# Thermal Hall conductance and a relative topological invariant of gapped two-dimensional systems

## TL;DR

This paper introduces a new formula for thermal Hall conductance in 2D lattice systems, defining a topological invariant linked to edge modes and establishing a bulk-boundary correspondence, with implications for free fermionic systems.

## Contribution

It derives an ambiguity-free Kubo-like formula for thermal Hall conductance and defines a relative topological invariant related to chiral central charges.

## Key findings

- The invariant matches the difference in chiral central charges.
- It vanishes for local commuting projector Hamiltonians.
- For free fermions, it relates to electric Hall conductance via Wiedemann-Franz law.

## Abstract

We derive a Kubo-like formula for the thermal Hall conductance of a 2d lattice systems which is free from ambiguities associated with the definition of energy magnetization. We use it to define a relative topological invariant of gapped 2d lattice systems at zero temperature. Up to a numerical factor, it can be identified with the difference of chiral central charges for the corresponding edge modes. This establishes the bulk-boundary correspondence for the chiral central charge. We also show that for any Local Commuting Projector Hamiltonian the relative chiral central charge vanishes, while for free fermionic systems it is related to the zero-temperature electric Hall conductance via the Wiedemann-Franz law.