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arXiv:cond-mat/0507394 · Journal of Physics A Mathematical and General

Monte Carlo study of the pure and dilute Baxter–Wu model

Nir Schreiber, Joan Adler

Published Aug 3, 2005• 26 citations
arXivHTMLPDF

Abstract

We studied the pure and dilute Baxter–Wu (BW) models using the Wang–Landau (WL) sampling method to calculate the density-of-states (DOS). We first used the exact result for the DOS of the Ising model to test our code. Then we calculated the DOS of the dilute Ising model to obtain a phase diagram, in good agreement with previous studies. We calculated the energy distribution, together with its first, second and fourth moments, to give the specific heat and the energy fourth order cumulant, better known as the Binder parameter, for the pure BW model. For small samples, the energy distribution displayed a doubly peaked shape, and finite size scaling analysis showed the expected reciprocal scaling of the positions of the peaks with L . The energy distribution yielded the expected BW α = 2/3 critical exponent for the specific heat. The Binder parameter minimum appeared to scale with lattice size L with an exponent θ B equal to the specific heat exponent. Its location (temperature) showed a large correction-to-scaling term θ 1 = 0.248 ± 0.025. For the dilute BW model we found a clear crossover to a single peak in the energy distribution even for small sizes and the expected α = 0 was recovered.

DOI
10.1088/0305-4470/38/33/004

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We studied the pure and dilute Baxter–Wu (BW) models using the Wang–Landau (WL) sampling method to calculate the density-of-states (DOS). We first used the exact result for the DOS of the Ising model to test our code. Then we calculated the DOS of the dilute Ising model to obtain a phase diagram, in good agreement with previous studies. We calculated the energy distribution, together with its first, second and fourth moments, to give the specific heat and the energy fourth order cumulant, better known as the Binder parameter, for the pure BW model. For small samples, the energy distribution displayed a doubly peaked shape, and finite size scaling analysis showed the expected reciprocal scaling of the positions of the peaks with L . The energy distribution yielded the expected BW α = 2/3 critical exponent for the specific heat. The Binder parameter minimum appeared to scale with lattice size L with an exponent θ B equal to the specific heat exponent. Its location (temperature) showed a large correction-to-scaling term θ 1 = 0.248 ± 0.025. For the dilute BW model we found a clear crossover to a single peak in the energy distribution even for small sizes and the expected α = 0 was recovered.
The critical behavior of the Baxter-Wu model belongs to the universality class of the four-state Potts model. While the introduction of annealed vacancies does not alter the criticality of the four-state Potts model, the dilute Baxter-Wu model has remained the subject of several competing scenarios. Here we investigate the phase diagram of the spin-1 Baxter-Wu model in the presence of a crystal field using transfer-matrix calculations and large-scale Monte Carlo simulations. Our results reveal a systematic evolution of the effective critical behavior with increasing crystal field, accompanied by increasingly strong finite-size corrections near the crossover to the first-order regime. Along the line of continuous transitions, the central charge remains close to c=1, while the scaling dimensions systematically deviate from the spin-1/2 limit as the crystal field increases, consistent with either continuously varying effective critical exponents or a slow crossover between competing critical behaviors. The first-order regime is independently characterized through multicanonical simulations, which confirm the expected finite-size scaling and interfacial behavior. Taken together, our results provide a unified picture of the dilute spin-1 Baxter-Wu model, substantially narrowing the range of possible scenarios for the crossover between continuous and first-order phase transitions.
We have used an importance-sampling Monte Carlo technique to study the Baxter-Wu model with various fractions of the lattice sites occupied by random, quenched, nonmagnetic, site impurities. We found the system had long-time fluctuations which were caused by the creation and motion of domain boundaries. For this reason our study required the use of a large number of Monte Carlo steps per spin. The data were analyzed using finite-size scaling to extract the infinite-lattice critical exponents and critical amplitudes from the Monte Carlo simulation of finite lattices. This analysis of the data for the pure Baxter-Wu model yielded results which agreed well with exact results and series-expansion predictions. The addition of quenched site impurities caused a dramatic change in the critical behavior. The pure-lattice critical exponents ($\ensuremath{\nu}=\frac{2}{3}$, $\ensuremath{\alpha}=\frac{2}{3}$, ${\ensuremath{\gamma}}_{m}\ensuremath{\simeq}1.17$) change upon the addition of only a few percent impurities to $\ensuremath{\nu}=1.00\ifmmode\pm\else\textpm\fi{}0.07$, $\ensuremath{\alpha}\ensuremath{\lesssim}0.0$, and ${\ensuremath{\gamma}}_{m}=1.95\ifmmode\pm\else\textpm\fi{}0.08$. The phase diagram as a function of impurity concentration and an estimate for the infinite-lattice percolation limit are also given.
We investigate the critical behavior of the two-dimensional spin-1 Baxter-Wu model in the presence of a crystal-field coupling Δ with the goal of determining the universality class of transitions along the second-order part of the transition line as one approaches the putative location of the multicritical point. We employ extensive Monte Carlo simulations using two different methodologies: (i) a study of the zeros of the energy probability distribution, closely related to the Fisher zeros of the partition function, and (ii) the well-established multicanonical approach employed to study the probability distribution of the crystal-field energy. A detailed finite-size scaling analysis in the regime of second-order phase transitions in the (Δ,T) phase diagram supports previous claims that the transition belongs to the universality class of the four-state Potts model. For positive values of Δ, we observe the presence of strong finite-size effects, indicative of crossover effects due to the proximity of the first-order part of the transition line. Finally, we demonstrate how a combination of cluster and heat-bath updates allows one to equilibrate larger systems, and we demonstrate the potential of this approach for resolving the ambiguities observed in the regime of Δ≳0.
Our research focuses on the creation of a 3D layered system model that adheres to a three-spin Baxter–Wu-like interaction. To address the phase transition problem, we use entropic sampling simulations to construct the density of states and analyze various quantities such as the total magnetization, average energy, magnetic susceptibility, energy probability distribution function, and inverse microcanonical temperature. Once we established that the system undergoes a first-order phase transition, we carried out a finite-size scaling procedure by carefully studying the critical behavior of the minimum temperature of cumulants for energy and magnetization. We also computed the pseudocritical temperature, Tc(L), which is the temperature where the energy density probability distribution has a double peak. Based on our analysis, we were able to obtain an estimation of the transition temperature as Tc≈3.94 and the latent heat value ΔE≈1.2 at this point.
The phase diagram and critical behavior of the spin-1 Baxter-Wu model with a crystal field in two dimensions is explored by renormalization group, conventional finite-size scaling, and conformal invariance techniques. We found that the phase diagram of this model is qualitatively the same as that of the dilute 4-states Potts model, presenting a multicritical point for a finite value of the crystal field, in disagreement with previous work based on finite-size calculations. However, our results indicate that the critical exponents vary continuously along the second-order transition line, differently from the expected behavior of the dilute 4-states Potts model.
Abstract Using a Monte Carlo simulation and the single histogram reweighting technique, we study the critical behaviors and phase transitions of the Baxter–Wu (BW) model on a two-layer triangular lattice with Ising-type interlayer couplings. Via the finite-size analysis, we obtain the transition temperatures and critical exponents at repulsive and attractive interlayer couplings. The data for the repulsive interlayer coupling suggest continuous transitions, and the critical behaviors are the same as those of the 2D BW model, belonging to the four-state Potts universality class. The reduced energy cumulants and the histograms reveal that attractive coupling leads to weak first-order phase transitions. The pseudocritical exponents with the existence of the interlayer couplings indicate that the first-order transition is very close to the critical point of the 2D standard BW model.