We consider the competition between the conventional s-wave and the triplet Balian-Werthamer or the B-phase pairings in the doped three dimensional narrow gap semiconductors, such as $\mathrm{ Cu}_x\mathrm{ Bi}_2\mathrm{ Se}_3$ and $\mathrm{ Sn}_{ 1-x}\mathrm{ In}_x\mathrm{ Te}$. 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 dynamic axionic state of matter. In contradistinction to the time-reversal invariant, topological B-phase, the $p + is$ state possesses gapped Majorana fermions as the 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 $\theta$ vacuum structure, where $\theta \neq 0$ or $\pi$.
Categories: cond-mat.supr-con cond-mat.mes-hall cond-mat.str-el hep-th quant-ph