The Physics of Elementary Excitations

I Families of Elementary Excitations.- 1. Crystals and Phonons.- 1.1 Order and Elementary Excitations.- 1.2 One-Dimensional Model.- 1.2.1 One-Dimensional Lattice.- 1.2.2 Lattice Vibration.- 1.2.3 Diatomic Crystals.- 1.3 Three-Dimensional Crystals.- 1.3.1 Lattice and Reciprocal Lattice.- 1.3.2 The Hamiltonian of the Harmonic Approximation.- 1.3.3 Periodic Crystals.- 1.4 Quantization of Lattice Vibrations.- 1.4.1 Phonons.- 1.4.2 Specific Heat of the Phonon Gas.- 1.4.3 Creation and Destruction Operators.- 1.4.4 Equations of Motion.- 1.5 Mössbauer Effect (Rigidity of Solids).- 1.5.1 Spectrum of Recoil Energy.- 1.5.2 Theorem of Bloch and De Dominicis.- 1.5.3 The Intensity of the Recoilless ?-Ray.- 1.6 Inelastic Scattering of Neutrons and Phonon Spectrum.- 1.6.1 Van Hove’s Formula.- 1.6.2 Dynamical Structure Factors in the Harmonic Approximation.- 1.7 Anharmonic Terms.- 1.7.1 General Definition of the Spectral Function.- 1.7.2 Retarded Green’s Functions.- 1.8 Temperature Green’s Functions and Perturbation Expansion.- 1.8.1 Thermal Green’s Functions.- 1.8.2 Pertubation Expansions.- 1.8.3 The Phonon Self-Energy.- 1.9 QuantumSolids.- 1.9.1 Nuclear Magnetism of Solid 3He.- 1.9.2 Defects in Quantum Solids.- 1.9.3 Self-Consistent Phonons.- 2. Polarization Waves and Dielectric Dispersion.- 2.1 Optical Lattice Vibrations and Dielectric Dispersion.- 2.1.1 Incorporation of Long-Range Interionic Forces into the Macroscopic Electric Field.- 2.1.2 Dielectric Dispersion.- 2.1.3 Lattice Vibrations in the Long-Wavelength Limit.- 2.2 Polarizability and Dielectric Constant.- 2.2.1 General Formula for Polarizability.- 2.2.2 The Relation between Polarizability and Dielectric Constant.- 2.2.3 Applications to Optical Lattice Vibrations.- 2.2.4 Plasma Oscillation and Screening Effect in Elerctron Gas.- 2.2.5 Absorption of Energy by Dielectrics.- 2.3 Exciton.- 2.3.1 Frenkel Exciton.- 2.3.2 Wannier-Mott Exciton.- 2.3.3 Excited States of the Many-Electron System.- 2.4 Excitons in the Optical Spectra.- 2.4.1 Fundamental Absorption Spectra.- 2.4.2 Spin-Orbit vs Exchange Interactions.- 2.4.3 The Observation of Translational Motion.- 2.4.4 Excitonic Molecule.- 2.4.5 Fission and Fusion of Excitions.- 3. Fermi Liquids.- 3.1 Models of Fermi Liquids.- 3.1.1 Hamiltonian of the System of Fermi Particles.- 3.1.2 The Electron-Gas Model.- 3.1.3 The Exchange Energy of the Electron Gas.- 3.1.4 rs Expansion.- 3.1.5 Systems with Short-Range Force.- 3.2 Stimulus to a Many-Body System and Its Response.- 3.2.1 Schrödinger Equation in the Presence of External Field.- 3.2.2 Linear Responses.- 3.2.3 Retarded and Temperature Green’s Functions.- 3.2.4 The Case of the Grand Canonical Distribution.- 3.3 The Electron Gas.- 3.3.1 Test Charge as the Electric Field.- 3.3.2 Dielectric Constants.- 3.3.3 The Correlation Energy.- 3.3.4 Dynamic Structure Factors.- 3.4 Individual Excitation and Collective Excitation.- 3.4.1 Density Fluctuation Due to the External Field.- 3.4.2 The Zeroth-Order Approximation for the Retarded Green’s Function.- 3.4.3 Individual Excitation and Collective Excitation.- 3.4.4 Plasma Oscillation.- 3.4.5 ZeroSound.- 3.5 General Property of Fermi Liquid.- 3.5.1 Energy of the Quasiparticle.- 3.5.2 Lifetime of the Quasiparticle.- 3.5.3 Existence of the Fermi Surface. Specific Heat and Magnetic Susceptibility at Low Temperatures.- 3.5.4 Dilute Solution of 3He in Liquid 4He.- 4. Phase Transitions and Elementary Excitations.- 4.1 Phase Transition and Broken Symmetry.- 4.2 Order Parameters.- 4.3 Magnons.- 4.3.1 Magnons.- 4.3.2 Spin Wave Approximation.- 4.3.3 Antiferromagnets.- 4.4 Hilbert Space of the Macroscopic Systems and Coherent States.- 4.4.1 HilbertSpace.- 4.4.2 Condensation of Magnons.