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arXiv:cond-mat/9607089 · Journal of Physics A Mathematical and General

Phase transitions and critical behaviour in one-dimensional non-equilibrium kinetic Ising models with branching annihilating random walk of kinks

Nóra Menyhárd, Géza Ódor

Published Dec 7, 1996• 52 citations
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Abstract

One-dimensional non-equilibrium kinetic Ising models evolving under the competing effect of spin flips at zero temperature and nearest-neighbour spin exchanges exhibiting a parity-conserving (PC) phase transition on the level of kinks are now further investigated numerically, from the point of view of the underlying spin system. Critical exponents characterizing its statics and dynamics are reported. It is found that the influence of the PC transition on the critical exponents of the spins is strong and the origin of drastic changes as compared to the Glauber - Ising case can be traced back to the hyperscaling law stemming from directed percolation. The effect of an external magnetic field, leading to directed percolation-type behaviour on the level of kinks is also studied, mainly via the generalized mean-field approximation.

DOI
10.1088/0305-4470/29/23/030

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One-dimensional non-equilibrium kinetic Ising models evolving under the competing effect of spin flips at zero temperature and nearest-neighbour spin exchanges exhibiting a parity-conserving (PC) phase transition on the level of kinks are now further investigated numerically, from the point of view of the underlying spin system. Critical exponents characterizing its statics and dynamics are reported. It is found that the influence of the PC transition on the critical exponents of the spins is strong and the origin of drastic changes as compared to the Glauber - Ising case can be traced back to the hyperscaling law stemming from directed percolation. The effect of an external magnetic field, leading to directed percolation-type behaviour on the level of kinks is also studied, mainly via the generalized mean-field approximation.
One-dimensional nonequilibrium kinetic Ising models evolving under the competing effect of spin - flips at zero temperature and nearest-neighbour spin exchanges exhibiting a parity-conserving (PC) phase transition on the level of kinks are investigated here numerically from the point of view of the underlying spin system. The dynamical persistency exponent and the exponent characterizing the two-time autocorrelation function of the total magnetization under nonequilibrium conditions are reported. It is found that the critical fluctuations at the PC transition have a strong effect on the spins: the behaviour becomes non-Markovian and the above exponents exhibit drastic changes as compared with the Markovian Glauber - Ising case. In this context the crucial importance of considering the global order parameter (instead of the local one) is emphasized.
A family of nonequilibrium kinetic Ising models, introduced earlier, evolving under the competing effect of spin flips at zero temperature and nearest-neighbour random spin exchanges, is further investigated. By increasing the range of spin exchanges and/or their strength the nature of the phase transition 'Ising-to-active' becomes of (dynamic) mean-field type and a first-order tricritical point is located at the Glauber ( delta =0) limit. Corrections to the mean-field theory are evaluated up to sixth order in a cluster approximation and found to give good results concerning the phase boundary and the critical exponent beta of the order parameter which is obtained as beta approximately=1.0.
Nonequilibrium kinetic Ising models evolving under the competing effect of spin flips at zero temperature and Kawasaki-type spin-exchange kinetics at infinite temperature T are investigated here in one dimension from the point of view of phase transition and critical behaviour. Branching annihilating random walks with an even number of offspring (on the part of the ferromagnetic domain boundaries), is a decisive process in forming the steady state of the system for a range of parameters, in the family of models considered. A wide variety of quantities characterize the critical behaviour of the system. Results of computer simulations and of a generalized mean field theory are presented and discussed.
Large-scale Monte Carlo simulations are used to explore the effect of quenched disorder on one-dimensional, nonequilibrium kinetic Ising models with locally broken spin symmetry, at zero temperature (the symmetry is broken through spin-flip rates that differ for "+" and "-" spins). The model is found to exhibit a continuous phase transition to an absorbing state. The associated critical behavior is studied at zero branching rate of kinks, through analysis spreading of + and - spins, and of the kink density. Impurities exert a strong effect on the critical behavior only for a particular choice of parameters, corresponding to the strongly spin-anisotropic kinetic Ising model introduced by Majumdar [Phys Rev. Lett 86, 2301 (2001)]. Typically, disorder effects become evident for impurity strengths such that diffusion is nearly blocked. In this regime, the critical behavior is similar to that arising, for example, in the one-dimensional diluted contact process, with Griffiths-like behavior for the kink density. We find variable cluster exponents, which obey a hyperscaling relation, and are similar to those reported by Cafiero [Phys Rev. E 57, 5060 (1998)]. We also show that the isotropic two-component AB --> [symbol: see text] model is insensitive to reaction disorder, and that only logarithmic corrections arise, induced by strong disorder in the diffusion rate.
Ordering of a one-dimensional stochastic Ising model evolving by a combination of spin flips and spin exchanges is investigated. The spin-flip rates satisfy detailed balance for the equilibrium state of the Ising model at temperature T, while the spin exchanges are random and of arbitrary range. Analytical methods and Monte Carlo simulations are used to show that, depending on the details of the spin-flip rate, finite-temperature phase transition may or may not occur in the system. When ordering occurs, it is of mean-field type and the scaling function describing the finite-size scaling of the magnetization fluctuations is found to be indistinguishable from that of an equilibrium Ising model with infinite-range interaction.