A course on many-body theory applied to solid states physics - chapters 1 and 2 9971503379

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Lecture Notes in Physics -

Vol. 11

A Course on Many-Body Theory Applied to Solid-State Physics

Charles P. Enz Department of Theoretica l Physics University of Geneva

r.r,

Il kové trdoj

, ...St.§

L. . . . ..( .::?......_ D.i l u m ..... ....5. ...L . .J~-

··:, ~( k• ............ .

h World Scientific fl' Singapore • New Jersey• London• Hong Kong

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PREFACE

Library of Concraa Cataloeing-in-PublicatJon Data

Enz, Clwles P. (Charles Paul), 1925A course on many-body theory applied to solid-state physics / Charles P. Enz cm. p. lncludes bibliographical references and index. ISBN 99715033360. -- ISBN 9971503379 (pbk.) 1. Solid state physics. 2. Many-body problem. I. Title QCI76.E58 1992 530.4'1--dc20 92-30895 CIP

Copyright

(í;)

1992 by World Scientific Publishing Co. Pte. Ltd.

Ali righls reserveil. This boolc, or parts thereof. may not be reproduced in any Jonn ~rbyany_lTU!QIU, electronic or mechanical, i11cluding photocopying, recording orany 111/omiallon storage and retrieval syslem now lcnown or to be invented without ' written pennission from the Publisher.

Printed by Singapore National Printers Ltd.

This book is based on a course on many-body theory I had been asked to give in the Postgraduate Teaching Program of Physics in French Switzerland (Troisieme Cycle de Physique en Suisse Romande) at the Federal Institute of Technology (Ecole Polytechnique Fédérale) in Lausanne during the summer terms of 1982 and of 1985. But some of the materials had already been presented at earlier invitations, 1962 in Leysin (IVe Cours de Perfectionnement de l' Association Vaudoise des Chercheurs en Physique) , 1965 in Lausanne (Troisieme Cycle, notes by Michel Romério) and in Geneva (University, notes by Fabio Barblan), 1967 in Trieste (International Centre for Theoretical Physics) , 1968/69 in Lausanne (Troisieme Cycle, notes by Michel Droz), 1969 in Majorca (International School of Physics) and 1971 in Geneva (Troisieme Cycle, notes by Dionys Baeriswyl). In comparing these different courses the evolution, both in scope and in sophistication appears striking to me. Interestingly, this evolution is both, my own and that of physics itself. The last decades have indeed brought exciting discoveries and interpretations in ever faster succession. For this reason I think it is a good time now to write such a book, in spite of the large number of already existing texts on many-body theory. On a more persona! level this timing also means a certain retrospective. My ambition and satisfaction in writing this book was to reach a selfcontained unity by giving as complete explanations as possible. This of course is a time-consuming and often frustrating task. But when I think of the disappointments in reading certain theoretical reviews which just repeat formulas without explaining them I become convinced that I did the right thing. But the question now is how well these explanations withstand criticism. Fortunately, in several instances I benefited from clarifying

vi o Preface

discussions with Daniel Loss, Ching Zhou and Ora. Entin-Wohlman which I acknowledge gratefully bere. Wanting to "leave nothing unexplained" of course sets other limits. So I had to leave out many exciting topics or cover them only with few words instead of detailed expositions. This, however, is the author's privilege which is comparable to that of a conductor's: it is he who selects the music to be performed. But he also works with the musicians to realise his interpretation, which may or may not be applauded in the concert hall. The performers in the orchestra, on the other band, have to work their daily exercises at home if they wish to belong to the orchestra or even become soloists. For the students of this book there is also a selection of exercises, all with detailed solutions and many of a similar kind just as the scales of the musicians all resemble each other. In this sense the book is indeed a course, as also in the exposition of the two introductory chapters, in particular the combinatorial toccata leading to Wick's theorems for a complex time path which is my composition. The material of the remaining three chapters, however, is selected more according to its research interest. Topics like localization by disorder or mesoscopic transport had greatly fascinated me because of their lack of intuitive evidence. Then of course came high-Tc superconductivity, resounding like a huge Richard Strauss orchestra; it still rings in the ears. But the topic of most concern to me had always been the problem of magnetism. I even had doubts of ever grasping the subject as a unity and I did not find much help in books or reviews. But what came out of a considerable effort as Chapter 5 reassures me that there may indeed be some unity and intelligibility in the matter. In all these efforts I was helped by Francine Gennai-Nicole in typing the manuscript in '!EX and by Jean-Gabriel Bosch in drawing the figures in MacDraw. The numerous computer problems I ran into were all solved with great competence and patience by Andreas Malaspinas. To all three I address my sincere thanks for their expert help. Many readers may find my style tiresome because of the obsession of "leaving nothing unexplained". I have no excuse for this because it is both persona! and eminently Swiss. Indeed, it is the attitude that made Swiss watch-makers famous and let the people of this country be concerned with the security of the Alpine passes ever since the foundation of the Confcederatio Helvetica 700 years ago. Since this book is ready just for Lady Helvetia's big celebration it gives me pleasure and pride to dedicate it to Her 700th anniversary. Geneva, summer 1991

Charles P. Enz

CONTENTS

v

PREFACE

Chapter 1. ELECTRONS AND PHONONS AND THEIR QUANTIZATION Electron and Phonon Modes in Perfect Crystals Many-body Description of One- and Two-body Forces Second Quantization 4. Groundstate Energy in Hartree-Fock Approximation Solutions to the Problems of Chapter 1 References to Chapter 1

l. 2. 3.

