Many-Particle Physics

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.

1. Introductory Material.- 1.1. Harmonic Oscillators and Phonons.- 1.2. Second Quantization for Particles.- 1.3. Electron - Phonon Interactions.- 1.4. Spin Hamiltonians.- 1.5. Photons.- 1.6. Pair Distribution Function.- Problems.- 2. Green's Functions at Zero Temperature.- 2.1. Interaction Representation.- 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.- 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.- 3.7. Real Time Green's Functions.- Wigner Distribution Function.- 3.8. Kubo Formula for Electrical Conductivity.- 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.- 4.2. Localized State in the Continuum.- 4.3. Independent Boson Models.- 4.4. Tomonaga Model.- 4.5. Polaritons.- Problems.- 5. Electron Gas.- 5.1. Exchange and Correlation.- 5.2. Wigner Lattice and Metallic Hydrogen.- Metallic Hydrogen.- 5.3. Cohesive Energy of Metals.- 5.4. Linear Screening.- 5.5. Model Dielectric Functions.- 5.6. Properties of the Electron Gas.- 5.7. Sum Rules.- 5.8. One-Electron Properties.- Problems.- 6. Electron-Phonon Interaction.- 6.1 Fröhlich Hamiltonian.- 6.2 Small Polaron Theory.- 6.3 Heavily Doped Semiconductors.- 6.4 Metals.- Problems.- 7. dc Conductivities.- 7.1. Electron Scattering by Impurities.- 7.2. Mobility of FröhlichPolarons.- 7.3. Electron-Phonon Interactions in Metals.- 7.4. Quantum Boltzmann Equation.- Problems.- 8. Optical Properties of Solids.- 8.1. Nearly Free-Electron System.- 8.2. Wannier Excitons.- 8.3. X-Ray Spectra in Metals.- Problems.- 9. Superconductivity.- 9.1. Cooper Instability.- 9.2. BCS Theory.- 9.3. Electron 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.- 10.2. 4He: Ground State Properties.- 10.3. 4He: Excitation Spectrum.- 10.4. 3He: Normal Liquid.- 10.5. Superfluid 3He.- Problems.- 11. Spin Fluctuations.- 11.1. Kondo Model.- 11.2. Anderson Model.- Problems.- References.- Author Index.