Quantum thermal Hall effect of Majorana fermions on the surface of superconducting topoloigcal insulators

We study the quantum anomalous thermal Hall effect in a topological superconductor which possesses an integer bulk topological number, and supports Majorana excitations on the surface. To realize the quantum thermal Hall effect, a finite gap at the surface is induced by applying an external magnetic field or by the proximity effects with magnetic materials or $s$-wave superconductors with complex pair-potentials. Basing on the lattice model Hamiltonian for superconducting states in Cu-doped Bi$_2$Se$_3$, we compute the thermal Hall conductivity as a function of various parameters such as the chemical potential, the pair-potential, and the spin-orbit coupling induced band gap. It is argued that the bulk topological invariant corresponds to the quantization rule of the thermal Hall conductivity induced by complex $s$-wave pair-potentials.