# 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 $k \cdot p$ 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.

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