Critical phenomena of nonequilibrium dynamical systems …...Recently, the branching annihilating...

PHYSICAL REVIEW E JUNE 1998VOLUME 57, NUMBER 6

Critical phenomena of nonequilibrium dynamical systems with two absorbing states

WonMuk Hwang*Department of Physics, Korea Military Academy, Seoul 139-799, Korea

Sungchul Kwon, Heungwon Park, and Hyunggyu Park†

Department of Physics, Inha University, Inchon 402-751, Korea~Received 22 December 1997!

We study nonequilibrium dynamical models with two absorbing states: interacting monomer-dimer models,probabilistic cellular automata models, nonequilibrium kinetic Ising models. These models exhibit a continu-ous phase transition from an active phase into an absorbing phase that belongs to the universality class of themodels with the parity conservation. However, when we break the symmetry between the absorbing states byintroducing a symmetry-breaking field, Monte Carlo simulations show that the system goes back to theconventional directed percolation universality class. In terms of domain wall language, the parity conservationis not affected by the presence of the symmetry-breaking field. So the symmetry between the absorbing statesrather than the conservation laws plays an essential role in determining the universality class. We also performMonte Carlo simulations for the various interface dynamics between different absorbing states, which yieldnew universal dynamic exponents. With the symmetry-breaking field, the interface moves, in average, with aconstant velocity in the direction of the unpreferred absorbing state and the dynamic scaling exponents appar-ently assume trivial values. However, we find that the hyperscaling relation for the directed percolationuniversality class is restored if one focuses on the dynamics of the interface on the side of the preferredabsorbing state only.@S1063-651X~98!06806-8#

PACS number~s!: 64.60.2i, 02.50.2r, 05.70.Ln, 82.65.Jv

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I. INTRODUCTION

Many nonequilibrium dynamical models show continuophase transitions similar to ordinary equilibrium models.fact, nonequilibrium models can supply much richer criticbehavior because their evolving dynamics do not requiredetailed balance. So the universality classes of nonequrium critical phenomena would be much more diverse awould be governed by various symmetry properties ofevolution dynamics.

An interesting example of nonequilibrium phase trantions is the absorbing phase transition. In this case, thexist some absorbing states in the configurational phspace. If the system gets into one of the absorbing statethe evolution dynamics, then the system is trapped insidthe absorbing states and no further dynamics occur to esout of the absorbing states. By controlling an external paraeter, one can observe a continuous phase transition fromactive steady-state phase into an inactive absorbing phRecently, various kinds of nonequilibrium models exhibitisuch an absorbing phase transition have been studied esively @1#. Most of the models investigated are found to blong to the directed percolation~DP! universality class@2–12#. A common feature of these models is that the absorbphase consists of a single absorbing state.

Only a few models have been studied that are not inDP universality class. Those are the modelsA and B of

*Address after September 1, 1998: Department of Physics, BoUniversity, Boston, MA 02215.

†Author to whom correspondence should be addressed. Electraddress: [email protected]

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probabilistic cellular automata~PCA! @13,14#, nonequilib-rium kinetic Ising models with two different dynamics~NKI !@15–17#, and interacting monomer-dimer models~IMD !@18,19#. Numerical investigations show that critical behaiors of these models are different from DP but formnon-DP universality class. These models share a commproperty that the absorbing phase consists of two equivaabsorbing states. By the analogy to the equilibrium Ismodel, which has two equivalent ground states, we callnon-DP universality class thedirected Ising~DI! universalityclass.

Recently, the branching annihilating random wal~BAW! with offspring have been studied intensively@20–28#. Even though the BAW model has a single absorbstate~vacuum!, its critical behavior depends on the paritythe number of offspring. It has been shown numerically tthe BAW models with an odd number of offspring~BAWo!belong to the DP class, while the BAW models with an evnumber of offspring~BAWe! belong to the DI class@22,26#.Dynamics of the BAWe models conserve the numberwalkers modulo 2, while the BAWo models evolve withoany conservation. The common feature of the PCA, NIMD, and BAWe models is that the number of particle~walkers in BAWe and kinks or domain walls in the othmodels! is conserved modulo 2. From this point of view,was suggested that the parity conservation is responsiblethe DI universality class. This is why the DI universaliclass is sometimes called as the PC~parity-conserving! uni-versality class.

