Probability-changing Cluster Research Articles

Two-size probability-changing cluster algorithm

We propose a self-adapted Monte Carlo approach to automatically determine the critical temperature by simulating two systems with different sizes at the same temperature. The temperature is increased or decreased by checking the short-time average of the correlation ratios of the two system sizes. The critical temperature is achieved using the negative feedback mechanism, which can be regarded as an Ehrenfest model for diffusion with a central force. Moreover, the thermal average near the critical temperature can be calculated precisely. The proposed approach is a general method to treat second-order phase transition, first-order phase transition, and Berezinskii–Kosterlitz–Thouless transition on the equal footing.

Large-Scale Monte Carlo Simulation of Two-Dimensional Classical XY Model Using Multiple GPUs

We study the two-dimensional classical XY model by the large-scale Monte Carlo simulation of the Swendsen-Wang multi-cluster algorithm using multiple GPUs on the open science supercomputer TSUBAME 2.0. Simulating systems up to the linear system size L =65536, we investigate the Kosterlitz–Thouless (KT) transition. Using the generalized version of the probability-changing cluster algorithm based on the helicity modulus, we locate the KT transition temperature in a self-adapted way. The obtained inverse KT temperature β KT is 1.11996(6). We estimate the exponent to specify the multiplicative logarithmic correction, -2 r , and precisely reproduce the theoretical prediction -2 r =1/8.

Novel Monte Carlo algorithms and their applications

We describe a generalized scheme for the probability-changing cluster (PCC) algorithm, based on the study of the finite-size scaling property of the correlation ratio, the ratio of the correlation functions with different distances. We apply this generalized PCC algorithm to the two-dimensional 6-state clock model. We also discuss the combination of the cluster algorithm and the extended ensemble method. We derive a rigorous broad histogram relation for the bond number. A Monte Carlo dynamics based on the number of potential moves for the bond number is proposed, and applied to the three-dimensional Ising and 3-state Potts models.

Finite-size scaling of correlation ratio and generalized scheme for the probability-changing cluster algorithm

We study the finite-size scaling (FSS) property of the correlation ratio, the ratio of the correlation functions with different distances. It is shown that the correlation ratio is a good estimator to determine the critical point of the second-order transition using the FSS analysis. The correlation ratio is especially useful for the analysis of the Kosterlitz-Thouless (KT) transition. We also present a generalized scheme of the probability-changing cluster algorithm, which has been recently developed by the present authors, based on the FSS property of the correlation ratio. We investigate the two-dimensional quantum XY model of spin 1/2 with this generalized scheme, obtaining the precise estimate of the KT transition temperature with less numerical effort.

Probability-Changing Cluster Algorithm: Study of Three-Dimensional Ising Model and Percolation Problem

We present a detailed description of the idea and procedure for the newly proposed Monte Carlo algorithm of tuning the critical point automatically, which is called the probability-changing cluster (PCC) algorithm [Y. Tomita and Y. Okabe, Phys. Rev. Lett. {\bf 86} (2001) 572]. Using the PCC algorithm, we investigate the three-dimensional Ising model and the bond percolation problem. We employ a refined finite-size scaling analysis to make estimates of critical point and exponents. With much less efforts, we obtain the results which are consistent with the previous calculations. We argue several directions for the application of the PCC algorithm.

Application of new Monte Carlo algorithms to random spin systems

We explain the idea of the probability-changing cluster (PCC) algorithm, which is an extended version of the Swendsen–Wang algorithm. With this algorithm, we can tune the critical point automatically. We show the effectiveness of the PCC algorithm for the case of the three-dimensional (3D) Ising model. We also apply this new algorithm to the study of the 3D diluted Ising model. Since we tune the critical point of each random sample automatically with the PCC algorithm, we can investigate the sample-dependent critical temperature and the sample average of physical quantities at each critical temperature, systematically. We have also applied another newly proposed algorithm, the Wang–Landau algorithm, to the study of the spin glass problem.

Probability-changing cluster algorithm for two-dimensionalXYand clock models

We extend the newly proposed probability-changing cluster (PCC) Monte Carlo algorithm to the study of systems with the vector order parameter. Wolff's idea of the embedded cluster formalism is used for assigning clusters. The Kosterlitz-Thouless (KT) transitions for the two-dimensional (2D) $\mathrm{XY}$ and q-state clock models are studied by using the PCC algorithm. Combined with the finite-size scaling analysis based on the KT form of the correlation length, $\ensuremath{\xi}\ensuremath{\propto}\mathrm{exp}(c/\sqrt{{T/T}_{\mathrm{KT}}\ensuremath{-}1}),$ we determine the KT transition temperature and the decay exponent $\ensuremath{\eta}$ as ${T}_{\mathrm{KT}}=0.8933(6)$ and $\ensuremath{\eta}=0.243(4)$ for the 2D $\mathrm{XY}$ model. We investigate two transitions of the KT type for the 2D q-state clock models with $q=6,8,12$ and confirm the prediction of $\ensuremath{\eta}{=4/q}^{2}$ at ${T}_{1},$ the low-temperature critical point between the ordered and $\mathrm{XY}$-like phases, systematically.

Crossover and self-averaging in the two-dimensional site-diluted Ising model: application of probability-changing cluster algorithm.

Using the newly proposed probability-changing cluster (PCC) Monte Carlo algorithm, we simulate the two-dimensional (2D) site-diluted Ising model. Since we can tune the critical point of each random sample automatically with the PCC algorithm, we succeed in studying the sample-dependent T(c)(L) and the sample average of physical quantities at each T(c)(L) systematically. Using the finite-size scaling (FSS) analysis for T(c)(L), we discuss the importance of corrections to FSS both in the strong-dilution and weak-dilution regions. The critical phenomena of the 2D site-diluted Ising model are shown to be controlled by the pure fixed point. The crossover from the percolation fixed point to the pure Ising fixed point with the system size is explicitly demonstrated by the study of the Binder parameter. We also study the distribution of critical temperature T(c)(L). Its variance shows the power-law L dependence, L(-n), and the estimate of the exponent n is consistent with the prediction of Aharony and Harris [Phys. Rev. Lett. 77, 3700 (1996)]. Calculating the relative variance of critical magnetization at the sample-dependent T(c)(L), we show that the 2D site-diluted Ising model exhibits weak self-averaging.