Calculus of Variations and Geometric Measure Theory

R. Alicandro - M. Cicalese - M. Ponsiglione

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

created by ponsiglio on 10 Jul 2009
modified by alicandr on 20 Nov 2013


Published Paper

Inserted: 10 jul 2009
Last Updated: 20 nov 2013

Journal: Indiana Univ. Math. J.
Volume: 60
Number: 1
Pages: 171-208
Year: 2011


We introduce and discuss discrete two-dimensional models for $XY$ spin systems and screw dislocations in crystals.

We prove that, as the lattice spacing epsilon 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 $
^2$ energetic regime.

Keywords: calculus of variations, Analysis of microstructure in crystals , Discrete-to-continuum limits, Topological singularities