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This textbook is for a course in advanced solid-state theory. It is aimed at graduate students in their third or fourth year of study who wish to learn the advanced techniques of solid-state theoretical physics. The method of Green's functions is introduced at the beginning and used throughout. Indeed, it could be considered a book on practical applications of Green's functions, although I prefer to call it a book on physics. The method of Green's functions has been used by many theorists to derive equations which, when solved, provide an accurate numerical description of many processes in solids and quantum fluids. In this book I attempt to summarize many of these theories in order to show how Green's functions are used to solve real problems. My goal, in writing each section, is to describe calculations which can be compared with experiments and to provide these comparisons whenever available. The student is expected to have a background in quantum mechanics at the level acquired from a graduate course using the textbook by either L. I. Schiff, A. S. Davydov, or I. Landau and E. M. Lifshiftz. Similarly, a prior course in solid-state physics is expected, since the reader is assumed to know concepts such as Brillouin zones and energy band theory. Each chapter has problems which are an important part of the lesson; the problems often provide physical insights which are not in the text. Sometimes the answers to the problems are provided, but usually not.

Many-Particle Physics

1. Introductory Material.- 1.1. Harmonic Oscillators and Phonons.- 1.2. Second Quantization for Particles.- 1.3. Electron - Phonon Interactions.- A. Interaction Hamiltonian.- B. Localized Electron.- C. Deformation Potential.- D. Piezoelectric Interaction.- E. Polar Coupling.- 1.4. Spin Hamiltonians.- A. Homogeneous Spin Systems.- B. Impurity Spin Models.- 1.5. Photons.- A. Gauges.- B. Lagrangian.- C. Hamiltonian.- 1.6. Pair Distribution Function.- Problems.- 2. Green's Functions at Zero Temperature.- 2.1. Interaction Representation.- A. Schrödinger.- B. Heisenberg.- C. Interaction.- 2.2. S Matrix.- 2.3. Green's Functions.- 2.4. Wick's Theorem.- 2.5. Feynman Diagrams.- 2.6. Vacuum Polarization Graphs.- 2.7. Dyson's Equation.- 2.8. Rules for Constructing Diagrams.- 2.9. Time-Loop S Matrix.- A. Six Green's Functions.- B. Dyson's Equation.- 2.10. Photon Green's Functions.- Problems.- 3. Green's Functions at Finite Temperatures.- 3.1. Introduction.- 3.2. Matsubara Green's Functions.- 3.3. Retarded and Advanced Green's Functions.- 3.4. Dyson's Equation.- 3.5. Frequency Summations.- 3.6. Linked Cluster Expansions.- A. Thermodynamic Potential.- B. Green's Functions.- 3.7. Real Time Green's Functions.- Wigner Distribution Function.- 3.8. Kubo Formula for Electrical Conductivity.- A. Transverse Fields, Zero Temperature.- B. Finite Temperatures.- C. Zero Frequency.- D. Photon Self-Energy.- 3.9. Other Kubo Formulas.- A. Pauli Paramagnetic Susceptibility.- B. Thermal Currents and Onsager Relations.- C. Correlation Functions.- Problems.- 4. Exactly Solvable Models.- 4.1. Potential Scattering.- A. Reaction Matrix.- B. T Matrix.- C. Friedel's Theorem.- D. Phase Shifts.- E. Impurity Scattering.- F. Ground State Energy.- 4.2. Localized State in the Continuum.- 4.3. Independent Boson Models.- A. Solution by Canonical Transformation.- B. Feynman Disentangling of Operators.- C. Einstein Model.- D. Optical Absorption and Emission.- E. Sudden Switching.- F. Linked Cluster Expansion.- 4.4. Tomonaga Model.- A. Tomonaga Model.- B. Spin Waves.- C. Luttinger Model.- D. Single-Particle Properties.- E. Interacting System of Spinless Fermions.- F. Electron Exchange.