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Unformatted text preview: 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 our courses for graduate and advanced undergraduate students at the Niels Bohr Institute, 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. We would like to thank the students who put up with the first versions of this book and for their enumerable and valuable comments and suggestions. We are particularly grateful to the students of Many-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 Research Council through Ole Rømer Grant No. 9600548. Ørsted Laboratory, Niels Bohr Institute 1 September, 2001 Karsten Flensberg 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 Ørsted Laboratory Niels Bohr Institute Henrik Bruus Mikroelektronik Centret Technical University of Denmark iii iv PREFACE Contents List of symbols xii 1 First and second quantization 1.1 First quantization, single-particle systems . . . . . . . . . . . . 1.2 First quantization, many-particle systems . . . . . . . . . . . . 1.2.1 Permutation symmetry and indistinguishability . . . . . 1.2.2 The single-particle states as basis states . . . . . . . . . 1.2.3 Operators in first quantization . . . . . . . . . . . . . . 1.3 Second quantization, basic concepts . . . . . . . . . . . . . . . 1.3.1 The occupation number representation . . . . . . . . . . 1.3.2 The boson creation and annihilation operators . . . . . 1.3.3 The fermion creation and annihilation operators . . . . 1.3.4 The general form for second quantization operators . . . 1.3.5 Change of basis in second quantization . . . . . . . . . . 1.3.6 Quantum field operators and their Fourier transforms . 1.4 Second quantization, specific operators . . . . . . . . . . . . . . 1.4.1 The harmonic oscillator in second quantization . . . . . 1.4.2 The electromagnetic field in second quantization . . . . 1.4.3 Operators for kinetic energy, spin, density, and current . 1.4.4 The Coulomb interaction in second quantization . . . . 1.4.5 Basis states for systems with different kinds of particles 1.5 Second quantization and statistical mechanics . . . . . . . . . . 1.5.1 The distribution function for non-interacting fermions . 1.5.2 Distribution functions for non-interacting bosons . . . . 1.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . 2 The electron gas 2.1 The non-interacting electron gas . . . . . . . . . . . . . . . . 2.1.1 Bloch theory of electrons in a static ion lattice . . . . 2.1.2 Non-interacting electrons in the jellium model . . . . . 2.1.3 Non-interacting electrons at finite temperature . . . . 2.2 Electron interactions in perturbation theory . . . . . . . . . . 2.2.1 Electron interactions in 1st order perturbation theory v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 4 5 6 7 9 10 10 13 14 16 17 18 18 19 21 23 24 25 28 29 29 . . . . . . 31 32 33 35 38 39 41 vi CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 44 45 46 48 49 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 52 53 53 56 59 61 63 64 4 Mean field theory 4.1 The art of mean field theory . . . . . . . . . . . . . . . . . . . . . 4.2 Hartree–Fock approximation . . . . . . . . . . . . . . . . . . . . . 4.3 Broken symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 The Heisenberg model of ionic ferromagnets . . . . . . . . 4.4.2 The Stoner model of metallic ferromagnets . . . . . . . . 4.5 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Breaking of global gauge symmetry and its consequences . 4.5.2 Microscopic theory . . . . . . . . . . . . . . . . . . . . . . 4.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 68 69 71 73 73 75 78 78 81 85 5 Time evolution pictures 5.1 The Schr¨odinger picture . . . . . . . . . . . . . . . . 5.2 The Heisenberg picture . . . . . . . . . . . . . . . . 5.3 The interaction picture . . . . . . . . . . . . . . . . . 5.4 Time-evolution in linear response . . . . . . . . . . . 