From hard spheres to hard-core spins

TL;DR

This paper introduces hardcore spin models inspired by hard spheres, revealing a Kosterlitz-Thouless transition between disordered and ordered phases driven by entropic effects, using advanced computational methods.

Contribution

It develops a new class of hardcore spin models with detailed analysis of phase transitions, generalizing Potts models and exploring their critical behavior.

Findings

Identified a Kosterlitz-Thouless transition in hardcore spin models.

Demonstrated order by disorder driven by entropic considerations.

Used height representations, cluster algorithms, and transfer matrix methods.

Abstract

A system of hard spheres exhibits physics that is controlled only by their density. This comes about because the interaction energy is either infinite or zero, so all allowed configurations have exactly the same energy. The low density phase is liquid, while the high density phase is crystalline, an example of "order by disorder" as it is driven purely by entropic considerations. Here we study a family of hard spin models, which we call hardcore spin models, where we replace the translational degrees of freedom of hard spheres with the orientational degrees of freedom of lattice spins. Their hardcore interaction serves analogously to divide configurations of the many spin system into allowed and disallowed sectors. We present detailed results on the square lattice in $d=2$ for a set of models with $\mathbb{Z}_n$ symmetry, which generalize Potts models, and their $U(1)$ limits, for ferromagnetic and antiferromagnetic senses of the interaction, which we refer to as exclusion and inclusion models. As the exclusion/inclusion angles are varied, we find a Kosterlitz-Thouless phase transition between a disordered phase and an ordered phase with quasi-long-ranged order, which is the form order by disorder takes in these systems. These results follow from a set of height representations, an ergodic cluster algorithm, and transfer matrix calculations.