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

Topologically non-trivial superconductivity has been predicted to occur in superconductors with a sizable spin-orbit (SO) 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 it is now well known, such systems in the form of a two-dimensional (2D) film or 1D nano-wires in a wire network can be used in topological quantum computation. When the external Zeeman splitting Gamma crosses a critical value Gamma(c), the system passes from a regular superconducting phase to a non-Abelian topological superconducting phase. In both cases (a) and (b) that we consider in this paper, the pair potential Delta is strictly s-wave in both the ordinary and the topological superconducting phases, which are separated by a topological quantum critical point at Gamma(c) = root Delta(2) + mu(2), where mu (>> Delta) is the chemical potential. On the other hand, since Gamma(c) >> Delta, the Zeeman splitting required for the topological phase (Gamma > Gamma(c)) far exceeds the value (Gamma similar to Delta) above which an s-wave pair potential is expected to vanish (and the system to become non-superconducting) in the absence of SO coupling. We are thus led to the situation that the topological superconducting phase appears to set in a parameter regime at which the system is actually non-superconducting in the absence of SO 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 SO coupling is the crucial parameter for the quantum transition into and the robustness of the topologically non-trivial superconducting phase realized for Gamma >> Delta.