Introduction to Quantum Field Theory in Condensed Matter Physics

410 Pages·2001·3.432 MB·English

by  Bruus, Flensberg.

Introduction to
Quantum (cid:12)eld theory in
condensed matter physics
Henrik Bruus and Karsten Flensberg
(cid:31)rsted Laboratory
Niels Bohr Institute
Copenhagen, 1 September 2001
ii
Contents
1 First and second quantization 3
1.1 First quantization, single-particle systems . . . . . . . . . . . . . . . . . . . 4
1.2 First quantization, many-particle systems . . . . . . . . . . . . . . . . . . . 6
1.2.1 Permutation symmetry and indistinguishability . . . . . . . . . . . . 6
1.2.2 The single-particle states as basis states . . . . . . . . . . . . . . . . 7
1.2.3 Operators in (cid:12)rst quantization . . . . . . . . . . . . . . . . . . . . . 9
1.3 Second quantization, basic concepts . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 The occupation number representation . . . . . . . . . . . . . . . . . 11
1.3.2 The boson creation and annihilation operators . . . . . . . . . . . . 12
1.3.3 The fermion creation and annihilation operators . . . . . . . . . . . 14
1.3.4 The general form for second quantization operators . . . . . . . . . . 16
1.3.5 Change of basis in second quantization . . . . . . . . . . . . . . . . . 17
1.3.6 Quantum (cid:12)eld operators and their Fourier transforms . . . . . . . . 19
1.4 Second quantization, speci(cid:12)c operators . . . . . . . . . . . . . . . . . . . . . 20
1.4.1 The harmonic oscillator in second quantization . . . . . . . . . . . . 20
1.4.2 The electromagnetic (cid:12)eld in second quantization . . . . . . . . . . . 21
1.4.3 Operators for kinetic energy, spin, density, and current . . . . . . . . 23
1.4.4 The Coulomb interaction in second quantization . . . . . . . . . . . 25
1.4.5 Basis states for systems with di(cid:11)erent kinds of particles . . . . . . . 26
1.5 Second quantization and statistical mechanics . . . . . . . . . . . . . . . . . 27
1.5.1 The distribution function for non-interacting fermions . . . . . . . . 30
1.5.2 Distribution functions for non-interacting bosons . . . . . . . . . . . 30
1.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 The electron gas 33
2.1 The non-interacting electron gas . . . . . . . . . . . . . . . . . . . . . . . . 34
2.1.1 Bloch theory of electrons in a static ion lattice . . . . . . . . . . . . 35
2.1.2 Non-interacting electrons in the jellium model. . . . . . . . . . . . . 37
2.1.3 Non-interacting electrons at (cid:12)nite temperature . . . . . . . . . . . . 40
2.2 Electron interactions in perturbation theory . . . . . . . . . . . . . . . . . . 41
st
2.2.1 Electron interactions in 1 order perturbation theory . . . . . . . . 43
nd
2.2.2 Electron interactions in 2 order perturbation theory . . . . . . . . 45
2.3 Electron gases in 3, 2, 1, and 0 dimensions . . . . . . . . . . . . . . . . . . . 46
iii
iv CONTENTS
2.3.1 3D electron gases: metals and semiconductors . . . . . . . . . . . . . 47
2.3.2 2D electron gases: GaAs/Ga1 xAlxAs heterostructures . . . . . . . . 48
(cid:0)
2.3.3 1D electron gases: carbon nanotubes . . . . . . . . . . . . . . . . . . 50
2.3.4 0D electron gases: quantum dots . . . . . . . . . . . . . . . . . . . . 51
3 Phonons; coupling to electrons 53
3.1 Jellium oscillations and Einstein phonons . . . . . . . . . . . . . . . . . . . 54
3.2 Electron-phonon interaction and the sound velocity . . . . . . . . . . . . . . 55
3.3 Lattice vibrations and phonons in 1D . . . . . . . . . . . . . . . . . . . . . 55
3.4 Acoustical and optical phonons in 3D . . . . . . . . . . . . . . . . . . . . . 58
3.5 The speci(cid:12)c heat of solids in the Debye model . . . . . . . . . . . . . . . . . 61
3.6 Electron-phonon interaction in the lattice model . . . . . . . . . . . . . . . 63
3.7 Electron-phonon interaction in the jellium model . . . . . . . . . . . . . . . 65
3.8 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 Mean (cid:12)eld theory 67
4.1 The art of mean (cid:12)eld theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Hartree{Fock approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3 Broken symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4.1 The Heisenberg model of ionic ferromagnets . . . . . . . . . . . . . . 75
4.4.2 The Stoner model of metallic ferromagnets . . . . . . . . . . . . . . 77
4.5 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5.1 Breaking of global gauge symmetry and its consequences. . . . . . . 80
4.5.2 Microscopic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5 Time evolution pictures 89
