Topological Insulators Avoid the Parity Anomaly

TL;DR

This paper explains how topological insulators host boundary states with an odd number of Dirac fermions that avoid parity anomaly issues through higher-dimensional bulk effects, preserving symmetries.

Contribution

It provides a dual perspective using topological quantization and regularization to show how boundary states avoid parity anomaly in topological insulators.

Findings

Boundary states can exist without breaking symmetries due to bulk effects.

Higher-dimensional bulk theory enables parity-invariant regularization.

Conditions for large gauge transformations on boundaries are clarified.

Abstract

The surface of a 3+1d topological insulator hosts an odd number of gapless Dirac fermions when charge conjugation and time-reversal symmetries are preserved. Viewed as a purely 2+1d system, this surface theory would necessarily explicitly break parity and time-reversal when coupled to a fluctuating gauge field. Here we explain why such a state can exist on the boundary of a 3+1d system without breaking these symmetries, even if the number of boundary components is odd. This is accomplished from two complementary perspectives: topological quantization conditions and regularization. We first discuss the conditions under which (continuous) large gauge transformations may exist when the theory lives on a boundary of a higher-dimensional spacetime. Next, we show how the higher-dimensional bulk theory is essential in providing a parity-invariant regularization of the theory living on the lower-dimensional boundary or defect.