Calculus of Variations and Geometric Measure Theory

M. Cicalese - G. Orlando - M. Ruf

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

created by orlando on 07 Apr 2020
modified by cicalese on 25 Jun 2022


Published Paper

Inserted: 7 apr 2020
Last Updated: 25 jun 2022

Journal: Interfaces and Free Boundaries
Volume: 23
Number: 3
Pages: 323-351
Year: 2021

ArXiv: 2004.02217 PDF


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.