Ferrni liquid theory.

(1) The elementary excitations of the system art quasi-particles and quasiholes, separated by a Ferrni sorface S p . The energy ~k of a quasiparticle is a continuous function when k masses S F , as is the gradient

V ~ E =J vFk -

386

MICROSC'O PIG DERIVATTCN

(2) The current carried by a quasi-particle is given by equation (32.71,

(3) When the system is compressed, the distortion a k F / a p of the Fermi surface is given by equation (31.9):

(4) The colIisionless transport equation for the quasi-pa~ticlesis given by equation (32.I1) :

In the course of verifying (1)-(41, we will also obtain a microscopic e x p r e s siuus for Ure i i ~erac t tioa function f k a m k l g , and the quasi-part icle distribution function dnk,(q, w ). The original derivation was given by Nozikres and Luttinger [28, 291, and is also presented in Nozikres [34]. This derivation is based on the socalled Ward identities, which are relations between self-energies and vertex functions. Here, we will closely follow the derivation given by Noziitres. Rickayzen [39] has given a somewhat shorter derivation of .the transport equation. His derivation does not explicit Iy use the Ward identities. The verification will proceed in the following way. We will first briefly review the diseussiorl about, quasi-particles in Chapter 26; Gorn which the hypothesis (1) follows immediately. Once we have firmly established the existence of the quasi-particles, we will proceed by defining quasi-particle creation and annihilation operators. These operators later be used t o form a microscopic expression for thc quasi-particle distribution function. Next, we will turn t o a rather lengthy and formal discussion of corzelation functions, which describe t h e response of the system to an cxtetnal perturbation. It is these correlation functions that we will cast in aform from which we can identify the propcr tics ('2)-(4) of thc Landau Fermi liquid theory. This will be done by examining the correIatioh functions on the Fermi surface and using the Ward identities. For simplicity, we wiH ignore the spin of the particles. This is jrist l o make the notation a little bit more comprehensible. In Chapter 26, we discussed the form of the Greerr's f i ~ n c t i n nTiear the Ferrni surface. We found that the low-lying excitations of the system, the

FERMT LIQUID THEORY

387

energies of which are given by the locations of the poles of the Green's function, behave essentially as particles - as we approach the Ferrni surface, the Lifetime of the excitations diverges and the width of the spectral density at the poles of the Green's fu~lctiongoes to zero. Because of the particlelike properties of the excitations, we call them quasi-particles or quasi-holes, depending on whether the real part of the location of the pole is above or below the Fermi surface. The energy of the excitations, however, is a smooth function of frequency as we pass t h ~ o u g hthe Fermi surface. This establishes the hypothesis (1). We establish this hypothesis formally by examining the Green's function near the Fermi surface. According to Chapter 26, the Green's function G(k, w) is

where r f ) = k z / 2 r n and M ( k , u ) is the self-energy. The self-energy is a smooth function at the Ferrni surface, where the imaginary part of M(k,w ) vanishes. The Fermi surface is determined by

To examine the law-lying excitations, it is convenient to change frequency variable w - p -+ Zi and to expand the denominator of G(k, G ) about G = 0 and k = kF with the result

The elementary excitations are at the energies at which the Green's funclion G(k,G) has poles. Wc may separate the contribution to the Green's functions from these poles from contributions due to multi-particle excitations. We have shown in Chapter 26 that the imaginary part of M (k,G) is quadratic in (k - kv) and 6 . All terrrls i t , equation (33.4) to first order in (k - kF) and J are thus real. Neglcc ting quadratic terms, we can then write

where r h is the residue of G(k,G) at k~ and G = 0,

MICROSCOPIC DERIVATION

388 and

Here the effective mass m* is given by

If

add the quadratic terms t o equation (33.51, we can write the Green's function near the Fermi surface as we

where the pole near the Ferrni surface has been separated out. This pole is due t o a quasi-particle excitation. The part Gin, is a slowly varying function which is regular on the Fermi surface, This part is the incoherent part of the Green's function. It arises from configurations of several elementary excitations a n d is of order i;;2 as W -+ 0. With this review, which clearly establishes the existence of the quasiparticles and quasi-holes, we may define a quasi-particle annihilation operator CkH in the Heisenberg picture by (see f291)

Here q h is an infinitesirrlal frequency such that v k < p and q k > l'k, where r;' is the lifetime of a quasi-particle ( k , a ) .The operator CkH and

t its Hermitian conjugate Ck,H are quasi-particle annihilation and creation operators. As Jkl approaches the Fermi surface, C tk ( t ) H I BO)behaves for longer and longer times as an exact normalized eigenstate of the Hamiltoniari H with an additional quasi-particle (k,a).Intuitively, this happcns because t exp(-ickt' rlk t ' ) ~ ~ ~ on ( tI 'q)o~)gives a distribution of single-particle states with momentum k. If the excitation energy of such a state is $, the integrand will oscillate with a frequency (EI; - tk). The only contribution from the integral will then be from the state with Fk = c k prnvided the lifetime of this excited state is much longer than time of averaging of thc int egrand, t has the properties of a quasiLet us now formally demonstrate that CkqH

+

par tide operator by showing that the thermal equilibrium value ( Ct ~ , ~ G ' ~ , ~ )

is prcciscIy thc equilibrium quasi-particle distribution function f ( c k ) . Let

FERMI LIQUID THEORY

389

be the exact eigenstates in the N-particle space of the Hamiltonian with energies By equation (33.101,the matrix elements of c : ,and ~ C k ,in ~ this basis are

EL^'.

t with the obvious definitions of the matrix elements (ck),,t and { c k ) , , t . Using equations (33.11) and (33.12) we can study the thermal equilibrium distribution function ,*) :

( c L , ~cL

where 3iNi and $ , (*-'I spectively, and

are eigenstates

with N and ( N - 1) particles, re-

with XG the partition function. We can cast equation (33.13) in a more transparent farm. To du so, we consider the finite-temperature spectral represent ation of the exact red-time single-partitle Green's function in the same basis yb,:

where the finite-temperature spectral function &( 0 if ~ k + ~> /p j~ which corresponds to a particle, and < 0 if tkL9 2 < p , which corresponds t o the propagation of a hole. In the equations invo vmg the product equation (33.341, we have ta integrate the particle hole propagator multiplied by sornc other function, such as I ( t , kt;p) or All(kl,q ) , bvcr particle frequencies J. 111 gerrctal the irltegrd over G can then be written

I

where the function F(k,q ) is regular as Iqi, w + 0. If we insert equation (:13,35) and the corresponding expression for G ( k - q / 2 ) in this integral,

MICROSCOPIC DEEtlVATION

396

we obtain four terms. Three of these terms contain at least one factor of the incoherent part of the Green's function. These terms have well-defined limits as q -+ 0. The fourth term is

X ,

w

'k- g/2

- (rk-q/2

$- w / 2

- p) + if1

(33.J6)

The structur'e of this term is complicated due to the fact that the two separate poles of each factor merge on the real axis as Iql,u -.0. In t h e limit of q + 0, we can write this integral as

