Topologically non-trivial superconductivity in spin-orbit coupled systems: Bulk phases and quantum phase transitions

Abstract

Topologically non-trivial superconductivity has been predicted to occur in superconductors with a sizable spin-orbit coupling in the presence of an external Zeeman splitting. Two such systems have been proposed: (a) s-wave superconductor pair potential is proximity induced on a semiconductor, and (b) pair potential naturally arises from an intrinsic s-wave pairing interaction. As is now well known, such systems in the form of a 2D film or 1D nano-wires in a wire-network can be used for topological quantum computation. When the external Zeeman splitting $Γ$ crosses a critical value $Γ_c$, the system passes from a regular superconducting phase to a non-Abelian topological superconducting phase. In both cases (a) and (b) we consider in this paper the pair potential $Δ$ is strictly s-wave in both the ordinary and the topological superconducting phases, which are separated by a topological quantum critical point at $Γ_c = \sqrt{Δ^2 + μ^2}$, where $μ(>> Δ)$ is the chemical potential. On the other hand, since $Γ_c >> Δ$, the Zeeman splitting required for the topological phase ($Γ> Γ_c$) far exceeds the value ($Γ\sim Δ$) above which an s-wave pair potential is expected to vanish (and the system to become non-superconducting) in the absence of spin-orbit coupling. We are thus led to a situation that the topological superconducting phase appears to set in a parameter regime at which the system actually is non-superconducting in the absence of spin-orbit coupling. In this paper we address the question of how a pure s-wave pair potential can survive a strong Zeeman field to give rise to a topological superconducting phase. We show that the spin-orbit coupling is the crucial parameter for the quantum transition into and the robustness of the topologically non-trivial superconducting phase realized for $Γ>> Δ$.