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

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 $E_{\varepsilon,δ}$, where $\varepsilon$ represent the coherence length (in the Ginzburg-Landau model) or the core-radius size (in the core-radius approach) and $δ$ denotes the periodicity scale. We carry out the $Γ$-convergence analysis of $E_{\varepsilon,δ}$ as $\varepsilon\to 0$ and $δ=δ_{\varepsilon}\to 0$ in the $|\log\varepsilon|$ scaling regime, showing that the $Γ$-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) $$ λ=\min\Bigl\{1,\lim_{\varepsilon\to0} {|\log δ_{\varepsilon}|\over|\log{\varepsilon}|}\Bigr\}, $$ we show that in a sense we always have a separation-of-scale effect: at scales less than $\varepsilon^λ$ we first have a concentration process around some vortices whose location is subsequently optimized, while for scales larger than $\varepsilon^λ$ the concentration process takes place "after" homogenization.