However, we recently showed for the IMD model thatexternal field that conserves the parity but breaks the smetry between two absorbing states forces the system bto the conventional DP universality class@29#. So we argued

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57 6439CRITICAL PHENOMENA OF NONEQUILIBRIUM . . .

that the symmetry between absorbing states rather thanconservation laws plays an essential role in determininguniversality class. Our argument was supported by recresults for generalized monomer-monomer models studby Bassler and Browne@30–32#, and for some stochastimodels by Hinrichsen@33#.

In this paper, we study the effect of a symmetry-breakfield in the IMD, PCA, and NKI models via stationary awell as dynamic Monte Carlo simulations. Stationary simlations and defect dynamics for all three models clearly shthat the DI universality class crosses over to the DP cunder a weak parity-conserving symmetry-breaking field.fact, the ratio of the number of stationary runs that fall inthe unpreferred absorbing state and the number of thosethe preferred state vanishes exponentially in system sizethe system with the symmetry-breaking field has in effecsingle absorbing state, which leads to the DP class.

We also introduce new types of interface dynamics tresult in different values of dynamic scaling exponenThese exponents are found to be universal. Withousymmetry-breaking field, the hyperscaling relation for theuniversality class is intact for various interface dynamiHowever, with the symmetry-breaking field, the interfamoves, in average, with a constant velocity in the directof the unpreferred absorbing state. The dynamic scalingponents apparently assume trivial values and violate theperscaling relation. However, we find that the hyperscalrelation for the DP universality class is restored if onecuses on the dynamics of the interface on the side ofpreferred absorbing state only.

In the next section, we report our numerical results forIMD model in various dynamic simulations. In Sec. III, theffect of the symmetry-breaking field in the IMD modeldicussed in detail via stationary and dynamic simulationsSec. IV, the numerical results for the PCA and NKI modare presented. Finally we conclude in Sec. V with a summand discussion.

II. DYNAMIC CRITICAL BEHAVIOR OF THE IMDMODEL

The interacting monomer-dimer~IMD ! model is a gener-alization of the simple monomer-dimer model on a catalysurface, in which particles of the same species have neaneighbor repulsive interactions@18#. Here we consider theone-dimensional IMD model with infinitely strong repusions between the same species. A monomer (A) cannot ad-sorb at a nearest-neighbor site of an already occupied mmer~restricted vacancy! but adsorbs at a free vacant site wino adjacent monomer-occupied sites. Similarly, a dimer (B2)cannot adsorb at a pair of restricted vacancies (B in nearest-neighbor sites! but adsorbs at a pair of free vacancies. Thare no nearest-neighbor restrictions in adsorbing particledifferent species. Only the adsorption-limited reactionsconsidered. Adsorbed dimers dissociate and a nearest nbor of the adsorbedA andB particles reacts, forms theABproduct, and desorbs the catalytic surface immediatWhenever there is anA adsorption attempt at a vacant sibetween an adsorbedA and an adsorbedB, we allow theA toadsorb and react immediately with the neighboringB, thusforming the AB product and desorbing the surface. If th

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process is not allowed, the IMD model possesses infinitmany absorbing states which will be discussed elsewh@34#.

The system has no fully saturated phases of monomerdimers, but instead two equivalent half-filled absorbistates. These states are comprised of only the monomethe odd- or even-numbered lattice sites, i.e., (A0A0•••) and(0A0A•••) where ‘‘A’’ represents a monomer-occupied siand ‘‘0’’ a vacant site. These two states are probabilisticaequivalent, unless we introduce a symmetry-breaking fidiscriminating the dynamics at the odd- and even-numbesites. In this section, we consider the IMD model withousymmetry-breaking field. This model can be parametrizedthe monomer adsorption-attempt probabilityp. The dimeradsorption-attempt probability is then given by 12p.

