Resistance of two-dimensional superconducting films

We consider the problem of finite resistance $R$ in superconducting films with geometry a strip width $W$ near zero temperature. The is generated by vortex configurations phase field. In first type process, quantum slip, worldline 2+1 dimensional space-time space-like (i.e., winds time and space). second type, tunneling, time-like two spatial directions) connects opposite edges film. For moderately disordered samples, processes favor train vortices, each which tunnels only across fraction sample. Optimization respect to number vortices yields tunneling distance order coherence length $\xi$, becomes equivalent slip. Based on this theory, we find $\ln R \sim -g W/\xi$, where $g$ dimensionless normal-state conductance. Incorporation fluctuations indicates transition an insulating state for $g \lesssim 1$.