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Modelo de Baxter-Wu em uma, duas e três dimensões para spin 1/2

Moallison F. Cavalcante

Published Jul 10, 2019• 0 citations
OpenAlex

Abstract

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.

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We have studied the critical, multicritical and first order behavior of the Spin-1 Baxter-Wu model in a crystal field. We have employed the mean field renormalization group approach and extensive Monte Carlo simulations using the single spin flip Metropolis algorithm and hystogram techniques. The global phase diagram is obtained and it is shown that the system belongs to the same universality class as the corresponding spin-1/2 version and presents a multicritical point separating a second order phase transition line from a first order line. The probability distribution of this pentacritical point is also determined.

•Dec 12, 2005•0 citations
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.
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 low-energy properties as well as quantum and thermal phase transitions of the Baxter-Wu model in a transverse magnetic field. Our study relies on stochastic series expansion quantum Monte Carlo and on series expansions about the low- and high-field limits at zero temperature using the quantum finite-lattice method on the triangular lattice. The phase boundary consists of a second-order critical line in the four-state Potts model universality class starting from the pure Baxter-Wu limit meeting a first-order line connected to the zero-temperature transition point (h≈2.4, T=0). Both lines merge at a tricritical point approximately located at (h≈2.3J, T≈J).
The thermodynamic properties of spin models on the lattice have been studied by employing phenomenological renormalization group techniques. The spin-1 Blume- Capel model and the spin-1 and spin-3/2 Baxter-Wu model have been treated. The Blume Capel model with spin-1, in its diluted version, on the simple cubic lattice has been studied by using the mean field renormalization group in the pair aproximation. We have obtained the critical temperature as a function of the vacant sites concentration as well a function of the crystal field anisotropy. We consider exchange e superexchange interactions between Fe atoms induzed by Al atoms and applied this model to FeAl alloys to describe its thermodynamics properties. We also studied the spin-1 and 3/2 Baxter-Wu model in a crystal field by employing the finite size renormalization group approach. The eigenvalues of the transfer matrix have been computed in order to get the corresponding phase diagram. Estimates of the firstand second-order transition temperatures have been obtained, as well as multicritical points, critical exponents and central charges, the latter two being obtained for some values of the crystal field. The Gibbs phase rule has been used to check the consistency of the phase diagram of some magnetic models, including those treated in this work. In this case it is necessary to take into account the symmetries carried by the magnetic Hamiltonians.
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 have studied the critical, multicritical and first order behavior of the Spin-1 Baxter-Wu model in a crystal field. We have employed the mean field renormalization group approach and extensive Monte Carlo simulations using the single spin flip Metropolis algorithm and hystogram techniques. The global phase diagram is obtained and it is shown that the system belongs to the same universality class as the corresponding spin-1/2 version and presents a multicritical point separating a second order phase transition line from a first order line. The probability distribution of this pentacritical point is also determined.