Emergence of concentration effects in the variational analysis of the $N$-clock model

Abstract

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.

References in corpus (2)

• Interfaces, modulated phases and textures in lattice systems
• A Ginzburg-Landau model with topologically induced free discontinuities