Baxter-Wu Model Research Articles

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

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

The phase diagram of the baxter-Wu model in the magnetic field and temperature plane

Using the Fisher zeros of the partition function of a finite sized cluster with ‘spherical’ boundary conditions an approximate phase diagram in the magnetic field—temperature plane is obtained for the Baxter-Wu model. The phase diagram compares favourably with previous approximations as well as the known exact results when the magnetic field is zero. The accuracy of the approximated phase diagram, especially for small values of the magnetic field, is dependent on the fact, as shown previously, that for zero magnetic field the Fisher zeros for this cluster lie on loci giving the exact zero field critical temperature.

Finite-size analysis in neural network classification of critical phenomena.

We analyze the problem of supervised learning of ferromagnetic phase transitions from the statistical physics perspective. We consider two systems in two universality classes, the two-dimensional Ising model and two-dimensional Baxter-Wu model, and perform careful finite-size analysis of the results of the supervised learning of the phases of each model. We find that the variance of the neural network (NN) output function (VOF) as a function of temperature has a peak in the critical region. Qualitatively, the VOF is related to the classification rate of the NN. We find that the width of the VOF peak displays the finite-size scaling governed by the correlation length exponent ν of the universality class of the model. We check this conclusion using several NN architectures-a fully connected NN, a convolutional NN, and several members of the ResNet family-and discuss the accuracy of the extracted critical exponents ν.

Two-dimensional dilute Baxter-Wu model: Transition order and universality.

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.

Pseudo-phase transitions of Ising and Baxter–Wu models in two-dimensional finite-size lattices

This article offers a detailed analysis of pseudo-phase transitions of Ising and Baxter–Wu models in two-dimensional finite-size lattices. We carry out Wang–Landau sampling to obtain the density of states. Using microcanonical inflection-point analysis with microcanonical entropy, we obtain the order of the pseudo-phase transitions in the models. The microcanonical analysis results of the second-order transition for the Ising model and the first-order transition for the Baxter–Wu model are consistent with the traditional canonical results. In addition, the third-order transitions are found in both models, implying the universality of higher-order phase transitions.

The location of the Fisher zeros and estimates of y T = 1/ν are found for the Baxter–Wu model

The Fisher zeros of the Baxter–Wu model are examined for the first time and for two series of finite-sized systems, with ‘spherical’ boundary conditions, their location is found to be extremely simple. They lie on the unit circle in the complex sinh[2βJ 3] plane. This is the same location as the Fisher zeros of the square lattice Ising model with nearest neighbour interactions and Brascamp–Kunz boundary conditions. The Baxter–Wu model is an Ising model with three-site interactions, J 3, on the triangle lattice. From the leading Fisher zeros, using finite-size scaling, accurate estimates of the critical exponent 1/ν are obtained and emphasis is placed on using different variables such as exp[−2βJ 3], exp[−4βJ 3], and sinh[2βJ 3] to enhance the accuracy of estimates. Furthermore, using the imaginary parts of the leading zeros versus the real part of the leading zeros, yields different results. This is similar to results of Janke and Kenna for the nearest neighbour, Ising model on the square lattice and extends this behaviour to a multisite interaction system in a different universality class than the pair-interaction cases.

Universality in the two-dimensional dilute Baxter-Wu model.

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.

Multicanonical simulations of the 2D spin-1 Baxter-Wu model in a crystal field

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.

An entropic simulational study of the spin-1 Baxter–Wu model in a crystal field

We investigate the critical behavior of the two-dimensional spin-1 Baxter–Wu model in a crystal field using entropic sampling simulations, based on the Wang–Landau method, with the joint density of states. We obtain the temperature–crystal field phase diagram, which includes a tetracritical line ending at a pentacritical point. A finite-size scaling analysis of the maximum of the specific heat, while changing the crystal field anisotropy, is used to obtain a precise location of the pentacritical point. Our results give the critical temperature and crystal field as Tpc=0.98030(10) and Dpc=1.68288(62). We also detect that at the first-order region of the phase diagram, the specific heat exhibits a double peak structure as in the Schottky-like anomaly, which is associated with an order–disorder transition.

Universality classes and machine learning

We formulate the problem of the universality class investigation using machine learning. We chose an example of the universality class of the two-dimensional 4-state Potts model. There are four known models within the universality class – the 4-state Potts model, the Baxter-Wu model, the Ashkin-Teller model, and the Turban model. All four of them together are not equivalent in the Hamiltonian representation, in the lattice symmetry, and the layout of spins on the lattice. We generate statistically independent datasets for all models using the same Monte Carlo technique. The machine learning methods will be used for the analysis of the universality class of models based on generated datasets.

Phase transitions of bilayer spin-1/2 Baxter-Wu model with repulsive interlayer interactions

Using a Monte Carlo simulation and the single histogram reweighting technique, we study the critical behaviors and phase transitions of the Baxter-Wu model on a two-layer triangular lattice with repulsive interlayer interactions. Via the finite-size analysis, we obtain the transition temperatures and critical exponents at different interlayer coupling strengths. The reduced energy cumulants and the histograms reveal pseudo-first-order phase transitions, and the critical exponents derivate from the standard Baxter-Wu model if the coupling is weak. As the increase of the coupling, the behavior of the pseudo-first order transition disappears and the critical exponents are in accordance with the ones known for the 2D 4-state Potts model.

