Vortex nucleation barriers in superconductors revisited

The knowledge of energetic barriers and free energy landscape is crucial for applications of superconductors. Examples include barriers for vortex nucleation and pinning in transmission lines, superconducting magnets, superconducting single-photon detectors, and superconductor-based qubits. Contrarily to the problem of finding energy minima and critical magnetic fields, currently, there are no controllable methods to explore the free energy landscape, identify saddle-points, and compute associated barriers. These barriers depend on coherence length, which makes the London-model-based approach by Bean and Livingston, in general, not reliable. Here we present a generalization of the string method to gauge field theories, which allows the calculation of free energy barriers in superconductors. We apply the technique to solve the full nonlinear problem of vortex nucleation in complicated geometries that previously were not amenable to analytical and numerical treatment. Then, we study the case of rough surfaces and the presence of pinning.