Coarse graining and large-$N$ behavior of the $d$-dimensional $N$-clock model

Abstract

We study the asymptotic behavior of the $N$-clock model, a nearest neighbors ferromagnetic spin model on the $d$-dimensional cubic $\epsilon$-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 $\epsilon \to 0$, 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.