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One-dimensional spin-anisotropic kinetic Ising model subject to quenched disorder

Large-scale Monte Carlo simulations are used to explore the effect of quenched disorder on one dimensional, non-equilibrium 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 et al. 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 et al. We also show that the isotropic two-component AB ->0 model is insensitive to reaction-disorder, and that only logarithmic corrections arise, induced by strong disorder in the diffusion rate.


INTRODUCTION
The study of non-equilibrium model systems has attracted great attention in recent years. A variety of phase transitions have been found characterized by critical exponents, both static and dynamic. Of special interest are transitions from a fluctuating active state into an absorbing one. A wide range of models with transitions into absorbing states was found to belong to the directed percolation (DP) universality class [1,2]. Another universality class of interest is the so-called parity conserving (PC) class [3,4,5,6]. The most extensively studied model in this class is branching annihilating random walks with an even number of offsprings (BARWe) in one dimension (1d). The first examples of models exhibiting a PC-type transition were two one-dimensional cellular automata studied by Grassberger [7]. The prototype spin-model for PC-type phase transitions, the non-equilibrium kinetic Ising model (NEKIM) was proposed by one of the authors [8]. In this model domain walls (kinks) between unlike spins follow BARWe dynamics.
One dimensional models showing DP transition, such as the contact process, exhibit pronounced changes in behavior under the effect of quenched spatial disorder, even for small impurity concentrations (see [9] and references there). On the other hand, as a recent computer simulation study [10] has shown with great accuracy, the parityconserving class appears to be highly resistant to impurities. The same study reported similar, or even stricter, negative results for the one-dimensional Glauber-Ising model [11].
Another aspect of the problem, however, was identified by Majumdar et al. [12], who introduced a specific, inherent spin anisotropy (kinetic disorder) in the Glauber-Ising model (Majumdar Dean Grassberger (MDG) model in the following), These authors found, both analytically and numerically, a slow logarithmic factor in the decay of the density of kinks ρ(t) for t → ∞.
The aim of the present study is to show that in certain Ising-like systems (possessing two absorbing states), local kinetic disorder may effectively remove one of the absorbing states. The resulting, single-absorbing state system is sensitive to spatial disorder, similar to models in the directed percolation class. To this end we start from a generalization of NEKIM, the nonequilibrium kinetic Ising model with anisotropy (NEKIMA) [13,14], with variable kinetic disorder, and add uncorrelated spatial impurities to the system. We recall that without spatial impurities and in the zero-branching limit considered throughout this paper, the exponent α, which governs the time dependence of the kink density, ρ(t) ∼ t −α , is invariably α = 1/2 (apart from possible logarithmic corrections). The characteristic cluster-exponents, which are more susceptible to kinetic disorder than those governing the kink density, differ however from their Glauber-Ising values. This phase will be called spin-anisotropic Glauber-Ising (SAGI). For definitions and details of the models referred to here, and the acronyms used, see Table I.
Our numerical studies show that the scaling behavior of the SAGI phase is resistant to quenched spatial impurities, of strength p 0 , in the small-impurity regime. Close to the incipient freezing of a diffusion channel, which occurs at about p 0 = 0.5, and which appears to be a dirty critical point [9], the exponent α jumps from 0.5 to 0.21. On increasing p 0 further, a Griffiths-like [15] phase is encountered, with α(p 0 ) increasing with p 0 . The cluster properties: the mean population size n(t) ∼ t η , the mean-square distance of spreading of spins R 2 (t) ∼ t z , and the survival probability, P (t) ∼ t −δ are also found to change from their SAGI expressions (Table II) to those characteristic of a Griffiths phase [15]. We find close similarities to the results on scaling in the impure contact pro- I: Summary of models. Here w denotes the spin transition probabilities, A = Γ(1 +δ), B = Γ/2(1 −δ) [16] in (almost) the whole plane of the corresponding phase diagram, with the hyperscaling law remaining valid. Here this behavior is found in an annihilating random walk-type model (ARW) i.e., without kink production, while production takes place on the level of spins, in one of the spin-channels ('+' spins in the following), as explained in Sect. II.
