Many-body quantum theory in condensed matter physics

352 Pages·2004·2.947 MB·English

by  Henrik Bruus, Karsten Flensberg

Introduction to
Many-body quantum theory in
condensed matter physics
Henrik Bruus and Karsten Flensberg
Ørsted Laboratory, Niels Bohr Institute, University of Copenhagen
Mikroelektronik Centret, Technical University of Denmark
Copenhagen, 15 August 2002
ii
Preface
Preface for the 2001 edition
This introduction to quantum field theory in condensed matter physics has emerged from
ourcoursesforgraduateandadvancedundergraduatestudentsattheNielsBohrInstitute,
University of Copenhagen, held between the fall of 1999 and the spring of 2001. We have
gone through the pain of writing these notes, because we felt the pedagogical need for
a book which aimed at putting an emphasis on the physical contents and applications
of the rather involved mathematical machinery of quantum field theory without loosing
mathematical rigor. We hope we have succeeded at least to some extend in reaching this
goal.
Wewouldliketothankthestudentswhoputupwiththefirstversionsofthisbookand
for their enumerable and valuable comments and suggestions. We are particularly grateful
to the students ofMany-particle Physics I & II, the academic year 2000-2001, and to Niels
Asger Mortensen and Brian Møller Andersen for careful proof reading. Naturally, we are
solely responsible for the hopefully few remaining errors and typos.
During the work on this book H.B. was supported by the Danish Natural Science Re-
search Council through Ole Rømer Grant No. 9600548.
Ørsted Laboratory, Niels Bohr Institute Karsten Flensberg
1 September, 2001 Henrik Bruus
Preface for the 2002 edition
After running the course in the academic year 2001-2002 our students came up with more
corrections and comments so that we felt a new edition was appropriate. We would like
to thank our ever enthusiastic students for their valuable help in improving this book.
Karsten Flensberg Henrik Bruus
Ørsted Laboratory Mikroelektronik Centret
Niels Bohr Institute Technical University of Denmark
iii
iv PREFACE
Contents
List of symbols xii
1 First and second quantization 1
1.1 First quantization, single-particle systems . . . . . . . . . . . . . . . . . . . 2
1.2 First quantization, many-particle systems . . . . . . . . . . . . . . . . . . . 4
1.2.1 Permutation symmetry and indistinguishability . . . . . . . . . . . . 5
1.2.2 The single-particle states as basis states . . . . . . . . . . . . . . . . 6
1.2.3 Operators in first quantization . . . . . . . . . . . . . . . . . . . . . 7
1.3 Second quantization, basic concepts . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 The occupation number representation . . . . . . . . . . . . . . . . . 10
1.3.2 The boson creation and annihilation operators . . . . . . . . . . . . 10
1.3.3 The fermion creation and annihilation operators . . . . . . . . . . . 13
1.3.4 The general form for second quantization operators . . . . . . . . . . 14
1.3.5 Change of basis in second quantization . . . . . . . . . . . . . . . . . 16
1.3.6 Quantum field operators and their Fourier transforms . . . . . . . . 17
1.4 Second quantization, specific operators . . . . . . . . . . . . . . . . . . . . . 18
1.4.1 The harmonic oscillator in second quantization . . . . . . . . . . . . 18
1.4.2 The electromagnetic field in second quantization . . . . . . . . . . . 19
1.4.3 Operators for kinetic energy, spin, density, and current . . . . . . . . 21
1.4.4 The Coulomb interaction in second quantization . . . . . . . . . . . 23
1.4.5 Basis states for systems with different kinds of particles . . . . . . . 24
1.5 Second quantization and statistical mechanics . . . . . . . . . . . . . . . . . 25
1.5.1 The distribution function for non-interacting fermions . . . . . . . . 28
1.5.2 Distribution functions for non-interacting bosons . . . . . . . . . . . 29
1.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 The electron gas 31
2.1 The non-interacting electron gas . . . . . . . . . . . . . . . . . . . . . . . . 32
2.1.1 Bloch theory of electrons in a static ion lattice . . . . . . . . . . . . 33
2.1.2 Non-interacting electrons in the jellium model . . . . . . . . . . . . . 35
2.1.3 Non-interacting electrons at finite temperature . . . . . . . . . . . . 38
2.2 Electron interactions in perturbation theory . . . . . . . . . . . . . . . . . . 39
2.2.1 Electron interactions in 1st order perturbation theory . . . . . . . . 41
v
vi CONTENTS
2.2.2 Electron interactions in 2nd order perturbation theory . . . . . . . . 43
2.3 Electron gases in 3, 2, 1, and 0 dimensions . . . . . . . . . . . . . . . . . . . 44
2.3.1 3D electron gases: metals and semiconductors . . . . . . . . . . . . . 45
2.3.2 2D electron gases: GaAs/Ga Al As heterostructures . . . . . . . . 46
1−x x
