Wolff Algorithm Research Articles

New Nonequilibrium-to-Equilibrium Dynamical Scaling and Stretched-Exponential Critical Relaxation in Cluster Algorithms

Nonequilibrium relaxation behaviors in the Ising model on a square lattice based on the Wolff algorithm are totally different from those based on local-update algorithms. In particular, the critical relaxation is described by the stretched-exponential decay. We propose a novel scaling procedure to connect nonequilibrium and equilibrium behaviors continuously, and find that the stretched-exponential scaling region in the Wolff algorithm is as wide as the power-law scaling region in local-update algorithms. We also find that relaxation to the spontaneous magnetization in the ordered phase is characterized by the exponential decay, not the stretched-exponential decay based on local-update algorithms.

Location of the Bicritical Point of the Anisotropic Heisenberg Model in a Crystal Field

Extensive Monte Carlo simulations are used to locate the bicritical point of the ferromagnetic anisotropic Heisenberg model on a simple cubic lattice in the presence of a crystal field. In order to achieve this goal, we employ a hybrid algorithm comprising Metropolis and Wolff algorithms along with field mixing. Multiple histogram techniques have also been used to obtain the probability distribution function of the conjugate extensive variables of interest. In conjunction with finite-size scaling analysis we computed the fourth-order cumulant of the order parameter along the first-order line and the corresponding bicritical point has been located from the crossings of the cumulants.

Dynamic scaling at classical phase transitions approached through nonequilibrium quenching

We use Monte Carlo simulations to demonstrate generic scaling aspects of classical phase transitions approached through a quench (or annealing) protocol where the temperature changes as a function of time with velocity $v$. Using a generalized Kibble-Zurek ansatz, we demonstrate dynamic scaling for different types of stochastic dynamics (Metropolis, Swendsen-Wang, and Wolff) on Ising models in two and higher dimensions. We show that there are dual scaling functions governing the dynamic scaling, which together describe the scaling behavior in the entire velocity range $v\ensuremath{\in}[0,\ensuremath{\infty})$. These functions have asymptotics corresponding to the adiabatic and diabatic limits, and close to these limits they are perturbative in $v$ and $1/v$, respectively. Away from their perturbative domains, both functions cross over into the same universal power-law scaling form governed by the static and dynamic critical exponents (as well as an exponent characterizing the quench protocol). As a by-product of the scaling studies, we obtain high-precision estimates of the dynamic exponent $z$ for the two-dimensional Ising model subject to the three variants of Monte Carlo dynamics: for single-spin Metropolis updates ${z}_{\mathrm{M}}=2.1767(5)$, for Swendsen-Wang multicluster updates ${z}_{\mathrm{SW}}=0.297(3)$, and for Wolff single-cluster updates ${z}_{\text{W}}=0.30(2)$. For Wolff dynamics, we find an interesting behavior with a nonanalytic breakdown of the quasiadiabatic and diabatic scalings, instead of the generic smooth crossover described by a power law. We interpret this disconnect between the two scaling regimes as a dynamic phase transition of the Wolff algorithm, caused by an effective sudden loss of ergodicity at high velocity.

Role of reduced geometry on critical spin-crossover behavior in molecular magnet: Monte Carlo simulation

In this work, the magnetic spin-crossover (SC) behavior was investigated using Monte Carlo simulation. The investigated SC structures were hexagonal films where each molecular magnetic spin was represented by the Ising spins interacting with neighboring spins with short-ranged exchange-type interaction and long-ranged mechano-elastic interaction. The SC spins were mobile but under spring-type interaction constraint. Monte Carlo simulation was used to attain the system mechanical equilibrium, and used to update spin configurations via the modified Wolff algorithm. With varying the films thickness, the radius ratio of the high-spin-state to the low-spin-state, and the system temperature, the thermal equilibrium magnetization was measured to extract the magnetic profiles as well as the fourth order cumulant of the magnetization. The critical temperature Tc was extracted via the cumulant-crossing extrapolated to the thermodynamic limit. From the simulation, the magnetization and magnetic susceptibility profiles were obtained as functions of temperature for films thickness ranging from one to four layers. The results show that with increasing films thickness the critical temperature increases due to stronger average short-ranged magnetic interaction. In addition, the greater radius ratio of the high-spin-state to the low-spin-state enhances the phase transition point since higher thermal energy level is required to compensate the larger thermodynamic work required by the volume changed. These results are in agreement with previous studies. The scaling relationship among the critical point, the radius ratio, and films thickness was also successfully proposed. With good statistical accuracy, this scaling formalism can then be used as qualitative guidelines to predict magnetic critical behaviors for further enhanced investigation or for designing desired functional SC applications.

