Superconductivity in Weyl metals

We report on a study of intrinsic superconductivity in a Weyl metal, i.e., a doped Weyl semimetal. Two distinct superconducting states are possible in this system in principle: a zero-momentum pairing BCS state, with point nodes in the gap function, and a finite-momentum FFLO-like state, with a full nodeless gap. We find that, in an inversion-symmetric Weyl metal, the odd-parity BCS state has a lower energy than the FFLO state, despite the nodes in the gap. The FFLO state, on the other hand, may have a lower energy in a noncentrosymmetric Weyl metal, in which Weyl nodes of opposite chirality have different energy. However, realizing the FFLO state is, in general, very difficult since the paired states are not related by any exact symmetry, which precludes a weak-coupling superconducting instability. We also discuss some of the physical properties of the nodal BCS state, in particular, Majorana and Fermi arc surface states.

Figure 1

Numerically calculated eigenstate dispersions for Eq. (61) along the $z$ direction in momentum space for a sample of finite width in the $x$ direction. Parameters in Eq. (61) are chosen as follows: $t_S=1,t_D=0.9,b=1,{\epsilon }_F=\mathrm{\Delta }=0.2$. Majorana modes are pinned to zero energy and exist in intervals of width of order $\left|\mathrm{\Delta }\right|/{\tilde{v}}_F$ around each Weyl node. Fermi arc edge modes are split into particle-hole antisymmetric and symmetric branches at $\pm {\epsilon }_F$.

Figure 2

Numerically calculated eigenstate dispersions for Eq. (61) along the $y$ direction in momentum space for a sample of finite width in the $x$ direction, at several values of $k_z$. Parameters in Eq. (62) are chosen as follows: $t_S=1,t_D=0.9,b=1,{\epsilon }_F=0.5,\mathrm{\Delta }=0.1$. (a) $k_z$ is outside the Fermi surface. Two pairs (corresponding to two surfaces) of chiral Fermi arc modes, crossing at $\pm {\epsilon }_F$ at $k_y=0$ are visible. (b) $k_z$ is just inside the Fermi surface. Two chiral Majorana modes are visible now, crossing at $k_y=0$ at zero energy. (c) $k_z$ is closer to a Weyl node location. Majorana modes directly connect with the Fermi arcs.

Figure 3

Same parameters as in Fig. 2, but values of $k_z$ are taken near (a) and beyond (b) the Weyl node location. Disappearance of the Fermi arcs necessitates a “flat band” at the transition point. The chirality of the Majorana modes changes sign at the transition. The transition itself occurs exactly at the Weyl node location in the limit $\mathrm{\Delta }/{\epsilon }_F\to 0$, but is shifted slightly away for any finite $\mathrm{\Delta }$.