Emergence of Concentration Effects in the Variational Analysis of the N-Clock Model

We investigate the relationship between the N-clock model (also known as planar Potts model or (Formula presented.) -model) and the XY model (at zero temperature) through a Γ-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 that may concentrate on geometric objects of various dimensions. This energy prevails over the usual vortex-vortex interaction energy.