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Emergence of limit-periodic order in tiling models.

Publication ,  Journal Article
Marcoux, C; Byington, TW; Qian, Z; Charbonneau, P; Socolar, JES
Published in: Physical review. E, Statistical, nonlinear, and soft matter physics
July 2014

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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Published In

Physical review. E, Statistical, nonlinear, and soft matter physics

DOI

EISSN

1550-2376

ISSN

1539-3755

Publication Date

July 2014

Volume

90

Issue

1

Start / End Page

012136

Related Subject Headings

  • Thermodynamics
  • Phase Transition
  • Monte Carlo Method
  • Molecular Conformation
  • Models, Molecular
  • Kinetics
  • Fluids & Plasmas
  • 09 Engineering
  • 02 Physical Sciences
  • 01 Mathematical Sciences
 

Citation

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Marcoux, C., Byington, T. W., Qian, Z., Charbonneau, P., & Socolar, J. E. S. (2014). Emergence of limit-periodic order in tiling models. Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics, 90(1), 012136. https://doi.org/10.1103/physreve.90.012136
Marcoux, Catherine, Travis W. Byington, Zongjin Qian, Patrick Charbonneau, and Joshua E. S. Socolar. “Emergence of limit-periodic order in tiling models.Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics 90, no. 1 (July 2014): 012136. https://doi.org/10.1103/physreve.90.012136.
Marcoux C, Byington TW, Qian Z, Charbonneau P, Socolar JES. Emergence of limit-periodic order in tiling models. Physical review E, Statistical, nonlinear, and soft matter physics. 2014 Jul;90(1):012136.
Marcoux, Catherine, et al. “Emergence of limit-periodic order in tiling models.Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics, vol. 90, no. 1, July 2014, p. 012136. Epmc, doi:10.1103/physreve.90.012136.
Marcoux C, Byington TW, Qian Z, Charbonneau P, Socolar JES. Emergence of limit-periodic order in tiling models. Physical review E, Statistical, nonlinear, and soft matter physics. 2014 Jul;90(1):012136.

Published In

Physical review. E, Statistical, nonlinear, and soft matter physics

DOI

EISSN

1550-2376

ISSN

1539-3755

Publication Date

July 2014

Volume

90

Issue

1

Start / End Page

012136

Related Subject Headings

  • Thermodynamics
  • Phase Transition
  • Monte Carlo Method
  • Molecular Conformation
  • Models, Molecular
  • Kinetics
  • Fluids & Plasmas
  • 09 Engineering
  • 02 Physical Sciences
  • 01 Mathematical Sciences