Axionic superconductivity in three-dimensional doped narrow-gap semiconductors

We consider the competition between the conventional s-wave and the triplet Balian-Werthamer or the B-phase pairings in doped three-dimensional narrow-gap semiconductors, such as CuxBi2Se3 and Sn1-xInxTe. When the coupling constants of the two contending channels are comparable, we find a simultaneously time-reversal and parity violating p+is state at low temperatures, which provides an example of a dynamic axionic state of matter. In contradistinction to the time-reversal invariant, topological B phase, the p+is state possesses gapped Majorana fermions as surface Andreev bound states, which give rise to an anomalous surface thermal Hall effect. The anomalous gravitational and electrodynamic responses of the p+is state can be described by the θ vacuum structure, where θ≠0 or π.