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Journal of Physics A Mathematical and General

Nonuniversal critical behaviour in the 1D BEG model with Kawasaki dynamics

José F. F. Mendes, Stephen J. Cornell, Michel Droz, Eduardo Lage

Published Jan 7, 1992• 4 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/25/1/012

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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.
Monte Carlo simulations and finite-size scaling were used to study the one-dimensional BEG (Blume-Emergy-Griffiths) model under the influence of two competing dynamics: single spin flip (Glauber dynamics at finite temperature T) and random exchange to two spins (Kawasaki dynamics at infinite temperature). The phase diagram shows first- and second-order phase transitions. The critical exponents are obtained for different values of the rates and the range of the exchanges. The authors find beta =0.325+or-0.020, gamma =0.996+or-0.004 and nu =1.71+or-0.01 for long-range exchanges and equal intrinsic rates. However, for different rates and the nearest-neighbour exchanges they obtain beta =0.07+or-0.03, gamma =1.60+or-0.06 and nu =1.53+or-0.08, thus exhibiting non-universal behaviour.
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.
Critical exponents of the Kawasaki dynamics in the Ising chain are re-examined numerically through the spectrum gap of evolution operators constructed both in spin and domain-wall representations. At low-temperature regimes the latter provides a rapid finite-size convergence to these exponents, which tend to z ≃ 3.11 for instant quenches under ferromagnetic couplings, while approaching to z ≃ 2 in the antiferro case. The spin representation complements the evaluation of dynamic exponents at higher temperature scales, where the kinetics still remains slow.
The critical behaviour of the three-dimensional Blume–Emery–Griffiths (BEG) model is investigated at D / J = 0, -0.25 and -1 in the range of −1⩽ K / J ⩽0 for J = 100. The simulations are carried out on a simple cubic lattice using the heating algorithm improved from the Creutz cellular automaton (CCA) under periodic boundary conditions. The universality of the model are obtained for re-entrant and double re-entrant phase transitions which occur at certain D / J and K / J parameters, with J and K representing the nearest-neighbour bilinear and biquadratic interactions, and D being the single-ion anisotropy parameter. The values of static critical exponents β, γ and ν are estimated within the framework of the finite-size scaling theory. The results are compatible with the universal Ising critical behaviour for all continuous phase transitions in these ranges.
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.
Abstract We use the Monte Carlo simulation technique to study the critical behavior of a three-state spin model, with bilinear and biquadratic nearest-neighbor interactions, known as the Blume–Emery–Griffiths model (BEG), in a square lattice. In order to characterize this model, we study the phase diagram, in which we identify three different phases: ferromagnetic, paramagnetic and quadrupolar, the later with one sublattice filled with spins and the other with vacancies. We perform our studies by using two algorithms: Metropolis update (MU) and Wang–Landau (WL). The critical scaling behavior of the model is complementary studied by applying results obtained by using both algorithms, while tricritical points and the tricritical scaling behavior is analyzed by means of WL measuring the joint density of states and using the method of field mixing in conjunction with finite-size scaling. Furthermore, motivated by the decoupling between spins observed within the quadrupolar phase, we further generalize the BEG model in order to study the behavior of the system by adding a next nearest neighbour (NNN) interaction between spins. We found that by increasing the strength of the (ferromagnetic) NNN interaction, a new ferromagnetic phase takes over that contains both quadrupolar and ferromagnetic order.