+

Considcr the case where G ( k 9/21 corresponds to the propagation of a particle. Then 9' > 0,and G(k - q / 2 ) correspnnds to the propagation of a hole, so qtl < 0. By using the idefitity

where

P

denotes pdncipal value, we can write

where the step-function ensures that k

+ qJ2 is outside the Ferrni surface,

The integral eqvation (33.37) then becomes

Consider now the case where G(k - q/2) corresponds t o the propagdion of a particle. Going through the same algebra that led l o equation (33.39), we

FERMI LIQUID THEORY

In these two equations, the functiah F ( k , q ) can b e evaluated at the Ferrni surface, since this function is ~egular. Combining equations (33.39) and (33.40) and using

as 141 4 0, we can finally write

Here G2(k) is the term obtained by first taking the limit q -+ 0 separately in the Green's functions, and then squaring:

This term contains the second-order poIes in equations (33.39) and (33.40), which are well-defined as q 4 0. Thus

where the singular part R ( k ; w )is

-

The singular part B(k;w ) enters only on the Fermi surface beCause of t h e two 6-functions. It is clear that this singular part has a limit q 0 which is not unique but depends an r = q / w . We will distinguish three limiting cases: Irl constant, so that 1q1 and w go to zero simultaneously; Irl = 0,and Irt = oo. These limits will be denoted by supkrscripts, Thus

398

MICROSCOPIC DEMVATION

-

We now examine the limit Iql 0, w + 0 of the scattering function, r, From equation (33.32) we see that the singuIarities that arise in r are due t o the product G ( k + q / 2 ) G ( k- q / 2 ) . Combining equations (33.32) and (33.43), we can write

~ ( kk") , { G 2 (k11) + ~'(k")) rT(k",k t ) . (33.47)

rr(k,kt) = ~ ( kkt) , + k"

In the case r = 0, we have

We manipulate these two equations. Equation (33.47) leads to

= I(k,k')

+ C ~ ( kt,1 ' ) ~ ' ( k " ) r 7 ( k t rk')., k"

We multiply this expression from the left, with [hill; - I ( E , k ) ~ ' ( k ) ,] and sum over k. The result is

r

,k

t

=

c [a,,,

the

inverse of

- r ( i ,k ) ~ 2 ( i )-' ]

k

kf) +

I

~ ( kk ,1 ' ) ~ ' ( k 1 ' ) r 7 ( k " k') , . k"

Fmm equation (33.48) wc obtain

-

[6k,krl ~ ( kk ", ) ~ ~ ( t " r0(k", )] k') = ~ ( kkt). , k"

We multiply this equation on the left with over k, which yields

sum [62,k- I ( & ,k ) ~ ~ ( k ) and ] -1

We insert equation (33.50) in equation (33.49) and relabel dummy indices E + k, k 4 k" to obtain

r' (k,E') -- rD(k,k') t

C rU(k,k " ) ~ ' ( k " ) ~ ' ( k "kt). , k'j

(33.51)

FERMI LIQUID THEORY Similarly, it is easy to verify that

ry(k,k t ) = r m ( k , kt)+

rm(k,kt') [R' (k") - ~

~ ( k "~'(k", ) ] b*).

On the Fermi surface, it is convenient to introduce

with k,k' on the Fermi surface. In terms of these functions, we can then write (33.51) as

Manipulating equation (33.31) in the same way as equation 133.471, and using equations (33.53) and (33.54), we can write the following equations for the vertex functiom on the Ferrni surface

We will need these results later at the final stage of the derivation of the Landau Fermi liquid theory. To complete this section and verify the Landau Fermi liquid theory, we will relate the values of the vertex functions $(k) and AF(kJ on the Fermi surface to the self-energy. These relationships are established by the Ward identi ties. We consider first the self-energy M(k,Z). A diagram for M(k,G) with atl intenla1 linc kt explicitly indicated is shown in figure 33.5. Tf we remove an internal line kt in all possible ways from M(k,G) we obtain the irreducible scattering functidn I ( b , k') (see figurc 33.5). Frorh this figure i t is clear that we can write M(k,3 ) = I ( k , k')G(kl). (33.57) kt

We want t o take the partial derivative af M ( k , G ) with respect to Z. This

40 1

FERMI LIQUrD THEORY

By comparing equation (33.59) and equation (33.60)

we arrive at

the first

Ward identity;

For the second Ward identity, we consider dM(k,5 ) / 1 3 k , . Now M ( k ,G) depends on k, only through the matrix elements of the interaction pdtential. These matrix elements are all invariant if all momenta are increased by the same amount. Therefore, instead of differentiating the right-hand side of equation (33.57) with respect ta k&, we may differentiate every internal line, given by G(k,G), with respect to E',. Thus

But - Ga ( ~ ' , E ' )

ak',

=

~ ( k '+ qa,zt)- ~ ( k 'z ,')

= lim q, 4 0

lirn

pak'a/m

4a+Q

Qru

+ M(k' + qa,w') - ~ ( k 'C'),

G(k/,w"t)G(kf

+ q,,i;l)7

Pa

The last factor is just the r = rn limit of ~ ( k ' , G ' ) ~ ( k ' , 6 ' so ) , we obtain

Thus

(33.62) R l ~ taccording to the

= m limit of eqliation (33.33)

By comparing (33.62) and (33.63],

we have

we obtain

For thc remaining two Ward identities, which involve A: and AT, \ye consider a simultaneous translation of the wavevector k and the Fcrmi surface by an amount q,. We denote by M ( k + q,,G;q,) the value of t h e self-energy u n d e ~such a translation. This is nothing but the self-energy in

MICROSCU PIC DERIVATION

402

a reference frame moving with a velocity. -q, /m. With this translation, we may define a 'total', or 'convective', de~ivative:

dM(k,w) dka

lim

Qa+O

1

[M(k + pa, 5;q,) Qcw

-M

(k,G)] .

(33.65)

Note that in this derivative, the Fermi surface follows the motion of k. It is now straightforward to evaluate dM/dC,. We have

Now

Tn the second factor, both k' and the Fenni surface are translated simultaneously so that their relative positions do not change, But then the poles of G(k', G') and GEL' + q,, G'; are on the same side of the Fermi surface, so according to (33.42) we have

lim ~ ( k 'G')G(L' ,

p a +O

+ qa, G';q,)

= ~ ~ ( kG'). ' ,

Thus,

If we compare this with the r = O lirriit of equation (33.33)

A: (k, J)=

m

+

l ( k , k ' ) (kt, ~ w"')~E(k', ~ G') k'

we see that

Far the last Ward identity, we consider an increase d p in the chemical potential ( d p > O), Due t o this increase, t h e volume enclosed by the Fer~rli surface must expand t o accommodate the increase in particles, hence k p must increase an amount d k (which ~ depends on direction in anisotropic systems), with

FERMI LIQUID THEORY

403

A quasi-particle is by definition an excitation on the Ferrni surface, so its energy is equal to the chemical potential. Thus, the energies of the quasiparticles increase when p increases. Consider a point A on the Fermi surface, As p increases, this point is translated to a point B , The energies of a quasipatticle at A and one at B must then satisfy

