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 parameter lambda = 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.

Automated summary

This study examines the emergence of topological singularities in periodic media within the Ginzburg-Landau model and the core-radius approach. Through Gamma-convergence analysis, it is found that there are finite number of vortex-like point singularities of integer degree at specific scales. Additionally, a separation-of-scale effect is demonstrated, where concentration processes occur around vortices at certain scales before subsequent optimization, followed by homogenization at larger scales.