Variational equivalence between Ginzburg-Landau, XY spin systems and screw dislocations energies

TL;DR

This paper establishes a rigorous mathematical connection between screw dislocations in crystals, XY spin systems, and Ginzburg-Landau theory, showing their energies are asymptotically equivalent in various regimes, which advances understanding of dislocation behavior.

Contribution

It introduces a notion of asymptotic variational equivalence and proves the equivalence of energies in different models, linking dislocations to vortices in superconductivity.

Findings

Variational equivalence of models in multiple regimes

New insights into screw dislocation asymptotics

Rigorous mathematical framework for dislocation-vortex analogy

Abstract

We introduce and discuss discrete two-dimensional models for XY spin systems and screw dislocations in crystals. We prove that, as the lattice spacing $\e$ tends to zero, the relevant energies in these models behave like a free energy in the complex Ginzburg-Landau theory of superconductivity, justifying in a rigorous mathematical language the analogies between screw dislocations in crystals and vortices in superconductors. To this purpose, we introduce a notion of asymptotic variational equivalence between families of functionals in the framework of $\Gamma$-convergence. We then prove that, in several scaling regimes, the complex Ginzburg-Landau, the XY spin system and the screw dislocation energy functionals are variationally equivalent. Exploiting such an equivalence between dislocations and vortices, we can show new results concerning the asymptotic behavior of screw dislocations in the $|\log\e|^2$ energetic regime.