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

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