5.1 The Schr(cid:127)odinger picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 The Heisenberg picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3 The interaction picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4 Time-evolution in linear response . . . . . . . . . . . . . . . . . . . . . . . . 93
5.5 Time dependent creation and annihilation operators . . . . . . . . . . . . . 93
5.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6 Linear response theory 97
6.1 The general Kubo formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2 Kubo formula for conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.3 Kubo formula for conductance . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.4 Kubo formula for the dielectric function . . . . . . . . . . . . . . . . . . . . 103
6.4.1 Dielectric function for translation-invariant system . . . . . . . . . . 105
6.4.2 Relation between dielectric function and conductivity . . . . . . . . 105
6.5 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
CONTENTS v
7 Green’s functions 107
7.1 \Classical" Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.2 Green’s function for the single particle Schr(cid:127)odinger equation . . . . . . . . . 108
7.3 The single-particle Green’s function of a many-body system . . . . . . . . . 111
7.3.1 Green’s function of translation-invariant systems . . . . . . . . . . . 112
7.3.2 Green’s function of free electrons . . . . . . . . . . . . . . . . . . . . 113
7.3.3 The Lehmann representation . . . . . . . . . . . . . . . . . . . . . . 114
7.3.4 The spectral function . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.3.5 Broadening of the spectral function. . . . . . . . . . . . . . . . . . . 117
7.4 Measuring the single-particle spectral function . . . . . . . . . . . . . . . . 117
7.4.1 Tunneling spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.4.2 Optical spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.5 The two-particle correlation function of a many-body system . . . . . . . . 122
7.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8 Equation of motion theory 127
8.1 The single-particle Green’s function . . . . . . . . . . . . . . . . . . . . . . 127
8.1.1 Non-interacting particles . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.2 Anderson’s model for magnetic impurities . . . . . . . . . . . . . . . . . . . 129
8.2.1 The equation of motion for the Anderson model . . . . . . . . . . . 131
8.2.2 Mean-(cid:12)eld approximation for the Anderson model . . . . . . . . . . 132
8.2.3 Solving the Anderson model and comparison with experiments . . . 133
8.2.4 Coulomb blockade and the Anderson model . . . . . . . . . . . . . . 135
8.3 The two particle correlation function . . . . . . . . . . . . . . . . . . . . . . 135
8.4 The Random Phase Approximation (RPA) . . . . . . . . . . . . . . . . . . 135
8.5 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
9 Imaginary time Green’s functions 139
9.1 De(cid:12)nitions of Matsubara Green’s functions . . . . . . . . . . . . . . . . . . 142
9.1.1 Fourier transform of Matsubara Green’s functions . . . . . . . . . . 143
9.2 Connection between Matsubara and retarded functions . . . . . . . . . . . . 144
9.2.1 Advanced functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9.3 Single-particle Matsubara Green’s function . . . . . . . . . . . . . . . . . . 146
9.3.1 Matsubara Green’s function for non-interacting particles . . . . . . . 146
9.4 Evaluation of Matsubara sums . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.4.1 Summations over functions with simple poles . . . . . . . . . . . . . 149
9.4.2 Summations over functions with known branch cuts . . . . . . . . . 150
9.5 Equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
9.6 Wick’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
9.7 Example: polarizability of free electrons . . . . . . . . . . . . . . . . . . . . 155
9.8 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
vi CONTENTS
10 Feynman diagrams and external potentials 159
10.1 Non-interacting particles in external potentials . . . . . . . . . . . . . . . . 159
10.2 Elastic scattering and Matsubara frequencies . . . . . . . . . . . . . . . . . 162
10.3 Random impurities in disordered metals . . . . . . . . . . . . . . . . . . . . 163
10.3.1 Feynman diagrams for the impurity scattering . . . . . . . . . . . . 165
10.4 Impurity self-average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
10.5 Self-energy for impurity scattered electrons . . . . . . . . . . . . . . . . . . 171