Moreover, in the present case the SAGI phase plays the role of a supercritical phase: active-like concerning '+'-cluster-behavior while frozen for the '−' spin phase, and of annihilating-random-walk type for the kink density ρ(t). The dual nature of global and cluster properties is not contradictory and can be traced back to the fact that the model is highly spin-anisotropic.
The NEKIMA model coincides, at some specific parameter values (MDG case, Fig.1) with the model of Majumdar et al. [12] mentioned above. The joint effect of kinetic and spatial impurities is even more pronounced at and in the vicinity of this point (see the phase diagram depicted in Fig.2, where the most spin-anisotropic point, the MDG-point is at the origin). In this region of the (p + , p 0 ) plane the impure scaling behavior is different: by increasing p 0 we have found a change in the behavior of the global quantity ρ(t): α switches from a value of 1/2 (apart from the very late time logarithmic correction) to α = 1. The cluster behavior is different for the '+' and '−' spins. For the '+' it appears to follow the above mentioned scheme [16] of the impure contact process: upon increasing p 0 a strong tendency towards the absorbing '+' phase can be seen with the hyperscaling law [17] satisfied. For the '−' cluster we find η = z/2 = 1, and δ = 0, with logarithmic corrections in time.
In summary, we find that by increasing spin anisotropy the sensitivity of critical behavior to quenched disorder is enhanced. (NEKIM → SAGI → MDG). It is worth noting that in order to maintain the symmetry of the NEKIMA model, the disorder affecting diffusion is anisotropic too, therefore it is different from the problem considered in [10], where for very strong disorder diffusion can be blocked completely..
In the Appendix a further, isotropic ARW model, AB → ∅, [1] is investigated via numerical simulations, under quenched disorder. In this model, which can be mapped onto the pure MDG model [2] the effect of impurities on the annihilation and diffusion rates are investigated separately: the density decay is unaffected in case of the former, while logarithmic corrections arise for impure diffusion rates for very long times.
The paper is organized as follows. In Sect. II. we define the models and the quantities to be investigated. Some previous results for the exponents of spreading from a localized source are recalled and commented on. In a subsection -to enlighten the different roles played by the '+' and '−' spins -our model is mapped onto a reactiondiffusion system. In Sect.III. the effect of spatial impurities is investigated for two different choices of the value of the anisotropy-parameter: the SAGI case (A) and the more spin-anisotropic MDG case (B). The results are discussed in Sect.IV., while in the Appendix a further ARW model, the AB → ∅ is investigated under the influence of quenched disorder.
In the one-dimensional Glauber model at temperature zero, the most general form for the flipping rate of spin s i is [11] (1) Usually the Glauber model is understood as the special caseδ = 0, Γ = 1 and we will use these parameter values in the following. Processes involving the reaction 1 kink → 3 kinks are introduced via the exchange rate (a kink is a '−+', or '+−' configuration: domain boundary between two oppositely magnetized regions). For negative values ofδ this model (called NEKIM), shows a line of PC-transitions in the (p ex ,δ)-plane [8]. The NEKIMA model is an extension with local symmetry breaking in the flipping rates of the '+' and '−' spins, as follows. Concerning the annihilation rates (of a spin in the neighborhood of oppositely oriented spins) the prescription in [12] is followed: Further spin symmetry breaking is introduced in the diffusion part of the Glauber transition rate as follows. The transition rates are taken as in Eq.
(1), while w(+; +−) and w(+; −+) may take smaller values: In this way, by locally favoring + spins, the effect of the other dynamically-induced field arising from the rates (Eq.3) is counterbalanced. The spin-exchange part of the NEKIM model remains unchanged, Eq.(2). Spreading from a localized source at criticality is usually described by the following three quantities where n(t) denotes the mean population size, R 2 (t) is the mean square spreading of particles (here spins) about the origin and P (t) is the survival probability. In most cases these quantities are defined for particles, in the present case, however, they will be used for spins. In NEKIMA '+' and '−' spins are not symmetric, therefore we have investigated two kinds of clusters. Namely, the development of the '−' cluster-seed was started from a wholly '+' environment while that of the '+' cluster from a sea of '−' spins. We will call them '−' cluster and '+' cluster, respectively. The cluster critical exponents for the NEKIMA were reported in [14]. Their values at a typical SAGI point (p + = 0.3, p = 0.5) and at the MDG point (p + = p = 0.5), for the ARW-case discussed here, are summarized on Table II. It should be noted here, however, that the precision of the simulations in the above cited paper does not exclude the possibility of the presence of some log(t) corrections.
