Quantum Condensed Matter Physics - Lecture Notes
Quantum Condensed Matter Physics - LectureNotesChetan NayakNovember 5, 2004
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ContentsIPreliminaries11 Conventions, Notation, Reminders1.1 Mathematical Conventions . . . . . . . . . . . . . . . . . . . .1.2 Plane Wave Expansion . . . . . . . . . . . . . . . . . . . . . .1.2.1 Transforms defined on the continuum in the interval[ L/2, L/2] . . . . . . . . . . . . . . . . . . . . . . . .1.2.2 Transforms defined on a real-space lattice . . . . . . .1.3 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . .1.4 Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . .44611II17Basic Formalism2 Phonons and Second Quantization2.1 Classical Lattice Dynamics . . . . . . . . . . . . . . . . . . .2.2 The Normal Modes of a Lattice . . . . . . . . . . . . . . . . .2.3 Canonical Formalism, Poisson Brackets . . . . . . . . . . . .2.4 Motivation for Second Quantization . . . . . . . . . . . . . .2.5 Canonical Quantization of Continuum Elastic Theory: Phonons2.5.1 Review of the Simple Harmonic Oscillator . . . . . . .2.5.2 Fock Space for Phonons . . . . . . . . . . . . . . . . .2.5.3 Fock space for He4 atoms . . . . . . . . . . . . . . . .3333191920212323232528
4CONTENTS3 Perturbation Theory: Interacting Phonons3.1 Higher-Order Terms in the Phonon Lagrangian .3.2 Schrödinger, Heisenberg, and Interaction Pictures3.3 Dyson’s Formula and the Time-Ordered Product3.4 Wick’s Theorem . . . . . . . . . . . . . . . . . .3.5 The Phonon Propagator . . . . . . . . . . . . . .3.6 Perturbation Theory in the Interaction Picture .313132333537384 Feynman Diagrams and Green Functions4.1 Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . .4.2 Loop Integrals . . . . . . . . . . . . . . . . . . . . . . . .4.3 Green Functions . . . . . . . . . . . . . . . . . . . . . . .4.4 The Generating Functional . . . . . . . . . . . . . . . . .4.5 Connected Diagrams . . . . . . . . . . . . . . . . . . . . .4.6 Spectral Representation of the Two-Point Green function4.7 The Self-Energy and Irreducible Vertex . . . . . . . . . .43434651535657595 Imaginary-Time Formalism5.1 Finite-Temperature Imaginary-Time Green Functions5.2 Perturbation Theory in Imaginary Time . . . . . . . .5.3 Analytic Continuation to Real-Time Green Functions5.4 Retarded and Advanced Correlation Functions . . . .5.5 Evaluating Matsubara Sums . . . . . . . . . . . . . . .5.6 The Schwinger-Keldysh Contour . . . . . . . . . . . .616164666769716 Measurements and Correlation Functions6.1 A Toy Model . . . . . . . . . . . . . . . .6.2 General Formulation . . . . . . . . . . . .6.3 The Fluctuation-Dissipation Theorem . .6.4 Perturbative Example . . . . . . . . . . .6.5 Hydrodynamic Examples . . . . . . . . .6.6 Kubo Formulae . . . . . . . . . . . . . . .6.7 Inelastic Scattering Experiments . . . . .6.8 Neutron Scattering by Spin Systems-xxx .6.9 NMR Relaxation Rate . . . . . . . . . . .75757881828486889090.7 Functional Integrals937.1 Gaussian Integrals . . . . . . . . . . . . . . . . . . . . . . . . 937.2 The Feynman Path Integral . . . . . . . . . . . . . . . . . . . 957.3 The Functional Integral in Many-Body Theory . . . . . . . . 97
CONTENTS7.47.57.65Saddle Point Approximation, Loop Expansion . . . . . . . . .The Functional Integral in Statistical Mechanics . . . . . . .7.5.1 The Ising Model and ϕ4 Theory . . . . . . . . . . . .7.5.2 Mean-Field Theory and the Saddle-Point Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . .The Transfer Matrix** . . . . . . . . . . . . . . . . . . . . . .99101101104105III Goldstone Modes and Spontaneous Symmetry Breaking1078 Spin Systems and Magnons8.1 Coherent-State Path Integral for a Single Spin .8.2 Ferromagnets . . . . . . . . . . . . . . . . . . . .8.2.1 Spin Waves . . . . . . . . . . . . . . . . .8.2.2 Ferromagnetic Magnons . . . . . . . . . .8.2.3 A Ferromagnet in a Magnetic Field . . . .8.3 Antiferromagnets . . . . . . . . . . . . . . . . . .8.3.1 The Non-Linear σ-Model . . . . . . . . .8.3.2 Antiferromagnetic Magnons . . . . . . . .8.3.3 Magnon-Magnon-Interactions . . . . . . .8.4 Spin Systems at Finite Temperatures . . . . . . .8.5 Hydrodynamic Description of Magnetic Systems8.6 Spin chains** . . . . . . . . . . . . . . . . . . . .8.7 Two-dimensional Heisenberg model** . . . . . .1091091141141151171181181191221221261271279 Symmetries in Many-Body Theory1299.1 Discrete Symmetries . . . . . . . . . . . . . . . . . . . . . . . 1299.2 Noether’s Theorem: Continuous Symmetries and Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329.3 Ward Identities . . . . . . . . . . . . . . . . . . . . . . . . . . 1349.4 Spontaneous Symmetry-Breaking and Goldstone’s Theorem . 1379.4.1 Order parameters** . . . . . . . . . . . . . . . . . . . 1419.4.2 Conserved versus nonconserved order parameters** . . 1419.5 Absence of broken symmetry in low dimensions** . . . . . . . 1419.5.1 Discrete symmetry** . . . . . . . . . . . . . . . . . . . 1419.5.2 Continuous symmetry: the general strategy** . . . . . 1419.5.3 The Mermin-Wagner-Coleman Theorem . . . . . . . . 1419.5.4 Absence of magnetic order** . . . . . . . . . . . . . . 1449.5.5 Absence of crystalline order** . . . . . . . . . . . . . . 144
* . . . . . . . . . . .Lack of order in the ground state**of existence of order** . . . . . . . .Infrared bounds** . . . . . . . . . .Critical Fluctuations and Phase Transitions14414414414414510 The10.110.210.310.410.5Renormalization Group and Effective Field Theories 147Low-Energy Effective Field Theories . . . . . . . . . . . . . . 147Renormalization Group Flows . . . . . . . . . . . . . . . . . . 149Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 152Phases of Matter and Critical Phenomena . . . . . . . . . . . 153Infinite number of degrees of freedom and the nonanalyticityof the free energy** . . . . . . . . . . . . . . . . . . . . . . . 15510.5.1 Yang-Lee theory** . . . . . . . . . . . . . . . . . . . . 15510.6 Scaling Equations . . . . . . . . . . . . . . . . . . . . . . . . . 15510.7 Analyticity of β-functions** . . . . . . . . . . . . . . . . . . . 15710.8 Finite-Size Scaling . . . . . . . . . . . . . . . . . . . . . . . . 15710.9 Non-Perturbative RG for the 1D Ising Model . . . . . . . . . 15910.10Dimensional crossover in coupled Ising chains** . . . . . . . . 16010.11Real-space RG** . . . . . . . . . . . . . . . . . . . . . . . . . 16010.12Perturbative RG for ϕ4 Theory in 4 Dimensions . . . . . 16010.13The O(3) NLσM . . . . . . . . . . . . . . . . . . . . . . . . . 16610.14Large N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17110.15The Kosterlitz-Thouless Transition . . . . . . . . . . . . . . . 17510.16Inverse square models in one dimension** . . . . . . . . . . . 18110.17Numerical renormalization group** . . . . . . . . . . . . . . . 18110.18Hamiltonian methods** . . . . . . . . . . . . . . . . . . . . . 18111 Fermions11.1 Canonical Anticommutation Relations . . . . . . . . . . .11.2 Grassmann Integrals . . . . . . . . . . . . . . . . . . . . .11.3 Solution of the 2D Ising Model by Grassmann Integration11.4 Feynman Rules for Interacting Fermions . . . . . . . . . .11.5 Fermion Spectral Function . . . . . . . . . . . . . . . . . .11.6 Frequency Sums and Integrals for Fermions . . . . . . . .11.7 Fermion Self-Energy . . . . . . . . . . . . . . . . . . . . .11.8 Luttinger’s Theorem . . . . . . . . . . . . . . . . . . . . .183183185188191195197198201
