Wolff Algorithm Research Articles

Critical dynamics of cluster algorithms in the random-bond Ising model.

In the present work, we present an extensive Monte Carlo simulation study on the dynamical properties of the two-dimensional random-bond Ising model. The correlation time τ of the Swendsen-Wang and Wolff cluster algorithms is calculated at the critical point. The dynamic critical exponent z of both algorithms is also measured by using the numerical data for several lattice sizes up to L=512. It is found for both algorithms that the autocorrelation time decreases considerably and the critical slowing-down effect reduces upon the introduction of bond disorder. Additionally, simulations with the Metropolis algorithm are performed, and the critical slowing-down effect is observed to be more pronounced in the presence of disorder, confirming the previous findings in the literature. Moreover, the existence of the non-self-averaging property of the model is demonstrated by calculating the scaled form of the standard deviation of autocorrelation times. Finally, the critical exponent ratio of the magnetic susceptibility is estimated by using the average cluster size of the Wolff algorithm.

Monte Carlo simulation with Wolff algorithm for scaling behavior of two dimensional XY model with KT phase transition

The property of XY model with its KT phase transition has been discussed in many aspects such as specific heat and magnetic susceptibility along with correlation function. Also, this is one of the original models for topological phase transition which gives a new type of phase transition without symmetry breaking. Since the calculation of these properties above can be achieved by Monte Carlo Method with Wolff algorithm, the visualization of vortex and antivortex is also an interesting task which is accomplished in this article. By successfully modeling the system and getting the step-by-step configuration, the relation between the generation of vortex and the variation of temperature can be concluded in a qualitative way. The calculation of spin correlation function shows the phase transition between disorder and quasi long-range order which represents the topological phase transition. As a reliable method, the Monte Carlo method with Wolff algorithm can be a choice of generating the training set for neural network which can calculated much more sophisticated case in a relatively short time and the transition between classical simulation and quantum simulation can be approached as mentioned in the end of this article.

Static universality of the Ising and Blume–Capel models on two-dimensional Penrose tiles

We study the phase transitions and the critical behavior of the Ising and Blume–Capel models on two kinds of two-dimensional Penrose tiles employing Monte Carlo methods and detailed finite-size scaling analysis. The Wolff algorithm is used for the Ising model whereas the Wang–Landau and Metropolis algorithms are used for the Blume–Capel model to obtain critical temperatures and exponents. The phase diagram of the Blume–Capel model reveals the presence of the double transition, the reentrant behavior, and the tricriticality in the vicinity of the critical single-ion anisotropy. Our results indicate that continuous phase transitions of the Ising and Blume–Capel models on the Penrose tiles belong to the two-dimensional Ising universality class. However, the scaling functions depend on the spin magnitude, anisotropy, and graph shape. Lastly, we verify that the critical Binder cumulant is also universal but depends on the shape of the graph.

Comparison of cluster algorithms for the bond-diluted Ising model.

Monte Carlo cluster algorithms are popular for their efficiency in studying the Ising model near its critical temperature. We might expect that this efficiency extends to the bond-diluted Ising model. We show, however, that this is not always the case by comparing how the correlation times τ_{w} and τ_{sw} of the Wolff and Swendsen-Wang cluster algorithms scale as a function of the system size L when applied to the two-dimensional bond-diluted Ising model. We demonstrate that the Wolff algorithm suffers from a much longer correlation time than in the pure Ising model, caused by isolated (groups of) spins which are infrequently visited by the algorithm. With a simple argument we prove that these cause the correlation time τ_{w} to be bounded from below by L^{z_{w}} with a dynamical exponent z_{w}=γ/ν≈1.75 for a bond concentration p<1. Furthermore, we numerically show that this lower bound is actually taken for several values of p in the range 0.5<p<1. Moreover, we show that the Swendsen-Wang algorithm does not suffer from the same problem. Consequently, it has a much shorter correlation time, shorter than in the pure Ising model even. Numerically at p=0.6, we find that its dynamical exponent is z_{sw}=0.09(4).

Geometric allocation approach to accelerating directed worm algorithm.

