Hierarchical Freezing in a Lattice Model

Byington, Travis W.; Socolar, Joshua E. S.

doi:10.1103/physrevlett.108.045701

Cited by 5 publications

(16 citation statements)

References 12 publications

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“…Key thermodynamic properties of a lattice model based on the 2D Socolar-Taylor tile were reported by Byington and Socolar, who presented strong evidence that the system undergoes an infinite sequence of phase transitions if cooled sufficiently slowly, identified a set of order parameters for the transitions, and found approximate (but highly accurate) scaling relations between the values of the order parameters in equilibrium at rescaled temperatures [18]. In the present paper, we review those results and extend them in several directions.…”

Section: Introductionmentioning

confidence: 93%

“…[18], the hierarchy of phase transitions can lead the system to fall out of equilibrium when quenched too rapidly. Roughly speaking, simultaneous attempts to establish order at two or more levels creates a competition resulting in defects that require extremely long times to heal due to their complex geometric and topological structures.…”

Section: Temporal Scaling and Kinetic Barriersmentioning

confidence: 99%

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Emergence of limit-periodic order in tiling models

Marcoux

Byington²,

Qian

et al. 2014

Phys. Rev. E

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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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Section: Introductionmentioning

confidence: 93%

Section: Temporal Scaling and Kinetic Barriersmentioning

confidence: 99%

Emergence of limit-periodic order in tiling models

Marcoux

Byington²,

Qian

et al. 2014

Phys. Rev. E

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“…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] .…”

Section: Introductionmentioning

confidence: 95%

Sparse phonon modes of a limit-periodic structure

Marcoux

Socolar

2016

Phys. Rev. B

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Limit-periodic structures are well ordered but nonperiodic, and hence have nontrivial vibrational modes. We study a ball and spring model with a limit-periodic pattern of spring stiffnesses and identify a set of extended modes with arbitrarily low participation ratios, a situation that appears to be unique to limit-periodic systems. The balls that oscillate with large amplitude in these modes live on periodic nets with arbitrarily large lattice constants. By studying periodic approximants to the limit-periodic structure, we present numerical evidence for the existence of such modes, and we give a heuristic explanation of their structure.

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“…Nonperiodic structures with long range order are of interest both theoretically and experimentally due to their novel (expected or observed) electronic, photonic, elastic and frictional properties [1]. Since the discovery of quasicrystals in the 1980's the definition of the term "crystal" has changed to include nonperiodic structures, and now refers to any structures with sharp diffraction peaks indicating long-ranged translational order [2].…”

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confidence: 99%

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

et al. 2017

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We have performed Monte Carlo (MC) simulations on two-dimensional systems of quadrupole particles confined to a triangular lattice in order to determine the conditions that permit the formation of a limit-periodic phase. We have found that limit-periodic structures form only when the rotations of the particles are confined to a set of six orientations aligned with the lattice directions. Related structures including striped and unidirectional rattler phases form when π/π66 rotations or continuous rotations are allowed. Order parameters signaling the formation of the limit-periodic structure and related structures are measured as a function of temperature. Our findings on the formation of the limit-periodic structure elucidate features relevant to the experimental creation of such a structure, which is expected to have interesting vibrational and electromagnetic modes.

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Hierarchical Freezing in a Lattice Model

Cited by 5 publications

References 12 publications

Emergence of limit-periodic order in tiling models

Emergence of limit-periodic order in tiling models

Sparse phonon modes of a limit-periodic structure

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

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