10.5.1 Lowest order approximation . . . . . . . . . . . . . . . . . . . . . . . 172
st
10.5.2 1 order Born approximation . . . . . . . . . . . . . . . . . . . . . . 172
10.5.3 The full Born approximation . . . . . . . . . . . . . . . . . . . . . . 175
10.5.4 The self-consistent Born approximation and beyond . . . . . . . . . 177
10.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
11 Feynman diagrams and pair interactions 181
11.1 The perturbation series for . . . . . . . . . . . . . . . . . . . . . . . . . . 181
G
11.2 The Feynman rules for pair interactions . . . . . . . . . . . . . . . . . . . . 183
11.2.1 Feynman rules for the denominator of (b;a) . . . . . . . . . . . . . 183
G
11.2.2 Feynman rules for the numerator of (b;a) . . . . . . . . . . . . . . 184
G
11.2.3 The cancellation of disconnected Feynman diagrams . . . . . . . . . 185
11.3 Self-energy and Dyson’s equation . . . . . . . . . . . . . . . . . . . . . . . . 187
11.4 The Feynman rules in Fourier space . . . . . . . . . . . . . . . . . . . . . . 188
11.5 Examples of how to evaluate Feynman diagrams . . . . . . . . . . . . . . . 190
11.5.1 The Hartree self-energy diagram . . . . . . . . . . . . . . . . . . . . 190
11.5.2 The Fock self-energy diagram . . . . . . . . . . . . . . . . . . . . . . 191
11.5.3 The pair-bubble self-energy diagram . . . . . . . . . . . . . . . . . . 192
11.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
12 The interacting electron gas 195
12.1 The self-energy in the random phase approximation . . . . . . . . . . . . . 195
12.1.1 The density dependence of self-energy diagrams . . . . . . . . . . . . 196
12.1.2 The divergence number of self-energy diagrams . . . . . . . . . . . . 197
12.1.3 RPA resummation of the self-energy . . . . . . . . . . . . . . . . . . 197
12.2 The renormalized Coulomb interaction in RPA . . . . . . . . . . . . . . . . 199
12.2.1 Calculation of the pair-bubble. . . . . . . . . . . . . . . . . . . . . . 200
12.2.2 The electron-hole pair interpretation of RPA . . . . . . . . . . . . . 202
12.3 The ground state energy of the electron gas . . . . . . . . . . . . . . . . . . 202
12.4 The dielectric function and screening . . . . . . . . . . . . . . . . . . . . . . 205
12.5 Plasma oscillations and Landau damping . . . . . . . . . . . . . . . . . . . 209
12.5.1 Plasma oscillations and plasmons . . . . . . . . . . . . . . . . . . . . 210
12.5.2 Landau damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
12.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
CONTENTS vii
13 Fermi liquid theory 215
13.1 Adiabatic continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
13.1.1 The quasiparticle concept and conserved quantities . . . . . . . . . . 217
13.2 Semi-classical treatment of screening and plasmons . . . . . . . . . . . . . . 219
13.2.1 Static screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
13.2.2 Dynamical screening . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
13.3 Semi-classical transport equation . . . . . . . . . . . . . . . . . . . . . . . . 222
13.3.1 Finite life time of the quasiparticles . . . . . . . . . . . . . . . . . . 225
13.4 Microscopic basis of the Fermi liquid theory . . . . . . . . . . . . . . . . . . 227
13.4.1 Renormalization of the single particle Green’s function . . . . . . . . 227
13.4.2 Imaginary part of the single particle Green’s function . . . . . . . . 229
13.4.3 Mass renormalization? . . . . . . . . . . . . . . . . . . . . . . . . . . 232
13.5 Outlook and summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
14 Impurity scattering and conductivity 235
14.1 Vertex corrections and dressed Green’s functions . . . . . . . . . . . . . . . 236
14.2 The conductivity in terms of a general vertex function . . . . . . . . . . . . 241
14.3 The conductivity in the (cid:12)rst Born approximation . . . . . . . . . . . . . . . 243
14.4 The weak localization correction to the conductivity . . . . . . . . . . . . . 246
14.5 Combined RPA and Born approximation . . . . . . . . . . . . . . . . . . . . 256
15 Transport in mesoscopic systems 257
15.1 The S-matrix and scattering states . . . . . . . . . . . . . . . . . . . . . . . 258
15.1.1 Unitarity of the S-matrix . . . . . . . . . . . . . . . . . . . . . . . . 261
15.1.2 Time-reversal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 262
15.2 Conductance and transmission coeÆcients . . . . . . . . . . . . . . . . . . . 263
15.2.1 The Landauer-Bu(cid:127)ttiker formula, heuristic derivation . . . . . . . . . 263
