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Physical review. B, Condensed matter

Monte Carlo renormalization-group study of the two-dimensional Glauber model

Sheldon L. Katz, J. D. Gunton, Chao Ping Liu

Published May 1, 1982• 39 citations
Publisher

Abstract

A recent Monte Carlo renormalization-group study of the two-dimensional kinetic Ising model is extended to obtain renormalized time correlation functions from initial lattices of 16 \ifmmode\times\else\texttimes\fi{} 16 and 32 \ifmmode\times\else\texttimes\fi{} 32 spins. Different ways to estimate the dynamical exponent $z$ are discussed, with our "best" estimate of $z\ensuremath{\simeq}2.23$ being consistent with the original study.

DOI
10.1103/physrevb.25.6008

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Journal of Physics A Mathematical and General•Jan 11, 1985•61 citations
A recent Monte Carlo renormalization-group study of the two-dimensional kinetic Ising model is extended to obtain renormalized time correlation functions from initial lattices of 16 \ifmmode\times\else\texttimes\fi{} 16 and 32 \ifmmode\times\else\texttimes\fi{} 32 spins. Different ways to estimate the dynamical exponent zzz are discussed, with our "best" estimate of z\ensuremath≃2.23z\ensuremath{\simeq}2.23z\ensuremath≃2.23 being consistent with the original study.
The critical dynamics of the two dimensional Kinetic Ising Model have been studied using Monte Carlo Real Space Renormalization Group techniques. Time dependent correlation functions were calculated for square lattices of 16 x 16 and 32 x 32 spins, using periodic boundary conditions wth 60,000 and 15,000 Monte Carlo steps per spin, respectively;The critical exponent z was estimated using a matching process for a variety of renormalization group operators. The estimates for z were surprisingly sensitive to small changes in the value of the renormalization group operator, with physically reasonable operators giving estimates for z less than the rigorous lower bound for z of z(,o) = 1.75. The estimates for z from a given renormalization group operator showed poor convergence properties. Attempts to estimate z from the scaling form of the correlation function were inconclusive, as were attempts to estimate the static exponent (eta);Possible explanations of these unexpected results are discussed. The most plausible source of the complex behavior seen in this work is the discrete nature of time in the Monte Carlo computer simulation, which generates phenomena not present in standard theoretical treatments of the Kinetic Ising Model.
The dynamic Monte Carlo renormalization group is applied to the two-dimensional Kawasaki and three-dimensional Glauber models. Our best "matching" results for the Kawasaki model yield a value of z=3.80z=3.80z=3.80 for the dynamical exponent, as compared with the exact value z=3.75z=3.75z=3.75. This provides strong confirmation for the validity of this method. The results for the Glauber model are less accurate, but our estimate of $z\ensuremath{\simeq}2.08$ is in reasonable agreement with the $\ensuremath{\epsilon}$ expansion.
Applying the dynamic Monte Carlo renormalisation group to the Glauber kinetic Ising model, the dynamical critical exponent z is found by simulation of up to 8192 2 and 512 3 spins on the vector computer CDC Cyber 205, using the new 'Method of 2 d colours' for the Monte Carlo part (update speed 22 megaspins/s). The two-dimensional result z=2.14-or+0.02 disagrees with Domany's hypothesis (1984). For three dimensions, a systematic trend in z with increasing blocksize leads to an extrapolated value z=1.965-or+0.010, which is consistent with a theoretically expected value 2.02.
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.