Chapter 2. GREEN'S FUNCTION FORMALISM IN COMPLEX TIME 5. 6. 7. 8. 9. 10. 11.

Schrodinger, Heisenberg and Interaction Representations Evolution in Complex Time and Global Equilibrium Averages Linear Response Functions and Green's Functions Wick's Theorem for Operator Products Wick's Theorem for Averages. Feynman Diagrams Thermodynamic Feynman Diagrams Dyson Equation,· Selfenergy Diagrams and Dressed Propagators 12. Generating Functional for Feynman Diagrams. Ward Identities vii

1 7 11

19 24

29

31 31 35 41

47 51

r,o

Contents o ix

viii o Contents

Solutions to the Problems of Chapter 2 References to Chapter 2

Chapter 3. TRANSPORT PHENOMENA AND DISORDER EFFECTS 13. 14. 15. 16.

85 94

97

Boltzmann-equation Description of Electron Transport Einstein Relation and Stationary Transport Coefficients Residua! Resistance and Dynamical Structure Factor Resistivity Due to Electron-phonon Interaction. Semiconductors 17. Low-temperature Resistivity Due to Electron-electron Interaction 18. Kubo and Mori Formulas. Evaluation for lmpurity Scattering 19. Disorder, Localization and Length Scaling Solutions to the Problems of Chapter 3 References to Chapter 3

98 105 112

Chapter 4. SUPERCONDUCTIVITY

177

20. Effective Electron Attraction. Cooper Pairs and Schafroth Pairs 21. BCS Groundstate, Gap Equation and Bogoljubov Transformation 22 . Binding Energy, Correlation Amplitudes and Coherence Length 23. Statistical Mechanics of Bogolons. The Transition Temperature 24. Thermodynamic Properties Near T = Te and Near T = O 25. Weak Static Magnetic Field. London Equation and Critical Field 26. Weak Electromagnetic Field. Ginzburg-Landau Equation 27. Type-I and -II Superconductors. Flux Quantum, Critical Current 28. Strong-coupling Theory and Coulomb Repulsion Solutions to the Problems of Chapter 4 References to Chapter 4

121 129 141 152 165 173

180 188 195 200 204 210 218 226 231 242 254

Chapter 5. MAGNETISM

W . Magnetism of Local Moments. Weiss' Molecular Field Theory :JO. Groundstate and Magnons of the Heisenberg Model :n. The Static Magnetic Susceptibility of Band Electrons :J2 . Itinerant-electron Magnetism. Stoner's Molecular Field Theory ;33, Groundstate, Excitations and Symmetries of the Hubbard Model 34. Weak Itinerant-electron Magnetism. Paramagnons :m. ltinerant-electron Ferromagnetism. Magnetization Fluctuations :J6. Itinerant-electron Antiferromagnetism. Spin-density Waves. Solutions to the Problems of Chapter 5 References to Chapter 5

257 262 271 281 291 301 312 328 344 353 368

AUTHORINDEX

373

SUBJECT INDEX

385

Chapter 1

ELECTR ONS AND PHONO NS AND THEIR QU ANTIZA TION

The subject of electrons and phonons in solids is treated in many books 1 of which we wish to mention particularly those by Ashcroft and Mermin ) 2 their and crystals ideal only chapter y 1111d by Ziman ). In this introductor 11xci tations, electrons and phonons, are considered. Invariance under trans1,~tions of the lattice then implies the fundamenta l property of Bloch func3 t,lons for the electrons, enunciated by Bloch in 1928 ). On the other hand, if be described by totally to have they present, 111tt.ny phonons or electrons are Hy mmetric or totally antisymmet ric wavefunctions since, as was shown in dotail by Pauli (Ref. 4, Section 14), these are the only acceptable permut.1\tion symmetries that the states describing many identical particles can luwe. In this spirit, second quantizatio n is introduced in a systematic way nud forms, together with the description of electrons and phonons and their l11tcractions , the basis for the subsequent chapters. 1. Electron and Phonon Modes in Perfect Crystals Except for the discussion of fluids and disordered solids in Chapter 3, the physical systems considered in this course are perfect crystals in 3dimensional r-space built up from a periodic lattice of unit cells of given point group symmetry. The lattice vectors 3

R=

L

miai ;

mi

= O,

±1, ±2, ...

