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
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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
By means of these transposition operators, equation (1.1) can be written
This equation must hold for all state functions in the Bilbert space under consideration. The equation can then be inserted on the right-hand side of the identity
which yields
(a I B
19)= (a I P;BFjk
I q)
for all ( j , E ) and for arbitrary wavefunctions @ and 3P in the Hilbert space. This implies the operator identity
B = PT aFjk 3k
far all ( j , 6 )
FUNDAMENTALS AND EXAMPLES
5
In particular, if we take I? to be the identity operator, i.e., that Multiplication from the right with
SO
= 1, it follows
ek
yields
that finally we obtain
Thus, t h e operators that correspond t o transpositions of particles are selfadjoint and unitary (so long as they operate on the space of state hnctions of identical particles) . If we multiply equation (1.2) from the left with pjk, we obtain
The operatom which represent the observables of a system of identical particles must therefore commute with all permutation operators;
[B, pi;.,]= o
for all
blk).
Vlfe will now calculate the eigenvalues of the transposition operators. ly be an eigenfunction of pjkwith eigenvdue a(jk) :
Let
I t follows that 2 Q = P jk Z @ = a (jk.1 2 ?Y9 soa(jk) =I.
Since the operators $jk are self-adjoint, their eigenvalues are real. Thus
In particular, equation (1.3) must hold if system under consideration:
is
the Harniltonian H of the
Hence, any solution af the Schriidinger equation with proper symmetries under particle-ifiterchange must be a simultaneous eigenfunetion of H and of all (and thus of all possible permutation operators P).
hi
SYSTEMS OF IDENTICAL PARTICLES
6
It seems reasonable to expect that if a function 9 is an eigenfunctiun of all pjjjkt the eigenvalues of all i j k must be identical:
pj
= a(,,]q
far all ( j ,i c ) .
Indeed, we can prove this by writing the transposition
Fjkas
so that 'jk~
= a t 1 j ) " 7 2 k 1 b ( l z ) ~= '(12)'
for all possible (j,k). We now make t h e following definitions,
If P j k q = +I for all ( j , k ) , w e s a y that V is syrnmelric.
Thus, the state function of a system of identical particles must be either symmetric or antisymmetric, is symmetric and IPA is antisymmetric, it is clear that for any If possible permutation F we have
and
Pe, = spn(p). BA where sgn(P) = +I if P contains an even number of transpositions, and sgn(P) = -1 if F contains an odd number of transpositions. Symmetric and ant isymmetric functions are always orthogonal:
( q AI Q S ) = 0, Up to this point we have implicitly assumed that there exist simultaneous eigenfunctions, i. e., symmetric or antisymmetric functions, of all PjkBecause the transposition o p e ~ a t o r sdo not commute, it is not clear that there exists a non-trivial space of functions in which all transposition operators are simultaneously diagonal. We will construct such functions by an explicit procedure below. We start by defining a symmetrization operator by so
s
FUNDAMENTALS ANLI EXAMPLES and an antisymmetrization operator A by
Each sum runs over all elements P of the permutation group S N . 'If f (q,. . . ,rN) is an arbitrary function of IV variables, we can use the operat ion *S(XI
I .
--
i
XN)
-- 3 f ( x l r . ..
7
EN)
t o construct a symmetric function; and through the operation
we can construct an antisymmetric function, since for any Pjk we have
and
and thus
for all (j,k). Using this pwcedure, we can construct both symmetric and ant isymmet ric eigenfun ctions of any given many-particle Harnilt onian. At this point, we introduce a principle of fundamental importance, generally known as the symmetry postulate:
The Wilbert space of state functions of a system of identical particles contains either only symmetric or only ant isymmet ric functions.
In the first case, the particles are called bosuns, and in t h e secand case f e m i o n s . The sucalled spin-statistiw theorem [I], which we will not prove
SYSTEMS OF IDENTICAL PARTELES
8
here, states that bosons only have integer spin and that ferrniorrs only have spin equal to half of an odd integer. To prove the symmetry postulate, we start with an eigenfunction quof the Hamiltoniab with definite symmetry, i. e., Qo is either-symmetric or antisymmetric. Any possible state function Qir of the system can be characterized by a non-vanishing overlap with at least one state BIPo that can be obtained by operating on Po by any possible operator B. With pjkXPo= pQo, where p = +l or p = -1, it follows from equation (1.3) that
i. e., we have
pjk@= pb for any function d which is not in the subspace orthogonal to gXPo.In other words, any such function @ must obey the same permutation-symmetry as
90. As
of non-interacting identical particles, which are acted upon by an external potential. The Hamiltonian for this system is ah example, we consider a system
We assume t h a t we have solved the associated one-particle problem
(We wiH always use Greek indices to denote the quantum states which characterize the single-particle orbitals, fop example v = (n,E,m).) We can then construct symmetric and antisymmetric solutions a d to the full problem by simply applying the symmetrization and antisymmetriz at ion operators s and A, which were introduced earlier, to a product of singleparticle orbitals:
FUNDAMENTALS AND EXAMPLES
9
The prefadors are chosen such that dS) and dA)are normalized. Here, each k (< N ) in ~ ( denotes ~ 1 a distinct orbital and nk denotes the number of particles that occupy each state vk, A corresponding additional prefactor has no effect on dA), since the determinant vanishes for nh > 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
FUNDAMENTALS AND EXAMPLES
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
FUNDAMENTALS AND EXAMPLES
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 <
Ek
~O~~OWS:
This corresponds to the folIowing operations: (1) interchange the row with c j = k with each tow above it until it is the top of the determinant, which gives a prefactor (-1)j-I; (2) cross out the first row and the: last column of the determinant;
(3) normalize the new determinant. Thus,
at
FUNDAMENTALS AND EXAMPLES
The operators are defined as Iznear operators on the Hilbert space 7 i ( N ) . The: action of Ek on a general many-particle wavefunction @ is then completely determined by the expansion of Q in Slater determinants, Q =
6
We define the creation operator as the adjoint of 8k. Hence, the action of on a Slater determinant follows from the definition of adjoint operators. If a, E H ( N - 1) and @ b E X ( I V ) , we have
21
@LQC
I as)
=
(Qc
(
IW
b )
1{
1@
)
if there is a j with other wise
t = bj
and from equation (1.71, this is
if there is a j such that k other wise
-
N = x(-l)i-lb~l bl * bq,
= bj
fikbibcib;+~
-
7
- 6 e ~ - 1b~
i=l
-
{
(-l)j-'bia
if there is a j such that
0
otherwise
cj-1
SECOND QUANTIZATION FOR FERMIQNS
20
whew C denotes the N-tuple which is the result of inserting k between cj-1 and c j in c , Furthermore, this is equal to
{-
1{
1 a,)
if there is a j for which
0
cj-1
c k < cj
otherwise.
We have thus shown that
= -
{o
1
if there is a j with
cj-1
< k < tj
otherwise.
This result clearly. means (if cp < k
< g+l)
The action of 13: thus 'creates' an additional particle in the state 4 ~ As . with E k , the action of Ekt on an arbitrary many-particle s t a t e B = f,aC is clear from the linearity of the adjoint operators:
x,
According to equations (2 .I) and (2.2) the creation and annihilation operators map Hilbert spaces of fixed particle numbers onto each other. Hontever, the formalism of second quantization is especially suited for problems with variable patticle numbers. Therefme, it is more appropriate to
FUNDAMENTALS AND EXAMPLES
21
form the product Hilbert space 3 1 @ E = 0 7 f ( N ) from t h e Ailbert spaces ( N ( N ) ,( 1 of antisymmetric wavefunctions of N particles. We define the scalar product on 3 to be the natural scalar product
IN)
We can then regard Ek and 2: as operators on the space 3,the so-called Fock-spa ce , instead of as collect ions of ope~atorswhich map different Hilbert spaces X ( N ] anto one another. In single-particle quantum mechanics the states are usually represented by the quantum numbers in the Dirac notation, e-g., the state 1 nlm) with position representation ( r I nf'm) = We can characterize the Slater determinants in an analogous way by their associated N-tuples of indices: I c ) with the position-spin representation (xl,. . . ,r N I e) = Oc(xl,. . .,xN). Another equivalent possibility is the so-called oeczlpalion-num ber representation :
1 C) =I n1, n z , a s , *) with ni = O i f i @{cl, ...,c N ) and ni = 1 if i E (q,. .. , CN). For example, suppose dl,43 and d4 are occupied, so c = [1,3,4).This gives We should keep in mind that the position-spin representation of such an abstract s t a t e vector is simply a normal Slater determinant, jhst as before. By using the occupation-number representation, however, we can write the the action of the creation and annihilation operators somewhat more compactly;
or, in surrirnary
with
Likewise, we have
SECOrVD Q UANTIZATIUN FOR FEIUMIONS
22 and hence
If we act with an annihilation operator Ek on a 'determinant' which occupied, a 'zero-row' determinant remains, In this only has t h e state determinant, there are no longer any occupied states. We define this state as the vacuum.
Obviously, we have ikIO}=O
for all k in N .
By applying the creation operators to the vacuum state, we can generate every possible N-particle state. We obtain the basis functions through
This correspondence between t,,c,, . . . c,, ( 0) and the determinant containing the elements dci (xj) cont aina m i h u s signs, since the factor (- l)j-I was not included directly in the definition of the operators 2k. We now arrive at the most important property of the fermion creation and annihilation operators: their anticommutation relations
The proof of the antitommutation relations runs as follows:
The factor if is just -Oh unless we have n k = 0, in which case the entire expression w i s h e s . Hence, we can write the last expression as
+
Thus, we have (Ek61 belc) 1 nl,., ,,nk,.. .) = O for any arbitrary Slater determinant, and the first af the anticommutation relations above follows.
FUNDAMENTALS AND EXAkfPLES
23
The proof of the remaining relations runs analogously. As an example, we will also show the proof of the last one for the case A!'= k :
Thus,
Because of the property E'E I nllnz,.:.) = n k k k introduce the number operator N through
I
nl,n2$...), we can
From this definition, it EoIIows that
i m Sbter determinants, If we expand an arbitrary many-particle state in Slater determinants (or completely symmetrized functions in the case of bosons) , we obtain:
Thus, the expectation value of the number of particles in the single- article state I in the s t a t e I i V ) is
Since C , lfCl2
= 1,it follows that
SECOND & UANTIZATTON FOR FERMIONS
24
for ferrnions (n:c) = O or 1). In the remainder of this chapter, we will express 'normal' operators (i, e., the operators that we are familiar with from f i s t quantization) in terms of the creation and annihilation operators. The result is known as the second quandization representation of the operators. We will show that we have:
(1) far single-particle operators
with (ilhlj) = J ~ ~ ( x ) ~ ( z ) $ ~ (and x)~x; (2) for local two-particle operators 1 $'= 2
N
.
t,3=1
1
00
-t-tB ( z ; , z ~=) (ij 16 1 k&)ciejc4Ek 2 a.] .kt= 1
(2.7)
8#3
with (ijjiil kt) =
~ ~ ~ ( r ) $ ; ( x ' ) vri)dt (x, (
L T ) ~ ~ dx ( Xdx'. ~
The action of the operators in second quantization defined by equations (2.6) and (2.7) is defined only on Slater determinants, Hence, to detetmine the action on an arbitrary many-fermion wavefunction, that wavefunction must first be expanded in Slater determinants. In contrast, the operators in first quantization act directly on the coordinates of the many-particle wavefunction. It is important to emphasize that equations (2.6) and (2,7) imply that the operators in first and second quantiza*tion are equal, i.e., their actions on an arbitrary many-fermion wavefunction lead to the same result, To prove this identity, we must show that the actions of the: operators are identical on the complete set of Slater determinants; i. e.
Ch(z;)I @J = C (i I h I j ) i ? fI ~a,)~ for all O, and 1
-
2
N
C u+i
1 G ( x ; , i j ) j a,) = 2
\CCI
C
-
(ij 16 I k ~ ?-tat )c;c~c~ 10 Z, b)for all a,.
dilcl=l
TOprove these identities, we once again expand the states on the left and on the right in the complete set
( a b )and show that the expansion coefficients
F77NDAMENTALS AND EXAMPLES are identical, a'-e., that the matrix dements satisfy
'
for all
a,
and
ah; and
that
for all G b and a,. We begin with the single-particle operator in the first quantization. From equation (1.6) we have
The ( N - 1)-fold product of integrals in which h(z;) does not appear, eontains at least one factor in which t h e indices of the single-particle orbitals are interchanged. As a result, this factor vanishes. It follows that the matrix element vanishes if two or more indices in the N-tuples b and c are interchanged. Thus, there are only two c a s e to consider:
The ( N - 1)-fold product of integrals which do not contain h ( x ; ) vanishes unless
$ECOND QUANTIZATION FOR FERMIONS
-
Hence, we must also have epli c i , so that we obtain a non-zero expression only for the identica permutations:
I
(2) 6 and
c
differ in one index; e.g., bk
6 c and ci # b, or in detaiI
In this c a e we c a n only obtain a non-zero value when the integral with h(zi)contains the two different orbitals #, and &. Thus, the sum over i only gives one contribution for the case i = 1 , whereas the sum over all permutations gives a non-zero contribution only for one particular permutation Po. This permutation is given by
'Inthis case we obtain
We must now show that the second-quantization representation of to the same matrix elements:
Po leads
FUNDAMENTALS AND EXAMPLES If there are some i = b, and j = c, , we have
where I, and tSindicate that the states b, and c, have been annihilated. The matrix element apparently vanishes if b and c differ in more than one index. We must then further consider the following two cases:
(0
(ti@, I Zj@,) = (ac1 ;Tij
1
= ni
1
=n
for i # j for i = j .
Thus w
(2) bk @ c and
ct
$ b. Tn this
case, we
(EiabI ?j@c)
= 1
{0
have if i = S k and j = CJ otherwise
where the respective signs correspond t o the sgn(Po) above, so that we obtain
(@a
I&/
at)
= sgn(Po)(bhI
I ccb
Thus, the statement is proven for single-particle operators, The calculations become rather lengthy for two-particle operators (particularly ih first quantizakion), In summary, one obtains
(2) br, # ch for one k, but b, = e , for s # k:
(3) 6)
# ck
and 61
#
ce for one
L and one &, but b, = c, for s # k,1:
SECOND QUANTIZATION FOR FERMIONS
28
(4) If the Slater determinants Ob and 0,differ in three or more indices, the matrix elements w i s h ;
It should be noted from the cases (2) and (3) that, at most, the overall sign changes if the diffeient orbitals are not in the same rows of the determinants. The proof that the matrix elements of the two-particle operators are identical in &st and second quantization will not be p r s u e d any further. The calculations a r e similar t o the case of single-particle ope$ators. As an example, we wilI show equation (2.9) above for second quantization, since we will use this equation later:
The second matrix element is only non-eero when we have
From this it follows that we must have either
( 1 ) k = j and L e i , or
In these cases we obtain with the help of the axticommutation relation equation (2.4):
c-t; -t c ~ E [ c ~a,-) = c-t~-t i i t I@,) j
I
2
3
t = 6ij.6ii.i
I O C ) - ~-; tc-e-. tc -
= 6ijYIi 1 @ c } - n i n j
% 3 f
1 @c)
I Q,)
FUNDAMENTALS AND EXAMPLES so, combining the two cases we obtain
Thus, we have seen that the sorcdled second quantization, within the present framework, is no more than an alternative formulation of quantum mechanics, and that this formulation is completely equivalent to the usuaI one. However, the second quantization provides a tool that permits a description of processes such as creation and annihilation of particles. Such processes cannot be discussed from the basis of the Scht6dinger equation, but only from within the framework of quamtum field theory.
Chapter 3 Second quantization for bosons Second quantization for bosons is anaIogous t o second quantization far fermions: the action of the creation and annihilation operators is defined on t h e complete system of symmetrized functions
Again, we represent these states by the owupat ion numbers n, in the Dirac notation. The occupation numbers declare that the single-particle state i appears n; times in (cl,. . . ,c N ) :
with the position-spin representation
In contrast to the case of the fermion wavefunctions, the occupation numbers are not restricted t o only the values 0 and 1. We then remind ourselves once more of t h e definition of the ferrnion creation and annihilation operators:
If we remove the prefactor Ok and the restrictions that the occupation numbers are less than or equal to unity from these definitions, we obtain almost immediately the definitions of the boson operators. Explidtly
SECOND Q UAiVTI2ATION FOR BOSOM
32
The square roots in these definitions (which are the only additions we had to make to the action of the ferrnion operators Ek and i.1) are chosen so that we can introduce the number operator just as in the case of the fermions
since we then have
By applying the creation operators on the vacuum, we can generate all the basis functions of the type in equation (3.1):
-t are the adjoints of one another, and the following Furthermore, bk and bk commutation reIations hold :
The representations of single-particle and two-particle operators for bosons are identical t o the representations of the fermion operators:
and
N
-C
00
C
1 1 ;t*t* = ;(xi, xj) = ( i j I C I ki?)bi b j b f "b k . 2 2,3=1 . 2 63.kP=l
(3.4)
Chapter 4
Unitary transformations and field operators The considerations up to this point have been based on a a fixed orthonormal basis of single-particle functions ( s p k ( x ) ). We will now discuss what happens when we transform to another orthonormal basis ( X h ( x ) } . We begin by expanding the old orthonorrnal basis in terms of the new one :
We are dealing with an orthonormal basis; hence, the transformation must be unitary: D ~ D = ~ = D D ~
D
The inverse transformations are
The wavefunction pk ( 3 ) is the x-representation of the abstract Rilbert space vector i.L I o), so
Thus, if we interpret the new basis functions ~j ( x ) as the x-representation of the Hilbert space vector I t.1 O), i.e., the operators B tj act relative to the 3
34
UNITARY TRANSFORMATIONS...
4act relative
basis { X j ( ~ ) in } the same way as the operators { p r ( x ) } , the new creation operators must satisfy
t o the basis
The corresponding inverse reIat ions are
The transformation t o a new single-particle basis thus corresponds to a unitary transformation of the creation operators. Such unitary transformations will play an extremely important role in later chapters. If we define the annihilation operators corresponding to htj as the adjoint operator of o- tj
we can immediately derive a relation between these and the annihilation operators Ek :
Hence
The inverse transformation of equation (4.5) is
From equation (4.6) we see that the transformation between the annihilation operators Ek and tk is unitary; it is the transpose of the transformation of creation operators. The fact that the new creation and annihilation operators obtained by a unitary transformation satisfy the same cornmut at ion relations as the originaI operators is of crucial importance. For example, it follows from [it,e:] = 61,k
F U N D A M E N T U S AND EXAMPLES that we also have
Furthermore, the representations of operators in terms of these sets of creation and annihilation operators are formally identical; for example
=(4
=If) by equation (4.2)
We will now consider a particular example of such a basis transformation: as our new basis we choose the eigenfunctions of the x-operators in the z-representation [remember that x = (r,s) = (position, spin)], which are 5-functions: xklY) corresponds t o 6(x - y).
The index k thus cozresponds to the continuous variable x. The functions 6(x - y) form a system of single-particle orbitals, which describe particles that are localized at the p o s i t i o ~ p i n - p o i n tx. This means that the probability of finding the particle at any other point y -# x vanishes. Since the system of &functions farms a basis, we can expand any possibIe singleparticle function in 6-functions:
Moteover, the 6-func t ions are orthogonal:
However, the 6-functions cannot normalized in the usual sense but this will not cause any problems in the present context. The transformation of basis introduced earlier
UNITARY TRANSFORMATIONS...
36
then becomes Again, the variable z takes on the roje of the index le. The corresponding transformation to the new creation operators
and for the new annihilation operatots we obtain
The operators 4t(r) and +(z) me traditionally called field operators. It is an importiifit fact that these operators satisfy the same commutation "t b;, - respectively, i.e., for relations as the original operators ii and bi, ferrnions
E!,
and for bosons
The representation of single-particle and two-particle operators by the field operators proceeds according to equations for ferrnions (2.6) and (2.7), and equations (3,3) and ( 3 , 4 ) for bosons, respectively:
We end this chapter by discussing a pictorial interpretation of the field t creates a particle in the state pi, which is charoperators. The operator ?i acterized by the quauturn number i. Similarly, the operator q t ( x ) creates
FUNDAMENTALS AND EXAMPLES
37
a particle which is localized at the point x. Thus, for the formalism to be consistent, the position-spin representation of the state 4t I 0 ) must be a 6-function centered at x , We give a brief verification that this is the case:
which yields
Chapter 5
Example: the Hamilt onian of t sanslationally invariant systems in second quantization The Hmiltonian of an interacting many-particle system is in first quahtization
For simplicity, we have assumed that the potentials are spin-independent. For a tranlsationally inva~iantsystem, the external potential must be const ant u(r) = u = constant and the particle-particle interactions can only depend on the relative positions of the particles v ( r , r I 1 = v ( r - rI 1.