2.3.3 1D electron gases: carbon nanotubes . . . . . . . . . . . . . . . . . . 48
2.3.4 0D electron gases: quantum dots . . . . . . . . . . . . . . . . . . . . 49
3 Phonons; coupling to electrons 51
3.1 Jellium oscillations and Einstein phonons . . . . . . . . . . . . . . . . . . . 52
3.2 Electron-phonon interaction and the sound velocity . . . . . . . . . . . . . . 53
3.3 Lattice vibrations and phonons in 1D . . . . . . . . . . . . . . . . . . . . . 53
3.4 Acoustical and optical phonons in 3D . . . . . . . . . . . . . . . . . . . . . 56
3.5 The specific heat of solids in the Debye model . . . . . . . . . . . . . . . . . 59
3.6 Electron-phonon interaction in the lattice model . . . . . . . . . . . . . . . 61
3.7 Electron-phonon interaction in the jellium model . . . . . . . . . . . . . . . 63
3.8 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4 Mean field theory 65
4.1 The art of mean field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 Hartree–Fock approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Broken symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4.1 The Heisenberg model of ionic ferromagnets . . . . . . . . . . . . . . 73
4.4.2 The Stoner model of metallic ferromagnets . . . . . . . . . . . . . . 75
4.5 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5.1 Breaking of global gauge symmetry and its consequences. . . . . . . 78
4.5.2 Microscopic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5 Time evolution pictures 87
5.1 The Schr¨odinger picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 The Heisenberg picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3 The interaction picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4 Time-evolution in linear response . . . . . . . . . . . . . . . . . . . . . . . . 91
5.5 Time dependent creation and annihilation operators . . . . . . . . . . . . . 91
5.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6 Linear response theory 95
6.1 The general Kubo formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2 Kubo formula for conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.3 Kubo formula for conductance . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.4 Kubo formula for the dielectric function . . . . . . . . . . . . . . . . . . . . 102
6.4.1 Dielectric function for translation-invariant system . . . . . . . . . . 104
6.4.2 Relation between dielectric function and conductivity . . . . . . . . 104
CONTENTS vii
6.5 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7 Transport in mesoscopic systems 107
7.1 The S-matrix and scattering states . . . . . . . . . . . . . . . . . . . . . . . 108
7.1.1 Unitarity of the S-matrix . . . . . . . . . . . . . . . . . . . . . . . . 111
7.1.2 Time-reversal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.2 Conductance and transmission coefficients . . . . . . . . . . . . . . . . . . . 113
7.2.1 The Landauer-Bu¨ttiker formula, heuristic derivation . . . . . . . . . 113
7.2.2 The Landauer-Bu¨ttiker formula, linear response derivation . . . . . . 115
7.3 Electron wave guides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.3.1 Quantum point contact and conductance quantization . . . . . . . . 116
7.3.2 Aharonov-Bohm effect . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.4 Disordered mesoscopic systems . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.4.1 Statistics of quantum conductance, random matrix theory . . . . . . 121
7.4.2 Weak localization in mesoscopic systems . . . . . . . . . . . . . . . . 123
7.4.3 Universal conductance fluctuations . . . . . . . . . . . . . . . . . . . 124
7.5 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8 Green’s functions 127
8.1 “Classical” Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.2 Green’s function for the one-particle Schr¨odinger equation . . . . . . . . . . 128
8.3 Single-particle Green’s functions of many-body systems . . . . . . . . . . . 131
8.3.1 Green’s function of translation-invariant systems . . . . . . . . . . . 132
8.3.2 Green’s function of free electrons . . . . . . . . . . . . . . . . . . . . 132
8.3.3 The Lehmann representation . . . . . . . . . . . . . . . . . . . . . . 134
8.3.4 The spectral function . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.3.5 Broadening of the spectral function . . . . . . . . . . . . . . . . . . . 136
8.4 Measuring the single-particle spectral function . . . . . . . . . . . . . . . . 137
8.4.1 Tunneling spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.4.2 Optical spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.5 Two-particle correlation functions of many-body systems . . . . . . . . . . . 141
8.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
9 Equation of motion theory 145
9.1 The single-particle Green’s function . . . . . . . . . . . . . . . . . . . . . . 145
