Resistance of two-dimensional superconducting films

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