Translationally invariant systems are necessarily infinite. This idealization implies that we are considering system where we can ignore surface effects. The first question we encoufiter when we wa~1.tt o express the Harniltonian of a translationally invariant system in second quantization, is the choice of a suitable single-particle basis. To obtain this, we consider the system of a large cube with sides L and volume fl = L ~ At . the end of the calculation we go t o t h e so-called thermodynamic limit N 4 m, R 4 oo while keeping NJR tonstant. To make the calculations t r anslationally invasiant , uTeimpose periodic boundary conditions on the edges of the cube. If we solve the single-particle Schrodinger equation without a n external potential with such
40
HAMILTONIAN OF TRANSLATIONALLY INVARIANT,.,
boundary conditions, the eigenfuactions will provide a suit able system of functions. The cigenfunctions are the plane waves
These functions are momentum eigenfunctions with the discrete momenta k = (27r/L) (n,e, nyel n,e,), where n,, ny,n, are any integers and e,, ey,e, are unit vectors in the r-, y- and %-directions. For spin-112 ferrnions, which we will hereafter restrict ourselves to, the ~ , ( s ) denote the usual Pauli-spinors:
+
+
1 for s = +1/2 0 for s = -1JZ
Q for s = +1/2 1 for s = -1/2, (The representation of x,[s) as functions of the double-valued variable s = f1/2 is equivalent to t h e traditional notation in the form of twbvectors
The countably infinite number of functions given by equation (5.1) form an orthonormal system:
By a theorem from the theory of Fourier series, the system (4kg}is complete in the space of square-integrable functions which are periodic with period S2. We will here only be interested in functions that are defined only on the volume R itself, but the completeness will of course apply here. From equation (2.6), we obtain the second-quantized representation of the kinetic energy-operator in the single-particle basis equation (5' 1 ):
Evaluation of the matrix elements yields
FUNDAMENTALS AND EXAMPLES
Thus, we obtain in all
Similarly, we obtain for t h e external (tonst ant) p ~ t e n t i athe l represent at ion
In order to calculate the interaction term, we expand the particle-particle potential in a Fourier series
with the components given by
We
can
write the interaction in terms of this Fourier series as
and, by using equation (2 -91, the second-quantized representat ion becomes
, kl,
rrl
$t
k2 n2 'k4~4
The matrix elements are
.. 'k303
Finally, when we insert this exp~essionfor the matrix elements into the represent ation of the interaction potential and rename the summation indices according to
we obtain
the result
From equation ( 5 , 6 ) , we see that there are two advantages plane waves as a single-particle basis.
in using the
(I) T h e kinetic energy is diagonal in this representation:
(2) The translational invariance of the interaction potential rfiakes it possible to use a Fourier transformation with respect to t he relative cuordinake (r r') only, that is, i b is not necessary to use two separate Fourier series with respect to both r and r'. This means that the number of independent sums in the second quantization representation of the interattioh potehtial can be reduced by three.
-
We will use the result equation (5.6) later in the discussion of the homogeneous electron gas.
Chapter 6 Density operators In this chapter we will discuss operators which have expectation values that correspond ta the particle density, Consider first a single particle. The probability of finding this particle at z in state v is given by
In this case, we then apparently have two possible ways of defining a density operator. (1)
~ ( x ) ~ ~ ~Thep~obabilitydensityp,(x)isthenjusttheexpec) ( x ~ . tation value of the operator p(z) in the state ( v):
(2)
? z Y ~ j v ) ( v Accardingtathisseconddefinitionafadensityaperaj. tor we obtain p, as the expectation value of ii, in the 2-eigenfuactions 1x1: (x f fiu 12) = (2 I V)(V I x) = p v C x ) ,
Even though these two density operators lead to identical expectation values, the operators themselves are not at all identical. The operatar F(x) is the projector on 1 x) and ii, is the projector on 1 v ) . For later purposes we will now once again calculate the x-representation Cposition-spin rep resent ation) of the operamtori ( z ) :
Thus, in this representation the operator b(z) is a local multiplicative operator. Just as for loeal pdentials,where we only need to write the diagonal
D E N S I T Y OPERATORS
44
elements and can imply a 6-function; for example, we write
instead or 6 = v(x1)d(xl - x2).
Similarly, we wiIl only write
for the density operator. We now turn the discussion to many-particle systems, If a system is in a (symmetric or antisymmetric) state Q ( z lI z ~. .,,,x N ) , the probability density of finding one particle at x is given by
6 t h e density is normalized on the particle number N . Again, there are two possibh ways of defining a density operator,
In equation (
(1) The natural many-particle generalization of the single-particle density operator discussed above is (in the $-representation)
that we have again a lacaI multiplicative operator in the zrepresentation. From this definition it follows that so
and because of the assumed symmetry or antisymmetry of expression is
a, this
We also write dowa the second-quantized representation of this-oper-
FUNDAMENTALS AND EXAMPLES so that the second-quantized represent atian can be written
where
Hence, we arrive at
This result is, perhaps, not too surprising, since we demonstrated in Chapters 4 and 5 that the operator (titL)in general counts the particles in state k. Because of t h e unique correspondence between these creation and annihilation operators and the field operators, the operator qt(e)$(z) then counts the number of particles in state +, i.e., the number of particles at position x with spin s. We should then expect that if we integrate over all x we will obtain the total particle number operator, equation (2.51, defined earlier:
(2) The second possible way of defining
where nkp = (P I Etl l k
a
density operator is through
I W). This definition yields
DENSITY OPERATORS
Whereas the v- and 2-representations were constructed in a formally similar way in the single-particle case, the correspondence between p(x) and nQ is not that obvious in the many-particle problem. In particular, the operator b(x) is an operator that acts on the many-particle Hilbert space, whereas nq acts on the single-particle Hi1bet.t space. However, the trace of i'i~ is equal to the particle number. This is easily verified in the x-repr'esentation:
On the other hand, the numbers nkf above are precisely the matrix elements of the operator iilyin the underlying single-par tick basis:
FurtIrermore, the matrix nkt is Hermitian, since
It follows that we can find a unitary transformation D which diagonalizes
The matrix elements on the diagonal are obtained from
FUNDAMENTALS AND EXAMPLES If we define
and
the operators iilt and
are adjoint, and we have
Thus, if we had started with ( x I t I 0) as the underlying single-particle basis, the matrix n k t would have been diagonal, and the operator iig would have had the following form:
The numbers nk have in this diagonal form a clear meaning: ak is the occupation number of the orbital I k) in the many-particle state a. We can write the expectation value of an arbitrary single-particle operator F, given by M3
in the many-particle
state
XP, in a more compact form as a trace with the
help of the aperators iiq:
We conclude this chapter by considering the so-called density matrix. This quantity is defined by (z,r')
= N J Q*(x',
r z , . . . , x ~ ) Q ( x 1, 2 , . . + ,z ~ ) d ~ - ' x -
(6.4)
Also in the case of the density matrix, we can define two different operators which have matrix elements given by equation (6.4):
DENSITY OPERATORS
48 (1)
N
6(x1 - x:)6(x - x;). The so-called 'density matrix operator' defined in this way is nan-local in the 2-represent ation; explicitly, this means that P(x, XI)
=
We can use this result to show that the density matrix is the expectation value of the operator @(ae, z') in the many-particle state Q:
We also write down the representation af the density matrix in second quantization :
where
PI
~
*
I jl
~
=
/ /
=
J dg J d g '
j
dy
dyt (i I i ~ g I &,,,I '
= 9: ( ~ ' 1~j
I Y) (P I j )
p:(y')b(rt - ylj'bjx - Y)P~(P)
(XI -
FUNDAMENTALS AND EXAMPLES Thus
&
=
P(Z, a
C ~ t t ~ ~ : (= d ) ~ ~ (XI
$+(X')~(Z).
(6.5)
i,j=l
(2) In the second possible represent ation of the density matrix, we use the operator Aq defined earlier to form a non-diagonal matrix element in the 2-represent ation:
We will now also prove an important statement about the density operator f i ? y : Let 9 be a fermion wavefunction. Then
A; = i i ~
9 is a Siater determinant.
Proof:
'+' We start with a single-particle basis in which the matrix ( n k 0 is diagonal. Then
which means that nk is either 0 or 1. This implies that Y is Slater determinant. -t -t .2L 1 O), so that 't' Let V) = clcz..
sa ni = n k ,
I
with Thus
a
DENSITY OPERATORS and :A = h a . For a Slater determinant, the density matrix then has the form
Chapter 7
The Hartree-Fock approximatIon The Hartre-Fock approximation (HF)is based an the hope t h a t we can approximately describe an interacting ferrnion system in terms of an efTec tive single-particle problem. The starting paint is the idea that each p r t i c l e moves in a mean-field potential, which is produced by all other particles and, in case there is one, by an external potehtial. For the sake of simplicity, we wilZ assume that t h e potential is local in the x-representation. Thus, the exact Hamiltonian of the system
becomes in this representation
Our goal is t o approximate this twa-particle operator by an effective singleparticle potential
where the effective single-particle potential GHF, the 'Hart ree-Fock' p otential, is unknown at the present. The solutions with the correct symmetry of the corresponding eigenvalue equation
52
THE H A RTREE-FOGK APPROXIMATION
are then simply Slater determinants
The single-particle orbitals in these Slater determinants are forrnalIy obtained from the equation h H F P j = 7pj. However, sa Zang as we do not know the Hartree-Fock potential, we, of course, cannot calculate the single-particle orbitals from this equation. To arrive at a way of determining these we make the ansatz of a Slater determinant of single-particle orbitals for the exact many-body wavefunctian. By Ritz' variational theorem, there is a rational way to optimize this anslaiz by minimizing the total energy of the system. Minimizing the total energy with respect to the single-particle orbitals will then yield the Hartree-Fock equation for these , Thus, we vary t ha expectation value -of the craet H a d t a n i a n in a later determinant of single-particle orbitals:'
The Lagrangian multipliers E,. in equation (7.1) arise from the constraint that the single-particle orbitals be normalized. For the total energy of the system to be a minimum with respect to the Slater determinant trial functions, it is necessary that the expression in brackets is stationary, i e . , that the first variation of this expression with respect to the trial functions vanishes, (We should at this point recaIl that it is possible t o vary (Rev,) and (Imp,) independently. This possibility can however be rewritten as formally independent variations of rp, and&. From the red-valued functiorral equation (7.1) above we then obtain, as usual, pairwise adjoint equations, It is sufficient to consider only one of these.) We calcuIate the expectation vaIue of k in a Slater determinant by introducing second quantization ( cf. equations (2.8) and (2.9 )J :
The expectation value of the inte~e~actian patential consists of 'two terms; the first term is traditionally called the direct -lerm, and the second the exchange termExplicitly, the variationaI equation (7.1) becomes
w...-z o m -
:!$. =I
a%,
g ' D m
-
w m y ,
z+Qm
g g Yg
v
%
+%+ r r m 25g 5 m ,
2'
0
Gg 2 g mCJ PI - 5
zj;- r *rt -; 5 E ' c l cc
zi g z+ 'zP -W
P a p ,
egg g r + D-
5.m
r
- m< * o
9 3 g
5 5-5
TEE HARTREE-POCK APPROXIMATION
54
In addition to the external potential u ( x ) and the classical potefitiaI energy dy p ( y ) u ( x , y), the Hartree-Fock potential contains a non-local part, the so-called exchange operator. As we would expect, the Bartree-Fock potential is identical for all single-particle orbitals - this corresponds to our notion of 'identical particles'. It is easy to verify that GHF is Hermitian. To do that, we first note that we have p* (x,y) = p(y , x) (cf, equation (6.6)) We then abt ain
1
Since GHF and h,, are Herrnitian operators, the Lagrangian parameter which entered in equation (7.3) as an eigenvalue, must be real, and the eigenfunctions p, must be mutually orthogonal. In retrospect, it is clear that it was not necessary to ehforce the orthogonality of the single-particle orbitals with an additional term - t;j J p3(zjiPj (z)ds in the variational principle. The direct term and the exchange term both contain the density matrix. Thus, both terms depend on the single-particle orbitals which we still have to determine. It is clear that we cannot solve t h e hat tree-Fock equation like a normal eigenvalue pi-olrlem. Instead, we must use a so-called self- co nsis-lent procedure.
~ z _ ~ I
We start with (0) an initial approl'dmation pi ( x ) , i = 1,. ,. ,N and then calculate the
(1) Hartree-Fock
CHF from the density matrix
I
I
I
1
FUNDAMENTALS AND EXAMPLES with which we solve the eigenvalue equation
In generaI, this equation has infinitely many s~lutions.The last step then consists of (3) selecting N orbitals cp,(new) with the N lowest eigenvalues .,f
Using these orbitals, we calculate a new Rartree-Fock potential by repeating step (1). We c a n then iterate the scheme above until 'self-consistency7s achieved, that is, until the procedure converges so that the new otbitals obtained in step (3) are the same as the ones used in step (1). In general, we obtain from the solution of the eigenvdue problem not only the occupied single-particle orbitals, which are necessary to determine p( x , 9 ) and CHF, but, itz principIe, also an infinite set of unoccupied singleparticle orbitals. All these orbitals together form the basis of the singleparticle Rilbert space. We now proceed and assume that we have solved the Hartree-Fock equations for a single-particle problem. We then obtain the Hartree-Fotk approximation of the total energy of the system by inserting the Slater determinant of the occupied orbitals in the expression equation (7.2) that we derived earlier:
Adding and subtracting the double sum once in this expression yields
If we insert the BartreeFock equation (7,3) in this expression, we obtain
THE HARTREE-FOCK APPROXLMATION
56
From the normalization of the single-particle orbitals, it finally follows
=
1
&i
that
({ij 1 6 1 ij} - {ij 1 6 1 j i } )
-6
The last equality is obtained by a completely analogous calculation, but with the simple sum split in two equal pieces instead of the double sum. We also write out explicitly the expression N
+x((ajI
/ i i a j ) - ( a j 16 1 ja))
= (a 1 i + Q/ a )
(7.51
j=l
which we used to derive equation (7.4). Equation ( 7 5 ) follows from the Hartree-Fock equation by multiplying by &(z) and integrating aver 2. At this point the question emerges as to why the Hartree-Fock energy does not equal the sum of the single-particle energies E ; , If we were t o stay within the initial assumption that we can approximate the exact Hamiltonian by an effective single-particle operator (i. e., H ~ ~we) would , have N obtained the sum E; as an approximation of the total energy, Bowever, this initial assumption was used later in the derivation as the basis for the Slater-determinant ansatz, which we used in the variational principle. The total energy obtained from this a d s a t z should then be better than It is easy to show with a perturbation calculation that this is indeed the case. To do this, we assume that we have solved the H a r t r e e Fock equations and thus have determined the self-consistent potential. If we
xi=1
xi€;.
then abbreviate the interaction term in the Hartree-Fock potential as the single-particle operator j, we can then write the exact Hamiltohian in the form
If we then treat theory
8' as a perturbation, weobtain in zeroth order perturbation N
FUNDAMENTALS AND EXAMPLES and in first order
From these expressions we see that
Thus, the energy EHFwhich is optimized by using the variational principle agrees with the energy obtained from first order perturbation theory, whereas the sum over the single-particle energies E i only agrees with zeroth order perturbation theory. The fact that the entire interaction energy is subtracted again from the sum of orbital energies in the expression for EFIF has furthermore a clear interpret at ion. According to equation (7.51, each individual r; cont aim
i.e., ti contains the energies of the interactions with all other occupied orbitals. For exampIe, €1 contains the interaction with partide 2, partick 3 and so on, up to particle N ; cz contains once again the energy of the interaction between partide 2 and particle 1, and so on. Thus, all the interaction energies will be counted twice in the sum over 6;:
This double counting is then compensated for in the expression fop EHF, equation (7.4). We will here nbw also give a second derivation of the Hartree-Fock equations. This derivation will give emphasis to another property of the
58
THE HARTREE-FOCK APPROXIMATION
Hartree-Fock opepator. We begin this derivation by writing the variation of the energy functional as
ahd restrict omselves to variations such that
(1) @ 3. q@ is, just as @, a Slater determinant, and (2) only one orbital will be varied:
For $ we will now take the elements of the (yet to be determined) complete orthonormal family consisting of the occupied orbitals pi, .. . , c p and ~ the unoccupied orbitals p ~ + l. ., .. The variation obviously vanishes for the first ones. We then replace q p ,where p 5 N , with pp qpq,where q > N . This meahs that we consider a state ] X) which is in some sense an excitation of the state I a):
+
I X) = ti
with
{ire
+
The variational equation then becomes
where we have used the orthonormality of the functions {pi). The result we obtain from t h e variation is
PtJRTDAMENTALS AND EXAMPLES
59
The BartreeFock equation follows indeed from this expression, for if we define the Harttee-Fock operator RnF as before, this equation yields
Thus, the Hartree-Fock operator has no non-zero matrix elements between occupied and unoccupied states. Hence, we can diagonalize the Bemitian operator hHF*with a unitary transformation e ) , which has the property that it only transforms occupied states into occupied ones, and unoccupied states into unoccupied ones:
unoccupied
It is important to understmd that the form of hHF is invariant under this diagonalization precisely because af this block-form of the transformation matrix. However, with this form invariance, the matrix equation above means that
in a suitable bases, This i s in principle the Hartree-Fock equation. The property that the matrix elements of fi between the Hartree-Fock wavefunction and the excited state Q q p vanish
i, very important in applications of the HartretpFack wavdunction and is frequently referred to as Brillouin's theorem. As an example, we will now consider the Hartree-Fock equation for atomic systems:
T H E HA RTREE-POCK APPROXIMATION
60
We have set p(r) = C s p ( x ) = Cs~ G sin) the direct term. It is impossible to perform an analogous spin-elimination in t he exchange term. This means that the Bastree-Fock potential is spin-dependefit, even if the exact Harniltonian of the system does not depend on spin. The direct part of the Hartree-Fock potential is clearly nothing but the electrostatic potential due to the charge density of all electmns. This fact seems at first t o be nonsensical, since the effective potential that a c t s on an orbital should not contain the electrostatic potential of this particular orbital, but only that of all the other electrons, Fortunately, this 'selfinteract ion' is canceled by a corresponding expression in the exchange potential:
(for o < N ) . We emphasize that this term is canceled only when ip,(s) is an occupied orbital. Hence, the unoccupied orbitals formally interact with N particles,
We will now also briefly
discuss to what extent one can treat ionized and excited states within the Hartree-Fock approximation. It should first be emphasized that because af the nature of the approximation, one cannot attach any exact physical meaning t o the orbital energies. However, in the case of atoms one c a n approximately relate the orbital efiergies to the ionization energies, This is because one can describe a singly ionized atomic state by a Slater determinant, which ~01Itaih~ all ground state orbitals except for themost weakly bound one. Thus, one starts with the assumption that the rest of the orbitals do not change when these electrons are removed. The expectation value of the energy in a determinant that contains all ground state orbitals except for p k is obtained from equation (7.2): states
L~ >.
,,
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.
FUNDAMENTALS AND EXAMPLES
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:
FUNDAMENTALS AND EXAMPLES
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
FUNDAMENTALS AND EXAMPLES
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
<
1
i
FUNDAMENTALS AND EXAMPEES so that from equations
(9.1)and (9.2) we obtain
If we insert this expression in equation (10.4), we obtain
We obtain another representation of this result, if, instead of expressing the energy per particle as a function of density, we express it as a function of the secalled Wagner-Seitz radius r,. This quantity is defined as the radius of the sphere that contains the volume per particle of the system:
If we use this d h i t i o n in equation (10.5) and furthermore express all lengths in units of the Bohr radius a. = h 2 / ( m c Z )we , obtain
From this general formula we can, among other things, conclude that the Hartree-Fock energy for the unpolarized electron gas is
and that for the completely polarized electron gas we have
when T S is l a g e r than 5 . 4 5 ~ Hence, ~. we should in this case expect a phase transition to a ferromagnetic state, Unfortunately, there are no metals with densities this low, so we cannot expect to observe such a phase transition, Nevertheless, the electron gas model yields the correct result that t h e alkali metals, for which the model was originally constructed, are not ferrornhgnetic. Ferramagnetism in other, more complex materials, is a very complicated phenomenon. If we plat the energy per particle (for an unpolarized system) against r,, we obtain the picture shown in figure 10-1.
T H E HOMOGENEOUS ELECTRON GAS IN. ..
Figure 10.1 Ground state energy af the homogeneous electron gas i n the Hartree-Fock approximation plotted against r,. Since on the one hand the Rartree-Pock energy is less than zero, and on the other hand is an upper bound for the exact ground state energy, we have proved that the system is bound. Thus, we have h u n d a model for the cohesion of alkali metals. The theoretically calculated values eF -gM/~)min IEH
= -1.29 e V ,
rp = 4.83a.
agrees surprisingly well with experiments, which, for example for sodium give ( E & ~ ~ / N )=~-1.13 : eV, rzxP = 3.96ao.
In Chapter 7 we showed that the Harttee-Fock energy corresponds to fist order perturbation theory
when the difference between the exact Harniltonian and the Hartree--Fock operator is treated as the perturbation:
In a translationally invariant system in the present sense, equation (10.9) also holds for the partitian
&='&
-2
,
and
fir=v3
I
I
85
FUNDAMENTALS AND EXAMPLES %
in which the entire particleparticle interact ion is regarded as a perturbation. Even if this assertion is quite obvious in view of the distussion above, we will now explicitly go through the calculation for the electron gas. T o do that, we use the representation equation (5.6) of the Ham;.Itonian in second quantization with Fourier components of the Coulomb potential given by equation (10.1):
The contribution from the Fourier component vq=a will be calculated separately by using the fermion ant i-commutation relations, equation (2.4)* The constant terms &-; and ~ ~ / ( 2 ~ )cancel v ~ mutually , ~ (cf. equations (10.1), (10.2) and (10.3)). The remaining constant expression
a,
can be neglected in the thermodynamic limit. To show this, we transform the integral t a an integr-a1 over the unit cube through the change of scale given by Rnew = J P l d / ~ .The result is
where e is a constant which does not depend on volume. The corresponding energy contribution per partitle will t h u s vanish as R - ~ /in~ the thermodynamic limit. In all, we are then left with
We will now use this representation of thc Hamiltonian to prove the statement that we made above: the Hartree-Fock energy of the electron gas corresponds precisely to the zeroth and first order terms in perturbhtion theory, if the entire interaction is regarded as a perturbation. Since thc zeroth order terms immediately give us the kinetic energy of the electron gas, we only need to show that the first-order matrix elements
THE HOMOGENEOUS ELECTRON GAS IN.,.
