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. Schrieffer, 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,PhysicalConstants 7 2.2 MathematicalConventions 7 2.3 QuantumMechanics 8 2.4 StatisticalMechanics 11 II Basic Formalism 14 3 Phonons and Second Quantization 15 3.1 ClassicalLatticeDynamics 15 3.2 TheNormalModesofaLattice 16 3.3 CanonicalFormalism,PoissonBrackets 18 3.4 MotivationforSecondQuantization 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 FockSpaceforPhonons 22 3.5.3 Fock space for He 4 atoms 25 4 Perturbation Theory: Interacting Phonons 28 4.1 Higher-OrderTermsinthePhononLagrangian 28 4.2 Schr¨odinger,Heisenberg,andInteractionPictures 29 4.3 Dyson’sFormulaandtheTime-OrderedProduct 31 4.4 Wick’sTheorem 33 4.5 The Phonon Propagator 35 4.6 PerturbationTheoryintheInteractionPicture 36 5 Feynman Diagrams and Green Functions 42 5.1 FeynmanDiagrams 42 5.2 LoopIntegrals 46 5.3 GreenFunctions 52 5.4 TheGeneratingFunctional 54 5.5 ConnectedDiagrams 56 5.6 SpectralRepresentationoftheTwo-PointGreenfunction 58 5.7 TheSelf-EnergyandIrreducibleVertex 60 6 Imaginary-Time Formalism 63 6.1 Finite-TemperatureImaginary-TimeGreenFunctions 63 6.2 PerturbationTheoryinImaginaryTime 66 6.3 AnalyticContinuationtoReal-TimeGreenFunctions 68 6.4 RetardedandAdvancedCorrelationFunctions 70 6.5 EvaluatingMatsubaraSums 72 6.6 TheSchwinger-KeldyshContour 74 iv 7 Measurements and Correlation Functions 79 7.1 AToyModel 79 7.2 GeneralFormulation 83 7.3 TheFluctuation-DissipationTheorem 86 7.4 PerturbativeExample 87 7.5 HydrodynamicExamples 89 7.6 KuboFormulae 91 7.7 InelasticScatteringExperiments 94 7.8 NMRRelaxationRate 96 8 Functional Integrals 98 8.1 GaussianIntegrals 98 8.2 TheFeynmanPathIntegral 100 8.3 TheFunctionalIntegralinMany-BodyTheory 103 8.4 SaddlePointApproximation,LoopExpansion 105 8.5 TheFunctionalIntegralinStatisticalMechanics 108 8.5.1 The Ising Model and ϕ 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-StatePathIntegralforaSingleSpin 114 9.2 Ferromagnets 119 9.2.1 SpinWaves 119 9.2.2 FerromagneticMagnons 120 9.2.3 AFerromagnetinaMagneticField 123 v 9.3 Antiferromagnets 123 9.3.1 The Non-Linear σ-Model 123 9.3.2 AntiferromagneticMagnons 125 9.3.3 Magnon-Magnon-Interactions 128 9.4 SpinSystemsatFiniteTemperatures 129 9.5 HydrodynamicDescriptionofMagneticSystems 133 10 Symmetries in Many-Body Theory 135 10.1DiscreteSymmetries 135 10.2 Noether’s Theorem: Continuous Symmetries and Conservation Laws . 139 10.3WardIdentities 142 10.4SpontaneousSymmetry-BreakingandGoldstone’sTheorem 145 10.5TheMermin-Wagner-ColemanTheorem 149 11 XY Magnets and Superfluid 4 He 154 11.1XYMagnets 154 11.2 Superfluid 4 He 156 IV Critical Fluctuations and Phase Transitions 159 12 The Renormalization Group 160 12.1Low-EnergyEffectiveFieldTheories 160 12.2RenormalizationGroupFlows 162 12.3FixedPoints 165 12.4PhasesofMatterandCriticalPhenomena 167 12.5ScalingEquations 169 12.6Finite-SizeScaling 172 12.7Non-PerturbativeRGforthe1DIsingModel 173 vi 12.8 Perturbative RG for ϕ 4 Theory in 4 − Dimensions 174 12.9 The O(3) NLσM 181 12.10Large N 187 12.11TheKosterlitz-ThoulessTransition 191 13 Fermions 199 13.1CanonicalAnticommutationRelations 199 13.2GrassmanIntegrals 201 13.3FeynmanRulesforInteractingFermions 204 13.4FermionSpectralFunction 209 13.5FrequencySumsandIntegralsforFermions 210 13.6FermionSelf-Energy 212 13.7Luttinger’sTheorem 214 14 Interacting Neutral Fermions: Fermi Liquid Theory 218 14.1ScalingtotheFermiSurface 218 14.2MarginalPerturbations:LandauParameters 220 14.3One-Loop 225 14.4 1/N andAllLoops 227 14.5QuarticInteractionsforΛFinite 230 14.6 Zero Sound, Compressibility, Effective Mass 232 15 Electrons and Coulomb Interactions 236 15.1GroundState 236 15.2Screening 239 15.3ThePlasmon 242 15.4RPA 247 15.5FermiLiquidTheoryfortheElectronGas 249 vii 16 Electron-Phonon Interaction 251 16.1Electron-PhononHamiltonian 251 16.2FeynmanRules 251 16.3PhononGreenFunction 251 16.4ElectronGreenFunction 251 16.5Polarons 253 17 Superconductivity 254 17.1 Instabilities of the Fermi Liquid 254 17.2Saddle-PointApproximation 255 17.3BCSVariationalWavefunction 258 17.4 Single-Particle Properties of a Superconductor 259 17.4.1GreenFunctions 259 17.4.2NMRRelaxationRate 261 17.4.3AcousticAttenuationRate 265 17.4.4 Tunneling . 266 17.5 Collective Modes of a Superconductor . 