Physics of a Limit-Periodic Structure

This study concerns the properties of physical systems that can spontaneously form non-periodic structures. It focuses on the Socolar-Taylor tiling model in which a single space-filling prototile forces a limit-periodic pattern–a state made up of the union of infinite levels of periodic structures with ever-increasing sizes–through local interaction rules. A two-dimensional lattice model, which possesses the SocolarTaylor tiling as its ground state is constructed. It is known that during a slow quench from an initial high-temperature, disordered phase, the ground state of the model emerges through an infinite sequence of phase transitions. As temperature is decreased, sublattices with periodic structures of increasing lattice constants become ordered. In this study, we construct a theory based on one-dimensional Ising model to explain the time scales required for equilibrium to be reached at a given temperature by sublattices of increasing lattice constants. We observe a discrepancy in the scaling behavior predicted by our theory and obtained from simulation of the tiling model, which is likely due to finite size effect of the tiling model. We find that during a rapid quench, the energy barriers created by competing domain walls cause the system to fall out of equilibrium. Two types of domain wall with different physical structures and energy costs are found in the system. The associated energetics of each type of domain wall is discussed, and a particular type of domain wall is identified as responsible for slowing down the ordering of the tiling system.