86
of the ground state Slater determinant I } precisely yield the exchange term, which we calculated above, of the Hartree-Fock energy. The occurring matrix elements can only be non-zero when the created states are the same as the annihilated ones, h.e., when either (k + 4,C ) = (k', d) and
(kt- q,nl) = [k,u )
or when
(k+ q , ~=) (k,a) and ( k f - s,rr)= ( k ' , a 1 ) The sccand possibility is ruled out, since it is equivalent t o q = O and was already excluded from the sum. The matrix element is then only non-zero when q = kt- k and u = (7'. We can then write
+
Because q # O and k' = k q, we always have that k' anti-commute the two operators in the middle to obtain
# k, and we
can
Thus, we have the first order energy contribution
If
we now choose the ground state occupation we discussed previously,
namely
we
-7) -
for P ='+' for a ='-I immediately obtain the previous exchange term u- =
B(k+ k
- k)
which was what we wanted to prove. At this point the question arises as t o whether we can obtain parts of the correlation energy in the same way by including higher order terms in the perturbation expansion. The question c a n indeed in principle be answered with a yes, but the calculation is anything but sirnpIe. It turns out that every order in perturbation theory beyond the present one diverges. Only the summation of the entire perturbation series, or at least infinite sub-series,
j
FUNDAMENTALS AND EXAMPLES
87
which is possible far the limiting case of r, 4 0 by diagrammatical methods, leaves a finite, sensible result- It is possible to overcome these problems, but only at a great effort until we have had a detailed discussion of Feynman diagrams. We will first discuss another possible way of considering at l e a t a part of the correlation energy. This possibility comes from the observation that the main reason for the divergence of the higher order perturbation terms originates in the long-range tail of the Coulomb potential. Hence, one thing that seems natural to do is to divide the total Coulomb interaction into a short-range and a long-range part* We can then calculate the correlation energy of the short-range part by, for example, perturbation theory, since this part should not give rise to divergent expressions, However, a nonpert urbative approach (or infinite order summation) is required to calculate the Long-range correlation energy. The division into a long-range and a short-range part can be done in the following way:
where
and
Here the limiting momentum kc is a suitably chosen constant. We will return later to how t o determine this constant. The Fourier transform of vS,
where
vs,-
shows that indeed is a short-range potential: it follows from the properties of the sine-integral, Si(x), that fop large y this function tends to zero in an oscillatory fashion and that the function value F ( 0 ) is unity. Thus, we obtain .a reasonable screened Coulomb potential in this case. The long-range part, which is just the difference between the exact Coulomb potential and the the short-range part Gs,, i.e.
88
THE BOMOGENEOUS ELECTRON GAS IN.. .
is finite for ti - rj = 0 , where it assumes the value (2/.rr.)k,. We conclude this chapter by stating the result for the short-range correlation energy from second-ordet perturbation theory. We have t o calculate
Here Go is the Slater determinant of plane waves in the ground state occupation and Eo is the corresponding kinetic energy [the expectation value of &). The summation runs in principle over the complete system of all configurations ah,which can be constructed frdm plane waves. However, the matrix element
vaishes whenever a, differs from a. in more than t h e configuration of two orbitals, This reduces drastically the number of terms in the sum. The remaining terms can be integrated approximately (see Raimes 131). The dorninaht cont rib-utions in ( k e/ k F ) are
We give a heuristic estimate due to Raimes [3] for the constaat .C
So long as the value of k, is not determined, this result is not very useful. The final determination of 6, will be carried out in the next chapter, where we wiIl discuss the long-.range correlation in terms of plasma oscillations,
Chapter 11 Long-range correlations in the elect ron-gas: plasmons A
gas of charged particles, which move against the background of oppositely charged particles, is calIed a neutral plasma. Tt is known from classical physics, for example from gas discharge tubes, that such systems can exhibit collective density oscillations, so-called plasma oscillations, The fundamental frequency of such oscillations c a n easily be derived from classical considerations: let us assume that we displace a part of the totaI charge a distance x. If n is t h e particle density of these c h a p , a polarization
P E nex will arise. This polarization will in its turn give rise to an electric field
This field acts as a restoring force e E on the displaced charges, Their motion can then be described by Newton's equation
This is the equation of a harmonic osciIlator with the frequency
which is generally called the plasma frequency. One would then expect that the electron gas in metals can also display such pIasrna oscillations, and there is indeed clear experimentd evidence for this: if the metal is bombarded with electrons of a fixed energy, the energy loss spectrum, ie., the fraction N ( A E ) / N of the eIec.trons which have lost
i
I
LONG-RANGE CURRELATTONS,,.
an energy AE, will show clear maxima at integer multiples of hwp. These electrons have excited one Or several quanta of plasma oscillation, which are called plasmons. If we use the definition equation (10,6) of the atomic radius r,, we tan write the plasma frequency in terms of this quantity:
For metallic densities, we obtain values of hup between 3 and 30 eV, Cornpared t o thermal excitations of electrons at room temperature, which are approximately of m energy of 0.02 eV, this is a very large value. Since such plasma oscillations can occur in the electron gas, it is natural to attempt to transform the Hamiltonian to aform which contains ap explicit harmonic-oscillator part of the form
with suitable colIective coordinates Qk and corresponding momenta fi* Such a procedure was first suggested by Bohm and Pines [5].The transformation of the Hamiltonian consists of several steps: first we find appropriate collective coordinates *.
&k
= Q ~ ( $ I ,..- .
~ N I
which ate written as functions of the electron coordinates, We do not gain much so long as we merely express a part of the Hamiltonian in these callective coordinates. Rather, the crucial point is t o introduce independ elat variables Q k , and impose the constraint that
-
Q~ corresponds t o Qk = Q k ( i i ) . A
We will see under what conditions we can replace the operators Qk with the operators Q ~ In. principle, we are looking for a solution of the SchrGdinger equation I?(ii, fii)q(ri) = Eg(ri)
such that Q(ri)is independeht of Qk,i , e . ,
Expressed in terms of the canonical momentum tion becomes
&$= 0.
?k
= -ih&
this condi(11.2)
FUNDAMENTALS AND EXAMPLES
9 1.
Under this condition, we cad transform the Schrodinger equation by adding zero to it:
We will finally bring this transformed Schrodinger equation t o the desired form by acting on it with a unitary transformation u so that the resulting eouat ion
r~(dk,&,r;,~i~
zQ(Qklri)
contains a harmonic-oscillatar part. The transformed wavehhctiotl depends explicitly on the collective coordinates. We are, however, interested in a solution such that it satisfies the transformed condition equation (11.2) r
We will solve the problem that we have obtained in this way by using perturbation theory. We will take the oscillator part and the remaining kinetic energy uf the electrorys as the unperturbed Wamiltonian Ifo, Thus, the zeroth order wavefunction will be constructed from the product of a Slater determinant of plane waves and oscillator wavefunctions in the colIective coordinates. In what now follows, we will discuss the individual steps of the transformation in detail. To dcfine the collective coordinates, we first consider the Fourier expansion of the density:
where
If we interpret the density as matrix elements of the density operator ~ ( r ) introduced earlier (cf. Chapter 6)
and introduce the operators $k by
we can also interpret the Fourier expansion as an operator equation:
LONG-RANGE CORRELATIONS...
92
Explicitly, the Fourier components are
These moment urn components correspond to small density packages which execute oscillatiohs according to the aforementioned classical consider at ions. It is then natural to choose these as collective coordinates. For convenience we will however include a constant and define
where
We have
which implies that
in agreement with the property of that it changes the momentum of a plane wave pY(ry)= exp(ik'. ru)/ fl by k:
4
gk
Hence, the operators are not self-adjoint. We can then formally rewrite the entire Coulomb interaction as
FUNDAMENTALS AND EXAMPLES
Since the plasma oscillations represent the motions of many electrorls, the oscillations are mostly due to the long-range part of the Coulomb interactions, Therefore, it seems reasonable to perform the transformation to collective coordinates only for this part of the interaction, and to leave the short-range part as a function of the electron coordinates. With the partition of the Coulomb potential interaction into a long-range part and a short-range part that we derived in the previous chapter (equations (10.11) and (10.12)), we obtain
Up to this point we have merely rewritten the Harniltonian. In the next step we will introduce QI, as independent coordinates. (In the end, these *
will of course replace the
Ok.)We dernand that Q~ satisfy
For the canonically conjugated operators, &, it follows from
that These cornmut ation relations are consistent only if
since then
LONG-RANGE CURRELATIONS,.,
94
We should also note that the collective operators Q~ of course must commute with one another just as the Q ~ . For the additional term, we choose the operator
which, in a field-theoretical language, describes a free field 4 and the interaction of this field with the electron field Bk. The operator fi is self-adjoint, which one readily confirms from the remarks above and by renaming the summation indices k i -k ( Q and ~ j kcommute, since they act on differ* *
A
ent sets of coordinates). Apparently k gives no contribution when acting on the solution function that we are looking for (see equation (11.2). For the transformation we take
u
= erp
I-:c
k< kc
The operator
u defined in this way is unitary, since
By using the identity
provided
we can quickly derive the following relations:
FUNDAMENTALS AND EXAMPLES
We write these relations once again in an abbreviated form: Q
~
W
- fold)
)
.(new)
= Qk
ri
.!old)
= S,
and
p p ) - - (old) -Pk
~(old/n.w)
+Qk
(ala/new)
-(new)
-
-(old) +
- Pi
pi
With the help of these relations we can immediately calculate the transformed Hamiltonian (cf. equation (11.3)):
*('I
N e-pl
-
2rn i = l O
LONG-RANGE CORRELATIO.N$~.,
96
Thus, we obtain one part which only depends on the electran coordinates pi and r; and which describes an electron gas with short-range interactions, Furthermore, we have an harmonic-oscillator part in the collective coordiriates, which reproduces the plasma oscillations, and, finally, we have two terms which contain both particle and collective coordinates and which consequently describe an-electron-plasmon interact ion. To understand the individual contributions better, we go over to second quantization- To do this, we first define the usual oscillator creation and annihilation operators (the nan-hermiticity of the operators and Q~ can easily be taken into account and does not lead to any ~dm~lications):
For these operators, we have
and furthermote, with
[Qk, " Pk- 1 = ifi, we have
Analogously, we obtain the relation
Fram these relations the commutation relation
follows. Thus, the plasmons are bosons, which is what we would expect of oscillator quanta. mrthermore, we have
FUNDAMENTALS AND EXAMPLES
In this last expression, we see that the part
QL
^t= vanishes if we insert the relations = -Q-k and Pk and take into account that for each vector k in the sphere with radius kc, the vectar -k is also contained in the sphere: *
--Pk
We can then write the part of the Hamiltonian that only contains the plasmom in the following form:
H:!;~
We can now write the operator in second quantization with respect to the electron coordinates. By using
we obtain
We invert the definition of the plasmon creation and annihilation operators,
and with this result wc obtain
98
LONG-RANGE CORRELATTOl\rS..,
This operator describes processes in which the momentum of ~II eleciron hcreases an amount k by absorbing (annihilating) a plasmoh with momentum k (the first term), or by emitting (creating) a plasman with momentum
-k (the second term). is usually omitThe second elect ron-plasmon interaction operat or ted far the reason that it contains the sum aver the phase factors
which vanishes for a random distribution of the electron coordinates rg , This approximation is known as the random phase a p p ~ o x i m a l i o n(RPA). If we write this operator in second quantization, we obtain
Thus, when we neglect this part, we neglect all processes where the rnnrnentlim of an electron increaes by an amount (k+ kt)ei tlzer by absorbing two plasmons (with momenta k,k') or by emitting two plasmons (with rrlornenta -k,-kt), or by absorbing arid emitting one plasrnons of cadi momentum &k and fk'. In summary, we have cast the miginal Hamiltonian into a farm that contains two different kinds of particles, eIec trons and plasmons, and where the electrans interact both with the pIasmons as well as with dne mother, Rnwever, the interactions between the elect runs are now only short-ranged. The question whether the transformation is useful can ultimately be answered in more detail beyond the simple arguments given so far, if we can show that the interaction term is small. This is indeed the case. The tedious calculations will not be carried through in detail here. I t suffices to say that this can be proved by another unitary transformation, after which the remainder is truly negligible. This transformation changes t t ~ eup till now large terms so that the electro~isacquires all effective k-dcpcndent mass and t h a t the plasnons acquire a k-dependent eigenfrequency, so that they both show dispersion: m -, m*(k) and wp + w(lc). In the foIlowing considerations, we will tot ally ignore these small efiec ts and calculate the ground s t ate energy in firs t-order perturbation theory.
FUNDA MENTASS AND EXAMPLES We choose the interaction-free term as Ho:
The wavefunction is then to zeroth order given by
where the collective part is a
of oscillator ground state functions
(C is a normalization constant), while the electron part is the usual Slater determinant of plane waves;
1 @(xi) = - det
flk < k p , a
)I
[e"jvqX 0J-( si
,
The zerot h-order energy is E(O)=
(!Po
fi? ICL+f~pl 1 P o ) = ( @ I C 2m g 10)+ 2m d
($0,
I;rP1, $0).
i
The kinetic energy has almady been ealeulated in the previous section (cf. equation (10.8)), with the result
For the plasmon contribution, we use the fact that since we are calculating the ground
state energy, we
only need the energy of the zero-point motions:
N and the definition With Ic; = 3an
farm
p
G
k , / b ~ ,this expression takes the
100
LONG-RANGE CORRELATIONS.. .
We regard the short-range part of the electron-electron interaction as a perturbation and obtain the first-order cor~ectionto the ground state energy. - ( 1 ) which we c a n readily express in second quantization, The operator He+, changes the plasmon number; thus, any matrix element between states of equal plasmon number will vanish. In particular, this holds for the expectation value involving I Qo), so that this part of the Hamiltonian does not contribute t o ~ ( ~ 1 .
This integral can be evaluated by elementary integrations (see, for example, Raimes [4]. The result is
With hwp = @/r,3/3 [ez / a o ] , we lhen oblairl in all
Thus, we have calculated the energy as a function of the so far unknown quantity P,or, equivalent lly, kc. The folIowing considerations will show that we may determine p by minimizing the energy E(') E(') with respect to p. We h a w
+
After integrating over the plasmon coordinates, the expectation value of the original Hamiltonian in the electron part of the wavefunction ~emains.This
FUNDAMENTALS A N D EXAMPLES
101
part is not the exact ground state, but only a n approximation of it. In particular, this wavefunction in this approximation depends on R,:
By Ritzy theorem,
we will obtain an upper bound for the ground state en-
ergy when we minimize with respect to kc. The very small contributian proportional to in equation (11.8) will then be negligible (a more careful consideration will show that this term is of the same order as contributions already neglected in the present approximation), and we obtain
which gives
p = 0 . 3 5 3112~ ~. As an example to illustrate the order of magnitude, we consider sodium, for which r, w 4 a , . This yields P = 0.71 and',k = 2.95 a o , so that the remaining short-range forces are indeed limited to only a few neighboring electrons. With /3 fured in this way, the energy, equation (11,8), is then finally d'e termin ed. I n this calculation, the constraint equation (11.2) was however disregarded. One can easily convince oneself that the transformed constraint
o =
-
" (aid) fi) fjty (old) (xj (U t P,
is not satisfied by the above wavefunction Bo (Qhrx j ) = p(Qk@(zj).However, it can be shown [6],that a small change in the wavefunction SO that it satisfies equation ( l l . 9 ) , has a negligible effkct on the ~ T - U U T L d$ a l e energy. However, a corresponding statement does not necessatily hold for other ohse-rvables of t h e system. It should be emphasized that the energy, equation (11,8), o't>trtirred hy this projection contains part of t h e long-ranged corrdation energy due to the t ransfor matior1 to collective coordinates that was carried through. To isolate this contribution, we subtract the Hartree-Fock cncrgy, aquation (10.8). Using the value determined above for 4,we olrtain per particle
102
LONG-RANGE CORRELATIONS. ..
We can now finally give a definite value for the short-ranged correlation energy, equation (10,13), that was calculated in the end of the previous chapter. We obtain per particle:
If we add the long-range and short-range contributions, we obtain in total for the correlation energy per particle:
We will later derive this a p p r o h a t ion for the correlation energy of the free electron gas by using t h e perturbation theory of Gell-Mann and Brueckner (Chapter 221.
Chapter 1 2 Phonons We have in the last two chapters approximated the ion lattice in a metaZ by a constant charge distribution. This is obviously a very rough approximation which only leads t o semi-reasonable quantitative results far a few simple metals, A better approximation is obtained by using the Hamiltonian
This is also an approximation, since we are treating the ions as particles without inner structure with a fixed interaclion potential W. We abbreviate the set of elect ran coordinates by
and the set af ion coordinates by
so that we can write the Hamiltunian as
We will search for solutions of the full Schrijdinger equation
where the ionic motion is coupled t o the electronic motions. A partial decoupling can be obtained by the so-called Born-Oppenheimer approxirrcaliun. This approximation is b a e d oh the large mass difference between the electrons and the ions. Because of this mass difference, the ions move much more slowly in their oscillations about their equilibrium positions than the electrons. Consequently, the electrons are approximately in a stationary st ate for each instantaneous configuration of the ions. Conversely, for each instantaneous configuration, the electron cloud will slightly modify the ionion interactions by screenihg them. Thus, we first of all solve the electronic Schrbdinger equation
for each fixed set of ion coordinates R. Far each fixed R, {qm(g, Ft)} is a complete system in t h e electron mmy-particle Hilber't space. For fixed B, we can then expand the exact many-body wavefunction:
We obtain a set of expansion coefficients x,, for each R. We insert this completely general representation of lVn(r,lX)in the full Schrijdinger equation to obtain
If we multiply this equation by qbk(r,Rj, Integrate over the electron coordinates d 3 N r , and use t h e ~rt~hogonality of $,, we arrive at
Here, the matrix elements (7,bk I f i ( F t ) 1 )$ , are functions of TL, since we integrated only over the electron coordinates. We result is a system of infinitely many coupled equatiofi . The Born-O ppen heimer approximat ion
accordihg t o additional point symmetry operations, such as rot at ions or reflections. Because of the simultaneous constraint of translational symmetry, the number of geometricdly different point lattices is severely reduced; for example, there are five in two dimensions, and in three dimensions there are 14 different Bravais lattices. If each point in the Bravais lattice is assigned a basis, ~ h i c hcan consist of one or several atoms or ions, we obtain the real crystal. It should be emphasized that the point symmetries of the Bravais lattice are not netessarily point symmetries of the crystal, since the basis can break these symmetries, if it consists of several a t o m , For example, the right-angle Bravais lattice pictured ih figure 12.1 is symmetric under rellection in horizontal and vestical lines cutting the unit cells in half, whereas the crystal with the indicated basis is not, The parallelepipedes spanhed by the vectors ai farm the unit cells of the lattice. The unit cells completely fill the entire space. However, most frequently one uses other unit ceUs, the so-called Wigner-Seilz cells. These are constructed by drawing all lines which are perpendicular bisectors of the lines from a particular latticg point to all its nearest neighbors, as shown in figure 12.2. The unit cells constructed in this way have the advantage
Figure 12.2 Censtructjon ofthe Wigner-Seitz cell for a hexagonal lattice, that they are invariant under the poinl ayrr~rr~ctry of thr: lattice. These symmetries are clearly seen, as in the hexagonal I3ravais lattice illustrated above. We now define new basis vectors by
The components of any vector in one or the other basis are called contra and covariant components, respectively, in vector calculus.
FITlVDAMENTALS A N D EXAMPLES
107
Given any vectors x = q a l + x z a l + x3a3 and y = y l h + it foIlows that the scalar product of them is
+ hb3,
~2b2
The union of the points
is called the reciprocal lattice (of t h e original lattice). The origin of the name comes from the fact that the basis vectors b; have the dimension of a reciprocallength. Asis easily verified, the lattices R:) and G , are muivally reciproca1, so that the reciprocal lattice of G , is the originaI (real-space) lattice R,(01 The Wigner-SeYtz cell of the reciprocal lattice G , is called the Bra'llouin zone, The scalar product of a lattice vector and a reciprocal lattice vector follows from the general scalar product above: R:)
- G , = 2n (rnlnl
+ m2n2+ rnsn3)
2nN
with N E 2 .
(12.3)
We will frequently use this relation. Let u s now consider a ~nacroscopicvolume GI with sides Llal, L2az a n d Lsaj, so that the volume contains L I . Lz - L3 unit cells. For the sake of simplicity, we w i l l in the remainder of this section consider only rnoriatomic crystals, in which the unit cell only corrtains a single atom or ion. The number of ion&,J , is then equal to the number of unit cells:
To describe the extensive properlies of the crystal, we will apply periodic boundary conditions in a11 three directions a; so that we imagine the system continued periodically ifi these direclions. (We recognize this procedu~efrom the discussion of t ranslat ionally invariant sys terns .) Each group of t ranslations along ai ( i = 1 , 2 , 3 $then becomes a cyclic group of order L;. Recause each such group then is an abelian group, it has only one-dimensional hreducible representations, i. e., the representations are numbers in the complex plane. From thc fact that each group is cyclic, it follows that these numbers are t h e LT,; roots of unity, i.e., complex numbers whose Lith power is unity. These numbers lie on the unit circle in the complex plane. The irreducible representations of each group can then be written exp(iq;b, . a;), where q i = 2 ~ n ; / L with ; n; E 2. T h e irreducible representations of the element (0) R, of the full translation group of the crystal, which is a product of t h e cyclic groups, are then ")
with
with n; E 3 .
