Theory of Charge and Heat Polarizations with the Keldysh Formalism

We investigate the heat polarization, a heat analog of the charge polarization, by using the gauge-covariant Keldysh formalism. In contrast to the charge-heat analogy naively expected, we find that the heat polarization does not appear spontaneously, since it consists not only of the heat-transfer contribution but of the heat-generation contribution, leading to the Mott rule. Nonetheless, it can be induced by a torsional magnetic field in $(3+1)$-D topological insulators and superconductors, which is described by the temporal part of the Nieh-Yan action.