Vortices and Magnetization in Kac’s Model

We consider a 2-dimensional planar rotator on a large, but finite lattice with a ferromagnetic Kac potential J _γ(i)=γ ² J(γ i), J with compact support. The system is subject to boundary conditions with vorticity. Using a gradient-flow dynamics, we compute minimizers of the free energy functional at low temperature, i.e. in the regime of phase transition. We have the numerical evidence of a vortex structure for minimizers, which present many common features with those of the Ginzburg-Landau functional. We extend the results to spins valued in S ² and compare with the celebrated Belavin and Polyakov model.