This equation shows how the energy of a quasi-particle changes with p , We will use this relation shortly. Now consider the self-energy M(k,w 1. As p increases to p d p , the selcenergy changes, We then have

+

Using the same construction as for the other Ward identities,

we arrive at

We examine the last facto~of this expression. With k' between the Fermi surfaces at JA and t p , this wavevector passes from the exterior of the Ferrni surface to the interior of the &rmi surface as p 4 p + d p , hence its pole crosses t h e real axis in the complex w',-,plahe. From equation (33.38), the last factar is in this case

+

+

If kfis not in the region between the Fermi suifaces at p and p dp, the lirrrit is sirr~plyG ' ( ~ ' , G ' ) ,Since the extra term appears only if k' is on the Fermi surface at p , we can write


404

which is just the r -* m limit of the product of the Green's functiohs. With this in (33.691, we have

If we compare this with the

P'

= mk

t of (33.33), we deduce that

All that remains is t o wtite down the Ward identities on the Fermi surface (k = k ~ ~h, 5 p , G = 0). We first note that the residue zk is

If we compare this with the first Ward identity, equation (33.611,

we see

that

Furthermore, the quasi-particle energy is at the pole of the Green's function. The location of this pole is given by

Tf we diffetcntiate this expression with respect to E,, we obtain

From the second Ward identity, equatiorl (33.641, and equation (33.721, we then have

A?(k,Z) =

1a€k

-

1

----fl.~,~ zk aka zk

If, an the other hand, we take the total derivative as dcfincd in equation (33.65) of (33.71) with respcct to E,, and use the third Wardidentity (33.661, we obtain

T o evaluate the final Ward identity on the Fermi surface, we differentiate (33.71) with respect to the chemical potential. The ~ e s u l tis

FERMI LIQUID TIIEORY

405

Using equation (33.681, the equation for the residue zk and the fourth Ward identity, equation (33.701,we obtain

But according t o (33.671, we have

which in (33.74) yields

We can evaluate dfk/dk, by transforming to a coordinate system moving with a uniform velocity -q, Jm. In this coordinate system, the Hamiltonian

If we let Ep denote the expectation value of the energy in a state 1 $), in this refere~lceframe, then

seen

From the definition (33 -65)we have

where JCyis the a-component of the total ewrenC dertsity. Hence, it follows that drk/dk, is the component jk,,of the current carried by the quasiparticle k- Jf the interactions are translationally invariant and thus conserve momentum, the current carried in the presence of a quai-particle in the inkacting system is the same as the current catried in the abscnce of interactions. This curte~ltis si~nplyk/m, so wr: have

406

MICROSCOPIC DEMVATION

We can then summarize the Ward identities as follows;

If we now use these forms of the Ward identities in equations (33.56), obtain

we

These two equations are precisely the properties (2)-(3) of the Landau Fermi liquid theory. We have then finished verifying these two properties, when we add the fact that it is obvious from perturbation theory that one qumi-particle contains precisely one bare particle. In addition, in the verification of these properties we have made the identification

Thus, we have obt a i n d the microscopic expression for fk k ~ . Equation (33.81) states that the Landau f-function is given by the scattering function in the limit of long wavelengths and low cnergiea. All that we have left to do is to verify the transport equation of the Landau Ferrni lipnid theory. To this end, wre consider the response of the quasi-particles t o a perturbation which couples to the density of the bystem. To be specific, we take the perturbation to be of the form

where Vq is an amplitude. The perturbatinn is macroscopic in the sense that q / k F < 1 and w / c F < 1. We seek thc response of the quasi-particle density, so we must find an expression for this quantity. Previausly, we introduced thc operat or C: ( t l H , which wo argued is a quasi-particle creation operator. In analogy with the padacle-density operator

we irrlroduce the quasi-particle density opcrator

FERMT LIQUID THEORY so

that

is the density of quasi-particks k at r . The response of this density t o the perturbation H I ( ¶ , 1 ) is then (see Chapter 27)

where q-is a positive infinitesimal which assures that we turn on the perturbation adiabatically, In ~articular,at t = 0 we have

If we define a renormalized ~ed-timecorrelation function Dd(k f - i';

q,w )

by

iE4(k,t - t'; q , wJ

we can write the quasi-particle response as

To show the equality between equations (33.83) ahd (33.85) wc must show under what conditions

for any two bosonic Heisenberg operators A(t)]{ and D(tIH (e-g., A =

t

ck-q/~C*+q/2

and B = P q ) . By iiisertirlg a complete set of eigenstales


408

$,

the left side of equation (33 -86) becomes

The right-hand side of equation (33.861, on the other hand, is

We define a function Sl( w ' ) by

and a function S2 (w ') by

Sz(wt) =

C

~ o n ~ n ( ou' b - (En-

Eo)) -

n

TIlesc functions are essent i d l y generalizations of thc spectral functions A(k, w ' ) and B(k,w') introduced in Chapter 15 (see eqwition (15.30)). Since En > E0,it follows t h a t Sl(w') and S2(wt) vanish if w' 0, $nd differ by the sign of the term ?rS2(-w) if w < 0. To avoid any complications, we will assume that w > 0. This is in any case not unreasonable, since w is the frequency w i t h which the perturbation is driving the system. The correlation function D4(k,t -1'; q , w ) is a real-time correlation function which is closely related t o the real-time correlation function D4(q) which we studied earlier. D4( q ) could be decomposed into a vertex part and a particle-hole propagator:

-

(k,B) of D(k, t - t'; q, w ) is a renormalized version The Fourier transform of D4(k, q ) which satisfies the same equatia~l,but with the two Green's functions replaced by mixed Green's functions g(k) and c(k) defined by

To cast this equation in a more transparent form, we st-art by writing out the zero-temperature form of the mixed Green's function g{k,t - t'):

=

C n

e i ~ ~ ~ N ~ - ~ ~ N + i ~ J ( t - t t l

( ~ k l o n(c:)

w)- EAN+'))

- ~ k- (Eo

i ~ k -#(t

- t')

+ iqk ~ i i ;

where we have used equations (33.11) and (33.12). On the other hand, the Lehmann representation for the Green's function G(k,S) is

MICROSCOPIC D E W A T I O N

410

Here, ~ ( kw", and B ( k ,w ' ) are the spectral functions

By comparing (33.93) and (33.941, we see that we have

iglk,t - t')

If k is near thc Fcrmi surface, we use A (k, w)

= A;,,@,

u )4- zkh(w

- f k ) e ( ~ k- 11)

B(k,u) = B h ~ c ~ ~ , ~ ) + ~ k ~ ( ~ + ~ k ) ~ ( ~ - ~ k )

whcrc Ai,,(k, w ) and B;,,jk, w ) are thc incohcrcnt parts of the spectral filnci,iorrs, We I1ave e ~ p l i c i t ~ lwritten y out the 0-functions to demonstrate that the two terms correspond to quwi-par ticle and quasi-hole excitations, respectively. These parts are analytic at the Fermi surface. The incoherent part of the spectral densities do not contribute anything as q k 4 0. In this limit, we the11 have

ig(k,t -

- t') e- - i ~ h ( t - - t ' )

&B(o

- g ) $ ( t - t')

- e -irk

(t- t')

- r k ) 8 ( t 1- 1 ) .

