Laughlin's argument and thermal Hall effect

We study the effective action of hydrostatic response to torsion in the absence of spin connections in gapped $(2+1)$-dimensional topological phases at low temperatures. In previous studies, a torsional Chern-Simons term with a temperature-squared ($T^2$) coefficient was proposed as an alternative action to describe the 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 a local bulk thermal response that 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 geometry dependence of local temperature, which precisely describes the edge thermal current. Furthermore, there are no local boundary diffeomorphism anomalies or bulk inflow thermal currents at equilibrium, and also there is no edge-to-edge adiabatic thermal current pumping upon changing the gravitational background. These results are consistent with exponentially suppressed thermal current for gapped phases.