Inserted: 6 aug 2019
Last Updated: 2 apr 2021
Journal: Comm. Pure Appl. Math.
We investigate the relationship between the $N$-clock model (also known as planar Potts model or $Z_N$-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.