Inserted: 3 may 2007
Last Updated: 16 dec 2009
Journal: Archive Rational Mech. Anal.
In this paper we consider the $XY$ ($N$-dimensional possibly anisotropic) spin type model and, by comparison with a Ginzburg-Landau type functional, we perform a variational analysis in the limit when the number of particles diverges. In particular we show how the appearance of vortex-like singularities can be described by properly scaling the energy of the system through a $\Gamma$-convergence procedure. We also address the problem in the case of long range interactions and solve it in $2$-dimensions.