CONTENTS712 Interacting Neutral Fermions: Fermi Liquid Theory12.1 Scaling to the Fermi Surface . . . . . . . . . . . . . . .12.2 Marginal Perturbations: Landau Parameters . . . . .12.3 One-Loop . . . . . . . . . . . . . . . . . . . . . . . . .12.4 1/N and All Loops . . . . . . . . . . . . . . . . . . . .12.5 Quartic Interactions for Λ Finite . . . . . . . . . . . .12.6 Zero Sound, Compressibility, Effective Mass . . . . . .205. 205. 207. 211. 214. 216. 21713 Electrons and Coulomb Interactions13.1 Ground State . . . . . . . . . . . . . . . .13.2 Screening . . . . . . . . . . . . . . . . . .13.3 The Plasmon . . . . . . . . . . . . . . . .13.4 RPA . . . . . . . . . . . . . . . . . . . . .13.5 Fermi Liquid Theory for the Electron Gas.237. 237. 237. 237. 237. 23914 Electron-Phonon Interaction14.1 Electron-Phonon Hamiltonian14.2 Feynman Rules . . . . . . . .14.3 Phonon Green Function . . .14.4 Electron Green Function . . .14.5 Polarons . . . . . . . . . . . .22322322522823323415 Rudiments of Conformal Field Theory24115.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24115.2 Conformal Invariance in 2D . . . . . . . . . . . . . . . . . . . 24215.3 Constraints on Correlation Functions . . . . . . . . . . . . . . 24415.4 Operator Product Expansion, Radial Quantization, Mode Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24615.5 Conservation Laws, Energy-Momentum Tensor, Ward Identities24815.6 Virasoro Algebra, Central Charge . . . . . . . . . . . . . . . . 25115.7 Interpretation of the Central Charge . . . . . . . . . . . . . . 25315.7.1 Finite-Size Scaling of the Free Energy . . . . . . . . . 25415.7.2 Zamolodchikov’s c-theorem . . . . . . . . . . . . . . . 25615.8 Representation Theory of the Virasoro Algebra . . . . . . . . 25815.9 Null States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26415.10Unitary Representations . . . . . . . . . . . . . . . . . . . . . 26815.11Free Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . 27015.12Free Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27515.13Kac-Moody Algebras . . . . . . . . . . . . . . . . . . . . . . . 28015.14Coulomb Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
8CONTENTS15.15Interacting Fermions . . . . . . . . . . . . . . . . . . . . . . . 28715.16Fusion and Braiding . . . . . . . . . . . . . . . . . . . . . . . 287VSymmetry-Breaking In Fermion Systems16 Mean-Field Theory16.1 The Classical Limit of Fermions . . . . . . .16.2 Order Parameters, Symmetries . . . . . . .16.3 The Hubbard-Stratonovich Transformation16.4 The Hartree and Fock Approximations . . .16.5 The Variational Approach . . . . . . . . . .289.17 Superconductivity17.1 Instabilities of the Fermi Liquid . . . . . . . . .17.2 Saddle-Point Approximation . . . . . . . . . . .17.3 BCS Variational Wavefunction . . . . . . . . .17.4 Condensate fraction and superfluid density** .17.5 Single-Particle Properties of a Superconductor17.5.1 Green Functions . . . . . . . . . . . . .17.5.2 NMR Relaxation Rate . . . . . . . . . .17.5.3 Acoustic Attenuation Rate . . . . . . .17.5.4 Tunneling . . . . . . . . . . . . . . . . .17.6 Collective Modes of a Superconductor . . . . .17.7 The Higgs Boson . . . . . . . . . . . . . . . . .17.8 Broken gauge symmetry** . . . . . . . . . . . .17.9 The Josephson Effect-xxx . . . . . . . . . . . .17.10Response Functions of a Superconductor-xxx .17.11Repulsive Interactions . . . . . . . . . . . . . .17.12Phonon-Mediated Superconductivity-xxx . . . .17.13The Vortex State*** . . . . . . . . . . . . . . .17.14Fluctuation effects*** . . . . . . . . . . . . . .17.15Condensation in a non-zero angular momentum17.15.1 Liquid 3 He*** . . . . . . . . . . . . . .17.15.2 Cuprate superconductors*** . . . . . .17.16Experimental techniques*** . . . . . . . . . . .291291292299300301. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .state***. . . . . . . . . . . . 21321321321321321321.18 Density waves in solids32318.1 Spin density wave . . . . . . . . . . . . . . . . . . . . . . . . . 32318.2 Charge density wave*** . . . . . . . . . . . . . . . . . . . . . 323
