Theory for conventional superconductivity at the surface of a Weyl semimetal

Recent experiments have seen intrinsic surface superconductivity in Weyl semimetals and thus, posed the question of whether the unusual Fermi arc states can support superconductivity without any proximity effect from the bulk. A conclusive answer is hindered by the absence of a well-defined surface Hamiltonian since the Fermi arcs merge with the bulk states at their end points. We circumvent this issue by adopting an alternate, Green's functions-based approach tailored to a layering model from which arbitrary Fermi arcs can be obtained by tuning phenomenological parameters. We find that Fermi arcs, indeed, can support a standard Cooper instability, and their leakage into the bulk has a negligible effect on the nature of the superconducting state if the bulk Weyl nodes are undoped. In the undoped limit, within mean-field theory, we find a finite critical temperature on the surface while the bulk critical temperature is zero, thus realizing a peculiar situation where the surface of a system orders while the bulk is disordered even though the latter has higher dimensionality.