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Physical review. B, Condensed matter

Phase transitions in centered-rectangular- and square-lattice-gas models

Kimmo K. Kaski, Wolfgang Kinzel, J. D. Gunton

Published Jun 1, 1983• 39 citations
Publisher

Abstract

Phase transitions in centered-rectangular- and square-lattice-gas models have been studied by transfer-matrix methods, with the use of finite-size scaling ideas. The transition lines and critical exponents have been obtained for the $p(2\ifmmode\times\else\texttimes\fi{}1)$ and $p(2\ifmmode\times\else\texttimes\fi{}2)$ structures for the centered-rectangular model with several pairwise interactions and a three-body interaction. This model is of interest in that the $p(2\ifmmode\times\else\texttimes\fi{}1)$ and $p(2\ifmmode\times\else\texttimes\fi{}2)$ transitions belong to the universality class of the $\mathrm{XY}$ model with cubic anisotropy and hence the critical exponents are nonuniversal. As well, it is a simple model of O on W(110). The transition lines have also been obtained for a square-lattice-gas model with competing interactions, for the (2\ifmmode\times\else\texttimes\fi{}1) and (2\ifmmode\times\else\texttimes\fi{}2) ordered structures. The importance of finite-size effects is demonstrated for both models.

DOI
10.1103/physrevb.27.6777

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Phase transitions in centered-rectangular- and square-lattice-gas models have been studied by transfer-matrix methods, with the use of finite-size scaling ideas. The transition lines and critical exponents have been obtained for the p(2\ifmmode×\else\texttimes\fi1)p(2\ifmmode\times\else\texttimes\fi{}1)p(2\ifmmode×\else\texttimes\fi1) and p(2\ifmmode×\else\texttimes\fi2)p(2\ifmmode\times\else\texttimes\fi{}2)p(2\ifmmode×\else\texttimes\fi2) structures for the centered-rectangular model with several pairwise interactions and a three-body interaction. This model is of interest in that the p(2\ifmmode×\else\texttimes\fi1)p(2\ifmmode\times\else\texttimes\fi{}1)p(2\ifmmode×\else\texttimes\fi1) and p(2\ifmmode×\else\texttimes\fi2)p(2\ifmmode\times\else\texttimes\fi{}2)p(2\ifmmode×\else\texttimes\fi2) transitions belong to the universality class of the XY\mathrm{XY}XY model with cubic anisotropy and hence the critical exponents are nonuniversal. As well, it is a simple model of O on W(110). The transition lines have also been obtained for a square-lattice-gas model with competing interactions, for the (2\ifmmode\times\else\texttimes\fi{}1) and (2\ifmmode\times\else\texttimes\fi{}2) ordered structures. The importance of finite-size effects is demonstrated for both models.
We present a transfer-matrix scaling study of a hard-square lattice-gas model with an exclusion shell extending up to second-nearest neighbors, and with a uniaxial finite repulsion active at third-nearest-neighbor separations. The phase diagram, as a function of temperature and chemical potential, shows three distinct regions bounded by two merging critical lines. At low temperature the system undergoes a transition to an ordered phase whose c(4\ifmmode\times\else\texttimes\fi{}2) structure is characterized---within this calculation---by the asymptotic degeneracy of the four largest eigenvalues of the transfer matrix. At high temperature this phase is reached from the disordered state via an intermediate, degenerate phase which, on average, exhibits long-range order in one dimension only. These two transition regimes are found to be separated by a multicritical point which is likely to arise from the competition between interaction anisotropy and thermal-disordering effects. Evidence of anisotropic critical behavior is found in the correlation lengths exponent ratios, ${\ensuremath{\nu}}_{?}$/${\ensuremath{\nu}}_{\ensuremath{\perp}}$, which appear to vary continuously along both transition lines. However, the accuracy of the predictions we make for the thermodynamics of the model is still limited by sensitive finite-size effects which are presumably emphasized by the relatively extended range of the interactions.
The critical exponents of the non-interacting hard-square lattice gas model are determined by means of series analysis and finite-size scaling. The series analysis leads to results for the exponents that are consistent with the expected Ising universal values. A scaling analysis is performed on different types of numerical finite-size data obtained by means of the transfer-matrix method. The results support Ising universality. A procedure based on conformal invariance reproduces the known Ising temperature and magnetic exponents within 10 -6 . Furthermore we estimate the critical density of the hard-square model as rho c =0.367 743 000 (5).
Using transfer matrix and finite-size scaling methods, we study the thermodynamic behavior of a lattice gas with two kinds of particles on the square lattice. Only excluded volume interactions are considered, so that the model is athermal. Large particles exclude the site they occupy and its four first neighbors, while small particles exclude only their site. Two thermodynamic phases are found: a disordered phase where large particles occupy both sublattices with the same probability and an ordered phase where one of the two sublattices is preferentially occupied by them. The transition between these phases is continuous at small concentrations of the small particles and discontinuous at larger concentrations, both transitions are separated by a tricritical point. Estimates of the central charge suggest that the critical line is in the Ising universality class, while the tricritical point has tricritical Ising (Blume-Emery-Griffiths) exponents. The isobaric curves of the total density as functions of the fugacity of small or large particles display a minimum in the disordered phase.
We present a finite-size-scaling study of a two-dimensional square-lattice-gas model with nearest-neighbor repulsion and next-nearest-neighbor attraction. This system may be used as a simplified model of adsorbates on transition-metal (100) surfaces. An accurate determination of the tricritical temperature is obtained by the utilization of the degeneracy of the three largest eigenvalues of the transfer matrix at a tricritical point. We obtain the phase diagram and critical and tricritical exponents, all quite consistent with previously known results. We also find that a recent conjecture relating the critical exponent $\ensuremath{\eta}$ to the correlation-length amplitude seems to work well even at the tricritical point.
Using a histogram Monte Carlo simulation method (HMCSM), Hu, Lin, and Chen found that bond and site percolation models on planar lattices have universal finite-size scaling functions for the existence probability Ep,{E}_{p},Ep​, the percolation probability P,P,P, and the probability Wn{W}_{n}Wn​ for the appearance of nnn percolating clusters in these models. In this paper we extend above study to percolation on three-dimensional lattices with various linear dimensions L.L.L. Using the HMCSM, we calculate the existence probability Ep{E}_{p}Ep​ and the percolation probability PPP for site and bond percolation on a simple-cubic (sc) lattice, and site percolation on body-centered-cubic and face-centered-cubic lattices; each lattice has the same linear dimension in three dimensions. Using the data of Ep{E}_{p}Ep​ and PPP in a percolation renormalization group method, we find that the critical exponents obtained are quite consistent with the universality of critical exponents. Using a small number of nonuniversal metric factors, we find that Ep{E}_{p}Ep​ and PPP have universal finite-size scaling functions. This implies that the critical Ep{E}_{p}Ep​ is a universal quantity, which is $0.265\ifmmode\pm\else\textpm\fi{}0.005$ for free boundary conditions and $0.924\ifmmode\pm\else\textpm\fi{}0.005$ for periodic boundary conditions. We also find that Wn{W}_{n}Wn​ for site and bond percolation on sc lattices have universal finite-size scaling functions.
We investigate the hard-square lattice-gas model by means of transfer-matrix calculations and a finite-size-scaling 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 correlation-length 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(-11) 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.