many-body quantum theory in condensed matter physics

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 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 1 st order perturbation theory . . . . . . . . 41 v vi CONTENTS 2.2.2 Electron interactions in 2 nd 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 1−x Al x As heterostructures . . . . . . . . 46 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-B¨uttiker formula, heuristic derivation . . . . . . . . . 113 7.2.2 The Landauer-B¨uttiker 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 1 st 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 [...]... development of quantum electrodynamics (QED) and quantum eld theory (QFT) in general By convention, the original form of quantum mechanics is denoted rst quantization, while quantum eld theory is formulated in the language of second quantization Regardless of the representation, be it rst or second quantization, certain basic concepts are always present in the formulation of quantum theory The starting point... characterized by the same quantum numbers such as mass, charge and spin, are in principle indistinguishable From the indistinguishability of particles follows that if two coordinates in an N particle state function are interchanged the same physical state results, and the corresponding state function can at most dier from the original one by a simple prefactor If the same two coordinates then are interchanged... principle stating that two fermions cannot occupy the same state, because if in Eq (1.15b) we let rj = rk then = 0 follows It thus explains the periodic table of the elements, and consequently the starting point in our understanding of atomic physics, condensed matter physics and chemistry It furthermore plays a fundamental role in the studies of the nature of stars and of the scattering processes in. .. be noted that the quantum label can contain both discrete and continuous quantum numbers In that case the symbol is to be interpreted as a combination of both summations and integrations For example in the case in Eq (1.5) with Landau orbitals in a box with side lengths Lx , Ly , and Lz , we have = =, n=0 Ly dky 2 Lz dkz 2 (1.9) In the mathematical formulation of quantum theory we shall often... and indistinguishability A fundamental dierence between classical and quantum mechanics concerns the concept of indistinguishability of identical particles In classical mechanics each particle can be equipped with an identifying marker (e.g a colored spot on a billiard ball) without inuencing its behavior, and moreover it follows its own continuous path in phase space Thus in principle each particle in. .. and (1.15b) are in Eq (1.21) hidden in the coefcients A1 , ,N A physical meaningful basis bringing the N coordinates on equal footing in the products 1 (r1 )2 (r2 ) N (rN ) of single-particle state functions is obtained by 1.2 FIRST QUANTIZATION, MANY-PARTICLE SYSTEMS 7 applying the bosonic symmetrization operator S+ or the fermionic anti-symmetrization dened by the following determinants and permanent:2... t) and A(r, t) The corresponding Hamiltonian is H= 1 2m 2 i r + eA(r, t) e(r, t) (1.2) An eigenstate describing a free spin-up electron travelling inside a box of volume V can be written as a product of a propagating plane wave and a spin-up spinor Using the Dirac notation the state ket can be written as |k, = |k, , where one simply lists the relevant quantum numbers in the ket The state function... basic concepts Many-particle physics is formulated in terms of the so-called second quantization representation also known by the more descriptive name occupation number representation The starting point of this formalism is the notion of indistinguishability of particles discussed in Sec 1.2.1 combined with the observation in Sec 1.2.2 that determinants or permanent of single-particle states form a... possible in quantum physics, the so-called bosons and fermions1 : (r1 , , rj , , rk , , rN ) = +(r1 , , rk , , rj , , rN ) (bosons), (1.15a) (r1 , , rj , , rk , , rN ) = (r1 , , rk , , rj , , rN ) (fermions) (1.15b) The importance of the assumption of indistinguishability of particles in quantum physics cannot be exaggerated, and it has been introduced due to overwhelming... permutations p on the set of N coordinates3 , and sign(p), used in the Slater determinant, is the sign of the permutation p Note how in the fermion case j = k leads to = 0, i.e the Pauli principle Using the symmetrized basis states the expansion in Eq (1.21) gets replaced by the following, where the new expansion coecients B1 ,2 , ,N are completely symmetric in their -indices, B1 ,2 , ,N S 1 (r1 )2 (r2 . e y leads to the Landau eigenstates |n, k y , k z , σ , where n is an integer, k y (k z ) is the y (z) component of k, and σ the spin variable. Recall that r|n, k y , k z , σ = H n (x/ k y )e − 1 2 (x/ k y ) 2 1 √ L y L z e i (k y y +k z z) χ σ (Landau. |r|ψ ν | 2 in the xy plane for (a) any plane wave ν = (k x , k y , k z , σ ), (b) the hydrogen orbital ν = ( 4, 2, 0, σ ), and (c) the Landau orbital ν = ( 3, k y , 0, σ). charged nucleus are considered, the. 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,