Thermal Hall response: Violation of gravitational analogs and Einstein relations

Abstract

The response of solids to temperature gradients is often described in terms of a gravitational analog: 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 noninteracting electron baths or phonons. We find that the thermal analog applied to theses cases is 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.