List of symbols ............................................... xiv
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 .................. 8
1.3 Second quantization, basic concepts .................... 10
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 .................................... 25
1.5 Second quantization and statistical mechanics .......... 26
1.5.1 Distribution function for non-interacting
fermions ........................................ 29
1.5.2 Distribution function for non-interacting
bosons .......................................... 29
1.6 Summary and outlook .................................... 30
2 The electron gas ............................................ 32
2.1 The non-interacting electron gas ....................... 33
2.1.1 Bloch theory of electrons in a static ion
lattice ......................................... 33
2.1.2 Non-interacting electrons in the jellium model .. 36
2.1.3 Non-interacting electrons at finite
temperature ..................................... 39
2.2 Electron interactions in perturbation theory ........... 40
2.2.1 Electron interactions in first-order
perturbation theory ............................. 42
2.2.2 Electron interactions in second-order
perturbation theory ............................. 44
2.3 Electron gases in 3, 2, 1 and 0 dimensions ............. 45
2.3.1 3D electron gases: metals and semiconductors .... 45
2.3.2 2D electron gases: GaAs/GaAlAs
heterostructures ................................ 47
2.3.3 ID electron gases: carbon nanotubes ............. 49
2.3.4 OD electron gases: quantum dots ................. 50
2.4 Summary and outlook .................................... 51
3 Phonons; coupling to electrons .............................. 52
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 ................... 54
3.4 Acoustical and optical phonons in 3D ................... 57
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 ....... 64
3.8 Summary and outlook .................................... 65
4 Mean-field theory ........................................... 66
4.1 Basic concepts of mean-field theory .................... 66
4.2 The art of mean-field theory ........................... 69
4.3 Hartree-Fock approximation ............................. 70
4.3.1 Hartree-Fock approximation for the homogenous
electron gas .................................... 71
4.4 Broken symmetry ........................................ 72
4.5 Ferromagnetism ......................................... 74
4.5.1 The Heisenberg model of ionic ferromagnets ...... 74
4.5.2 The Stoner model of metallic ferromagnets ....... 76
4.6 Summary and outlook .................................... 78
5 Time dependence in quantum theory ........................... 80
5.1 The Schrodinger picture ................................ 80
5.2 The Heisenberg picture ................................. 81
5.3 The interaction picture ................................ 81
5.4 Time-evolution in linear response ...................... 84
5.5 Time-dependent creation and annihilation operators ..... 84
5.6 Fermi's golden rule .................................... 86
5.7 The T-matrix and the generalized Fermi's golden rule .. 87
5.8 Fourier transforms of advanced and retarded functions .. 88
5.9 Summary and outlook .................................... 90
6 Linear response theory ...................................... 92
6.1 The general Kubo formula ............................... 92
6.1.1 Kubo formula in the frequency domain ........... 94
6.2 Kubo formula for conductivity .......................... 95
6.3 Kubo formula for conductance ........................... 97
6.4 Kubo formula for the dielectric function ............... 99
6.4.1 Dielectric function for translation-invariant
system ......................................... 100
6.4.2 Relation between dielectric function and
conductivity ................................... 101
6.5 Summary and outlook ................................... 101
7 Transport in mesoscopic systems ............................ 103
7.1 The S-matrix and scattering states .................... 104
7.1.1 Definition of the S-matrix ..................... 104
7.1.2 Definition of the scattering states ............ 107
7.1.3 Unitarity of the S-matrix ...................... 107
7.1.4 Time-reversal symmetry ......................... 108
7.2 Conductance and transmission coefficients ............. 109
7.2.1 The Landauer formula, heuristic derivation ..... 110
7.2.2 The Landauer formula, linear response
derivation ..................................... 112
7.2.3 The Landauer-Biittiker formalism for
multiprobe systems ............................. 113
7.3 Electron wave guides .................................. 114
7.3.1 Quantum point contact and conductance
quantization ................................... 114
7.3.2 The Aharonov-Bohm effect ....................... 118
7.4 Summary and outlook ................................... 119
8 Green's functions .......................................... 121
8.1 "Classical" Green's functions ......................... 121
8.2 Green's function for the one-particle Schrodinger
equation .............................................. 121
8.2.1 Example: from the S-matrix to the Green's
function ....................................... 124
8.3 Single-particle Green's functions of many-body
systems ............................................... 125
8.3.1 Green's function of translation-invariant
systems ........................................ 126
8.3.2 Green's function of free electrons ............. 126
8.3.3 The Lehmann representation ..................... 128
8.3.4 The spectral function .......................... 130
8.3.5 Broadening of the spectral function ............ 131
8.4 Measuring the single-particle spectral function ....... 132
8.4.1 Tunneling spectroscopy ......................... 133
8.5 Two-particle correlation functions of many-body
systems ............................................... 136
