Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Finite-size scaling study of the equilibrium cluster distribution of the two-dimensional Ising model

Raúl Toral, Charles Edward Wall

Published Oct 1, 1987• 10 citations
Publisher

Abstract

The authors use a very fast and efficient algorithm to study by Monte Carlo methods the equilibrium cluster distribution C l (L) and the mean number of clusters per lattice site containing l particles in a square lattice of L 2 sites of the two-dimensional Ising model at the critical point. Finite-size scaling theory is then used to analyse the scaling ansatz C l (L)=l - tau f(l s /L), tau and s being critical exponents. The second moment of the cluster distribution p 2 (L)= Sigma l 2 C l is shown to behave as p 2 (L) approximately L theta with theta =1.895+or-0.010. The effect of corrections to scaling is also discussed.

DOI
10.1088/0305-4470/20/14/032

Related papers

Single-cluster Monte Carlo study of the Ising model on two-dimensional random lattices

Wolfhard Janke E, Mohammad W. Katoot, Ramon Villanova

We use the single-cluster Monte Carlo update algorithm to simulate the Ising model on two-dimensional Poissonian random lattices with up to 80 000 sites which are linked together according to the Voronoi-Delaunay prescription. In one set of simulations we use reweighting techniques and finite-size scaling analysis to investigate the critical properties of the model in the very vicinity of the phase transition. In the other set of simulations we study the approach to criticality in the disordered phase, making use of improved estimators for measurements. From both sets of simulations we obtain clear evidence that the critical exponents agree with the exactly known exponents for regular lattices, i.e., that (lattice) universality holds for the two-dimensional Ising model.

arXiv:hep-lat/9402008•Physical review. B, Condensed matter•Apr 1, 1994•60 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.

arXiv:cond-mat/0106154•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 27, 2001•32 citations

A finite-size scaling study of the 4D Ising model

Eduardo Sánchez-Velasco

The author studies the finite-size scaling behaviour of the susceptibility and of the second derivative of the susceptibility with respect to the magnetic field at the critical point for the four-dimensional Ising model with zero magnetic field using Monte Carlo techniques. The finite-size scaling exponents are found to be in good agreement with the form predicted by mean-field theory critical exponents.

Journal of Physics A Mathematical and General•Oct 1, 1987•9 citations

New Method to Study Critical Phenomena–Mean-Field Finite-Size Scaling Theory

Masuo Suzuki, Makoto Katori

A new method is proposed to study critical phenomena, which is a unification of Weiss' mean-field theory and Fisher's finite-size scaling theory. A systematic series of clusters is considered and generalized mean-fields are applied to the boundary spins of such clusters. From the size-dependence of the coefficients of mean-field singularities systematically obtained, the static critical exponents α, β and γ, and critical slowing down exponent Δ can be estimated. The critical point T C can be also evaluated from the size-dependence of mean-field critical points obtained in a systematic series of clusters. Some applications to the two- and three-dimensional Ising models are given explicitly to obtain T C =(2.24(0.036)) J / k B and γ=1.7(0.05) in the two-dimensional square lattice in our preliminary calculations.

Journal of the Physical Society of Japan•Jan 15, 1986•45 citations

Cluster-size heterogeneity in the two-dimensional Ising model

Woo Seong Jo, Sudo Yi, Seung Ki Baek +1 more

We numerically investigate the heterogeneity in cluster sizes in the two-dimensional Ising model and verify its scaling form recently proposed in the context of percolation problems [Phys. Rev. E 84, 010101(R) (2011)]. The scaling exponents obtained via the finite-size scaling analysis are shown to be consistent with theoretical values of the fractal dimension d(f) and the Fisher exponent τ for the cluster distribution. We also point out that strong finite-size effects exist due to the geometric nature of the cluster-size heterogeneity.

arXiv:1209.2568•Physical Review E•Sep 10, 2012•11 citations

The mean size of Ising clusters: a confluent singularity analysis

David S. Gaunt, R Brak

The authors re-examine, using a confluent singularity analysis, existing low-temperature series for the mean size of finite clusters for the pure spin- 1 / 2 nearest-neighbour Ising model in two dimensions. Although the triangular lattice is problematical, results for the honeycomb and square lattices may be interpreted as indicating a dominant exponent theta =1.87+or-0.04 (consistent with the theoretical prediction theta = gamma + beta ), together with competing analytic and non-analytic ( Delta 1 =1.35+or-0.25) correction-to-scaling terms.

Journal of Physics A Mathematical and General•Oct 1, 1985•2 citations
The authors use a very fast and efficient algorithm to study by Monte Carlo methods the equilibrium cluster distribution C l (L) and the mean number of clusters per lattice site containing l particles in a square lattice of L 2 sites of the two-dimensional Ising model at the critical point. Finite-size scaling theory is then used to analyse the scaling ansatz C l (L)=l - tau f(l s /L), tau and s being critical exponents. The second moment of the cluster distribution p 2 (L)= Sigma l 2 C l is shown to behave as p 2 (L) approximately L theta with theta =1.895+or-0.010. The effect of corrections to scaling is also discussed.
We use the single-cluster Monte Carlo update algorithm to simulate the Ising model on two-dimensional Poissonian random lattices with up to 80 000 sites which are linked together according to the Voronoi-Delaunay prescription. In one set of simulations we use reweighting techniques and finite-size scaling analysis to investigate the critical properties of the model in the very vicinity of the phase transition. In the other set of simulations we study the approach to criticality in the disordered phase, making use of improved estimators for measurements. From both sets of simulations we obtain clear evidence that the critical exponents agree with the exactly known exponents for regular lattices, i.e., that (lattice) universality holds for the two-dimensional Ising model.
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.
A new method is proposed to study critical phenomena, which is a unification of Weiss' mean-field theory and Fisher's finite-size scaling theory. A systematic series of clusters is considered and generalized mean-fields are applied to the boundary spins of such clusters. From the size-dependence of the coefficients of mean-field singularities systematically obtained, the static critical exponents α, β and γ, and critical slowing down exponent Δ can be estimated. The critical point T C can be also evaluated from the size-dependence of mean-field critical points obtained in a systematic series of clusters. Some applications to the two- and three-dimensional Ising models are given explicitly to obtain T C =(2.24(0.036)) J / k B and γ=1.7(0.05) in the two-dimensional square lattice in our preliminary calculations.
We numerically investigate the heterogeneity in cluster sizes in the two-dimensional Ising model and verify its scaling form recently proposed in the context of percolation problems [Phys. Rev. E 84, 010101(R) (2011)]. The scaling exponents obtained via the finite-size scaling analysis are shown to be consistent with theoretical values of the fractal dimension d(f) and the Fisher exponent τ for the cluster distribution. We also point out that strong finite-size effects exist due to the geometric nature of the cluster-size heterogeneity.
The authors re-examine, using a confluent singularity analysis, existing low-temperature series for the mean size of finite clusters for the pure spin- 1 / 2 nearest-neighbour Ising model in two dimensions. Although the triangular lattice is problematical, results for the honeycomb and square lattices may be interpreted as indicating a dominant exponent theta =1.87+or-0.04 (consistent with the theoretical prediction theta = gamma + beta ), together with competing analytic and non-analytic ( Delta 1 =1.35+or-0.25) correction-to-scaling terms.