Critical behavior of two-dimensional spin systems under the random-bond six-state clock model

The critical behavior of the clock model in two-dimensional square lattice is studied numerically using Monte Carlo method with Wolff algorithm. The Kosterlitz-Thouless (KT) transition is observed in the six-state clock model, where an intermediate phase exists between the low-temperature ordered phase and the high-temperature disordered phase. The bond randomness is introduced to the system by assuming a Gaussian distribution for the coupling coefficients with the mean μ=1 and different values of variance, from σ2=0.1 to σ2=3.0. An abrupt jump in the helicity modulus at the transition, which is the key characteristic of the KT transition, is verified with a stability argument. The critical temperature Tc for both pure and disordered systems is determined from the critical exponent η(Tc)=1/4. The results showed that a small amount of disorder (small σ) reduces the critical temperature of the system, without altering the nature of transition. However, a larger amount of disorder changes the transition from the KT-type into that of non-KT-type.

Investigation of Phase Transitions in the Site-Diluted Three-Dimensional Potts Model

The phase transitions and critical phenomena in three-dimensional (3D) site-diluted 3-and 4-state Potts models is investigated by Monte-Carlo method based on the highly efficient Wolff algorithm. The systems with linear sizesL=20-44 at spin concentrationsp=1.00, 0.95, 0.90, 0.80, 0.70, 0.65 are explored. The second-order phase transition is shown to occur in the three-dimensional 3-state Potts model with nonmagnetic impurities. In the 4-state Potts model there are observed first-order phase transitions in weakly diluted state, when the model is strongly diluted the first-order phase transitions change to the second-order one. On the basis of the finite-size scaling theory static critical exponents of specific heatα, susceptibilityγ, magnetizationβ, and exponent of correlation radiusνfor the systems under study are calculated.

Dynamical behavior of the Niedermayer algorithm applied to Potts models

In this work, we make a numerical study of the dynamic universality class of the Niedermayer algorithm applied to the two-dimensional Potts model with 2, 3, and 4 states. This algorithm updates clusters of spins and has a free parameter, E0, which controls the size of these clusters, such that E0=1 is the Metropolis algorithm and E0=0 regains the Wolff algorithm, for the Potts model. For −1<E0<0, only clusters of equal spins can be formed: we show that the mean size of the clusters of (possibly) turned spins initially grows with the linear size of the lattice, L, but eventually saturates at a given lattice size L˜, which depends on E0. For L≥L˜, the Niedermayer algorithm is in the same dynamic universality class of the Metropolis one, i.e, they have the same dynamic exponent. For E0>0, spins in different states may be added to the cluster but the dynamic behavior is less efficient than for the Wolff algorithm (E0=0). Therefore, our results show that the Wolff algorithm is the best choice for Potts models, when compared to the Niedermayer’s generalization.

Influence of aperiodic modulations on first-order transitions: Numerical study of the two-dimensional Potts model

We study the Potts model on a rectangular lattice with aperiodic modulations in its interactions along one direction. Numerical results are obtained using the Wolff algorithm and for many lattice sizes, allowing for a finite-size scaling analyses to be carried out. Three different self-dual aperiodic sequences are employed, which leads to more precise results, since the exact critical temperature is known. We analyze two models, with 6 and 15 number of states: both present first-order transitions on their uniform versions. We show that the Harris-Luck criterion, originally introduced in the study of continuous transitions, is obeyed also for first-order ones. Also, we show that the new universality class that emerges for relevant aperiodic modulations depends on the number of states of the Potts model, as obtained elsewhere for random disorder, and on the aperiodic sequence. We determine the occurrence of log-periodic behavior, as expected for models with aperiodic modulated interactions.

Monte Carlos studies of critical and dynamic phenomena in mixed bond Ising model

The phase transition of a random mixed-bond Ising ferromagnet on a cubic lattice model is studied both numerically and analytically. In this work, we use the Metropolis and Wolff algorithm with histogram technique and finite size scaling theory to simulate the dynamics of the system. We obtained the thermodynamic quantities such as magnetization, susceptibility, and specific heat. Our results were compared with those obtained using a new technique in effective field theory that employs similar probability distribution within the framework of two-site clusters.