In other words, the f~inctiong(k, E - 1'3 is a renormalized convehtional unperturbed propagator with the unperturbed energies replaced by the exact quasi-particle energies. Going through thc s a m c analysis for F(k,t - t'), wc find precisely the same result. B y Fourier transforming g(k,t - 1') and g(k,.l - t') and inserting the results in equation (33.02) we arrive a t

-

In this expression, the poles of the integrand are on different s i d a of the real G-axis, and the contributions from the poles dominate the integral. In the limit 1q1 -, 0, w -+0 (the T-limit)+the factor behaves precisely as the r-limit of G(k t q/2)G(k - q / 2 ) , except for the

+

-1/2

fact that g(k q / 2 ) and ji(k - 9/21 are renormalized by a factor of z E each compared to the interacting Green's functions G(k f q / 2 ) . Hence, this r-limit is the r-limit of G(k q / 2 ) G ( L- q / 2 ) divided by r k , so the result is

+

This result inserted in equation (33.85)yields

By using equation (33.52) and the Ward identity equation (33.751, we

tan

write

We can eliminate A; between these two equations. By inserting equation (33.99) in (33.981, we obtain

(33.101)

On the ather hand, if we relabel k fkjk' and sum aver k', we have

4

k' in equation (33.98), multiply by

Inserting equation (33.102) in (33.1 01) yidds

If we introduce the amplitude

F of the force created by the perturbation,

Equation (33.103) takes the form

This is precisely the ttarispart equation ih ^the absence of collisio-~~s of the Landau Ferrni liquid theory and the verification crf the Landau Fermi Iiquid theory is complete. Wc conclude with a comment on the expression (33.83) for linkCq,w ) :

This expression shows that Snk(q,w) can be interpreted as a probability amplitude for finding a quai-particle-quasi-hole pair of morrit:ntlrm q and energy w . Note that the expression is n o t positive dcfinite, so that an interpretation as a pmbabiliiy is not permissible.

Chapter 34

Application to the Kondo problem A standard result due to Bloch in transport theory ~ r e d i c t sthat the tesistivity p(T) of metals should vary as 'T5at low temperatures, due to normal elec tron-phonon scattering. In real metals, even the simplest ones, such as potassium and sodium, the resistivity exhibits a more complex temperature dependence than this, AltIiough the normal electron-phonon scattering term can in some cases be identified, there are: also contributions to the ~esistivityfrom many other scattering processes, such as electron-cllectron scattering and electron-impurity scattering, The contribution to the resistivity from these two processes is proportional to T~ at low ternpc~atures and low impurity concentrations (for an extensive review, scc Bass et al. [40]). If a small amount of magnetic impurities, such as iron, chromium or mmganeae, is added to the metal, the resistivity exhibits a minimum. The temperature at which this minimum occurs seerns to vary with the density n; of the impurity, and the depth of the minimum, p(03 - p ( T ~ , ) is , proportional to n;. Since p ( 0 ) itself ifi proportional to n;, the relative minimum in the resistivity is roughly independent of ni, and is usually equal t o about ten percent of' p(O). The explanation for thc cffecl, but by no means a complete theory of the temperature dependence uf the system, was provided by Kondci in 1964. He considered the problem of a singlc magnetic impurity in the sea of canduction electrons. The corduction electrons may suffer a spin-flip as they scatter off thc impurity. This makes the problem a very difficult many-body problem, sincc the PatlZi exclusion pri~lciplehas to be taken into account when the scattering rates off thc intcrnal degrees of freedom of the impurity are calculated. It is found that a low-temperature regime is entered when the temperature is much less khan a characteristic temperature, the Kondo temperature TK of the system. Exactly what happens near this temper-

XONDO PROBLEM

414

ature could not be sorted out until renormalization group techniques were applied to t h e problem by Wilson 1421. He showed that when T goes below TK,the system enters into an effective strong-coupling regime, where the system behaves as though the coupling J between the impurity and the conduction electrons goes to infinity. Once in this strong coupling regime, Nozikres [31] showed that Fermi liquid theory could adequately describe the system. We will here b r i d y review this application af Fermi liquid theory by Nozikres. We consider a single magnetic spin- irnpurity in the sea of N Bloch electrons, The appropriate Hamiltonian for this problem is the Kondo Hamiltenian, which consists of the kinetic energy of t h e conduction electrons and an anti-ferromagnetic coupling to the spin of the impurity:

4

Rere, S is the spin operator of the impurity, s,,r is the spin operator of the conduction electrons, and J > 0. For our purposes here, it is better to write t h e Hamiltonian in a real-space represent at ion. We imagine that t creates an electron of spin a at site the system is on a lattice, so that ci0 i with a spatial wavefunction $(r - R i ) . By a simple transformation, the Hamiltonian (34.1) can then be written

whcre 0 is the impurity site. The T*j are the hopping integrals given by

with HBloch thc Bloch Bamiltonian of the conduction electtons. We will follow Nw~ieres and apply Femni liquid thcory to the lowtenlperaturc regime of this problem. This means that we start by assuming that at low temperatures, the coupling J is very largc compared to a11 other dynamic degrees of freedom. A single electron can then substantially lower its energy by binding to t h e impurity in a singlet s t a t e . As J -t a, the binding energy goes t o infinity, so that it requires an infinite amount of encrgy t o remove the electron from the singlet state. Ilence, the impurity will trap B single conduction electron into a singlet statate. By t h e Pauli exclusion ptinciplc, n o mare electrons can occupy the singlet state, so the impurity rrow acts .as an irlfi~litelyrep~llsivesite for thc lV -1 remaining electrons. We will identify this state of the systcrri -asour idcrzl 'non-inter:rncting"refcrcnce system of the Fermi liquid theory. A t large, hut finite J , it is still impossible t o brcsk the singlet state, but virtual excitations of the singlct, stnt,t: krt: possible. 'I'he impurity site

KONDO PROBLEM

416

where we have cartied out the expansion only to first ordet. The quantities So, a and 4,,, are numbers which ~ e t r n i ta phenomenological description of the low-temperature behavior .of the system. These numbers csrrespond to the parameters of the conventional Fermi liquid theory. Only the four quantities bO, a and (bb,fo = $Sf4a enter. Here we have defined symmetric and antisymmetric parts of $,,I, in analogy with f a and f3. The total number of electrons is constant and 4' never shows up. If we define

and sum over quasi-particles in equation (39.41, we obhin

A theorem due

Fumi [dl] relates the total energy of the impurity due to its ihteractions with the conduction electrons to the scattering phase shifts. From our point of view, the impurity causes an effective electronelectton interaction, so the energy of the impurity is expressed as a shift in the energies af the conduction electron, such that the sum of these shifts is equal to the total energy of the impurity. Fumi's theorem then states that the energy shifts are -60/[~v'(0)]. Here, ~ ' ( 0 )is the density-of-states at the Fermi surface for one spin direction for an irnpasity-free system, i.e., the number d electron states per unit energy at the Ferrni surface (as opposed t o our definition of v(O) from before, which is the number of states per unit to

errergy and uni-l v o k m e ) . The quasi-particle energies due to the interactions

are then

As a result of scattering off the impurity, new states will appear in thc electron spectrnm, for example, the bound singlet state. Consequently, there will be a change 5u'CO) in the density-of-state at the Fe~misurface, By combining equation (34.7) and (34.81, we obtain this change as