In our previous stationary and dynamic Monte Casimulations@18,19#, it was found that the system undergoescontinuous phase transition from a reactive phase intoabsorbing phase, which belongs to the DI universality claThe kink representation of the IMD model is complicatdue to its multicomponent nature. Three types of kinks cbe defined between lattice sites occupied by a dimer andimer, by a dimer and a vacancy, and by a vacancy anvacancy. No conservation law is associated with each typkink, but the total number of kinks is conserved moduloSo, in a broad sense, one can say that the IMD moevolves by the parity-conserving dynamics like the BAWmodel. In this section, we discuss the dynamic critical bhavior of the IMD model via Monte Carlo simulations. Somof the results reported previously@19# are much improvedusing efficient algorithms and some results for various intface dynamics are presented.

A. Defect dynamics

We start with a lattice occupied by monomers at alterning sites except at the central vacant site, i.(•••A0A000A0A0•••), where0 represents a defect at thcentral site. Then the system evolves along the dynamicof the model. After one adsorption attempt on the averaper lattice site~one Monter Carlo time step!, the time isincremented by one unit. A number of independent rutypically 53105, are made up to 33104 time steps for vari-ous values ofp near the critical probabilitypc . Most runs,however, stop earlier because the system enters into onthe absorbing states. We measure the survival probabP(t) ~the probability that the system is still active at timet),the number of dimersN(t) averaged over all runs, and thmean square distance of spreading of the active regionR2(t)averaged over surviving runs. At criticality, the valuesthese quantities scale algebraically in the long time limit@2#,

P~ t !;t2d,

N~ t !;th, ~1!

R2~ t !;tz

and double-logarithmic plots of these values against tishow straight lines at critiality. Off criticality, these plotshow curvatures. More precise estimates for the dyna

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6440 57HWANG, KWON, PARK, AND PARK

scaling exponents can be obtained by examining the loslopes of the curves. The effective exponentd(t) is definedas

2d~ t !5log@P~ t !/P~ t/b!#

log~b!~2!

and similarly forh(t) and z(t). In this paper, we plot theeffective exponents against 10/t with b510. Off criticality,these plots show positive or negative curvatures. The scaexponents can be extracted by taking the asymptotic vaof the effective exponents at criticality.

Our estimates for the critical probability and the dynamscaling exponets are pc50.5322(3), d50.290(5),h50.00(1), andz51.135(5) ~see Fig. 1!. Note that the es-timate forz is much improved compared with our previouresult:z51.34(20)@19#. These values are in excellent agrement with those of the DI universality class such asBAWe model@26#.

FIG. 1. Plots of the effective exponents against 10/t for thedefect dynamics of the symmetric IMD model. Three curves frtop to bottom in each panel correspond top50.5315, 0.5322, and0.5329.

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B. Interface dynamics

For the interface dynamics, we start with a pair of vacacies placed at the central sites of a lattice and with monomoccupied at alternating sites, i.e., (•••A0A00A0A•••)where the interface between two different absorbing stateplaced in the middle of two central vacancies00. In this case,the system never enters an absorbing state, so that thevival probability is always equal to 1 and the exponentd50. Even though the values ofd andh vary with the typesof dynamics, their sumd1h, which is responsible for thegrowth of the number of kinks~or dimers! in surviving runs,is known to be universal@35,19#. This guarantees that thgeneralized hyperscaling relation is always satisfied;b/n i1d1h5dz/2 whered is the spatial dimension andb andn iare steady-state exponents explained in the next sectiond50 in this type of interface dynamics, it does not suppany new information about dynamic critical behavior of tsystem.

In this section, we introduce three different types of inteface dynamics that may give nontrivial scaling of the survial probability P(t). For convenience, the ordinary interfacdynamics as above is called astype-A interface dynamics.Our previous study for the type-A interface dynamics foundthat h50.285(20) andz51.14(2) as expected@19#.

