M.Sc Thesis
M.Sc Student Nikola Nikolai Stastistical Mechanics of an Aperiodic Tiling Model Department of Physics Professor Dov Levine

Abstract

The equilibrium statistical mechanics of a tiling model based on the aperiodic set of Wang tiles by Kari & Culik (KC) are numerically investigated. The KC set is the smallest known aperiodic set of Wang tiles, and is set apart from most other aperiodic sets in that its tilings are not produced by an inflation rule. The statistical mechanics model based on the KC set has ground states that are identical to finite tilings of the KC set, and as such the KC model has extensive ground state entropy, with ground states that exhibit quasi-periodic symmetry in the vertical direction and are disordered in the horizontal direction. It is found that the transition from the quasi-periodic ground state to the high temperature disordered phase proceeds through a cascade of periodic phases, separated by first order transitions, such that the length of the period increases, on average, as the temperature decreases, and the characteristic frequencies of the periodic phases are rational approximations of the irrational characteristic frequency of the quasi-periodic ground-state. We argue that the cascade of periodic phases can be explained by an analogy to the ground state configurations of the Frenkel-Kontorova model. We show that the free energy of the Kari-Culik model can be expressed in a form similar to the zero temperature Hamiltonian of the Frenkel-Kontorova model, with two incommensurate competing length scales, such that the temperature plays the role of the substrate potential amplitude. In this analogy, the quasi-periodic phase at low temperature corresponds to the incommensurate phase of the Frenkel-Kontorova model, while the periodic phases at higher temperatures correspond to the commensurate phases, and are imposed on the system by the entropy part of the free energy. A possible finite temperature commensurate-incommensurate transition is indicated by numerical results, with long range quasi-periodic order persistent at low positive temperatures.