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 $\Gamma$ 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) 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 ${\Gamma }_c=\sqrt{{\Delta }^2+{\mu }^2}$, 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 ($\Gamma \sim \Delta$) 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 $\Gamma >>\Delta$.

as published in the focus issue on Topological Quantum Computation, New J. Phys. 13 (2011)

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