Since this represent ation rnat~ixis o~e-dimensional,it is identical to its group characters, From the orthogonality of characters we readily obtain the following two results:
Li C
I , , , . ,
(01
=
m,=I
J
~OTR;)=O
0
otherwise
J
for q a reciprocal lattice vector G otherwixe.
0
In the first of these two equations, the summation over n* = 1,... , L; includes all q-vectors which lie in the unit cells spanned by the vectors bi, Instead, we can of course sum over the Brillouin zone (BZ). If we also use
only if m = ( m lmz, , m3)= (0,0,O), we can use the shorter notation
In equation (12.5), we have used A to derlote a Kronecker-ddta modulus reciprocal lattice vector G . That concludes the short digression on crystal lattices, We will now further analyze the Schrodinger equation, equation (12.25, for the ionic motion. To do so, we begin by collecting all potential energy t e r m into one tot a1 patent ial, E ( R.~ . ,,Rj) , and for ease of not ation we will not explicitly indicate the dependence of on the electronic state rjlk. The SchPijdinger equation then hecomes
.
W
!
pi +I)&(*
xt(BJ = E ~ x ~ ( Q .
Because the ions in a solid body essentially stay a€ t h e lattice sites, we can (01 expand about these equilibrium positions. With Rj Rj + u j , we obtain
=
,
R
=
~ ( R Y ' ,. . . ,RY']
+ ~ u m m n *-v ~ , w J
m=l
~ ( -
0 )
FUNDAMENTALS AND EXAMPLES
The coefficients are derived from the ion-ion potential, for example
The linear term in this: expansion vanishes, since each coefficient is precisely the sum of all forces acting on each ion, and this force must be zero from the definition of the equilibrium position. Thus, we can approximately write the Hamiltonian for the ionic motion as
where
and
z(~(')),
The first term, is constant, and is rrot interesting for the discussion on ionic motion. We will now examine fiph a little more closely. To do so, we will, just as we did for the plasmons, make a. transformation, to collective oscillator coordinates, so-called normal coorda~tates:
Here the 811131 extends over all lattice points in t h e systenz volume S1. This definition holds for arbitrary vectors q. Frorn equation (12,3), however, we see that for any reciprocal lattice vector G
us+c= u q to discuss the normal coordinates only for vettors q in t h e first Brillonin zonc. Furthermore, it is convenient to restrict the discussion to the discrete set of vectors q ni 4 = q l b l + gzbz + ~ 3 b 3 with q; = 2n-, n; E 2. so it is customary
Li
This set of vectors is quasi-continuous. We c a n with the help of the theorem (1 2.4) derive the inverse relation
We introduce the operators
which are conjugate t o the normal coordinates pj tan then be expressed in t e r m af P:
u:. The momentum operator
l%e inverse relation can be obtaihed from equation (12.5) as
Far the coordinates defined in this way, w e apparently have
Next, we express the kinetic energy, equation (12.6),111the new coordinates:
FUNDAMENTALS A N 0 EXAMPLES
111
do'
+
Instead of summing over all = m1a1+ m2aZ m3a3 in here summed over all integer vectors rn = (ml , m2,m3). An analogous calculation for the potential energy yields
II, we have
We can simplify this expression by writing
for the difference between lattice vectors. To further simplify the expression for the potential energy, we examine in which way the matrix
displays the translational invariance of the oroblem, For a constant vector C and Xn,E R, - C we haw B / ~ R L= a/axk, and because of the translational invariance of the ion-iali potential
W ( R ~. ,. . ,Rj) = W ( X ~.,. . ,x J ) we also have
In particular, if we choose C = R?', we obtain
pf=
aE
axpxf
Xa= 0 (0)
- A O,b-a &q~
(01--RbTa 01
Xb=Rb -R,
We can then sum the expression
over the difference k
conditions:
n- m for e a d ~ fixed rn by using the periodic boundary
With equations (12,8) and (12,9), the potential energy can be written
If
we
use a matrix notation, this expression can be written
We now proceed to shdw that the 3 x 3-matrix A, is symmetric for each q, First of all, we have since
Furthermore, we can iritkrchange 'the order in which the two derivatives are taken. so that
If we mrnbine equations (12.111, (12.8) and c12.10), we o b h h
so
that we finally arrive at
The matrix hq is symmetric, so we can diagonalize it with an orthogonal t ransfor matian Yq:
We now define n e w ctlurdirlates through
FUNDAMENTALS A N D EXAMPLES
In terms of these, the potentiaI energy takes the form
It is easy to confirm that the kinetic energy remains jnvariant under the coordinate transfotmation Yq. For
we have
so that in rr~atrixnotation we can write
Since the scalar product is invariant under the orthogonal transform at loll ' Yq, it follows that
The Bamiltonian for the lattice motion then finally becomes
We have then succeeded in showing that the ionic motion can be approximately transformed to a system of independent oscillators with eigenfrequencies wq7,. The quanta of these oscillators are called phonons. The only approximation t h a t we have done in this, is t o omit the higher order terms
in the Taylor expansion about the equilibrium positions. We will, however, briefly discuss these terms later. The eigenvalues satisfy uq,, = w , ~ , ~ This . follows from combining the definition (12.9) with equation (12.10):
The matrices Aq and are thus identical, hence also (by suit able enumeration) the eigenvalues w & ~ , It ~ .also foUows directly from the definition equation (12.9) of the matrix hq that r~g,, = W ~ + G , ~ .Consequently,we only need to calculate the dispersion relation ~ ( q in ) the first Brillouin zone. The exact calculation of such a dispersion relation is quite tedious, since the matrix Aq, from which we obtain the eigenvalues w ~ by, diago~ nalization, consists of the Fourier transform of the second derivative of the interaction potential of all ions, In most cases, it suffices t o take into account only nearest neighbors or next nearest neighbors. TypicaIly, one then obtains the picture shown in figure 12.3.
F ~ r s t Brrllouin tone
Figure 123 The optical a d acoustic phanon brmches are depicted schematically.
If the basis consists of only one ion, one obtains a dispersion curve which is linear in q for small q : w ( q 0) = cp. Small q means Iarge wavelengths, sa these modes are the usual sound waves in condensed matter. This means that
is the sound velocity of t h e mediurn, Phonons with a dispersion curve of this k-hd are called acoustic phonons,
115
FUNDAMENTALS AND EXAMPLES
If the cryst a1 contains several different atoms, csptical p honon branches will appear in the dispersion curve. The quanta are called optical phonons because they can be excited by photons in the visible region. We have up t o this point not said anything about the polarization, ire., the direction of oscillation of the three different modes wq,#, p = 1 , 2 , 3 . However, we have already calculated these implicitly. The three vector components Q:, Q: , Q i of the oscillations with frequencies wq,l, uq,2,hq,3, are precisely the projections of t h e original displacement vectors Uq of the particles on the row vectors of the orthogonal matrix Yq:
Thus, the column vectors of the matrix Y + ,which are the row vectors of the matrix Y , give the polarization directions of the three modes wqlcl. The polarization directions depend, of course, on q, just as the frequen-
c,
tqIp
cies do, Depending on the direction of oscillation relative to the direction of propagation q, one distinguishes between trmsversa and longitudinal phonons:
for purely transverse phonons purely logitudinal phonone.
C ~ , P. q ' ItqlC, 1 . lql far
We cast H~~i n second quantization by introducing crcation and annihilation operators (analogously t o how we did with the plasmons):
It can be shown that these operators satisfy t h e boson commutation reIations, and that H~~ can be written in the form
This is the BamiIt.onian of a system of non-interacting bosans. The anharmonic terms, i. t , , the higher terms in the Taylor expansion about the equilibrium positions, which we have neglected up t o this point, can also be interpreted in terms of creation and annihilation operators. Ih order to write these terms explicitly, we use the inverse transformation
PC,,. =
[ MU,,,] fi
112 (h,P
4-
gLq,p) ,
This yields
so
that
Here, ([, denotes the pth component of the poIarization vector If we insert these expressions in the third-order term
and also use
k
g
and finally define
we obtain for the phonon-phanon interaction:
i&.
FUNDAMENTALS AND EXAMPLES
117
When, this expression is muItipIied out, it yields a sum of three-phonon processes which are subjected to momentum conservation q + q' + 9'' = G , Such processes dbviously play an importmt role when many phonons are excited, as is the case at high temperatures, They are important for the description of thermal conductivity and thermal expansion of crystal lattices. After these rdrnarks on the ionic motion, we now continue with the electronic problem. The Taylor expansion of the electron-ion potential leads to first order to
We then arrive at the following expression far the total Harniltonian for the dectrmiir: problem, equation (12.1):
where
Tn this expression, we have only kept the Grsl-order term of the higher-order terms in the Taylor expansion:
The solution of the ~ ~ " - ~ r o b l e ri .ne , , the calculation of the electron states in a fixed potehtial which is periodic with the lattice, is usually done with
the so-called pseud~potentialmethod. The electron-electron potential is then usually regarded only as a screening of the ion potential. The electron states calculated in this way aze called B l ~ c hstates. They artre of the form
where pk (r 4-
do)) = PI,(4
for all lattice vectors R,Yo>. In what follows, we: will assume that these single-particle functions are known. If and Ek, are the creation and annihilation operators of these one-electron states, we can write t h e operator He--ph,which eventually will have the form of electron-phdnon interaction, in second quantization ats
6:.
Since the Bloch functions satisfy dko
='b(k+~)o
it suffices to carry out the double sum over the first Brilloai~~ zone. To proceed, we expand the ekctton-ion potential in a Fourier series:
and
This yields
By using equation (12.15) we obtain
FUNDAMENTALS AND EXAMPLES
so, in total
Here we see that the electron-phonon interaction apparently vanishes for purely tramverse phonons, since for these we have q E = 0. It should further be remarked that the sum over q, which s t e m from the Fourier transform of the electron-ion potential, extends over all vectors q, and is not restricted only to those in the first Brillauin zone, However, the phonons created only have morrlenta which are contained in the first a
Rrillouin zone, since
This follows immediately from the definition of t h e hornla1 coordinates 0; and P: , from which the creation and annihilation operators were formed by linear combirlations.
Since each allowed vcctot q can be written unambiguously as the sum of a vector in the first Brillouin zone and a reciprocal lattice vector, we can split up the sum over q in the following way:
With wq,cu
we then finally obtain
= W ( s + ~ ) , and a
b1a
=E(s+~),a
The only part in this expression that depends on r is the exponentid. To write in second quantization, we need the matrix elements of the exponential and the loch functions: .
In the calculations of the integrals, we can use the fact that the functions pk(r) are periodic with the lattice and that the lattice constants are small compared to the scale with which the exponential functions change appreciably. Consequently, it is a very good approximation t o substitute the unit eel1 average value ip;,rpk for the product p;,pk in each unit cell (and thus for the entire volume R). The matrix element then becomes
In particular, for the case k' = k + q
+ G we have the exact result
1
= 5/,~3~~i+q+C(r~~ unit * (cell r ~d =3 J r ~ ~ q + ~ ( ~ ) ~ ~ (
where the overlap integral is evaluated over one unit cell, Hence, the unit cell average ab ovc is for the case k' = k g G exactly equal to the overlap
+ +
integral:
normalized with respect to the unit cell volume, We then obtain in total
Due t o the Kronecker-delta, not only does the surn aver
k' reduce to a
single term, but the sum over reciprocd lattice vectors G is also reduced, For k,q E BZ there can only be a single reciprocal lattice vector G(k, q) such that k1also lies in the first Rrillouin gone. This is clearly illustrated in figure 12.4. We have 0
if k + q is i n the first I32
GO otherwise
FUNDAMENTALS AND EXAMPLES
-
@ First
k I \
ar
+
b Go
First ez
Figure 12.4 if k q is not in the first BZ, we can write k' = k t q where k' is in the first B2,and Go is a recipracd lattice vector.
+ Go,
The entite electron-phonon interaction is then
The interaction thus consists of processes in which the momentum of an electron changes by q+ G , either by absorption of a phonon of mornenluln q or by emission of a phonon of momelit urn (-q) . If also G = 0, the process is called n normal process, and if G = G o (see figure 12.4),the ptocess is called an Umklapp process. (Such a distinction between normal and UrnkIapp processes can also be made for the case of the phonon-phonon interaction discussed previously.) The electron phonorr interact ion cquat ion (12.16) plays a role for example in the description of the electrical conductivity.
Chapter 13 Superconductivity The most striking consequence of the elect ton-phonon interaction is superconductivity: below a certain terriperature, the transition temperature, the electrical resistivity of many materials disappears. We cannot here go into the phenomenology of superconductivity, and can only address the basic concepts af the theory of Bardeen, C o o p e ~and Schrieffer (BCS)[7] and of Bogoliubov 181 and of Valat in [9].Our goal is just to show how helpful manyparticle theory is to the understanding of superconductivity. Furthermore, we should get a hint of the importance of symmetry-breaking and anomalous expectation values. We wiIl in this chapter only d k u s s phonon-mediated superconductivity within the weak-coupling BCS theory. The first clue as to the nature of the phenomenon is offered by the fact that the transition temperature is different for different isdopes of the same material. This means that the motion of the nuclei, i. e., the phonons, plays a role. The system of electrons and phonons will be described as in Chapter 12 by a Barniltonian of the form
Here
already contains the efrfIect of the rigid, periodic lattice. IIence, we will in this chapter by 'electrons' mean Rlach states. Ebr later use, we introduce the ground state of fiio': all states with single-particle energies ~k below the Fermi energy E F ,which depends on the number of electrons, are occupied; all others are empty (we will assume that € I , = E - ~ ) :
SUPERCONDUCTIVITY The operator
describes the phonons, which we assume do not interact with one another. For simplicity, we also restrict ourselves to one polarization. The electrmphonon interaction, Hedph,results in an effective electron-elect ron interaction by the exchange of phonons. We write &-ph as
where we for simplicity have ignored Umklapp processes and have taken the interaction Vq to depend only of the magnitude of q. The operator
bigq,
with the phonon-number operator, Cq but does commute with the electron number operator, Ck, i.ig&,.Hence, a state with a fixed number of phonons cannot be an eigenstate of H . T o get some understandihg of how this affects the time-evolution of a single electron under H , we consider a small time-increment 6 from an initial time t = 0
does not commute
and Taylor-expand:
At time t o = 0, the irlitial state is
and at t = 6 , the state is
This tim+evalution shows that the electron wilI emit a phonon with a certain probability, and Vq gives the correspondil~gprobability amplitude. In the further tirrle evolution, the possibility of reabsorbing phonons enters in addition t o the possibility of emitting a second phonon, This simple argument demonstrates that: (1) the electron is surrounded by a phonon cloud due to the interaction (2) a a consequence, the properties of the electron, in persion reIation r k , will be changed.
articular its
dis-
FUNDAMENTALS AND EXAMPLES
125
We can treat the electron-phonon interaction as a perturbation and calculate the e l e e t r o ~energies ~ using perturbation theory. In second order we obtain a seduction in the electron energy. We use the standard formula
The sum runs over all eigenstates I d,, ) of trons and nph phonons, respectively. If we set
(HL')
+ &h)
with n, elec-
we obtain
and since bq
I vac) = b = Spa I vac) , we furthermore obtain
only the s t a t e
with energy w , 1
= fktq
+W-q
= t k + g -tW q
leavcs a non-vanishing conk ributian, The energy shift then becomes
FUNDAMENTALS AND EXAMPLES
127
other phonons, and that such a t e m also appears as an inte~mediatestate in higher order perturbation theory. In these cases, processes for which
are particularly import ant ( i ~ lthe example above, equation (13.3) holds emc tly in second order perturbation theory for the nan-vanishing contributions). Equation (13.2) becomes with the approximation equation (13,3$
For the two electrons to affect one another through this indirect scattering process, the final states (k+q, 0 ) and (kf- q,d ) must be unoccupied. Right at the Fermi energy c ~ occupied , and unoccupied states lie closely together and c a n therefore easily be mixed, and the effective interaction equation (13.4)becomes relatively large and attractive:
The picture of the superconducting mechanism is hinted at in this attractive interact ion. An electron deforms the cryst a1 lattice in its surrounding, which implies that the electron emits phonans; the deformation and polarization acts as an attraction on other electrons. The elementary calculations above can be more elegantly formulated with the methods that we will discuss in later chapters. Here, we only wanted to demonstrate that the electron-phonon coupling can lead to an effective elec tron-electran interaction. To nnderstand super conductivity, it is impartant tu realize that a weak attractive interaction bctwcen two fermions alone does not necessarily lead to a bound state. We can illustrate this fact by the following example. Tf we transform t o center-of-mass and relative coordinates for the two particles under consideration, we bbtain the problem of a single particle in an attractive potential, In th~ee-dimensionalspace, this potential must have a certain minimum strength to lead to a bound state, A thorough discussion on the connection between density-of-stat es, dimensianality of space and the appearance of bound states can b c found in Economou [lo]. As a model, we consider a three-dimensional potential well with a characksistic length R of the range of the potential, and an effective depth Vo, The spatial extent of the wavefunction is characterized by a Iength a. The potential energy is approximately
Figure 13.1 A wavefunction of extent a/R in a potential wen of depth & and width R. and the kinetic energy is
The total energy is depicted as a function of a in figure 13.2.
Figure 13.2 Total energy as function of a far the simple model above.
With this simpIe estimate weubtain a bound state, i.e., Epot+Ek;, < 0, only for Vo > h 2 / ( 2 r n ~ ' ) In . this case the extent of the wavefunction is approximately equal to the extent of the well (I? a ) . Thus, for Vo < hh2/(2rnfi2) the reduction in interaction energy does not make u p for the increase in kinetic energy, and it is energetically favorable for the t w o ferrnions to movc independently in states with small kinetic energy, We now consider two electrons on the Fermi surface. They would like to lower their energy by farming a bound state of a relatively large extent from
FUNDAMENTALS AND EXAMPLES
129
the states available to them. Such states only have momenta at least of the order of k F , because of the Pauli principle. This is a many-body effect in so far as the other electrons are necessary t o occupy states with lower momenta. If the effective interaction is at tractive even only over small distances, the two electrons ~H arrange their spins to be parallel (otherwise, they cannot cbme near each other) and form a spin singlet. The understanding of this so-called Cooper instability of the electron gas by the formation of pairs in the presence of a weak particle-particle interaction at low temperatures (we consider here T = O and start out from a sharp Fermi surface at E ~ will ) of course not entirely clarify the question of the nature of the superconducting state. First of all, we have inadmissibly distinguished the two electrrrrns from all the others, Secondly, we have exttapolated the concept that 'every two electrons farm a Cooper pair' to the paradoxical situation that eventually no unpettnrbed electrons will remain within the Ferrni surface, 2. e., below the Ferrni energy. Therefore, t h e argument for the pair formation becomes questionabie, However, from this discussion we can understand that theelectrons tend to form singlet pairs in the presence of a weak attractive interact ion. However, in a many-electron system, this picture is complicated by the fact that we have t o consider how electron form pairs in the presence of other electrons, which also form pairs. Hence, the problem have to be solved self-consistently, with the electron pairing and the background of t h e other electrons determined simultaneously. It is impossible t o calculate the coupled electron-phonon system so precisely that we can obtain the tiny energy difference between super condue t ifig and normal states as differentea betwee11 total energies. (The energy scale of the transition temperatarex noltzrnann's constant 2 5 x eV, which sets the scale for t h e desired energy differences.) We are not interested in attempting to calculate details such as the dispersion relation fk of the Bloch electrons and the electron-phonon interaction to such an %accuracy,Rather, i t is important t o underst and the di#e re n ce between the normal-conductor and the superco~lductorphases. For this, it suffices to consider the piece of the Hamiltonian which is responsible for the difference between thesse t w o phases. We investigate the situation which k associated with Cooper pairs at rest. We imagine that we can cast the phonon degrees of freedam in t h e form of an effective electrorl-electron interaction, and that we in this i~iteraetianonly keep t h e part which contains two Cooper pairs:
We optimize the wavefunction of Bardecn, Cooper and Schrieffer with respect to this Harniltonian. This wavefunction is written as
SUPERCONDUCTIVITY
130
We quickly see something new and surprising in this harmless-looking tansatx The particle number is not a good quantum number for the general BCS function; parts with different particle numbers will be mixed. It should be rerna~kedthat the limiting case
lvkI = 1 and uk = 0 for rk < EF vk = 0 and IukI = 1 otherwise leads to the ground state of the unperturbed problem fii0) (cf. equation
(
The requirement of a determined particular particle number can o d y be. satisfied on the average, This is accomplished as usual by t h e introduction of a chemical potential p (= fF].To normalize the state, we choose
and uk real and positive for a11 k. This constraint can be satisfied by taking
in the expression which is to be minimized:
The result is
x (2 sin Bkr cos =
C ( r k - pj (1+ k
ci(dkj-dk
1 1 + C ;Vkl;1 sin(28k) sin(20k )
COS(~O~)J
ci(4kt-4kl
kk'
(13.7)
FUNDAMENTALS AND EXAMPLES The condition of stationarity can be satisfied by
with a an arbitrary constant, independent of k, and tan 2dk =
-(
e
~
~/(fk ~ -+ ~Pk )
)
where
Hence, we have t o investigate the equatioh
where we have assumed that Vkkt< 0. This equation has the form of an integral equation in the continuum limit of k and k':
+
with the abbreviation Ek G d l 12 ~ ( f k~ - pI2. The solution of equation (13.11) is pa~ticularlysimple if we take into consideration the fact that the effective interaction is particularly strong near the Fermi energy (et, % p ) and can be rnodeled by the sirnple approxirrlation if Ick
- p1 < W D
m d 1ckf- flI
< WD
otherwise, Here w~ is a cut-off parameter of the order of a typical phonon energy, a few meV. With this model interaction, the k-dependence of Ak and Bk simplifies :
A
fbr
[ E ~ - ~
otherwise
s i n ( 2 0 ~ )=
{ 0IA
1 / ~ k for
I E -~ PI
otherwise
1
LJD
1 for c k - p < -WD 1 w ~ ~ 2 ~ = - ~ + c ~ ( ~ o ~ ) ] f =a r {~ t~k [- -Ip -c r~o ] (13"11) 0 for CI,- p > w ~ . Then A is the solutioh of a szngle equation
The phase factor A/IA1 = ea" remains undetermined. In the so-called weak-coupling limit, which means that the product of the strength V of the effe~tiveinteraction with the density-of-states at the Fermi energy ~ ( 0 = ) ;1r dN a ~; is small compared to the Fermi energy, and that the denisty-of-states in the enetgy range o f t h e integration can be taken to be v(O), the sum in equation (13.15) can he replaced by an integral which yields
The difference in total energy (see equations (13.71,(13 .I),and the definition (13.10)) between the BCS state and the normal state is then
which in weak-coupling limit becomes, with equations (13.10), (13.131, (13.14) and finally equation (13.16)
FUNDAMENTALS AND EXAMPLES
In s u m m a y , we conclude that in the formation of the BCS-state (1) energetically deep-lyin-g states remain occupied,
Ivkl
I for
fk
< p - [A\;
(2) energetically high-lying states remain unoccupied, w k S 0 fur
ck
>p+]A].
