  We investigate the relationship between the $N$-clock model (also known as
planar Potts model or $\mathbb{Z}_N$-model) and the $XY$ model (at zero
temperature) through a $\Gamma$-convergence analysis of a suitable rescaling of
the energy as both the number of particles and $N$ diverge. We prove the
existence of rates of divergence of $N$ for which the continuum limits of the
two models differ. With the aid of Cartesian currents we show that the
asymptotics of the $N$-clock model in this regime features an energy which may
concentrate on geometric objects of various dimensions. This energy prevails
over the usual vortex-vortex interaction energy.
