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.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.

arXiv:cond-mat/0203454Journal of the Physical Society of JapanJun 15, 200215 citations

Generalized Probability-Changing Algorithm

Yusuke Tomita, Yutaka Okabe

Springer proceedings in physicsJan 1, 20041 citation

Framework and properties of cluster algorithms

Werner Kerler A

Nuclear Physics B - Proceedings SupplementsMar 1, 19932 citations

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

Yusuke Tomita, Yutaka Okabe

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.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.

arXiv:cond-mat/0106154Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsAug 27, 200132 citations

Probability-changing cluster algorithm for two-dimensional XY and clock models

Yusuke Tomita, Yutaka Okabe

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.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.

arXiv:cond-mat/0202161Physical review. B, Condensed matterApr 12, 200291 citations

\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.