Thus, the. occupations diffkr appreciably from those of t h e normalcond~lctingstate a, (see equation (13.1)) bnly in a very small region of order [ A ] , This leads to a reduction in the total energy of the order of exp{-2/(v(0J~)}, from equation (13.17) w i t h equation (13.16). This expression is not analytic in thc strength V af the potential and cannot be obtained from usual perturbation theory, ie., a power series in Ir. I n view of this fact we will now make a digression and discuss this fact in some detail. If we examine the calcuIationa above in some detail, we will see that they lead t o the following possible interpret at ion:
+
-t ;t-kl) is optimized with respect to the operator For caeh k,(uk ukckT
and the expectation value
is determined self-consistently.
This is analogous to the HartrerFock operator (here expressed through field operators in the shorthand z $(r)]:
4,
with a self-consistently determined expectation value
Just as the direct Coulomb potential and the so-called exchange field of all other electrons enter as an effective potential in the Har tree-Fock approximation, the so-called anomalaus expectation values of two creation or annihilation operators, i. e., ( E - ~ ~ Eplays ~ ~ )the role of an effective mean-field in the theory of supe~conductivity. The similarity with the Hartree-Fock approximation becomes clearer if we form t he following effective Hamil tonian by taking expectatioh values of the field operators above in the extremely short-ranged model interact ion
(ordinary expectation values and constant terms will be ignored, since they
are unihteres t ing) . This yields
with the expectation value
t o be determined self-consistently It should be emphasized that the expeclation value of in the state qBCS is not a simple sum of the expectation values of ~ ( k The ) ~ relation between them is shown most quickly by rearranging (moment urn- and spin-indices are supprmued) :
kd
FUNDA MEfNTALS AND EXAMPLES
135
The first two terms are the same as with the operator ~ ( k ) The . third term is a real number: it is the ubiquitous term which must be present in the calcuIation of the total energy in order to avoid a double-counting of the interaction energy (cf. the discussion of the Hartree-Fock total energy in Chapter 7). The expectation vaIue of the fourth term vanishes for the YBCs-onsaiz;this term i s bilinear in the difference af the expectation values and will henceforbthbe ignored, We obtain a meaning of A as a 'gap-function', i. e., the energy gap in the excitation spectrum, if we consider the elementary excitations in the BCS-state. This is most elegantly shown, if one convinces oneself that the operators defined by
so
that
satisfy
The newly introduced operators satisfy the usud anti-commutation relations, even though these operators are mixtures of fermion creation- and annihilat inn-op eratars:
By means of eqnation (13.21) and equation (13.221, these operators can then be interpret~das creation and annihilation operators of fermion quasiparticles on the ground state or on a new 'vacuum state', 1 qBCS), Using the definitions equations (13.51, (13.19) and (13.201, one can easily calculate the result
SUPERCONDUCTIVITY
136
[The relation is not exact in the absence on the right-hand side of the terms discussed above bilinear in the fluctuations i?d - (tttt))and (2; - (CE)], respectively.] The smallest possible excitation energy is for the simplified interaction equation (13&12)thus
The excitation spectrum of the BCS-state thus exhibits a gap. The Hamiltonian which, for example, contains the interaction of one electron with an electromagnetic field has the structure (electromagnetic field)
?tc final Einitid
state
(...+r +...?)
.
(..-4' +-..i.)
Since no quasi-particles are present in the ground state, the factors which contain at least one ;j. vanish when they act an ] qBcs), so the mean-field must then excite jwo quasi-particles; in experiments on optical absorption one finds stronger absorption only for photon energies larger than 21A1. This absorption lies in the far infrared (10" - 1012~z). The spatially homogeneaus state I QB cS), which we calculated within the approximation equatio11 (13.121, is characterized by a complex order parameter, which is the exp eet at ion value
If we irlsert the transformation to momentum-annihilation 0pera;tars and use the adjoint equation of equation (13.20), we obtain furthermore 1
A(r) = VE
C e"k'-')-'
('HCS
I e-klek~r I QBCS)
kk'
= y
F
c kk'
.-i(k1-k)-r
*
U ~ V P
/
I
(QBCS ? l t ~ ? l QBCB) ~~
\
Y
.'
Different vahles of the phase factor eia are degenerate in energy. It is abvious that we cannot start with a wavefunction for the zlormal state where ($l$T) = 0 (in this state the symmetry of u is not broken) and do a perturbation expansion in an operator which is invariant under thig symmetry and t hch oh tain a s t a t e in which the symmetry is broken. This means that the perturbation series cannot converge for an arbitrary m, This clarifies the breakdown of usual perturbation theory when it is used t o try t o obtain
FUNDAMENTALS AND EXAMPLES
137
the result equation f 13.17) for the gain in energy. The problem of particle number conservation of the BCS wavefunction can be removed by superposing wavefunctions with fixed phases to obtain a wavefunction with a fixed particle number:
with lukl and Ivkl given by equation (13.14).
B y this argument one
the following paint, which is essential for a deeper understanding of superconductivity. For example, we can imagine a piece of copper cut into two pieces and our experience teIls us that both pieces can be described by independent electron distribution. The number of electrons in one piece I s hot obtained exactly, The situation can be described in the grand canonical ensemble, calculated with a c o m m o n electrochemical potential. In this way, density matrices of different particle numbers add. Correlations between electrons in the different pieces are restricted t o at most a small boundam layer. If we apply an electrostatic potential, charge conservation will enforce a correlation betwem the two pieces: the total current through both picces is the same. Thesc long-range correlations t h ~ i soriginate in a canservation law, the law of charge conscrvation. In the BCS theory, a new rnacrobccrpic variable emerges with the complex phase ACT), We cannut imagine simply cutting a homogeneous superconductor with a fixed phase into two independent pieces, since both pieces have the same phase. Besides conservation laws, breaking of a syrnrnebtrg can thus also lead to long-range correlations. The situation is similar to a fernornagnet, in which a piece with homogeneous rnagnetizatioh cannot be thought to be cut into two independent pieces, but in both pieces the full rotational syrnmetry is euqally broken by singling out a magnetization direction. Energetically, st w t ~ swith different magnetization directions are degenerate in the absence of external fields, just as the ftmctions qBCS of different eta are degenerate in energy. All the above considerations corlccrn the case associated with Cooperpairs 'at rest'. We will here do without the discussion of 'moving' Cooperpairs and the connected (rather tedious) proof of the statemcnt that the BCS wavefunction really is a supercondacling state. Another limitation of the discussion here was the restriction t o spatially homogeneous systems. It is plausible that we can discuss a situation of a spatially inhbmogeneous, but slowly varying, system through a local BCS-picture with a local, spatially slowly varying complex ghp-parameter A(r) and that many phenomena p ~ o duce macroscopic equations for Air). This is preciesly what is done in thc does however easily overlook
theory of Ginzburg and Landau [ll] and Gorkov [12f. As a representative example of severaI ather effects, we will discuss here the effects of the fact that ${(r)liif(t.)is closely related to fermion operators, but possesses a macroscopic expectation value. Xn the presence of a spatially slowly-varying vector potential A(r), the wavefunction of the electrons changes to lowest order only through a space-dependent phase fact or:
This factor drops out of the density n(r) = C, ($! (r)$, (r)),b u t not from the order-parameter/gap function A(r) -.. { r ) where it enters twice. Thus, the order parameter A behaves in mahy ways like a macroscopic wavefunction. If we follow the wavefunction and A around a homogeneous superconducting ring, A can only change by a factor
[ f
1 = exp[2~il]= exp -2-
A ( s l d a ] with ( E 3,
because of the single-valuedness of the expectation value. Thus, the line integral f A . ds and also the enclosed magnetic flux J B . dS can only change by integral numbers of 'flux quanta' (2.rrhc/2e). This is the origin of the Josephson effect. We will with this digression of the phenomenology of superconductivity erid this part on the overview of the most important many-body effects and quasi-particles of solid statc physics. Next, we will turn to the development of a systematic, and,above all, diagrammatic met hod of calculation of manyparticle systems.
Part I1
Green's Functions
Chapter 14 Pictures The abservables of a quantum mechanical system are characterized by linear operators in a Hilbert space, The fact that these operators themselves do not represent any measurable quantities is crucial to the definition of the so-called pictures. One cah merely measure the expect ation values of the operators in addition to the square of the absolute value of sums of transition amplitlrdes
(.lcl l o 14). (For ease of notation, we will hereafter only write operators with the caret when the context requires an unambiguous distinction.) Lct A(t) be in general a tirne-dependentoperator, which is unitary at each time t :
We then have for a general matrix element
Thus, if we trarlsforrn all wavefunctions and all operators according to
all measutable quantities will rernairr unchanged:
Such a t~ansforrnation is called a picture transformation. Each timedependent unitary operator d e h s a particular picture. I n each picture,
PICTURES
142
the 'normal' wavefunctions and operators have what we may call an addit ional artificial time-independence. However, t h e structure of the stationary Schrijdinger equation as an eigenvalue equation is retained in every picture:
'We now consider a few examples:
(1) The Schriidinger Picture: A(t)= 1. We denote by the Schrijdinger picture the usual quantum mechanical picture in which the operators and wavefunctions have their 'natural' timedependence. We will hereafter for the sake of clarity denote by a subscript S operators and wavefunctions in the SchtBdinger picture. The timedependent Schrijdinger equation is in this notation
We will now discuss the so-called timeevolution operator in the Schrijdinger picture. This operator is defined by the equation
[ IV(tIl))in the Bilbcrt space of the problem in question, Inserting equation (14,1$ into the Schrijdinger equation yields for an arbitrary initial function
Since this holds for any initial function, the operator differential equation
together with thc initial condition U ( t o , t o )-- 1 follows. We will now pmve that the operator U ( t ,t o ) S is unitary. If we use the defining equation (14.1)twice, we arrive at
It follows that '
w,-lo39 = U ( t ,t']sU(tt, t&.
For t = to we have 1 = upo,i ' ) s ~ ( t f , . t o ) s .
GREEN'S FUNCTIONS Hence ~ ( t ,
= U(to,t)s.
Furthermore, from the operator differential equation (14.2) and its adjoint equation
it fcrlIows that
which implies that
Since U ( t a , t o ) s = 1, we have ~ t ( t ~ = , t1, ~so )~ ~t ( t , i o ~ ~ ~ (=t ,1.t a ) ~ We already have s h m n that US is invertible; hence, the unitarity follows:
If the Ramiltonian is explicitly time-independent in the Schrodinger picthre, we can readily solve the operator differential equation (14.2). The solution is i (14.3) U ( ~ , L O= ) SCXP [-%Hs(t - t o ) ]
(2) The Heisenberg picture This picture is defined by choosing
The wavefunctions are constant in time in the Heisenberg picture:
I Q(t)),
= A(t)
1 Q(t))s = U ( t o , i ) s I *(i))s =I
V'(to))s= eonstant. (14,4)
The operators in the Heisenberg picture ate given by
If HS is explicitly time-independent,
we have
from equation (14.3) (if we
also fix t o = 0)
~ ( t= )e+ ~f ~ ~ ' ~ ( t ) ~ ~ - t ~ s ~ (14.6) ,
PICTURES
144
Since the wavefunctions are constant in the Reisenberg picture, the entire dynamics is contained in the operators, We now derive an equation of mot ion for operators in the Heisenberg picture, d dt
ih-0(tIH = ih-
d
dt
(u~o~u~)
Tf we then use and
we can write
If we furthermore set
we
finally obtain the s+izalled Reisenberg equation of mation:
The formulation of quantum mechanics in the Heisenberg picture formally resembles classical nred~anics.To realize this, we consider two canonically conjugated variables q and p. For these, we have [qj $k] = i f i b j k .
I t follows that
GREEN'S FUNCTIONS Analoguousl y,
we
obtain
If an operator has the form of
a. general Laurent series
it folO11ms that
PI]
[!?j1 ~ ( 9 ,
a
= ih -0(q, ap.7
P)
and
In particular, we have in the Heisenber.g picture
so that
and, analogausty
Thus, Ha~rdtcln'sequations of mot ion of classical mechanics hold formally for canonicalIy conjugate Heisenberg operators.
(3) The interaction picture Here, we start by partitioning the Harnittonian according tn
where we assume that the time-independent problem given by We choose the unitary operator
is solved.
PICTURES as the transformation to this pictute. This yields
and
~ ( t= )e i~ ~
i*-+
o ' ~ ( t ) ~0 t~ -
(14.9)
In particular, the operator Ho is the same in both the Sch~ijdingerpicture and the interaction picture:
The partitiming of the ful1 Ramiltonian into Ho and V is in principle completely arbitrary (so long as Ho is time-independent). We will, however, hereafter always let Hg describe a system of independeni particles (if necessary in an external potential), and V denote the interactions between the particles. As an example, we will now transform the creation and annihilation operators to the interaction picture. We assume that we have solved the problem given by Ho and that cjt and C j are the creation and annihilation operators in the Schrijdinger picture of the corresponding single-particle states. We will show that only a phase-factor is added in the interaction picture:
For proof, we use the operator Ho in diagonal form
and then show that the usual representation of the apcrators i n the interaction picture
on the compIete system of Slater determinants
leads to the result equation ( 14-10],
G R E E N S FUNCTIONS First of all, we have
c ..
Ho
1
= Wa I go} where W, =
nla'r;.
But c j I 4,) is of course also an eigenfunction of Ho for arbitrary j and a:
From c!cicj
= -C+Cr 3'ci= - 6 i j c i
t + cjeiei
we obtain
From this the statement to be proved follows:
The proof for cjt runs analogously. Both the operators a well as the wavefunctions are the-dependent in the interaction picture. It follows that there are equations of motions for both quantities. For the operators, this equatiofi is (the derivation is completely analogous t o the derivation of the Beisenberg equation of motion) :
with
Next, we will derive the equation of motion for the wavefunction in the interaction picture. We s t art with t h e Schrijditlger equation
PICTURES
148
The second-quantized representation of the particle-par ticle interaction is, with equation (14.9)
and with equation (14.10), this can be written
Equation (14.12) is known as the Tomonaga-Schzuingcr equation. It is ih its structure similar to the SchrSdinger equation. The part due t o the Hgmotion, which is assumed from the outset, is separated out so that only the particleparticle interaction is left (hence the nRme 'interaction picture'). Analogous to the SchrGdinger picture, we now define a time-evolution operator in the interaction picture through
Just as for US,we can prove the following proper ties:
u ,t
= U(L,tl')u(t'f, tt) uyt,t t ) = u-yt, t') = u(tt,t ) .
Consequently, we have the following pelat ion between the time-evolut ion operators in the Schrodinger picture and in the interaction picture:
~ ( t') t ,= e ( i l h ) H O t ~t(l ltS, e - (i/h)&
t'
(14,151
GREEN'S FUNCTIONS
149
Here we have to be a little careful - a moment's thought shows that this is not a picture transformation, since the time arguments are different in the two exponential functions. Thus, U(1,it) is not U ( t ,t')S transformed t o the interaction picture. If H S is explicitly time-dependent, we obtaih from equation (14.3)
Equation (14.15) yields a simple relation between operators in the interaction picture and in the Heisenberg picture: if we use equation (14.5) and set to = 0 we obtain
We wi1I now further investigate the time-evolution operator. If we insert the definition equation (14.14) in the Tomonaga-Schwinger equation, we obtain an equation of motion for U :
a
i h - U ( t , t') = V ( t ) I u ( t ,t') dt
(14.17)
with the initial condition ~ ( t ' , t ' = ) 1. This differential equation with its initial condition is equivalent to the following integral equation;
which is readily shown by substitution. This integral equation is suitable for sucte~sivleapproximations:
Correspondingly, we obtain for the N ' ~approximation
PICTURES where
By construction, we always have Tf the successive approximations converge, we finally arrive at
~ ( t') t , = 1+
C ~ ( " ) ( ttf,) .
This se~iescan be interpreted as a perturbation expansion in the potential V . We will now make an additional simplification of the formula for ~ ( ~ 1 . We will show it here in detail for
The simplification consists of removing the variable integration limit tl so that both integrations will run over the interval [t', t ] . This clearIy means that we will replace the triangular integration region shown in figure 14.1 by the entire square. If the V('t;)Iwere c-numbers, we could without any problem evaluate the integral aver the square and finany divide the result by 2. However, since we are dealing with functions of operators, the potentials at different times will in general not commute, i. e . ,
Hence, the evaluation of the iritegral is somewhat complicated. We start by
setting
The two cxprcssion are identical, as one can see in figure 14.1. In the first exprestisioh, the integration over t l runs from t' to t , and for each t l , tz runs from t1 to the the bisectris ( t z = t l ) . In the second expression, on the other hand, only the sequence of points of the integration is changed, The integration over t z runs from t' to t and for each t*,the integration over t l runs from the bisectris Itl = t 2 ) to i . By renaming the integration variables in the second exptkssion, the integral can be written
GREEN'S FUNCTIONS
Figure 14.1 The integrals in ~ ( over ~ tl1 and t z can be divided into two separate integrals; om over tl first, the other over t z first.
For further formal simplificatioh we define the time-ordered product of two operators through
Thus, we finally arrive at
The time-ordered product is not defined for tr = t 2 , However, this causes no problem in our case, since for t l = t 2 both operators are identical and thus commute with each other. With an analogous definition of the time-ordered product of 1.2 operators we obtain
This is the central equatioh for perturbation calculations an many-partide systems. In a11,we obtain far the time-evolution operator
152
We can formally write the expression for U (t ,f') as
PICTURES
Chapter 15 The single-particle Green's function Let I !Pi) denote the ground state in the Heisenberg picture of an inkratting many-particle system. In particular , 1 qO)is then time-independent , and we have HI @o) = E o l 90). Furthermore, lei $ ( x , tIH denote t h e field operator in the Heisenberg picture. This operator annihilates a field quantum at space-spin point x at time t , and $t(x, i) is the corresponding creation operator. Occasionally, we will consider the spin-dependence separately, in which case we write
and
The definition of the single-particle Green's function is
or, if we want to emphasize the spin-dependency
We define the time-ordered product of two creation or annihilation operators b~
THE SIiVGAE-PARTICLE GREEN'S FUNCTION
154
with plus-sign for bosons and minus-sign for ferrnions. This is an extension of the definition given in equation (14.20). The definition equatiofi (15.2) contains am additional minus sign for fermions. Hawever, this makes no difference in equation (14.20), since the interaction potential always is represented by an even number of creation and annihilation operators (cf. equation (14.13)). The definition given here for the single-particle Green's function holds bath for bosofis and fermions. In the following considerations we will however restrict ourselves t o fermions. The single-particle Green's function is in this case
where
is the standard step-function. Up to this point, we have only considered the Green's function in spacespin representation, If we transform t o creation and annihilation operators of momentum eigenfunctions by a basis transformation ( cf. equaLions (4.7) and (4.8))
the Gteen's function becomes
It is then natural to define the momentum Green" function by the quantity
iG,@(kt,k't') =
1 (@a P o l Qo)
1
HI!
( ~ I H C : ~ ~ $0)
+
(15,4)
GREEN'S FUNCTIONS
155
Tn general, the Green's function in any single-particle b.asis pepresent at ion is defined by
We will now derive a number of properties of the single-particle Green's function. First, we will prove that if the Harniltonian is explicitly timeindependent in the Schrodinger representation, and therefore also in the Heisenberg representation, the single-particle Green's function depends only on the time-difference (-1 - t'), a. e,, H # H ( t ) =+ G(Xt, ~ ' 2 ' ) = G(A(t - t') , x'o).
