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Lectures on the Quantum Hall Effect

David Tong

TL;DR

This work surveys how topology and geometry underpin the quantum Hall effect, connecting microscopic wavefunctions with long-wavelength effective theories and edge physics. It builds from Landau level quantization and Berry phases to the integer quantized Hall effect, highlighting the Kubo formalism and TKNN/Chern-number invariants as topological anchors. It then extends to the fractional regime through Laughlin states, plasma analogies, and toy models, illustrating how interactions give rise to topological order and anyonic excitations. Finally, it discusses lattice realizations such as Chern insulators and the Hofstadter problem, demonstrating the universality of topological transport across continuum and lattice systems and the central role of edge modes in robust quantized conductance.

Abstract

The purpose of these lectures is to describe the basic theoretical structures underlying the rich and beautiful physics of the quantum Hall effect. The focus is on the interplay between microscopic wavefunctions, long-distance effective Chern-Simons theories, and the modes which live on the boundary. The notes are aimed at graduate students in any discipline where $\hbar=1$. A working knowledge of quantum field theory is assumed. Contents: 1. The Basics (Landau levels and Berry phase). 2. The Integer Quantum Hall Effect. 3. The Fractional Quantum Hall Effect. 4. Non-Abelian Quantum Hall States. 5. Chern-Simons Theories. 6. Edge Modes.

Lectures on the Quantum Hall Effect

TL;DR

This work surveys how topology and geometry underpin the quantum Hall effect, connecting microscopic wavefunctions with long-wavelength effective theories and edge physics. It builds from Landau level quantization and Berry phases to the integer quantized Hall effect, highlighting the Kubo formalism and TKNN/Chern-number invariants as topological anchors. It then extends to the fractional regime through Laughlin states, plasma analogies, and toy models, illustrating how interactions give rise to topological order and anyonic excitations. Finally, it discusses lattice realizations such as Chern insulators and the Hofstadter problem, demonstrating the universality of topological transport across continuum and lattice systems and the central role of edge modes in robust quantized conductance.

Abstract

The purpose of these lectures is to describe the basic theoretical structures underlying the rich and beautiful physics of the quantum Hall effect. The focus is on the interplay between microscopic wavefunctions, long-distance effective Chern-Simons theories, and the modes which live on the boundary. The notes are aimed at graduate students in any discipline where . A working knowledge of quantum field theory is assumed. Contents: 1. The Basics (Landau levels and Berry phase). 2. The Integer Quantum Hall Effect. 3. The Fractional Quantum Hall Effect. 4. Non-Abelian Quantum Hall States. 5. Chern-Simons Theories. 6. Edge Modes.

Paper Structure

This paper contains 89 sections, 727 equations, 4 figures.

Table of Contents

  1. The Basics
  2. Introduction
  3. The Classical Hall Effect
  4. Classical Motion in a Magnetic Field
  5. The Drude Model
  6. Quantum Hall Effects
  7. Integer Quantum Hall Effect
  8. Fractional Quantum Hall Effect
  9. Landau Levels
  10. Landau Gauge
  11. Turning on an Electric Field
  12. Symmetric Gauge
  13. Berry Phase
  14. Abelian Berry Phase and Berry Connection
  15. An Example: A Spin in a Magnetic Field
  16. ...and 74 more sections

Figures (4)

  • Figure 1: The spectral flow for the energy states of a particle moving around a solenoid.
  • Figure 2: The map from Brillouin zone to Bloch sphere
  • Figure 3: The Hofstadter butterfly
  • Figure 4: The composite fermion picture describes a hierarchy of plateaux around, starting with $\nu=1/3$, in terms of the integer quantum Hall effect for electrons bound to two vortices.