8.6 Summary and outlook ................................... 139
9 Equation of motion theory .................................. 140
9.1 The single-particle Green's function .................. 140
9.1.1 Non-interacting particles ..................... 142
9.2 Single level coupled to continuum ..................... 142
9.3 Anderson's model for magnetic impurities .............. 143
9.3.1 The equation of motion for the Anderson model .. 145
9.3.2 Mean-field approximation for the Anderson
model .......................................... 146
9.4 The two-particle correlation function ................. 149
9.4.1 The random phase approximation ................ 149
9.5 Summary and outlook ................................... 151
10 Transport in interacting mesoscopic systems ................ 152
10.1 Model Hamiltonians .................................... 152
10.2 Sequential tunneling: the Coulomb blockade regime ..... 154
10.2.1 Coulomb blockade for a metallic dot ............ 155
10.2.2 Coulomb blockade for a quantum dot ............. 158
10.3 Coherent many-body transport phenomena ................ 159
10.3.1 Cotunneling .................................... 159
10.3.2 Inelastic cotunneling for a metallic dot ....... 160
10.3.3 Elastic cotunneling for a quantum dot .......... 161
10.4 The conductance for Anderson-type models .............. 162
10.4.1 The conductance in linear response ............. 163
10.4.2 Calculation of Coulomb blockade peaks .......... 166
10.5 The Kondo effect in quantum dots ...................... 169
10.5.1 From the Anderson model to the Kondo model ..... 169
10.5.2 Comparing the Kondo effect in metals and
quantum dots HS(2) ..............................173
10.5.3 Kondo-model conductance to second order in
HS(2) .......................................... 174
10.5.4 Kondo-model conductance to third order in Hys .. 175
10.5.5 Origin of the logarithmic divergence ........... 180
10.5.6 The Kondo problem beyond perturbation theory ... 181
10.6 Summary and outlook ................................... 182
11 Imaginary-time Green's functions ........................... 184
11.1 Definitions of Matsubara Green's functions ............ 187
11.1.1 Fourier transform of Matsubara Green's
functions ...................................... 188
11.2 Connection between Matsubara and retarded functions ... 189
11.2.1 Advanced functions ............................. 191
11.3 Single-particle Matsubara Green's function ............ 192
11.3.1 Matsubara Green's function for non-
interacting particles .......................... 192
11.4 Evaluation of Matsubara sums .......................... 193
11.4.1 Summations over functions with simple poles .... 194
11.4.2 Summations over functions with known branch
cuts ........................................... 196
11.5 Equation of motion .................................... 197
11.6 Wick's theorem ........................................ 198
11.7 Example: polarizability of free electrons ............. 201
11.8 Summary and outlook ................................... 202
12 Feynman diagrams and external potentials ................... 204
12.1 Non-interacting particles in external potentials ...... 204
12.2 Elastic scattering and Matsubara frequencies .......... 206
12.3 Random impurities in disordered metals ................ 208
12.3.1 Feynman diagrams for the impurity scattering ........ 209
12.4 Impurity self-average ................................. 211
12.5 Self-energy for impurity scattered electrons .......... 216
12.5.1 Lowest-order approximation ..................... 217
12.5.2 First-order Born approximation ................. 217
12.5.3 The full Born approximation .................... 220
12.5.4 The self-consistent Born approximation and
beyond ......................................... 222
12.6 Summary and outlook ................................... 224
13 Feynman diagrams and pair interactions ..................... 226
13.1 The perturbation series for  ......................... 227
13.2 The Feynman rules for pair interactions ............... 228
13.2.1 Feynman rules for the denominator of (b,a) .... 229
13.2.2 Feynman rules for the numerator of (b,a) ...... 230
13.2.3 The cancellation of disconnected Feynman
diagrams ....................................... 231
13.3 Self-energy and Dyson's equation ...................... 233
13.4 The Feynman rules in Fourier space .................... 233
13.5 Examples of how to evaluate Feynman diagrams .......... 236
13.5.1 The Hartree self-energy diagram ................ 236
13.5.2 The Fock self-energy diagram ................... 237
13.5.3 The pair-bubble self-energy diagram ............ 238
13.6 Cancellation of disconnected diagrams, general case ... 239
13.7 Feynman diagrams for the Kondo model .................. 241
13.7.1 Kondo model self-energy, second order in J ..... 243
13.7.2 Kondo model self-energy, third order in J ...... 244
13.8 Summary and outlook ................................... 245
14 The interacting electron gas ............................... 246
14.1 The self-energy in the random phase approximation ..... 246
14.1.1 The density dependence of self-energy
diagrams ....................................... 247
14.1.2 The divergence number of self-energy diagrams .. 248
14.1.3 RPA resummation of the self-energy ............. 248
14.2 The renormalized Coulomb interaction in RPA ........... 250
14.2.1 Calculation of the pair-bubble ................. 251
14.2.2 The electron-hole pair interpretation of RPA ... 253
14.3 The groundstate energy of the electron gas ............ 253
14.4 The dielectric function and screening ................. 256
14.5 Plasma oscillations and Landau damping ................ 260
14.5.1 Plasma oscillations and plasmons ............... 262
14.5.2 Landau damping ................................. 263
14.6 Summary and outlook ................................... 264
15 Fermi liquid theory ........................................ 266
