Topological response of gapped fermions to a $\text{U}(1)$ Gauge field

TL;DR

This paper derives the topological electromagnetic response of gapped fermions in various dimensions, linking the response to Chern-Simons terms and topological invariants, and discusses boundary effects.

Contribution

It provides a detailed path integral derivation of topological responses in 2+1 and higher dimensions, connecting them to characteristic classes and boundary phenomena.

Findings

Derivation of the Hall response via Chern-Simons term in 2+1D.

Extension of topological response to higher odd dimensions.

Discussion of bulk-boundary correspondence in topological phases.

Abstract

We present a detailed path integral derivation of the topological response of gapped free fermions, in $2+1$ dimensions, to an external $\text{U}(1)$ Gauge field. The well-known Hall response is obtained by identifying the Chern-Simons term in the effective action with the correct coefficient. We extend the result to $2d+1$ dimensions in which the response is associated to a Chern-Simons term with a coefficient related to a characteristic class coming from topological band theory. We comment on the bulk-to-boundary principle which arises naturally when one considers the theory on a manifold with boundary.