We will prove this only for the spatial Gwen's function, since the proof is quite similar for other representations. First, we transform the Heisenberg operators with equation (14.6) t o the Schrodinger picture
-
(From here on, we will always use units in which h = 1.) For t > 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
>. '<<
(they appear only for even n )
The proof follows directly from the theorem for usual products:
wherc t p ( l ) > tp(2j > .. . 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
- rj)]
5 0, the diagram is
dtl
e"1
1 I
( j ~ v rnn) [ - ( - i j 6 , , 0 ( ~ ~
- cj)]
[ - ( - i ) b k ~ ( r F- E
jkmn
5
- --
( j t lu ( j k ) / ' j!ci
dt1."'-
-03
This expression gives the direct term of the energy shift:
~ ) ]
CALCULATION OF T H E VACUUM AMPLITUDE
244
Analogously, we obtain the exchange energy from the diagram .gP) (figure 21.7 ( b ) ) :
We now consider a term of second order; the diagram gives a tont ributian
C C J1 dll Jt' jkmn Pqr*
i2
- 22
-m
dl2 e d r ~ t 1 1 t ~ t 2 1() j i1
u
gp'
at figure 21.3
1 rnn}
-P3
C C(jkl v l mnJ(m l
d t 2 e + ~ ( ' l+'2 -00
j kmn Pqrs
where we used t h e fact that each hole-line gives an additional ~ r i n u vsign t o arrive at the last expression. A t this point wc calculate the general Iirnit needed fur the energy shift:
dt, -cw
evaluated at t = 0. From
.it follows that
e(ct
'PI )ti , ( r + i h ) t ,
- . ,,(r+iA)t,
I
PERTURBAT7UN THEORY and by induction q e arrive at
= lim i
-
~ - + 0 [F
-
I
t iPn][2f + i(0,-I -tP,)]. . . [(n - 1 ) + ~ i ( P 2 + - . . + Pn)] (- iIn-2
(21.7)
d
Pn(Pn-1 + P n ) . - - ( P z + . - + + P n ) '
This gives t h e result for the second-order diagram
above:
The family of the two topologicaIIy equivalent diagrams in; figure 21-5 then contributes
>
.
For the sacalled second-order exchange diagrams (see figure 2 1.8) w1: obtain in an andogous manner
CALCULATION OF THE VACUUM AMPLITUDE
Figure 21.8 The second-order exchange diagrams ) :JI
(2) and gZ2
We now finally consider an arbitrary connected diagram of nth order:
s(nl =
(-1'-
in 2"
(jk 1 u 1 rnn) ...( z y 1 v 1 w r )
-.. .
lkmn
qwx
The dia ram contains h hole-lines and consequently 2n - h particle-lines. Here, G6)-lines with two equal-time arguments must be interpreted as holelines, on the basis of the limit-prescription rule (4). All summation indices that appear belong to either particle-lines or hole-lines. Tf we now use
(at,, D t b ) = bo0e -
i~(') and
s-ic.(t.-ta)8
=-
G
-]
-
cF
(
OD
for particles
- E ~ )
forholes
so are the sum over the particle-indices and the sums over hole-indices to be perfarmed abwe and below ep , respectively, because of the step-functions. Each ~('I-line give- rise to a factor (- i), and from each hole-line we abt ain an additional factor af (-1):
sCnI = (-)f
i"(-i)2n
2"
Hh
C
particle cncrgica
> CF
( j b 1 v 1 m n ) ... (xy 1 v
1 WZ)
We rearrange the integrand to perform the time-integration. Consider a particular tirne t j . Only four lines give a contribution to this tirne -the ones that are attached t o the endpaints of the jth interaction line. For
PERTURBATION THEORY
247
the Iin,e with index, say, k pointing away from the interaction line at t j ,the ti-dependent contribution is exp(ickfj), independent of whether the line is a particle line ar a hole-line, since
and
The t j-dependent contribution from the line with index m pointing away from the interaction line is exp(-iemtj). If we index the jth interaction line according to figure 21.9, the tj-dependent factor in the integrand is
Figure 21.9 Indexing of the jth interaction line at time t j
I€we set
Aj
tmj
+ cn, - ck, -
61,
for j=:I, ..., n
(21.1 0)
we can write the entire integrand as
By using the general formula equation (21.71, we can then write the contribution to the energy shift from the diagrams g ( n ) as
CALCULATION OF THE VACUUM AMPLITUDE
If we aIso use the fact that 2n and 2n-1 diagrams, respectively, leave identical contributions, we obtain for the entire family:
+ topologically equivalent diagrams -
(jk I
(_]C+h
2"
particles above r~ holes b e l o w r~
n,,
l mn). . . IXYI v l ~ $ 1
,(o).,~.,
6*p
Aa(Afi+An-~)---(An+An-~+--m+A~) (21.11)
This is the main result of the present chapterc. With it, it is very simple to calculate the contribution t o the energy shift from a family of topologically equivalent diagrams. If the family is symmetric under reflection in a vertical linc in the middle, s = 1, If the diagrams are symmetric, s = 0. The Krwnecker-6 that appears can be read of€by inspec tion of a representative of thefamily, and the reductionof indices can then be performed. Furthermore, there is an additionaI trick that can be used to determine the individual factors of the denominatar (see figure 21.10). Imagine ( n-1) horizontal lines hetween the n interaction lines. Each of these lines leavcs a factor to the denominator, by adding the energies d the hole-lines ahci then subtracting the energies of the particIe-lincs that are cut by this horixontal line. So long as no contractions appear (see figure 21.11 ( 4 ) ) ) this is easy to see since the four endpoints of each layer of interaction lines just contributes one term to the sum (A, + A,-I . . ,). If a ont traction appears, as in figure 21-11 ( b ) , one particle line a n d one hole line has t o be omitted per contraction. In this c a e , the corresponding particle and hole energies cancel out in the sum (An+ An-1 . . ,), since they are of equal magnitude but of opposite sign.
+
+
i
i
PERTURBATION THEORY
Figure 2 1.10 Horizon t a1 lines are inserted bet ween the in terac tinn lines.
Figure 21.11 In example (a), no contractions appear which directly connect the two interaction lines, whereas the resnlt of t h e conlractiorr~in (b) is to eliminate one particle and one hole line.
As an example, we will now calculate the cnntsibu tion of the family of ring diagrams of third order shown in figure 21.5. As a representative of the family we choose the diagram in figure 21.12, for which
CALCULATION OF THE VACUUM A MPLZTUDE
Figute 21.12 We choose the following indexing for our representative thirdorder ring diagram.
The energy shift from this diagram is
With the help of the above representation of the denominator, it is easiy to see that none bf these fattots can vanish: since all hole energies E A satisfy c h <: tp,and all particle energies E~ satisfy e p > t ~ we, must have C ch C c, # 0. The value zero can be obtained at the most for disconnected diagrams, as shown in figure 21.13. Rut the disconnected di~gramswere already summed out by the linked-cluster theorem. The factors that appear in the denominators can clearly be interpreted as excitation energies of intermediate states, e. g.,
where the W; are energies of the unperturbed system. Wo corresponds t o the ground state, wheteas the intermediate state with energy Wl is obtained by occupying the levels pl and p2 with energies above EF instead of the levels hl and h 2 , which are occupied in the ground state:
PERTURBATION TREORY
Figure 21.13 The contribution to the energy shift can be zero only for disconnected diagram,
Similarly, we obtain far the other factors
where
Since none of the factors is zero, the ground state itself can never appear as an intermediate state. The energy shift from a family of topologically equivalent diagrams g can theh be written as
To further interpret this formula, we derive yet another equation for the energy shift. We start with the representation
Furthermore, we use
CALCULATION OF T H E VACUUM AMPLITUDE
252
Since 0 > 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
'<
k'-1
s ( k + , ) , k t b ~ , ~vr 6 k , ( k ' - q ) b f l , ~ '
Ft'
6q,(k' - k ) d , , d
= exchange energy as in Chapter 10. Here, b d h propagator lines are equal-time lines. i,e., hole lines with k , 6' < E F , acco'rding to Feynman rule (4)All higher-order diagrams which contain the elements in figure 22.4 also vanish, because b(k+g),kt = bq,(kl -k) in the d o ) - l i n e above, from which we
Figure 22.4 Diagrarns which contain these parts as factors also vanish. obtain the indices shown in figure 22,5. However, this cannot, occur since one of the free lines eatesponds to a hole (sum over k < k F ) , and the
and
kl-k
I\ kCT.
k
I/ k€f
Figure 22.5 Labeling of the ~(')-lines in figure 22.4.
other t o a particle (sum aver k > 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
ELECTRON GAS CORRELATION ENERGY
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
ELECTRON GAS CORRELATION ENERGY
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
ELECTRON GAS CORRELATION ENERGY
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
in agreement with the expression equalioll (22.10) that we obtained earlier for (AE)!:;. We will not here further pursue the proof of equation (22.14) for arbitrary n. The correctness of the hypothesis is fairly c l e a ~ ,on the bais
ELECTRON GAS CORRELATION ENERGY
270
of the analogy between the function Fq(t) and t h e Feynman propagator of quantvmfield theory (see [16]), However, we should note that in the fourth order, different topologically non-equivalent types of ring diagram start t o appear. Thcse lead t o new energy denominators. In f o u ~ t horder, we have for example a. contribution from the diagrams shown in figure 22.15:
Figure 22.15 Ln fourth order, there are topologically non-~quivalent diagrams.
-
"'I / 9/ 1J jpil
-47r5 " -
&I
d3p2d3p3d3p4
f a r *=I,..,4
We will here-after assume that the hypothesis equation (22.14) is correct, We insert the represelltation
in the definikion of J,. This yields
where
PERTURBATION THEORY
The total contribution from the ring diagrams is then
Strictly speaking, the series in the integrand converges only for
The snlall q-values aTe interesting. We calculate the vaIue of Q q ( u )for q -4 0. With
-as q
4
0,we obtain from equation (22.15)
ELECTRON GAS CORRELATION ENEfiGY
272
It is clear that we may end up outside the convergence radius of the logarithm for small q . Nevertheless, we will
(1) further use the logarithm-function, and
(2) approximate Qq by Qo since the smail q-values dominate the contribution to the integral. In this approximation, we obtain
for the resulting multi-dimensional integral. Together with t h e earlier result
we obtain
E,,,/iV
= {0.062211~(~-~-,) - 0.094
+ 0 ( r , ln(r,))}
Ry.
(22-17)
This ia t h e final result of thc long calculation. Gell-Mann and Brueckner I161 provided the correction to the approximation (2) for the second-order term in figure 22.8 ( a ) , which can be calculated cxactly. The result is a small corr~ctionterm which is independent of r,, and which has been incorpurated in the value 0.142 in cquation (22.16). Further corrections correspond to powers of r,, which have been consisLenlly neglected here. The 'approximation' and the procedure (I) can be shown to hold exactly. This is done with the so-called Sawada method 1181, which we will now briefly indicate. The idea is the following: Once it is known that the ring diagrams dominale irr the limiting case of high density, one can try to construct a Hamiltonian, for which the pert~lrhationexpansion from the outset only contains the ring diagrams, ot a related class of diagrams. Thc complete fernlion interaction potential is (with conservation of mnonlentum taken into account):
We obtain
if we instead of a l l rnon~cntum-conserving processes, as shown in figure 22,16, only include the foIlowing: a drastic reduction
PERTURBATION THEORY
Figure 22.16 A general diagram for the complete Cuulomb interaction.
This apelatar can be illustrated diagrammatically as shown in figure 22.17 (where t h e direction relative to the imagined time axis is of importance, in contrast to earlier). The perturbation expansion of these operators contains diagrams other than the ring diagrams, en$., the diagram shown in figure 22.18. Howcver, the contribution of them diagrams vanishes in the limit Y8 4
0.
Figure 22.17 T h e effective Sewada interaction cant ains only theses diagrams. The pic.toria1 representation lets us assume that the Sawada p u t o e ~ ~ t i a l can be expressed in terms of an e1r:ctrori-hole pair creation operator dq (k,
t
a k , n u k + ~ , c*
We then obtain
The IIamiltonian with this interaction can then be trcatcd e x a c t l y . In all fairness, it must be said that in spite of the elegance nf the idca of the
ELECTRON GAS CORRELATION ENERGY
Figure 22.18 In addition to dl ring diagrams, the Sawada interaction also gives rise to diagrams such as this one. exact treatment of this Hamiltonian, the important application is in the Gell-Mann-Bruetkne~ summation, However, the exact treatment of 'ii does
give additional information on the excited states. In the case of the ground s t ate, the Sawada~methodconfirm the correctness of replacing the series by the logarithm (as done by Gell-Mann and Brueckner). We close this chapter by comparing the result equation (22.17) with the resuit from pl~slnohtheory, equation (11.10):
The coefficient of the logarithm is identical in both expressions. This is not surprising, since this term comes from the short-range part in the second order, which is treated correctly by both methods. The rg-dependent term diffefs clearly. Even thaugh the Gell-Mann-Brueckner calculation is doubtlessly correct for r, -, 0, it does give worse values fot the correlation energy fur ~ e ametallic l densities than the plasmon theory.
Chapter 23 Diagrammatic calculation of the single-particle Green's function: Dyson's equation The representation
which we proved in Chapter 18, provides the basis for the diagrammatic analysis of the single-particle Green's function. If we replace the underlined field operators by other creation and annihilation operators, for example ~ ~ , ( t ) ~ and ~t ~ , ~ ( we t ' obtain ) ~ , the representations for other Green's functions, in
this example GD8(kt,k't'). The denominator in this ~cpresentationof the Green's functions corresponds to the vacuum amplitude, the diagrammatic calculation of which was discussed in Chaptet 21. If the second-quantized representation of the interaction ~ ( t ;is) inserted ~ and Wick's theorem is used, the numerator tan also be represented diagrammatically with the Feynman rules af Chapter 20. In first order, we obtain the six diagrams of figure 23.1. The two arguments of the Green's function appear as external endpoints of the diagrams. We have here only depicted the diagrams for particle propagation. The corresponding diagrams for hole propagation are obtained by interchanging the endpoint indices and changing the ditections of all arrows. The diagrams (3) and (41, as well as the diagrams (5) and (6) in figure 23.1 are topologically. equivalent, in the sense of our earlier definition, i , e . , they can be obtained from one another by interchanging the
DYSON'S EQUATION
P
'If,
Figure 23.1 The first-order diagrams of the numerator of the Green's function.
indices on the vertices of a n t interaction line. The diagrams (11 and (2) are disconnected. It is clear, after translation to formal language, that such diagrams factorize into contributions from connected sub-diagram. In what follows, we will provc that all terms in the numerator of the single-particle Green" function are obtained from the product of all connected digrarns which connect the t w o external endpoints wilh all diagrams of the vacuum amplitude (see figure 23.2). A11 there is t o do is simply to combirie all diagrams which arc connected between the two external endpoirlt,~with all othcr loose pieces, the sum total of which is the vacuum amplitude, In this
+
o l l other diagmrns cannecfq hofh
external ~ n points d
1 r
1+
+
@+
&+
all other diagrams o f +he v a ~ u r n
ampl~tude
Figure 23.2 Diagrammatic represent atjon of the numerat or of the Green 's function.
PERTURBATION THEORY
Figure 23.3 Six diagrams of the n urnerator of the Green's function.
way, we obtain all topologically distinct diagrams. However, it also remains to be shown that the prefactors agree. As an example, we first consider
the six diag~amsin figure 23.3, which all contribute to the perturbation expansion of the numerator, These diagrams an give the same contribution, namely (in a transparent notation):
O n the other hand, the
product of g$Z), and
and
gp)(figure 23.4) is
Thus, we obtain the sum of the diagrams in figure 23.3. We now have t o generalize this calculation to diagrams of arbitrary order. Once we have succeeded with that task, the staicrrrent i~ proved.
.
DYSON'S EQUATION
Figure 23.4 All diagrams of figure 23.3 can be produced by the connected diagrams g$2), and gd(11 .
We consider a general term of order v in the numerator of G.This term will consist of a uniquely determined connected piece G of nth order, which connects both endpoints, and furthermore of a loose piece t~("-") of order ( u - n). This latter piece is also uniquely determined. This one m y be partitioned into connected sub-diagrams, as in the example above.; this is hawever meaningless for the present proof. There are precisely passibilities t o choose n interaction lines out of the v ones connected t o the times t l , . .. t , in order t o construct the connected diagrams The remaining (Y - n) interaction lines result in the disconnected vacuum amplitude diagrams g ( k - n ) . Each possible combination apparently corresponds t o a diagram which contributes to the numerator of the Green's function. All these diagrams give the same contribution. Thus, the sum of them contributes
(L)
GP).
(E)
We have here used the easily shown fact, that the number of fermion hole lines (denoted by f, and t' - t l ) is additive, The hypothesis is then proved, We can cancel the denorninato~of the Green's function with the vacuum amplitude factor of the numerator and obtain the important result (see figure 23.5)
PERT 111rnATIUN THEORY
Figure 23.5 In the series expansion for the Green's function, only diagrams connecting the external endpoints need to be included,
The index L means that only the connected diagrams are to be summed aver. Equation (23.1) is, if we wish, the 'linked-cluster theorem' for the single-particle Green's function. In the remaining terms of the perturbation expansion, many degeneracies in the diagtams appear (I) through the interchange of entire interaction lines (figure 23.6). This gives in all n! permutation pwsibiIities for nth mder diagrams.
(2) Ry topological degeneracy (figure 23.7). There are two such possibilities per interaction line. This means an additional degeneracy factor of Zn f o ~nth order diagrams. In contrast to the vacuum amplitude, there are in the present case only asymmettic diagram, so that we do not count any diagrams tjwice,
Figure 23.6 Diagrams which can be obtained from one another by interchanging entire interaction l i i ~ e sleave iden tical con trjbu tions.
280
DYSOlV'S EQUATION
Figure 23.7 These diagrams are topologicalIy equivalent and give identical tontribu tions.
At this point, we agree t o represent thk entire family of (n!2")degcne~ate diagrams by asingle diagram without indices. At the same time, we establish a new Feynmah rule. Feynman rule (7) for t h e prefactor of the unindexed connected Green's function diagrams: prefactor = in. The degeneracy factor ( ~ 4 ! 2cancels ~) out with the denominator of the earlier shown prefactors of the indexed diagrams. All conncc tcd second-arder diagrams for the Green" function are shown in figure 23.8, Each of these unindexed diagrams represents a family of 222! diagrams. Some of these unindexed diagram clearly vanish in thc momcnturn reprcsentatian of a translationally invariant system.
Figure 23.8 All setan $-order can nected diagrams for the Green 's function.
We now have a rough idea of what the perturbation expansion of the sibgle-particle Green's function looks like. As the illustration of the secondorder diagrams in f i g ~ r e23.8 shows, many diagrams are constructed from a few simple 'building blocks'. This is the basis for what is mayhe the rrio~t,
PERTURBATION THEORY
281
important equation in the entire perturbation theory: Dyson's equation. We will now derive this equation. If the full Green's function is depicted by a thick line, the pertu~bationseries can be summarized by figure 23.9.
C-
Figare 23.9 The self-energy M allows for a compact representation of the perturbation series far G .
-
The self-energy M denotes the sum of all self-energy insertions (figure 23'10). These are any part of a diagram which is connected to the rest
Figure 23.10 ?'he sclf-energy is obtained by summifig all sdflenetgy insertions, as shown here. of the diagram only by one incoming and one outgoing line. Furthernlorc, we classify self-energy insertions as reducible if they can br: partitioned into lower-order self-energy insertions by cutting a single ~('1-line. For example, the diagrams on the tap row in figure 23.11 are irreducible, whereas the d i a g r a m on the bottom row are reducible. The sum of of all irreducible self-energy insertions is called the proper self-energy, and is denoted by M (without a -) , It is clear that we obtain the total self-energy from the sum of all possible repetitions of the proper self-energy connected with Green's furlctian line, as shown diagrarnrnatically in figure 23.12. Formally, figure 23.12 can bc written
Convergence factors e-'It] which may possibly appear have been omitted from these definitions of M and %, respectively.
DYSON'S EQUATION
,
,
,
~redu~~blr
nnd
Figure 23.11 Examples of irreducible and reducible self-energy insertions. The reducible ones can be cut in to lower-order diagrams by cutting a single ~ ( ' ) - ~ i has e indicated by the dashed line.
Figure 23.12 The selfenergy energy.
% is obtained as a sericv in
t 6 c proper self-
PERTURBATION THEORY
283
The equation for the Green's function Cah then be represented as shown in figure 23.13.
ti:::
tl - -A;--
% '
Figure 23.13 Dyson's equation for the Green 's f h n c tion,
This is Dysan's equataon. F ~ r r n d l yit~is written
At first sight this equation seems astonishing, since we have seen earlier that the single-particle Green's function is calculated from the two-particle Green's function, and so on, and here it appears that we obtain the singleparticle Green's function from an integral equation in which there is a seemingly independent quantity, the self-energy. However, we could have shown before that the single-particle Green's function satisfies such an equation. In Chapter 16,we fourid the equation of motion for the Green's functions:
[ia+ a:] at
and
2m
rtt') = 6(t - t')b(r - r ' )
~("((r,
In this approach, M is introduced by
This equation shows the relation between M and G2. We must now identify the M introduced in this way as the proper self-energy. We can do this by convincing ourselves that the equation of motion
-1
a + v; 8i 2rm
[i-
~ ( r ti,t')
is satisfied by the solution of Dyson's equation
Chapter 24 Diagrammatic analysis of the Green's function G(k, w) We start by rewriting the perturbation expansion of the Green's function In the space-representat ion in a manifestly cavariant not ation. As is customary in relativistic problems, we define
and use the 'Minkowski scalar product'
The usc of the. single letter k for (k,w ) should not lead to any canfusior~. Hawever, as a precaution we will only abbre~ittt~e Ikl with k in unambiguous caucs, such as in tornbinatio~~ with d3k. Up to this point, we had established that each interaction line gives a
contribution (see figure 24.1)
If we now set
DUGITAMMATIC ANALYSIS OF THE GREENS FUNCTTIQN
Figure 24.1 A general interaction line with its vertices.
the diagrammatic representation in figure 24.2 b ecornes
by modifying Feynman r ~ l e( 5 ) (which says that all indices must be integrated over). This agrees of course with our eatlier expression,
The convergence fator e-'ltl will hereafter be omitted far ease of notation; it must of course be reinserted when any expression is actually calculated explicit1y,
Figure 24.2 T h e vertices ri and xfl' of an interaction line are relabeled in the covariant nutatio~~. It is evident that this notation makes a covariant furmulation for relativistic systems possible, where the interaction h~ a finite speed of propagation and is no longer described rncrcly by a delta-function in time. In this context, the time does not play a special role any longer; we can omit Feynman rule (1) as well as the requirement that the interaction lines must be horizontal. The diagrams that we have studied so far are frequentIy drawn in a different way; some examples are shown in figure 24.3. From here on, we will make use of both representations.