15.1 Adiabatic continuity .................................. 266
15.1.1 Example: one-dimensional well .................. 267
15.1.2 The quasiparticle concept and conserved
quantities ..................................... 268
15.2 Semi-classical treatment of screening and plasmons .... 269
15.2.1 Static screening ............................... 270
15.2.2 Dynamical screening ............................ 271
15.3 Semi-classical transport equation ..................... 272
15.3.1 Finite lifetime of the quasiparticles .......... 276
15.4 Microscopic basis of the Fermi liquid theory .......... 278
15.4.1 Renormalization of the single particle
Green's function ............................... 278
15.4.2 Imaginary part of the single-particle Green's
function ....................................... 280
15.4.3 Mass renormalization? .......................... 283
15.5 Summary and outlook ................................... 283
16 Impurity scattering and conductivity ....................... 285
16.1 Vertex corrections and dressed Green's functions ...... 286
16.2 The conductivity in terms of a general vertex
function .............................................. 291
16.3 The conductivity in the first Born approximation ...... 293
16.4 Conductivity from Born scattering with interactions ... 296
16.5 The weak localization correction to the conductivity .. 298
16.6 Disordered mesoscopic systems ......................... 308
16.6.1 Statistics of quantum conductance, random
matrix theory .................................. 308
16.6.2 Weak localization in mesoscopic systems ........ 309
16.6.3 Universal conductance fluctuations ............. 310
16.7 Summary and outlook ................................... 312
17 Green's functions and phonons .............................. 313
17.1 The Green's function for free phonons ................. 313
17.2 Electron-phonon interaction and Feynman diagrams ...... 314
17.3 Combining Coulomb and electron-phonon interactions .... 316
17.3.1 Migdal's theorem ............................... 317
17.3.2 Jellium phonons and the effective electron-
electron interaction ........................... 318
17.4 Phonon renormalization by electron screening in RPA ... 319
17.5 The Cooper instability and Feynman diagrams ........... 322
17.6 Summary and outlook ................................... 324
18 Superconductivity .......................................... 325
18.1 The Cooper instability ................................ 325
18.2 The BCS groundstate ................................... 327
18.3 Microscopic BCS theory ................................ 329
18.4 BCS theory with Matsubara Green's functions ........... 331
18.4.1 Self-consistent determination of the BCS
order parameter Δк ............................. 332
18.4.2 Determination of the critical temperature Tc ... 333
18.4.3 Determination of the BCS quasiparticle
density of states .............................. 334
18.5 The Nambu formalism of the BCS theory ................. 335
18.5.1 Spinors and Green's functions in the Nambu
formalism ...................................... 335
18.5.2 The Meissner effect and the London equation .... 336
18.5.3 The vanishing paramagnetic current response
in BCS theory .................................. 337
18.6 Gauge symmetry breaking and zero resistivity .......... 341
18.6.1 Gauge transformations .......................... 341
18.6.2 Broken gauge symmetry and dissipationless
current ........................................ 342
18.7 The Josephson effect .................................. 343
18.8 Summary and outlook ................................... 345
19 ID electron gases and Luttinger liquids .................... 347
19.1 What is a Luttinger liquid? ........................... 347
19.2 Experimental realizations of Luttinger liquid
physics ............................................... 348
19.2.1 Example: Carbon Nanotubes ...................... 348
19.2.2 Example: semiconductor wires ................... 348
19.2.3 Example: quasi 1D materials .................... 348
19.2.4 Example: Edge states in the fractional
quantum Hall effect ............................ 348
19.3 A first look at the theory of interacting electrons
in 1D ................................................. 348
19.3.1 The "quasiparticles" in 1D ..................... 350
19.3.2 The lifetime of the "quasiparticles" in 1D ..... 351
19.4 The spinless Luttinger-Tomonaga model ................. 352
19.4.1 The Luttinger-Tomonaga model Hamiltonian ....... 352
19.4.2 Inter-branch interaction ....................... 354
19.4.3 Intra-branch interaction and charge
conservation ................................... 355
19.4.4 Umklapp processes in the half-filled band
case ........................................... 356
19.5 Bosonization of the Tomonaga model Hamiltonian ........ 357
19.5.1 Derivation of the bosonized Hamiltonian ........ 357
19.5.2 Diagonalization of the bosonized Hamiltonian ... 360
19.5.3 Real space representation ...................... 360
19.6 Electron operators in bosonized form .................. 363
19.7 Green's functions ..................................... 368
19.8 Measuring local density of states by tunneling ........ 369
19.9 Luttinger liquid with spin ............................ 373
19.10 Summary and outlook .................................. 374
A Fourier transformations .................................... 376
A.l Continuous functions in a finite region ................ 376
A.2 Continuous functions in an infinite region ............. 377
A.3 Time and frequency Fourier transforms .................. 377
A.4 Some useful rules ...................................... 377
A.5 Translation-invariant systems .......................... 378
Exercises ..................................................... 380
Bibliography .................................................. 421
Index ......................................................... 424
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