*Accepted Paper*

**Inserted:** 21 dec 2021

**Last Updated:** 7 may 2022

**Journal:** Nonlinear Analysis

**Year:** 2021

**Abstract:**

We consider a classical Heisenberg system of $\mathbb{S}^2$ spins on a square lattice of spacing $\varepsilon$. We introduce a magnetic anisotropy by constraining the out-of-plane component of each spin to take only finitely many values. Computing the $\Gamma$-limit of the energy functional as $\varepsilon\to 0$ 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 and 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.

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