PERTURBATlON THEORY
Figure 24.3 When Feynamn rule ( 1 ) can be omitted, 'time loses i t s special meaning and many diagrams cah be drawn differently by allowing for the interaction line not to be horizontal.
In what folloes, we will restrict ourseIves to translationally invariant systems, which do not explicitly depend on time. We said in Chapter 15 that in this case, the single-particle Green's function only depends on the difference between the space and time coordinates, and that we c a n usc bur-dimensional Fourier transforms given by
and
We Fourier ttmsform the interaction potential in thc same way:
where the Fourier transform
for nun-relativistic systems becorrles
DIAGRAMMATIC ANALYSIS OF T H E GREEN'S FUNCTION
In order t o derive the Feynman ruled for the Green'e function G a p ( k ) , we must first Fourier transform the diagrams which appear in the spacerepresentation. We do that in the example shown in figure 24.4. The contribution from this diagram is
Figure 24.4 A first-order diagram for the Green 's function. We replace the time limit by
where we agree ta take the limit q 4 O+ at the end bf the calculation. After this manipulation we can,for esthetic reasons, insert a new additional factor exp [- iup{t - 1: since this factor is unity anyhow, due t o the delta
)I,
PERTURBATION THEORY
289
function 6(t1 - t i ) contained in u ( q , x i ) . This transforms the underlined usual scalar product t o a Minkowski scalar product, and we can then write for the whole expression
Thus, w e obtain a four-momentum delt a-functidn at each of the two vertices at XI and x i . This results in
Since thc full Green's function is obtained from the Fourier transform
we
can interpret the expression
as thc diagram in figure 24.5, which contributes t o G a p ( k )
DIAGRAMMATIC ANALYSIS OF T H E GREEN'S FUNCTION
Figure 24.5 After Fourier transforming the contribution from the diagram in figure 24.4, the contribution can berepresented by this diagram for G a p ( k ) . This example lets us deduce the contribution of a general diagram t o the Green's function:
(1) For each u(m,x i ) , there is a u(q) (2) Each
do) ( r,y)
gives a
do) (1).
( 3 ) The prefac tors and signs arisidg from the topological structure and the number of interaction linm remain unchanged by the t'ourier transform, (4) In the example, we momentum was conserved at the internal vertices. It is easy t o prove the assumyt ion that this holds for general diagrams. For example, the diagram in figure 24.6 gives
Figure 24.6 A general vertex of an interaction line. The integration variable x does not appear in any Green's function, and the integral over x can be performed with the result
PERTURBATION THEORY
291
An integrations d4z which belong t o internal vertices can then be performed explicitly, and leave a momentum- and energy-conserving delta function d4)at the corresponding vertices. We note that precisely the exponential factors that are necessary for the corresponding calculation of the adjacent ve~ticesremain in this equation. (5) Because of the 6-functions, se-veral of the momentum integrals which arise from the Fourier transforms can be performed. A few integrals which are independent from one another, for example integrals over momentum running through particle loops, remain and yield each a factor of ( 6 ) In each diagram, there is a sequence of do)-lines which connect the external endpoints with one another. Since all vertices inbe tween conserve momentum, perfarming the integrations over the 6-functions at the endpoints results in the same momentum an the external legs. The two remaining exponential factors can then be combined. Hence, all diagrams that contribute to t h e Green's function have a common factor of
(This dso holds for the diagram that consists of a single ~ t ~ ) - l i n a . ) Since thc sum of a11 these diagrams yields the full Green's function
we obtain by comparison with the Fourier coriiponents the sought-for perturbation exparlsion of G(k, w).
Aftm these considerations, we can formulate the Feynman rules f o r G(k, w ) : (0) (1) The Green's function Gmg(k) is represented by a line labeled by the mornentutn k and with the spin-indices ahd P labeling the end--points [figure 24.7). Each such linc gives a factor
292
DIAGM MMATIC ANALYSIS OF THE GREEN'S FUNCTION where, for example
$1
= tklZ/2rn.
(0) Figure 24.7 The Green 's function Ga4(k)-
(2) An interaction line is labeled by the momentum q transferred by the interaction, and the vertices are labeled by spin-indices a,a' (first vertex) and p, p' (second vertex) (see figure 24.8). Each interaction line contributes a factor u ( q ), . PP'
Figure 21.8 Labeling of an interaction line and its vertices,
(3) ~(")-lincswhich begin and end on the same irltecactior~line arc interpreted as eiuq (k,w ), where the limit, --+ O+ is taken at the end of the calculation (figure 24.9).
Figurc 24.9 Examples of diagrams in which the Green's function has its endpointb an the same interaction line.
(4) Morncnturn is conserved at each vertex. This i s illustratcd in figure 24.10.
PERTURBATION THEORY
Figure 24.10 A moment urn q is transferred by the interaction to the Green's function connected at one vertex.
(5) A11 four-momenta that appear (after satisfying momentum consmvation) are integrated over. When the integrals are performed, each b multiplied by a phase-space factor:
(The four-mamenturn k, which enters with the Green's function G ( k ) at one endpoint and leaves at the other, is of course not integrated over .) ( 6 ) Thc sign of the diagram is givon by (-)', closed G'(')-loops.
where l is the numher of
(7) Thc prefact or of a diagram of order n is
I" 2" n!
( (i)"
for an indexed diagram of order n
for an mlindexed diagram of order n
With these rules, we can easily translate the connected Green's function diagram of first order shown ih figure 24.11. The result is
--
Becau~eof momentum conse~vation,only the q 0 Fourier component of the interaction potential appears. (In the case of the electron gas, this one is explicitly omitted, since it is canceled by the positive ionic background.) We also note that the two ~(')(k)-factors ordy couple to the third onc through the spin-indices. Hence, the Green's functions factor out for spin-independent in teractiuns.
294
DIAGRAMMATIC ANALYSTS OF T H E GREEN'S FUNCTION
Ta
hfl
Figure 24 ,I1 The so-called tadpole diagram of the Green's function.
Next, we consider Dyson's equation fur translationally invariant ~ysterns. In the space-representat ion, this equation is
We emphasize through the notation G(x - y) eic., that for such systems, G, do) and consequently also M , only depend on the difference between the arguments. A simple direct calculation, or the convolution theorem for Fourier transforms, show that by Fou~iertransforming, the (convolution) int cgraIs become ordinary products:
where M A B ( k )is the four-dimensional Fourier transform of MA,(z - y). and M have the form If the interaction is spin-independent, G ,
In this case, byson's equation has the simple form
+
~ ( k=) G ( ~ ) ( ~~J ( ~ ) ( k ) ~ ( k ) ~ ( k ) from which wc can solve directly for G:
GCk) =
(k)
1
1 - ~ ( ' ) ( k ) ~ ( k )[do) ( k ) ] - l - M (k)'
The function G(') is given by
'i'he inverse is
-
(24.4)
[ ~ ( ' ) ( k w, ) ]
-'= w -
irrespectively of whether Ikl tant repmentation
tk
where, for example c 0) i = Iklz/2rn
> EF
or
Ikl < kF. We then obtain the impor-
This is an exact representation of the full Green's function for translationally invariant, spin-independent and not explicitly time-dependent systems. Since we know that the poles of the Green's function give the exact excitation-energies of the system, the problem of finding these is reduced to soIving the equat isn w - $1 - M ( k , w ) = 0.
It is important to emphasize that any approximation that is used for the seM-energy means a sunmation of infinitely many diagrams for the Gwen's function. If we far example take only the first-order diagrams and g,(1) in the proper self-energy, M ~ ( ' 1 , we obtain the approximation far the G~een'sfunction shown in figure 24,12.
=
Figure 24.12 Even the sirnplest first-order apprxirnatinn for the self-energy means that an infinity of diagrams are included in the Green's iunc tion. Finally, we will aho express the tot a1 chergy of an interacting system through the self-energy. In Chapter 15, we derived the equation
,
Chapter 25
Self-consist ent perturbation theory, an advanced perspective on Hartree-Fock theory In what follows, a seIf-energy diagram that does not contain any self-energy insertions other than itself will be called a skeletut~,Exarnplcs of skeletons and diagrams that are not skeletons are give11 in figure 25.1. These skeletons are 'dressed' by inserting all possible self-energy insertions in each d o ) - l i n e and somrning (see figure 25.2). Finally, by thc a~mmatianall ~('I-lines are replaced by full G-lines. O l ~ ecan casily convince oneself that thc sum of all
Skelefon 1
* * a
Not s h e l e t u n
Figure 25.1 The diagrams i n the tap row are skeleton diagrams. The diagrams i n the bv t tom row consist of several self-energy insertions and are not skeletons.
SELF-CONSISTENT PERTURBATION THEORY
298
such dressed skeletons produces the full irreducible self-energy:
M=
C all drds3ed sktl6tons
(25.1)
- -..a
Figure 25.2 The skclctons are dressed by inserting all pasible self-energy insertion3 at aJI lines.
-
Tf the sum is truncated aftcr a finite number of dressed skeletons, for exam-
II
ple after the two first-order dressed skeletons, we obtain through Dyson's equation an integral equation for the Green's function:
This equation must then be solved self-consistently; such approximations are called %elf-consistent perturbation thenry '. Irr what follows, we let
,
PERTURBATION THEORY
299
(Here x denotes both space and spin variables as before; x = (r, s).) The Hamiltonian H = Ho V does not depend on time. Therefore, it is suit able to.go over to a Fourier representation with respect to (t - 1'):
+
If these expressions are inserted in the space-pin representation of Dyson 's equation, we obtain a corresponding Dyson's equation for the Fourier camponents:
a, ~
'
7
4
We will now prove that Dysonk equation in this approximation, equation (25.31, is equivalent to the Battree-Fock procedure derived in Chapter 7, The apptoxirnation for the self-eaergy can be represmted as shown in figure 25.3. These diagrams are equivalent to the equatiorl
I
+ i v ( x , x ' ) ~ ( x tr, t t + ) . If we use G ( t , t t ) = G(0, ( t + - t ) ) = G(O, O+), and perform the trivial Fourier transform with respect to ( t - 3 ' ) , we obtain the foIlo w i n g expression for thc irreducible self-energy :
M ( r ,t') = b ( r - r ) -o(z, x ' )
J dxl
[ - ~ G ( I ~ o , ~ ~ D +t )(]t ,. I )
[-iG(x0, rtO+)]
which is frequency-independent Furthermore, from the relatior1 equation (15.15) between the Green's function and t h e density matrix we obtain
SELF-CONSISTENT PERTURBATION THEORY
300
In this approximation, the irreducible self-energy corresponds precisely t o the Hartree-Fock potential, equation (7.3) (disregarding the external part uIx3).
Figure 25.3 Self consis tent firs t-order approximation for the self energy.
With the operator
v2
h o ( 4 = -2;;E
+4x1
the Green's function G(') (zt,rtt') satisfies the equation of motion (cf. equations (16.8) and (16.9)):
By Fourier transforming with respect to ( t - t') it follows that
I', w ) = S(z - x') .
[w - ho(x)]G(')(I,
(25.5)
If wc operate with (w - h o ( x ) ) on Dyaon's equation equation (25.3) and use equation (25.5$,we obtain
J
If we insert the approximation equation (25.4) in this equation, we obtain
This is the eqllation of motion for the Green's function for a system of bndependent particles, which move in an effcct ivc, self-consistently determined potential, which is precisely the Hartree-Fock potential. Thus, the s t a t e ment is proved, For the sake of campletenas, we will now also express the Greens' functions obtained from equation (25 -73, which we will denote by Girl,., in terms of the Hartree- Fock singlc-particle orbitals. We have If we denote by c 5 and c j the creation and annihilation operators for these single-phrticle or itals, we can express t h e field operators through these
:r
PERTURBATION THEORY
301
In the Heisenberg picture, which is identical to the interaction picture for independent particles, we then have
and, correspondingly,
The Green's function is then
Thus, if the orbitals in the ground-state Slater dcterminants arc occupied up to the maximum single-particle energy r F , we obtain for the Bartree-Fock Green's function:
Finally, we obtain by Fouricr transforming
It can be verified by direct substitution that this Green's function satisfies the equation of motion equation (25.7). The only thing needed is the completeness of the Hart~ee-Fock single-particle basis.
302
SELF-CONSISTENT PERTURBATION THEORY
The expansion, of M in a series of dressed skeletons presents an interesting possibility for generalizing the usual Hartree-Fock procedure by including higher orders, for example as shown in figure 25.4. This diagram corresponds to
Figure 25.4 This second-order diagram can be included in the equation for M to yield a higher-order Hartree-hck approximation, In this case, M does really depend on the time difference [ t - t ' ) , i.e., the Fourier transform will, in contrast to the Rartree-Fock approximation equation (25.4), dcpend on the frequency w . It should be noted, that by such an approximation the integrd operator J MCx, y, w ) d y in general is not hermitian, so that the corresponding term in equation (25.6) cannot readily be interpreted as an effective non-local single-particle potential. In spite of this, we can also in this case obtain a representation af the Green's function analogous to q u a t i o n (259). To obtain this, the eigenvalue problems
are first solved for arbitrary w . A simple calculation 1191,shows that
In contrast to the eigenvalues (cf. equation (25.12)), the left and right eigenvectors in gcncral are not complex conjugate:
PERTURBATIUN THEORY The Green's function then has the representation
which is readily demonstrated by substitution in equation (25.6)and using equations (25.10). and (25.14). If the equation
has solutions w,, for fixed n, the Green's function has, according to equation (25,15), poles at w,, . These poles are in general complex. In the next chapter we will thoroughly d i s c u s uader which conditions the real and imaginary parts of these poles c a n be interpreted as excitation energies and inverse life-times of quasi-particles. The equation (25.15) is an exact representation of the Green's function if the eract irreducible self-energy is inserted in equatiansC25.10) and (25.11). In this case, we can compare with the (also exact) Lehrnann-~epresentation of the Green's function, which was derived in Chapter 15 for translationally invariant systems (cf equation (15.28)):
Since the poles in equations (25.17) and (25.151 agree, the values ,w , = s,(w,,) G trn obtained from equation (25.16) must correspond to the cxwt excitation energies, reckoned from the chemical potential, for the ( N 1)particle system
+
tm = p
N+l+(Ern
EF+I)
- jv
(25.18)
or for the ( N - 1)-particle system
-
Similarly, for the corresponding aigenvactors f , ( x , w,,) f n ( run,) , G lp,(x), we lnnst have ~ r ( xn )
=
Fm{x) =
(@fI $(XI I 9,N+ 1)
{c+= 1 $+(4 I q) = &(")
for c,
= >p
P, (x) and
((25.20)
304
SELF-CONSISTENT PERTURBATION THEORY
and
We then obtain the following representation for the Green" function:
If we consider equation (25.10) at the frequencies w = r , (= ~
obtain )
~ we 0
This 'non-linear eigenvalue prob1em"ves an important re-for mulation of Dyaon's equation. It was first obtained by Schwinger 1201.
~
Chapter 26
The quasi-particle concept We recall that in Part 1 the first part of this book, which was concerned with the foundations of many- particle physics, we several times used various mathematical tricks to go from an interacting many-particle system t o a sys tern of non-interacting 'quasi-particles'. At this point it is now possible for us t o introduce these in a systematic way and to understand them correctly. The Green's function for free particles is (cf. equation 115.23))
i ~ ( ' ) ( kt), '-.
-
--
.
= B(t) [ e - i c k ' ~ ( ( k l- kv)] (particle contribution) -8(-tJ
-
[ ~ - ' ' ~ ' # ( k ~lkl)] (holc contribution).
This Green's function consists essentially of a time-dependent exponential function cxp(-irkt), wIiich says that a particle, ar a holc, with a well-defined energy ~k propagates in time. If a particle which has been added to the ground s t a t e of a many-particle system interacts with all other particles, the Green's fi~nctionwill in general be very complicated. However, we may ask under what conditions the time development of such a system can be interpreted as the propagation of a 'particle' with a reasonably well defined energy and a sufficiently long life-time. In other words, we ask under what conditions the propagator has the form
Thc dispersion relation f(k) and the life-time l/l'(k) thcn reflect the intcractions with the other particles; the effect of these interactions can be called thc polarization of the medium. The initially 'hare' particle tu a certain extent drags a polarization cloud with it and so becomes a 'dresscd' particle with R different dispersion relation and life-time. We have already learnt about one such quasi-particle: the Ha~tree-Fock electron, which drags with it the exchange holc and consequently Iias a modified dispersion relation
&UASP-PARTICLES
306
To answer the questkn
whether the Green's function of an interacting system can be interpreted as a quasi-particle propagator, we must backtrack a little. We consider once again the Lehmann-representation of the Green's function for translationally invariant system ('cf. equation (15.28)):
G(k,u) =
1q+o+
[F I
PiN+',(k1 I
The first term in the expression for
w-p-w
G has
I2
I ~3 I(*: I *F) ( N + U + irl n,k
a pole at
and the second term has a pole at
Figure 26.1 Location of the poles of the Green's function in the complex w -plane. We introduce the reiarded and
advohced
'Green's functions;
where the curly brackets denote the anti-commutator of the field operators, as usual. The Lehrnann representation of G and G ~ only I differ ~ in the sign of the small imaginary part in the denominator:
and w
with poles
IN-1)
s p - w,,-k
- iq;
at
and In contrast t o G , the Green's functions G~ and G~ are analytic in the upper and bwer w-plane, respectively. We note that
e R ( k ,w)* = ~ ~ ( k , wfor) w E 72.
h r p # u and p E R
(26.3)
we can in each case calculate the limit
4
O with the
result P<W
: G ( ~ , U ) = G ~ (P~ E, ~ R)
(26.4)
P>U
: ~ ( k , w ) = ~ ~ ( k ,~w E)
(26.5)
We insert these relations in
R .
&UA SP-PArtTICLES
0
A
0
I
P
Cr
xxxxxx
X X X X
XXXXXXXXXXX
Figure 26.2 Location of the pales ,of the retarded and advanced Green's function.
which yields
We use the integration path in the complex &-plane shown in figure 26.3 to fur the^ evaluate these integals.
Figure 26.3 Integration path for cqr~atian$26.6). The integration path is divided into two pieces C1and Cz with circular arcs 71 and yz, respectively.
We have
since ~ ~ (w )khas , no pales below the real axis. The exponential makes the integral over the circular arc go to zero in thc lirriit; ofinfinite radius so that
PERTURBATION THEORY the first k r n for G(k, t ) becomes
Lm
ge-iut GA (k,U S=
J
p-ioo
-e dw 2~
-iwt
eA(k,w).
We rewrite the second term of G(k, t ) ih equation (26.6)in a corresponding way, but now we have t o take into account the fact that G~ may have poles at w, with residues r , in the enclosed area:
Thus,
from these expressions, we obtain the following result for the Green's function C(k,w) after separating w , in real and imaginary parts w, = r , - irk:
Suppose nuw that the pole at wo = f o - i F o is closest to the real axis of all poles, i.e., To < I?, for u # 0. For sufficiently large times, we can thcn write
Of couruc, we cannot wait
that the contribution from this poke dies out; thus, we consider times of the order so long
For such times, the Green's function G(k, t ) is
Q UASI-PARTICLES
310
A11 that remains is to investigate under what conditions the integral may be neglected , The quantity (co - p) > 0, i.e., the distance from the Fermi energy to the quasi-particle energy, is a measure of the energy and hence of the deviation f r o m the ground state t o which the quasi-particle is related. In order to guarantee a reasonably well-defined energy, the quasi-particle must propagate at least for a time t > l / ( ~ p), according t o the energy-time uncertainty principle, We will now show that this condition together with equation (26+7),i. e.,
are suficent conditions t o let us neglect the integral term, The condition ro << c - p is contained implicitly in this inequality, a. e . , the life-time of the quasi-particle must be large enough for an accurate measu~ementof the quasi-particle t o be at all possible. Hence, the pole must be very close to the real axis. The integrand vanishes far large imaginary parts of w . Hence, only the contributions close to the real axis play any role. For an estimate of the integrand, we can then take w to b e real
Under the assumption discussed above, i.e., that the pole is very close to the real axis, the poles dominate the integrals for real values of w , so we have
and by using eq~iation(26.3):
Hence, the integral in equation (26.8) is approximately
and with the substitution .y 1 becomes
i(w
w = p - iy, dw = -i dy
the integral
PERTURBATION THEORY
311
- p ) , thesl exp(-yt) is different from zero essentially only If now i > if y << ED - p . However, in this region the denominator is
for t w
l/ro. Thus, the integral is much smaller than 1% 1 under the given as-
sumptions, whereas exp(-Tot) w 1 in the time interval under consideratioh, so that
I
-;zoedico e -rot
I
f?
