Three Lectures On Topological Phases Of Matter

TL;DR

This set of notes surveys topological phases described by free-fermion band theory and their connection to effective field theories. It develops the classification of gapless modes via Weyl/Dirac points, Berry curvature, and the Nielsen-Ninomiya constraints, then connects bulk band topology to boundary phenomena through Chern-Simons theory and anomaly inflow. The lectures culminate with fractional quantum Hall physics, emergent gauge fields, and topological order, and culminate in Haldane’s graphene model as a concrete Chern-insulator realization without a net magnetic field. Together, the work links microscopic band structure, topological invariants, and robust edge excitations to establish a unified framework for 2D and 3D topological phases with both integer and fractional quantum Hall phenomenology.

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.

Figures (43)

• Figure 1: In one dimension, the single-particle energy $\varepsilon(p)$ generically crosses the Fermi energy at an isolated momentum $p_0$.
• Figure 2: In one dimension, for every value of the momentum at which $\varepsilon(p)$ increases above $\epsilon_F$, there is another point at which it decreases below $\varepsilon_F$.
• Figure 3: Generically, quantum mechanical energy levels do not cross as a parameter is varied.
• Figure 4: A pair of bands described by a chiral Dirac Hamiltonian.
• Figure 5: The interior of the hexagon symbolizes the Brillouin zone $\mathcal{B}$. We consider a band Hamiltonian that is gapped except at finitely many points, which are indicated by dots.