Modern Condensed Matter Physics

Contents
Preface
Acknowledgments
1 Overview of Condensed Matter Physics
	1.1 Definition of Condensed Matter and Goals of Condensed Matter Physics
	1.2 Classification (or Phases) of Condensed Matter Systems
		1.2.1 Atomic Spatial Structures
		1.2.2 Electronic Structures or Properties
		1.2.3 Symmetries
		1.2.4 Beyond Symmetries
	1.3 Theoretical Descriptions of Condensed Matter Phases
	1.4 Experimental Probes of Condensed Matter Systems
2 Spatial Structure
	2.1 Probing the Structure
	2.2 Semiclassical Theory of X-Ray Scattering
	2.3 Quantum Theory of Electron–Photon Interaction and X-Ray Scattering
	2.4 X-Ray Scattering from a Condensed Matter System
	2.5 Relationship of S(q±) and Spatial Correlations
	2.6 Liquid State versus Crystal State
3 Lattices and Symmetries
	3.1 The Crystal as a Broken-Symmetry State
	3.2 Bravais Lattices and Lattices with Bases
		3.2.1 Bravais Lattices
		3.2.2 Lattices with Bases
		3.2.3 Lattice Symmetries in Addition to Translation
	3.3 Reciprocal Lattices
	3.4 X-Ray Scattering from Crystals
	3.5 Effects of Lattice Fluctuations on X-Ray Scattering
	3.6 Notes and Further Reading
4 Neutron Scattering
	4.1 Introduction to Neutron Scattering
	4.2 Inelastic Neutron Scattering
	4.3 Dynamical Structure Factor and f -Sum Rule
		4.3.1 Classical Harmonic Oscillator
		4.3.2 Quantum Harmonic Oscillator
	4.4 Single-Mode Approximation and Superfluid 4He
5 Dynamics of Lattice Vibrations
	5.1 Elasticity and Sound Modes in Continuous Media
	5.2 Adiabatic Approximation and Harmonic Expansion of Atomic Potential
	5.3 Classical Dynamics of Lattice Vibrations
6 Quantum Theory of Harmonic Crystals
	6.1 Heat Capacity
	6.2 Canonical Quantization of Lattice Vibrations
	6.3 Quantum Dynamical Structure Factor
	6.4 Debye–Waller Factor and Stability of Crystalline Order
	6.5 Mössbauer Effect
7 Electronic Structure of Crystals
	7.1 Drude Theory of Electron Conduction in Metals
	7.2 Independent Electron Model
	7.3 Bloch’s Theorem
		7.3.1 Band Gaps and Bragg Reflection
		7.3.2 Van Hove Singularities
		7.3.3 Velocity of Bloch Electrons
	7.4 Tight-Binding Method
		7.4.1 Bonds vs. Bands
		7.4.2 Wannier Functions
		7.4.3 Continuum Limit of Tight-Binding Hamiltonians
		7.4.4 Limitations of the Tight-Binding Model
		7.4.5 s–d Hybridization in Transition Metals
	7.5 Graphene Band Structure
	7.6 Polyacetylene and the Su–Schrieffer–Heeger Model
		7.6.1 Dirac electrons in 1D and the Peierls instability
		7.6.2 Ground-State Degeneracy and Solitons
		7.6.3 Zero Modes Bound to Solitons
		7.6.4 Quantum Numbers of Soliton States and Spin–Charge Separation
	7.7 Thermodynamic Properties of Bloch Electrons
		7.7.1 Specific Heat
		7.7.2 Magnetic Susceptibility
	7.8 Spin–Orbit Coupling and Band Structure
	7.9 Photonic Crystals
	7.10 Optical Lattices
		7.10.1 Oscillator Model of Atomic Polarizability
		7.10.2 Quantum Effects in Optical Lattices
8 Semiclassical Transport Theory
	8.1 Review of Semiclassical Wave Packets
	8.2 Semiclassical Wave-Packet Dynamics in Bloch Bands
		8.2.1 Derivation of Bloch Electron Equations of Motion
		8.2.2 Zener Tunneling (or Interband Transitions)
	8.3 Holes
	8.4 Uniform Magnetic Fields
	8.5 Quantum Oscillations
	8.6 Semiclassical E± × B± Drift
	8.7 The Boltzmann Equation
	8.8 Boltzmann Transport
		8.8.1 Einstein Relation
	8.9 Thermal Transport and Thermoelectric Effects
9 Semiconductors
	9.1 Homogeneous Bulk Semiconductors
	9.2 Impurity Levels
