Torsion, energy magnetization and thermal Hall effect

TL;DR

This paper investigates the effective action for thermal Hall effects in gapped 2+1D topological phases, showing that bulk torsional terms do not produce local thermal responses, aligning with the expectation of suppressed thermal currents.

Contribution

It demonstrates that the proposed torsional Chern-Simons term does not induce local bulk thermal responses in gapped phases, clarifying the role of boundary effects and invariance properties.

Findings

Bulk $T^2$ torsional term is invariant up to boundary terms.

No boundary diffeomorphism anomalies or bulk inflow thermal currents at equilibrium.

Thermal currents are exponentially suppressed in gapped phases.

Abstract

We study the effective action of hydrostatic response to torsion in the absence of spin connections in gapped $\left(2+1\right)$-dimensional topological phases. In previous studies, a torsional Chern-Simons term with a temperature-squared ($T^2$) coefficient was proposed as an alternative action to describe thermal Hall effect with the idea of balancing the diffusion of heat by a torsional field. However, the question remains whether this action leads to local bulk thermal response which is not suppressed by the gap. In our hydrostatic effective action, we show that the $T^2$ bulk term is invariant under variations up to boundary terms considering the back reaction of the geometry on local temperature, which precisely describes the edge thermal current. Furthermore, there is no boundary diffeomorphism anomalies and bulk inflow thermal currents at equilibrium and therefore no edge-to-edge adiabatic thermal current pumping. These results are in consistent with exponentially suppressed thermal current for gapped phases.