Quantum Theory of Heat Magnetization

We give the thermodynamic definition of the heat magnetization, and calculate with use of the Keldysh formalism in a curved spacetime. As the charge current is coupled to a vector potential, the heat current is coupled to a part of a vielbein. Consequently, as we define the orbital magnetization by a magnetic field, we can define the heat magnetization by a torsional magnetic field induced by a vielbein. Such heat magnetization, together with the Kubo formula for the thermal conductivity calculated by a torsional electric field, leads to the proper thermal Hall conductivity satisfying the Wiedemann-Franz law. Our results indicate that the quantum thermal Hall effect in (2$+$1)-D time-reversal-broken topological insulators or superconductors is described by the Chern-Simons action of a vielbein. \\[4pt] [1] A. Shitade, arXiv:1310.8043.\\[0pt] [2] A. Shitade, arXiv:1310.8046.