  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.
