Inserted: 10 jul 2009
Last Updated: 20 nov 2013
Journal: Indiana Univ. Math. J.
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