Numerical bifurcation study of superconducting patterns on a square

TL;DR

This study numerically investigates vortex pattern bifurcations in superconductors modeled by Ginzburg-Landau equations on a square, revealing stability transitions and solution branch connections through detailed bifurcation analysis.

Contribution

It introduces a numerical approach combining Newton's method and gauge-invariant discretization to analyze bifurcations in superconducting vortex patterns.

Findings

Identification of bifurcation points as magnetic field varies

Insights into stability and instability transitions

Mapping of solution branch connections

Abstract

This paper considers the extreme type-II Ginzburg-Landau equations that model vortex patterns in superconductors. The nonlinear PDEs are solved using Newton's method, and properties of the Jacobian operator are highlighted. Specifically, it is illustrated how the operator can be regularized using an appropriate phase condition. For a two-dimensional square sample, the numerical results are based on a finite-difference discretization with link variables that preserves the gauge invariance. For two exemplary sample sizes, a thorough bifurcation analysis is performed using the strength of the applied magnetic field as a bifurcation parameter and focusing on the symmetries of this system. The analysis gives new insight in the transitions between stable and unstable states, as well as the connections between stable solution branches.