The change in the density-of-states leads t o a change &C, in ihe specific

Measurement of the specific heat will thus yield ry. Note that the change in C, change is of order I J N , since we are studying the effects of a single irnpur i ty. In the prcscnce of an applied external magnetic field 11, the quasi-particle energies will shift, depending on the orientation of the spin relative to the

LANDd U FERMI LXQ UID TIIEORY direction of the magnetic field (see equations (34.7) and (34.8)):

To first order in the magnetic field strength, the chemical potential remains unchanged, so the Fermi levels for the two spin directions must be equal to p . The density of electrons with spin up induced by the magnetic field is

and the induced ,down-spin density is

In these two expressions, we have taken into account the change in the density-of-states due t o the impurity. We obtain the magnetization m from

with the resu1.t

where we used equation (34.9). The spin susceptibility is then

This result is easily understood. T h e terms in the bracket give thc enhancenleilt of the susceptibility. The terrri r u / [ n - ~ ' ( O ) ] is an crrhancernenl d u e b u t hc increased density-of-states at the Fermi surface (equation (34,9)), and the term 24' JT comes from the impurity-mediated electron-electron interaction.

KONDO PROBLEM

418

We have now obtained two expressions containing the t w o parameters a and # a . We would like to obtain an expression for a third property, which can then test the predictive powers of the theory. To this end, we will now calculate the zeretemperatute conductivity of the system. In the end, we also want to calculate the low-temperature conductivity of the system, to verify that we obtain a conductivity which increases with temperature, consistent with a minimum in the resistivity at a finite temperature. Because we are only including s-wave scattering, every collision restores angular momentum & = 0. The cullision integral that enters the Boltzmann equation then takes the simple form

where nl = n - no(T)is the current carrying departure from thermal equilibrium and W, (t) is the total relaxation rate of n, (€1,which inchdes relaxation both due t o elastic and inelastic scattering. The conductivity is then directly found from the Boltamann transport theory [37] as

is the Ferrni distribution function, At zero temperature, only elaqtic. scattcting is possible, For elastic s-wave scattering, the relaxation ratc can be found fairly easily in the following way. The scattering problem can be formulated i11 terms of the secalled T-matrix [43], defined as Here,

f(6)

w h e ~ eykrg,is the incoming Bloch state, is the exact wavcfunction, and V,,,t{r) is the scattering potential. With outgoing-wave boundary conditions, the diagonal T-matrix is given in terms of the phase-shifts as

The self-energy is related to the scattering matrix hy

where ni is the impurity concentration. Thc relaxation rate is the magnitude of the imaginary part of the self-enetgy, so eq~iat~ios~ (34.11) gives

LANDAU FERMI LIQUID THEORY In the absence of magnetic fields, 6,(p) = 60, so we obtain

We now consider the low-temperature conductivity a(T). At low, but finite, temperatures, the F e d function in the expression (34.10) broadens over a Tange of width T.We consider only elastic scattering. We can use equation (34.10) directly, and we expand sin2 6(t) t o order (s - p ) 2 . Unless 6 e 7 ~ 1 2,such , an expansion involves coefficients of higher o r d c ~than given in equation (34.5). However, in the case of strorig coupling and particle-hole symmetry, one can show that b0 n/2,so that the expansion yields

and we obtain for the conductivity

The result (34.12) shows that the conductivity increases as the temperature increases from T = 0,consistent with a minimum in the resistivity at a finite temperature, For a rigorous result, we: s h o ~ I dalso consider inelastic scattering. In this case, processes in which one electron scatters off the impurity and excites one or several electron-hole pairs have t o be included, We will not go through the actual calculation here. The result is the same as equatiorr (34.12) for the case a0 5, with a11 additional term proportional to T ~which , arises from inelastic particle-particle scattering ( ~ e eNozikres [31] for details).

KONDO PROBLEM We have then concluded our discussions of interacting many-pa~ticlesyst e m . Our discussions have been on an introductory level, aimed at making the reader familiar with basic ideas and, by now, classical results, such as the Gell-Mann-Brueckner theory of the electron gas. There are, of course, many topics that we have not discussed at all. One important topic is density functional theory, which is a method (for extended systems essentially the only method) which can deal with inhomogeneous systems in a systematic way (see, for example, Dreizler and Gross [44]), Another topic is highly correlated electron systems, which are systems in which electron cotrelations domihate. As a consequence, approximations such as the random-phase approximation a e inadequate. 0 t her methods, for example variational methods using trial wavefunctions, provide a viable approach. The importance of such systems is illustrated by pointing but that they include the fractional quantum Ha11 effect (see Chakraborty and Pietilainen [45]), heavy fermions and, most likely, high-temperature superconductors. It is OUT hope that the interested reader h a s at this point achieved a sufficient understanding of basic many-body theories to be able t o turn to these more recent and still developing areas.

References [I] Pauli W Phys. Rctr. 58 716 (1940) 121 See, for example, Szaba A and Ostlund N S Modem Quantum Chemistry (New York: McGraw-Hill, 1989) [3] Raimes S Many Electron Theory (Amsterdam: Nor t h-Holland 1972) [4] Raimes S The Wave Mechanics of Electrons in MeiaIs (Amsterdam: Nos t h-Holland 1961)

f51 Bohm D and Pines 13 Phys. Rev. 82 62511951); 85 338 (1952) [6] Bohm D, Ruang K and Pines D Phys. Rev. 107 71 (1957)

[7] Basdeen J , Cooper TA N and SchriefFer J R Phys. Rev. 108 1175 (1957) [8] Bogoliubov N N Nuovo Cirnento 7 794 (1958); Zh. Eksperim. i Teor. Fir. 34 (1958) (Engl. Transl. S o v ~ e tPhys JETP 7 41 (1958))

[lo] Economou E N Green's Fvmctkons in Quaniurn Physics 2nd edn (Berlin: Springer-Verlag 1983)

Ill] Ginzburg V L and Landau L D Zh. Ekspehm. 4 Teor. Fix. 20 1064 (1950)

[I21 Gorkov L P Zh. Ekspen'm, i T e o ~ Fir. . 36 1918 (1959) (Engl. TransL Soviei Phys. JETP 9 1384 (1959)) [13] Gell-Mann M and T,ow F Phys. &eu. 84 350 (1951)

[14] Wick G C Yhys, Rev. 80 268cI950) 1151 Goldstone J Pmc. Roy, Soc. A239 267 (1957)

[16] Gcll-Manth M and Brueckner K A Phys. REV, 106 364 (1957)