	9.3 Optical Processes in Semiconductors
		9.3.1 Angle-Resolved Photoemission Spectroscopy
	9.4 The p–n Junction
		9.4.1 Light-Emitting Diodes and Solar Cells
	9.5 Other Devices
		9.5.1 Metal–Oxide–Semiconductor Field-Effect Transistors (MOSFETs)
		9.5.2 Heterostructures
		9.5.3 Quantum Point Contact, Wire and Dot
	9.6 Notes and Further Reading
10 Non-local Transport in Mesoscopic Systems
	10.1 Introduction to Transport of Electron Waves
	10.2 Landauer Formula and Conductance Quantization
	10.3 Multi-terminal Devices
	10.4 Universal Conductance Fluctuations
		10.4.1 Transmission Eigenvalues
		10.4.2 UCF Fingerprints
	10.5 Noise in Mesoscopic Systems
		10.5.1 Quantum Shot Noise
	10.6 Dephasing
11 Anderson Localization
	11.1 Absence of Diffusion in Certain Random Lattices
	11.2 Classical Diffusion
	11.3 Semiclassical Diffusion
		11.3.1 Review of Scattering from a Single Impurity
		11.3.2 Scattering from Many Impurities
		11.3.3 Multiple Scattering and Classical Diffusion
	11.4 Quantum Corrections to Diffusion
		11.4.1 Real-Space Picture
		11.4.2 Enhanced Backscattering
	11.5 Weak Localization in 2D
		11.5.1 Magnetic Fields and Spin–Orbit Coupling
	11.6 Strong Localization in 1D
	11.7 Localization and Metal–Insulator Transition in 3D
	11.8 Scaling Theory of Localization and the Metal–Insulator Transition
		11.8.1 Thouless Picture of Conductance
		11.8.2 Persistent Currents in Disordered Mesoscopic Rings
		11.8.3 Scaling Theory
		11.8.4 Scaling Hypothesis and Universality
	11.9 Scaling and Transport at Finite Temperature
		11.9.1 Mobility Gap and Activated Transport
		11.9.2 Variable-Range Hopping
	11.10 Anderson Model
	11.11 Many-Body Localization
12 Integer Quantum Hall Effect
	12.1 Hall-Effect Transport in High Magnetic Fields
	12.2 Why 2D Is Important
	12.3 Why Disorder and Localization Are Important
	12.4 Classical and Semiclassical Dynamics
		12.4.1 Classical Dynamics
		12.4.2 Semiclassical Approximation
	12.5 Quantum Dynamics in Strong B Fields
	12.6 IQHE Edge States
	12.7 Semiclassical Percolation Picture of the IQHE
	12.8 Anomalous Integer Quantum Hall Sequence in Graphene
	12.9 Magnetic Translation Invariance and Magnetic Bloch Bands
		12.9.1 Simple Landau Gauge Example
	12.10 Quantization of the Hall Conductance in Magnetic Bloch Bands
13 Topology and Berry Phase
	13.1 Adiabatic Evolution and the Geometry of Hilbert Space
	13.2 Berry Phase and the Aharonov–Bohm Effect
	13.3 Spin-1/2 Berry Phase
		13.3.1 Spin–Orbit Coupling and Suppression of Weak Localization
	13.4 Berry Curvature of Bloch Bands and Anomalous Velocity
		13.4.1 Anomalous Velocity
	13.5 Topological Quantization of Hall Conductance of Magnetic Bloch Bands
		13.5.1 Wannier Functions of Topologically Non-trivial Bands
		13.5.2 Band Crossing and Change of Band Topology
		13.5.3 Relation Between the Chern Number and Chiral Edge States: Bulk–Edge Correspondence
	13.6 An Example of Bands Carrying Non-zero Chern Numbers: Haldane Model
	13.7 Thouless Charge Pump and Electric Polarization
		13.7.1 Modern Theory of Electric Polarization
14 Topological Insulators and Semimetals
	14.1 Kane–Mele Model
	14.2 Z2 Characterization of Topological Insulators
	14.3 Massless Dirac Surface/Interface States
	14.4 Weyl Semimetals
		14.4.1 Fermi Arcs on the Surface
		14.4.2 Chiral Anomaly
	14.5 Notes and Further Reading
15 Interacting Electrons
	15.1 Hartree Approximation
	15.2 Hartree–Fock Approximation
		15.2.1 Koopmans’ Theorem
