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Journal of Physics C Solid State Physics

Nonuniversal critical dynamical behavior in 1D spin systems

Eduardo Lage

Published Sep 10, 1987• 14 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/20/25/023

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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.
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.
The critical dynamics of a disordered Ising ferromagnetic chain with two coupling constants (J 1 >or=J 2 >0) is studied for Glauber dynamics. Using a domain wall argument the dynamical critical exponent z is found to be non-universal but independent of the disorder, namely z=1+J 1 /J 2 . The problem is formulated in terms of diffusion in a random medium. The diffusion is shown to be normal. Relationships with apparently very different diffusion problems, like the diffusion in hierarchically structured media, are established.
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 authors investigate the one-dimensional Blume-Emery-Griffiths model with Kawasaki dynamics, using domain-wall arguments (DWA) and Monte Carlo simulations (MCS) (conventional and Gillespie algorithm), after a quench from a disordered state to low temperatures. They observe that this model exhibits domain scaling behaviour, controlled by a universal exponent (x=1/3) as in other dimensions for model B. However, they find also that the critical exponent z is not universal and depends on the coupling constants of the Hamiltonian. The results of DWA are consistent with those of MCS.