9.1.1 Non-interacting particles . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.2 Anderson’s model for magnetic impurities . . . . . . . . . . . . . . . . . . . 147
9.2.1 The equation of motion for the Anderson model . . . . . . . . . . . 149
9.2.2 Mean-field approximation for the Anderson model . . . . . . . . . . 150
9.2.3 Solving the Anderson model and comparison with experiments . . . 151
9.2.4 Coulomb blockade and the Anderson model . . . . . . . . . . . . . . 153
9.2.5 Further correlations in the Anderson model: Kondo effect . . . . . . 153
9.3 The two-particle correlation function . . . . . . . . . . . . . . . . . . . . . . 153
9.3.1 The Random Phase Approximation (RPA) . . . . . . . . . . . . . . 153
viii CONTENTS
9.4 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
10 Imaginary time Green’s functions 157
10.1 Definitions of Matsubara Green’s functions . . . . . . . . . . . . . . . . . . 160
10.1.1 Fourier transform of Matsubara Green’s functions . . . . . . . . . . 161
10.2 Connection between Matsubara and retarded functions . . . . . . . . . . . . 161
10.2.1 Advanced functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
10.3 Single-particle Matsubara Green’s function . . . . . . . . . . . . . . . . . . 164
10.3.1 Matsubara Green’s function for non-interacting particles . . . . . . . 164
10.4 Evaluation of Matsubara sums . . . . . . . . . . . . . . . . . . . . . . . . . 165
10.4.1 Summations over functions with simple poles . . . . . . . . . . . . . 167
10.4.2 Summations over functions with known branch cuts . . . . . . . . . 168
10.5 Equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
10.6 Wick’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
10.7 Example: polarizability of free electrons . . . . . . . . . . . . . . . . . . . . 173
10.8 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
11 Feynman diagrams and external potentials 177
11.1 Non-interacting particles in external potentials . . . . . . . . . . . . . . . . 177
11.2 Elastic scattering and Matsubara frequencies . . . . . . . . . . . . . . . . . 179
11.3 Random impurities in disordered metals . . . . . . . . . . . . . . . . . . . . 181
11.3.1 Feynman diagrams for the impurity scattering . . . . . . . . . . . . 182
11.4 Impurity self-average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
11.5 Self-energy for impurity scattered electrons . . . . . . . . . . . . . . . . . . 189
11.5.1 Lowest order approximation . . . . . . . . . . . . . . . . . . . . . . . 190
11.5.2 1st order Born approximation . . . . . . . . . . . . . . . . . . . . . . 190
11.5.3 The full Born approximation . . . . . . . . . . . . . . . . . . . . . . 193
11.5.4 The self-consistent Born approximation and beyond . . . . . . . . . 194
11.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
12 Feynman diagrams and pair interactions 199
12.1 The perturbation series for G . . . . . . . . . . . . . . . . . . . . . . . . . . 199
12.2 infinite perturbation series!Matsubara Green’s function . . . . . . . . . . . . 199
12.3 The Feynman rules for pair interactions . . . . . . . . . . . . . . . . . . . . 201
12.3.1 Feynman rules for the denominator of G(b,a) . . . . . . . . . . . . . 201
12.3.2 Feynman rules for the numerator of G(b,a) . . . . . . . . . . . . . . 202
12.3.3 The cancellation of disconnected Feynman diagrams . . . . . . . . . 203
12.4 Self-energy and Dyson’s equation . . . . . . . . . . . . . . . . . . . . . . . . 205
12.5 The Feynman rules in Fourier space . . . . . . . . . . . . . . . . . . . . . . 206
12.6 Examples of how to evaluate Feynman diagrams . . . . . . . . . . . . . . . 208
12.6.1 The Hartree self-energy diagram . . . . . . . . . . . . . . . . . . . . 209
12.6.2 The Fock self-energy diagram . . . . . . . . . . . . . . . . . . . . . . 209
12.6.3 The pair-bubble self-energy diagram . . . . . . . . . . . . . . . . . . 210
12.7 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
CONTENTS ix
13 The interacting electron gas 213
13.1 The self-energy in the random phase approximation . . . . . . . . . . . . . 213
13.1.1 The density dependence of self-energy diagrams . . . . . . . . . . . . 214
13.1.2 The divergence number of self-energy diagrams . . . . . . . . . . . . 215
13.1.3 RPA resummation of the self-energy . . . . . . . . . . . . . . . . . . 215
13.2 The renormalized Coulomb interaction in RPA . . . . . . . . . . . . . . . . 217
13.2.1 Calculation of the pair-bubble . . . . . . . . . . . . . . . . . . . . . . 218
13.2.2 The electron-hole pair interpretation of RPA . . . . . . . . . . . . . 220
13.3 The ground state energy of the electron gas . . . . . . . . . . . . . . . . . . 220
13.4 The dielectric function and screening . . . . . . . . . . . . . . . . . . . . . . 223
13.5 Plasma oscillations and Landau damping . . . . . . . . . . . . . . . . . . . 227
