Vortex density models for superconductivity and superfluidity

TL;DR

This paper develops mathematical models for vortex line density in superconductors and Bose-Einstein condensates, providing insights into critical conditions for vortex formation and a novel nonlocal obstacle problem characterization.

Contribution

It introduces new vortex density functionals derived via Gamma-convergence and characterizes vortex density using a nonlocal obstacle problem framework.

Findings

Identification of critical magnetic field and forcing thresholds for vortex formation.

Characterization of vortex density through a nonlocal obstacle problem.

Derivation of models applicable to general 3D domains.

Abstract

We study some functionals that describe the density of vortex lines in superconductors subject to an applied magnetic field, and in Bose-Einstein condensates subject to rotational forcing, in quite general domains in 3 dimensions. These functionals are derived from more basic models via Gamma-convergence, here and in a companion paper. In our main results, we use these functionals to obtain descriptions of the critical applied magnetic field (for superconductors) and forcing (for Bose-Einstein), above which ground states exhibit nontrivial vorticity, as well as a characterization of the vortex density in terms of a non local vector-valued generalization of the classical obstacle problem.