5.5 Time dependent creation and annihilation operators 5.6 Summary and outlook . . . . . . . . . . . . . . . . . 2.3 2.2.2 Electron interactions in 2nd order perturbation theory . Electron gases in 3, 2, 1, and 0 dimensions . . . . . . . . . . . . 2.3.1 3D electron gases: metals and semiconductors . . . . . . 2.3.2 2D electron gases: GaAs/Ga1°x Alx As heterostructures . 2.3.3 1D electron gases: carbon nanotubes . . . . . . . . . . . 2.3.4 0D electron gases: quantum dots . . . . . . . . . . . . . 3 Phonons; coupling to electrons 3.1 Jellium oscillations and Einstein phonons . . . . . 3.2 Electron-phonon interaction and the sound velocity 3.3 Lattice vibrations and phonons in 1D . . . . . . . 3.4 Acoustical and optical phonons in 3D . . . . . . . 3.5 The specific heat of solids in the Debye model . . . 3.6 Electron-phonon interaction in the lattice model . 3.7 Electron-phonon interaction in the jellium model . 3.8 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 87 88 88 91 91 93 6 Linear response theory 6.1 The general Kubo formula . . . . . . . . . . . . . . . . . . . . 6.2 Kubo formula for conductivity . . . . . . . . . . . . . . . . . 6.3 Kubo formula for conductance . . . . . . . . . . . . . . . . . 6.4 Kubo formula for the dielectric function . . . . . . . . . . . . 6.4.1 Dielectric function for translation-invariant system . . 6.4.2 Relation between dielectric function and conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 95 98 100 102 104 104 . . . . . . . . . . . . . . . . . . . . . . . . vii CONTENTS 6.5 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 7 Transport in mesoscopic systems 7.1 The S-matrix and scattering states . . . . . . . . . . . . . . . . . . 7.1.1 Unitarity of the S-matrix . . . . . . . . . . . . . . . . . . . 7.1.2 Time-reversal symmetry . . . . . . . . . . . . . . . . . . . . 7.2 Conductance and transmission coefficients . . . . . . . . . . . . . . 7.2.1 The Landauer-B¨ uttiker formula, heuristic derivation . . . . 7.2.2 The Landauer-B¨ uttiker formula, linear response derivation . 7.3 Electron wave guides . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Quantum point contact and conductance quantization . . . 7.3.2 Aharonov-Bohm effect . . . . . . . . . . . . . . . . . . . . . 7.4 Disordered mesoscopic systems . . . . . . . . . . . . . . . . . . . . 7.4.1 Statistics of quantum conductance, random matrix theory . 7.4.2 Weak localization in mesoscopic systems . . . . . . . . . . . 7.4.3 Universal conductance fluctuations . . . . . . . . . . . . . . 7.5 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 . 108 . 111 . 112 . 113 . 113 . 115 . 116 . 116 . 120 . 121 . 121 . 123 . 124 . 125 8 Green’s functions 8.1 “Classical” Green’s functions . . . . . . . . . . . . . . . . 8.2 Green’s function for the one-particle Schr¨ odinger equation 8.3 Single-particle Green’s functions of many-body systems . 8.3.1 Green’s function of translation-invariant systems . 8.3.2 Green’s function of free electrons . . . . . . . . . . 8.3.3 The Lehmann representation . . . . . . . . . . . . 8.3.4 The spectral function . . . . . . . . . . . . . . . . 8.3.5 Broadening of the spectral function . . . . . . . . . 8.4 Measuring the single-particle spectral function . . . . . . 8.4.1 Tunneling spectroscopy . . . . . . . . . . . . . . . 8.4.2 Optical spectroscopy . . . . . . . . . . . . . . . . . 8.5 Two-particle correlation functions of many-body systems . 8.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 127 128 131 132 132 134 135 136 137 137 141 141 144 9 Equation of motion theory 9.1 The single-particle Green’s function . . . . . . . . . . . . . . . . . . . 9.1.1 Non-interacting particles . . . . . . . . . . . . . . . . . . . . . . 9.2 Anderson’s model for magnetic impurities . . . . . . . . . . . . . . . . 9.2.1 The equation of motion for the Anderson model . . . . . . . . 9.2.2 Mean-field approximation for the Anderson model . . . . . . . 