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 Condensed Matter

Suppressing critical slowing down in two-dimensional Ising model simulations by the multigrid Monte Carlo method

E. Stoll

Published Oct 2, 1989• 8 citations
Publisher

Abstract

Statistically independent equilibrium configurations of interacting systems can easily be generated with little computing effort by multigrid coarse-to-fine transforms. The very slow long-wavelength motions close to critical points are thereby accelerated by many orders of magnitude. This is demonstrated with a novel updating algorithm from small to large two-dimensional Ising systems specifying effective interactions according to renormalisation group considerations, corrected for non-linearities and saturation effects.

DOI
10.1088/0953-8984/1/39/006

Related papers

Inverse Monte Carlo Renormalization Group Transformations for Critical Phenomena

Dorit Ron, Robert H. Swendsen, Achi Brandt

We introduce a computationally stable inverse Monte Carlo renormalization group transformation method that provides a number of advantages for the calculation of critical properties. We are able to simulate the fixed point of a renormalization group for arbitrarily large lattices without critical slowing down. The log-log scaling plots obtained with this method show remarkable linearity, leading to accurate estimates for critical exponents. We illustrate this method with calculations in two- and three-dimensional Ising models for a variety of renormalization group transformations.

Physical Review Letters•Dec 18, 2002•32 citations

Multigrid Monte Carlo in the Sine Gordon model

Grabenstein, M, Mikeska, B

We pose two questions about the dynamical critical behavior of multigrid Monte Carlo: Will a multigrid Monte Carlo simulation of the two dimensional Sine Gordon model exhibit critical slowing down, as expected by a theoretical analysis of Metropolis acceptance rates? Can we reduce critical slowing down caused by decreasing acceptance rates on large blocks by performing more updates on coarser lattices? To this end we simulate the model with a W-cycle (gamma = 2) and a higher cycle with gamma = 4 using piecewise constant interpolation. The answer to the first question is positive, the answer to the second one is negative.

•Nov 18, 1993•3 citations

Simulations without critical slowing down: Ising and three-state Potts models

Daniel Kandel, Eytan Domany, Achi Brandt

We have developed a novel simulation method that combines a multigrid technique with a stochastic blocking procedure. Our algorithm eliminates critical slowing down completely, as we demonstrated previously by simulating the two-dimensional Ising model at criticality. Here we describe results of much more extensive simulations of the two-dimensional Ising and three-state Potts models and provide a heuristic argument to explain why our method works.

Physical review. B, Condensed matter•Jul 1, 1989•66 citations

Simulations without critical slowing down

Daniel Kandel, Eytan Domany, Dorit Ron +2 more

We have developed a novel simulation method that combines a multigrid technique with a stochastic blocking procedure. Our algorithm eliminates critical slowing down completely, as demonstrated by simulations of the two-dimensional Ising model at criticality.

Physical Review Letters•Apr 18, 1988•97 citations

Variational Approach to Monte Carlo Renormalization Group

Yantao Wu, Roberto Car

We present a Monte Carlo method for computing the renormalized coupling constants and the critical exponents within renormalization theory. The scheme, which derives from a variational principle, overcomes critical slowing down, by means of a bias potential that renders the coarse grained variables uncorrelated. The two-dimensional Ising model is used to illustrate the method.

arXiv:1707.08683•Physical Review Letters•Nov 28, 2017•13 citations

Collective Monte Carlo Updating for Spin Systems

Ulli Wolff A

A Monte Carlo algorithm is presented that updates large clusters of spins simultaneously in systems at and near criticality. We demonstrate its efficiency in the two-dimensional $\mathrm{O}(n)$ $\ensuremath{\sigma}$ models for $n=1$ (Ising) and $n=2$ ($x\ensuremath{-}y$) at their critical temperatures, and for $n=3$ (Heisenberg) with correlation lengths around 10 and 20. On lattices up to ${128}^{2}$ no sign of critical slowing down is visible with autocorrelation times of 1-2 steps per spin for estimators of long-range quantities.

Physical Review Letters•Jan 23, 1989•2,558 citations
Statistically independent equilibrium configurations of interacting systems can easily be generated with little computing effort by multigrid coarse-to-fine transforms. The very slow long-wavelength motions close to critical points are thereby accelerated by many orders of magnitude. This is demonstrated with a novel updating algorithm from small to large two-dimensional Ising systems specifying effective interactions according to renormalisation group considerations, corrected for non-linearities and saturation effects.
We introduce a computationally stable inverse Monte Carlo renormalization group transformation method that provides a number of advantages for the calculation of critical properties. We are able to simulate the fixed point of a renormalization group for arbitrarily large lattices without critical slowing down. The log-log scaling plots obtained with this method show remarkable linearity, leading to accurate estimates for critical exponents. We illustrate this method with calculations in two- and three-dimensional Ising models for a variety of renormalization group transformations.
We pose two questions about the dynamical critical behavior of multigrid Monte Carlo: Will a multigrid Monte Carlo simulation of the two dimensional Sine Gordon model exhibit critical slowing down, as expected by a theoretical analysis of Metropolis acceptance rates? Can we reduce critical slowing down caused by decreasing acceptance rates on large blocks by performing more updates on coarser lattices? To this end we simulate the model with a W-cycle (gamma = 2) and a higher cycle with gamma = 4 using piecewise constant interpolation. The answer to the first question is positive, the answer to the second one is negative.
A Monte Carlo algorithm is presented that updates large clusters of spins simultaneously in systems at and near criticality. We demonstrate its efficiency in the two-dimensional O(n)\mathrm{O}(n)O(n) $\ensuremath{\sigma}$ models for n=1n=1n=1 (Ising) and n=2n=2n=2 ($x\ensuremath{-}y$) at their critical temperatures, and for n=3n=3n=3 (Heisenberg) with correlation lengths around 10 and 20. On lattices up to 1282{128}^{2}1282 no sign of critical slowing down is visible with autocorrelation times of 1-2 steps per spin for estimators of long-range quantities.