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arXiv:cond-mat/9206004 · Physical review. B, Condensed matter

Dynamic Monte Carlo renormalization-group method

Martin-D. Lacasse, Jorge Viñals, Martin Grant

Published Mar 1, 1993• 37 citations
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Abstract

The dynamical critical exponent of the two-dimensional spin-flip Ising model is evaluated by a Monte Carlo renormalization-group method involving a transformation in time. The results agree very well with a finite-size scaling analysis performed on the same data. The value of z=2.13\ifmmode\pm\else\textpm\fi{}0.01 is obtained, which is consistent with most recent estimates.

DOI
10.1103/physrevb.47.5646

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Europhysics Letters (EPL)•Jan 20, 1995•16 citations
The dynamical critical exponent of the two-dimensional spin-flip Ising model is evaluated by a Monte Carlo renormalization-group method involving a transformation in time. The results agree very well with a finite-size scaling analysis performed on the same data. The value of z=2.13\ifmmode\pm\else\textpm\fi{}0.01 is obtained, which is consistent with most recent estimates.
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.
We introduce a novel variance-reducing Monte Carlo algorithm for accurate determination of correlation times. We apply this method to two-dimensional Ising systems with sizes up to $15\ifmmode\times\else\texttimes\fi{}15$, using single-spin flip dynamics, random site selection, and transition probabilities according to the heat-bath method. From a finite-size scaling analysis of these correlation times, the dynamic critical exponent zzz is determined as z=2.1665(12)z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2.1665(12)z=2.1665(12).
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.
The critical relaxation of the magnetization in a two-dimensional Ising model with quenched random non-magnetic impurities has been studied by numerical simulation. A squared lattice of size 400 2 with spin concentrations p=1.0; 0.95, 0,9, 0.85, 0.8, 0.75 and 0.7 was considered. The dynamical critical exponent z was determined by the Monte Carlo method combined with the dynamical renormalization group method. The following values were received for z(p): z(1)=2.24+or-0.07, z(0.95)=2.24+or-0.06, z(0.9)=2.24+or-0.06, z(0.85)=2.38+or-0.05, z(0.8)=2.51+or-0.06, z(0.75)=2.66+or-0.07 and z(0.7)=2.88+or-0.06. A singular dynamic scaling behaviour with z=A' mod ln(p-p c ) mod +B' was found for systems with spin concentrations p<or=0.85 near the percolation threshold p c , where A'=0.56+or-0.07 and B'=1.62+or-0.07 are temperature-independent constants.
The critical relaxation of the magnetization in a 2D Ising model with quenched random non-magnetic impurities is studied by numerical simulation. A square lattice of size 400 2 and spin concentrations p = 1.0, 0.95, 0.9, 0.85, 0.8, 0.75 and 0.7 are considered. The dynamical critical exponent z is determined by the dynamical renormalization group Monte Carlo method. The values for z(p) are: z (1) = 2.24 ± 0.07, z (0.95) = 2.24 ± 0.06, z (0.9) = 2.24 ± 0.06, z (0.85) = 2.38 ± 0.05, z (0.8) = 2.51 ± 0.06, z (0.75) = 2.66 ± 0.07 and z (0.7) = 2.88 ± 0.06. A singular dynamic-scaling behaviour with z(p) = A '|ln( p − p c )| + B ' is found for systems with spin concentrations near the percolation threshold p c , where A ' = 0.56 ± 0.07 and B ' = 1.62 ± 0.07 are temperature-independent constants.