Electromagnetic response of topological superconductors

We resolve several puzzles related to the electromagnetic response of topological superconductors in 3+1 dimensions. In particular we show by an analytical calculation that the interface between a topological and normal superconductor does not exhibit any quantum Hall effect as long as time reversal invariance is preserved. We contrast this with the analogous case of a topological insulator to normal insulator interface. The difference is that in the topological insulator the electromagnetic vector potential couples to a vector current in a theory with a Dirac mass, while in the superconductor a pair of Weyl fermions is gapped by Majorana masses and the electromagnetic vector potential couples to its axial currents.

Figure 1

Diagrammatic illustration of the process resulting in the current Eq. (15).

Figure 2

The Hall coefficient ${\sigma }_{yx}$ as (a) a function of $\sigma /m_0$ for small negative $\sigma$ and (b) as a function of $m_0/\sigma$ for large $\sigma$. In (b) we also show the effective theory result $I_{\mathrm{total}}=\frac{\mathrm{sgn}\left(\sigma \right)}{12{\pi }^2}{tan}^{-1}\left(\frac{m_0}{\sigma }\right)$ in yellow. In both panels, $e=\hslash =1$.