269 17.6RepulsiveInteractions 272 V Gauge Fields and Fractionalization 274 18 Topology, Braiding Statistics, and Gauge Fields 275 18.1TheAharonov-Bohmeffect 275 18.2ExoticBraidingStatistics 278 18.3Chern-SimonsTheory 281 18.4GroundStatesonHigher-GenusManifolds 282 viii 19 Introduction to the Quantum Hall Effect 286 19.1Introduction 286 19.2TheIntegerQuantumHallEffect 290 19.3TheFractionalQuantumHallEffect:TheLaughlinStates 295 19.4FractionalChargeandStatisticsofQuasiparticles 301 19.5FractionalQuantumHallStatesontheTorus 304 19.6TheHierarchyofFractionalQuantumHallStates 306 19.7FluxExchangeand‘CompositeFermions’ 307 19.8EdgeExcitations 312 20 Effective Field Theories of the Quantum Hall Effect 315 20.1Chern-SimonsTheoriesoftheQuantumHallEffect 315 20.2Dualityin2+1Dimensions 319 20.3TheHierarchyandtheJainSequence 324 20.4K-matrices 327 20.5FieldTheoriesofEdgeExcitationsintheQuantumHallEffect 332 20.6Dualityin1+1Dimensions 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 23.1ImpurityStates 345 23.2AndersonLocalization 345 23.3 The Physics of Metallic and Insulating Phases 345 ix 23.4TheMetal-InsulatorTransition 345 24 Field-Theoretic Techniques for Disordered Systems 346 24.1Disorder-AveragedPerturbationTheory 346 24.2TheReplicaMethod 346 24.3Supersymmetry 346 24.4TheSchwinger-KeldyshTechnique 346 25 The Non-Linear σ-Model for Anderson Localization 347 25.1 Derivation of the σ-model 347 25.2 Interpretation of the σ-model 347 25.3 2 +  Expansion 347 25.4TheMetal-InsulatorTransition 347 26 Electron-Electron Interactions in Disordered Systems 348 26.1PerturbationTheory 348 26.2 The Finkelstein σ-Model 348 x [...]... elementary particle physics, in nuclear physics, and in classical statistical mechanics In elementary particle physics, large numbers of real or virtual particles can be excited in scattering experiments The principal distinguishing feature of elementary particle physics – which actually simplifies matters – is relativistic invariance Another simplifying feature is that in particle physics one often considers... described in terms of the physics of non-interacting electrons in a periodic potential The methods described in this course will allow us to go beyond this and tackle the complex and profound phenomena which arise from the Coulomb interactions between electrons and from the coupling of the electrons to lattice distortions The techniques which we will use come under the rubric of many-body physics or quantum... actually simplifies matters – is relativistic invariance Another simplifying feature is that in particle physics one often considers systems at zero-temperature – with applications of particle physics to astrophysics and cosmology being the notable exception – so that there are quantum fluctuations but no thermal fluctuations In classical statistical mechanics, on the other hand, there are only thermal... in, say, a field theory course is the physical scale We will be concerned with: • ω, T 1eV • |xi − xj |, 1 q 1˚ A as compared to energies in the MeV for nuclear matter, and GeV or even T eV , in particle physics Special experimental techniques are necessary to probe such scales • Thermodynamics: measure the response of macroscopic variables such as the energy and volume to variations of the temperature,... shows, many-particle systems exhibit various phases – such as ice and water – which are not, for the most part, usefully described by the microscopic equations Instead, new low-energy, long-wavelength physics emerges as a result of the interactions among large numbers of particles Different phases are separated by phase transitions, at which the low-energy, long-wavelength description becomes non-analytic... energy of the elastic medium must be translationally and rotationally invariant (at shorter distances, these symmetries are broken to discrete lattice symmetries, but let’s focus on the long-wavelength physics for now) Translational invariance implies V [u + u0 ] = V [u], so V can only be a function of the derivatives, ∂i uj Rotational invariance implies that it can only be a function of the symmetric... whose number is fixed), we will have to develop the formalism of second quantization 3.5 1 Canonical Quantization of Continuum Elastic Theory: Phonons 3.5.1 Review of the Simple Harmonic Oscillator No physics course is complete without a discussion of the simple harmonic oscillator Here, we will recall the operator formalism which will lead naturally to the Fock space construction of quantum field theory . elementary particle physics, in nuclear physics, and in classical statistical mechanics. In elementary par- ticle physics, large numbers of real or virtual particles can be excited in scattering experiments Units,PhysicalConstants 7 2.2 MathematicalConventions 7 2.3 QuantumMechanics 8 2.4 StatisticalMechanics 11 II Basic Formalism 14 3 Phonons and Second Quantization 15 3.1 ClassicalLatticeDynamics 15 3.2. GeV or even TeV,in particle physics. Special experimental techniques are necessary to probe such scales. • Thermodynamics: measure the response of macroscopic variables such as the energy and volume