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

The International Journal of High Performance Computing Applications

A Fast Algorithm To Simulate the Microcanonical Dynamics of the Ising Model

Richard C. Brower, K.J.M. Moriarty, Pablo Tamayo

Published Dec 1, 1989• 0 citations
Publisher

Abstract

We describe a computer program that simulates the microcanonical Ising model at a peak speed of 100 mil lion spin updates per second on a single processor of a Class VII supercomputer. The program is vectorized and multispin coded for lattices of all sizes and dimensions by introducing multiple copies for lattices smaller than 128 on a side. Autocorrelations are measured to monitor the effects of critical slowing down and to calculate dy namical critical exponents.

DOI
10.1177/109434208900300407

Related papers

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

Critical slowing down in microcanonical Ising dynamics

Richard C. Brower, K.J.M. Moriarty, Eric Myers +2 more

We describe measurements of autocorrelation times in a version of Creutz's microcanonical dynamics for the nearest-neighbor Ising system on a 128\ifmmode\times\else\texttimes\fi{}129\ifmmode\times\else\texttimes\fi{}128 lattice. The dynamical critical exponent z is found to be z=2.26\ifmmode\pm\else\textpm\fi{}0.05, which is comparable to the dynamical critical exponent for Monte Carlo algorithms.

Physical review. B, Condensed matter•Dec 1, 1988•5 citations

Monte Carlo investigation of critical dynamics in the three-dimensional Ising model

Stephan Wansleben, D. P. Landau

We report the results of a Monte Carlo investigation of the (equilibrium) time-displaced correlation functions for the magnetization and energy of a simple cubic Ising model as a function of time, temperature, and lattice size. The simulations were carried out on a CDC CYBER 205 supercomputer employing a high-speed, vectorized multispin coding program and using a total of 5\ifmmode\times\else\texttimes\fi{}${10}^{12}$ Monte Carlo spin-flip trials. We used L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L lattices with periodic boundary conditions and L as large as 96. The short-time and long-time behaviors of the correlation functions are analyzed by fits to a sum of exponential decays, and the critical exponent z for the largest relaxation time is extracted using a finite-size-scaling analysis. Our estimate z=2.04\ifmmode\pm\else\textpm\fi{}0.03 resolves an intriguing contradiction in the literature; it satisfies the theoretical lower bound and is in agreement with the prediction obtained by \ensuremath{\epsilon} expansion. We also consider various small systematic errors that typically occur in the analysis of relaxation functions and show how they can lead to spurious results if sufficient care is not exercised.

Physical review. B, Condensed matter•Mar 1, 1991•129 citations

Vectorized Monte Carlo simulation of large Ising models near the critical point

Masatoshi Mori, Yoshinori Tsuda

The relaxation of the two-dimensional Ising model above ${T}_{c}$ for lattices with size up to 10 240\ifmmode\times\else\texttimes\fi{}10 240 is studied by the Monte Carlo simulation in vectorized super-spin-coding on the HITAC S-810/20 supercomputer. The present estimation of the dynamic critical exponent z gives z=2.076\ifmmode\pm\else\textpm\fi{}0.005, which is equal to the critical exponent ${\ensuremath{\Delta}}^{(l)}$ of the linear relaxation time in a two-dimensional system. Also, the critical exponent, ${\ensuremath{\Delta}}^{(\mathrm{nl})}$ of the nonlinear relaxation time is obtained: ${\ensuremath{\Delta}}^{(\mathrm{nl})=1.932\ifmmode\pm\else\textpm\fi{}0.018}$. These results support the Racz scaling law ${\ensuremath{\Delta}}^{(l)}$=${\ensuremath{\Delta}}^{(\mathrm{nl})+\mathrm{\ensuremath{\beta}}}$ (\ensuremath{\beta} is the critical exponent of the order parameter).

Physical review. B, Condensed matter•Apr 1, 1988•32 citations

Microcanonical renormalization-group simulation of Ising systems

Adauto J. F. de Souza, F. G. Brady Moreira

We report the results of a microcanonical simulation of the two- and three-dimensional Ising models at criticality. We also present a microcanonical algorithm that allows a simultaneous simulation of a lattice spin system at different energies, in our case the number of different energies is 32. The critical behavior of the systems was studied via a recently proposed microcanonical renormalization-group technique that yields independent estimates for the critical energy and the critical temperature, as well as for three critical exponents providing a direct test of hyperscaling. Our results in two dimensions are in good agreement with exact results. In the three dimensions our quoted values are consistent with Monte Carlo estimates recently reported in the literature. We obtain ${\mathit{u}}_{\mathit{c}}$=-0.991(1), ${\mathit{T}}_{\mathit{c}}$=4.5112(3), \ensuremath{\beta}/\ensuremath{\nu}=0.541(1), \ensuremath{\nu}=0.630(3), and \ensuremath{\alpha}=0.109(1).