- 4.5. Polaritons.- A. Semiclassical Discussion.- B. Phonon-Photon Coupling.- C. Exciton-Photon Coupling.- Problems.- 5. Electron Gas.- 5.1. Exchange and Correlation.- A. Kinetic Energy.- B. Direct Coulomb.- C. Exchange.- D. Seitz' Theorem.- E. ?(2a).- F. ?(2b).- G. ?(2c).- H. High-Density Limit.- I. Pair Distribution Function.- 5.2. Wigner Lattice and Metallic Hydrogen.- Metallic Hydrogen.- 5.3. Cohesive Energy of Metals.- 5.4. Linear Screening.- 5.5. Model Dielectric Functions.- A. Thomas-Fermi.- B. Lindhard, or RPA.- C. Hubbard.- D. Singwi-Sjölander.- 5.6. Properties of the Electron Gas.- A. Pair Distribution Function.- B. Screening Charge.- C. Correlation Energies.- D. Compressibility.- 5.7. Sum Rules.- 5.8. One-Electron Properties.- A. Renormalization Constant ZF.- B. Effective Mass.- C. Pauli Paramagnetic Susceptibility.- D. Mean Free Path.- Problems.- 6. Electron-Phonon Interaction.- 6.1 Fröhlich Hamiltonian.- A. Brillouin-Wigner Perturbation Theory.- B. Rayleigh-Schrödinger Perturbation Theory.- C. Strong Coupling Theory.- D. Linked Cluster Theory.- 6.2 Small Polaron Theory.- A. Large Polarons.- B. Small Polarons.- C. Diagonal Transitions.- D. Nondiagonal Transitions.- E. Dispersive Phonons.- F. Einstein Model.- G. Kubo Formula.- 6.3 Heavily Doped Semiconductors.- A. Screened Interaction.- B. Experimental Verifications.- C. Electron Self-Energies.- 6.4 Metals.- A. Phonons in Metals.- B. Electron Self-Energies.- Problems.- 7. dc Conductivities.- 7.1. Electron Scattering by Impurities.- A. Boltzmann Equation.- B. Kubo Formula: Approximate Solution.- C. Kubo Formula: Rigorous Solution.- D. Ward Identities.- 7.2. Mobility of Fröhlich Polarons.- A. Single-Particle Properties.- B. ??1 Term in the Mobility.- 7.3. Electron-Phonon Interactions in Metals.- A. Force-Force Correlation Function.- B. Kubo Formula.- C. Mass Enhancement.- D. Thermoelectric Power.- 7.4. Quantum Boltzmann Equation.- A. Derivation of the Quantum Boltzmann Equation.- B. Gradient Expansion.- C. Electron Scattering by Impurities.- D. T2 Contribution to the Electrical Resistivity.- Problems.- 8. Optical Properties of Solids.- 8.1. Nearly Free-Electron System.- A. General Properties.- B. Force-Force Correlation Functions.- C. Fröhlich Polarons.- D. Interband Transitions.- E. Phonons.- 8.2. Wannier Excitons.- A. The Model.- B. Solution by Green's Functions.- C. Core-Level Spectra.- 8.3. X-Ray Spectra in Metals.- A. Physical Model.- B. Edge Singularities.- C. Orthogonality Catastrophe.- D. MND Theory.- E. XPS Spectra.- Problems.- 9. Superconductivity.- 9.1. Cooper Instability.- 9.2. BCS Theory.- 9.3. Electron Tunneling.- A. Tunneling Hamiltonian.- B. Normal Metals.- C. Normal-Superconductor.- D. Two Superconductors.- E. Josephson Tunneling.- 9.4. Infrared Absorption.- 9.5. Acoustic Attenuation.- 9.6. Excitons in Superconductors.- 9.7. Strong Coupling Theory.- Problems.- 10. Liquid Helium.- 10.1. Pairing Theory.- A. Hartree and Exchange.- B. Bogoliubov Theory of 4He.- 10.2. 4He: Ground State Properties.- A. Off-Diagonal Long-Range Order.- B. Correlated Basis Functions.- C. Experiments on nk.- 10.3. 4He: Excitation Spectrum.- A. Bijl-Feynman Theory.- B. Improved Excitation Spectra.- C. Superfluidity.- 10.4. 3He: Normal Liquid.- A. Fermi Liquid Theory.- B. Experiments and Microscopic Theories.- C. Interaction between Quasiparticles: Excitations.- D. Quasiparticle Transport.- 10.5. Superfluid 3He.- A. Triplet Pairing.- B. Equal Spin Pairing.- Problems.- 11. Spin Fluctuations.- 11.1. Kondo Model.- A. High-Temperature Scattering.- B. Low-Temperature State.- C. Kondo Temperature.- 11.2. Anderson Model.- A. Collective States.- B. Green's Functions.- C. Spectroscopies.- Problems.- References.- Author Index.