Energy magnetization and transport in systems with a non-zero Berry curvature in a magnetic field

TL;DR

This paper develops a formalism linking Berry curvature, energy magnetization, and transport currents, providing new insights into energy circulation and Hall responses in topological systems with magnetic fields.

Contribution

It introduces a novel approach to derive energy magnetization from transport relations and solves the Boltzmann equation for systems with Berry curvature in magnetic fields.

Findings

Energy magnetization relates to circulating energy currents in Chern insulators.

The formalism recovers known expressions for energy magnetization.

The distribution function explains regular Hall response in time-reversal invariant systems.

Abstract

We demonstrate that the well-known expression for the charge magnetization of a sample with a non-zero Berry curvature can be obtained by demanding that the Einstein relation holds for the electric transport current. We extend this formalism to the transport energy current and show that the energy magnetization must satisfy a particular condition. We provide a physical interpretation of this condition, and relate the energy magnetization to circulating energy currents in Chern insulators due to chiral edge states. We further recover the expression for the energy magnetization with this alternative formalism. We also solve the Boltzmann Transport Equation for the non-equilibrium distribution function in 2D for systems with a non-zero Berry curvature in a magnetic field. This distribution function can be used to obtain the regular Hall response in time-reversal invariant samples with a non-zero Berry curvature, for which there is no anomalous Hall response.