Time-reversal anomaly and Josephson effect in time-reversal-invariant topological superconductors

Topological superconductors are gapped superconductors with protected Majorana surface/edge states on the boundary. In this paper, we study the Josephson coupling between time-reversal-invariant topological superconductors and s-wave superconductors. The Majorana edge/surface states of time-reversal-invariant topological superconductors in all physical dimensions 1, 2, and 3 have a generic topological property which we name the time-reversal anomaly. Due to the time-reversal anomaly, the Josephson coupling prefers a nonzero phase difference between topological and trivial superconductors. The nontrivial Josephson coupling leads to a current-flux relation with a half period in a superconducting quantum interference device geometry, and also a half-period Fraunhofer effect in dimensions higher than 1. We also show that an in-plane magnetic field restores the ordinary Josephson coupling, as a sharp signature that the proposed effect is a consequence of the unique time-reversal property of the topological edge/surface states. Our proposal provides a simple and general approach to experimentally verify whether a time-reversal-invariant superconductor is topological.