Time-reversal-invariant topological superconductivity in doped Weyl semimetals

Time-reversal-invariant topological superconductors are a new state of matter which have a bulk superconducting gap and robust Majorana fermion surface states. These have not yet been realized in solid state systems. In this paper, we propose that this state can be realized in doped Weyl semimetals or Weyl metals. The Fermi surfaces of a Weyl metal carry Chern numbers, which is a required ingredient for such a topological superconductor. By applying the fluctuation-exchange approach to a generic model of time-reversal-invariant Dirac and Weyl semimetals, we investigate what microscopic interactions can supply the other ingredient, viz., a sign change of the superconducting gap function between Fermi surfaces with opposite Chern numbers. We find that if the normal state is inversion symmetric, on-site repulsive and exchange interactions induce various nodal phases as well as a small region of topological superconductivity on the phase diagram. Unlike in the ${}^3\text{He}$-B topological superconductor, the phase here does not rely on any special momentum dependence of the pairing amplitude. Breaking inversion symmetry precludes some of the nodal phases and the topological superconductor becomes much more prominent, especially at large ferromagnetic interaction. Our approach can be extended to generic Dirac or Weyl metals.

Figure 1

Schematic illustration of the pairing interaction and the gap functions described in Sec. 2. The dotted circles denote the normal-state FSs with index $i$ as discussed in the text and Chern numbers $C=\pm 1$, while the solid lines represent the gaps ${\Delta }_{1,2}$ in the superconducting state. ${\overline{\chi }}_{ij}$ is the average Cooper scattering amplitude from Kramers conjugate states on the $(i,-i)$ FSs to the $(j,-j)$ FSs.

Figure 2

The energy dispersion (a) and the corresponding FS contour (b) in the $k_x\text{−}{k}_z$ plane, for the $\mathcal{I}$-breaking term given by Eq. (14). The red and blue contours in (b) correspond to spin-up and -down FSs, respectively.

Figure 3

Diagrams that contribute to the effective interaction to second order in the bare interactions. The wavy lines represent any one of the five interactions $U_s$, $U_p$, $V$, $J_z$, and $W$.

Figure 4

Phase diagram up to second order in the interactions at $W=0$ in the presence of $\mathcal{I}$ symmetry. The gray phase at negative $J_z$ and intermediate $U$ or $V$ is the $\mathcal{T}$-symmetric topological superconductor. The phases were determined numerically, using $\mu =0.1$ and $\Lambda =1.0$. Details of all the phases are in Table 1.

Figure 5

Phase diagram up to second order in the interactions at $W=0$ in the absence of $\mathcal{I}$ symmetry. The black phase at negative $J_z$ is the $\mathcal{T}$-symmetric TSC, which occupies a larger part of the phase diagram compared to Fig. 4 when $\mathcal{I}$ symmetry is present. The phases were determined numerically, using $\mu =0.1$ and $\Lambda =1.0$. The $J_z=0$ cross section is identical to the one in Fig. 4. Details of all the phases are in Table 1.