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

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