Critical phenomena of the hard-sphere lattice gas on the simple cubic lattice

Yamagata, Atsushi

doi:10.1016/0378-4371(95)00282-0

Cited by 10 publications

(6 citation statements)

References 29 publications

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“…The values of the critical exponents are consistent with those of the hardsphere lattice gas on the simple cubic lattice, β/ν = 0.313 (9) and γ/ν = 2.37(2) [19]. They are different from the corresponding values of the Ising model, β/ν = 0.518 (7) and γ/ν = 1.9828(57) [21].…”

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confidence: 54%

Universality in the three-dimensional hard-sphere lattice gas

Yamagata

1996

Physica A: Statistical Mechanics and its Applications

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Abstract.We perform Monte Carlo simulations of the hard-sphere lattice gas on the body-centred cubic lattice with nearest neighbour exclusion. We get the critical exponents, β/ν = 0.311(8) and γ/ν = 2.38(2), where β, γ, and ν are the critical exponents of the staggered density, the staggered compressibility, and the correlation length, respectively. The values of the hard-sphere lattice gas on the simple cubic lattice agree with them but those of the threedimensional Ising model do not. This supports that the hard-sphere lattice gas does not fall into the Ising universality class in three dimensions.In this letter we study the hard-sphere lattice gas whose atoms interact with infinite repulsion of nearest neighbour pairs. The grand partition function iswhere z is an activity and Z V (N) is the number of configurations in which there are N atoms in the lattice of V sites. There are many studies on: the square lattice [1,2,3,4,5,6,7,8,9,10,11,12,13], the triangular lattice [3,6,9,10,14,15,16], the honeycomb lattice [17], the simple cubic lattice [1,3,9,14,18,19], the body-centred cubic lattice [3,9,14,20], and the face-centred cubic lattice [3]. On the simple cubic lattice the author has estimated the critical exponents, β/ν = 0.313(9) and γ/ν = 2.37(2), where β, γ, and ν are the critical exponents of the staggered density, the staggered compressibility, and the correlation length, respectively [18]. The corresponding values of the Ising model are β/ν = 0.518(7) and γ/ν = 1.9828(57) [21]. It does not seem that the hard-sphere lattice gas falls into the Ising universality class in three dimensions. The purpose of this letter is to obtain another piece of evidence of that. We estimate the critical exponents of the hard-sphere lattice gas on the body-centred cubic lattice.We carry out Monte Carlo simulations [22,23] of the hard-sphere lattice gas (1) on the body-centred cubic lattice of V sites, where V = 2×L×L×L (L = 2 × n, n = 2, 3, . . . , 30), under fully periodic boundary conditions [11,18]. We measure the staggered density,and the staggered compressibility,where R = 2 (N A − N B )/V and N A (N B ) is the number of the atoms in the A (B)-sublattice. Each run is divided into ten or twelve blocks. · · · is an expectation in a block and · · · is one over blocks. All the simulations are done at the critical activity, z c = 0.7223, [20] over 12 × 10 5 Monte Carlo steps per site (MCS/site) or 10 × 10 5 MCS/site after discarding 5 × 10 4 MCS/site to attain equilibrium. We have checked that simulations from the ground state configuration (The atoms occupy all the sites of one sublattice and the other 1 is vacant.) and no atom one gave consistent results. The pseudorandom numbers are generated by the Tausworthe method [24,25].We estimate a critical exponent and an amplitude by using the finite-size scaling [26,27,28] Figure 2 shows the size dependence of χ †′ at z = z c . The solid line indicates 0.037L 2.38 . The values of the critical exponents are consistent with those of the hardsphere lattice gas on the simple cubic...