Numerical simulation study of the dynamical behavior of the Niedermayer algorithm

We calculate the dynamic critical exponent for the Niedermayer algorithm applied to the two-dimensionalIsing and XY models, for various values of the free parameterE0. ForE0 = − 1 we regain the Metropolis algorithm and forE0 = 1 we regain the Wolffalgorithm. For − 1 < E0 < 1, we show that the mean size of the clusters of (possibly) turned spins initially grows with the linear size ofthe lattice, L, but eventually saturates at a given lattice size , which depends on E0. For , the Niedermayer algorithm is equivalent to the Metropolis one, i.e., they have the same dynamic exponent.For E0 > 1, the autocorrelation time is always greater than forE0 = 1 (Wolff) and, more important, it also grows faster than a power ofL. Therefore, we show that the best choice of cluster algorithm is the Wolffone, when comparing against the Niedermayer generalization. We also obtainthe dynamic behavior of the Wolff algorithm: although not conclusively, wepropose a scaling law for the dependence of the autocorrelation time onL.

Monte Carlo analysis of critical phenomenon of the Ising model on memory stabilizer structures

We calculate the critical temperature of the Ising model on a set of graphs representing a concatenated three-bit error-correction code. The graphs are derived from the stabilizer formalism used in quantum error correction. The stabilizer for a subspace is defined as the group of Pauli operators whose eigenvalues are $+1$ on the subspace. The group can be generated by a subset of operators in the stabilizer, and the choice of generators determines the structure of the graph. The Wolff algorithm, together with the histogram method and finite-size scaling, is used to calculate both the critical temperature and the critical exponents of each structure. The simulations show that the choice of stabilizer generators, both the number and the geometry, has a large effect on the critical temperature.

Single-cluster dynamics for the random-cluster model

We formulate a single-cluster Monte Carlo algorithm for the simulation of the random-cluster model. This algorithm is a generalization of the Wolff single-cluster method for the q-state Potts model to noninteger values q>1. Its results for static quantities are in a satisfactory agreement with those of the existing Swendsen-Wang-Chayes-Machta (SWCM) algorithm, which involves a full-cluster decomposition of random-cluster configurations. We explore the critical dynamics of this algorithm for several two-dimensional Potts and random-cluster models. For integer q, the single-cluster algorithm can be reduced to the Wolff algorithm, for which case we find that the autocorrelation functions decay almost purely exponentially, with dynamic exponents z(exp)=0.07 (1), 0.521 (7), and 1.007 (9) for q=2, 3, and 4, respectively. For noninteger q, the dynamical behavior of the single-cluster algorithm appears to be very dissimilar to that of the SWCM algorithm. For large critical systems, the autocorrelation function displays a range of power-law behavior as a function of time. The dynamic exponents are relatively large. We provide an explanation for this peculiar dynamic behavior.

Locally converging algorithms for determining the critical temperature in Ising systems

We introduce a class of algorithms that converge to criticality automatically, in a way similar to the invaded cluster algorithm. Unlike the invaded cluster algorithm which uses global percolation as a test for criticality, these local algorithms use an average over local observables, specifically the number of satisfied bonds, in a feedback loop which drives the system toward criticality. Two specific algorithms are introduced, the average algorithm and the locally converging Wolff algorithm. We apply these algorithms to study the Ising square lattice and the Ising Bethe lattice. We find reasonable convergence to the critical temperature for both systems under the locally converging Wolff algorithm. We also re-examine the phase diagram of the dilute two-dimensional (2D) Ising model and find results supporting our previously reported conclusions regarding the existence of a local regime of magnetization below the percolations threshold. In addition, the presented algorithms are computationally more efficient than the invaded cluster algorithm, requiring less CPU time and memory.

On Potts Model Clustering, Kernel K-Means and Density Estimation

Many clustering methods, such as K -means, kernel K -means, and MNcut clustering, follow the same recipe: (i) choose a measure of similarity between observations; (ii) define a figure of merit assigning a large value to partitions of the data that put similar observations in the same cluster; and (iii) optimize this figure of merit over partitions. Potts model clustering represents an interesting variation on this recipe. Blatt, Wiseman, and Domany defined a new figure of merit for partitions that is formally similar to the Hamiltonian of the Potts model for ferromagnetism, extensively studied in statistical physics. For each temperature T, the Hamiltonian defines a distribution assigning a probability to each possible configuration of the physical system or, in the language of clustering, to each partition. Instead of searching for a single partition optimizing the Hamiltonian, they sampled a large number of partitions from this distribution for a range of temperatures. They proposed a heuristic for choosing an appropriate temperature and from the sample of partitions associated with this chosen temperature, they then derived what we call a consensus clustering: two observations are put in the same consensus cluster if they belong to the same cluster in the majority of the random partitions. In a sense, the consensus clustering is an “average” of plausible configurations, and we would expect it to be more stable (over different samples)than the configuration optimizing the Hamiltonian.The goal of this article is to contribute to the understanding of Potts model clustering and to propose extensions and improvements: (1) We show that the Hamiltonian used in Potts model clustering is closely related to the kernel K -means and MNCutcriteria. (2) We propose a modification of the Hamiltonian penalizing unequal clustersizes and show that it can be interpreted as a weighted version of the kernel K -meanscriterion. (3) We introduce a new version of the Wolff algorithm to simulate configurations from the distribution defined by the penalized Hamiltonian, leading to penalized Potts model clustering. (4) We note a link between kernel based clustering methods and nonparametric density estimation and exploit it to automatically determine locally adaptive kernel bandwidths. (5) We propose a new simple rule for selecting a good temperature T.As an illustration we apply Potts model clustering to gene expression data and compare our results to those obtained by model based clustering and a nonparametric dendrogram sharpening method.