	15.3 Hartree–Fock Approximation for the 3D Electron Gas
		15.3.1 Total Exchange Energy of the 3DEG in the Hartree–Fock Approximation
	15.4 Density Functional Theory
	15.5 Kohn–Sham Single-Particle Equations
	15.6 Local-Density Approximation
	15.7 Density–Density Response Function and Static Screening
		15.7.1 Thomas–Fermi Approximation
		15.7.2 Lindhard Approximation
	15.8 Dynamical Screening and Random-Phase Approximation
	15.9 Plasma Oscillation and Plasmon Dispersion
		15.9.1 Plasma Frequency and Plasmon Dispersion from the RPA
		15.9.2 Plasma Frequency from Classical Dynamics
		15.9.3 Plasma Frequency and Plasmon Dispersion from the Single-Mode Approximation
	15.10 Dielectric Function and Optical Properties
		15.10.1 Dielectric Function and AC Conductivity
		15.10.2 Optical Measurements of Dielectric Function
	15.11 Landau’s Fermi-Liquid Theory
		15.11.1 Elementary Excitations of a Free Fermi Gas
		15.11.2 Adiabaticity and Elementary Excitations of an Interacting Fermi Gas
		15.11.3 Fermi-Liquid Parameters
	15.12 Predictions of Fermi-Liquid Theory
		15.12.1 Heat Capacity
		15.12.2 Compressibility
		15.12.3 Spin Susceptibility
		15.12.4 Collective Modes, Dynamical and Transport Properties
	15.13 Instabilities of Fermi Liquids
		15.13.1 Ferromagnetic Instability
		15.13.2 Pomeranchuk Instabilities
		15.13.3 Pairing Instability
		15.13.4 Charge and Spin Density-Wave Instabilities
		15.13.5 One Dimension
		15.13.6 Two-Dimensional Electron Gas at High Magnetic Field
	15.14 Infrared Singularities in Fermi Liquids
		15.14.1 Perfect Screening and the Friedel Sum Rule
		15.14.2 Orthogonality Catastrophe
		15.14.3 Magnetic Impurities in Metals: The Kondo Problem
	15.15 Summary and Outlook
16 Fractional Quantum Hall Effect
	16.1 Landau Levels Revisited
	16.2 One-Body Basis States in Symmetric Gauge
	16.3 Two-Body Problem and Haldane Pseudopotentials
	16.4 The ν =1 Many-Body State and Plasma Analogy
		16.4.1 Electron and Hole Excitations at ν =1
	16.5 Laughlin’s Wave Function
	16.6 Quasiparticle and Quasihole Excitations of Laughlin States
	16.7 Fractional Statistics of Laughlin Quasiparticles
		16.7.1 Possibility of Fractional Statistics in 2D
		16.7.2 Physical Model of Anyons
		16.7.3 Statistics Angle of Laughlin Quasiholes
	16.8 Collective Excitations
	16.9 Bosonization and Fractional Quantum Hall Edge States
		16.9.1 Shot-Noise Measurement of Fractional Quasiparticle Charge
	16.10 Composite Fermions and Hierarchy States
		16.10.1 Another Take on Laughlin’s Wave Function
		16.10.2 Jain Sequences
	16.11 General Formalism of Electron Dynamics Confined to a Single Landau Level
		16.11.1 Finite-Size Geometries
	16.12 Relation between Fractional Statistics and Topological Degeneracy
	16.13 Notes and Further Reading
17 Magnetism
	17.1 Basics
	17.2 Classical Theory of Magnetism
	17.3 Quantum Theory of Magnetism of Individual Atoms
		17.3.1 Quantum Diamagnetism
		17.3.2 Quantum Paramagnetism
		17.3.3 Quantum Spin
	17.4 The Hubbard Model and Mott Insulators
	17.5 Magnetically Ordered States and Spin-Wave Excitations
		17.5.1 Ferromagnets
		17.5.2 Antiferromagnets
	17.6 One Dimension
		17.6.1 Lieb–Schultz–Mattis Theorem
		17.6.2 Spin-1/2 Chains
		17.6.3 Spin-1 Chains, Haldane Gap, and String Order
		17.6.4 Matrix Product and Tensor Network States
	17.7 Valence-Bond-Solid and Spin-Liquid States in 2D and Higher Dimensions
		17.7.1 Z2 Topological Order in Resonating Valence-Bond Spin Liquid
