Table Of Content

Many-Body Physics Chetan Nayak Physics 242 University of California, Los Angeles January 1999 Preface Some useful textbooks: A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics G. Mahan, Many-Particle Physics A. Fetter and J. Walecka, Quantum Theory of Many-Particle Systems S. Doniach and Sondheimer, Green’s Functions for Solid State Physicists J. R. Schrie(cid:11)er, Theory of Superconductivity J. Negele and H. Orland, Quantum Many-Particle Systems E. Fradkin, Field Theories of Condensed Matter Systems A. M. Tsvelik, Field Theory in Condensed Matter Physics A. Auerbach, Interacting Electrons and Quantum Magnetism A useful review article: R. Shankar, Rev. Mod. Phys. 66, 129 (1994). ii Contents Preface ii I Preliminaries 1 1 Introduction 2 2 Conventions, Notation, Reminders 7 2.1 Units, Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Mathematical Conventions . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 II Basic Formalism 14 3 Phonons and Second Quantization 15 3.1 Classical Lattice Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 The Normal Modes of a Lattice . . . . . . . . . . . . . . . . . . . . . 16 3.3 Canonical Formalism, Poisson Brackets . . . . . . . . . . . . . . . . . 18 3.4 Motivation for Second Quantization . . . . . . . . . . . . . . . . . . . 19 3.5 Canonical Quantization of Continuum Elastic Theory: Phonons . . . 20 3.5.1 Review of the Simple Harmonic Oscillator . . . . . . . . . . . 20 iii 3.5.2 Fock Space for Phonons . . . . . . . . . . . . . . . . . . . . . 22 3.5.3 Fock space for He4 atoms . . . . . . . . . . . . . . . . . . . . . 25 4 Perturbation Theory: Interacting Phonons 28 4.1 Higher-Order Terms in the Phonon Lagrangian . . . . . . . . . . . . 28 4.2 Schro¨dinger, Heisenberg, and Interaction Pictures . . . . . . . . . . . 29 4.3 Dyson’s Formula and the Time-Ordered Product . . . . . . . . . . . . 31 4.4 Wick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.5 The Phonon Propagator . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.6 Perturbation Theory in the Interaction Picture . . . . . . . . . . . . . 36 5 Feynman Diagrams and Green Functions 42 5.1 Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.2 Loop Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.3 Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.4 The Generating Functional . . . . . . . . . . . . . . . . . . . . . . . . 54 5.5 Connected Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.6 Spectral Representation of the Two-Point Green function . . . . . . . 58 5.7 The Self-Energy and Irreducible Vertex . . . . . . . . . . . . . . . . . 60 6 Imaginary-Time Formalism 63 6.1 Finite-Temperature Imaginary-Time Green Functions . . . . . . . . . 63 6.2 Perturbation Theory in Imaginary Time . . . . . . . . . . . . . . . . 66 6.3 Analytic Continuation to Real-Time Green Functions . . . . . . . . . 68 6.4 Retarded and Advanced Correlation Functions . . . . . . . . . . . . . 70 6.5 Evaluating Matsubara Sums . . . . . . . . . . . . . . . . . . . . . . . 72 6.6 The Schwinger-Keldysh Contour . . . . . . . . . . . . . . . . . . . . . 74 iv 7 Measurements and Correlation Functions 79 7.1 A Toy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7.2 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.3 The Fluctuation-Dissipation Theorem . . . . . . . . . . . . . . . . . . 86 7.4 Perturbative Example . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.5 Hydrodynamic Examples . . . . . . . . . . . . . . . . . . . . . . . . . 89 7.6 Kubo Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.7 Inelastic Scattering Experiments . . . . . . . . . . . . . . . . . . . . . 94 7.8 NMR Relaxation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 96 8 Functional Integrals 98 8.1 Gaussian Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.2 The Feynman Path Integral . . . . . . . . . . . . . . . . . . . . . . . 100 8.3 The Functional Integral in Many-Body Theory . . . . . . . . . . . . . 103 8.4 Saddle Point Approximation, Loop Expansion . . . . . . . . . . . . . 105 8.5 The Functional Integral in Statistical Mechanics . . . . . . . . . . . . 