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 Eε,δ, where ε 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 Eε,δ as ε→ 0 and δ= δε→ 0 in the | log ε| 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 λ=min{1,limε→0|logδε||logε|}(upon extraction of subsequences), we show that in a sense we always have a separation-of-scale effect: at scales smaller than ελ we first have a concentration process around some vortices whose location is subsequently optimized, while for scales larger than ελ the concentration process takes place “after” homogenization.