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

TL;DR

This paper investigates how strong spin-orbit coupling enables the survival of s-wave superconductivity under high Zeeman fields, leading to topologically non-trivial phases crucial for quantum computing.

Contribution

It demonstrates that spin-orbit coupling is essential for stabilizing topological superconductivity in systems with large Zeeman splitting, even when the pair potential is strictly s-wave.

Findings

Topological phase transition occurs at ca0c2a0c2a0ca0ca0=ca0c2a0c2a0ca0ca0^2+ca0c2a0c2a0ca0ca0^2

Spin-orbit coupling is crucial for the emergence and stability of topological superconducting phases at high Zeeman fields

The topological phase can exist even when the system would be non-superconducting without spin-orbit coupling.

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 $\Gamma >> \Delta$.