Emergence of limit-periodic structure without matching rules

We study the emergence of nonperiodic order in a tiling model based on a certain 2D hexagonal prototile with nearest-neighbor interactions. The model is closely related to the Taylor-Socolar tiling model~[1], but with a simpler Hamiltonian with a degenerate class of ground states that includes both periodic and limit-periodic structures. We present the results of a lattice Monte Carlo simulation of the orientational degrees of freedom of a system of the prototiles. We find that the limit-periodic structure emerges from a sufficiently slow quench through the same infinite sequence of second-order phase transitions observed in the full Taylor-Socolar model. A related 3D model with a simple cubic prototile exhibits similar behavior, but with first-order transitions and a more complex set of limit-periodic ground states.\\[4pt] [1] T. W. Byington and J. E. S. Socolar, {\it Phys. Rev. Lett.} {\bf108}, 045701 (2012).