Torsional Anomalies, Hall Viscosity, and Bulk-boundary Correspondence in Topological States

TL;DR

This paper investigates the transport properties of topological insulators, focusing on Hall viscosity and anomalies, revealing how torsion and boundary effects encode topological phase differences and edge phenomena.

Contribution

It introduces a detailed analysis of Hall viscosity in topological phases, including torsional contributions and anomaly inflow, linking bulk properties to edge anomalies and gravitational actions.

Findings

Hall viscosity difference is well-defined and linked to the mass gap.

Torsional contributions modify the covariant diffeomorphism anomaly.

Edge spectral flow is influenced by torsional dislocations.

Abstract

We study the transport properties of topological insulators, encoding them in a generating functional of gauge and gravitational sources. Much of our focus is on the simple example of a free massive Dirac fermion, the so-called Chern insulator, especially in 2+1 dimensions. In such cases, when parity and time-reversal symmetry are broken, it is necessary to consider the gravitational sources to include a frame and an independent spin connection with torsion. In 2+1 dimensions, the simplest parity-odd response is the Hall viscosity. We compute the Hall viscosity of the Chern insulator using a careful regularization scheme, and find that although the Hall viscosity is generally divergent, the difference in Hall viscosities of distinct topological phases is well-defined and determined by the mass gap. Furthermore, on a 1+1-dimensional edge between topological phases, the jump in the Hall viscosity across the interface is encoded, through familiar anomaly inflow mechanisms, in the structure of anomalies. In particular, we find new torsional contributions to the covariant diffeomorphism anomaly in 1+1 dimensions. Including parity-even contributions, we find that the renormalized generating functionals of the two topological phases differ by a chiral gravity action with a negative cosmological constant. This (non-dynamical) chiral gravity action and the corresponding physics of the interface theory is reminiscent of well-known properties of dynamical holographic gravitational systems. Finally, we consider some properties of spectral flow of the edge theory driven by torsional dislocations.