The N-Clock Model: Variational Analysis for Fast and Slow Divergence Rates of N

We study a nearest neighbors ferromagnetic classical 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 the 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 as the classical XY model (also known as planar rotator 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.