108 8.5.1 The Ising Model and ’4 Theory . . . . . . . . . . . . . . . . . 108 8.5.2 Mean-Field Theory and the Saddle-Point Approximation . . . 111 III Goldstone Modes and Spontaneous Symmetry Break- ing 113 9 Spin Systems and Magnons 114 9.1 Coherent-State Path Integral for a Single Spin . . . . . . . . . . . . . 114 9.2 Ferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 9.2.1 Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 9.2.2 Ferromagnetic Magnons . . . . . . . . . . . . . . . . . . . . . 120 9.2.3 A Ferromagnet in a Magnetic Field . . . . . . . . . . . . . . . 123 v 9.3 Antiferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 9.3.1 The Non-Linear (cid:27)-Model . . . . . . . . . . . . . . . . . . . . . 123 9.3.2 Antiferromagnetic Magnons . . . . . . . . . . . . . . . . . . . 125 9.3.3 Magnon-Magnon-Interactions . . . . . . . . . . . . . . . . . . 128 9.4 Spin Systems at Finite Temperatures . . . . . . . . . . . . . . . . . . 129 9.5 Hydrodynamic Description of Magnetic Systems . . . . . . . . . . . . 133 10 Symmetries in Many-Body Theory 135 10.1 Discrete Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10.2 Noether’s Theorem: Continuous Symmetries and Conservation Laws . 139 10.3 Ward Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 10.4 Spontaneous Symmetry-Breaking and Goldstone’s Theorem . . . . . . 145 10.5 The Mermin-Wagner-Coleman Theorem . . . . . . . . . . . . . . . . 149 11 XY Magnets and Superﬂuid 4He 154 11.1 XY Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 11.2 Superﬂuid 4He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 IV Critical Fluctuations and Phase Transitions 159 12 The Renormalization Group 160 12.1 Low-Energy E(cid:11)ective Field Theories . . . . . . . . . . . . . . . . . . 160 12.2 Renormalization Group Flows . . . . . . . . . . . . . . . . . . . . . . 162 12.3 Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 12.4 Phases of Matter and Critical Phenomena . . . . . . . . . . . . . . . 167 12.5 Scaling Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 12.6 Finite-Size Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 12.7 Non-Perturbative RG for the 1D Ising Model . . . . . . . . . . . . . . 173 vi 12.8 Perturbative RG for ’4 Theory in 4−(cid:15) Dimensions . . . . . . . . . . 174 12.9 The O(3) NL(cid:27)M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 12.10Large N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 12.11The Kosterlitz-Thouless Transition . . . . . . . . . . . . . . . . . . . 191 13 Fermions 199 13.1 Canonical Anticommutation Relations . . . . . . . . . . . . . . . . . 199 13.2 Grassman Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 13.3 Feynman Rules for Interacting Fermions . . . . . . . . . . . . . . . . 204 13.4 Fermion Spectral Function . . . . . . . . . . . . . . . . . . . . . . . . 209 13.5 Frequency Sums and Integrals for Fermions . . . . . . . . . . . . . . . 210 13.6 Fermion Self-Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 13.7 Luttinger’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 14 Interacting Neutral Fermions: Fermi Liquid Theory 218 14.1 Scaling to the Fermi Surface . . . . . . . . . . . . . . . . . . . . . . . 218 14.2 Marginal Perturbations: Landau Parameters . . . . . . . . . . . . . . 220 14.3 One-Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 14.4 1=N and All Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 14.5 Quartic Interactions for (cid:3) Finite . . . . . . . . . . . . . . . . . . . . . 230 14.6 Zero Sound, Compressibility, E(cid:11)ective Mass . . . . . . . . . . . . . . 232 15 Electrons and Coulomb Interactions 236 15.1 Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 15.2 Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 15.3 The Plasmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 15.4 RPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 15.5 Fermi Liquid Theory for the Electron Gas . . . . . . . . . . . . . . . 249 vii 16 Electron-Phonon Interaction 251 16.1 Electron-Phonon Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 251 16.2 Feynman Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 16.3 Phonon Green Function . . . . . . . . . . . . . . . . . . . . . . . . . 