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.
Type
Journal articleSubject
KineticsModels, Molecular
Molecular Conformation
Monte Carlo Method
Phase Transition
Thermodynamics
Permalink
https://hdl.handle.net/10161/12616Published Version (Please cite this version)
10.1103/PhysRevE.90.012136Publication Info
Marcoux, Catherine; Byington, Travis W; Qian, Zongjin; Charbonneau, Patrick; & Socolar,
Joshua ES (2014). Emergence of limit-periodic order in tiling models. Phys Rev E Stat Nonlin Soft Matter Phys, 90(1). pp. 012136. 10.1103/PhysRevE.90.012136. Retrieved from https://hdl.handle.net/10161/12616.This is constructed from limited available data and may be imprecise. To cite this
article, please review & use the official citation provided by the journal.
Collections
More Info
Show full item recordScholars@Duke
Patrick Charbonneau
Professor of Chemistry
Professor Charbonneau studies soft matter. His work combines theory and simulation
to understand the glass problem, protein crystallization, microphase formation, and colloidal
assembly in external fields.
Joshua Socolar
Professor of Physics
Prof. Socolar is interested in collective behavior in condensed matter and dynamical
systems. His current research interests include:
Limit-periodic structures, quasicrystals, packing problems, and tiling theory;
Self-assembly and phases of designed colloidal particles;
Shear jamming and stick-slip behavior in dry granular materials;
Organization and dynamics of complex networks;
Topological elasticity of mechanical lattices.
Alphabetical list of authors with Scholars@Duke profiles.

Articles written by Duke faculty are made available through the campus open access policy. For more information see: Duke Open Access Policy
Rights for Collection: Scholarly Articles
Works are deposited here by their authors, and represent their research and opinions, not that of Duke University. Some materials and descriptions may include offensive content. More info