Heat asymmetries in nanoscale conductors: The role of decoherence and inelasticity

We investigate the heat flow between different terminals in an interacting coherent conductor when inelastic scattering is present. We illustrate our theory with a two terminal quantum dot setup. Two types of heat asymmetries are investigated: electric asymmetry $\Delta_E$, which describes deviations of the heat current in a given contact when voltages are exchanged, and contact asymmetry $\Delta_C$, which quantifies the difference between the power measured in two distinct electrodes. In the linear regime, both asymmetries agree and are proportional to the Seebeck coefficient, the latter following at low temperature a Mott-like formula with a dot transmission renormalized by inelasticity. Interestingly, in the nonlinear regime of transport we find $\Delta_E\neq\Delta_C$ and this asymmetry departure depends on the applied bias configuration. Our results may be important for the recent experiments by Lee \textit{et al.} [Nature \textbf{498}, 209 (2013)], where these asymmetries were measured.