251 16.4 Electron Green Function . . . . . . . . . . . . . . . . . . . . . . . . . 251 16.5 Polarons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 17 Superconductivity 254 17.1 Instabilities of the Fermi Liquid . . . . . . . . . . . . . . . . . . . . . 254 17.2 Saddle-Point Approximation . . . . . . . . . . . . . . . . . . . . . . . 255 17.3 BCS Variational Wavefunction . . . . . . . . . . . . . . . . . . . . . . 258 17.4 Single-Particle Properties of a Superconductor . . . . . . . . . . . . . 259 17.4.1 Green Functions . . . . . . . . . . . . . . . . . . . . . . . . . 259 17.4.2 NMR Relaxation Rate . . . . . . . . . . . . . . . . . . . . . . 261 17.4.3 Acoustic Attenuation Rate . . . . . . . . . . . . . . . . . . . . 265 17.4.4 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 17.5 Collective Modes of a Superconductor . . . . . . . . . . . . . . . . . . 269 17.6 Repulsive Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 272 V Gauge Fields and Fractionalization 274 18 Topology, Braiding Statistics, and Gauge Fields 275 18.1 The Aharonov-Bohm e(cid:11)ect . . . . . . . . . . . . . . . . . . . . . . . . 275 18.2 Exotic Braiding Statistics . . . . . . . . . . . . . . . . . . . . . . . . 278 18.3 Chern-Simons Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 18.4 Ground States on Higher-Genus Manifolds . . . . . . . . . . . . . . . 282 viii 19 Introduction to the Quantum Hall E(cid:11)ect 286 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 19.2 The Integer Quantum Hall E(cid:11)ect . . . . . . . . . . . . . . . . . . . . 290 19.3 The Fractional Quantum Hall E(cid:11)ect: The Laughlin States . . . . . . 295 19.4 Fractional Charge and Statistics of Quasiparticles . . . . . . . . . . . 301 19.5 Fractional Quantum Hall States on the Torus . . . . . . . . . . . . . 304 19.6 The Hierarchy of Fractional Quantum Hall States . . . . . . . . . . . 306 19.7 Flux Exchange and ‘Composite Fermions’ . . . . . . . . . . . . . . . 307 19.8 Edge Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 20 E(cid:11)ective Field Theories of the Quantum Hall E(cid:11)ect 315 20.1 Chern-Simons Theories of the Quantum Hall E(cid:11)ect . . . . . . . . . . 315 20.2 Duality in 2+1 Dimensions . . . . . . . . . . . . . . . . . . . . . . . 319 20.3 The Hierarchy and the Jain Sequence . . . . . . . . . . . . . . . . . . 324 20.4 K-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 20.5 Field Theories of Edge Excitations in the Quantum Hall E(cid:11)ect . . . . 332 20.6 Duality in 1+1 Dimensions . . . . . . . . . . . . . . . . . . . . . . . 337 21 P;T-violating Superconductors 342 22 Electron Fractionalization without P;T-violation 343 VI Localized and Extended Excitations in Dirty Systems344 23 Impurities in Solids 345 23.1 Impurity States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 23.2 Anderson Localization . . . . . . . . . . . . . . . . . . . . . . . . . . 345 23.3 The Physics of Metallic and Insulating Phases . . . . . . . . . . . . . 345 ix 23.4 The Metal-Insulator Transition . . . . . . . . . . . . . . . . . . . . . 345 24 Field-Theoretic Techniques for Disordered Systems 346 24.1 Disorder-Averaged Perturbation Theory . . . . . . . . . . . . . . . . 346 24.2 The Replica Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 24.3 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 24.4 The Schwinger-Keldysh Technique . . . . . . . . . . . . . . . . . . . . 346 25 The Non-Linear (cid:27)-Model for Anderson Localization 347 25.1 Derivation of the (cid:27)-model . . . . . . . . . . . . . . . . . . . . . . . . 347 25.2 Interpretation of the (cid:27)-model . . . . . . . . . . . . . . . . . . . . . . 347 25.3 2+(cid:15) Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 25.4 The Metal-Insulator Transition . . . . . . . . . . . . . . . . . . . . . 347 26 Electron-Electron Interactions in Disordered Systems 348 26.1 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 26.2 The Finkelstein (cid:27)-Model . . . . . . . . . . . . . . . . . . . . . . . . . 348 x