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Journal of Physics A Mathematical and General

Vectorised dynamics Monte Carlo renormalisation group for the Ising model

Claus Kalle

Published Oct 1, 1984• 81 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/17/14/014

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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.
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.
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.
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 $z\ensuremath{\simeq}2.23$ being consistent with the original study.
A novel dynamic Monte Carlo renormalization-group method is devised. Application to the kinetic Ising model in one dimension yields a value of zzz, the dynamic critical exponent, which is in excellent agreement with the exact result. In two dimensions we find $z=2.22\ifmmode\pm\else\textpm\fi{}0.13$ with the most probable value z=2.17z=2.17z=2.17.