Many-Body Theory of Solids: An Introduction

332 Pages·1984·4.996 MB·English

by  John C. Inkson (auth.)

MANY-BODY
THEORY
OF SOLIDS
An Introduction
MANY-BODY
THEORY
OF SOLIDS
An Introduction
JOHN C. INKSON
Cavendish Laboratory
University of Cambridge
Cambridge, England
PLENUM PRESS· NEW YORK AND LONDON
Library of Congress Cataloging in Publication Data
Inkson, John c., dafe-
Many-body theory of solids.
Includes bibliographies and index.
1. Solid state physics. 2. Many-body problem. 3. Green's functions. 1. Title.
QCI76.I44 1983 530.4'1 83-17771
ISBN 978-1-4757-0228-6 ISBN 978-1-4757-0226-2 (eBook)
DOl 10.1007/978-1-4757-0226-2
©1984 Plenum Press, New York
Softcover reprint of the hardcover 1s t edition 1984
A Division of Plenum Publishing Corporation
233 Spring Street, New York, N.Y. 10013
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted
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PREFACE
T
here exists a gap in the present literature on quantum mechanics
and its application to solids. It has been difficult to find an intro
ductory textbook which could take a student from the elementary quan
tum mechanical ideas of the single-particle Schrodinger equations,
through the formalism and new physical concepts of many-body theory,
to the level where the student would be equipped to read the scientific
literature and specialized books on specific topics. The present book,
which I believe fills this gap, grew out of two courses which I have given
for a number of years at the University of Cambridge: "Advanced Quan
tum Mechanics," covering the quantization of fields, representations,
and creation and annihilation operators, and "Many Body Theory," on
the application of quantum field theory to solids. The first course is a
final-year undergraduate physics course while the second is a joint first
year postgraduate physics course and fourth-year undergraduate math
ematics course. In an American context this would closely correspond to
a graduate course at the masters level.
In writing this book I have tried to stress the physical aspects of the
mathematics preferring where possible to introduce a technique by using
a simple illustrative example rather than develop a purely formal treat
ment. In order to do this I have assumed a certain familiarity with solid
state physics on the level of a normal undergraduate course, but the book
should also be useful to those without such a background.
The book is divided into three basic parts. The first four chapters
introduce the basic tools and concepts of advanced quantum mechanics
in a simple context. In this way the reader will have already been intro
duced to the Green's functions, diagrams, etc., before the complicated
formalism is developed. The second part introduces the Green's function
self-energy techniques. These form the backbone of the application of
quantum mechanics to interacting systems. The method used is the so
called "equation of motion" method which I believe is more physically
understandable than the use of Wick's theorem (which leads directly to
v
PREFACE
Feynman diagrams) but the use of the diagrammatic interpretation of the
resulting iterative solution is stressed. The final section consists of the
application of the techniques to a number of examples. These have been
chosen to bring out particular physical aspects rather than for their
inherent importance. A number of examples and a bibliography have
been provided at the end of each chapter for readers wishing to look
further into the areas treated.
I'd like to express my appreciation to my wife Pam for all her patient
hours of typing, and to the staff of Plenum Press, especially Ken Derham
and Steve Pisano, for their help in the preparation of this book. Finally,
I'd like to dedicate this book to my family, Pam, Andrea, Beverley, and
Jonathan, for making it all worthwhile.
JOHNINKSON
vi
CONTENTS
CHAPTER 1. THE INTERACTING SYSTEM
1.1. The Basic Problem ............................... 1
1.2. The Jellium Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3. Hartree Theory-The Sommerfeld Model. . . . . . . . . . . 5
1.4. Hartree-Fock.................................... 10
1.5. Exchange and Correlation Holes. . . . . . . . . . . . . . . . . . . 15
1.6. Correlation Effects and the Thomas-Fermi Model. . . . 17
Bibliography .................................... 24
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
CHAPTER 2. GREEN'S FUNCTIONS OF THE SINGLE-
PARTICLE SCHRODINGER EQUATION
2.1. Green's Functions of the Schrodinger Equation. . . . . . 27
2.2. Green's Functions and Perturbation Theory. . . . . . . . . 30
2.3. Time-Dependent Green's Functions. . . . . . . . . . . . . . . . 33
2.4. Green's Function Diagrams. . . . . . . . . . . . . . . . . . . . . . . . 36
2.5. Green's Functions or Wave Functions? . . . . . . . . . . . . . 38
Bibliography .................................... 40
ProF 'Us........................................ 41
CHAPTER 3. QUANTIZATION OF WAVES (SECOND
QUANTIZATION)
3.1. Waves and Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2. The Linear Chain of Atoms ....................... 44
3.3. The General Quantization of a Wave System . . . . . . . . 51
3.4. Quantization of the Electromagnetic Field .......... 55
3.5. Elementary Excitations and "Particles". . . . . . . . . . . . . . 59
3.6. Perturbations and the Elementary Excitations ....... 61
3.7. Summary....................................... 63
Bibliography .................................... 64
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
vii
CONTENTS
CHAPTER 4. REPRESENTATIONS OF QUANTUM
MECHANICS
4.1. Schrodinger Representation. . . . . . . . . . . . . . . . . . . . . . . 68
4.2. Heisenberg Representation. . . . . . . . . . . . . . . . . . . . . . . . 68
4.3. Interaction Representation . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4. Occupation Number Representation. . . . . . . . . . . . . . . . 73
4.5. Interaction between Waves and Particles. . . . . . . . . . . . 81
4.6. Field Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Bibliography .................................... 88