{1.1)

i=l

are generated by the primitive translations ai which span the unit cell C centered at R= O whose volume is v= a 1 · (82 x a 3 ). gene,ates: reciprocal lattice in Pe,iodicity in

r-space

k-spac: 1

J. Electron and Phonon Modes in Perfect Crystals ◊ 3

2 ◊ Chapter 1. Electrcms an d Phon on s and Their' Qua11tiz11tio11

the basic relation

eiK-R

=1

(1.2)

where the reciprocal lattice vectors K may be expressed in terms of primitive translations bj in analogy to (1.1),

'l'li lH Hliows that Lis also the number ofk-points contained in Z. Combining l,hnHo uxplicit values (1.8) with Eqs. (1.1) , (1.4) and (1.6), it is easy to derive 126 t,lin t,wo relations (Problem 1.1) ' ' )

L1

'°' ik-R _ '°' 8 - L., k,K L., e

K= Lnibi; ni =0, ±l, ±2, ... .

(1.3)

wltnro k is not restricted to Z, and

j=l

It is easily seen that the basic relation (1.2) is satisfied by choosing the bj according to the conditions (1.4) These conditions are satisfied with b 1 = 27ra2 x a 3 /v, etc. Excitations in a perfect crystal may be described by a field on the 5 lattice ), J(R), which has a Fourier representation

f (R) =

L g(k)eik-R .

(1.5)

k EZ

In writing this equation discrete wavevectors were assumed which is most convenient for calculat ions. Also, because of the basic relation (1.2) the ksum may be restricted to the unit cell Z of the reciprocal lattice, centered at K = O. This domain is called the first Brillouin zone or reduced zone, and 16 k E Z is called a crystal momentum • ) divided by 1i (h = 27l'n is Planck's constant). Explicitly, Z is defined as the semi-open domain 3

1 '°'11: -b . · -k= L.,11' 2

1

+-2 < 11:J. < -

(1.6)

j=l

whose boundary {}Z is formed by the median planes of the rays from K= O to all neighboring K-vectors necessary to close Z. Discreteness of the k-vectors in Eq. (1.5) is achieved by imposing on the crystal the periodic boundary conditions

f(R + 2Liai) = f (R) ; i = 1, 2, 3

(1.9)

K

R EV

3

12 ' )

~ 1 '°' eik•R -_ uao L.,

-

(1.10)

'

LkEZ

w liorn R is not restricted to V. These relations express, respectively, m t,lionormality modulo a vector K and closure in Z of the plane wave lloldH. J\ pplied to Eq. (1.5), relation (1.9) yields the Fourier amplitude

g(k)

L /(R)e-ik-R = g(k + K)

=½

(1.11)

REV

wliid, is seen to have the periodicity of the reciprocal lattice. F'ree conduction electrons or holes in a perfect crystal are described by ,~ mu~-body Hamiltonian 2

W

(1.12) +U(r) =~ 2m momenelectron's the being p , R) + = U(r 1

wl t.lt a periodic potential U(r) t. 11111 and m its mass. Thus H el is invariant under the group of transla1,lom; by R . Since this group is abelian , i.e. all its elements commute with 1 rni.ch other, the wavefunctions 1/; in the Schrodinger equation He 1/; = c'I/; may be which group this of form a one-dimensional unitary representation pnrnmetrized by a wavevector k E Z . Thus the action of the translation by ll IH

(1.13)

1,,.om this representation it is easily seen that e-ik•r'l/;k(r) is a periodic f1111ct ion so that Bloch 's theorem l) (1.14)

(1. 7)

where Li are large positive integers. This means that the crystal volume is V= Lv with L = 8L 1 L 2 L 3 . Equations (1.5) , (1.7) imply that ki -LiaJ7r = v i are integers, and with Eqs. (1.4), (1.6) one finds that

liolds. Insertion of (1.14) into the Schrodinger equation shows that the pe1 rlodic part uk(r) satisfies a modified Schrodinger equation H~ uk = €kuk 2 1 unit cell C the in solved be may which U(r) + wlth H~ = (p + nk) /2m hy imposing the periodic boundary conditions (1.14) . Quantum mechanics thcn tells us that the,e is a countably infinite set of solutions which

im

J. Electron and Phonon Modes in Perfect Crystals o 5

4 o Chapter 1. Electrons and Phonons and Th eir Quan tization

be labeled by a band index n. Since the electron has spin 1/2 there is an additional degree of freedom c, = +, - , indicating the spin direction up or down and which is represented by the column vectors

l+)=G); H=G)·

(1.16)

where k is a composite mode index such that ±k= (n , ±k, u). The eigenstates are lk) = "Pnklu) where "Pnk are Bloch functions of the form (1.14) and the eigenvalues ek = enk are the energy bands. Orthonormality in the crystal vol ume V, (1.17) combined with Eqs. (1.9), (1.13) and (1.14) implies the orthonormality in the unit cell C, (1.18)

While the Bloch functions are extended over the whole crystal volume V , t he Wannier fun ctions 1 )

L "Pnk(r + R)

'ť h c

(1.15)

Thus the Schrodinger equation may finally be written in the form

wn(r +R)=

= m, the slope is zero. fin al remark concerning free electrons is that often relativistic cor1 11ť tlo 111; to the Hamiltonian {1.12) cannot be neglected, the most important l111l11g the spin-orbit interaction 7,B) wli ll, , for a single band, n

Us.o.

