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

arXiv:cond-mat/9304008 · International Journal of Modern Physics C

CLUSTER VS. SINGLE-SPIN ALGORITHMS—WHICH ARE MORE EFFICIENT?

N. ITO, G. A. Kohring

Published Feb 1, 1994• 11 citations
arXivHTMLPDF

Abstract

A comparison between single-cluster and single-spin algorithms is made for the Ising model in 2 and 3 dimensions. We compare the amount of computer time needed to achieve a given level of statistical accuracy, rather than the speed in terms of site updates per second or the dynamical critical exponents. Our main result is that the cluster algorithms become more efficient when the system size, Ld, exceeds, L~70–300 for d=2 and l~80–200 for d=3. The exact value of the crossover is dependent upon the computer being used. The lower end of the crossover range is typical of workstations while the higher end is typical of vector computers. Hence, even for workstations, the system sizes needed for efficient use of the cluster algorithm is relatively large.

DOI
10.1142/s0129183194000027

Related papers

Comparison of cluster algorithms for the bond-diluted Ising model

Arnold H. Kole, G. T. Barkema, Lars Fritz

Monte Carlo cluster algorithms are popular for their efficiency in studying the Ising model near its critical temperature. We might expect that this efficiency extends to the bond-diluted Ising model. We show, however, that this is not always the case by comparing how the correlation times τ_{w} and τ_{sw} of the Wolff and Swendsen-Wang cluster algorithms scale as a function of the system size L when applied to the two-dimensional bond-diluted Ising model. We demonstrate that the Wolff algorithm suffers from a much longer correlation time than in the pure Ising model, caused by isolated (groups of) spins which are infrequently visited by the algorithm. With a simple argument we prove that these cause the correlation time τ_{w} to be bounded from below by L^{z_{w}} with a dynamical exponent z_{w}=γ/ν≈1.75 for a bond concentration p<1. Furthermore, we numerically show that this lower bound is actually taken for several values of p in the range 0.5<p<1. Moreover, we show that the Swendsen-Wang algorithm does not suffer from the same problem. Consequently, it has a much shorter correlation time, shorter than in the pure Ising model even. Numerically at p=0.6, we find that its dynamical exponent is z_{sw}=0.09(4).

arXiv:2107.08534•Physical review. E•Jan 31, 2022•4 citations

MONTE CARLO METHODS IN SEQUENTIAL AND PARALLEL COMPUTING OF 2D AND 3D ISING MODEL

M. Diaconu, R. Puscasu, Alexandru Stancu +1 more

Because of its complexity, the 3D Ising model has not been given an exact analytic solution so far, as well as the 2D Ising in non zero external field conditions. In real materials the phase transition creates a discontinuity. We analysed the Ising model that presents similar discontinuities. We use Monte Carlo methods with a single spin change or a spin cluster change to calculate macroscopic quantities, such as specific heat and magnetic susceptibility. We studied the differences between these methods. Local MC algorithms (such as Metropolis) perform poorly for large lattices because they update only one spin at a time, so it takes many iterations to get a statistically independent configuration. More recent spin cluster algorithms use clever ways of finding clusters of sites that can be updated at once. The single cluster method is probably the best sequential cluster algorithm. We also used the entropic sampling method to simulate the density of states. This method takes into account all possible configurations, not only the most probable. The entropic method also gives good results in the 3D case. We studied the usefulness of distributed computing for Ising model. We established a parallelization strategy to explore Metropolis Monte Carlo simulation and Swendsen-Wang Monte Carlo simulation of this spin model using the data parallel languages on different platform. After building a computer cluster we made a Monte Carlo estimation of 2D and 3D Ising thermodynamic properties and compare the results with the sequential computing. In the same time we made quantitative analysis such as speed up and efficiency for different sets of combined parameters (e.g. lattice size, parallel algorithms, chosen model).

•Jan 1, 2003•0 citations

HIGH SPEED ALGORITHMS AND SIMULATIONS OF THE CRITICAL DYNAMICS OF THE ISING MODEL

Hans-Otto Heuer

We review the development of Monte Carlo algorithms for the simulation of the Ising model with local dynamics. The multi lattice coding algorithm for the pure and for the site-disordered Ising model is described in detail. Application of these algorithms on fast vector computers have led to new progress concerning the critical dynamics of the pure Ising model leading to the critical exponent z = 2:05. The importance of disorder for the critical dynamics of the Ising model has been investigated recently. We review the numerical results and compare with theoretical predictions. A crossover scenario is found as for the static behavior with an effective dynamical exponent z = 1:75 fl. 1 Introduction The Ising model is a standard model in statistical physics for the description of a ferromagnet in an extremely simplified manner. It is assumed that quantum spin operators may be replaced by classical spins S i = \\Sigma 1 which are oriented parallel or antiparallel to a magnetic filed H: H...

WORLD SCIENTIFIC eBooks•May 1, 1996•1 citation

Properties of cluster-size heterogeneity near the phase transition in the two-dimensional Ising model

Renée Kauppi

Two different definitions of cluster-size heterogeneity are investigated as well as correlation time of different quantities using the Metropolis algorithm and the Wolff algorithm. It is confirmed that the correlation time multiplied by the computation time is lower for the Wolff algorithm in an area around the critical temperature. It is also confirmed that one definition of the heterogeneity has a local maximum at the critical temperature where as the other has an abrupt change in derivative. The local maximum appears with L ≥ 64 and it is predicted but not verified that systems with L > 43 have such a maximum. The relationship between the number of distinct cluster sizes for clusters with spin-up and spin-down is investigated and it is observed that these transition from being significantly different at lower temperatures to being mostly similar at higher temperatures. The point of transition appears to be near the critical temperature.