IzD1
The contribution from the integral in equation (26.8) can then be neglected. Only the contributions from the poles next t o the re,al axis remain:
In this equation, the dependence of the position and residue of the pole an k are emphasized by a transparent change in notation. Tn the eonsiderations above, it was assumed that t > 0. For t < 0, one obiains a campletely analogous expression for the propagation of 'quasiholes'. In the derivation, the integration path nmgt in this case clme in the upper complex weplane, In all, the final result is
whcre all terms discussed which are negligible uhdcr the condition equation (26.9) are included in I(k, t ) , Any approximation for the irreducible self-energy gives an appraximation for the Green's function. From the representation given in equation (24.5)
the pole, and consequently the dispe~siontclation and the life-time of the resulting quasi- pa^ ticle, can bc lead off imrrlediately. Wc will now investigate the approximation hf B ~ ( ~ 1 (26.12)
Figure 26.4 The self-energy diagrams approximation for M .
and
in the Hartree-Fock
t~?'
where M(') is the self-energy in the Hartree-Fock approximation (see figure 26.4), in detai1. Since the Aar tree-Fock equation for t~anslatimallyinvariant systems is satisfied by plane waves, it is clear from the considerations in the previous chapters that the approximation equation (26.12) will give the dispersion relation and life-time for the Hartree-Fock quasi-particles, In what follows, we will explicitly prove this, The first diagram, t~t('), only gives a contribution for the p = 0 Fourier component, because of momentum conservation:
= -ivCg=O)
lim (r-r1)--*O
1 \
I
) ~ tq ,yt = O +1
ix32id-~p-r
J
Y
=G(0)(r,rl,t=O+)
where in the last step we used the rclation between the Green's function and the particle density, which we proved in Chapter 15, The second diagram, g,(1), yields (for instantaneous, i . e . , frequencyindependent, interactions)
Insertion of the explicit expression cquation (15.26) for yields c Z W T
2~
lim
E+ot
do) in this equation
PERTURBATION T H E W
313
By making the substitution u(new) c60) - w(oLd) together with the representation equation (15.24) of the step-function we finally obtain
We obtain in total
Thus, the pole of t h e Greens' function equation (26.11) is at
i.e., we have a qn-i-particle
with
- a dispersion relation
This i s p~eciselythe dispersion relation cqtlatian (9.4) [for k+ = t- = k F ) that we derived for translationally invariant system in the Hattree'" Pock approximation; -
an imaginary patt r(k) qu asi-particle is ihfinite.
==
0, ire., the life-time of the Hartree -Pock
We end this exarrrple by quoting from Chapter 10 the value of the intcgral in equation (26.13) for the electron gas:
Up to this point, we have bee11 able t o say that we can, under certain candilions, interpret t hc full Green's function as a quasi-particlc propagator.
QUASI- PARTICLES
3 14
The example of the Hartree-Fock electron showed that such quasi-particles can indeed appear. The question now arises if such quasi-gar ticles can .appear in any ~ystem,or if they are very singular phenomena. The answer is:
In any many-particle system for which perturbat ion theory converges, there are quasi-particles in the vicinity of the chemical potential p . To prove this statement, we use the folIowing fact:
lim Irn[M(k, LJ)]= sgn(p - w ) c ( k ) ( w-
W+P
2
(26.15)
>
where c(k) 0; c(k) E R. We will now outline the proof (for details, see 1211). lir the first step of the proof, one shows that
In the next step, one shows that the contribution from the corresponding dressed diagram behaves in the same way:
Figure 28.5 The lowest-order contrjbution t o the fife-time of a quasi-particle comes E m the skeleton diagram IJ!)'. 'I'hecorresponding diagram with the 4 2 ) drcsscd Green 's function is gg, To do this, one uses the representation equation (15.29) for the Green's
function:
The dependence of the integrand oh w corresponds to thc frequency dependence of G ( O ) ( ~ w , ) , which allows for using the estirrlate (1) in (2).
PERTURBATION THEORY
315
Finally, in the last step one shows that as w 4 p , the imaginary part of an arbitrary dressed skeleton either behaves as (2) or vanishes fastet than ( u - ~ )According ~ . t o equation (25.11 the hteducible self-energyis obtained from the sum of all dressed skeletons, from which equation (26.14) follows for any system for which perturbation theory converges. The poles ~ ( k=) ~(k) - ir(k) of the Green's function are determined by the equation
On the basis
of equation (26.15)' we will now show that for excitation energies close to the Fermi surface (i,.e,? for ~ ( k z)i p ) , the existence condition
for quasi-particles, r(k) < ~ ( k ) p , must be satisfied. To obtain an estimate for r(k) (for the case ~ ( k )= p ) , we expand equation (26.16) about the exact r e d part of ~ ( k ) :
In the expansion of the imagainary part of M , only the lowest-order cantributions will be considered, since the imaginary part is ih any case very small in the vicinity of w k p z ~(k).We obtain the following estimate for T(k):
r
For quasi-particles, i.e., for ~ ( k r ) p , is positive a n d is near the k r n i surface negligibly small compared to r(k) - p, due t o the factor (r(k) - p ) 2 . This is precisely what we wanted to show, For quasi-holes, is negative, which in view of the form e-r(k)' for t c 0 is quite sensible. Thus, wc conclude that r(k) changes sign at the Fe~mi sutface ((k) = Near the Fermi surface we also havc for ~ ( k < ) p:
r
For calculations and t o gain physical insight, it is often useful tc~assign an effeclive mass to quasi-particles (and quasi-holes). To do so, essentially two different possible definitions are used:
(1) For non-interacting systenls, we have
It is then natural t o define an effective-mass tensor at a wavevector kO for an arbitrary dispersion relation f(k) by
(2) If m are interested in a dispersion relatian ~ ( kin ) the neighborhood of a particular point ko (for example a 'band edge' in solid state physics), we can expand ~ ( k about ) ko to first order
and compare this with the corresponding expansion for a free particle
"!'
k2
(k)= 2m
k;
1
-+-kom(k-ko) 2m rn
1 =f;)+-ko.(k-ko) rn
and demand that
The simplest choice is
For the quasi-particles and quasi-holes discussed here, the effective mass is closcly related to the irreducible self-erlergy. This relation is w o ~ t hremembering. We have now comc to an application of the quasi-particle concept: T h e givcs u s thr: possibility of considering definition of the effective mass an 'cffcctive Newton's equation of motion' Tor the quasi-particles, To this end, we construct a wavepacket of qnasi-particles, which movcs with a group velocity vg, = V k ~ ( k ) .The work At done by an external force F on the quasi-particle in a time At is A6 = F - vp,At. We also have AE= Vkc(k) Ak = vg, . Ak. B y combining these expressinn we uhtain (as At -+ a):
PERTURBATION THEORY
317
This is a 'quasi-Newtonian equation': if vH could be canceled out, we would obtain dk/dt = F. We obtain for the ith component of the acceleration of the quasi-particles:
With this expression, the motion of quasi-particles can in principle be calculated. This is particularly instructive for the case of so-called crystalelectrons, the dispersion reIation of which contains the interaction with the crystal lattice, This dispersion relation has roughly the structute shown in figure 26.6: At the edge crf the Brillouin zone,the curvature and hence the effective mass for several bands are negative. This is a reflactioh of the'iact that the crystal-electrons experience Bragg-reflections at thismpoint. ,
.
F ~ r s 82 t edge
Figure 26.6 Schematic graph of the electron energies in a crystal. The electron states at the edge of the Brillouin zone undergo Bragg scattering; which opens up an energy gap,
We obtain another i n t ~ r e s t ~ i application r~g if w t also derive the 'Newton's equation for quasi-particles' for the case of a (veloci ty-dependerit) Lormtz force. The cyclotron resonance frequency
can be determined very accurately expe~irnentaIly,In Mlis way, one obtains information about the dispersion relation ~(k) close to the Fcrmi surface, since only crystal-electrons near the Fermi surface take part in cyclotron resonance.
la1 Non - i r l t e r n c t ~ n g s y s t e m
It 1
(bl Normol s y s t e m s
Superconductor
Figure 26.7 In the non-interacting system (a), the momentum distribution function drops disc~ntiauous~y to zero at the Fermi surface, A finite amount of this discontinuity remains in the interacting system (b), For a superconductor (c), f i r which conventional perturbation theory does not converge, the moment urn distribution function goes continuously to zero.
In the discussions above, we have termed the surface ~ ( k = ) p the F e m i surface of interacting systems, and have learnt of a few properties, for example the change of sign of r, of this surface. In the remaining part of this chapter, we wiIl show that this surface also possesses the property which is primarily connected with the concept 'Fermi edge" namely, a sharp drop in the distribution function. In non-interacting systems the momentum distribution function ( n ( k ) )= (@o / I
CLC~
has a sharp edge, as shown in figure 26.7(a), Since the exact interacting ground state can be represented as superp ositimis of non-interac t ing configurations, we expect that the distribution function will be somewhat smeared out for interacting system. We will show that for 'normal' systems, for which usual perturbation theory converges (in contrast to, for example, superconductors), this distribution exhibits a finite, non-zero discontinuity as depicted in figure 26,7(b), To ?rove this, we insert the expression equation (28,10) f o r t < 0
derived previously, into equation (15.31):
(gl(k)) ,=-a lirn t-o GCL, t )
-
PERTURBATION THEORY
319
and consider two sequences (k+), and (k-), of vectors which converge to a vector ko on the Ferrni surface from the outside and the inside, respectively, i . e , , we have
r(k;) < p = ~ ( k o < ) ~(k:). It folIows that
G(k;,t) Iim G(k,,t) t-40-
= iz(k,)e- i f k e r k + k = iz(k,)+I(k,,O-)
~ ( k t ), = ~ ( k ; ft ,)
(for t
t (for t < 01,
< 0)
lim ~ ( k ; ,t ) = ~ ( k ,f O-). t-0-
We make the plausible assumption that I is continuous as a function of k, at least near the Fermi surface . (If this assumption is incorrect, there will be an additional contribution [~(k+--r ka,0-1 - I(k- -t ko,0-)] i in the result below. Since I is small, this would be a small correction to s(ko) 1 and only slightly affects the size of the discontinuity.) Altogether, we obtain
We have then shown that the momenturrl distribution has a discontinuity of (apptoximate) magnitude r ( k o )at each point ko on the Fermi surface, and in fact i t then looks approximately a we sketched in figure 26.7.
Chapter 27 Diagrammatic calculation o f the two-particle Green's function and the polarization propagator Let us begin by reminding ourselves of the definition of the two-particle Green's function:
of natation.) T h g first step in the diagrammatic calculation is again the adiabatic switching procedure. Analogously to the case of the single-particre Green's function, it can be shown that (icf. equation (18.11) ( f i r e will
in this chapter omit the spin indices for
i2~(rAtA r g, t B ,relCIr D t D )= lim ~ 4
1 (a0 0 1 Uc(-=,
ease
00)
I Qo)
This represent ation allows for a diagrammatic analysis with essentially the same Feynman-rules as those we derived in Chapter 20. We draw the full two-particle Green's function as a structure with four legs. The external endpoints correspond to the four arguments of thc Green" fuunction. Thus, as the single-particle Green's function G(rAt A , rg i for cach
322
TWO-PA RTICLE GRE;%N'SFUNCTIONS
possible time-ordering represented either particle-propagation ( t A > 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 ) ) :
TWO-PARTICLE GREEN'S FUNCTIONS
=
[ 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
TWO-PARTICLE GREEN'S FUNCTIONS
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
la I $+q
andp<2
338
TWO-PA RTKLE GREEN'S FUNCTIONS
We wiIl learn an application of this result in the next chapter when we discuss the effective interaction in the electron gas.
Chapter 28
Effective interaction and dressed vertices, an advanced perspective on plasrnons The polarization propagator has an import ant application in the calculation of the smcalled effective, or dressed, interaction. One automatically comes t o this definition if one tries to partially. sum the perturbation expansion fop the Green's function by successive insert ions of polarization ioser tions in the interaction lines (see figure 28J). The summation of all polarization inser-
Figure 28.1 Summation of all polarization insertions yields an effective interaction.
tions gives, by definition, the full polarization propagator, so the effective interaction is given by
as illustrated in figure 28.2, In this definition, a dressed interaction line, as well as a usual bare one, contribute a factor i in the translation to formal
language. We obtain
EFFECTIVE INTERACTION
Figure 28.2 Diagrammatic representation a£ the effective interaction.
With the definition of the irreducible polarization propagator Q , we can d e ~ i v eDyson's equation for the effective interaction: .
as illustrated in figure 28.3. This equation too is particularly simple for
Figure 28.3 Bysoon 's eqrratirln for the effective interaction. translationally invariant systems with spin-indep enderrt inst ant ancous interactions: (28.3) u e f f ( % w ) = u(q) u(q)Q(q,w)u,fi('~, 4We solve this equation IOT the effective intcsaetion and define the (in general frequency and momentum-dependent ) dielectric function f(q, w ):
+
Thus, in contrast to the barc particle-particle interact,inn, the effective interaction takes into account t h e polarization of the medium ia the sensc of
linear response theory. The frequency dependence means, after Fourier back-transforming , that we now have a time-dependent interaction. This can clearly be understood as a certain inertia of the polarization cloud. From equation (28.4), we obtain the following equation: a
which together with equation (27-13) yields a relation between the full polarization prupagat or and t he dielectric function
We will now discuss an example for Coulomb systems ( u ( ~ = ) 4 ~ e ~ / ~ the sc~calledrandom-phase approximation (RPA). This is basically nothing but the summation of ring diagrams just as in the Gell-Mann-Brueckner theory. We approximate the Green's function by the diagrams shown in figure 28.4. As a consequence, the approximation for the effective interaction
' ) :
t.,: I + b+D+P+V---Figure 28.4 Random-phase approximation for the Green's function.
We can interpret this approximation in the following way. In the perturbation expansion of the effective interaction, the zeroth-order polarhation propagator is inserted instead of the full irreducible polarization:
is as shown in figure 28.5.
RPA: 4 z 11('), i . e . ,
Figure 28.5 Random-phase approximation fm the eflectivc interaction.
EFFECTIVE JNTERACTION
This yields (RPA)(q, LJ)
ueff
=
4 q)
1 - u(s)n(O)(n,w)
and
dRPA) (q, u) = 1 - ~ ( ~ )( O1) (q, 1 w)
(28.8)
respectively, The name RPA was originally coined within the framework of plasmon theory, where, under the assumption of randomly distributed phase-factors, one can neglect a part of the electron-electron interaction, As WE have seen, the plasmon theory gives a result similar to the ringdiagram approximation by G e l l - M a n and Brueckner. Thus, the term RPA has come to be applied also for the latter theory. In this context, the physical reason for the name is however ncrt quite clear, nos is the ring-diagram approximation in its original sense identical to the RPA. From t h e explicit repesentdion, equations (27.25) and (27,26), of the Lindhard function, we obtain following limits:
and
Equation (28.9) yields the following result for the effective interaction in the limit of g/kk*<< 1 u ~ ) ( ~ = , w 0) c
h e 2 92
+ A2
where A' =
6+(~/!2)ez EF
(28.11)
which, after Fourier back-transforming, gives the following form for the static effective interaction for rkF >> 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 &(<) is defined by
From cquation (33.151, we see that
C p n b ( ~+ Bnr
'*-I'
-
EA~))(~L),~~
( ~ k ) ~ t T= -'
&(Of ( E )
(33.16)
n?bt
where f ((1 is the Ferrni distribution function
Equation (33.16) in equation (33.13) then yields
Near the Femzi snrface, we have
the spectral function & ( E ) can be written as a sum of a singular term due tb the quasi-particle and a smooth incoherent term: so here
With equation (33.18) inserted i-n (33.171, we obtain
The last term in equation (33.19) vanishes as we move onto the Fermi surface because Akqinc([)is smooth, f(<)is bounded and r l : / [ ( c k - ~ ) 2 + $ ] vanishes as q k when q k 4 0. Hence, we conclude that
which is precisely the quasi-particle equilibrium distribution function. We will now turn t o the verification of the properties (2) to (4) of the Landau Fermi liquid theory. These properties are all related to how the system responds to an external perturbation. It is then natu~althat our starting point is t o discuss this response. Once we have done so, we will examihe this expression in the limit of long wavelengths and low energies of the perturbation, from which we can verify the properties of the Landau Fermi liquid theory. In Chapter 27, we derived an expression for the linear response of ah observable A(t) due t o an externd perturbatioh H,,t(t')H turned on a t time t = to. The response is
where Wo is the interacting ground state, (We assume in this chapter that all states are normalized.) In terms of the retarded correlation function
wc can write
6(A)('t)as
/
t
a(A)(t) =
d t t ~ ~ t') ( t ,
to
In most cases, ~ ~ t') ( depends t , only on the difference t - i f , in which we can set t' = 0 in equation (33.22) and write
The function ~
case
~ (defined t ) by equation (33.24) is called a retarded correlation function, since it describes an after-effect. In analogy to this function, we may define an advanced correlation function D* ( t ) by
392
MICROSCOPIC DERIVATION
We will see later in this section that the retarded and the advanced cotrelation functions are intimately related to t h e real-time correlation function, which is defined by
In fact, one may show that the real-time correlation function completely determines the retarded and advanced correlation functions (see, for e x a m ple, Fetter arid Walecka [22]). The real-time correlation function is usually more convenient to use in zeretemperature calculations than the retarded correlation function. The reason far this is clear if we ~ernindourselves of the definition of the Green's function G(r, t ) :
The structure of the real-time correlation function is then similar to that of the Green's function. In the cases of interest to us, the real-time correlation functions will be proportional to the two-particle Green's function. This means that dl the techniques for calculating the Green's functions, such as Feynrnan diagrams, can be applied t o the calculation of the real-time correlation functions. Once the real-time correlation has been calculated, it is easy to obtain the retarded correlation function. If we consider the response of the density t o a perturbation which couples to the density we are thus led to study the real-time density-density corrclation function
The density operator pq(tIH is in the plane-wavc representation
If, on the other hand, we want to calculate the response of the density t o a perturbation which couples to the current density, we are led t o study the correlation function
where a = 1,2,J denotes the three Cartesian components. The current operator is in the plane-wave representation
With t,hese density and current operators in equations (33.27) and (33.28) we see that these correlation functior~sare proportional to s u m s over twoparticle Green's functions.
FERMI LIQUID THEORY
393
We will use the Fourier-transforms of the cotrelation functions (33.27) and (33.28) with respect t o time:
For convenience, we have here introduced the notation p = 1 , 2 , 3 , 4 . At this point, it is also convenient to introduce a four-vector notation. We will denote (k,G ) by k, where k and G are particle wavevectors and energies, respettively, and we will denote the wavevector q and frequency w of the external perturbation by q . We also introduce a four-vector defined by
vi
T h e diagrammatic representa'tion of D,Cq, w ) is given in figure 33.1.
Figure 33.1 The carrelation function D, (q,w ) is ob t i n e d by s~lrnmingall these polarization diagrams over k.
With the four-vcctor notation introduced above, U P t q , w )is given by t h e equation iDphT) = ~ ( k9 / 2 ) G ( k - ¶ / 2 ) h ( k ; p) (33.30)
x
+
k
where
In equation (33.301, Ap{k;q ) is thc so-calIed vertex function. This function has the diagrammatic representatio~rgiven in figure 33.2, which yields the equation
Here, r(k,k l ; . q )is the scattering function, which is the collection of all the
MICROSCOPIC DERIVATION
Figure 33.2 Diagrammatic represent a tion of the vertex function A(k; g) in equation (33.29). connected diagrams shown in figure 33.3- This function is the sum of dl
Figure 33.3 The scat terihg function r ( k , k t ; q ) cunsists of all possible ways a particle-hde pair of momentum q can scatter.
possible ~ c a t t ~ e r i nevents g between a particle and a hole. The scattering function may be decomposed into irreducible parts I(k,6'; q ) and products of article and hole propagators. The irreducible parks are represented by diagrams which cannot be separated by cutting one particle and one hole line only. From figure 33 -4, we see that T(k, k'; q ) satisfies
Figure 33.4 The scat Eering firrl ctjon can he separated into irreducible parts and products of particle and hole propagators.
FERMT LIQUID THEORY
r(k k t ; 4) = ~ ( kk';, p)
+ C ~ ( kkt';, q ) ~ ( k ' '+ q/2)G(kM- 9 / 2 ) r ( k " , k'; q ) . k"
(33.32) Using equation (33.32), we can then write the foEIawing integral equation for the vertex function:
In the equation fur the correl&tion function, equation (33.301, and in equations (33.33) ahd (33.32) we see that the quantity
erlters. This quantity describes the propagatioh of a dressed particle-hole pair. As we are interested in perturbations with long wavelengths and low energies, which has as a consequence that the excitations of the system will be on the Fermi surface, we will ultimately take the Iirrlit tql 4 0 and w 4 0 of the correlation function. 'I'o do so, we rriust k t ~ o whow thc product equation (33.341 behaves in this limit. Near the Fermi surface, we can writc
where t k can be taken t o he real, since the damping of the propagator vanishcs as we approach the Fermi surface. Irere, 17 > 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
MICROSCOPIC DERIVATION
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
MICROSCOPIC DERIVATION
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. Equation (33.87) can then be written as
and equation (33.88) can he written as
Using thc by now familiar identity
FEEMI LIQ UID THEORY these two equations become
and
respectively. These two expressions are identical 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
