Odd-parity topological superconductor with nematic order: Application to ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$

${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ was recently proposed as a promising candidate for time-reversal-invariant topological superconductors. In this work, we argue that the unusual anisotropy of the Knight shift observed by Zheng and co-workers (unpublished), taken together with specific heat measurements, provides strong support for an unconventional odd-parity pairing in the two-dimensional $E_u$ representation of the $D_{3d}$ crystal point group, which spontaneously breaks the threefold rotational symmetry of the crystal, leading to a subsidiary nematic order. We predict that the spin-orbit interaction associated with hexagonal warping plays a crucial role in pinning the two-component order parameter and makes the superconducting state generically fully gapped, leading to a topological superconductor. Experimental signatures of the $E_u$ pairing related to the nematic order are discussed.

Figure 1

(a) Crystal structure of ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ viewed from the $c$ axis. Note that the $x$ axis is normal to a mirror plane. (b) Hexagonal Fermi contour at $k_z=0$ for the Hamiltonian (8). (c) and (d) show the angle dependence of the anisotropic superconducting gap over the $k_z=0$ Fermi contour for the $E_u$ order parameters $V_x$ and $V_y$, respectively, defined in (6). The presence of nodes in (c) and the full gap in (d) are robust and model independent.