•Jan 1, 2020•0 citations

Replica-exchange cluster algorithm

Wolfhard Janke E, Elmar Bittner

In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature/energy range around the critical point.By combining the replica-exchange algorithm with cluster updates and an adaptive routine to find the range of interest, we introduce a new flexible and powerful method for systematic investigations of critical phenomena.As a result, we gain two further orders of magnitude in the performance for 2D and 3D Ising models in comparison with the recently proposed Wang-Landau recursion for cluster algorithms based on the multibondic algorithm, which is already a great improvement over the standard multicanonical variant.

arXiv:1108.0354•International Symposium on Lattice Field Theory (LATTICE)•Jun 23, 2010•0 citations

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
A comparison between single-cluster and single-spin algorithms is made for the Ising model in 2 and 3 dimensions. We compare the amount of computer time needed to achieve a given level of statistical accuracy, rather than the speed in terms of site updates per second or the dynamical critical exponents. Our main result is that the cluster algorithms become more efficient when the system size, Ld, exceeds, L70–300 for d=2 and l80–200 for d=3. The exact value of the crossover is dependent upon the computer being used. The lower end of the crossover range is typical of workstations while the higher end is typical of vector computers. Hence, even for workstations, the system sizes needed for efficient use of the cluster algorithm is relatively large.
Monte Carlo cluster algorithms are popular for their efficiency in studying the Ising model near its critical temperature. We might expect that this efficiency extends to the bond-diluted Ising model. We show, however, that this is not always the case by comparing how the correlation times τ_{w} and τ_{sw} of the Wolff and Swendsen-Wang cluster algorithms scale as a function of the system size L when applied to the two-dimensional bond-diluted Ising model. We demonstrate that the Wolff algorithm suffers from a much longer correlation time than in the pure Ising model, caused by isolated (groups of) spins which are infrequently visited by the algorithm. With a simple argument we prove that these cause the correlation time τ_{w} to be bounded from below by L^{z_{w}} with a dynamical exponent z_{w}=γ/ν≈1.75 for a bond concentration p<1. Furthermore, we numerically show that this lower bound is actually taken for several values of p in the range 0.5<p<1. Moreover, we show that the Swendsen-Wang algorithm does not suffer from the same problem. Consequently, it has a much shorter correlation time, shorter than in the pure Ising model even. Numerically at p=0.6, we find that its dynamical exponent is z_{sw}=0.09(4).
Because of its complexity, the 3D Ising model has not been given an exact analytic solution so far, as well as the 2D Ising in non zero external field conditions. In real materials the phase transition creates a discontinuity. We analysed the Ising model that presents similar discontinuities. We use Monte Carlo methods with a single spin change or a spin cluster change to calculate macroscopic quantities, such as specific heat and magnetic susceptibility. We studied the differences between these methods. Local MC algorithms (such as Metropolis) perform poorly for large lattices because they update only one spin at a time, so it takes many iterations to get a statistically independent configuration. More recent spin cluster algorithms use clever ways of finding clusters of sites that can be updated at once. The single cluster method is probably the best sequential cluster algorithm. We also used the entropic sampling method to simulate the density of states. This method takes into account all possible configurations, not only the most probable. The entropic method also gives good results in the 3D case. We studied the usefulness of distributed computing for Ising model. We established a parallelization strategy to explore Metropolis Monte Carlo simulation and Swendsen-Wang Monte Carlo simulation of this spin model using the data parallel languages on different platform. After building a computer cluster we made a Monte Carlo estimation of 2D and 3D Ising thermodynamic properties and compare the results with the sequential computing. In the same time we made quantitative analysis such as speed up and efficiency for different sets of combined parameters (e.g. lattice size, parallel algorithms, chosen model).
We review the development of Monte Carlo algorithms for the simulation of the Ising model with local dynamics. The multi lattice coding algorithm for the pure and for the site-disordered Ising model is described in detail. Application of these algorithms on fast vector computers have led to new progress concerning the critical dynamics of the pure Ising model leading to the critical exponent z = 2:05. The importance of disorder for the critical dynamics of the Ising model has been investigated recently. We review the numerical results and compare with theoretical predictions. A crossover scenario is found as for the static behavior with an effective dynamical exponent z = 1:75 fl. 1 Introduction The Ising model is a standard model in statistical physics for the description of a ferromagnet in an extremely simplified manner. It is assumed that quantum spin operators may be replaced by classical spins S i = \\Sigma 1 which are oriented parallel or antiparallel to a magnetic filed H: H...
Two different definitions of cluster-size heterogeneity are investigated as well as correlation time of different quantities using the Metropolis algorithm and the Wolff algorithm. It is confirmed that the correlation time multiplied by the computation time is lower for the Wolff algorithm in an area around the critical temperature. It is also confirmed that one definition of the heterogeneity has a local maximum at the critical temperature where as the other has an abrupt change in derivative. The local maximum appears with L ≥ 64 and it is predicted but not verified that systems with L > 43 have such a maximum. The relationship between the number of distinct cluster sizes for clusters with spin-up and spin-down is investigated and it is observed that these transition from being significantly different at lower temperatures to being mostly similar at higher temperatures. The point of transition appears to be near the critical temperature.
In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature/energy range around the critical point.By combining the replica-exchange algorithm with cluster updates and an adaptive routine to find the range of interest, we introduce a new flexible and powerful method for systematic investigations of critical phenomena.As a result, we gain two further orders of magnitude in the performance for 2D and 3D Ising models in comparison with the recently proposed Wang-Landau recursion for cluster algorithms based on the multibondic algorithm, which is already a great improvement over the standard multicanonical variant.
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.