A classical S² spin system with discrete out-of-plane anisotropy: Variational analysis at surface and vortex scalings

We consider a classical Heisenberg system of S² spins on a square lattice of spacing ɛ. We introduce a magnetic anisotropy by constraining the out-of-plane component of each spin to take only finitely many values. Computing the Γ-limit of a suitable scaling of the energy functional as ɛ→0 we prove that, in the continuum description, the system concentrates energy at the boundary of sets in which the out-of-plane component of the spin is constant. In a second step we analyze a different scaling of the energy and we prove that, in each of such phases, the energy can further concentrate on finitely many points corresponding to vortex-like singularities of the in-plane components of the spins.