A heat bath approach to anomalous thermal transport: interplay of Berry curvature and inelastic dissipation

We present results for the entire set of anomalous charge and heat transport coefficients for metallic systems in the presence of finite temperature. In realistic physical systems

this necessitates the inclusion of inelastic dissipation; relatively little is known theoretically about its effects on anomalous transport. We show that inelastic scattering properties are strongly intertwined with Berry curvature physics. Our calculations are made possible by the introduction of a Leggett-Caldeira reservoir which allows us to avoid the artificial device of the pseudogravitational potential. Using our formulae we characterize the finite temperature behavior of the important anomalous Wiedemann Franz ratio. We show that, despite previous expectations, this ratio can exhibit either an upturn or a downturn as temperature increases away from zero and we emphasize that this arises from a \textit{competition} between Berry curvatures having different signs in different regions of the Brillouin zone.  Our work demonstrates that identifying an inelastic scattering mechanism from this ratio in experiments is much more challenging than previously thought. We explore similar effects on the anomalous transverse Mott relation and generalize our approach to finite magnetic field Hall transport.