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

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 p...

Full description

Saved in:
Bibliographic Details
Published inInterfaces and free boundaries Vol. 23; no. 3; pp. 323 - 351
Main Authors Cicalese, Marco, Orlando, Gianluca, Ruf, Matthias
Format Journal Article
LanguageEnglish
Published European Mathematical Society Publishing House 01.09.2021
Subjects
Convergence (Mathematics)
Ferromagnetism
Lattice theory
Mathematical models
Particle spin
Switzerland
Online AccessGet full text

Cover

More Information
Summary: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.
ISSN:1463-9963
1463-9971
DOI:10.4171/ifb/456