# From the $N$-clock model to the $XY$ model: emergence of concentration effects in the variational analysis

created by orlando on 06 Aug 2019
modified on 07 Aug 2019

[BibTeX]

Preprint

Inserted: 6 aug 2019
Last Updated: 7 aug 2019

Year: 2019

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 as both the number of particles and $N$ diverge. By suitably rescaling the energy of the $N$-clock model, we illustrate how its thermodynamic limit strongly depends on the rate of divergence of~$N$ with respect to the number of particles. The $N$-clock model turns out to be a good approximation of the $XY$ model only for $N$ sufficiently large; in other regimes of $N$, we show with the aid of cartesian currents that its asymptotic behavior can be described by an energy which may concentrate on geometric objects of various dimensions.