RE~v

{1.19)

½L wn (r + R) e- ik-R = "Pnk+K{r) .

11

1

L

+2

{1.20)

{1.22)

u.,(R)

R ' E~v'

x C 1111 , (R - R')u.,, (R')}

are localized around r = -R. They may be viewed as fields (1.5) with intern a! degrees of freedom n and r EC (they are not, however, observables in t he quantum mechanical sense) . We may thus calculate their Fourier ampli t ude according to {1.11) and find , with use of Eqs. (1.9) and {1.13) , that

=

2 (R) {p;M

L

?-{_Ph=

kEZ

"Pnk{r)

2{ď X VU)· p =~ C 4m 2

wl,m·o ď = (u''", c,Y, c, z ) are the Pauli spin matrices acting on the states ( I. I li}. Since mv = p = Ti'v /i, Us.o. is of the order (v/c)2U (c is the light v,11111:ity). The main effect of Us.o. is to mix band and spin states, so that now IA·) L:0nu,n'u' (k)1Pn'klu') . This gives rise to double group representations · nť t,ho crystal symmetry7 ' 8 ) . Phonons in a perfect crystal are described by the displacement vectors 11 .,( R) of the basis atoms (or ions in the case of a conductor) v= l, ... '. B ťro 111 their equilibrium position R+r 11 in the unit cell at Rand by the conJu1'.fLl.1· 111omenta p.,(R) = M„ů.,{R) , M„ being the mass of the ato~ic ~pecies ,, 'ťl111s the cartesian components u„i{R) and P„i (R) are fields w1th 3B 1111,ornal degrees of freedom. Pree phonons are defined by the harmonie ILpproximation of the interatomic potential, i.e. by the Hamiltonian

= L Hf..\

{1.23)

R ,v

wlu!re the C„ 111 , are real 3 x 3 force-constant matrices with the crystal poriodicity {l. 7) and Hft\ is the one-body Hamiltonian of the atom with ,,qu ili brium position R r 11 • The Hamiltonian {1.23) is diagonalized by 1,m111,forming to normal-mode amplitudes or phonon coordinates Qq, Pq ,u·cording to

+

R EV

This periodicity with the reciprocal lattice is of course also shared by the energy bands enk . Since the Hamiltonian (1.12) is a real operator, the Schrodinger equation is time-reversal invariant which implies Kramers ' theorem 7 ) u~-k = unk, en-k = enk· Combining this reflection symmetry of enk with the periodicity enk+K = enk one easily finds that two bands n i= m crossing at a border point of the reduced zone Z have symmetric slopes, (1.21)

p.,(R)

= Í:(M„wq/LB) 1 l 2 eq(v)eiq•Rp_q

(1.24)

q

where q is a crystal momentum, q E Z. Here q is a composite mode index 1, 2, 3 l, ... , B is the branch index, j ,mch that ±q (µ, ±q, j). µ U1e polarization index, w /Ti the {real) phonon frequency and eq(v) is a (complex) polarization v~tor. In Eq. {1.24) reality of u„ and P„ implies Q_q

=

=

=

=

Q;, P_q

=P;, = w_q

wq, e _q

=e/ .

(1.25)

f . Many -body

Quaritizatior1 6 o Chapter 1. Electrons and Phonons and Their

eq(v) are The cartesia n compon ents eq(v, i) ofthe polariza tion vectors 125 eigenfunctions of the dynamical matrix • • ) 2 (1.26) (MI/Ml/,)- 112 cl/l/,(R )e-iq•R D1111'(q) = h

L

REV

with eigenvalues

w:, (1.27)

D 11i,v'i'(q)eq(v', i')= w:eq(v, i).

L v',i'

of the Schrodinger This eigenvalue equation for the phonon s is the analog eq(v, i) are nctions eigenfu the r, Howeve s. equatio n (1.16) for the electron are elements of elements of a 3B-dimensional vector space while "Pnk(r) rmality relation a Hilbert space. This analogy also extends to the orthono (1.18) which in the phonon case takes the form

½~ eµ:i,/v) • eµ'qj' (v)= 8µµ'8jj' .

