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arXiv:0803.3334 · Physical Review E

Monte Carlo study of the triangular lattice gas with first- and second-neighbor exclusions

Wei Zhang, Youjin Deng

Published Sep 2, 2008• 36 citations

Abstract

We formulate a Swendsen-Wang-like version of the geometric cluster algorithm. As an application, we study the hard-core lattice gas on the triangular lattice with the first- and second-neighbor exclusions. The data were first analyzed by finite-size scaling without including logarithmic corrections. We determine the critical chemical potential as mu_{c}=1.75682(2) and the critical particle density as rho_{c}=0.180(4) . From the Binder ratio Q and susceptibility chi , the thermal and magnetic exponents are estimated as y_{t}=1.51(1) approximately 32 and y_{h}=1.8748(8) approximately 158 , respectively, while the analyses of energylike quantities yield y_{t} ranging from 1.440(5) to 1.470(5). Nevertheless, the data for energylike quantities are also well described by theoretically predicted scaling formulas with logarithmic corrections and with exponent y_{t}=32 . These results are very similar to the earlier study for the four-state Potts model on the square lattice [J. Stat. Phys. 88, 567 (1997)], and strongly support the general belief that the model is in the four-state Potts universality class. The dynamic scaling behavior of the Metropolis simulation and the combined method of the Metropolis and the geometric cluster algorithm is also studied; the former has a dynamic exponent z_{int} approximately 2.21 and the latter has z_{int} approximately 1.60 .

DOI: 10.1103/physreve.78.031103

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We formulate a Swendsen-Wang-like version of the geometric cluster algorithm. As an application, we study the hard-core lattice gas on the triangular lattice with the first- and second-neighbor exclusions. The data were first analyzed by finite-size scaling without including logarithmic corrections. We determine the critical chemical potential as mu_{c}=1.75682(2) and the critical particle density as rho_{c}=0.180(4) . From the Binder ratio Q and susceptibility chi , the thermal and magnetic exponents are estimated as y_{t}=1.51(1) approximately 32 and y_{h}=1.8748(8) approximately 158 , respectively, while the analyses of energylike quantities yield y_{t} ranging from 1.440(5) to 1.470(5). Nevertheless, the data for energylike quantities are also well described by theoretically predicted scaling formulas with logarithmic corrections and with exponent y_{t}=32 . These results are very similar to the earlier study for the four-state Potts model on the square lattice [J. Stat. Phys. 88, 567 (1997)], and strongly support the general belief that the model is in the four-state Potts universality class. The dynamic scaling behavior of the Metropolis simulation and the combined method of the Metropolis and the geometric cluster algorithm is also studied; the former has a dynamic exponent z_{int} approximately 2.21 and the latter has z_{int} approximately 1.60 .

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 finite-size-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.

Using phenomenological renormalization (transfer-matrix scaling), we have reexamined the phase transition of a triangular lattice gas with particles having both nearest- and second-nearest-neighbor exclusions. Widely accepted classical studies indicated that disordering of the ordered ["$p(2\ifmmode\times\else\texttimes\fi{}2)$"] state is first order. In contradiction, we show that the transition is second order; its exponents are consistent with the four-state Potts model universality class, in accord with its Landau-Ginzburg-Wilson Hamiltonian classification.

By introducing a small amount of non-magnetic impurities into an antiferromagnetic (AF) two-dimensional (2D) Potts model on a triangular lattice it is that the impurities in spin systems described by this model result in the change of a first order to a second-order phase transition. The systems with linear sizes L × L = N, L = 9-144 are considered. Investigations are performed using the standard Metropolis algorithm along with Monte-Carlo single-cluster Wolff algorithm. On the basis of the theory of finite-size scaling, critical exponents (CE) are calculated: the heat capacity α, the susceptibility γ, the order parameter β, and the CE of the correlation radius ν.

The phase transitions in 2D ferro- and antiferromagnetic Potts models with number of spin states q = 3 on a triangular lattice are investigated by the cluster and classical Monte Carlo methods. Systems with linear sizes L = 20–120 are considered. Fourth-order Binder cumulants and histogram data analysis are used to show that second- and first-order phase transitions are observed in the ferromagnetic and antiferromagnetic Potts models, respectively. The static critical indices are calculated for specific heat α, susceptibility γ, magnetization β, and correlation length ν on the basis of finite-size scaling theory for a ferromagnetic Potts model.

The critical behavior of the disordered two-dimensional antiferromagnetic Potts model with the number of spin states q= 3 on a triangular lattice with disorder in the form of nonmagnetic impurities is studied by the Monte Carlo method. The critical exponents for the susceptibility γ, magnetization β, specific heat α, and correlation radius ν are calculated in the framework of the finite-size scaling theory at spin concentrations p = 0.90, 0.80, 0.70, and 0.65. It is found that the critical exponents increase with the degree of disorder, whereas the ratios and do not change, thus holding the scaling equality

\frac{{2\beta }}{\nu } + \frac{\gamma }{\nu } = d

. Such behavior of the critical exponents is related to the weak universality of the critical behavior characteristic of disordered systems. All results are obtained using independent Monte Carlo algorithms, such as the Metropolis and Wolff algorithms.

We investigate the spatial overlap of successive spin configurations in Markov chain Monte Carlo simulations using the local Metropolis algorithm and the Swendsen-Wang and Wolff cluster algorithms. We examine the dynamics of these algorithms for models in different universality classes: Ising model, Potts model with three components, and four-state Potts model. The overlap of two successive Wolff clusters reflects critical behavior and can be used as an order parameter for the algorithm's dynamics. In the case of the Swendsen-Wang algorithm, similar behavior is demonstrated by the variance of the overlap of two consecutive lattice configurations, which behaves like an order parameter. Nothing similar is observed for the Metropolis algorithm, where the dynamics in the critical region are determined by the spin-flip frequency, which is equivalent to the acceptance rate. Thus, the critical behavior of the Wolff cluster overlap and of the variance of the configuration overlap in the Swendsen-Wang algorithm are naturally related to the critical behavior of geometric objects -- Fortuin-Kasteleyn clusters. Interestingly, in all cases the geometric quantity -- the configuration overlap or its variance -- reflects the thermodynamics of the phase transition.