[17] See Onsager I,, Mittag L and Stephen (Leipzig 1966)

M J

Ann. der Physik 18 71

1181 For details, see Sawada K S h y s . Rev. 106 372 (1957)

[19] See, for example, Morse P M and Feshbach H Methods of Theoretical Physics vol I (New York: McGraw-Hill, 1953) [20] Schwibger J Pmc, Nai* Acad. Sci. USA 37 452 (1951)

[21] Luttinger J

M Phys. Rev. 121 942 (1961)

1221 Fetter A L and Walecka J D Qurzniztna Theory of Many-Par-licle Systems p 1588 (New York: McGraw-Hill, 1971)

[23] Falicov L Adv. Phys 10 57 (1961) [24] Matsubara T Prog. Theoret. Phys. 14 351 (1955) [25] Gaudin M Nuci. Phys. 15 89 (1960) [26] Kohn W and Luttinger J M Phys, Beu. 118 41 (1960)

1271 Landau L D Zh. Eksperim. i Teor. Fiz. 30 I058 (1956) (Engl*TransI. Soviet Yhys. JETP 3 920 (1956)); Zh. Fkspera'm. i Teor. Fez. 32 59 (1957) (Engl. Tkansl. Souicl Phys. JETP 5 101 (1957)) [28] Noziires P and Luttinger J M Phys. Rev. 127 1423 (1962) 1291 Luttinger J M and Nozihes P PAys, Rev. 127 1431 (1962)

[30] Rice M Ann, Phys. 31 100 (1965) [31] Nozitres P J.

Lo111

Tetnp. Phys. 17 31 (1974)

[32] See, for example, Varma C M, Littlewoad P B, Schmitt-Rink S, Abraharns E and Ruckenstein A E YAys, R ~ QLett. . 63 1032 (19893, and Engclbreeht J R and Kandcria M Phys. Rev. L e f t , 65 1032 (1990)

[33] Pines D and Nozikres F The T h e o y of Quantum Liquids vols I and TI (New York: Addis~n-l+~ealey, 1989) /34] Nozikres P Interacting Fcrmi Sysiems (New York, Nk': Benjamin, 1964)

1351 Luttinger J M Phys, Rev. 121 942 (1961) 1361 Heinonen O and Kohn W Phys. Rev. H36 3565 (19871, and Beinonen 0 Phys. Rev. B40 7'298 (1589) 1371 Zirnan 3 Elecirr.oras rand Phanons (Oxford: 'l'hc Clarendon Press, 1960)

[38] Abel W

R,Anderson A C

and Wheatley J

C Phys. Rev. Left. 17 74

(1966)

1391 Rickayzen G Green's Functions and Condensed Matter (London: Academic Press, 1980) [40] Bass J, Pratt W P and Sch~oedmP A Rev, Mod. Phys. 62 645 (1996) [41] Fumi

F G Philos. Mag. 46 1007 (1955)

[42] Wilson

K G Rev, Mud. Phys. 47 773 (1975)

[43] Mahan G D Many-Particle Physics, 2nd edn (New York: Plenum, 1990)

[44] D~eizlerR M and Gross E K U Density Functional T h e o ~ g(New York: Springer Verlag , 1990)

[45] Chakrabo~tyT and Pietilainen P The F r a ~ i i o n a lQuantum Hall E$ect, vol85 of Springer Series i n Solid State Sciences (Berlin: Springer, 1989)

Suggested further reading General many-particle theory Abrikosov A A , Gorkov L P and Dzyaloshinski I E Methods of Quantum Field Theory in Statistical Physics (New York: Dover, 1963) Bloch C Dingfim Expahsion in Quantum Stahsiicul Mechanacsvol3 of Studies in Stafisfical Mecha.mics de Boer J and Uhlenbeck eds (Amstetdam: North-Holland, 1965) Bonch-Bruevich V L and Tyablikov S V The Green's Funciion Method in St adistical Mechanics (Amsterdam: North-Holland, 1962) Brout R and Carruthers P Leciums on the Many-Electron Prvhltm (New York: Interscience, 1963) Brown G E Many-Body Pr.ablerns (Amsterdam: North-IIollarrd, 1972) Economou E N Green's Functions in Qaantum Physics 2nd edn (Betlia: Springer, 1983) Fetter A L and Walecka J D Qvanizlm Theory of Many-Particle Systems (New York: McGraw-Hill, 1971) Fulde P Elcctmn Correlata'ons in Molecules a n d Solids vol 100 of Springer Solid Siate Series (New York: Springer Verlag 1991) Dreizler R M and Gross E K U Density Function a1 Th e o y (Berlin: Springer Verlag, 1990) Hugenholtz N M R e p , Prog. Phys. 28 201 (1965) Kadanoff L Y and Baym G Quaniurn Staiislical Mechanics (New 'York:Benjamin, 1962) Kirzhnits 23 A Field Theareiical Methods an Many-Body Sysfems (Oxford: Pergamon 1967) Kohn W and Luttinger J M Phys. Rev. 118 41 (1960) l~lktingerJ M Phys. Rev. 121 942 (1961j Luttinger J M and Ward 3 C Phys. Rcw 118 1417 (1960) March N M, Young W H and Sarnpanlhar S The Mafig-Body Problem in Q u a n i u m Mechanics (London: Cambridge 1967) Martin P C and Schwinger J Phys. Rev. 115 1342 (1959)

426

F URTEER READING

Mattuck R D A Gzside t o Feynman Diagrams z'n the Many-Bod3 Problem 2nd edn (New York; McGraw-Bill, 1976) Mills R Propagators for M a n y - P a d c l e Systems (New York: Gordon and Bteach, 1969) Negele J W and Orland H Quantum Many-Particle Systems (New York: Ad dison-Wesley, 1988) Nozikres P Theory of Interacting F e m i Systems (New York: Benjamin, 1964) Patry W E eF a1. The Many-Body Problems (Oxford: Clarendon, 1973) Pines D The Many-Body ProbIem (New York: Benjamin, 1961) Popov V N Fvnctional Integrals i n Quantum Field The6l-y and Sfatis-tacal Physics (Dordrecht: D Reidel Publishing Co, 1983) Raimes S Many-Electron Theory (Amsterdam: North-Holland, 1972) Schirmer J , Cederbaum LS and Waiter 0 Phys. Rev. A28 1237 (1982) SchuIman, L S Techniques and Applications of Path Integration (New Ymk: Wiley, 1981) Schultx T D Quantum Field Theory and ihe Many-Body Problem (New York: Gordon and Breach, 1964) Van Hove L, Hugenholtz N and Howland L Quantum Theory of ManyP ~ r i z e l eSystems (New Yurk: Benjamin, 1961) Zabolitzky J G Prop. an Part. a n d Nucl. Phys. 16 103 (1985)

