Finite-size analysis of the hard-square lattice gas

Abstract

We investigate the hard-square lattice-gas model by means of transfer-matrix calculations and a finite-sizescaling analysis. Using a minimal set of assumptions we find that the spectrum of correction-to-scaling exponents is consistent with that of the exactly solved Ising model, and that the critical exponents and correlationlength amplitudes closely follow the relation predicted by conformal invariance. Assuming that these spectra are exactly identical, and conformal invariance, we determine the critical point, the conformal anomaly, and the temperature and magnetic exponents with numerical margins of 10 Ϫ11 or less. These results are in a perfect agreement with the exactly known Ising universal parameters in two dimensions. In order to obtain this degree of precision, we included system sizes as large as feasible, and used extended-precision floating-point arithmetic. The latter resource provided a substantial improvement of the analysis, despite the fact that it restricted the transfer-matrix calculations to finite sizes of at most 34 lattice units.

Lattice gas with nearest- and next-to-nearest-neighbor exclusion

“…However, the statistical error listed in Table II has still to be revised to include the effect of the uncertainty in the critical point, which we quote as μ c = 4.584(2) on the basis of the results in Table I. This leads to our final estimate y h = 1.877 (9).…”

“…(3). This situation is quite unlike, e.g., the nearest-neighbor exclusion model, which allows [9] the very accurate determination of some universal parameters. It thus seemed necessary to apply Monte Carlo calculations, which can handle much larger system sizes, to analyze the model of Eq.…”

“…The model is thus a hard-square model. It should not be confused with other hard-square models, such as Fisher's [1] and Baxter's [2,3], and the nearest-neighbor exclusion hard-square model without further interactions [4][5][6][7][8][9]. The latter model, as well as that of Eq.…”

“…The next-tonearest-neighbor exclusion, which applies to sites on different rows, does not impose further restrictions on the configuration of an L-site circular row, besides the restrictions imposed by the nearest-neighbor exclusion. The applicable type of coding was already employed and described in the literature [8,9]. However, the application of sparse-matrix techniques also requires the coding of particle configurations as obtained after adding a part of the next circular row of sites on top of an L-site circular row.…”

Monte Carlo simulations of two-dimensional hard core lattice gases

“…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,…”