CONTENTS918.3 Density waves with non-trivial angular momentum-xxx . . . . 32318.4 Incommensurate density waves*** . . . . . . . . . . . . . . . 323VIGauge Fields and Fractionalization19 Topology, Braiding Statistics, and Gauge Fields19.1 The Aharonov-Bohm effect . . . . . . . . . . . .19.2 Exotic Braiding Statistics . . . . . . . . . . . . .19.3 Chern-Simons Theory . . . . . . . . . . . . . . .19.4 Ground States on Higher-Genus Manifolds . . . .325.20 Introduction to the Quantum Hall Effect20.1 Introduction . . . . . . . . . . . . . . . . . . . . . .20.2 The Integer Quantum Hall Effect . . . . . . . . . .20.3 The Fractional Quantum Hall Effect: The Laughlin20.4 Fractional Charge and Statistics of Quasiparticles .20.5 Fractional Quantum Hall States on the Torus . . .20.6 The Hierarchy of Fractional Quantum Hall States .20.7 Flux Exchange and ‘Composite Fermions’ . . . . .20.8 Edge Excitations . . . . . . . . . . . . . . . . . . . . . . . . .States. . . . . . . . . . . . . . . .327. 327. 330. 332. 333.337. 337. 340. 344. 349. 352. 353. 354. 35721 Effective Field Theories of the Quantum Hall Effect36121.1 Chern-Simons Theories of the Quantum Hall Effect . . . . . . 36121.2 Duality in 2 1 Dimensions . . . . . . . . . . . . . . . . . . . 36421.3 The Hierarchy and the Jain Sequence . . . . . . . . . . . . . 36921.4 K-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37121.5 Field Theories of Edge Excitations in the Quantum Hall Effect37521.6 Duality in 1 1 Dimensions . . . . . . . . . . . . . . . . . . . 37922 Frontiers in Electron Fractionalization38522.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38522.2 A Simple Model of a Topological Phase in P, T -Invariant Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38622.3 Effective Field Theories . . . . . . . . . . . . . . . . . . . . . 38922.4 Other P, T -Invariant Topological Phases . . . . . . . . . . . . 39122.5 Non-Abelian Statistics . . . . . . . . . . . . . . . . . . . . . . 393
10CONTENTSVII Localized and Extended Excitations in Dirty Systems39923 Impurities in Solids40123.1 Impurity States . . . . . . . . . . . . . . . . . . . . . . . . . . 40123.2 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40323.2.1 Anderson Model . . . . . . . . . . . . . . . . . . . . . 40323.2.2 Lifschitz Tails . . . . . . . . . . . . . . . . . . . . . . . 40523.2.3 Anderson Insulators vs. Mott Insulators . . . . . . . . 40623.3 Physics of the Insulating State . . . . . . . . . . . . . . . . . 40723.3.1 Variable Range Hopping . . . . . . . . . . . . . . . . . 40823.3.2 AC Conductivity . . . . . . . . . . . . . . . . . . . . . 40923.3.3 Effect of Coulomb Interactions . . . . . . . . . . . . . 41023.3.4 Magnetic Properties . . . . . . . . . . . . . . . . . . . 41223.4 Physics of the Metallic State . . . . . . . . . . . . . . . . . . 41623.4.1 Disorder-Averaged Perturbation Theory . . . . . . . . 41623.4.2 Lifetime, Mean-Free-Path . . . . . . . . . . . . . . . . 41823.4.3 Conductivity . . . . . . . . . . . . . . . . . . . . . . . 42023.4.4 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 42323.4.5 Weak Localization . . . . . . . . . . . . . . . . . . . . 42923.4.6 Weak Magnetic Fields and Spin-Orbit Interactions:the Unitary and Symplectic Ensembles . . . . . . . . . 43423.4.7 Electron-Electron Interactions in the Diffusive FermiLiquid . . . . . . . . . . . . . . . . . . . . . . . . . . . 43423.5 The Metal-Insulator Transition . . . . . . . . . . . . . . . . . 43923.5.1 Percolation . . . . . . . . . . . . . . . . . . . . . . . . 43923.5.2 Mobility Edge, Minimum Metallic Conductivity . . . . 44123.5.3 Scaling Theory for Non-Interacting Electrons . . . . . 