Applications in condensed matter physics Andersolr P W Basic Notions in Condensed Matter Physics (Menlo Park: Benjamii~/Cunfings,1984) Bogoliubov N N Leetures on Quanlplm Statistics vol 1 (New Yotk: Gordon and Breach, 1967) Bogoliubov N N , Tolrnachev V V and Shirkov D V A New Me thud an the Theory of Swpzm.conductiwitg (New York: Consultants Bureau Inc., 1959) Chakraborty T ahd Pietilainen P The Fmctional Quantum H ~ l Efled, l 85 of Springer S e r i e s in Solid State Sciences (Berlin: Springer, 1989) Correlataon Functions and Quasi- Particle Interactions in Condensed Mutter Woods Halley J ed (New York: Ylenllm, 1978') Doniac11.S and Sondheirrier F, II Green's Functicuns for Solid State PI~ysicisis (Reading, MA: Benjamin, 1974) DuBois D F Ann. Phgs. 7 174 (1959) Forster D Hydrodynamic +F'r'luctuutioas, Broken Symmetry a n d Correlation Functions (Reading, MA: Benjamin, 1975) Fradkio E Field Theara'as: of Condensed Matter Systems (Rcdwood City, CA: Addison-Wesley 1991) Fuldc P Electron Correlataons in Moleczllcs and Solids, vol 100 of ,i;pringer Series in Solid Shte Sciences (Rerlin: Springer Verlag, 1991) Gorhochenko V D and Maksimov E G Sow. Phys. Usp, 23 35 (1980)

FURTHER READING Hedin L Phys. Rev. 139 A796 (1965) Hedin L and 1,undqvis.t S Solid State Phys. 23 1 (1969) Khalatnihv P M An Infmduction t o the T h e m y of Superflaidity (New York: Benjamin 1965) Kohn W Phys. Reu. 105 509 (1956) Kuper C G Theury of Superconductivity (Oxford: Clarendon, 1969) Landau L D, Lifshitz E M and Pitaevski L P Sta-listical Physics, part 2: T h e ~ of y the Condensed Sfaie 2nd edn, vol 9 of A Course ia T h ~ o r e t i c a l Physics (New York: Pergamon, 1986) Maddung 0 Solid Sfaie Physics vols I, I1 and TIT (New York;Springer, 1981) Mahan G D Many-ParticJe Physics 2nd edn (New York: Plenum, 1990) March N M and Parrinelln M Colleclive Efects in Solids and Liquids (Bristol: Adam Hilger , 1982) March N M and Tosi M P Coulomb L i p i d s (London: Academic, 1984) Nambu Y Phys. Rev, 117 648 (1960) Pines D Elementary Excitations in Solids (New York: Benjamin, 1963) Pines I3 and Nazikres P The Theory of Quamtum Liquids vols 1 and 2 (New York: Addison-Wesley, L 989) Rarnmer J and Smith H Rev. Mod. Phys. 58 323 (lg88) Rickayzen G Green's Functions and Condensed Ma-1-ler(London: Academic, 1980) Rickayze~rG TIt e n y oj Supercondu clivity (New York: Wiley, 1965) Sthricffer J R Theory of Supereenduciivity (New York: Benjarrfi11, 1964) Singwi K S arid Tosi M P Solzd Sta-le Phys. 36 177 (1981) Supe~*coad*ucta'vily vols 1 and 11, Parks R D ed (New York: Marcel Dekker, Tnc., 1969) Tinkham M Introduction. t o S2~pe~cond~ctiviiy (New York: McGraw-Hill, 1975)

Zirnan J M Electrons and Phonons (Oxford: OxFord University Press, 1960)

Applications in nuclear physics Rethe H A Phys. Re*. 103 1353 (1956) Bohr A and Mottclson B R Nvclenr S1ructure vol 1 (New York: Benjamin, 1969); vol 2 (Reading, M A: Benjamin, 1975) Brown G E: l'heory of Nacletzr Matter (Copenhagen: NOR.DITA Lecture N d e s , 1964) Clark J W Prog. in Part. nmd flacl. P h y s . 2 89 (1979) Easlca B H Lectures nn i h e Many-Body Problem and its Relation t o Nuclear Physics (Pittsburgh: University of Pittsburgh, 1963) Kumar I< Perturbafion Theory a n d the Nuclcnr Many-Body Problem (Arr~sterdarn: North-Holland, 1962) Mar~halekE R and Weneser 3 A n n a b uf Phys. 63 569 (1Y6Y)

428

FURTHER READING

NegeIe J W Rev. Mod. Phys. 54 913 (19821 Ring P ahd Schuck P The Nuclear Man y-Body Problem (New York: Springer, 1980) Thouless D J R e p , Pro$. Phys. 27 53 (1964)

Applications in atomic physics and quantum chemistry Corelation Efects ia Molecules Lefebre R and Moser C eds vol 14 of Advances in Chemacal Physics (London:Interscience, 1969) Hurley A C Electron Correlation in Smali Molecules (New York: Academic, 1976) Linderberg J and d h r n Y Propagators in Quantum Chemistry (London: Academic, 1973) Lindgren I and Morrison J Aiomic Many-BoHy Theory (Berlin: Springer, 1982) Many-Body Theory j o r Aiomic Systems (Proceeding of Nobelsgmpesiwx 461, Lindgren I and Lundqvkt S eds Phys, S c ~ i p t a21 (1980) McWeeny R in The New World of Quantum Chemastry Pullman B and Parr R eds, p 3 (Dordrecht: D b i d e l Publishing, 1976) von Niessen W, Cederbaum L S and Domcke W in Excited Stafes in QaantLm Chemistry Nicolaides C A and Beck D R eds, p 183 (Dordrecht: D Reidel Publishing, 1978) voa Niessen W, Cederbaum L S, Dorncke W and Schirmer J in Carnputationoi Meihads in Chemisiq I3argon J ed (New Ymk: Plenum, 1980) 6 h r n Y in The New World o j Quantum Chemistry Pullman R and Parr eds, p 57 (Dordrecht: D Reidel Publishing, 19761 Robb M A in Computlational Techniques in Quantum Chernisiy and Molecular Physics Diercksen G H F, Sutcliffc B T and Vcillard A eds (Dordrecht: D Reidel Publishing, 1975) Three Approaches t o Electron Correlations in Atoms Sinanoglu 0 and Brueckner K A eds (New Haven: Yale University Press, 1970)

Index adiabatic tutning-on of the interaction 195 annihilation operator for bosons 3 1 for fermions 17 unitary transformation 34 anomalous diagrams 356 anomalous expectation valve 134 ant i-symmetrization operator 7, 8 anti-symmetric wavefunct ion 6 Bardeen-Cooper-Schrieffer (B CS*) theory 123 BCS wavefunction 129 Bloch density operator 349 Bloch function 1 18 Bloch equation 352 Boltzmann t~ansporttheory 379, 4 18