The worm algorithm is a versatile technique in the Markov chain Monte Carlo method for both classical and quantum systems. The algorithm substantially alleviates critical slowing down and reduces the dynamic critical exponents of various classical systems. It is crucial to improve the algorithm and push the boundary of the Monte Carlo method for physical systems. We here propose a directed worm algorithm that significantly improves computational efficiency. We use the geometric allocation approach to optimize the worm scattering process: worm backscattering is averted, and forward scattering is favored. Our approach successfully enhances the diffusivity of the worm head (kink), which is evident in the probability distribution of the relative position of the two kinks. Performance improvement is demonstrated for the Ising model at the critical temperature by measurement of exponential autocorrelation times and asymptotic variances. The present worm update is approximately 25 times as efficient as the conventional worm update for the simple cubic lattice model. Surprisingly, our algorithm is even more efficient than the Wolff cluster algorithm, which is one of the best update algorithms. We estimate the dynamic critical exponent of the simple cubic lattice Ising model to be z≈0.27 in the worm update. The worm and the Wolff algorithms produce different exponents of the integrated autocorrelation time of the magnetic susceptibility estimator but the same exponent of the asymptotic variance. We also discuss how to quantify the computational efficiency of the Markov chain Monte Carlo method. Our approach can be applied to a wide range of physical systems, such as the |ϕ|^{4} model, the Potts model, the O(n) loop model, and lattice QCD.

Chaotic modes of a non-linear fractional oscillator

A non-linear fractional oscillator is a generalization of a classical non-linear oscillator in consideration of the hereditary or the memory effect. The memory effect is a property of a dynamic system in which its current state depends on a finite number of its previous states. Therefore, a non-linear fractional oscillator can be mathematically described using integro-differential equations with difference kernels or fractional order derivatives. In this paper, a fractional non-linear oscillator has been investigated to identify chaotic oscillatory modes. The quantitative measure of chaotic regimes is the largest (maximal) Lyapunov exponents. For calculating the maximal Lyapunov exponents, the Wolff algorithm based on the Gram-Schmidt orthogonalization procedure has been selected, using both numerical solution for an initial fractional dynamical system by using variational equations. The Wolff algorithm also makes it possible to plot the spectrum of Lyapunov exponents as a function of control parameters for the initial dynamical system. It has been shown that some spectra of Lyapunov exponents contain positive values indicating the existence of chaotic modes, which are also confirmed by the corresponding phase trajectories.

Cluster size effects in the magnetic properties of [formula omitted]-[formula omitted] alloys

A spin-1/2 Ising model, defined in the body centered cubic lattice, is used to describe some of the thermodynamic properties of Fep-Alq alloys, with p+q=1. The model assumes, besides the nearest-neighbor exchange coupling, the existence of further next-nearest-neighbor superexchange interactions, where the latter ones depend on the aluminum atoms cluster size. The Ising system so considered is studied by employing Monte Carlo simulations, using a hybrid algorithm consisting of one single-spin Metropolis move together with one single-cluster Wolff algorithm allied, in addition, with single histograms procedures and finite-size scaling techniques. Quite good fits to the experimental results of the ordering critical temperature, as a function of Al concentration in the range 0⩽q<0.7, are obtained and compared to more recent theoretical approaches done on the same alloys.

Compensation temperature in spin-[formula omitted] Ising trilayers: A Monte Carlo study

We study the magnetic and thermodynamic properties of a spin-1∕2 Ising system containing three layers, each of which is composed exclusively of one out of two possible types of atoms, A or B. The A-A and B-B bonds are ferromagnetic while the A-B bonds are antiferromagnetic. The study is performed through Monte Carlo simulations using the Wolff algorithm and the data are analyzed with the aid of the multiple-histogram reweighting technique and finite-size scaling tools. We verify the occurrence of a compensation phenomenon and obtain the compensation and critical temperatures of the model as functions of the Hamiltonian parameters. The influence of each parameter on the overall behavior of the system is discussed in detail and we present our results in the form of phase diagrams dividing the parameter space in regions where the compensation phenomenon is present or absent. Our results may provide invaluable information for experimentalists seeking to build materials with desired characteristics.