On the level of '+' -spins the SAGI phase (1/2 = p > p + ) is an active one as shown by the '+'-cluster exponents on Table II. For producing '+' spins the rate in Eq.(4) with p = 1/2 is responsible, while p + annihilates them. Crudely speaking the probability of the '+' spingenerating transition process, p − p + , can be thought of as the quantity corresponding to the particle production rate in the contact process.
As one can see due to the anisotropies the '−' and '+' spins follow different dynamics. To understand the peculiar behavior of the species we transform their motion into the language of reaction-diffusion models, making a connection with the basic models and classes [2].
A. Particle picture of the spin-processes Let X be a particle and ∅ its absence. For the '−' spins as X-es and + spins as ∅-s Eq. (3) gives creation, while Eq.(5) leads to the creation From the above three, the net process remains as Similarly Eq.(4) means the coagulation processes From eqs. (9) and (10) the generic reactions follow. This corresponds to the recently studied reversible model, [18,19,20,21], where transition to an absorbing phase can occur only for zero branching rate, which is excluded in our case, hence we don't expect an active-absorbing phase transition of the '−' spins. On the other hand for the '+' spins as X's we get from Eq.(3) the spontaneous decay and from Eq.(5) the coagulation while Eq. (4) gives the creation by a neighbor These give again XX ↔ X but according to Eq. (12) here the spontaneous annihilation of X, X → ∅ is also possible due to Eq. (3), which is a necessary condition for a DP class transition. Hence for '+' spins a phase transition (at finite reaction rates) is not excluded and as we will see later it emerges as the effect of the disorder. The corresponding critical behavior is similar to that of the disordered DP (albeit an anisotropic one).

III. EFFECTS OF QUENCHED IMPURITIES
At each site i the diffusion rates of kinks p = w(−; +−) = w(−; −+) in Eq.(4) are modified by adding quenched, uncorrelated impurities with uniform probability distribution of the form where ǫ(i) is randomly distributed in the interval 0 ≤ ǫ(i) ≤ 1 and 0 ≤ p 0 ≤ 0.5. On the other hand the rates, which fix the (a)symmetry of the model w(+; −−) = 1, w(−; ++) = 0 and p + of Eq.(15) are kept uninfluenced. In this way kinetic impurities and quenched local impurities are let to act separately, though simultaneously, resulting in a rich phase diagram. Two characteristic impure behaviors are exemplified by the following two choices of the fixed parameters: (1) p + = 0.3 (to be called the SAGI-case) and (2) p + = 0.5 (to be called the MDG-case).
The phase diagram of the impure NEKIMA is shown schematically on Fig.1. At the MDG point the '−' phase gets dominating (Section III.B.). This absorbing '−'phase is separated from the SAGI phase by the MDGline, a line of critical points characterized by the cluster exponents given in Table II and where α = 1/2. This line is depicted here only tentatively (especially for higher p 0 values), as its neighborhood is not the subject of our further detailed study. The diffusion-block transition line corresponds to the limiting value: p 0 = 1/2, where the only non-fixed spin-flip probability can become zero, hence diffusion blocking starts. The SAGI phase in the vicinity of this point will be discussed in the next Section. It is worth emphasizing again that in the present model only one transition probability gets impure (one spin-flip rate) in contrast to a previous investigation of NEKIMCA with impurities [10], where all four transition rates (spin-flip+annihilation) have been complemented with a term like that in Eq. (15). We have also looked into the effect of impurities in one of the annihilation rates of the two fixing the model's anisotropy: no effect shows up in the vicinity of p 0 = 0.5 in this case as can be expected.
