Calculus of Variations and Geometric Measure Theory

M. Cicalese - G. Orlando - M. Ruf

The N-clock model: variational analysis for fast and slow divergence rates of N

created by cicalese on 02 Apr 2021
modified on 07 May 2022


Accepted Paper

Inserted: 2 apr 2021
Last Updated: 7 may 2022

Journal: Archive for Rational Mechanics and Analysis
Year: 2020

ArXiv: 2012.09548 PDF


We study a nearest neighbors ferromagnetic spin system on the square lattice in which the spin field is constrained to take values in a discretization of the unit circle consisting of $N$ equi-spaced vectors, also known as $N$-clock model. We find a fast rate of divergence of $N$ with respect to the lattice spacing for which the $N$-clock model has the same discrete-to-continuum variational limit of the $XY$ model, in particular concentrating energy on topological defects of dimension 0. We prove the existence of a slow rate of divergence of $N$ at which the coarse-grain limit does not detect topological defects, but it is instead a $BV$-total variation. Finally, the two different types of limit behaviors are coupled in a critical regime for $N$, whose analysis requires the aid of Cartesian currents.