The Physics of Elementary Excitations

This book is an introduction to the physics of elementary excitations in condensed matter with emphasis on basic concepts and their mathematical representations. The nature of the book is mainly determined by the fact that it was originally written, in Japanese, as one volume of Iwanami Series of Fundamental Physics supervised by Professor H. Yukawa. Our task was to portray the theory of condensed matter from a unified point of view for the student looking for his own research field and also for more senior readers interested in fundamentals of contemporary physics. As our point of view, we chose the concept of elementary excitation, which we believe to be one of the most fruitful concepts discovered by the quantum theory of matter. The present English edition has been translated by the authors themselves from the second, revised Japanese edition published in 1978, six years after publication of the first edition. In translating, we have introduced no major modifications; only the list of references has been made more suitable to overseas readers. In the English as well as in the Japanese editions, Chaps. 1,4, and part of 6 were written by Nakajima, Chaps. 2, 5, and 7 by Toyozawa, and Chaps. 3 and part of 6 by Abe. Finally we should like to thank Professor P. Fulde for kind help and Dr. H. Lotsch, SpriIiger-Verlag, for patient cooperation in making this English edition a reality.

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.