A. The SAGI case (p+ = 0.3) We will investigate the effect of impurities below the MDG point i.e. for p + < 0.5 at a typical parameter value p + = 0.3 (see Fig.1) by simulating a system of size L = 3 × 10 4 with periodic boundary conditions up to t max = 10 6 Monte Carlo steps (MCS) (throughout the paper time is measured by MCS). For small impurity rates p 0 the behavior is found unperturbed. Correspondingly, the critical exponent of the density of kinks as a function of time is α = 1/2 and the exponents of the ′ + ′ and ′ -′ clusters are as seen on Table II. By increasing p 0 , the rate of disorder, however, the spin flip rate p i in Eq. (15) can become zero in the vicinity of p 0 = 0.5 and above. To interpret the results in this region it is important to recall [14] that the critical behavior of the kink density and that of the '+' and '−' clusters are not connected in the present model (thus no scaling law connects the critical dynamical exponent Z and the cluster size exponent z). While there is no particle production on the level of kinks (ARW-process), on the level of spins there is production as mentioned in the previous section. The spin-transition rate p, for the process (− − +) → (− + +) leads to the increase of + spins. This +-spin production is maximal at p 0 = 0. The SAGI phase is an active one, more precisely a '+'-active one (the '−'-spin phase is frozen). At the same time in the usual sense of Glauber-Ising-type models, on the level of kinks, there is no active phase as the kink production p ex in Eq.(2) is zero.
By increasing p 0 the system stays in this '+'-active phase, neither the kink-decay exponent α = 1/2 nor the cluster exponents of Table II   vicinity of p 0 = 0.5 the local exponent values defined as (where we used m = 4) behave as on Fig.3. The fixed point value of the density of kinks at p 0 = 0.5 can be fitted with a power of ln(t) withα = 1.45943 as shown on Fig. 4. This exponent differs from that of the known value of the infinite randomness fixed point of the 1+1d DP (α DP = 0.38197) [24], not surprisingly kinks follow an anisotropic dynamics here. The excellent fit with Eq. (17) suggests that this point is a dirty critical point, which is reached from the 'active' phase (here from the '+'-active phase). The value of α is a new critical exponent. For higher values of impurity strength a Griffiths-like phase with is entered. Though in this region the present formulation of the impure problem already goes beyond its range of validity, however the tendency of the impure behavior is well exhibited in our simulations (e.g. α(p 0 ) increases steadily as expected in a Griffiths phase, etc.). Nevertheless, we refrain from giving more details. The time dependences of the ′ + ′ and ′ -′ cluster quantities (Eqs. 6) also show a behavior characteristic of an impure phase transition. The behavior of the ′ -′ cluster shows an effect of 'melting': η and z/2 move from 0 to some finite values, upon increasing p 0 , while δ = 0 remains the same. The ′ + ′ cluster behavior however, shows a scenario of a transition to an absorbing state, similarly to the contact process investigated by Cafiero et al. [16]. However the direction is opposite: in [16] for small values of disorder (called p in their paper) an absorbing phase is present, while the active is reached at p = 1. We plot on Fig. 5 the '+' cluster behavior, namely the local exponent values of the survival probability defined as δ ef f (t) = − ln [P (t)/P (t/m)] ln(m) (19) for different values of disorder. One can clearly see a turn from the SAGI-active behavior (δ = 0) towards a Griffiths-phase-like non-universal cluster scaling by increasing δ in the vicinity of p 0 = 0.5. At p 0 = 0.5 (dirty critical point of the '+' spins) an ultra-slow, activated scaling of the form Eq.(17) can be observed in the cluster data (see Fig. 4). Fitting with the activated scaling forms for the '+' data resulted in:δ = 0.060(6),z = 3.88(1) andη = 3.37 (2). These values again differ from those of the strong disorder fixed point of DP, probably due to the anisotropies of the processes. Our simulation data for the mean size of the '+'-cluster R(t) depicted on a double-logarithmic scale is shown on Fig.6. The linear SAGI behavior crosses over to decreasing values of z/2 by reaching the p 0 = 0.5 transition point (sub-linear behavior, for similar behavior see Tables I. and II. in [16]) The MDG point can be found at the origin of the phase diagram (Fig.1). At these parameter values the impure system decays into a fully compact state, where all spins are flipped to '−' but different types of kinks exert marginal perturbation on each other [25]. In more detail [12] found analytically that the magnetization decays as while the kink density behaves as (where b is constant). For the time being -since much stronger (power-law-type) effects enter due to the quenched spatial impurities -we disregard the ln(t)-type of correction and show only the leading singular behavior of the cluster properties (see Table II). By introducing impurities of strength p 0 = 0.3, the space-time evolution of kinks looks as can be seen on Fig.7. The absorbing nature of the '+' phase is apparent. The fully '−' spin configuration sets in quickly (compare this figure with Fig. 1. of [14], where single '−' spins move via random walk in the see of '+'-es.).
