Emergence of limit-periodic order in tiling models.

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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.





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Marcoux, Catherine, Travis W Byington, Zongjin Qian, Patrick Charbonneau and Joshua ES Socolar (2014). Emergence of limit-periodic order in tiling models. Phys Rev E Stat Nonlin Soft Matter Phys, 90(1). p. 012136. 10.1103/PhysRevE.90.012136 Retrieved from https://hdl.handle.net/10161/12616.

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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.

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