High-Activity Expansion for the Columnar Phase of the Hard Rectangle Gas

Nath, Trisha; Kundu, Joyjit; Rajesh, R.

doi:10.1007/s10955-015-1285-y

Cited by 21 publications

(26 citation statements)

References 56 publications

(117 reference statements)

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“…The existence of two transitions was rationalized by deriving a high density expansion about the ordered columnar phase. Columnar phases have a sliding instability in which a defect created by removing a single particle from the fully packed configuration splits into fractional defects that slide independently of each other along some direction [15,37,52]. The high density expansion for the 4-NN model showed that the sliding instability is present in only certain preferred sublattices.…”

Section: Introductionmentioning

confidence: 99%

The high density phase of thek-NN hard core lattice gas model

Nath¹,

Rajesh²

2016

J. Stat. Mech.

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The k-NN hard core lattice gas model on a square lattice, in which the first k next nearest neighbor sites of a particle are excluded from being occupied by another particle, is the lattice version of the hard disc model in two dimensional continuum. It has been conjectured that the lattice model, like its continuum counterpart, will show multiple entropy-driven transitions with increasing density if the high density phase has columnar or striped order. Here, we determine the nature of the phase at full packing for k up to 820302. We show that there are only eighteen values of k, all less than k = 4134, that show columnar order, while the others show solid-like sublattice order.

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Section: Introductionmentioning

confidence: 99%

The high density phase of thek-NN hard core lattice gas model

Nath¹,

Rajesh²

2016

J. Stat. Mech.

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“…An activity z is associated with each rectangle. The system is disordered at low densities and shows columnar order at high densities [48]. We call the phase odd (even) in which the majority of heads (bottom left corner) of the rectangles are in odd (even) rows.…”

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confidence: 99%

Stability of columnar order in assemblies of hard rectangles or squares

Nath¹,

Dhar²,

Rajesh³

2016

EPL

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Lattice theory and statistics PACS 64.60.De -Statistical mechanics of model systems PACS 64.60.Bd -General theory of phase transitionsAbstract -A system of 2 × d hard rectangles on square lattice is known to show four different phases for d ≥ 14. As the covered area fraction ρ is increased from 0 to 1, the system goes from low-density disordered phase, to orientationally-ordered nematic phase, to a columnar phase with orientational order and also broken translational invariance, to a high density phase in which orientational order is lost. For large d, the threshold density for the first transition ρ * 1 tends to 0, and the critical density for the third transition ρ * 3 tends to 1. Interestingly, simulations have shown that the critical density for the second transition ρ * 2 tends to a non-trivial finite value ≈ 0.73, as d → ∞, and ρ * 2 ≈ 0.93 for d = 2. We provide a theoretical explanation of this interesting result. We develop an approximation scheme to calculate the surface tension between two differently ordered columnar phases. The density at which the surface tension vanishes gives an estimate ρ * 2 = 0.746, for d → ∞, and ρ * 2 = 0.923 for d = 2. For all values of d, these estimates are in good agreement with Monte Carlo data.

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“…Thus, for larger k, the layered-columnar transition would appear to be similar to the disorderedcolumnar transition in k×k hard squares. For this model, high density expansions suggest that the critical density tends to an asymptotic value that is less than one for large k [61]. If this were true, the continuum problem should have two transitions.…”

Section: Discussionmentioning

confidence: 82%

Phase diagram of a system of hard cubes on the cubic lattice

et al. 2019

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We study the phase diagram of a system of 2 × 2 × 2 hard cubes on a three dimensional cubic lattice. Using Monte Carlo simulations, we show that the system exhibits four different phases as the density of cubes is increased: disordered, layered, sublattice ordered, and columnar ordered. In the layered phase, the system spontaneously breaks up into parallel slabs of size 2 × L × L where only a very small fraction cubes do not lie wholly within a slab. Within each slab, the cubes are disordered; translation symmetry is thus broken along exactly one principal axis. In the solid-like sublattice ordered phase, the hard cubes preferentially occupy one of eight sublattices of the cubic lattice, breaking translational symmetry along all three principal directions. In the columnar phase, the system spontaneously breaks up into weakly interacting parallel columns of size 2 × 2 × L where only a very small fraction cubes do not lie wholly within a column. Within each column, the system is disordered, and thus translational symmetry is broken only along two principal directions. Using finite size scaling, we show that the disordered-layered phase transition is continuous, while the layered-sublattice and sublattice-columnar transitions are discontinuous. We construct a Landau theory written in terms of the layering and columnar order parameters, which is able to describe the different phases that are observed in the simulations and the order of the transitions. Additionally, our results near the disordered-layered transition are consistent with the O(3) universality class perturbed by cubic anisotropy as predicted by the Landau theory.

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High-Activity Expansion for the Columnar Phase of the Hard Rectangle Gas

Cited by 21 publications

References 56 publications

The high density phase of thek-NN hard core lattice gas model

The high density phase of thek-NN hard core lattice gas model

Stability of columnar order in assemblies of hard rectangles or squares

Phase diagram of a system of hard cubes on the cubic lattice

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