Finite-temperature effective boundary theory of the quantized thermal Hall effect

TL;DR

This paper derives a microscopic effective boundary theory for the quantized thermal Hall effect in two-dimensional insulators and superconductors, linking boundary free energy to bulk thermal Hall conductivity via gravitational field coupling.

Contribution

It presents a microscopic derivation of the boundary effective free energy functional for the quantized thermal Hall effect, connecting it to the bulk Chern number and gravitational field.

Findings

Boundary theory explains quantized thermal Hall conductivity.

Effective boundary free energy derived from Dirac fermions in gravitational field.

Consistent bulk-boundary relationship established.

Abstract

A finite-temperature effective free energy of the boundary of a quantized thermal Hall system is derived microscopically from the bulk two-dimensional Dirac fermion coupled with a gravitational field. In two spatial dimensions, the thermal Hall conductivity of fully gapped insulators and superconductors is quantized and given by the bulk Chern number, in analogy to the quantized electric Hall conductivity in quantum Hall systems. From the perspective of effective action functionals, two distinct types of the field theory have been proposed to describe the quantized thermal Hall effect. One of these, known as the gravitational Chern-Simons action, is a kind of topological field theory, and the other is a phenomenological theory relevant to the St\v{r}eda formula. In order to solve this problem, we derive microscopically an effective theory that accounts for the quantized thermal Hall effect. In this paper, the two-dimensional Dirac fermion under a static background gravitational field is considered in equilibrium at a finite temperature, from which an effective boundary free energy functional of the gravitational field is derived. This boundary theory is shown to explain the quantized thermal Hall conductivity and thermal Hall current in the bulk by assuming the Lorentz symmetry. The bulk effective theory is consistently determined via the boundary effective theory