Baxter-Wu Model Research Articles

Monte Carlo study of the Spin-1 Baxter-Wu model in a crystal field

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.

Efficient energy cumulants for the Baxter–Wu model

In this paper, using both analytic methods and Monte Carlo simulations with our triangle cluster algorithm, we illustrate the scaling behavior of two possible 4th-order connected energy cumulants across the well-known second and first-order phase transitions of the Baxter–Wu model under zero external magnetic field. It is found that 4th-order connected energy cumulant introduced by Janke provides a very good theoretical tool for finding and distinguishing phase transitions, especially in case the order parameter of a particular model is hardly known. Also, the physical importance of the cumulants’ local minima and maxima is investigated, showing that they are finite-size scaling effects.

Baxter-Wu model in a transverse magnetic field.

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).

Critical energy distribution function of the Baxter–Wu model

In this work we examine the critical finite-size scaling behavior of the energy probability distribution function and its corresponding Binder cumulant at critical point. Based on the results of Monte Carlo simulations at zero external magnetic field using the recently developed triangle-cluster algorithm, we calculate the energy distribution function’s exponents and we derive the scaling relations for the energy Binder cumulant. Finally, we predict the exact form of the scaled energy distribution function. Most of our conclusions seem applicable not only to the Baxter–Wu model but also to other Ising-like models.

Robustness of a topological phase: Topological color code in a parallel magnetic field

The robustness of the topological color code, which is a class of error-correcting quantum codes, is investigated under the influence of a uniform magnetic field on the honeycomb lattice. Our study relies on two high-order series expansions using perturbative continuous unitary transformations in the limit of low and high fields, exact diagonalization, and a classical approximation. We show that the topological color code in a single parallel field is isospectral to the Baxter-Wu model in a transverse field on the triangular lattice. It is found that the topological phase is stable up to a critical field beyond which it breaks down to the polarized phase by a first-order phase transition. The results also suggest that the topological color code is more robust than the toric code in the parallel magnetic field.

Baxter–Wu model in the presence of an external magnetic field

In this work we study an unusual phase transition of the Baxter–Wu model in the presence of an external magnetic field. The model is pure Baxter–Wu, which means that only three-spin interactions are taken into account. We construct a phase diagram on the temperature–field plane based mainly on the singularities of the specific heat. These singularities are more clearly observed than those of the magnetic susceptibility which are used in existing works. We establish a discontinuity in the critical exponents when the field is changed from zero to negative.

The Hintermann–Merlini–Baxter–Wu and the infinite-coupling-limit Ashkin–Teller models

We show how the Hintermann–Merlini–Baxter–Wu model (which is a generalization of the well-known Baxter–Wu model to a general Eulerian triangulation) can be mapped onto a particular infinite-coupling-limit of the Ashkin–Teller model. We work out some mappings among these models, also including the standard and mixed Ashkin–Teller models. Finally, we compute the phase diagram of the infinite-coupling-limit Ashkin–Teller model on the square, triangular, hexagonal, and kagome lattices.

Sampling the ground-state magnetization ofd-dimensionalp-body Ising models

We demonstrate that a recently introduced heuristic optimization algorithm [Phys. Rev. E 83, 046709 (2011)] that combines a local search with triadic crossover genetic updates is capable of sampling nearly uniformly among ground-state configurations in spin-glass-like Hamiltonians with $p$-spin interactions in $d$ space dimensions that have highly degenerate ground states. Using this algorithm we probe the zero-temperature ferromagnet to spin-glass transition point ${q}_{c}$ of two example models, the disordered version of the two-dimensional three-spin Baxter-Wu model [${q}_{c}=0.1072(1)$] and the three-dimensional Edwards-Anderson model [${q}_{c}=0.2253(7)$], by computing the Binder ratio of the ground-state magnetization.

Scaling properties of the Baxter–Wu model

We use a recently developed cluster algorithm for the Baxter–Wu model to study characteristic features of its behavior. Magnetic scaling properties of the model are investigated for second-order phase transitions. We improve significantly the accuracy of Monte Carlo simulation results and we provide new results for the behavior of the Binder cumulants for temperatures below, above and at the critical point.

Critical amplitude ratios of the Baxter–Wu model

A Monte Carlo simulation study of the critical and off-critical behavior of the Baxter–Wu model, which belongs to the universality class of the 4-state Potts model, was performed. We estimate the critical temperature window using known analytical results for the specific heat and magnetization. This helps us to extract reliable values of universal combinations of critical amplitudes with reasonable accuracy. Comparisons with approximate analytical predictions and other numerical results are discussed.

A new cluster algorithm for the Baxter–Wu model

We propose a new cluster algorithm for the Baxter–Wu model that significantly reduces critical slowing down. We examine the behavior of the created clusters as we vary the temperature and then specify dynamic exponents. Comparison is made with the Metropolis algorithm and with the other existing cluster algorithm.

Phase transitions in self-dual generalizations of the Baxter–Wu model

We study two types of generalized Baxter–Wu models, by means of transfer-matrix and Monte Carlo techniques. The first generalization allows for different couplings in the up- and down-triangles, and the second generalization is to a q-state spin model with three-spin interactions. Both generalizations lead to self-dual models, so that the probable locations of the phase transitions follow. Our numerical analysis confirms that phase transitions occur at the self-dual points. For both generalizations of the Baxter–Wu model, the phase transitions appear to be discontinuous.

