Symmetry Breaking in Annular Domains for a Ginzburg-Landau Superconductivity Model

Abstract

A minimization problem for the Ginzburg-Landau functional is considered on an annulus with thickness R in the class of functions \( \mathcal{F}^d\) where only the degree (the winding number), d, on the boundary is prescribed. The existence of a critical thickness R c which depends on the Ginzburg-Landau parameter δ and d is established. It is shown that if R>Rc then any function in separated form in polar coordinates can not be a global minimizer in the class \( \mathcal{F}^d\) and therefore the functions in a minimizing sequence can not be of this form. Hence for a sufficiently thick annulus, the variational Ginzburg-Landau problem is not a regular perturbation, in δ, of the corresponding problem for harmonic maps. The proof is based on an explicit construction of a sequence of admissible functions with vortices approaching the boundary. The key point of the proof is that the Ginzburg-Landau energy of the functions in this sequence approaches a constant independent of R and δ. By contrast, in the class of functions with separated form in polar coordinates and the same boundary conditions, there exists a unique minimizer whose energy depends on R. This minimizer is rotationally invariant and converges uniformly to the solution of the corresponding harmonic map problem as δ→0.KeywordsCritical ThicknessBlaschke ProductUnique MinimizerSeparate FormRegular PerturbationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.