- 4.4.3 Coherent States.- 4.5 Coherence of de Broglie Wave and Superfluidity.- 4.5.1 Coherent States of the de Broglie Wave.- 4.5.2 Ginzburg-Landau Theory of Superconductivity.- 4.5.3 Josephson Effect.- 4.6 Broken Symmetry and Elementary Excitation.- 4.6.1 The Heisenberg Ferromagnet.- 4.6.2 The Spin Model of Liquid 4He.- 4.6.3 Classical Crystals.- 4.7 Goldstone’s Theorem.- 4.7.1 Conditions for the Theorem to Apply.- 4.7.2 The Case of Superconductivity.- 4.8 SoftModes.- 4.8.1 Ferroelectrics with Hydrogen Bonds.- 4.8.2 Soft Mode and Central Peak.- 4.9 Mean Field Approximation.- 4.9.1 Stoner’s Model of Ferromagnetic Metals.- 4.9.2 BCS Model of Superconductors.- 4.9.3 Excitonic States.- 4.9.4 Electron-Hole Metals.- 4.10 Fluctuations.- 4.10.1 Low-Dimensional Systems.- 4.10.2 Critical Phenomena.- 4.10.3 Superconductor and Superfluid 3He.- 4.10.4 Ferromagnetic Metals.- II Interaction Between Elementary Excitations.- 5. Linear Interactions and Coupled Modes.- 5.1 Linear Interaction.- 5.2 Carrier Plasma Coupled to the Optical Mode of Lattice Vibrations in Polar Semiconductors.- 5.3 The Plasma Model of Metal.- 5.4 Polariton.- 5.4.1 Polariton and Dielectric Dispersion.- 5.4.2 Spatial Dispersion and Optical Processes.- 6. Renormalization and Damping — Centering Around Electron-Phonon Interaction.- 6.1 Electron-Phonon Interaction in an Ionic Crystal.- 6.1.1 Optical Lattice Vibration in the Presence of an Electron.- 6.1.2 Electron-Phonon Interaction.- 6.2 Polaron.- 6.2.1 Renormalization of Mass (Perturbation Calculation of SecondOrder).- 6.2.2 PhononCloud.- 6.2.3 Damping.- 6.2.4 Numerical Values of ?.- 6.3 Intermediate Coupling Method and Method of Path Integral.- 6.3.1 Intermediate Coupling Method.- 6.3.2 Pathlntegral.- 6.3.3 Elimination of Phonon Variables.- 6.3.4 Feynman’s Variational Principle.- 6.3.5 Application to the Polaron.- 6.4 Electron-Phonon Interaction in Metals.- 6.4.1 Hamiltonian.- 6.4.2 Electron Self-Energy.- 6.5 Temperature Green’s Function and Spectral Function.- 6.6 Pertubation Expansion and Partial Summation.- 6.6.1 Diagrams and Rules of Calculation.- 6.6.2 Self-Energy.- 6.7 Migdal Approximation and Electron Self-Energy.- 6.7.1 Migdal Approximation.- 6.7.2 One-Electron Spectral Function.- 6.73 The Solution of Dyson’s Equation.- 6.7.4 Limitation of the Quasiparticle Picture.- 6.8 Electron-Phonon Interaction and Superconductivity.- 6.8.1 Divergence of the Vertex Function.- 6.8.2 Nambu Representation.- 7. Interaction Between Elementary Excitations and Spectral Line Shapes.- 7.1 What Happens with Nonlinear Interactions?.- 7.2 The Absorption and Emission Spectra of a Localized Electron in the Phonon Field.- 7.2.1 A Variety of Localized Electrons.- 7.2.2 The Generating Function for the Optical Spectra and Their Moments.- 7.2.3 A Model Calculation of the Generating Function.- 7.2.4 Phonon Sidebands and Zero-Phonon Line.- 7.2.5 Strong-Coupling Limit and Configuration-Coordinate Model.- 7.2.6 A Model Calculation of Coupling Strength.- 7.2.7 The Effect of Curvature Difference in the Adiabatic Potentials.- 7.3 Excition-Phonon Interaction and Fundamental Absorption Spectra.- 7.3.1 The Generating Function for the Fundamental Absorption Spectra.- 7.3.2 Spectral Narrowing Due to the Translational Motion of the Exciton.- 7.3.3 Direct and Indirect Transitons with Their Interference.- 7.3.4 Renormalization of Exciton-Phonon Interaction.- 7.3.5 Phonon Structures in the Absorption Spectra.- 7.4 Final-State Interaction.- 7.4.1 Exciton-Phonon Bound State.- 7.4.2 The Edge Anomalies in the Soft x-Ray Absorption Spectra of Metals.- 7.5 Self-Trapping.- 7.5.1 Polaronvs Self-Trapped Electron.- 7.5.2 Free Exciton vs Self-Trapped Excitons.- 7.5.3 The Electron Bubble and Exciton Bubble in Liquid Helium.