*Accepted Paper*

**Inserted:** 7 apr 2020

**Last Updated:** 2 apr 2021

**Journal:** Interfaces and Free Boundaries

**Year:** 2020

**Abstract:**

We study the asymptotic behavior of the $N$-clock model, a nearest neighbors ferromagnetic spin model on the $d$-dimensional cubic $\varepsilon$-lattice in which the spin field is constrained to take values in a discretization $\mathcal{S}_N$ of the unit circle~$\mathbb{S}^{1}$ consisting of $N$ equispaced points. Our $\Gamma$-convergence analysis consists of two steps: we first fix $N$ and let the lattice spacing $\varepsilon \to 0$, obtaining an interface energy in the continuum defined on piecewise constant spin fields with values in $\mathcal{S}_N$; at a second stage, we let $N \to +\infty$. The final result of this two-step limit process is an anisotropic total variation of $\mathbb{S}^1$-valued vector fields of bounded variation.

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