15.2.2 The Landauer-Bu(cid:127)ttiker formula, linear response derivation. . . . . . 265
15.3 Electron wave guides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
15.3.1 Quantum point contact and conductance quantization . . . . . . . . 266
15.3.2 Aharonov-Bohm e(cid:11)ect . . . . . . . . . . . . . . . . . . . . . . . . . . 270
15.4 Disordered mesoscopic systems . . . . . . . . . . . . . . . . . . . . . . . . . 271
15.4.1 Statistics of quantum conductance, random matrix theory . . . . . . 271
15.4.2 Weak localization in mesoscopic systems . . . . . . . . . . . . . . . . 273
15.4.3 Universal conductance (cid:13)uctuations . . . . . . . . . . . . . . . . . . . 274
15.5 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
16 Green’s functions and phonons 277
16.1 The Green’s function for free phonons . . . . . . . . . . . . . . . . . . . . . 277
16.2 Electron-phonon interaction and Feynman diagrams . . . . . . . . . . . . . 278
16.3 Combining Coulomb and electron-phonon interactions . . . . . . . . . . . . 281
16.3.1 Migdal’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
16.3.2 Jellium phonons and the e(cid:11)ective electron-electron interaction . . . 282
16.4 Phonon renormalization by electron screening in RPA . . . . . . . . . . . . 283
viii CONTENTS
16.5 The Cooper instability and Feynman diagrams . . . . . . . . . . . . . . . . 286
17 Superconductivity 289
17.1 The Cooper Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
17.2 The BCS groundstate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
17.3 BCS theory with Green’s functions . . . . . . . . . . . . . . . . . . . . . . . 289
17.4 Experimental consequences of the BCS states . . . . . . . . . . . . . . . . . 289
17.4.1 Tunneling density of states . . . . . . . . . . . . . . . . . . . . . . . 289
17.4.2 speci(cid:12)c heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
17.5 The Josephson e(cid:11)ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
18 One-dimensional electron gas and Luttinger liquids 291
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
18.2 First look at interacting electrons in one dimension . . . . . . . . . . . . . . 292
18.2.1 One-dimensional transmission line analog . . . . . . . . . . . . . . . 292
18.3 The Luttinger-Tomonaga model - spinless case . . . . . . . . . . . . . . . . 292
18.3.1 Interacting one dimensional electron system . . . . . . . . . . . . . . 292
18.3.2 Bosonization of Tomonaga model-Hamiltonian . . . . . . . . . . . . 292
18.3.3 Diagonalization of bosonized Hamiltonian . . . . . . . . . . . . . . . 292
18.3.4 Real space formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 292
18.3.5 Electron operators in bosonized form . . . . . . . . . . . . . . . . . . 292
18.4 Luttinger liquid with spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
18.5 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
18.6 Tunneling into spinless Luttinger liquid . . . . . . . . . . . . . . . . . . . . 292
18.6.1 Tunneling into the end of Luttinger liquid . . . . . . . . . . . . . . . 292
18.7 What is a Luttinger liquid? . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
18.8 Experimental realizations of Luttinger liquid physics . . . . . . . . . . . . . 292
18.8.1 Edge states in the fractional quantum Hall e(cid:11)ect . . . . . . . . . . . 292
18.8.2 Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
A Fourier transformations 293
A.1 Continuous functions in a (cid:12)nite region . . . . . . . . . . . . . . . . . . . . . 293
A.2 Continuous functions in an in(cid:12)nite region . . . . . . . . . . . . . . . . . . . 294
A.3 Time and frequency Fourier transforms . . . . . . . . . . . . . . . . . . . . 294
A.4 Some useful rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
A.5 Translation invariant systems . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Exercises 297
Preface
This introduction to quantum (cid:12)eld 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 (cid:12)eld theory without loosing
mathematical rigor. We hope we have succeeded at least to some extend in reaching this
goal.
We wouldliketothankthestudentswhoputupwiththe(cid:12)rstversionsofthisbookand
for theirenumerableandvaluablecomments andsuggestions. We are particularlygrateful
to the students of Many-particle Physics I & II, theacademic year 2000-2001, andto Niels
Asger Mortensen and Brian M(cid:28)ller 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 R(cid:28)mer Grant No. 9600548.
(cid:31)rsted Laboratory, Niels Bohr Institute Karsten Flensberg
1 September, 2001 Henrik Bruus
1
2 PREFACE