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

Europhysics Letters (EPL)

The Cluster Updating Monte Carlo Algorithm Applied to the 3 d Ising Problem

F. Livet

Published Sep 7, 1991• 24 citations
Publisher

Abstract

The cluster updating Monte Carlo algorithm is applied to the 3 d simple cubic Ising model. The experimental results are shown to have no critical slowing-down. A simple method predicts the limits of any Monte Carlo calculation. These limits are closely approached. The information treatment is then discussed in calculating the transition temperature. By the simple “fourth-order cumulant method” of Binder, one obtains β c ≃ 0.221 654 4(10).

DOI
10.1209/0295-5075/16/2/003

Related papers

Pushing the limits of Monte Carlo simulations for the three-dimensional Ising model

Alan M. Ferrenberg, Jiahao Xu, David Paul Landau

While the three-dimensional Ising model has defied analytic solution, various numerical methods like Monte Carlo, Monte Carlo renormalization group, and series expansion have provided precise information about the phase transition. Using Monte Carlo simulation that employs the Wolff cluster flipping algorithm with both 32-bit and 53-bit random number generators and data analysis with histogram reweighting and quadruple precision arithmetic, we have investigated the critical behavior of the simple cubic Ising Model, with lattice sizes ranging from 16^{3} to 1024^{3}. By analyzing data with cross correlations between various thermodynamic quantities obtained from the same data pool, e.g., logarithmic derivatives of magnetization and derivatives of magnetization cumulants, we have obtained the critical inverse temperature K_{c}=0.221654626(5) and the critical exponent of the correlation length ν=0.629912(86) with precision that exceeds all previous Monte Carlo estimates.

arXiv:1806.03558•Physical review. E•Apr 4, 2018•191 citations

Cluster-update acceleration of interface roughening in the 3D Ising model

Martin Hasenbusch B, Steffen Meyer

We present a new cluster-update Monte Carlo algorithm for a fluctuating interface in the 3D Ising model. Critical slowing down is drastically reduced and the dynamical critical exponent at the roughening temperature of the 3D Ising model is found to be ${\mathit{z}}_{\mathrm{cl}}$=0.44\ifmmode\pm\else\textpm\fi{}0.03. For the Metropolis algorithm we obtain ${\mathit{z}}_{\mathrm{Met}}$=2.0\ifmmode\pm\else\textpm\fi{}0.1 while the Swendsen-Wang algorithm performs even worse than local updating. A new method which suppresses fluctuations in the two bulk phases is introduced to determine moments of the interfacial thickness.

Physical Review Letters•Feb 4, 1991•36 citations

MONTE CARLO SIMULATIONS OF THE ISING MODEL ON THREE-DIMENSIONAL RANDOM LATTICE USING THE CLUSTER ALGORITHM

J Hang

In order to reduce the critical slowing down in Monte Carlo simulations, a numerical study for the Ising model on 3-D random lattice using Swendsen-Wang algorithm is performed. The critical point obtained isβc=0. 075?. 001. Moreover, to analyse the dynamical property of Swendsen-Wang method, the dynamical critical exponent z is estimated. The result is z =0. 74 ±0. 03. This indicates that the Swendsen-Wang method can greatly reduce the critical slowing down for the Ising model on 3-D random lattice.

Acta Physica Sinica•Jan 1, 1993•0 citations

MONTE CARLO SIMULATIONS OF THE ISING MODEL ON THREE-DIMENSIONAL RANDOM LATTICE USING THE CLUSTER ALGORITHM

JI DA-REN, J. B. Zhang, (1)浙江大学,杭州310027; (2)浙江近代物理中心,杭州310027

In order to reduce the critical slowing down in Monte Carlo simulations, a numerical study for the Ising model on 3-D random lattice using Swendsen-Wang algorithm is performed. The critical point obtained is βc=0. 075±0. 001. Moreover, to analyse the dynamical property of Swendsen-Wang method, the dynamical critical exponent z is estimated. The result is z =0. 74 ±0. 03. This indicates that the Swendsen-Wang method can greatly reduce the critical slowing down for the Ising model on 3-D random lattice.

Acta Physica Sinica•Jan 1, 1993•0 citations

Critical exponents of the classical three-dimensional Heisenberg model: A single-cluster Monte Carlo study

Christian Holm, Wolfhard Janke E

We have simulated the three-dimensional Heisenberg model on simple cubic lattices, using the single-cluster Monte Carlo update algorithm. The expected pronounced reduction of critical slowing down at the phase transition is verified. This allows simulations on significantly larger lattices than in previous studies and consequently a better control over systematic errors. In one set of simulations we employ the usual finite-size scaling methods to compute the critical exponents \ensuremath{\nu},\ensuremath{\alpha},\ensuremath{\beta},\ensuremath{\gamma},\ensuremath{\eta} from a few measurements in the vicinity of the critical point, making extensive use of histogram reweighting and optimization techniques. In another set of simulations we report measurements of improved estimators for the spatial correlation length and the susceptibility in the high-temperature phase, obtained on lattices with up to ${100}^{3}$ spins. This enables us to compute independent estimates of \ensuremath{\nu} and \ensuremath{\gamma} from power-law fits of their critical divergencies.

arXiv:hep-lat/9301002•Physical review. B, Condensed matter•Jul 1, 1993•232 citations

