  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.
