Torsion, Parity-odd Response and Anomalies in Topological States

TL;DR

This paper investigates the electromagnetic and gravitational responses, including torsion effects, of topological states like insulators and Weyl semimetals, using anomaly polynomials and dimensional reduction techniques.

Contribution

It derives the parity-odd responses and effective actions for topological insulators considering torsion and curvature, extending understanding of anomalies in these systems.

Findings

Derived parity-odd responses for Dirac fermions in 2+1 and 4+1 dimensions.

Analyzed covariant anomalies and edge state inflow mechanisms.

Established effective actions for 3+1 topological insulators with torsion and curvature.

Abstract

We study the response of a class of topological systems to electromagnetic and gravitational sources, including torsion and curvature. By using the technology of anomaly polynomials, we derive the parity-odd response of a massive Dirac fermion in $d=2+1$ and $d=4+1$, which provides a simple model for a topological insulator. We discuss the covariant anomalies of the corresponding edge states, from a Callan-Harvey anomaly-inflow, as well as a Hamiltonian spectral flow point of view. We also discuss the applicability of our results to other systems such as Weyl semi-metals. Finally, using dimensional reduction from $d=4+1$, we derive the effective action for a $d=3+1$ time-reversal invariant topological insulator in the presence of torsion and curvature, and discuss its various physical consequences.