In the type-B interface dynamics, we stop the evolution ifthe interface collapses to its initial configuration, i.e., twvacant sites between the absorbing states. Then this rutreated as a dead one. This dynamics is originally introduby Bassler and Browne for a three species monommonomer model@30#. At criticality, we measureP(t), N(t),andR2(t). P(t) now represents not a true survival probabity but a probability of avoiding a collapse. In Fig. 2, we plthe effective exponents against 10/t. Our estimates for thedynamic scaling exponents ared50.73(1), h520.41(1),andz51.16(2). These values satisfy the generalized hypscaling relation of the DI universality class and are in exclent accord with those reported by Bassler and Browne@30#.To see how much these exponents are robust, we changcriteria for stopping the evolution from two active sitesfour and six active sites. We find that there is no essenchange in the values of the exponents. Therefore the ntrivial value of d in the type-B dynamics is believed to beuniversal.

In the type-C interface dynamics, we focus only on theprofile between the central site and the leftmost site~the leftfront! of the active region. We stop the evolution when tleft front of the active region comes back to the center~initialposition! and treat this run as a dead one~see Fig. 3!. SoP(t) represents a probability of avoiding a collapse of tactive region in the left side with respect to the initial loction of the interface. We measureN(t) as the number ofdimers only in the left side of the center andR2(t) as themean square spreading of the active region in the left sidthe center. The type-C interface dynamics is useful when onneeds to distinguish the behavior of the left and right froof the interface. So it is especially important to consider ttype of dynamics for the interface between unequivalentsorbing states~see Sec. III!. The effective exponents again10/t at criticality are plotted in Fig. 4. Our estimates ared50.395(5), h520.10(1), andz51.150(5), which also sat-

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isfy the generalized hyperscaling relation of the DI univsality class. The value ofd is different from those for thedefect and other interface dynamics. It would be usefucheck that this exponent is also universal for other modelthe DI universality class. In Sec. IV, we study the typeCdynamics for the PCA model and find this exponent is uversal.

Finally, we introduce thetype-D interface dynamics.Thisdynamics is similar to the type-C dynamics in measuringphysical quantities. But we do not stop the evolution whthe left front of the active region hits the center. Howevwhile its left front wanders in the right side of the center, wtreat this run as a dead one temporarily, and setN(t) andR2(t) to zero. When it comes back to the left side of tcenter, we treat this run as a surviving run again and meaN(t) andR2(t) as usual in the type-C dynamics.P(t) rep-resents a probability that the active region covers the left sof the center and is expected to converge to a nonzerostant less than 1. The effective exponents against 10/t atcriticality are plotted in Fig. 5. Our estimates ared50.010(5), h50.29(1), andz51.15(1). As expected,d isnearly zero andP(t) converges to 0.69~1! ~see Fig. 6!. So

FIG. 2. Plots of the effective exponents against 10/t for the type-B interface dynamics of the symmetric IMD model at criticali(pc50.5322).

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the type-D interface dynamics yields the same exponentsin the type-A ~ordinary! dynamics. The type-A dynamicsdoes not yield the correct scaling exponents for the interfbetween unequivalent absorbing states, so in this casetype-D dynamics can be employed instead~see Sec. III C!.

III. THE IMD MODEL WITH A SYMMETRY-BREAKINGFIELD

We introduce a symmetry-breaking field that makessystem prefer one absorbing state over the other@29#. Thiscan be done by differentiating the monomer adsorptiattempt probabilityp at an odd-numbered vacant site andan even-numbered one. If a monomer attempts to adsoran even-numbered free vacant site, the adsorption attemrejected with probabilityh (0<h<1). The caseh50 corre-sponds to the ordinary IMD model discussed in the previosection. For finiteh, the monomers tend to adsorb morean odd-numbered site than an even-numbered one. Soabsorbing state with odd-numbered sites occupied by momers is probabilistically preferable to the other absorbstate. However, the kink dynamics of this model still coserves the parity in terms of the total number of kinks. In tsection, we show that the symmetry-breaking field forcessystem back to the conventional DP universality classstationary and dynamic simulations. Therefore one can cclude that not the parity conservation but the symmetrytween the absorbing states is essential in determininguniversality class of the absorbing phase transitions.