Born-Oppenheimer approximation 104 bosun3 commutation relations 32 creation and annihilation operators 31 definition 7 Bravais lattick 105 Hrillouin zone 107 Brillouitl's theorem 59 chemical potential 173 cohesion of alkali metals 84 collective coordihates for lattice oscil~ations109 fop plasma oscillations 90

cdlision frequency 379 collision integral 379 conservation of four-momentum 289 contraction 214 Cooper instability 129 Cooper-p air 129 correlation energy 65 of dense electron gas 255 of electron gas, long-range part 101 of electron gas, short-range part 87, 102 correlation functivn advanced 391 for locd observahles 179 for non-local observables 181 real-time 392 retarded 391 creation operator for bosons 31 for ferrnions 17

uni2ary transformation 34 crystal electron 317

crystal lattice 105 current density operator 39 2 of quasi-particle 380,386, 406 cyclic group 1117 densi ty-density response function 336 density-density correlation function 180, 392 density matrix 47

INDEX in the HartreeFock approximation 53 relation to Green's functions 163 spin-independent BS in translationally invariant systems 75 density-of-s t ates 370 at the Fermi surface 132 density operator 43, 91 quasi-part icle 406 dielect sic constant 340 in RPA 342 d i m t term 52, 74 dispersion relation 75 crystal electron 317 electrons interacting with phonons 124 general quasi-particle 305 Hartwe-Fock electron 313 phonon 114 plasmon 343 zero sound 383 Dyson's equation 283 for the effective irlt erac tion 340 as a nun-linear eigenvalue problem 304 for the polarization. propagat or 333 for t ranslat ionally invariant systems 294 effective .mass 3 15, 368,388

effective interaction 339 electron gas free 79 homogeneous 79 spin-polarized 82 elect ron-electron irlt erac tin11 i:ffcctjvc

thro~xgh

phonon-exchange 126 long-range part 87, 93, 378

short-range part 87, 93, 267, 378 electron-hole pair creation operator 273 electron-phonan interaction 118, 124 electron-plasmon interaction 95 energy-loss spectr urn 89 ensemble average 349 exchange energy of the homogeneous electrah gas 81 of tpanslationally invariant systems 76 exchange charge '70 exchange hole 70 exchange term 52, 69, 74 Fermi energy 123 Fermi liquid 362 marginal 363 normal 385 Fernli mamentlzm 78 Fermi surface 318 Permi temperature 361

fermians anti-cornmutat ion relation 22 creation and annihilation operators 17 definition 7 Feynman rules 220, 280, 291, 325 field operator 36 Fock space 21 Fumi's theorem 416 gap function 135

GeIl-Mann-Brueckner theory 255 Gell-Mann-Low theorem 197 Goldstone's theorem 237 grand canonical erlfielnble 349

IIartree approximation 6 1 Hartre(:-Fock (HF) 5 1 approximation for irreducible self-energy 299

ap proxirnatioh for two-par t icle

Luttinger" theorem 336

Green's function 327 atomic systems 59 calculation of excitation

rnean-field 51 Minkodski scalar product 285

energies 62 calculation of ionization en ergies 6 1 potential 53, 62 restricted H F 67 restricted procedure

(RHF) 71 Roothaan procedure 71 un~estrictedRE' (UHF) 68 variational principle 52, 63 Heisenberg equation of motion 144 identical particles 3 indistinguishability of identical particles 3 internal energy 353 ionization energy 61 jellium 79 Josephsoh effect 138

Kondo effect 362, 413 Kondo Bamiltonian 414 Koopman" theorem 61 Landau f-function 369 Lehmann representation 174, 303, 409 Lindhard function 336 linear response theory 334, 391 linear respmse of quasi-par t icles 406 Iinked cluster theorem for single-partide Green's function 279 fnr t wo-particle Green's function 324 for vacuum amplitude 237 loop theorem 227

notrnal-ordered product 207 normal coordihates 109 n-particle Green's function 185

occupation-numbel representation 21, 3 1

pairing 207 particle and hole operators 205 particle number operator 23, 32, 45 partition function 350 Pauii principle 10 Padi spinor 40 Pauli spin susceptibility 374, 417 periodic boandary condition 39, 80 permutation gr-oup 7 permutation operator 4 perturb alion expansion BriLlouin-Wigncr expansion 192 at finite temperatures 349, 352 of the eaetgy shift 200 Rayleigh-Schrijdinger expansion 191 of single-particle Green's function 204 of t h e time evolution Operator 1 5 1 phase shift in the Kondo problem 415 phonon 103 acoustic 114 longitudinal and transverse polarieations 115 optical 115 phonon-phonon interaction 116 picture transformation 1 4 1

Reisenberg picture 143, 352 interaction picture 145 Schrodinger picture 142 plasma fpequency 89 plasma oscillation 89 plasmon 89 polarization insertion irreducible 333 polarization propagator 179, 268 diagrammatic expansion 329 retarded 336

dispersion relation 177

quasi-Newton equation 317 quasi-particles 305 creation and annihilation operators 388 distribution function 388

random phase approxlmatioa (RPA) 98 for the effective interaction 341 reduced Hamiltmian 129 reciptocal lattice 107 ring digrams 242, 249, 260 Sawada method 272 scattering function 393 diagrammatic expansion 394 irreducible 394 second quantization

for bowns 31 for fermions 17 operator representation 24,

32, 36 self-energy 281 irreducible 281 self-cunsislent perturbation theory 297 self-consistent procedu~e54, 65 single-par t icle Green's function 153 advanced 306 diagrammatic expansion 275, 286, 356

.

equation of motion 184, 327 Rartree-Fock approximation 301 incoherent 388 mixed particle and quasiparticle 409 for non-interacting fermians 166 spectral representation 174, 389 retatded 306 temperature dependent 352 for t r anslationally invariant systems 159, 281 skeleton 297 dressed 297 Slater determinant 9 completeness theorem 13 sound absorption 383 sound velocity 114 specific heat 370 of ~ e r liquid d 370 in the Koado problem 416 spectral function 175 gcncrslized 4b8 finite-temperature 390 spin-statistic theorem 7 spin waves in Fetmi liquid 383 step fdnct ion, representation of 167 sum rule 175 superconducting transition temperature 123 superconductivity 123 high Tc 363 symmetry breaking 136 symmetty dilemma of t; he Har tree-Fock prodedure 67, 71 symmetry pas tulate 7 symmetric wavefunction 6 symme t riaation operator 6

INDEX

433

thermodynamic limit; 39, 77, 78 time-evolution operator in the interaction picture 148

in the SchrSdinget picture 142 time-ordered product 153, 182,

time-ordered products 215 Wigner-Seitz radius 83 Wignet-Seitz cell 106 for

zero sound 381

352 T-rnatrh 418 Tomonaga-Schwinger

for normal products 212

equation

148

topological equivalence of diagrams 242, 275, 279 translation group group character 107 irreducible represent ation 107 translationaIly invariant systems 39 in Hartree-Fock approximation 73 transposition operator 4 tweparticle Green's function 179 diagrammatic expansion 321 relation with irreducible self-energy 283 Umklapp process 121 uncertainty principle 3

vacuum amplitude 224. diagrammatic calcnlat ion 237 vacuum s t a t e 22 for BCS quasi-particles 135 vertex 222 d~essed344 vertex insertion 344 vertex function 393 on the Fermi surface 399 irreducible 345 virial t heorern 63

Ward identities 386, 399 an the Fermi surface 404 Wiek's theorem 205

Many-Particle Theory,

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