Physical review. B, Condensed matter•Oct 1, 1993•11 citations

Microcanonical cluster Monte Carlo simulation

Michael J. Creutz

I propose a numerical simulation algorithm for statistical systems which combines a microcanonical transfer of energy with global changes in clusters of spins. The advantages of the cluster approach near a critical point augment the speed increases associated with multispin coding in the microcanonical approach. The method also provides a limited ability to tune the average cluster size.

arXiv:hep-lat/9204005•Physical Review Letters•Aug 17, 1992•37 citations
We describe a computer program that simulates the microcanonical Ising model at a peak speed of 100 mil lion spin updates per second on a single processor of a Class VII supercomputer. The program is vectorized and multispin coded for lattices of all sizes and dimensions by introducing multiple copies for lattices smaller than 128 on a side. Autocorrelations are measured to monitor the effects of critical slowing down and to calculate dy namical critical exponents.
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...
We report the results of a Monte Carlo investigation of the (equilibrium) time-displaced correlation functions for the magnetization and energy of a simple cubic Ising model as a function of time, temperature, and lattice size. The simulations were carried out on a CDC CYBER 205 supercomputer employing a high-speed, vectorized multispin coding program and using a total of 5\ifmmode\times\else\texttimes\fi{}1012{10}^{12}1012 Monte Carlo spin-flip trials. We used L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L lattices with periodic boundary conditions and L as large as 96. The short-time and long-time behaviors of the correlation functions are analyzed by fits to a sum of exponential decays, and the critical exponent z for the largest relaxation time is extracted using a finite-size-scaling analysis. Our estimate z=2.04\ifmmode\pm\else\textpm\fi{}0.03 resolves an intriguing contradiction in the literature; it satisfies the theoretical lower bound and is in agreement with the prediction obtained by \ensuremath{\epsilon} expansion. We also consider various small systematic errors that typically occur in the analysis of relaxation functions and show how they can lead to spurious results if sufficient care is not exercised.
The relaxation of the two-dimensional Ising model above Tc{T}_{c}Tc​ for lattices with size up to 10 240\ifmmode\times\else\texttimes\fi{}10 240 is studied by the Monte Carlo simulation in vectorized super-spin-coding on the HITAC S-810/20 supercomputer. The present estimation of the dynamic critical exponent z gives z=2.076\ifmmode\pm\else\textpm\fi{}0.005, which is equal to the critical exponent ${\ensuremath{\Delta}}^{(l)}$ of the linear relaxation time in a two-dimensional system. Also, the critical exponent, ${\ensuremath{\Delta}}^{(\mathrm{nl})}$ of the nonlinear relaxation time is obtained: ${\ensuremath{\Delta}}^{(\mathrm{nl})=1.932\ifmmode\pm\else\textpm\fi{}0.018}$. These results support the Racz scaling law ${\ensuremath{\Delta}}^{(l)}$=${\ensuremath{\Delta}}^{(\mathrm{nl})+\mathrm{\ensuremath{\beta}}}$ (\ensuremath{\beta} is the critical exponent of the order parameter).
We report the results of a microcanonical simulation of the two- and three-dimensional Ising models at criticality. We also present a microcanonical algorithm that allows a simultaneous simulation of a lattice spin system at different energies, in our case the number of different energies is 32. The critical behavior of the systems was studied via a recently proposed microcanonical renormalization-group technique that yields independent estimates for the critical energy and the critical temperature, as well as for three critical exponents providing a direct test of hyperscaling. Our results in two dimensions are in good agreement with exact results. In the three dimensions our quoted values are consistent with Monte Carlo estimates recently reported in the literature. We obtain uc{\mathit{u}}_{\mathit{c}}uc​=-0.991(1), Tc{\mathit{T}}_{\mathit{c}}Tc​=4.5112(3), \ensuremath{\beta}/\ensuremath{\nu}=0.541(1), \ensuremath{\nu}=0.630(3), and \ensuremath{\alpha}=0.109(1).