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Physical Review Letters

Critical Slowing Down in the Kinetic Ising Model; Evidence for the Failure of the Dynamical Scaling Hypothesis

T. Schneider, E. Stoll, Kurt Binder

Published Oct 16, 1972• 18 citations
Publisher

Abstract

We report computer simulations of critical slowing down in a two-dimensional kinetic Ising system consisting of a square $n\ifmmode\times\else\texttimes\fi{}n$ lattice with periodic boundary conditions. The results for the divergence of the relaxation times disagree with the predictions of the dynamical scaling hypothesis and with recent estimates obtained from a high-temperature expansion method.

DOI
10.1103/physrevlett.29.1080

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Critical dynamics of the two-dimensional kinetic Ising model: High-temperature series analysis of the autorelaxation time

Jos Rogiers, Joseph O. Indekeu

For the two-dimensional nearest-neighbor kinetic Ising model without conservation laws, we derive on the square lattice the high-temperature series expansion of the autorelaxation time, to the eleventh order. With use of ratio methods and Pad\'e approximants, we obtain for the dynamical critical exponent ${\mathrm{\ensuremath{\Delta}}}_{\mathit{A}}$ the value 2.09\ifmmode\pm\else\textpm\fi{}0.03, which implies the value 2.34\ifmmode\pm\else\textpm\fi{}0.03 for the linear relaxation exponent z. This value is about 10% larger than several other recent estimates. This discrepancy is possibly due to the relatively small number of nonzero coefficients in the series expansion.

Physical review. B, Condensed matter•Apr 1, 1990•22 citations
We report computer simulations of critical slowing down in a two-dimensional kinetic Ising system consisting of a square n\ifmmode×\else\texttimes\finn\ifmmode\times\else\texttimes\fi{}nn\ifmmode×\else\texttimes\fin lattice with periodic boundary conditions. The results for the divergence of the relaxation times disagree with the predictions of the dynamical scaling hypothesis and with recent estimates obtained from a high-temperature expansion method.
The kinetic Ising model on a square lattice is studied by means of the finite-size scaling with the help of the Monte Carlo method. The results obtained are consistent with the universality hypothesis that the value of the dynamic critical expotent is universal for certain types of transition probabilities. The present results also suggest that the dynamic critical exponents for the relaxation times of the magnetization and energy are the same. Our estimation of the dynamic critical exponent z gives that z \cong2.2 ±0.1.
A simple dynamical model of interacting Ising spins is discussed near the critical point. It is shown that all the moments are positive and never vanish even at the critical point. Lower and upper bounds of the susceptibility are obtained, which yield the non-vanishing of the real part of the susceptibility. This gives a simple proof for the poly-dispersive nature of our system. The method of high temperature expansions for the dynamical susceptibility is presented and is applied to the two-dimensional Ising model up to the ninth order, the results of which show that the critical index of slowing down (the relaxation time τ (1) ∝( T - T c ) - Δ ; Δ =2±0.05) is different from that of the static susceptibility.
Extending the Monte Carlo method to dynamic critical phenomena we investigated the time-dependent correlation functions in the two-dimensional one-spin-flip Ising model and the critical behavior of the associated relaxation times. These relaxation times are the following: ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}$, characterizing the approach of the order parameter to equilibrium after a change of temperature $\ensuremath{\Delta}T$ of the system; ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}$ characterizing the slowing down of the order-parameter correlation and autocorrelation functions, respectively; ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}$, characterizing the slowing down of the energy correlation and autocorrelation functions; and finally ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}$, characterizing the cross-correlation function. We give estimates for the associated exponents ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}1.90\ifmmode\pm\else\textpm\fi{}0.10$, and ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}\ensuremath{\approx}1.60\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0.95\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0$, which are consistent with the dynamic scaling hypothesis and with exact inequalities. A detailed comparison with recent high-temperature-expansion estimates is performed, and the reliability of the Monte Carlo results is carefully analyzed.
Results are presented for the computer simulation of the time-dependent behavior of the Ising model on a 100 \ifmmode\times\else\texttimes\fi{} 100 lattice. Groups of particles are regarded as samples from an infinite lattice and the interactions between the sample particles and those outside are expressed through new boundary conditions based on extending the concept of mean field in a self-consistent way. These new boundary conditions are explored analytically for a few small samples and a sharp critical point is found in contrast to periodic boundary conditions. The relaxation times for long-range order and nearest-neighbor correlation (energy) for the 100 \ifmmode\times\else\texttimes\fi{} 100 system have been computed and the corresponding critical exponents estimated. The results are ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}}=2.30\ifmmode\pm\else\textpm\fi{}0.30$ and ${\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}}=0.38\ifmmode\pm\else\textpm\fi{}0.04$; these are compared with other recent results from computer simulations and series calculations.
For the two-dimensional nearest-neighbor kinetic Ising model without conservation laws, we derive on the square lattice the high-temperature series expansion of the autorelaxation time, to the eleventh order. With use of ratio methods and Pad\'e approximants, we obtain for the dynamical critical exponent ${\mathrm{\ensuremath{\Delta}}}_{\mathit{A}}$ the value 2.09\ifmmode\pm\else\textpm\fi{}0.03, which implies the value 2.34\ifmmode\pm\else\textpm\fi{}0.03 for the linear relaxation exponent z. This value is about 10% larger than several other recent estimates. This discrepancy is possibly due to the relatively small number of nonzero coefficients in the series expansion.