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arXiv · 2004.02217

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

Marco Cicalese·Gianluca Orlando·Matthias Ruf
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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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Marco Cicalese, Gianluca Orlando, Matthias Ruf. 2020-04-05. Coarse graining and large-$N$ behavior of the $d$-dimensional $N$-clock model. https://arxiv.org/abs/2004.02217

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