Husimi-lattice solutions and the coherent-anomaly-method analysis for hard-square lattice gases

Rodrigues, Nathann T

doi:10.1103/physreve.103.032153

Cited by 7 publications

(9 citation statements)

References 100 publications

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“…In the 1-NN model, the four nearest neighbors of a particle are excluded from being occupied. As density is increased, the system is known to undergo a continuous transition from a disordered fluid phase to an ordered sublattice phase (see [27,57] and references within for the large body of work on this model). The transition is expected to belong to the Ising universality class: γ /ν = 7/4, β/ν = 1/8, α/ν = 0, and ν = 1 [27,49].…”

Section: A 1-nn Model In Two Dimensionsmentioning

confidence: 99%

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Rejection-free cluster Wang-Landau algorithm for hard-core lattice gases

et al. 2021

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We introduce a rejection-free, flat histogram, cluster algorithm to determine the density of states of hardcore lattice gases. We show that the algorithm is able to efficiently sample low entropy states that are usually difficult to access, even when the excluded volume per particle is large. The algorithm is based on simultaneously evaporating all the particles in a strip and reoccupying these sites with a new appropriately chosen configuration. We implement the algorithm for the particular case of the hard-core lattice gas in which the first k next-nearest neighbors of a particle are excluded from being occupied. It is shown that the algorithm is able to reproduce the known results for k = 1, 2, 3 both on the square and cubic lattices. We also show that, in comparison, the corresponding flat histogram algorithms with either local moves or unbiased cluster moves are less accurate and do not converge as the system size increases.

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Section: A 1-nn Model In Two Dimensionsmentioning

confidence: 99%

“…The intractability of the model has resulted in many attempts to obtain the critical density and chemical potential using systematic expansions and approximate methods. These include high activity expansions [33,62,63], estimates of surface tension between ordered phases [64][65][66], and limits of Husimi tree [57].…”

Section: B 2-nn Model In Two Dimensionsmentioning

confidence: 99%

Rejection-free cluster Wang-Landau algorithm for hard-core lattice gases

et al. 2021

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“…1. The solution of a given model in the core of the infinite tree (i.e., in the thermodynamic limit), known as Husimi lattice (HL) [26], can be seen as the first level of approximation for its behavior on the square lattice [25,27]. We may index the levels of approximation by the number of sites shared between each pair of adjacent clusters, which is L = 1 for this lattice.…”

Section: Ordinary Husimi Latticementioning

confidence: 99%

“…For instance, we may use, e.g., the Hoshen-Kopelman algorithm [37] to identify the possible clusters of empty sites in the interior of the RBB and, then, try to fully cover these clusters (if they exist) with k-mers through an exact enumeration process to find m kL (i; l, t, r). It turns out however that this complicated procedure can become very computationally demanding already for relatively small L's; at one hand, because N kL becomes large for large k and, on the other hand, because there are much bulk configurations for small k. Another possibility is the use of the RRs for the case L − 1 to obtain those for L, as recently done for hard squares in [25]. In fact, as illustrated in Fig.…”

Section: A Preliminariesmentioning

confidence: 99%

“…These differences lead us to inquire whether solutions on improved treelike lattices may provide more reliable approximations to the behavior of rods on the square lattice. For instance, in a recent paper [25], hard square lattice gases were investigated on a sequence of generalized Husimi lattices [26] (built with diagonal square lattice cells which share L sites with each of their four neighboring cells) and accurate estimates for the critical density and fugacity for the models on the square lattice were obtained from extrapolations to L → ∞ of the numerically exact results for increasing L. This is in agreement with previous findings by Monroe [27], who introduced and successfully applied this approach to determine the critical parameters of the Ising and other spin models. In this paper, we study fully-packed rods on these Husimi lattices, as well as on another sequence proposed by Kobayashi and Suzuki [28], where the diagonal square cells are replaced by regular square lattice cells that share L sites and L − 1 edges with neighboring ones.…”

Section: Introductionmentioning

confidence: 99%

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Entropy of fully-packed rigid rods on generalized Husimi trees: a route to the square lattice limit

Rodrigues,

Stilck,

Oliveira

2021

Preprint

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Although hard rigid rods (k-mers) defined on the square lattice have been widely studied in the literature, their entropy per site, s(k), in the full-packing limit is only known exactly for dimers (k = 2) and numerically for trimers (k = 3). Here, we investigate this entropy for rods with k ≤ 7, by defining and solving them on Husimi lattices built with diagonal and regular square lattice clusters of effective lateral size L, where L defines the level of approximation to the square lattice.Due to an L-parity effect, by increasing L we obtain two systematic sequences of values for the entropies s L (k) for each type of cluster, whose extrapolations to L → ∞ provide estimates of these entropies for the square lattice. For dimers, our estimates for s(2) differ from the exact result by only 0.03%, while that for s(3) differs from best available estimates by 3%. In this paper, we also obtain a new estimate for s(4). For larger k, we find that the extrapolated results from the Husimi tree calculations do not lie between the lower and upper bounds established in the literature for s(k). In fact, we observe that, to obtain reliable estimates for these entropies, we should deal with levels L that increase with k. However, it is very challenging computationally to advance to solve the problem for large values of L and for large rods. In addition, the exact calculations on the generalized Husimi trees provide strong evidence for the fully packed phase to be disordered for k ≥ 4, in contrast to the results for the Bethe lattice wherein it is nematic, thus providing evidence for a high density nematic-disordered transition in the system of k-mers with vacancies.

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Shape-interaction dualism: unraveling complex phase behavior in triangular particle monolayers

Akimenko,

Gorbunov,

Myshlyavtsev

et al. 2024

J. Phys.: Condens. Matter

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This paper examines the effect of finite attractive and repulsive interactions on the self-assembly of triangular-shaped particles on a triangular lattice. The ground state analysis of the lattice model has revealed an infinite sequence of ordered structures, a phenomenon referred to as the "devil's staircase" of phase transitions. The model has been studied at finite temperatures using both the transfer-matrix and tensor renormalization group methods. The concurrent use of these two methods lends credibility to the obtained results. It has been demonstrated that the initial ordered structures of the "devil's staircase" persist at non-zero temperatures. Further increase of the attraction between particles or a decrease of the temperature induces the appearance of subsequent ordered structures of the "devil's staircase". The corresponding phase diagram of the model has been calculated. The phase behavior of our model agrees qualitatively with the phase behavior of trimesic acid adsorption layer on single crystal surfaces.

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Husimi-lattice solutions and the coherent-anomaly-method analysis for hard-square lattice gases

Cited by 7 publications

References 100 publications

Rejection-free cluster Wang-Landau algorithm for hard-core lattice gases

Rejection-free cluster Wang-Landau algorithm for hard-core lattice gases

Entropy of fully-packed rigid rods on generalized Husimi trees: a route to the square lattice limit

Shape-interaction dualism: unraveling complex phase behavior in triangular particle monolayers

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