# Thermal conductance of an Edge mode

## TL;DR

This paper investigates the thermal conductance of edge modes in 2D and 3D models, analyzing effects of magnetic fields and backscattering, and confirms experimental results for topological insulators.

## Contribution

It provides a theoretical framework for thermal conductance in edge modes considering magnetic fields and backscattering, aligning with experimental observations.

## Key findings

- Thermal conductivity agrees with bosonization results for non-interacting fermions.
- Backscattering reduces conductance in magnetic fields.
- Thermal, electric, and thermoelectric conductances decrease with increasing magnetic field.

## Abstract

Thermoelectric conductance of an edge mode is investigated. The edge modes of a $2D $ and $3D$ two band model with parabolic dispersion is considered. For the the one dimensional non interacting fermions the thermal conductivity computed agrees with the result known from $Bosonization$ computations. In the presence of a magnetic field, backscattering is allowed and controls the value of the thermal conductivity. The thermal conductivity is obtained from the continuity equation of thermal current energy conservation. The thermal conductivity is computed introducing the $Scattering$ matrix for particles and anti-particles. At finite temperatures the backscattering is allowed, the electric conductance, the thermoelectric conductance and the thermal conductance decrease with the increase of the magnetic field. At finite temperatures, weak localization effects are small and can be ignored. We confirm the experimental results in a magnetic field for a $3D$ Topological Insulator. An experimental set-up was proposed to test our theory.