Critical current as a signature of bulk topological phase transition in a quasi-1D Rashba nanowire

In a normal phase of a Rashba spin-orbit coupled semiconductor nanowire, the measurement of conductance can probe the bulk helical gap opened by a Zeeman field along the wire. However, for a nanowire with proximity-induced superconductivity, we cannot rely on the measurement of two-terminal conductance to probe a bulk topological phase transition. Here, we consider a Rashba spin-orbit coupled superconductor in a Zeeman field in different confinement potentials that allow the system to have single or multiple channels. To study the helical topological gap in this superconducting system, we propose to use another physical observable, the critical current. We consider the limit where the critical current is determined by depairing conditions (closing of quasiparticle excitation gap). We use a low-energy model to calculate the critical current analytically and a tight-binding approximation to verify and go beyond the analytical calculation. Our results quantify how the topological gap can be estimated from the critical current as a function of chemical potential and magnetic field.