[Paper Review] Three Lectures On Topological Phases Of Matter

Edward Witten|arXiv (Cornell University)|Oct 26, 2015

Topological Materials and PhenomenaPhysics and Astronomy32 references168 citations

TL;DR

This paper provides a pedagogical introduction to topological phases of matter using free fermion band theory and effective field theory, focusing on relativistic low-energy excitations in crystals, Weyl and Dirac fermions, and the quantum Hall effect. It demonstrates how topological invariants—such as Chern numbers and Berry phases—emerge from band structure, leading to robust gapless edge states and quantized transport, with key results including the stability of Dirac points in graphene under symmetry-preserving perturbations and the emergence of the integer and fractional quantum Hall effects from Chern-Simons field theory.

ABSTRACT

These notes are based on lectures at the PSSCMP/PiTP summer school that was held at Princeton University and the Institute for Advanced Study in July, 2015. They are devoted largely to topological phases of matter that can be understood in terms of free fermions and band theory. They also contain an introduction to the fractional quantum Hall effect from the point of view of effective field theory.

Motivation & Objective

• To explain topological phases of matter using noninteracting fermion band theory and effective field theory.
• To clarify the role of topology in stabilizing gapless edge modes and quantized transport in systems like graphene and quantum Hall states.
• To introduce the Chern-Simons effective action and its connection to quantized Hall conductivity in 2+1D systems.
• To analyze the stability of Dirac points in graphene under symmetry-preserving perturbations using group theory and Berry phases.
• To provide a field-theoretic perspective on the integer and fractional quantum Hall effects, emphasizing anomaly inflow and edge states.

Proposed method

• Derives the relativistic dispersion relation near band crossings in 1D and 3D, showing how linearization leads to chiral fermions.
• Uses the Berry connection and curvature to define topological invariants such as the Chern number in momentum space.
• Applies the Nielsen-Ninomiya theorem to explain the necessity of equal numbers of right- and left-moving modes in 1D periodic systems.
• Constructs the Chern-Simons effective action in 2+1D to describe the quantized Hall conductivity and its relation to band topology.
• Analyzes edge states via anomaly inflow, showing how bulk topological invariants constrain gapless edge modes.
• Uses a tight-binding model with nearest-neighbor hopping on a honeycomb lattice to derive the Dirac Hamiltonian and locate Dirac points at the Brillouin zone corners.

Experimental results

Research questions

• RQ1 How do topological invariants like the Chern number emerge from band structure in free fermion systems?
• RQ2 Why are gapless edge modes protected in topological insulators and how do they relate to bulk topology?
• RQ3 What is the role of symmetry in stabilizing Dirac points in graphene and preventing gap opening?
• RQ4 How does the Chern-Simons effective action explain the quantization of the Hall conductivity in 2+1D?
• RQ5 How does the fractional quantum Hall effect arise from effective field theory, and what is the role of anyonic statistics?

Key findings

• In 1D, the periodicity of momentum space forces an equal number of right- and left-moving modes, preventing chiral anomalies and ensuring anomaly cancellation.
• In 3D, the Nielsen-Ninomiya theorem forbids a single Weyl fermion, requiring at least two Weyl points of opposite chirality.
• Dirac points in graphene’s honeycomb lattice are protected by time-reversal and point-group symmetries, remaining gapless under small symmetry-preserving perturbations.
• The two Dirac points in graphene are degenerate in energy due to a 2π/6 rotation symmetry that exchanges them.
• The Chern-Simons effective action correctly reproduces the quantized Hall conductivity σxy = νe²/h, with ν an integer for the integer quantum Hall effect.
• The spin quantum Hall effect in graphene arises when spin-orbit coupling is included, leading to a topological invariant of 2 for spin-up and spin-down channels.