Condensed Matter Theory

A rigorous graduate-level course on condensed matter theory โ€” from Landau Fermi liquid theory through Bose-Einstein condensation, superfluidity, BCS superconductivity, Ginzburg-Landau theory, and topological phases of matter โ€” with full derivations, MathJax equations, and Python/Fortran simulations.

Course Overview

Condensed matter physics studies the emergent properties of macroscopic quantum systems. When $10^{23}$ particles interact through quantum mechanics, entirely new phenomena arise โ€” superfluidity, superconductivity, topological order โ€” that have no analogue in single-particle physics. This course develops the theoretical framework for understanding these collective quantum phenomena, following the tradition of Abrikosov, Gor'kov & Dzyaloshinski, Fetter & Walecka, and Altland & Simons.

What You Will Learn

  • - Landau Fermi liquid theory and quasiparticles
  • - Bose-Einstein condensation and superfluidity
  • - Cooper pairing and BCS superconductivity
  • - Ginzburg-Landau phenomenology
  • - Type-II superconductors and vortex lattices
  • - Josephson effects and macroscopic quantum tunneling
  • - Berry phase and band topology
  • - Quantum Hall effect and topological insulators

Prerequisites

Key Constants

  • $k_B = 1.381 \times 10^{-23}$ J/K
  • $\hbar = 1.055 \times 10^{-34}$ Jยทs
  • $m_e = 9.109 \times 10^{-31}$ kg
  • $e = 1.602 \times 10^{-19}$ C

Part I: Fermi Liquid Theory

Landau's revolutionary insight: interacting fermions can be described as a gas of quasiparticles โ€” long-lived excitations near the Fermi surface that carry the same quantum numbers as free electrons but with renormalized properties.

1. Free Fermi Gas2. Quasiparticles3. Landau Fermi Liquid Theory4. Landau Parameters5. Zero Sound & Collective Modes

Part II: Bose-Einstein Condensation & Superfluidity

Below a critical temperature, bosons macroscopically occupy the ground state. In interacting systems this produces superfluidity โ€” flow without viscosity, quantized vortices, and a two-fluid hydrodynamics described by the Gross-Pitaevskii equation.

1. Bose-Einstein Condensation2. Superfluid Helium3. Two-Fluid Model4. Gross-Pitaevskii Equation5. Quantized Vortices

Part III: Superconductivity

Cooper pairing of electrons near the Fermi surface opens a gap in the excitation spectrum, producing zero electrical resistance and the Meissner effect. BCS theory provides the microscopic foundation; Ginzburg-Landau theory captures the macroscopic phenomenology.

1. Cooper Pairs2. BCS Theory3. Ginzburg-Landau Theory4. Type-II Superconductors5. Josephson Effect

Part IV: Topological Phases

Beyond symmetry breaking: topological invariants classify quantum phases of matter. The Berry phase, Chern numbers, and Zโ‚‚ indices explain the quantum Hall effect, topological insulators, and protected edge states.

1. Berry Phase in Solids2. Quantum Hall Effect3. Topological Insulators

Video Lectures: Quantum Matter Series (22 lectures)

A curated series of 22 lectures covering Fermi liquids, BEC, superfluidity, BCS superconductivity, and topological phases โ€” organized to align with the course structure.

Watch Video Lectures โ†’

Central Equations

Landau Energy Functional

$$E[\delta n_{\mathbf{k}\sigma}] = \sum_{\mathbf{k}\sigma} \epsilon_{\mathbf{k}} \delta n_{\mathbf{k}\sigma} + \frac{1}{2V}\sum_{\mathbf{k}\mathbf{k}'} f_{\mathbf{k}\mathbf{k}'} \delta n_{\mathbf{k}} \delta n_{\mathbf{k}'}$$

Gross-Pitaevskii Equation

$$i\hbar \frac{\partial \Psi}{\partial t} = \left(-\frac{\hbar^2}{2m}\nabla^2 + V_{\text{ext}} + g|\Psi|^2\right)\Psi$$

BCS Gap Equation

$$\Delta_{\mathbf{k}} = -\sum_{\mathbf{k}'} V_{\mathbf{k}\mathbf{k}'} \frac{\Delta_{\mathbf{k}'}}{2E_{\mathbf{k}'}} \tanh\frac{E_{\mathbf{k}'}}{2k_BT}$$

Berry Phase

$$\gamma_n = \oint \langle n(\mathbf{R})| \nabla_{\mathbf{R}} |n(\mathbf{R})\rangle \cdot d\mathbf{R}$$