Thermodynamics of a Tiling Model

TL;DR

This study investigates the thermodynamic properties of a two-dimensional Wang tile tiling model at finite temperature using Monte Carlo simulations, revealing a possible continuous phase transition and aging phenomena.

Contribution

It introduces a Hamiltonian for Wang tiles with a large degeneracy and analyzes its thermodynamic behavior, including phase transition and aging, through numerical simulations.

Findings

Evidence of a continuous phase transition at non-zero temperature

Presence of aging properties below the critical temperature

Violation of the fluctuation-dissipation theorem in the model

Abstract

A particular, two-dimensional, tiling model, composed by the so called Wang tiles has been studied at finite temperature by Monte Carlo numerical simulations. In absence of any thermal bath the Wang tiles give the opportunity of building a very large number of non-periodic tilings. We can construct a local Hamiltonian such that only perfectly matched tilings are ground states with zero energy. This Hamiltonian has a very large degeneracy. The thermodynamic behaviour of such a system seems to show a continuous phase transition at non zero temperature. An order parameter with non-trivial features is proposed. Under the critical temperature the model exhibits aging properties. The fluctuation-dissipation theorem is violated.