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 ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ 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 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 $\theta$ vacuum structure, where $\theta \ne 0$ or $\pi$.

Figure 1

A cut of the zero temperature phase diagram for $g_s\rho \left(\mu \right)=1$, showing the normalized pairing amplitudes, as a function of the ratio $g_t/g_s$, where $\rho \left(\mu \right)$ is the density of states at the Fermi level. The amplitudes are normalized by ${\Delta }_s^0$, the $s$-wave amplitude, when the triplet channel is turned off. The $s$-wave and the $B$-phase amplitudes are respectively shown as the red and the black lines.

Figure 2

The dimensionless quantity $\hslash {\kappa }_{xy}/\left(k_B^{2}T\right)$ for the gapped SABS, as a function of $T/T_F$, where ${\kappa }_{xy}$ and $T_F=\mu /k_B$ are, respectively, the thermal Hall conductivity and the Fermi temperature. We have used the mean-field gap amplitudes for $g_s\rho \left(\mu \right)=1$ and $g_t/g_s=0.99895$. This quantity saturates to the universal number $\pi /24$ in the zero temperature limit. Inset: The dimensionless thermal Hall conductivity $\hslash {\kappa }_{xy}/\left(k_B\mu \right)$ for the gapped SABS, as a functions of $T/T_F$.