44323.6 The Integer Quantum Hall Plateau Transition . . . . . . . . . 44724 Non-Linear σ-Models for Diffusing Electrons and AndersonLocalization44924.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44924.2 The Replica Method . . . . . . . . . . . . . . . . . . . . . . . 45124.3 Non-Interacting Electrons . . . . . . . . . . . . . . . . . . . . 45224.3.1 Derivation of the σ-model . . . . . . . . . . . . . . . . 45224.3.2 Interpretation of the σ-model; Analogies with Classical Critical Phenomena . . . . . . . . . . . . . . . . . 46024.3.3 RG Equations for the NLσM . . . . . . . . . . . . . . 46224.4 Interacting Electrons . . . . . . . . . . . . . . . . . . . . . . . 463
CONTENTS1124.5 The Metal-Insulator Transition . . . . . . . . . . . . . . . . . 46824.6 Mesoscopic fluctuations*** . . . . . . . . . . . . . . . . . . . 468
12CONTENTS
Part IPreliminaries1
CHAPTER1Conventions, Notation, Reminders1.1Mathematical ConventionsVectors will be denoted in boldface, x, E, or with a Latin subscript xi , Ei ,i 1, 2, . . . , d. Unless otherwise specified, we will work in d 3 dimensions.Occasionally, we will use Greek subscripts, e.g. jµ , µ 0, 1, . . . , d where the0-component is the time-component as in xµ (t, x, y, z). Unless otherwisenoted, repeated indices are summed over, e.g. ai bi a1 b1 a2 b2 a3 b3 a·bWe will use the following Fourier transform convention:! dω f (ω) e iωt1/2(2π)! dtf (t) eiωtf (ω) 1/2(2π) f (t) 1.2(1.1)Plane Wave ExpansionA standard set of notations for Fourier transforms does not seem to exist.The diversity of notations appear confusing. The problem is that the normalizations are often chosen differently for transforms defined on the realspace continuum and transforms defined on a real space lattice. We shalldo the same, so that the reader is not confused when confronted with variedchoices of normalizations.3
41.2.1CHAPTER 1. CONVENTIONS, NOTATION, REMINDERSTransforms defined on the continuum in the interval[ L/2, L/2]Consider a function f (x) defined in the interval [ L/2, L/2] which we wishto expand in a Fourier series. We shall restrict ourselves to the commonlyused periodic boundary condition, i. e., f (x) f (x L). We can write,1 "f (x) fq eiqx ,L q(1.2)Because the function has the period L, q must be given by 2πn/L, where theinteger n 0, 1, 2, . . . Note that n takes all integer values between and . The plane waves form a complete orthogonal set. So the inverseis! L/21dxeiqx f (x).(1.3)fq L L/2Let us now take the limit L , so that the interval between thesuccessive values of q, q 2π/L then tend to zero, and we can convert theq-sum to an integral. For the first choice of the normalization we get ! dqf (x) lim Lfq eiqx ,(1.4)L 2πand! lim Lfq L dxeiqx f (x).(1.5) If we define f (q) limL Lfq , everything is fine, but note the asymmetry: the factor (1/2π) appears in one of the integrals but not in the other,although we could have arranged, with a suitable choice of the normalizationat the very beginning,so that both integrals would symmetrically involve a factor of (1/ 2π). Note also thatlim Lδq,q! 2πδ(q q ).L (1.6)These results are simple to generalize to the multivariable case.1.2.2Transforms defined on a real-space latticeConsider now the case in which the function f is specified on a periodiclattice in the real space, of spacing a, i. e., xn na; xN/2 L/2, x N/2
1.2. PLANE WAVE EXPANSION5 L/2, and N a L . The periodic boundary condition implies that f (xn ) f (xn L). Thus, the Fourier series now readsf (xn ) 1"fq eiqxn .L q(1.7)Note that the choice of the normalization in Eq. (1.7) and Eq. (1.2) aredifferent. Because of the periodic boundary condition, q is restricted toq 2πm,N
Quantum Condensed Matter Physics - Lecture Notes Chetan Nayak November 5, 2004
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