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arXiv:1808.06702 · Physical review. E

Three-dimensional Baxter-Wu model

L. N. Jorge, L. S. Ferreira, Álvaro A. Caparica

Published Sep 27, 2019• 7 citations
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Abstract

A classic three-dimensional spin model, based upon the Baxter-Wu scheme, is presented. It is found, by entropic sampling simulations, that the behavior of the energy and magnetization fourth-order cumulants points to a first-order phase transition. A finite-size procedure was performed, confirming that the system scales with the dimensionality d=3, yielding a high-resolution estimate of the transition temperature as T_{c}=11.377485(29).

DOI
10.1103/physreve.100.032141

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A classic three-dimensional spin model, based upon the Baxter-Wu scheme, is presented. It is found, by entropic sampling simulations, that the behavior of the energy and magnetization fourth-order cumulants points to a first-order phase transition. A finite-size procedure was performed, confirming that the system scales with the dimensionality d=3, yielding a high-resolution estimate of the transition temperature as T_{c}=11.377485(29).
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 two-dimensional Baxter-Wu model with spin-1, in the presence of a crystal field, is studied by using Monte Carlo simulations. The standard single-spin-flip Metropolis algorithm is used to generate the configurations from which the order parameter, specific heat and magnetic susceptibility are measured. Single histogram techniques are used to get the results close to the phase transitions. The finite-size scaling procedure is employed in order to obtain the critical behavior. The simulations have shown that the critical exponents are different from the spin-1/2 case and are crystal field dependent.
The two-dimensional spin-1 Baxter-Wu model is studied by using Monte Carlo simulations. The standard single-spin-flip Metropolis algorithm is used to generate the configurations from which the order parameter, specific heat and magnetic susceptibility are measured. The finite-size scaling procedure is employed in order to get the critical behavior. Extensive simulations show that the critical exponents are different from those of the spin-1/2 model suggesting that the spin-1 model is in a different universality class.
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.
This work consists in investigating the phases and transition between them in a magnetic model, namely the Baxter-Wu (B-W) model, in different spatial dimensions. Although the model was originally proposed for a two-dimensional triangular lattice, we will here make a generalization for one- and three-dimensional lattices as well. In one-dimension, the model can be treated by means of the transfer matrix technique. Although it is an exact procedure, in which the system does not present any phase transition, the results can not be written in a closed analytical form for non-zero external field, and are given by the eigenvalues of a non-Hermitian matrix. For zero external field, the partition function is calculated. In two and three dimensions the model is studied using the Mean Field Theory (MFT), using the scheme based on the Bogoliubov inequality. In this treatment, we use blocks of a single spin, a triangle of spins, and in two and three dimensions the exact results for the one-dimensional network. Although the values of the transition temperature are close to the expected values for each lattice, the transition is always of first order, whereas the exact results for the two-dimensional triangular lattice confirm a second-order transition.
We study the question of universality in the two-dimensional spin-1 Baxter-Wu model in the presence of a crystal field Δ. We employ extensive numerical simulations of two types, providing us with complementary results: Wang-Landau sampling at fixed values of Δ and a parallelized variant of the multicanonical approach performed at constant temperature T. A detailed finite-size scaling analysis in the regime of second-order phase transitions in the (Δ,T) phase diagram indicates that the transition belongs to the universality class of the four-state Potts model. Previous controversies with respect to the nature of the transition are discussed and attributed to the presence of strong finite-size effects, especially as one approaches the pentacritical point of the model.