(1.28)

?-{Ph= ~

Lwq(P ; Pq

q

of mode q. This wl1111 n t,lio term wq/2 represe nts the zero-point energy introdu ced by Dirac to quantiz e 11111111111 ·,ll Jield quantiz ation was first 9 tely analogous. 111.,, t,11 uly11n.mics ). The case of phonon s is seen to be comple

Forces '-1 , Mn11y- body Descri ption of One- and Two-b ody t) w~ich may F(r, field l extema an by ed produc are ( >110-hody forces particle j = 1; ... , N d11p„11d 011 t.ime and which acts on an observable Oj of Pj• These forces are (111,., I 11111 or ion) with position rj and momen tum ,111,dl1Ll,oc l by an N-body interact ion energy N

1

~ {F(rj, t), Oj}=

/11( 1 111 1111 t. hť

2 I

(1.30)

where the bracket means the commu tator [A, B] = AB

tation relation s which, togethe r with (1.30), implies the usual commu (1.32)

all other commu tators being zero, the phonon number Nq

= b:bq

-

1

V LF(q , t)d(-q ). q

(2.1)

(see the is necessary in case Oj does not commu te with r j 11H 11111plcH below), and 111111

w li ich

N

1

L 2{8(r - rj), Oj}

(2.2)

j=l

IHt,ho rlensity associa ted to the observable Oj 1111, H,urier-transformed density

5

).

In the last expression (2.1)

N

BA.

fields p 11 (R), Equatio n (1.30) is, in fact, a mode quantization of the quantu m mechanical u 11 (R) which therefore become observables in the rs b: and bq by the sense. Introdu cing creation and annihilation operato prescrip tion (1.31)

[bq , b~] = 8qq' ,

-

d3 rF(r,t) d(r) =

trizannticom mutator {A, B} = AB+B A takes care ofthe symme

d(r) =

s (which have been Since Pq and Qq are canonically conjuga te variable is n conditio ation defined dimensionless), the quantiz

[

Ji V

(1.29)

+ Q;Qq) .

q

o/ One- arid Two-boáy Forces o 7

Hamilto nian (1.29) 11lt111 l111i•o1110H nu observable. With Eqs. (1.31), (1.33) the form ed quantiz the K' 11 11 11v11r l11to 1 (1.34) JiPh = Lwq(N q + 2)

J

ation (1.9), one Using Eqs. (1.24)- (1.28) and crystal momen tum conserv (1.23) simplifies shows that, after some algebra, the phonon Hamilto nian m 1.2), to a sum of harmonie oscillator Hamilto nians (Proble

De11cňp tfor1

(1.33)

d(q) = [ d3 rd(r)e- iq•r =

Ív

L ~{e-iq•r;, Oj}

(2.3)

j=I

the inverse Fourier lti l11troduced, and the externa l field is assume d to have 111proHOn tation ' (2.4) - , t)e•q-r. 1 ""' L..,F(q F(r, t) = V q

r, the discrete It should be noted that because of the continu ity of icted values of unrestr with (1.8) , (1.6) Eqs. by defined bere 11 voctors are ns (1.7) conditio ry bounda c periodi t,ho integers vi. As a consequence, the to ILl'll goneralized

(2.5)

8 o Chapter 1. Electrons and Phonons and Their Quantization

2. Many-body Description of One- and Two-body Forces o 9

and Eqs. (1.9}, (1.10} are replaced, respectively, by _.!_ { V Jv

d3reiq·r

111 t lu, 1Ll,omic species v then are

=8

n.,(q) =

(2.6)

q ,o

R

and

1111d

"' . Vl 'L.,,e'q•r = 8(r)

(2.7)

j.,(q)

q

~l) .

{e - iq·(R+rv+uv(R)),

p.,(R)}:

(2.13)

v

(ca ) may be measured by X-ray or by elastic neutron scattering (see 1,1011 15). On the other hand, Lv M„j.,(0) = La, 11 P.,(R) = L Lv M„c y, 111 , 1 o c is the center-of-mass velocity of the crystal. Because of the ~eriodic lu11111clmy conditions (2.5), c = O which means that wi~h t~~se .boundary , ,1111 llt.lo11s recoils of the crystal as a whole are not descnbed . 1 By taking for Oj the one-body Hamiltonians and H_~\ of l•'q,i ( 1.12) and (1.23) , respectively, one obtains the energy dens1ties of 1111•1t.rons and phonons, 11 ,

(2.8) hel(q)

h ph( Q )

j

which couples to an electric potential eU(r, t) (e is the elementary charge) , the current density

(2.11)

j

which couples to a magnetic field gµ 8 H(r, t), (g ~ 2 is the spectroscopic splitting factor and µ 8 = eh/2mc the Bohr magneton). This description by densities ii(r) and Jn(r) leads naturally to the notion of an electron fluid described by hydrodynamic equations (see Section 13) . The surprising fact which recently emerged from an analysis of size effects in ohmic conduction is that the electron fluid in metals like copper has a viscosity comparable to water (for copper 17(300K) = 0.6cP) 10). In the case of the atoms or ions forming a perfect crystal the r j are the positions R+ r 11 + u., (R), v = 1, .. . , B . The number and current densities

(2.14)

_ '"'!{e-iq-(R+rv+ uv(R)) Hph } - ~ , R,11 ' R,11

2

(2.15)