Improving Random Number Generators in the Monte Carlo simulations via twisting and combining

Problems for various random number generators accompanying the Wolff algorithm [U. Wolff, Phys. Rev. Lett. 62 (1989) 361; U. Wolff, Phys. Lett. B 228 (1989) 379] are discussed, including the hidden errors first reported in [A.M. Ferrenberg, D.P. Landau, Y.J. Wong, Phys. Rev. Lett. 69 (1992) 3382]. A general (though simple) method of twisting and combining for improving the performance of these generators is proposed. Some recent generators motivated by such a twisting and combining method with extremely long period are discussed. The proposed method provides a novel and simple way to improve RNGs in its performance.

Schramm–Loewner evolution in the three-state Potts model—a numerical study

The scaling limit of the spin cluster boundaries of the Ising model withdomain wall boundary conditions is Schramm–Loewner evolution (SLE) withκ = 3. We hypothesize that the three-state Potts model with appropriate boundary conditionshas spin cluster boundaries which are also SLE in the scaling limit, but withκ = 10/3. To test this, we generate samples using the Wolff algorithm and test them againstpredictions of SLE: we examine the statistics of the Loewner driving function, estimate thefractal dimension and test against Schramm’s formula. The results are in support of ourhypothesis.

Monte Carlo studies of the cubic lattice mixed-bond Ising model

In this work, we used the Cluster Wolff algorithms to simulate the dynamics of the system. We obtained the thermodynamic quantities such as magnetization and susceptibility. Critical temperatures were estimated at the maximum of the susceptibility and with these values we made the phase diagram, T C versus p (the concentration of magnetic bonds) for different α values, being α the competition parameter. Our results were compared to those obtained by using a new technique in the effective field theory which employs similar probability distribution within the framework of two-site clusters.

Using duration models to reduce fragmentation in audio segmentation

We investigate explicit segment duration models in addressing the problem of fragmentation in musical audio segmentation. The resulting probabilistic models are optimised using Markov Chain Monte Carlo methods; in particular, we introduce a modification to Wolff's algorithm to make it applicable to a segment classification model with an arbitrary duration prior. We apply this to a collection of pop songs, and show experimentally that the generated segmentations suffer much less from fragmentation than those produced by segmentation algorithms based on clustering, and are closer to an expert listener's annotations, as evaluated by two different performance measures.

The critical temperature and shift exponent of the layered Ising lattice with surface exchange modification

Using Monte Carlo simulation with the Wolff algorithm and single histogram method, the critical behavior of the ferromagnetic Ising thin films with thickness ranging from n = 3 to 15 layers, has been studied. With varying the ratio of surface interaction to bulk one, κ = J s / J B , we found a special point κ sp at which all the film have a unique critical temperature independent of film thickness n . In the region that κ is less than κ sp , the shift exponent λ is independent of κ in the limit n → ∞ , but the strength of surface coupling strongly affects the gradient of asymptotic curve λ vs 1 / n . When κ is larger than κ sp , however, the clear power law behavior is not found.

Dynamic critical exponents for Swendsen–Wang and Wolff algorithms obtained by anonequilibrium relaxation method

Using a nonequilibrium relaxation method, we calculate the dynamic critical exponentz of the two-dimensional Ising model for the Swendsen–Wang and Wolffalgorithms. We examine dynamic relaxation processes following a quenchfrom a disordered or an ordered initial state to the critical temperatureTc, and measure the exponential relaxation time of the system energy. For theSwendsen–Wang algorithm with an ordered or a disordered initial state,and for the Wolff algorithm with an ordered initial state, the exponentialrelaxation time fits well to a logarithmic size dependence up to a lattice sizeL = 8192. For the Wolff algorithm with a disordered initial state, we obtain an effective dynamic exponentzexp = 1.19(2) upto L = 2048. For comparison, we also compute the effective dynamic exponents through the integratedcorrelation times. In addition, an exact result of the Swendsen–Wang dynamic spectrum ofa one-dimensional Ising chain is derived.