Results of high-precision simulations are shown on Fig.8. For p 0 < 0.5 one can see α → 1 in the long time limit. This behavior can be ascribed to the choice of the impurity, which acts against one type of diffusion (which favors '+' spins). As a consequence the '−'-es can spread better than in the impure case. The effect of impurities on the '−' clusters is illustrated on Fig. 9 for the η and z/2 exponents. The local slope curves of η and z/2 are almost the same, they approach the asymptotic value η = z/2 = 1 from below with an upward curvature (δ = 0 remains constant). This kind of effective exponent behavior usually corresponds to logarithmic correction to scaling. In case of the general form for N (t) the effective exponent behaves as Therefore a logarithmic fit of the form a + b/ ln(t) (25) has been applied for the local exponent curves and the values obtained both for η and z/2 are shown in Table   0  III. The clusters are compact, of course, thus the hyper-  scaling law is satisfied.
To summarize, instead of the weak logarithmic tendency towards the dominance of the '−'-es as for the pure MDG model the impurity pushes the system into a more anisotropic decay, characterized by the kink decay exponent α = 1. The depletion of the '+'-cluster is illustrated on Figs.10 and 11. The '+'-cluster dies out apparently in a power-law manner with the exponents η = −1, δ = 1 and z/2 = 0. These scaling behaviors suggest a transition from one absorbing phase (MDG) to an other one as the effect of quenched impurities, which increases anisotropy.
It is worth noting that by switching on the spinexchange term (p ex ) in the pure case a quicker dominance of the '−'-es takes place for the same parameter values of the model as here. In that case, no power-law regime is visible (except for some early time transient region and for very low values of the kink production p ex ) and the absorbing phase sets in exponentially fast typically (see [13]) due to the faster destruction of metastable '+' domains.

IV. DISCUSSION AND CONCLUSIONS
In this paper the problem of the effect of quenched impurities on one-dimensional non-equilibrium Glauber-Ising-type models has been investigated numerically. Our previous study of the same problem, without spinanisotropy, led to the conclusion of a minimal of effect [10]. Here the emphasis has been on introducing spinanisotropy, and to give insight into the the joint effect of kinetic and quenched spatial anisotropies. We have restricted our studies to the zero branching rate limit of the model (ARW-case). Depending on the ratio of anisotropies in the diffusion and annihilation channels (rates) a variety of behaviors appear when the quenched randomness is turned on, with the inference that some amount of kinetic anisotropy can significantly enhance the influence of quenched spatial anisotropies.
This behavior is different from that found for isotropic NEKIM [10], where reaction and diffusion are completely blocked for very strong disorder. The effects of such blocking are investigated in the context of the AB → ∅ model; only the diffusive ones are found to be marginally relevant.
Summarizing the results for exponents η, δ, and z (Table II), for '+' spins in the case p = p + = 1/2 (MDG+) we have the (0, 1/2, 1) spreading fixed point exponents of the XX → X diffusive coagulation process, because due to Eq.(3) the XX ↔ X is unbalanced, favoring XX → X annihilation of '+'-es and the proliferation of '−'-s. In the p + < p = 1/2 kinetic disorder case (SAGI+) we found [14] a novel kind of anisotropic fixed point (1, 0, 2), which is the same what one obtains if quenched disorder is added to (MDG-). This can be understood by looking closer on the effect of these disorders on the spin species. Both kind of disorders enhance the production of one species at the expense of the other. While in case of (SAGI+) the '−'-es are suppressed, for the (diso MDG-) the same thing happens with the '+'-es (due to Eq. (15) and the other spin domains grow linearly with η = z/2 = 1.
