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Using the Kubo linear-response formalism we derive expressions to calculatethe heat current generated by magnetization dynamics in magnets with brokeninversion symmetry and spin-orbit interaction (SOI). The effect of producingheat currents by magnetization dynamics constitutes the Onsager reciprocal ofthe thermal spin-orbit torque (TSOT), i.e., the generation of torques on themagnetization due to temperature gradients. We find that the energy currentdriven by magnetization dynamics contains a contribution from theDzyaloshinskii-Moriya interaction (DMI), which needs to be subtracted from theKubo linear response of the energy current in order to extract the heatcurrent. We show that the expressions of the DMI coefficient can be derivedelegantly from the DMI energy current. Guided by formal analogies between theBerry phase theory of DMI on the one hand and the modern theory of orbitalmagnetization on the other hand we are led to an interpretation of the latterin terms of energy currents as well. Based on \textit{ab-initio} calculationswe investigate the heat current driven by magnetization dynamics in Mn/W(001)magnetic bilayers. We predict that fast domain walls drive strong ITSOT heatcurrents.

The inverse thermal spin-orbit torque and the relation of the Dzyaloshinskii-Moriya interaction to ground-state energy currents