Fermion Path Integrals And Topological Phases

TL;DR

This paper explores the connection between fermionic symmetry-protected topological phases and anomalies, focusing on free fermion systems like topological insulators and superconductors, and clarifies the significance of the electromagnetic theta angle in 3D topological insulators.

Contribution

It provides a detailed interpretation of fermionic SPT phases in terms of anomalies and clarifies the meaning of the theta angle in 3D topological insulators within band theory.

Findings

Analysis of fermionic SPT phases via anomalies

Clarification of the theta angle in 3D topological insulators

Application to time-reversal invariant topological phases

Abstract

Symmetry-protected topological (SPT) phases of matter have been interpreted in terms of anomalies, and it has been expected that a similar picture should hold for SPT phases with fermions. Here, we describe in detail what this picture means for phases of quantum matter that can be understood via band theory and free fermions. The main examples we consider are time-reversal invariant topological insulators and superconductors in 2 or 3 space dimensions. Along the way, we clarify the precise meaning of the statement that in the bulk of a 3d topological insulator, the electromagnetic $\theta$-angle is equal to $\pi$.