13.5.1 Plasma oscillations and plasmons . . . . . . . . . . . . . . . . . . . . 228
13.5.2 Landau damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
13.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
14 Fermi liquid theory 233
14.1 Adiabatic continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
14.1.1 The quasiparticle concept and conserved quantities . . . . . . . . . . 235
14.2 Semi-classical treatment of screening and plasmons . . . . . . . . . . . . . . 237
14.2.1 Static screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
14.2.2 Dynamical screening . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
14.3 Semi-classical transport equation . . . . . . . . . . . . . . . . . . . . . . . . 240
14.3.1 Finite life time of the quasiparticles . . . . . . . . . . . . . . . . . . 243
14.4 Microscopic basis of the Fermi liquid theory . . . . . . . . . . . . . . . . . . 245
14.4.1 Renormalization of the single particle Green’s function . . . . . . . . 245
14.4.2 Imaginary part of the single particle Green’s function . . . . . . . . 248
14.4.3 Mass renormalization? . . . . . . . . . . . . . . . . . . . . . . . . . . 251
14.5 Outlook and summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
15 Impurity scattering and conductivity 253
15.1 Vertex corrections and dressed Green’s functions . . . . . . . . . . . . . . . 254
15.2 The conductivity in terms of a general vertex function . . . . . . . . . . . . 259
15.3 The conductivity in the first Born approximation . . . . . . . . . . . . . . . 261
15.4 The weak localization correction to the conductivity . . . . . . . . . . . . . 264
15.5 Combined RPA and Born approximation . . . . . . . . . . . . . . . . . . . . 273
16 Green’s functions and phonons 275
16.1 The Green’s function for free phonons . . . . . . . . . . . . . . . . . . . . . 275
16.2 Electron-phonon interaction and Feynman diagrams . . . . . . . . . . . . . 276
16.3 Combining Coulomb and electron-phonon interactions . . . . . . . . . . . . 279
16.3.1 Migdal’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
16.3.2 Jellium phonons and the effective electron-electron interaction . . . 280
16.4 Phonon renormalization by electron screening in RPA . . . . . . . . . . . . 281
16.5 The Cooper instability and Feynman diagrams . . . . . . . . . . . . . . . . 284
x CONTENTS
17 Superconductivity 287
17.1 The Cooper instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
17.2 The BCS groundstate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
17.3 BCS theory with Green’s functions . . . . . . . . . . . . . . . . . . . . . . . 287
17.4 Experimental consequences of the BCS states . . . . . . . . . . . . . . . . . 288
17.4.1 Tunneling density of states . . . . . . . . . . . . . . . . . . . . . . . 288
17.4.2 specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
17.5 The Josephson effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
18 1D electron gases and Luttinger liquids 289
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
18.2 First look at interacting electrons in one dimension . . . . . . . . . . . . . . 289
18.2.1 One-dimensional transmission line analog . . . . . . . . . . . . . . . 289
18.3 The Luttinger-Tomonaga model - spinless case . . . . . . . . . . . . . . . . 289
18.3.1 Interacting one dimensional electron system . . . . . . . . . . . . . . 289
18.3.2 Bosonization of Tomonaga model-Hamiltonian . . . . . . . . . . . . 289
18.3.3 Diagonalization of bosonized Hamiltonian . . . . . . . . . . . . . . . 289
18.3.4 Real space formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 289
18.3.5 Electron operators in bosonized form . . . . . . . . . . . . . . . . . . 289
18.4 Luttinger liquid with spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
18.5 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
18.6 Tunneling into spinless Luttinger liquid . . . . . . . . . . . . . . . . . . . . 290
18.6.1 Tunneling into the end of Luttinger liquid . . . . . . . . . . . . . . . 290
18.7 What is a Luttinger liquid? . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
18.8 Experimental realizations of Luttinger liquid physics . . . . . . . . . . . . . 290
18.8.1 Edge states in the fractional quantum Hall effect . . . . . . . . . . . 290
18.8.2 Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
A Fourier transformations 291
A.1 Continuous functions in a finite region . . . . . . . . . . . . . . . . . . . . . 291
A.2 Continuous functions in an infinite region . . . . . . . . . . . . . . . . . . . 292
A.3 Time and frequency Fourier transforms . . . . . . . . . . . . . . . . . . . . 292
A.4 Some useful rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
A.5 Translation invariant systems . . . . . . . . . . . . . . . . . . . . . . . . . . 293
B Exercises 295
C Index 326