Thermal Hall response: violation of gravitational analogues and Einstein relations

TL;DR

This paper challenges the gravitational analogue approach for describing thermal responses in quantum Hall states, showing it fails to accurately predict energy currents and violates Einstein relations in certain conditions.

Contribution

It demonstrates the invalidity of the gravitational analogy and Einstein relations in describing thermal responses of topological states under realistic conditions.

Findings

Space-dependent gravitational potential induces transverse energy currents.

Temperature profiles do not produce expected energy currents in weak coupling limit.

The Einstein relation between electrostatic and chemical potentials does not hold in this context.

Abstract

The response of solids to temperature gradients is often described in terms of a gravitational analogue: the effect of a space-dependent temperature is modeled using a space dependent metric. We investigate the validity of this approach in describing the bulk response of quantum Hall states and other gapped chiral topological states. To this end, we consider the prototypical Haldane model in two different cases of (i) a space-dependent electrostatic potential and gravitational potential and (ii) a space-dependent temperature and chemical potential imprinted by a weak coupling to non-interacting electron baths and phonons. We find that the thermal analogue is \textit{invalid}; while a space dependent gravitational potential induces transverse energy currents proportional to the third derivative of the gravitational potential, the response to an analogous temperature profile vanishes in limit of weak coupling to the thermal bath. Similarly, the Einstein relation, the analogy between the electrostatic potential and the internal chemical potential, is not valid in such a setup.