Automatic Prostate Lesions Detection on MR Images Based on the Ising Model

Prostate cancer is the second most prevalent type of cancer in the male population worldwide. Prostate imaging tests have adopted for the prevention, diagnosis, and treatment. It is known that early detection increases the chances of an effective treatment, improving the prognosis of the disease. This paper proposes an automatic methodology for prostate lesions detection. It consists of the following steps: Extracting candidates for lesions with Wolff algorithm; feature extraction using the Ising model measures and finally the uses support vector machine in the classification of a lesion or healthy tissue. The methodology was validated using a set of 28 exams containing the lesion markings and obtained a sensitivity of 95.92%, specificity of 93.89% and accuracy of 94.16%. These are promising since they were more significant than other methods compared.

Cluster representations and the Wolff algorithm in arbitrary external fields

We introduce a natural way to extend celebrated spin-cluster Monte Carlo algorithms for fast thermal lattice simulations at criticality, such as the Wolff algorithm, to systems in arbitrary fields, be they linear magnetic vector fields or nonlinear anisotropic ones. By generalizing the ``ghost spin'' representation to one with a ``ghost transformation,'' global invariance to spin symmetry transformations is restored at the cost of an extra degree of freedom which lives in the space of symmetry transformations. The ordinary cluster-building process can then be run on the representation. We show that this extension preserves the scaling of accelerated dynamics in the absence of a field for Ising, Potts, and $\mathrm{O}(n)$ models and demonstrate the method's use in modeling the presence of novel nonlinear fields. We also provide a c++ library for the method's convenient implementation for arbitrary models.

Weak Universality in the Disordered Two-Dimensional Antiferromagnetic Potts Model on a Triangular Lattice

The critical behavior of the disordered two-dimensional antiferromagnetic Potts model with the number of spin states q= 3 on a triangular lattice with disorder in the form of nonmagnetic impurities is studied by the Monte Carlo method. The critical exponents for the susceptibility γ, magnetization β, specific heat α, and correlation radius ν are calculated in the framework of the finite-size scaling theory at spin concentrations p = 0.90, 0.80, 0.70, and 0.65. It is found that the critical exponents increase with the degree of disorder, whereas the ratios and do not change, thus holding the scaling equality $$\frac{{2\beta }}{\nu } + \frac{\gamma }{\nu } = d$$ . Such behavior of the critical exponents is related to the weak universality of the critical behavior characteristic of disordered systems. All results are obtained using independent Monte Carlo algorithms, such as the Metropolis and Wolff algorithms.

Research dynamic modes of stick-slip effect with the account of hereditarity

In study with the help of the spectrum of maximal Lyapunov exponents, dynamic regimes of the stick-slip effect were studied with allowance effect of hereditarity. Spectrum of the Lyapunov exponents were constructed using the Wolff algorithm with Gram-Schmidt orthogonalization depending on the values of the control parametersfriction and adhesion coefficients, as well as fractional index values, which determine the heredity of the dynamical system under consideration. The existence of an area of positive values of the maximum Lyapunov exponents is shown, which indicates the presence of chaotic regimes. Oscillograms and phase trajectories are constructed.

Cluster size distribution to characterize the first–order phase transitions of equilibrium systems

Through Monte Carlo simulations we study cluster size distribution (CSD) of equilibrium systems. The system is simulated by applying the Wolff algorithm or single cluster Monte Carlo update algorithm. Our results show that CSD is a good tool in the identification of first–order transitions.

Critical properties of 2d disordered 3-state antiferromagnetic potts model ON TRIANGULAR LATTICE

By introducing a small amount of non-magnetic impurities into an antiferromagnetic (AF) two-dimensional (2D) Potts model on a triangular lattice it is that the impurities in spin systems described by this model result in the change of a first order to a second-order phase transition. The systems with linear sizes L × L = N, L = 9-144 are considered. Investigations are performed using the standard Metropolis algorithm along with Monte-Carlo single-cluster Wolff algorithm. On the basis of the theory of finite-size scaling, critical exponents (CE) are calculated: the heat capacity α, the susceptibility γ, the order parameter β, and the CE of the correlation radius ν.

