Cross-Correlated Responses of Topological Superconductors and Superfluids

We study nontrivial responses of topological superconductors and superfluids to the temperature gradient and rotation of the system. In two-dimensional gapped systems, the Strěda formula for the electric Hall conductivity is generalized to the thermal Hall conductivity. Applying this formula to the Majorana surface states of three-dimensional topological superconductors predicts cross-correlated responses between the orbital angular momentum and thermal polarization (entropy polarization). These results can be naturally related to the gravitoelectromagnetism description of three-dimensional topological superconductors and superfluids, analogous to the topological magnetoelectric effect in ${\mathbb{Z}}_2$ topological insulators.

Figure 1

Electronic responses in (a) two-dimensional (2D) and (c) three-dimensional (3D) topological insulators (TIs) (left) and thermal and mechanical (rotating) responses in (b) 2D and (d) 3D topological superconductors (TSCs) (right). In (b), the arrow along the boundary indicates the chiral Majorana edge channel with dispersion $E=v{p}_{\theta }$. In the 3D TSC (d), temperature gradient induces the surface thermal Hall current ${\mathit{j}}_E$. A uniform mass gap is induced in the surface fermion spectrum by doping magnetic impurities near the surface of the 3D TI and TSC such that spins are all pointing out or in (arrows on the surface).