Anomalous thermal Hall effect in a disordered Weyl ferromagnet

Thermal Hall effect is a heat analog of the Hall effect, namely, the heat current flows perpendicular to a temperature gradient. According to the Wiedemann-Franz law, the Lorenz ratio $L^{ij}\equiv \kappa^{ij}/T\sigma ^{ij}$ goes to the universal Lorenz number $L_{0} \equiv \pi^{2}k_{B}^{2} /3e^{2}$ as $T\to 0$, in which $\sigma^{ij}$ and $\kappa^{ij}$ are the electric and thermal (Hall) conductivities and $T$ is temperature. At finite temperature, we can investigate effects of inelastic scattering by the breakdown of the Wiedemann-Franz law. In spite of its usefulness, it is theoretically difficult to calculate $T\kappa^{xy}$ because it is not expressed by the Kubo formula $T\tilde{{\kappa }}^{xy}$ alone but is corrected by the heat magnetization $2M_{Qz} $. Recently, I found a gravitational vector potential coupled to the energy current and established the Keldysh formalism to calculate $T\tilde{{\kappa }}^{xy}$ and $2M_{Qz} $ even in disordered or interacting systems [1]. Here I apply this formalism to a disordered Weyl ferromagnet which exhibits the anomalous (thermal) Hall effect. I first quantum-mechanically calculate $\sigma^{ij}$ and $T\kappa^{ij}$ on an equal footing and reproduce the Wiedemann-Franz law. This is the first step towards a unified theory of the anomalous Hall effect at finite temperature, in which inelastic scattering by magnons is relevant. [1] A. Shitade, Prog. Theor. Exp. Phys. 2014, 123I01 (2014).