Universal ratios of critical amplitudes in the Potts model universality class

Monte Carlo (MC) simulations and series expansions (SE) data for the energy, specific heat, magnetization, and susceptibility of the three-state and four-state Potts model and the Baxter–Wu model on the square lattice are analyzed in the vicinity of the critical point in order to estimate universal combinations of critical amplitudes. We also form effective ratios of the observables close to the critical point and analyze how they approach the universal critical-amplitude ratios. In particular, using the duality relation, we show analytically that for the Potts model with a number of states q ⩽ 4 , the effective ratio of the energy critical amplitudes always approaches unity linearly with respect to the reduced temperature. This fact leads to the prediction of relations among the amplitudes of correction-to-scaling terms of the specific heat in the low- and high-temperature phases. It is a common belief that the four-state Potts and the Baxter–Wu model belong to the same universality class. At the same time, the critical behavior of the four-state Potts model is modified by logarithmic corrections while that of the Baxter–Wu model is not. Numerical analysis shows that critical amplitude ratios are very close for both models and, therefore, gives support to the hypothesis that the critical behavior of both systems is described by the same renormalization group fixed point.

Nonequilibrium critical dynamics of the two-dimensional Ising model quenched from a correlated initial state

The universality class, even the order of the transition, of the two-dimensional Ising model depends on the range and the symmetry of the interactions (Onsager model, Baxter-Wu model, Turban model, etc.), but the critical temperature is generally the same due to self-duality. Here we consider a sudden change in the form of the interaction and study the nonequilibrium critical dynamical properties of the nearest-neighbor model. The relaxation of the magnetization and the decay of the autocorrelation function are found to display a power law behavior with characteristic exponents that depend on the universality class of the initial state.

Continuous phase transitions with a convex dip in the microcanonical entropy

The appearance of a convex dip in the microcanonical entropy of finite systems usually signals a first order transition. However, a convex dip also shows up in some systems with a continuous transition as, for example, in the Baxter-Wu model and in the four-state Potts model in two dimensions. We demonstrate that the appearance of a convex dip in those cases can be traced back to a finite-size effect. The properties of the dip are markedly different from those associated with a first order transition and can be understood within a microcanonical finite-size scaling theory for continuous phase transitions. Results obtained from numerical simulations corroborate the predictions of the scaling theory.

Baxter–Wu model with fixed magnetization

We study Baxter–Wu triangular model with fixed magnetization in the framework of canonical and microcanonical ensemble, constructing the density of states by Wang–Landau algorithm. We use an approximation similar to a recently developed scheme (critical minimum energy subspace). In this scheme the sampling is not extended to the whole energy spectrum for the given magnetization but it is restricted in a properly determined part of it. We have studied lattices with linear size up to L = 840 . Our conclusions corroborate analogous conclusions for the square Ising model: An unconventional temperature-driven first-order phase transition take place for lattice sizes above some threshold but this transition ceases to exist at the thermodynamic limit.

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

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. Finite size scaling analysis showed as expected reciprocal scaling of the positions of the peaks with L. The energy distribution yielded the expected $\alpha=2/3$ critical exponent for the specific heat. The Binder parameter minimum appeared to scale with lattice size L with an exponent $\theta_B$ equal to the specific heat exponent. Its location (temperature) showed a large correction-to-scaling term $\theta_1=0.248\pm 0.025$. 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 $\alpha=0$ was recoverd.

Short-time scaling via Monte Carlo single spin-flip algorithms for the Baxter–Wu model

The short-time critical dynamics of the Baxter–Wu model is investigated via Monte Carlo simulations using single spin-flip algorithms. The critical dynamic exponents z and θ are estimated and it is shown that the N-fold way provides a reliable estimate for the ratio of the static exponents ( 2 β / ν ) and a good confirmation of the short-time scheme. However, we find that the Metropolis and heat-bath algorithms are less satisfactory when used in the Baxter–Wu model to observe the short-time scaling.

Finite-size scaling at first- and second-order phase transitions of Baxter–Wu model

Using a finite-size phenomenological theory we investigate the behavior of the Baxter–Wu model for both first- and second-order transitions. In order to distinguish between the two kinds of transition we study the finite-size scaling behavior of the order parameter and the susceptibility of the model. At the critical temperature the known critical exponents of the system govern the scaling behavior, while below the critical temperature the scaling exponents are the dimensionality of the system. The presented scaling theory for the Baxter–Wu model is an appropriate generalization of the corresponding theory of Binder and Landau for the Ising model [Binder and Landau, Phys. Rev. B 30 (1984) 1477]. We perform numerical simulations using standard Monte Carlo techniques and we find that our results are in good agreement with theoretical predictions.

Finite-size scaling analysis of the critical behavior of the Baxter–Wu model

We use the recently developed critical minimum energy subspace (CrMES) approximation scheme to study the critical behavior of the Baxter–Wu model. This scheme uses only a properly determined part of the energy spectrum and allows us to obtain high accuracy for relatively large systems with considerably reduced computational effort. The density of states is constructed by a multi-range Wang–Landau sampling and from this we obtain the critical properties of specific heat and of the “reduced Binder cummulant”. The good agreement of our results with existing exact solutions demonstrates the accuracy of our approximation technique.