9.2.3 Solving the Anderson model and comparison with experiments 9.2.4 Coulomb blockade and the Anderson model . . . . . . . . . . . 9.2.5 Further correlations in the Anderson model: Kondo effect . . . 9.3 The two-particle correlation function . . . . . . . . . . . . . . . . . . . 9.3.1 The Random Phase Approximation (RPA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 145 147 147 149 150 151 153 153 153 153 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii CONTENTS 9.4 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 10 Imaginary time Green’s functions 10.1 Definitions of Matsubara Green’s functions . . . . . . . . . . . 10.1.1 Fourier transform of Matsubara Green’s functions . . . 10.2 Connection between Matsubara and retarded functions . . . . . 10.2.1 Advanced functions . . . . . . . . . . . . . . . . . . . . 10.3 Single-particle Matsubara Green’s function . . . . . . . . . . . 10.3.1 Matsubara Green’s function for non-interacting particles 10.4 Evaluation of Matsubara sums . . . . . . . . . . . . . . . . . . 10.4.1 Summations over functions with simple poles . . . . . . 10.4.2 Summations over functions with known branch cuts . . 10.5 Equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Wick’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Example: polarizability of free electrons . . . . . . . . . . . . . 10.8 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 . 160 . 161 . 161 . 163 . 164 . 164 . 165 . 167 . 168 . 169 . 170 . 173 . 174 11 Feynman diagrams and external potentials 11.1 Non-interacting particles in external potentials . . . . . . . 11.2 Elastic scattering and Matsubara frequencies . . . . . . . . 11.3 Random impurities in disordered metals . . . . . . . . . . . 11.3.1 Feynman diagrams for the impurity scattering . . . 11.4 Impurity self-average . . . . . . . . . . . . . . . . . . . . . . 11.5 Self-energy for impurity scattered electrons . . . . . . . . . 11.5.1 Lowest order approximation . . . . . . . . . . . . . . 11.5.2 1st order Born approximation . . . . . . . . . . . . . 11.5.3 The full Born approximation . . . . . . . . . . . . . 11.5.4 The self-consistent Born approximation and beyond 11.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 . 177 . 179 . 181 . 182 . 184 . 189 . 190 . 190 . 193 . 194 . 197 12 Feynman diagrams and pair interactions 12.1 The perturbation series for G . . . . . . . . . . . . . . . . . 12.2 infinite perturbation series!Matsubara Green’s function . . . 12.3 The Feynman rules for pair interactions . . . . . . . . . . . 12.3.1 Feynman rules for the denominator of G(b, a) . . . . 12.3.2 Feynman rules for the numerator of G(b, a) . . . . . 12.3.3 The cancellation of disconnected Feynman diagrams 12.4 Self-energy and Dyson’s equation . . . . . . . . . . . . . . . 12.5 The Feynman rules in Fourier space . . . . . . . . . . . . . 12.6 Examples of how to evaluate Feynman diagrams . . . . . . 12.6.1 The Hartree self-energy diagram . . . . . . . . . . . 12.6.2 The Fock self-energy diagram . . . . . . . . . . . . . 12.6.3 The pair-bubble self-energy diagram . . . . . . . . . 12.7 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 199 199 201 201 202 203 205 206 208 209 209 210 211 ix CONTENTS 13 The interacting electron gas 13.1 The self-energy in the random phase approximation . 13.1.1 The density dependence of self-energy diagrams 13.1.2 The divergence number of self-energy diagrams 13.1.3 RPA resummation of the self-energy . . . . . . 13.2 The renormalized Coulomb interaction in RPA . . . . 13.2.1 Calculation of the pair-bubble . . . . . . . . . . 13.2.2 The electron-hole pair interpretation of RPA . 13.3 The ground state energy of the electron gas . . . . . . 13.4 The dielectric function and screening . . . . . . . . . . 13.5 Plasma oscillations and Landau damping . . . . . . . 13.5.1 Plasma oscillations and plasmons . . . . . . . . 13.5.2 Landau damping . . . . . . . . . . . . . . . . . 13.