1c•1ipoct,ively. The limit q = O in (2.14) and (2 .15} produces the many-body ll11111lltonians of free electrons and free phonons,

W 1 = he1(0) = LHJ'

(2.10) which couples to a vector potential (e/c)A(r, t), and the spin density

= L ~{e-iq•r,, H_;'} j

L

~

~

1111

The corresponding expressions (2.3) and fields F(r, t) are 1): the number density n(q) = e-iq·r, (2.9)

- -iq•r, s(q ) -_ '"' ! aje 2

2

H'J

a+=~(ax+iaY)= (~ ~); a-=~(ax-iay)= (~ ~)

= (~

=L R

where the last expression is the Dirac function . The most important examples of electronic densities are obtained by inserting into Eq. (2.2} the observables Oj = 1, Pj/m, a1 /2 where p is the electron momentum and ď are the Pauli spin matrices introduced in Eq. (1.22),

az

(2.12)

I : e- iq·(R+rv+uv(R))

(2.16)

11 11d

(2.17) R,11

'1'1111 lnst relation is nothing else than Eq. (1.23) which is thus recognized as 11lrc111.dy being a many-body Hamiltonian with respect to the atoms. This is 1111 liccause (1.23) is a function of the phonon fields p.,(R) and u.,(R). Since, 1111 1,1 10 other hand the electron fields, namely the Wannier functions (1.19), 11 111 not observables, the many-body Hamiltonian of the free electrons must ht1 l,uilt up from the observables H;' of individual particles (electrons). Two-body forces among electrons and ions are, in essence, shielded ( loulomb forces in Born-Oppenheimer approximation 6 ) . This is an adiabatic "l'Proximation in which the electrons follow instantaneously the ionic mo1,11111 1 which is justified since vionfvel Jm/M., 0.01. The two-body

~

~

9. S econd Quantization o 11

10 o Chapter 1. Electrons and Phonons and Th eir Quantization

forces among the conduction electrons are described by the two-body interaction Hamiltonian 1-lel-el

=

~

t

(j =/- l)(rj - r1)

(2.18)

j ,l=l

where ef,(r) is a shielded Coulomb potential. Similarly, the two-body interactions between electrons and ions are described by a shielded Coulomb potential which, however, depends on the atomic species v, wl-ion=

L

{ef,jR+rv+uv(R)-rj)-v(R+rv-rj)}. (2.19)

REV,v,j

Developing this expression in powers of the displacements u„ and retaining 26 7 only the first term, one obtains the usual electron-phonon interaction ' ' ) . Making use of Eqs. (1.9), (1.24) and (2.9) it may be cast into the form (Problem 1.3) 6 ) wl-ph

=

LL K

LÓq+q',K'Y/q')n(q')Qq

(2.20)

~ ~(LBM„wq)- 112 q' · eq(1.1)i

11

(q')eiq'·ru = 'Y.:_q(-q')

(2.21)

is the coupling function and i.,(q) is the Fourier transform of the potential ef>v(r) defined as in Eq. (2.4). In Eq. (2.20) q E Z but q' is unrestricted so that momentum conservation expressed by the Kronecker-8 may require a K =/- O. Such a momentum transfer to the lattice is called an Umklapp 2 process 1 ' ' 6 ' 7 ) (German for flip over). This non-rigorous conservation modulo a K reflects the discreteness of the translation group of a perfect crystal which is a broken symmetry as compared to the continuous translation group of a fluid. Two-body elastic forces among the atoms (or ions) are contained in the potential energy term of the Hamiltonian (1.23). Terms of higher order in the displacements u„ may also play a role, in particular in insulators. Such anharmonic terms involving 3 or more phonons give rise to phonon-phonon interactions2 ' 5 ' 6 ' 7 ) 1fPh-ph n

=~ ~ ~ n! ~ ~ K

15

t11 wh 11H thormal expansion, heat conduction or second sound ' ). I 1!111 1,1 11 t11\rt.lc11larly important for light .a toms for which M„ is of the order 1111h11 1111111,11 11111.H:;, :;ince in this case the displacements u„ are comparable 111 t li1 1l111l,l1•11 Hp11.ei11gs, even at low temperatures. This is the case of quantum w/1,I" I ) whol'O tuuneling of the atoms to vacant sites takes place. I11 t.111 1 c1u;o of the electron-ion interaction (2.19), higher order terms 111 I ht1 d lt1pln.comcnts u„ have also been considered in the literature. Such Will/I 11/wuou 7>rocesses 2 ' 11 ) were once invoked to explain some temperature 1u111 11 11il v l11 t,ho carrier mobility of semiconductors (see Section 16); they cer7 h•litl v pl 1w I L role in the optical spectra of semiconductors ). Very recently 111,11 l111 vn liťo ll invoked in the problem of high-temperature superconducll \ li v ( 111111 Uliapter 4) where anharmonicity (e.g. double-well potentials) 11l11 y 1111 l111portant role (see, e.g. Ref. 12).

li

H111 •c1111I

Quantization

q'

where

'Yq(q') =

A,tl 111111H111lc offccts play a role in the thermal properties of dielectric 1 1 1 ,1 t 11111

8Q1+ ... +qn ,Kc(n) QQ1 ... QQn ; n > 3. Q1 -- -Qn -

(2.22)

Q1- --Qn

Here q 1 ... qn are crystal momenta and, as in Eq. (2.20), the Kronecker-8 expresses momentum conservation modulo a K. Note that Umklapp processes K =/- O are only possible for n 2': 3; i.e. for interaction terms.

