Emergence of limit-periodic order in tiling models

Abstract

A two-dimensional (2D) lattice model defined on a triangular lattice with nearest-and next-nearest-neighbor interactions based on the Taylor-Socolar monotile is known to have a limit-periodic ground state. The system reaches that state during a slow quench through an infinite sequence of phase transitions. We study the model as a function of the strength of the next-nearest-neighbor interactions and introduce closely related 3D models with only nearest-neighbor interactions that exhibit limit-periodic phases. For models with no next-nearest-neighbor interactions of the Taylor-Socolar type, there is a large degenerate class of ground states, including crystalline patterns and limit-periodic ones, but a slow quench still yields the limit-periodic state. For the Taylor-Socolar lattic model, we present calculations of the diffraction pattern for a particular decoration of the tile that permits exact expressions for the amplitudes and identify domain walls that slow the relaxation times in the ordered phases. For one of the 3D models, we show that the phase transitions are first order, with equilibrium structures that can be more complex than in the 2D case, and we include a proof of aperiodicity for a geometrically simple tile with only nearest-neighbor matching rules.

Assembly of hard spheres in a cylinder: a computational and experimental study

“…Although it typically receives less attention, their assembly pathways can be just as important [6,7]. While failure to assemble optimal structures often results in gel or glass formation, ordered suboptimal outcomes, including limit periodic structures [8] and quasicrystals [9], are also observed. Because formulating a general theory of out-of-equilibrium dynamics is challenging, insights into self-assembly are often first obtained through numerical simulations.…”

Sparse phonon modes of a limit-periodic structure

“…It has also been shown in simulations that a collection of identical achiral units with only nearest neighbor interactions can spontaneously form a hexagonal limit-periodic structure when slowly cooled 15,16 . With recent advances in colloidal particle synthesis, the fabrication of particles with the necessary interactions for formation of the LP structure seems experimentally feasible [17][18][19][20] .…”

“…Though no naturally occurring LP structures have been discovered, a recent result in tiling theory shows that local interactions among tiles that are identical up to reflection symmetry can favor the production of twoor three-dimensional hexagonal LP structures 14,15 . It has also been shown in simulations that a collection of identical achiral units with only nearest neighbor interactions can spontaneously form a hexagonal limit-periodic structure when slowly cooled 15,16 .…”

“…The structure is completely homogeneous in the sense that it consists of a uniform density of identical tiles. We note that in a statistical mechanical lattice model of this system this LP structure forms spontaneously in a slow quench from a state of disordered tile orientations 15 .…”

Formation of limit-periodic structures by quadrupole particles confined to a triangular lattice

“…For the systems considered here, the periodic subsets comprising the structure have densities that scale like 2n ξ − , which is reflected in the fact that peak amplitudes decay exponentially with n for i i j m δ = with any j . An example of a limit-periodic structure and its diffraction pattern can be seen in the work by Byington and Socolar in which kagome lattices of increasing size overlap to form a single limit-periodic structure [1,11]. Such structures have been shown to be ground states of systems with physically plausible Hamiltonians and have been observed to form through a sequence of phase transitions in simulations of slow cooling from a random initial state in several models [1,11,12].…”