Monte Carlo study of an anisotropic Ising multilayer with antiferromagnetic interlayer couplings

We present a Monte Carlo study of the magnetic properties of an Ising multilayer ferrimagnet. The system consists of two kinds of non-equivalent planes, one of which is site-diluted. All intralayer couplings are ferromagnetic. The different kinds of planes are stacked alternately and the interlayer couplings are antiferromagnetic. We perform the simulations using the Wolff algorithm and employ multiple histogram reweighting and finite-size scaling methods to analyze the data with special emphasis on the study of compensation phenomena. Compensation and critical temperatures of the system are obtained as functions of the Hamiltonian parameters and we present a detailed discussion about the contribution of each parameter to the presence or absence of the compensation effect. A comparison is presented between our results and those reported in the literature for the same model using the pair approximation. We also compare our results with those obtained through both the pair approximation and Monte Carlo simulations for the bilayer system.

Phase transitions and thermodynamic properties of triangular strongly-diluted antiferromagnetic Potts model

By introducing a substantial amount of non-magnetic impurities into an antiferromagnetic 2d Potts model on a triangular lattice it is that the impurities in spin systems described by this model result in the change of a first-order to a second-order phase transition. The systems with linear sizes L×L=N, L=18−48 are considered. Investigations are performed using the standard Metropolis algorithm along with Monte-Carlo single-cluster Wolff algorithm.

Nonequilibrium behaviors of the three-dimensional Heisenberg model in the Swendsen-Wang algorithm.

Recently, it was shown [Y. Nonomura, J. Phys. Soc. Jpn. 83, 113001 (2014)JUPSAU0031-901510.7566/JPSJ.83.113001] that the nonequilibrium critical relaxation of the two-dimensional (2D) Ising model from a perfectly ordered state in the Wolff algorithm is described by stretched-exponential decay, and a universal scaling scheme was found to connect nonequilibrium and equilibrium behaviors. In the present study we extend these findings to vector spin models, and the 3D Heisenberg model could be a typical example. To evaluate the critical temperature and critical exponents precisely using the above scaling scheme, we calculate nonequilibrium ordering from the perfectly disordered state in the Swendsen-Wang algorithm, and we find that the critical ordering process is described by stretched-exponential growth with a comparable exponent to that of the 3D XY model. The critical exponents evaluated in the present study are consistent with those in previous studies.

Static and dynamic critical behavior of thin magnetic Ising films

This work presents a study of the effect of film thickness on the static and dynamic critical behavior of thin magnetic Ising films. Monte Carlo simulations using the Wolff algorithm were performed to determine the static and dynamic critical exponents of the films. A dimensionality crossover from 2D to 3D (due to the finiteness of the films) in the static and dynamic critical behavior was observed as the film thickness increases. In addition, a slight increase in the effective dimension deff and a considerable increase in the critical temperature Tc(∞) were found. Small values for the dynamic critical exponents were obtained, indicating that the Wolff algorithm is a very efficient method for these magnetic systems.

Bicritical universality of the anisotropic Heisenberg model in a crystal field.

The bicritical properties of the three-dimensional classical anisotropic Heisenberg model in a crystal field are investigated through extensive Monte Carlo simulations on a simple cubic lattice, using Metropolis and Wolff algorithms. Field-mixing and multidimensional histogram techniques were employed in order to compute the probability distribution function of the extensive conjugate variables of interest and, using finite-size scaling analysis, the first-order transition line of the model was precisely located. The fourth-order cumulant of the order parameter was then calculated along this line and the bicritical point located with good precision from the cumulant crossings. The bicritical properties of this point were further investigated through the measurement of the universal probability distribution function of the order parameter. The results lead us to conclude that the studied bicritical point belongs in fact to the three-dimensional Heisenberg universality class.

Rényi information flow in the Ising model with single-spin dynamics.

The n-index Rényi mutual information and transfer entropies for the two-dimensional kinetic Ising model with arbitrary single-spin dynamics in the thermodynamic limit are derived as functions of ensemble averages of observables and spin-flip probabilities. Cluster Monte Carlo algorithms with different dynamics from the single-spin dynamics are thus applicable to estimate the transfer entropies. By means of Monte Carlo simulations with the Wolff algorithm, we calculate the information flows in the Ising model with the Metropolis dynamics and the Glauber dynamics, respectively. We find that not only the global Rényi transfer entropy, but also the pairwise Rényi transfer entropy, peaks in the disorder phase.