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 . 213 . 214 . 215 . 215 . 217 . 218 . 220 . 220 . 223 . 227 . 228 . 230 . 231 14 Fermi liquid theory 14.1 Adiabatic continuity . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 The quasiparticle concept and conserved quantities . . 14.2 Semi-classical treatment of screening and plasmons . . . . . . 14.2.1 Static screening . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Dynamical screening . . . . . . . . . . . . . . . . . . . 14.3 Semi-classical transport equation . . . . . . . . . . . . . . . . 14.3.1 Finite life time of the quasiparticles . . . . . . . . . . 14.4 Microscopic basis of the Fermi liquid theory . . . . . . . . . . 14.4.1 Renormalization of the single particle Green’s function 14.4.2 Imaginary part of the single particle Green’s function 14.4.3 Mass renormalization? . . . . . . . . . . . . . . . . . . 14.5 Outlook and summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 . 233 . 235 . 237 . 238 . 238 . 240 . 243 . 245 . 245 . 248 . 251 . 251 15 Impurity scattering and conductivity 15.1 Vertex corrections and dressed Green’s functions . . . 15.2 The conductivity in terms of a general vertex function 15.3 The conductivity in the first Born approximation . . . 15.4 The weak localization correction to the conductivity . 15.5 Combined RPA and Born approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 . 275 . 276 . 279 . 279 . 280 . 281 . 284 . . . . . . . . . . . . . . . . . . . . 16 Green’s functions and phonons 16.1 The Green’s function for free phonons . . . . . . . . . . . . . . . . . . 16.2 Electron-phonon interaction and Feynman diagrams . . . . . . . . . . 16.3 Combining Coulomb and electron-phonon interactions . . . . . . . . . 16.3.1 Migdal’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.2 Jellium phonons and the effective electron-electron interaction 16.4 Phonon renormalization by electron screening in RPA . . . . . . . . . 16.5 The Cooper instability and Feynman diagrams . . . . . . . . . . . . . . . . . . . . 253 254 259 261 264 273 x 17 Superconductivity 17.1 The Cooper instability . . . . . . . . . 17.2 The BCS groundstate . . . . . . . . . 17.3 BCS theory with Green’s functions . . 17.4 Experimental consequences of the BCS 17.4.1 Tunneling density of states . . 17.4.2 specific heat . . . . . . . . . . . 17.5 The Josephson effect . . . . . . . . . . CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 . 287 . 287 . 287 . 288 . 288 . 288 . 288 18 1D electron gases and Luttinger liquids 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 First look at interacting electrons in one dimension . . . 18.2.1 One-dimensional transmission line analog . . . . 18.3 The Luttinger-Tomonaga model - spinless case . . . . . 18.3.1 Interacting one dimensional electron system . . . 18.3.2 Bosonization of Tomonaga model-Hamiltonian . 18.3.3 Diagonalization of bosonized Hamiltonian . . . . 18.3.4 Real space formulation . . . . . . . . . . . . . . . 18.3.5 Electron operators in bosonized form . . . . . . . 18.4 Luttinger liquid with spin . . . . . . . . . . . . . . . . . 18.5 Green’s functions . . . . . . . . . . . . . . . . . . . . . . 18.6 Tunneling into spinless Luttinger liquid . . . . . . . . . 18.6.1 Tunneling into the end of Luttinger liquid . . . . 18.7 What is a Luttinger liquid? . . . . . . . . . . . . . . . . 18.8 Experimental realizations of Luttinger liquid physics . . 18.8.1 Edge states in the fractional quantum Hall effect 18.8.2 Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 289 289 289 289 289 289 289 289 289 290 290 290 290 290 290 290 290 A Fourier transformations A.1 Continuous functions in a finite region . . A.2 Continuous functions in an infinite region A.3 Time and frequency Fourier transforms . A.4 Some useful rules . . . . . . . . . . . . . . A.5 Translation invariant systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...
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