Nucleation of bulk superconductivity close to critical magnetic field

TL;DR

This paper analyzes the behavior of bulk superconductivity near the critical magnetic field in a 2D Ginzburg-Landau model, providing energy estimates and linking to the Abrikosov lattice problem.

Contribution

It offers precise energy estimates for superconducting states near the second critical field and introduces a new $L^ Infty$-bound technique.

Findings

Identifies a threshold magnetic field for bulk superconductivity contribution.

Relates the energy of bulk states to the Abrikosov lattice problem.

Provides a novel $L^ Infty$-bound of independent interest.

Abstract

We consider the two-dimensional Ginzburg-Landau functional with constant applied magnetic field. For applied magnetic fields close to the second critical field $H_{C_2}$ and large Ginzburg-Landau parameter, we provide leading order estimates on the energy of minimizing configurations. We obtain a fine threshold value of the applied magnetic field for which bulk superconductivity contributes to the leading order of the energy. Furthermore, the energy of the bulk is related to that of the Abrikosov problem in a periodic lattice. A key ingredient of the proof is a novel $L^\infty$-bound which is of independent interest.