	17.8 An Exactly Solvable Model of Z2 Spin Liquid: Kitaev’s Toric Code
		17.8.1 Toric Code as Quantum Memory
	17.9 Landau Diamagnetism
18 Bose–Einstein Condensation and Superfluidity
	18.1 Non-interacting Bosons and Bose–Einstein Condensation
		18.1.1 Off-Diagonal Long-Range Order
		18.1.2 Finite Temperature and Effects of Trapping Potential
		18.1.3 Experimental Observation of Bose–Einstein Condensation
	18.2 Weakly Interacting Bosons and Bogoliubov Theory
	18.3 Stability of Condensate and Superfluidity
	18.4 Bose–Einstein Condensation of Exciton-Polaritons: Quantum Fluids of Light
19 Superconductivity: Basic Phenomena and Phenomenological Theories
	19.1 Thermodynamics
		19.1.1 Type-I Superconductors
		19.1.2 Type-II Superconductors
	19.2 Electrodynamics
	19.3 Meissner Kernel
	19.4 The Free-Energy Functional
	19.5 Ginzburg–Landau Theory
	19.6 Type-II Superconductors
		19.6.1 Abrikosov Vortex Lattice
		19.6.2 Isolated Vortices
	19.7 Why Do Superconductors Superconduct?
	19.8 Comparison between Superconductivity and Superfluidity
	19.9 Josephson Effect
		19.9.1 Superconducting Quantum Interference Devices (SQUIDS)
	19.10 Flux-Flow Resistance in Superconductors
	19.11 Superconducting Quantum Bits
20 Microscopic Theory of Superconductivity
	20.1 Origin of Attractive Interaction
	20.2 BCS Reduced Hamiltonian and Mean-Field Solution
		20.2.1 Condensation Energy
		20.2.2 Elementary Excitations
		20.2.3 Finite-Temperature Properties
	20.3 Microscopic Derivation of Josephson Coupling
	20.4 Electromagnetic Response of Superconductors
	20.5 BCS–BEC Crossover
	20.6 Real-Space Formulation and the Bogoliubov–de Gennes Equation
	20.7 Kitaev’s p-Wave Superconducting Chain and Topological Superconductors
	20.8 Unconventional Superconductors
		20.8.1 General Solution of Cooper Problem
		20.8.2 General Structure of Pairing Order Parameter
		20.8.3 Fulde–Ferrell–Larkin–Ovchinnikov States
	20.9 High-Temperature Cuprate Superconductors
		20.9.1 Antiferromagnetism in the Parent Compound
		20.9.2 Effects of Doping
		20.9.3 Nature of the Superconducting State
		20.9.4 Why d-Wave?
Appendix A. Linear-Response Theory
	A.1 Static Response
	A.2 Dynamical Response
	A.3 Causality, Spectral Densities, and Kramers–Kronig Relations
Appendix B. The Poisson Summation Formula
Appendix C. Tunneling and Scanning Tunneling Microscopy
	C.1 A Simple Example
	C.2 Tunnel Junction
	C.3 Scanning Tunneling Microscopy
Appendix D. Brief Primer on Topology
	D.1 Introduction
	D.2 Homeomorphism
	D.3 Homotopy
	D.4 Fundamental Group
	D.5 Gauss–Bonnet Theorem
	D.6 Topological Defects
Appendix E. Scattering Matrices, Unitarity, and Reciprocity
Appendix F. Quantum Entanglement in Condensed Matter Physics
	F.1 Reduced Density Matrix
	F.2 Schmidt and Singular-Value Decompositions
	F.3 Entanglement Entropy Scaling Laws
	F.4 Other Measures of Entanglement
	F.5 Closing Remarks
Appendix G. Linear Response and Noise in Electrical Circuits
	G.1 Classical Thermal Noise in a Resistor
	G.2 Linear Response of Electrical Circuits
	G.3 Hamiltonian Description of Electrical Circuits
		G.3.1 Hamiltonian for Josephson Junction Circuits
Appendix H. Functional Differentiation
Appendix I. Low-Energy Effective Hamiltonians
	I.1 Effective Tunneling Hamiltonian
	I.2 Antiferromagnetism in the Hubbard Model
	I.3 Summary
Appendix J. Introduction to Second Quantization
	J.1 Second Quantization
	J.2 Majorana Representation of Fermion Operators
References
Index