However due to Eq.(3) the handicapped species end up in a different way. In case of SAGI the '−'-s can survive resulting in (0, 0, 0) as t → ∞, slowing down the domain wall decay (resulting in α = 0.5). In case of the diso-MDG the lonely '+'-s can die out completely, characterized by η = −1 mass and δ = 1 survival exponents, enabling the linear growth of kinks. Therefore α = 1 can be observed here.

V. ACKNOWLEDGMENTS
Support from the Hungarian Research Fund OTKA (Grant No. T-046129) during this study is gratefully acknowledged. G.Ó. thanks the access to the NIIFI Cluster-GRID, LCG datagrid and the Supercomputer Center of Hungary.

VI. APPENDIX
In this appendix the effect of quenched disorder in a further ARW type model, the AB → ∅ one, will be investigated. Unlike SAGI, this model is free of anisotropies. In our previous paper [10] we investigated the inactive phase of the cellular automaton version of the nonequilibrium kinetic Ising model (NEKIMCA) model with quenched disorder. In that model annihilating random walk of domain walls (kinks) AA → ∅ is the dominant process. We have not found any effect of the weak disorder on the density decay of kinks. If the disorder was so strong that diffusion could be blocked at certain sites (the system is split into independent blocks of finite sizes) a new fluctuating phase with finite steady state density emerged as the consequence of the parity conservation of kinks. Along and just below the blocking transition line the disorder caused logarithmic slow-down corrections to the ρ ∝ t −1/2 impure law of the ARW process [26] and continuously changing exponents (α < 0.5) for strong disorder. Those results are in agreement with analytic results for diffusion disorder [27] and annihilation disorder [28] of the ARW. The anisotropic reactions of the NEKIMA model [13] result in a two component reaction dynamics of kinks destroying the PC class critical behavior. Without kink production (p ex = 0) one can map onto the AB → ∅, however, since A and B species are domain walls of the spins they are arranged alternately along the 1d system. Consequently they can always meet their nearest neighbors (no hard-core reactions take place as in [29]) and one observes the usual ρ ∝ t −1/2 decay instead of the ρ ∝ t −1/4 law of the two component model [30].
In this appendix we show results for the quenched disordered AB → ∅ model to complement results for the quenched disorder NEKIMA model. To our knowledge this has not been studied before. We have investigated the effect of disorder in diffusion and annihilation separately. A random sequential simulation program has been run in 1d systems of size L = 5 × 10 5 with periodic boundary conditions and single particle occupation restriction. We have investigated the density decay for initial conditions with randomly (RI) and pair-wisely (PI) distributed A-s and B-s of full lattice occupancy.
One elementary MC step is built up as follows. A particle and a direction is chosen randomly. If the nearest neighbor (nn) site in the selected direction was empty the particle is moved there with probability 1 − ǫ D (where ǫ D is the diffusion disorder strength). If the nn site was occupied with a particle of a different type we removed both particles with probability 1 − ǫ A . The time (t) is incremented by ∆t = 1/n p (where n p is the total number of particles) and the density decay was followed up to t max = 10 8 MCS.
In case of ǫ D = 0 we did not see any effect of the disorder in the annihilation up to ǫ A ≤ 1 both for RI and PI conditions (see Fig.12 for RI). The density decay is completely insensitive to the quenched randomness of the annihilation reaction rate.
On the other hand for disorder in the diffusion (hopping) probabilities (ǫ A = 0, ǫ D ≤ 1) one can see logarithmic corrections to the scaling of the impure case for ǫ D → 1 (see Fig. 13 for RI). Following a sharp initial decay, when local correlations are built up the evolution tends towards the ρ ∝ t −0.25 law, but as in case of the ARW model logarithmic slow-down appears for very late times and strong disorder (already without diffusion blocking). It has been tested by simulating on different sizes (L) that the turns in the curves are not artifacts of finite system sizes. Since the basic mechanism of annihilating random walk here AB → ∅ is the same as for the AA → ∅ model, we think that the relevancy of diffusion disorder compared to the annihilation disorder can be observed in the inactive phase of the NEKIMCA [10] as well.