Topological singularities in periodic media: Ginzburg-Landau and core-radius approaches

Abstract

We describe the emergence of topological singularities in periodic media within the Ginzburg-Landau model and the core-radius approach. The energy functionals of both models are denoted by $E_{\epsilon ,\delta }$, where $\epsilon$ represent the coherence length (in the Ginzburg-Landau model) or the core-radius size (in the core-radius approach) and $\delta$ denotes the periodicity scale. We carry out the $\Gamma$-convergence analysis of $E_{\epsilon ,\delta }$ as $\epsilon \to 0$ and $\delta ={\delta }_{\epsilon }\to 0$ in the $|\log \epsilon |$ scaling regime, showing that the $\Gamma$-limit consists in the energy cost of finitely many vortex-like point singularities of integer degree. After introducing the scale parameter (upon extraction of subsequences) $$\lambda =\min \{1,\underset{\epsilon \to 0}{\lim }\frac{|\log {\delta }_{\epsilon }|}{|\log \epsilon |}\},$$ we show that in a sense we always have a separation-of-scale effect: at scales less than ${\epsilon }^{\lambda }$ we first have a concentration process around some vortices whose location is subsequently optimized, while for scales larger than ${\epsilon }^{\lambda }$ the concentration process takes place "after" homogenization.