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Journal of Physics A Mathematical and General

Critical dynamics for one-dimensional models

F. Leyvraz, Naeem Jan

Published Mar 11, 1986• 10 citations
Publisher

Abstract

A plausible argument using the concepts of dynamic renormalisation group is used to derive the dynamical exponent z of the one-dimensional q=2, 3 and 4 Potts and four-state clock models. The exact result (z=2) for the Ising model with Glauber dynamics is recovered and in addition the authors find that the q=3, 4 Potts and four-state clock models belong to the same universality class. Moreover they find that z=3.05+or-0.1 for the (q=2) Potts model with Kawasaki dynamics in agreement with the z=4- eta prediction of Halperin, Hohenberg and Ma (1974). A simple qualitative derivation is also given.

DOI
10.1088/0305-4470/19/4/022

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A plausible argument using the concepts of dynamic renormalisation group is used to derive the dynamical exponent z of the one-dimensional q=2, 3 and 4 Potts and four-state clock models. The exact result (z=2) for the Ising model with Glauber dynamics is recovered and in addition the authors find that the q=3, 4 Potts and four-state clock models belong to the same universality class. Moreover they find that z=3.05+or-0.1 for the (q=2) Potts model with Kawasaki dynamics in agreement with the z=4- eta prediction of Halperin, Hohenberg and Ma (1974). A simple qualitative derivation is also given.
Recent results concerning non-universality in the dynamical critical behavior of the one-dimensional Ising model, are generalised to several formulations of the Potts model, using arguments based on domain wall motion. By applying real-space renormalisation group methods, the authors also study the dynamical Ising model, with single flips, both for a period distribution of nearest-neighbor exchanges and for the random chain (with nonzero couplings). They show that the critical dynamical exponent is z=1+J M /J m where J M (J m ) is the greatest (smallest) of the exchanges.
The one-dimensional kinetic Potts models are studied for both Glauber and Kawasaki dynamics with different transition rates. Rigorous lower bounds for the dynamical exponent (z) are obtained from the initial response rate and the scaling hypothesis for the relaxation time. It is shown that the z exponent is non-universal for several periodic models with a basic unit cell (J 1 ,J 2 ,. . .,JM) containing different coupling constants. The critical dynamics of the alternating bond model is also studied by a simple argument about the movement of domain walls.
The two-dimensional Ising model with competing Glauber and Kawasaki dynamics is studied by Monte Carlo simulations. We show that the model exhibits the phenomenon of self-organization when the Kawasaki dynamics is the dominant one. For this model we show that the values of the critical exponents calculated at the stationary states are in accordance with the exact ones known for the equilibrium Ising model. These results give support to the idea that the equilibrium and nonequilibrium Ising models, which exhibit up-down symmetry, belong to the same universality class. \textcopyright{} 1996 The American Physical Society.
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.