Theoretical Solid-State Physics Course, University of Basel, Fall Semester 2012

Lecturer: Dr. Vladimir M. Stojanović

Course Description

This course covers basic many-body theory of condensed-matter systems. It is intended for master students and requires knowledge of quantum mechanics at an advanced undergraduate level, as well as familiarity with basic concepts of solid-state physics. Approximately half ot the lecture material is based on the book "Many-body quantum theory in condensed matter physics" by H. Bruus and K. Flensberg, while the rest is contained in the notes prepared by the lecturer.

Problem Sessions

Teaching Assistants: Patrick Hofer, Samuel Aldana, Dr. Gerson Ferreira, Dr. Rakesh Tiwari

Problem Sheets (in PDF format)

Problem Sheet 0 ( Hellman-Feynman theorem and its application to Bloch electrons )
Problem Sheet 1 ( Useful operator identities; Fermionic Bogoliubov transformation )
Problem Sheet 2 ( Tight-binding electrons on a checkerboard lattice; Critical points in the density-of-states )
Problem Sheet 3 ( Spin-polarized electron gas; Second-order perturbation theory for the electron gas )
Problem Sheet 4 ( Spin-orbit interaction in the 2DEG; Tight-binding model with Rashba spin-orbit coupling )
Problem Sheet 5 ( Spectral function for bosons; Single level coupled to the continuum )
Problem Sheet 6 ( Green's function for topological insulators; Momentum dependence of electron-phonon coupling )
Problem Sheet 7 ( Properties of thermal two-time correlation functions; Quantum diffusion formalism and optical conductivity )
Problem Sheet 8 ( Zero-sound collective mode in charge-neutral Fermi gases; Plasmon dispersion in an interacting electron gas )
Problem Sheet 9 ( Excitations in a two-dimensional electron gas )
Problem Sheet 10 ( Hubbard sectors; From the t-J model to the Heisenberg model; Holstein-Primakoff transformation )

Lecture Notes

  • Electrons in periodic potentials; Lattice models  
  • The electron gas  
  • Spin-orbit coupling  
  • Self-consistent electron-nuclear dynamics in solids  
  • Lattice dynamics in solids  
  • Electron-phonon coupling  
  • Two-particle correlation functions: example of the Lindhard polarization function  
  • RPA and polarization function of an interacting electron gas  
  • Zero-sound collective mode in charge-neutral Fermi gases  
  • Plasmons, screening, and Friedel oscillations  
  • Strongly-correlated systems: the Hubbard model  
  • The Hubbard model at half-filling and the Mott-Hubbard insulators  
  • Ferromagnetic and antiferromagnetic orders: similarities and differences, low-energy excitations  
  • Spin-wave quantization and quantum fluctuations in the Neel state  
  • Spin ordering at weak coupling: spin density waves  

    Tentative Course Outline

    I. Introduction
    • Second quantization
    • Electrons in periodic potentials
    • Lattice models
    II. The electron gas
    • Jellium approximation; non-interacting electrons in the jellium model
    • Electron-electron interactions in Rayleigh-Schroedinger perturbation theory
    • Spin-polarized electron gas and its region of stability
    • Failure of second-order perturbation theory
    III. Spin-orbit interaction
    • Physical origins of the spin-orbit interaction; implications for the bulk band structure of semiconductors within the framework of the method
    • Two-dimensional electron gas (2DEG)
    • Rashba and Dresselhaus-type spin-orbit interactions
    IV. Green's functions
    • Green's function for the one-particle Schroedinger equation
    • Single-particle Green's functions for many-body system
    • Equation-of-motion theory for Green's functions
    V. Phonons and electron-phonon interaction
    • Born-Oppenheimer approximation; the self-consistent electron-nuclear problem
    • Lattice dynamics in the discrete (atomistic) model; quantization into phonons; acoustic and optical phonon modes
    • Non-adiabatic corrections: electron-phonon coupling; inelastic scattering rates
    • Polaron: the concept and generic features
    VI. Response functions with applications to the electron gas
    • The general Kubo linear-response formalism; Kubo formula for the dielectric function
    • Lindhard's polarization function for a non-interacting electron gas
    • The random phase approximation (RPA): example of the polarization function of an interacting Fermi gas
    • Zero-sound collective mode
    • Plasmon mode in Fermi systems with Coulomb interaction
    • Static screening in an interacting electron gas; Friedel oscillations
    VII. Fields, broken symmetry, and collective properties
    • Notion of the continuum and fields
    • Long-wavelength modes: example of lattice dynamics in the continuum approach
    • Broken continuous symmetry and Goldstone modes
    VIII. Interacting electron systems in different dimensions
    • Three dimensions: Fermi liquid theory
    • Microscopic basis of Fermi liquid theory
    • Interacting electrons in one dimension
    • The spinless Luttinger-Tomonaga model
    IX. Strongly correlated systems
    • Examples of strongly-correlated electron systems; the Hubbard model
    • The Hubbard model at half-filling and the Mott-Hubbard insulators
    • Ferromagnetic and antiferromagnetic orders: similarities and differences, low-energy excitations
    • Quantization of spin waves: the Holstein-Primakoff transformation; quantum fluctuations in the Neel state
    • Spin ordering at weak coupling: spin density waves

    Useful literature for further reading:

  • Band structure: G. Grosso and G.P. Parravicini, Solid State Physics, Academic Press, 2000.  
  • The electron gas: A. L. Fetter, J. D. Walecka, Quantum Theory of Many-Particle Systems, Courier Dover Publications, 1971.  
  • Spin-orbit interaction: T. Ihn, Semiconductor Nanostructures: Quantum States and Electronic Transport, Oxford University Press, 2010.  
  • Electron-phonon interaction: G. D. Mahan, Many Particle Physics, Plenum Publishers, 2000.  
  • Fermi-liquid theory: A.A. Abrikosov, L.P. Gorkov, and I. Dzyaloshinskii, Methods of Quantum Field Theory in Statistical Physics, Prentice-Hall, 1963.  
  • Interacting electrons in one-dimensional systems: T. Giamarchi, Quantum Physics in One Dimension, Oxford University Press, 2003.  
  • Strongly-correlated electron systems: P. Fazekas, Lecture Notes on Electron Correlation and Magnetism, World Scientific, 1999.