'l'h, , ubjccts of many-body theory, as of field theory, are processes in wltl, li nxri tn.tions of the system appear or disappear. This means that the 11 11111/u •ť 11/ O, is g_iven by

with the condition

+oo

GA1(z ) =

j dtGA1(t) eizt = r AB(z ) ± (rA+B+(-z*))* .

{7.20)

Therefore, knowledge of the propagator {7.17) allows to calculate in turn r AB through {7.18), G'.;1 through (7.20) and hence the response function through (7.16). In view of the imaginary-time formalism to be used in subsequent chapters the following i magi nary-time propagatordefined on the path {6.18) is of particular importance (our sign convention is the same as in Refs. 7 and 8):

- QAB(T)

= (T{A{-iT)B{0)}) = 0(T)(A( - iT)B(0)) ± 0(-T)(B(0)A(-iT))

.

(7.21 )

,e·

(7.22 )

The justification runs as follows (see Section 8 of Ref. 7): The dynamics is determined by the eigenstates and eigenvalues of 1í which may be obtained from the extremum condition 8(1Pl1íl1P) = O, supplemented wit h orthogonality conditions and with the requirement of fixed particle number 8(1PINl1P) = O. In terms of a Lagrange parameter µ , the chemical potential, the problem reduces to 8(1Pl1íµl 1P) = O, plus orthogonality conditions. Defining the Heisenberg representation (5.7) with {7.22) , not hing i~ changed in what has been discussed so far , except for the following relatio11 which was not true before, whose derivation is analogous to t.l mt of Eq. (7.9) , (B(ť)A(t

+ i{:J))

.

= 13-1L

(1.2:1)

(7.24)

. (7.25)

gAB(iv) e-irrr

(7.26)

e-iv/3 = ±1 ; I/= 1/± '

. 5) which determines the Matsubara freq uencies

v+ 11 _

= 2mr / /3

; boso~ic. } operators ; n = 0, ±1 , ±2, . -•

= (2n+ 1)'rr / f3

ferm1omc

;

Hl nce

(7.27) (7.28)

13-1 1 /3 dTei(v-v')-r = 8.,.,,

1111' inverse of (7.25) becomes

gAB(iv)

Here the last term is actually zero because of {6.18). However, a continuation of gAB beyond this interval is desirable in order to have a Fourier transform. This requires a relation between (A(-iT)B(0)) and (B(0)A(-iT)) . Such a relation is easily obtained if the dynamics of the system is also = 1í - µN, as is the statistics through (6.4) and {6.8) . In defined with other words, the Heisenberg time evolution (5.9) should be redefined as

(A(t)B( t,' )) =

= ±QAB(T + /3); -/3 < T < o

YAB(T)

which for T = O expresses a periodic/antiperiodic boundary_ condi~ion on the interval (6.18). It is thus possible to write YAB as a Four1er _senes

= Jo{

13

(7.29)

·

dTYAB(T) e'",,. .

In order to be able to compare with the retarded Green's f~nction we 1) for T > O in terms of the eigenstates \n) and e1genvalues E q. (7 ·2 I '11 of 1íµ , (7.30) - QAB(T) = L e/3(!1- En)(n\A\n')(n'\B\n )e(En- En1)T

I - llfCSS

nn'

111111 \11sert into (7.29). Making use of (7.26) the result is ± e{3 (En -En,) -1

g AB (iv)

=L

ef3(!1- En)(n\A\n')(n'\B\nJ

En - En ,

+ iv

(7.31)

nn'

Witi! li may also be written as 9 Aa (±i\vl) = L ef3!1{e-f3En =f e- f3 En'}(n\Aln')(n'\B\n) nn'

±oo

±oo

x (- i )

J

dti (E,. - En,±ilv\)t

=-i

J

dte=i=lvlt([A(t), B(0)]::i=) . (7.32)

o · lds by analytic continuation . w1. th (7.15 ) , (7 ·20) yie 1liw• c·o111 p1mson (7.33) z >0 ' g- /\D (Z ) -- G- ret( AB z) ·, Im o

8. Wick's Theorem for Operotor Products o 47

46 o Chapter 2. Green's Function Formalism in Comple,; Time

so that the imaginary-time propagators are seen also to contain all the relevant physical information. Of particular importance in what follows are the unperturbed electron and phonon propagators of the form (7.21), Q0 (k; r)

= -(T{ak(-ir)at})

0

(7.34)

and

are the Fermi and Bose distribution functions , respectively. Taking the Fourier transform (7.29) of Eqs. (7.38) and (7.39) , mak:ing use of (7.26), one finally obtains 1 (7.44) Q0 (k; i11_) = -.- - - w_ -€k+µ and _0

(7.35)

which will be represented by a straight line with an arrow and by a wavy line, respectively, running in time from O to -ir. In order to calculate them explicitly we first determine the time dependence of the operators ak and bq, integrating the equation of motion (5.18) with the unperturbed electron and phonon Hamiltonians (3.27) and (1.34), respectively, and mak:ing use of the relations (3.13) and (3.42). The result is (7.36)

and b~(t)

=

e-iw•tbq .