On the order of the phase transition in the spin-1 Baxter–Wu model

In this work we investigate the order of the phase transition of the spin-1 Baxter–Wu model. We used extensive entropic simulations to describe the behavior of quantities which reveal the order of the phase transition. We applied finite-size scaling laws for continuous and discontinuous phase transitions. Our results show that this system exhibits an indeterminacy regarding the order of the phase transition, i.e., the results are conclusive for both transitions, whether continuous or discontinuous. In such a scenario we carried out a study of the configurations in the region of the phase transition, which confirmed that the model seems to undergo a tetracritical transition, with the coexistence of a ferromagnetic and three ferrimagnetic configurations, suggesting that it may be a multicritical point, where the continuous and the discontinuous phase transitions may coexist reflecting different features of the system. We propose a new way of carrying out the entropic simulations where two densities of states are generated simultaneously, for the ferro- and the ferrimagnetic states. The behavior of the energy probability distribution, the inverse temperature, and the magnetizations for each state confirms the coexistence of the discontinuous and the continuous phase transitions.

Three-dimensional Baxter-Wu model.

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).

Baxter–Wu model according to a mean-field approach

The Baxter–Wu model in one, two, and three dimensions is studied according to a mean-field-like approach. The thermodynamics and the phase transition behavior are obtained by considering different types of clusters. The mean-field solution shows that the model presents a first-order phase transition in any dimension, contrary to the exact result of a second-order transition in the two-dimensional model. Although the one-dimensional version has no phase transition at finite temperatures, the Baxter–Wu model on a three-dimensional lattice may indeed undergo a first-order transition.

Finite-Size Scaling for the Baxter-Wu Model Using Block Distribution Functions

In the present work, we present an alternative way of applying the well-known finite-size scaling (FSS) theory in the case of a Baxter-Wu model using Binder-like blocks. Binder’s ideas are extended to estimate phase transition points and the corresponding scaling exponents not only for magnetic but also for energy properties, saving computational time and effort. The vast majority of our conclusions can be easily generalized to other models.

Wang–Landau sampling: Saving CPU time

In this work we propose an improvement to the Wang–Landau (WL) method that allows an economy in CPU time of about 60% leading to the same results with the same accuracy. We used the 2D Ising model to show that one can initiate all WL simulations using the outputs of an advanced WL level from a previous simulation. We showed that up to the seventh WL level (f6) the simulations are not biased yet and can proceed to any value that the simulation from the very beginning would reach. As a result the initial WL levels can be simulated just once. It was also observed that the saving in CPU time is larger for larger lattice sizes, exactly where the computational cost is considerable. We carried out high-resolution simulations beginning initially from the first WL level (f0) and another beginning from the eighth WL level (f7) using all the data at the end of the previous level and showed that the results for the critical temperature Tc and the critical static exponents β and γ coincide within the error bars. Finally we applied the same procedure to the 1/2-spin Baxter–Wu model and the economy in CPU time was of about 64%.

Universality and phase diagrams of the Baxter-Wu Model in a Crystal Field: spin-1 and spin-3/2

Conventional finite-size scaling and conformal invariance theory are used in order to study the critical behavior of the spin-1 and spin-3/2 Baxter-Wu model. For spin-1 the results are similar to the Blume-Capel model. However, for spin-3/2, the phase diagram is much richer, and presents, besides a pentacritical point, an additional multicritical endpoint. In both cases, the universality class is the same as the spin-1/2 model, even at the multicritical points.

Scaling in the vicinity of the four-state Potts fixed point

We study a self-dual generalization of the Baxter–Wu model, employing results obtained by transfer matrix calculations of the magnetic scaling dimension and the free energy. While the pure critical Baxter–Wu model displays the critical behavior of the four-state Potts fixed point in two dimensions, in the sense that logarithmic corrections are absent, the introduction of different couplings in the up- and down triangles moves the model away from this fixed point, so that logarithmic corrections appear. Real couplings move the model into the first-order range, away from the behavior displayed by the nearest-neighbor, four-state Potts model. We also use complex couplings, which bring the model in the opposite direction characterized by the same type of logarithmic corrections as present in the four-state Potts model. Our finite-size analysis confirms in detail the existing renormalization theory describing the immediate vicinity of the four-state Potts fixed point.

Critical behavior of the spin-1 and spin-3/2 Baxter-Wu model in a crystal field.

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.

Critical Behavior of the Spin-1/2 Baxter-Wu Model: Entropic Sampling Simulations

In this work we use a refined entropic sampling technique based on the Wang-Landau method to study the spin-$1/2$ Baxter-Wu model. The static critical exponents were determined as $\alpha=0.6545(68)$, $\beta=0.0818(30)$, $\gamma=1.18193(77)$, and $\nu=0.66341(47)$. The estimate for the critical temperature was $T_c=2.269194(45)$. We compare the present results with those obtained from other well established approaches and we find a startling closeness with the exact values, besides the high precision reached for the critical temperature. We also calculate the coefficients $a$ and $b$ for the divergence of the microcanonical inverse temperature at the ground state achieving an excellent agreement in comparison with the simulation estimates.