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
CHAPTER 5. INTERACTING SYSTEMS AND
QUASIPARTICLES
5.1. Single-Particle States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2. Absorbing Media ................................ 94
5.3. Exact and Approximate Eigenstates . . . . . . . . . . . . . . . . . 96
5.4. Landau Quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Bibliography .................................... 110
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
CHAPTER 6. MANY-BODY GREEN'S FUNCTIONS
6.1. Definition of the Many-Body Green's Function. . . . . . 113
6.2. Relationship to Single-Particle Green's Function. . . . . 115
6.3. Energy Structure and the Green's Function ......... 116
6.4. The Lehman Representation and Quasiparticles ..... 120
6.5. Expectation Values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.6. Equation of Motion for the Green's Function. . . . . . . . 126
6.7. Hartree and Hartree-Fock Approximations. . . . . . . . . . 128
6.8. The Self-Energy. . . . . . . .. ........................ 131
Bibliography .................................... 133
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
CHAPTER 7. THE SELF-ENERGY AND PERTURBATION
SERIES
7.1. Functional Derivatives and the Calculation of G
and ~.. . . .. . ..... . .... . ... . .... . ... . ...... ..... . 135
7.2. Iterative Solution for the Green's Function and Self-
Energy... . . .. . ...... ..... .... . ..... ............. 140
7.3. Screening and the Perturbation Series. . . . . . . . . . . . . . 144
7.4. The Screened Interaction and Selective Summations 150
viii
CONTENTS
7.5. The Uniform System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Bibliography .................................... 155
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
CHAPTER 8. DIAGRAMMATIC INTERPRETATION OF
THE GREEN'S FUNCTION SERIES'
8.1. Diagrammatic Interpretation of the Perturbation
Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.2. Diagrammatic Expansion. . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.3. Infinite Series and Irreducible Diagrams. . . . . . . . . . . . 166
804. The Hartree Potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
8.5. The Uniform System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.6. Rules for Evaluating Diagrams. . . . . . . . . . . . . . . . . . . . . 174
8.7. Selective Summations ............................ 177
8.8. Practical Aspects of Diagrammatics. . . . . . . . . . . . . . . . . 180
Bibliography .................................... 181
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
CHAPTER 9. THE NORMAL SYSTEM
9.1. The Jellium Solid Response Function. . . . . . . . . . . . . . . 184
9.2. The Self-Energy (Physical Considerations) . . . . . . . . . . 190
9.3. Evaluation of the Self-Energy and Quasiparticle
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
904. Landau Quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
9.5. Insulating Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
9.6. Surfaces......................................... 211
Bibliography .................................... 215
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
CHAPTER 10. THERMAL EFFECTS ON THE GREEN'S
FUNCTION
10.1. The Density Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
10.2. Statistical Mechanics ............................ 221
10.3. The "Thermal" Heisenberg Representation ........ 225
lOA. Evaluation of the Perturbation Expansion. . . . . . . . . . 227
10.5. Periodicity of 9 and the Extension to Energy
Dependency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
10.6. Real-Time Thermal Green's Functions. . . . . . . . . . . . . 233
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
ix
CONTENTS
CHAPTER 11. BOSON PARTICLES
11.1. Collective Excitations in Solids. . . . . . . . . . . . . . . . . . . 239
11.2. Electron-Phonon System ........................ 243
11.3. Plasmons and the Total Interaction. . . . . . . . . . . . . . . . 250
11.4. Boson Systems with a Condensate. . . . . . . . . . . . . . . . . 253
Bibliography .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
CHAPTER 12. SPECIAL METHODS
12.1. The Density Functional Method (Nearly Uniform
Electron Gases) ................................. 265
12.2. Highly Localized Systems (Anderson-Hubbard
Models) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
12.3. Canonical Transformations. . . . . . . . . . . . . . . . . . . . . . . 276
12.4. Mean-Field Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Bibliography ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
CHAPTER 13. SUPERCONDUCTIVITY
13.1. Cooper Pairs. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 290
13.2. Canonical Transformations. . . . . . . . . . . . . . . . . . . . . . . 295
13.3. Propagator Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
Bibliography ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
APPENDIX: LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . 311
INDEX ............................................. 325
x
CHAPTER ONE
THE INTERACTING
SYSTEM
1.1. THE BASIC PROBLEM
The calculation of the motion of one particle is, under classical or
quantum mechanics, a well-defined problem with a well-defined solu
tion. True, in the latter case we must then be careful as to how we inter
pret our solution and remember that we work with probabilities, but in
principle the solution can be achieved to the Schrodinger equation for
that particle. In the usual form we have
Hif; = Eif; [1.1]
with
h2 +
H = - - \72 V(r) [1.2]
2m
and, with its boundary conditions, it forms a well-defined computational
problem. If we consider two or three particles, we may consider the prob
lem soluble whether they have a mutual interaction or not. Once we get
into the realm of large numbers of particles, however, sheer computa
tional difficulty prevents a solution. The only exception is the case of a
noninteracting set of particles, for then the total Hamiltonian is of the form
[1.3]
and splits into a set of independent single-particle Hamiltonians
[l.4]
1