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BK

Corrected version

Oxford : Oxford University Press, 2016

xix, 437 stran : ilustrace ; 25 cm

ISBN 978-0-19-856633-5 (vázáno)

Oxford graduate texts

Obsahuje bibliografii na stranách 423-425 a rejstřík

Popsáno podle dotisku z roku 2020

001640454

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 Quant urn 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 particles 25 // 1.5 Second quantization and statistical mechanics 26 // 1.5.1 The distribution function for non-interacting fermions 29 // 1.5.2 The 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 lst-order perturbation theory 42 // 2.2.2 Electron interactions in 2nd-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 0D 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 ID 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 H-F 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 Schrödinger 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 98 // 6.4.1 Dielectric function for translation-invariant system 100 // 6.4.2 Relation between dielectric function and conductivity 100 // 6.5 Summary and outlook 101 //

7 Transport in mesoscopic systems ?2 // 7.1 The 5-matrix and scattering states ?? // 7.1.1 Definition of the 5-matrix // 7.1.2 Definition of the scattering states 106 // 7.1.3 Unitarity of the 5-matrix 106 // 7.1.4 Time-reversal symmetry ?7 // 7.2 Conductance and transmission coefficients 108 // 7.2.1 The Landauer formula, heuristic derivation 109 // 7.2.2 The Landauer formula, linear response derivation 111 // 7.2.3 Landauer-Büttiker formalism for multiprobe systems 112 // 7.3 Electron wave guides 113 // 7.3.1 Quantum point contact and conductance quantization 113 // 7.3.2 The Aharonov-Bohm effect 117 // 7.4 Summary and outlook 118 // 8 Green’s functions 120 // 8.1 “Classical” Green’s functions 220 // 8.2 Green s function for the one-particle Schrödinger equation 120 // 8.2.1 Example: from the 5-matrix to the Green’s function 123 // 8.3 Single-particle Green’s functions of many-body systems 124 // 8.3.1 Green’s function of translation-invariant systems 125 // 8.3.2 Green’s function of free electrons 225 // 8.3.3 The Lehmann representation 227 // 8.3.4 The spectral function 229 // 8.3.5 Broadening of the spectral function 230 // 8.4 Measuring the single-particle spectral function 131 // 8.4.1 Tunneling spectroscopy 132 // 8.5 Two-particle correlation functions of many-body systems 135 // 8.6 Summary and outlook 13g // 9 Equation of motion theory 13g // 9.1 The single-particle Green’s function I39 // 9.1.1 Non-interacting particles 141 // 9.2 Single level coupled to a continuum 141 // 9.3 Anderson’s model for magnetic impurities 142 // 9.3.1 The equation of motion for the Anderson model 144 // 9.3.2 Mean-field approximation for the Anderson model 145 // 9.4 The two-particle correlation function 148 // 9.4.1 The random phase approximation 148 // 9.5 Summary and outlook 15q //

10 Transport in interacting mesoscopic systems 151 // 10.1 Model Hamiltonians 151 // 10.2 Sequential tunneling: the Coulomb blockade regime I53 // 10.2.1 Coulomb blockade for a metallic dot I54 // 10.2.2 Coulomb blockade for a quantum dot I57 // 10.3 Coherent many-body transport phenomena 158 // 10.3.1 Cotunneling 158 // 10.3.2 Inelastic cotunneling for a metallic dot 159 // 10.3.3 Elastic cotunneling for a quantum dot 160 // 10.4 The conductance for Anderson-type models 161 // 10.4.1 The conductance in linear response 162 // 10.4.2 Calculation of Coulomb blockade peaks 165 // 10.5 The Rondo effect in quantum dots 168 // 10.5.1 From the Anderson model to the Rondo model 168 // 10.5.2 Comparing Rondo effect in metals and quantum dots 173 // 10.5.3 Rondo-model conductance to second order in 173 // 10.5.4 Rondo-model conductance to third order in 174 // 10.5.5 Origin of the logarithmic divergence 179 // 10.5.6 The Rondo 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, 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 Self-consistent full Born approximation and beyond 222 // 12.6 Summary and outlook 224 // 13 Feynman diagrams and pair interactions 226 // 13.1 The perturbation series for Q 227 // 13.2 The Feynman rules for pair interactions 228 // 13.2.1 Feynman rules for the denominator of Gib, a) 229 // 13.2.2 Feynman rules for the numerator of Gib, 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 Hart ree 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 Renormalizationofthesingle-particleGreen’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  // 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 332 // 18.4.2 Determination of the critical temperature Tc 333 // 18.4.3 Determination of 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 Nambu formalism 335 // 18.5.2 The Meissner effect and the London equation 336 // 18.5.3 Zero 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 346 // CONTENTS xiii // 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 ID materials 348 // 19.2.4 Example: Edge states in fractional quant urn Hall effect 348 // 19.3 A first look at the theory of interacting electrons in ID 348 // 19.3.1 The “quasiparticles" in ID 350 // 19.3.2 The lifetime of the “quasiparticles” in ID 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 423 // Index 426