(7.37)

Noting that the time evolution of the propagators (7.34), (7.35) is given by 1i~ defined in (6.5) one finds, for T > O, Q0 (k; r)

= -e-(e:k-µ.)r (akat)

0

(7.38)

and, using Eqs. (1.31) and the fact that for phonons the chemical potential is zero, 1 V 0 (q·' r) = --{e-W•T(b (7.39) 2 qb+) q o + e+wqT(b+ -q b-q ) o } · Here the averages are easily evaluated in the number representation, (1.33) for the phonons and similarly for the electrons, to be (Problem 2.2) (atak)o

=1-

(akat)o

=f

0

(€k - µ)

(7.40)

and (7.41)

where (7.42)

and 1 n 0 (w) = -1- n 0 (-w) = -"'-= -21 [ coth -(3w -1 ] e,.,w - 1 2

(7.43)

V

(

q; ÍII +)

w

1(

= (ÍII+ / - w2q = 2

1 iv+ - wq -

1

ÍII

+ + wq

)

(7.45)

8. Wick's Theorem for Operator Products The aim of this and the next section is to develop the calculational scheme of Feynman diagrams or graphs to evaluate expressions of the form (6. 19) or (6.30). In this procedure it is usual to invoke a "Wick's theorem" . ln its original form introduced by Wick 17) this was an algebraic identity Lo develop chronological operator products into a sum of normal-ordered products containing c-number factors arising from certain operator pairlngs (see definitions (8.4) and (8.2) below). The key feature of these normal products being that they are annihilated by the unperturbed vacuum projťctor (6.22), the result for an expression (6.30) then simply is a sum of products of such c-number pairings, which gives rise to zero-temperature l•'oynman diagrams. Such diagrams were introduced for the first time in q11antum electrodynamics and in meson theory by Feynman to calculate 18 v1tcuum polarization and selfenergy effects ) Since normal products are not annihilated by the unperturbed finitetrnnperature density operator (6.7), the hope was that a Wick's-theorem l11rl,orization would remain valid at least for the ensemble averages (6.19) . M1tLsubara5 ) showed by a suitable redefinition of the annihilating part of I111• operators in the normal product that this is indeed the case (see also ll11f. 19) . Unfortunately, however, the physical meaning of creation and 111111ihilation operators (see Section 3) is lost by Matsubara's trick. But 20 t lilH loss is actually unnecessary. Indeed, it has been shown by Enz ) that, 11111Lrary to common belief (see Ref. 7, Section 12.2 and Ref. 8, Section 24), I (111 ťnsemble average of ordinary normal products also leads to a perfectly pliyHical Wick's-theorem type factorization which, together with the , 1111111ber pairings mentioned above, lead to the Matsubara factorization. I hlti formalism will be developed in the next section while the present , , I ion is concerned with the operator form of Wick's theorem.

48 o Chapter 2. Green 's F\mction Formalism in Complex Tim e

8. Wick's Theorem for Operotor Products o 49

The definition of normal product is based on the division into positiveand negative-frequency parts A(±) of irreducible operators A which occur in the interactions 'Hint· These objects, called "simple" factors in Ref. 17, are linear combinations of creation and annihilation operators (3.8), A= A(+)+ AH ; AHl4>o)

= (4>olA(+) = O

(8.1)

where 14>0 ) is the groundstate defined in (6.22) . In the case of bosons/ fermions the A(±) satisfy commutation/anticommutation relations (3.10),

= é_,B = c - number [AH , Blem 2.1: Eqs. {7.13), {7.14) . {12.40)

Taking time-Fourier transforms in t and ť according to Eq. (7.19) and making use oftime-translation invariance (7.9), we obtain the Ward identity

~ ~

Jo(q; 0)óq,O

.,,ul V = vL.

where the density of states takes care of the precise integration limit:;. Taking the T-derivative of the expression for n 0 , one obtains the conditioll

Taylor-expand N{c x = (3c this yields

(nV)2

1Yl1lrh reduces to Eq. {10.12) by taking into account Eqs. (7.45), {10.11)

8µ] / • fo =-T1 [€k-µ-T8T

Since lnf0 /{l - f 0 ) = -(3(ck - µ) one finds , making use of {10.4),

cv 0 =

iv+)e-iv+(r--r')óq,o