Topological Singularities in Periodic Media: Ginzburg-Landau and Core-Radius Approaches

We describe the emergence of topological singularities in periodicmediawithin the Ginzburg-Landau model and the core-radius approach. The energy functionals of both models are denoted by E-epsilon,E- delta, where e 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,E- delta as epsilon -> 0 and delta = delta(epsilon) -> 0 in the vertical bar log epsilon vertical bar 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 parameterlambda = min{1, lim (epsilon -> 0) vertical bar log delta(epsilon)vertical bar/vertical bar log epsilon vertical bar}(upon extraction of subsequences), we show that in a sense we always have a separation-of-scale effect: at scales smaller 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.