Modelling quasicrystals at positive temperature

TL;DR

This paper investigates whether nonperiodic quasicrystal structures persist at positive temperatures using a 2D lattice model with Wang tiles, combining theoretical analysis and Monte Carlo simulations.

Contribution

It introduces a lattice model for quasicrystals with uncountably many ground states and demonstrates nonperiodicity at low temperatures through numerical methods.

Findings

Order parameter is zero at high temperatures

Order parameter becomes nonzero at low temperatures

Numerical evidence supports persistence of nonperiodicity

Abstract

We consider a two-dimensional lattice model of equilibrium statistical mechanics, using nearest neighbor interactions based on the matching conditions for an aperiodic set of 16 Wang tiles. This model has uncountably many ground state configurations, all of which are nonperiodic. The question addressed in this paper is whether nonperiodicity persists at low but positive temperature. We present arguments, mostly numerical, that this is indeed the case. In particular, we define an appropriate order parameter, prove that it is identically zero at high temperatures, and show by Monte Carlo simulation that it is nonzero at low temperatures.