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

• Published in: Archive for rational mechanics and analysis 2022-02, Vol.243 (2), p.559-609
• Main Authors: Alicandro, Roberto, Braides, Andrea, Cicalese, Marco, De Luca, Lucia, Piatnitski, Andrey
• Format: Article
• Language: English
• Subjects: Classical Mechanics; Coherence length; Complex Systems; Digital Object Identifier; Energy; Energy costs; Fluid- and Aerodynamics; Homogenization; Mathematical and Computational Physics; Phase transitions; Physics; Physics and Astronomy; Scale effect; Singularities; Theoretical; Topology; Vortices
• ISSN: 0003-9527; 1432-0673; 1432-0673
• DOI: 10.1007/s00205-021-01731-7

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