Rotational isotropy breaking as proof for spin-polarized Cooper pairs in the topological superconductor Cu${}_x$Bi${}_2$Se${}_3$

In the promising candidate for topological superconductors, Cu${}_x$Bi${}_2$Se${}_3$, we propose a way to exclusively determine the pairing symmetry. The proposal suggests that the angle dependence of the thermal conductivity in the basal $ab$ plane shows a distinct strong anisotropy only when the pairing symmetry is an odd-parity spin-polarized triplet below the superconducting transition temperature ($T_{\mathrm{c}}$). Such striking isotropy breaking below $T_{\mathrm{c}}$ is explicitly involved in the Dirac formalism for superconductors, in which the spin-orbit coupling is essential. We classify possible gap functions based on the Dirac formalism and clarify an origin of the isotropy breaking.

Figure 1

Schematic of the calculation target system, which is discretized along the $z$ direction. Thus the quasiparticles are localized around the edge surface plane at $z=0$.

Figure 2

Temperature dependence of the thermal conductivity carried by the edge bound states for the gap functions (a) ${\Delta }_2$ and (b) ${\Delta }_{4a}$. $N_z=64$.

Figure 3

(Left) Distributions of quasiparticle energies in in-plane momentum space with $N_z=64$ for the gap functions (a) ${\Delta }_2$ and (b) ${\Delta }_{4a}$, respectively. (Right) Contour plots of the minimum eigenvalues regarded as the edge states for (a) ${\Delta }_2$ and (b) ${\Delta }_{4a}$, respectively. Contours are drawn at regular intervals ($\delta \Delta ={\Delta }_0/20$).