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confidence: 54%

Universality in the three-dimensional hard-sphere lattice gas

Yamagata

1996

Physica A: Statistical Mechanics and its Applications

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“…Note that the maximum density is 1/2. In higher dimensions [24,30,31,54,55,56,57,58] and other geometries (see [40] and references therein), the same kind of transition is observed, also belonging to the Ising universality class.…”

Section: A Nearest Neighbor Exclusion (1nn)mentioning

confidence: 76%

“…This system, which can also interpreted either as 45 o tilted hard-squares of linear size λ = √ 2 or as hard disks of radius √ 2/2, has been extensively studied and here we present some results for the sake of both completeness and comparison. Many different approaches have been used to describe its properties on a square lattice: series expansions [3,5,11,12,13], cluster variational and transfer matrix methods [5,14,15,16,17,18,19,20,21,22,23], renormalization group [24,25], Monte Carlo simulations [26,27,28,29,30,31,32,33], Bethe lattice [5,34,35,36,37,38], and more recently density functional theory [39]. Moreover, this model has also been considered because of its interesting mathematical [40,41,42] and dynamical [43,44,45,46,47,48,49,50,51,…”

Section: A Nearest Neighbor Exclusion (1nn)mentioning

confidence: 99%

Monte Carlo simulations of two-dimensional hard core lattice gases

Fernandes

Arenzon

Levin

2007

The Journal of Chemical Physics

134

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Monte Carlo simulations are used to study lattice gases of particles with extended hard cores on a two dimensional square lattice. Exclusions of one and up to five nearest neighbors (NN) are considered. These can be mapped onto hard squares of varying side length, λ (in lattice units), tilted by some angle with respect to the original lattice. In agreement with earlier studies, the 1NN exclusion undergoes a continuous order-disorder transition in the Ising universality class. Surprisingly, we find that the lattice gas with exclusions of up to second nearest neighbors (2NN) also undergoes a continuous phase transition in the Ising universality class, while the Landau-Lifshitz theory predicts that this transition should be in the universality class of the XY model with cubic anisotropy. The lattice gas of 3NN exclusions is found to undergo a discontinuous order-disorder transition, in agreement with the earlier transfer matrix calculations and the Landau-Lifshitz theory. On the other hand, the gas of 4NN exclusions once again exhibits a continuous phase transition in the Ising universality class -contradicting the predictions of the Landau-Lifshitz theory. Finally, the lattice gas of 5NN exclusions is found to undergo a discontinuous phase transition.

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“…Lattice gases with nearest-neighbour exclusion have been studied on a number of different lattices by means of a variety of approaches (see, e.g., [3][4][5][6][7][8][9][10][11][12][13][14][15][16] and references therein). The universal classification of phase transitions in hard-core lattice gases is thus known to depend on the dimensionality, the presence of further interactions and the way in which the lattice can be partitioned into sublattices.…”

Section: Introductionmentioning

confidence: 99%

“…Next we consider higher numbers of dimensions, but restrict ourselves to particles on bipartite lattices which have, besides nearest-neighbour exclusion, no further interactions. Rather surprisingly, non-Ising behaviour has been reported for the simple-cubic [12] and the body-centred-cubic [13] lattice gases. However, some uncertainty exists due to the absence of corrections to scaling in these analyses.…”

Section: Introductionmentioning

confidence: 99%

High-dimensional lattice gases

Heringa

Blöte

Luijten

2000

J. Phys. A: Math. Gen.

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We investigate the critical behaviour of hard-core lattice gases in four, five and six dimensions by means of Monte Carlo simulations. In order to suppress critical slowing down, we use a geometrical cluster Monte Carlo algorithm. In particular, nearest-neighbour-exclusion lattice gases on simple hypercubic lattices are investigated. These models undergo Ising-like ordering transitions where the majority of the lattice-gas particles settle on one of two sublattices. A finitesize-scaling analysis of the simulation data confirms that these lattice gases display classical critical behaviour. The results agree with the renormalization predictions at and above the upper critical dimensionality. In particular, the predicted value of the Binder cumulant is confirmed.

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Critical phenomena of the hard-sphere lattice gas on the simple cubic lattice

Cited by 10 publications

References 29 publications

Universality in the three-dimensional hard-sphere lattice gas

Universality in the three-dimensional hard-sphere lattice gas

Monte Carlo simulations of two-dimensional hard core lattice gases

High-dimensional lattice gases

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