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arXiv · 2012.09548

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

Marco Cicalese·Gianluca Orlando·Matthias Ruf
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Abstract

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.

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Marco Cicalese, Gianluca Orlando, Matthias Ruf. 2020-12-17. The $N$-clock model: Variational analysis for fast and slow divergence rates of $N$. https://arxiv.org/abs/2012.09548

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