Cluster Monte Carlo simulation of the transverse Ising model

Henk W. J. Blöte, Youjin Deng

We formulate a cluster Monte Carlo method for the anisotropic limit of Ising models on (d+1)-dimensional lattices, which in effect, are equivalent with d-dimensional quantum transverse Ising models. Using this technique, we investigate the transverse Ising models on the square, triangular, Kagome, honeycomb, and simple-cubic lattices. The Monte Carlo data are analyzed by finite-size scaling. In each case we find, as expected, that the critical behavior fits well in the (d+1)-dimensional Ising universality class. For the transverse Ising model on the square lattice, we determine the Binder cumulant of the classical counterpart for a range of aspect ratios between the system sizes in the third or "classical" direction and that in the other two directions. Matching this universal function with the case of the isotropic Ising model yields the length ratio relating the isotropic Ising model with the anisotropic limit. The efficiency of the present algorithm is reflected by the precision of its results, which improves significantly on earlier analyses.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Dec 10, 2002•282 citations
The cluster updating Monte Carlo algorithm is applied to the 3 d simple cubic Ising model. The experimental results are shown to have no critical slowing-down. A simple method predicts the limits of any Monte Carlo calculation. These limits are closely approached. The information treatment is then discussed in calculating the transition temperature. By the simple “fourth-order cumulant method” of Binder, one obtains β c ≃ 0.221 654 4(10).
While the three-dimensional Ising model has defied analytic solution, various numerical methods like Monte Carlo, Monte Carlo renormalization group, and series expansion have provided precise information about the phase transition. Using Monte Carlo simulation that employs the Wolff cluster flipping algorithm with both 32-bit and 53-bit random number generators and data analysis with histogram reweighting and quadruple precision arithmetic, we have investigated the critical behavior of the simple cubic Ising Model, with lattice sizes ranging from 16^{3} to 1024^{3}. By analyzing data with cross correlations between various thermodynamic quantities obtained from the same data pool, e.g., logarithmic derivatives of magnetization and derivatives of magnetization cumulants, we have obtained the critical inverse temperature K_{c}=0.221654626(5) and the critical exponent of the correlation length ν=0.629912(86) with precision that exceeds all previous Monte Carlo estimates.
We present a new cluster-update Monte Carlo algorithm for a fluctuating interface in the 3D Ising model. Critical slowing down is drastically reduced and the dynamical critical exponent at the roughening temperature of the 3D Ising model is found to be zcl{\mathit{z}}_{\mathrm{cl}}zcl​=0.44\ifmmode\pm\else\textpm\fi{}0.03. For the Metropolis algorithm we obtain zMet{\mathit{z}}_{\mathrm{Met}}zMet​=2.0\ifmmode\pm\else\textpm\fi{}0.1 while the Swendsen-Wang algorithm performs even worse than local updating. A new method which suppresses fluctuations in the two bulk phases is introduced to determine moments of the interfacial thickness.
In order to reduce the critical slowing down in Monte Carlo simulations, a numerical study for the Ising model on 3-D random lattice using Swendsen-Wang algorithm is performed. The critical point obtained isβc=0. 075?. 001. Moreover, to analyse the dynamical property of Swendsen-Wang method, the dynamical critical exponent z is estimated. The result is z =0. 74 ±0. 03. This indicates that the Swendsen-Wang method can greatly reduce the critical slowing down for the Ising model on 3-D random lattice.
In order to reduce the critical slowing down in Monte Carlo simulations, a numerical study for the Ising model on 3-D random lattice using Swendsen-Wang algorithm is performed. The critical point obtained is βc=0. 075±0. 001. Moreover, to analyse the dynamical property of Swendsen-Wang method, the dynamical critical exponent z is estimated. The result is z =0. 74 ±0. 03. This indicates that the Swendsen-Wang method can greatly reduce the critical slowing down for the Ising model on 3-D random lattice.
We have simulated the three-dimensional Heisenberg model on simple cubic lattices, using the single-cluster Monte Carlo update algorithm. The expected pronounced reduction of critical slowing down at the phase transition is verified. This allows simulations on significantly larger lattices than in previous studies and consequently a better control over systematic errors. In one set of simulations we employ the usual finite-size scaling methods to compute the critical exponents \ensuremath{\nu},\ensuremath{\alpha},\ensuremath{\beta},\ensuremath{\gamma},\ensuremath{\eta} from a few measurements in the vicinity of the critical point, making extensive use of histogram reweighting and optimization techniques. In another set of simulations we report measurements of improved estimators for the spatial correlation length and the susceptibility in the high-temperature phase, obtained on lattices with up to 1003{100}^{3}1003 spins. This enables us to compute independent estimates of \ensuremath{\nu} and \ensuremath{\gamma} from power-law fits of their critical divergencies.
We formulate a cluster Monte Carlo method for the anisotropic limit of Ising models on (d+1)-dimensional lattices, which in effect, are equivalent with d-dimensional quantum transverse Ising models. Using this technique, we investigate the transverse Ising models on the square, triangular, Kagome, honeycomb, and simple-cubic lattices. The Monte Carlo data are analyzed by finite-size scaling. In each case we find, as expected, that the critical behavior fits well in the (d+1)-dimensional Ising universality class. For the transverse Ising model on the square lattice, we determine the Binder cumulant of the classical counterpart for a range of aspect ratios between the system sizes in the third or "classical" direction and that in the other two directions. Matching this universal function with the case of the isotropic Ising model yields the length ratio relating the isotropic Ising model with the anisotropic limit. The efficiency of the present algorithm is reflected by the precision of its results, which improves significantly on earlier analyses.