  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 $\Gamma >>
\Delta$.
