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arXiv:2407.05387 · Journal of Statistical Mechanics Theory and Experiment

Universal energy and magnetisation distributions in the Blume–Capel and Baxter–Wu models

A. R. S. Macêdo, J. A. Plascak, Alexandros Vasilopoulos, Nikolaos G. Fytas, M. Akritidis, Martin Weigel

Published Oct 25, 2024• 8 citations
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Abstract

Abstract We analyse the probability distribution functions of the energy and magnetisation of the two-dimensional Blume–Capel (BC) and Baxter–Wu (BW) models with spin values S ∈ { 1 / 2 , 1 , 3 / 2 } in the presence of a crystal field Δ. By employing extensive single-spin flip Monte Carlo simulations and a recently developed method of studying the zeros of the energy probability distribution we are able to probe, with a good numerical accuracy, several critical characteristics of the transitions. Additionally, the universal aspects of these transitions are scrutinised by computing the corresponding probability distribution functions. The energy distribution has been underutilised in the literature when compared to that of the magnetisation. Somewhat surprisingly, however, the former appears to be more robust in characterising the universality class for both models upon varying the crystal field Δ than the latter. Finally, our analysis suggests that in contrast to the BC ferromagnet, the BW model appears to suffer from strong finite-size effects, especially upon increasing Δ and S, that obscure the application of traditional finite-size scaling approaches.

DOI
10.1088/1742-5468/ad784e

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Abstract We analyse the probability distribution functions of the energy and magnetisation of the two-dimensional Blume–Capel (BC) and Baxter–Wu (BW) models with spin values S ∈ { 1 / 2 , 1 , 3 / 2 } in the presence of a crystal field Δ. By employing extensive single-spin flip Monte Carlo simulations and a recently developed method of studying the zeros of the energy probability distribution we are able to probe, with a good numerical accuracy, several critical characteristics of the transitions. Additionally, the universal aspects of these transitions are scrutinised by computing the corresponding probability distribution functions. The energy distribution has been underutilised in the literature when compared to that of the magnetisation. Somewhat surprisingly, however, the former appears to be more robust in characterising the universality class for both models upon varying the crystal field Δ than the latter. Finally, our analysis suggests that in contrast to the BC ferromagnet, the BW model appears to suffer from strong finite-size effects, especially upon increasing Δ and S, that obscure the application of traditional finite-size scaling approaches.
The Blume-Capel model on a square lattice and the Baxter-Wu model on a triangular lattice have been studied through extensive Monte Carlo simulations using the single spin update according to the Metropolis algorithm. Both models have been treated with spin S=1S=1S=1 and spin S=3/2S=3/2S=3/2 in the presence of crystal field interactions. The transition temperature and correlation length critical exponent have been computed, as a function of the crystal field interaction, by employing the recent developed method of the zeros of the energy probability distribution. The important question of the universality class along the second-order transition lines could be resolved by looking at the corresponding universal probability distributions of the energy and magnetizations. Mixing field arguments turned out to be quite important to decide the possible change in the universality class of the models as the crystal field varies. Single histograms techniques have also been utilized in order to obtain simulational data close to the second-order transitions and multicritical points. As expected, the Blume-Capel model has first- and second-order transitions, where a tricritical point occurs for spin S=1S=1S=1, while an isolated double-critical-endpoint is present for spin S=3/2S=3/2S=3/2. Although the topology of the phase diagram of the Baxte-Wu model for spin S=1S=1S=1 is similar to that of the Blume-Capel model, with the tricritical point being replaced by a pentacritical point, the Baxter-Wu model has critical exponents that change along the tetracritical line as the crystal field is increased. On the other hand, the Baxter-Wu model with spin S=3/2S=3/2S=3/2 has a phase diagram that is completely different from the one obtained for the Blume-Capel, with the presence of a pentacritical point and a tetracritical endpoint. The present data also show that, despite the change of the probability distributions as the crystal field varies, the models with different spin values belong to the same universality class.
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.
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.
We report on numerical simulations of the two-dimensional spin-1 Blume-Capel ferromagnet embedded in a triangular lattice. Utilizing a range of Monte Carlo and finite-size scaling techniques, we explore several critical aspects along the crystal field–temperature ( Δ , T ) transition line. Wang-Landau simulations measuring the joint density of states in combination with the method of field mixing allow us to probe the phase coexistence curve in high resolution, determining the tricritical point ( Δ t , T t ) with improved accuracy and verifying the tricritical exponents. Extensive multicanonical simulations identifying transition points across the phase diagram characterize the Ising universality class for Δ < Δ t with precise determination of thermal and magnetic critical exponents expected in the second-order regime. On the other hand, for Δ > Δ t , a finite-size scaling analysis is dedicated to revealing the first-order signature in the surface tension that linearly increases upon lowering the temperature deeper into the first-order transition regime. Finally, a comprehensive picture of the phase diagram for the model is presented, collecting transition points obtained from the combined numerical approach in this study and previous estimates in the literature.
The phase diagram and the critical behavior of the spin-1 and the spin-3/2 two-dimensional Baxter-Wu model in a crystal field are studied by conventional finite-size scaling and conformal invariance theory. The phase diagram of this model, for the spin-1 case, is qualitatively the same as those of the diluted 4-states Potts model and the spin-1 Blume-Capel model. However, for the present case, instead of a tricritical point one has a pentacritical point for a finite value of the crystal field, in disagreement with previous work based on finite-size calculations. On the other hand, for the spin-3/2 case, the phase diagram is much richer and can present, besides a pentacritical point, an additional multicritical end point. Our results also support that the universality class of the critical behavior of the spin-1 and spin-3/2 Baxter-Wu model in a crystal field is the same as the pure Baxter-Wu model, even at the multicritical points.
Abstract We investigate aspects of universality in the two-dimensional (2D) spin-1 Baxter-Wu model in a crystal field Δ using a parallel version of the multicanonical algorithm employed at constant temperature T. A detailed finite-size scaling analysis in the continuous regime of the Δ – T phase diagram of the model indicates that the transition belongs to the universality class of the 4-state Potts model. The presence of first-order-like finite-size effects that become more pronounced as one approaches the pentacritical point of the model is highlighted and discussed.