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

We demonstrate that the well-known expression for the charge magnetization of a sam-ple 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 trans-port energy current and show that the energy magnetization must satisfy a particular condition. We provide a physical interpretation of this condition, and relate the en-ergy magnetization to circulating energy currents in Chern insulators due to chiral edge states. We further recover the expression for the energy magnetization with this alterna-tive 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 anoma-lous Hall response.