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, where ? represent the coherence length (in the Ginzburg-Landau model) or the core-radius size (in the core-radius approach) and ? denotes the periodicity scale. We carry out the ? -convergence analysis of E as ?-> 0 and ?= ?-> 0 in the | log ?| scaling regime, showing that the ? -limit consists in the energy cost of finitely many vortex-like point singularities of integer degree. After introducing the scale parameter ?=min{1,lim?->0|log??||log?|}(upon extraction of subsequences), we show that in a sense we always have a separation-of-scale effect: at scales smaller than ? we first have a concentration process around some vortices whose location is subsequently optimized, while for scales larger than ? the concentration process takes place "after" homogenization.