A. Stationary simulations

We run stationary Monte Carlo simulations starting wan empty lattice with sizeL. Then the system evolves alonthe dynamic rule with the symmetry-breaking field using priodic boundary conditions. We set the value of tsymmetry-breaking fieldh50.5 for convenience. After asufficiently long time, the system reaches a quasisteady sfirst and stays for a reasonably long time before finally etering into an absorbing state. We measure the concentraof dimers in the quasisteady state and average over mindependent runs that have not yet entered into an absor

FIG. 3. The type-C interface dynamics in the IMD model. Grayblack, and white dots represent monomers, dimers, and vacanrespectively.

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state. The number of independent runs varies from 53104

for the system sizeL532 to 33103 for L5512.Elementary scaling theory combined with the finite-s

scaling theory@2,36# predicts that the average concentratiof dimersr at criticality in the steady state scales with sytem sizeL as

r~L !;L2b/n', ~3!

whereb is the order parameter exponent andn' is the cor-relation length exponent in the spatial direction. In the retive phase (p,pc), the concentrationr remains finite in thelimit L→`, but it should vanish exponentially with systesize in the absorbing phase (p.pc).

At pc , we expect the ratio of the concentrations of dimefor two successive system sizesr(L/2)/ r(L)52b/n', ignor-ing corrections to scaling. This ratio converges to 1 forp,pc and approaches 2 forp.pc in the limit L→`. We plotthe logarithm of this ratio divided by log102 as a function ofp for L564,128,256, and 512 in Fig. 7. The crossing poibetween lines for two successive sizes converge to the pat pc.0.413(1) andb/n'.0.23(2). Thecritical probabilitycan be more accurately estimated from the defect dynamsimulations;pc50.4138(3)~see Sec. III B!. We run station-ary simulations at this value ofp and find the better estimatfor b/n'50.243(8) ~see Fig. 8!. This value is consistenwith the standard DP value of 0.2524~5! @26#.

By analyzing the decay characteristics of dimer conctrations at criticality, we can extract information about t

FIG. 4. Plots of the effective exponents against 10/t for the type-C interface dynamics of the symmetric IMD model at criticality.

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relaxation time exponentn i . From the elementary scalintheory, one can expect the short time behavior of the dimconcentration at criticality as

r~ t !;t2b/n i. ~4!

The characteristic~relaxation! time t for a finite system is

FIG. 5. Plots of the effective exponents against 10/t for the type-D interface dynamics of the symmetric IMD model at criticality.

FIG. 6. P(t) vs 10/t for the type-D interface dynamics of thesymmetric IMD model.

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t;Ln i /n'. ~5!

In Fig. 9, we plotr at pc versus time for various systemsizes. Investigating the slopes in this double-logarithmic pwe estimateb/n i50.165(10). The double-logarithmic plo~Fig. 10! for the characteristic timet versus the system sizL shows a straight line from which we obtainn i /n'

51.45(10), which is consistent with the above results aagrees reasonably well with the standard DP value1.580~2!.

Similar results are obtained for the system with a weasymmetry-breaking field. Therefore we conclude thatsymmetry-breaking field in the system with two absorbistates is relevant in determining the universality classmakes the system behave like having a single absorstate. In fact, the number of stationary runs falling into tunpreferred absorbing state compared with the numbethose into the preferred state vanishes exponentially intem size~see Sec. IV A!. The preferred absorbing state bhaves as a unique absorbing state of the system.

FIG. 7. Plots of log10@ r(L/2)/ r(L)#/ log10(2) vs p for theasymmetric IMD model with the symmetry-breaking fieldh50.5for various system sizesL564, 128, 256, and 512.

FIG. 8. The average concentration of dimersr at criticality inthe quasisteady state against the system sizeL in a double-logarithmic plots for various system sizesL5322512 for theasymmetric IMD model withh50.5. The solid line is of slope20.243.

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B. Defect dynamics

When the symmetry between the absorbing states isken, the defect dynamics are sensitive to the initial confirations. One can start with a lattice with a defect, either inpreferred absorbing state or in the unpreferred absorbstate. The latter case does not show critical spreading ofactive region. The domain of the preferred absorbing sgrows at the center of the active region while the regionthe unpreferred absorbing state recedes with a consspeed. This dynamics is much like two interface dynambetween the preferred and unpreferred absorbing stwhere two interfaces move in the opposite direction. We wdiscuss the interface dynamics in the next subsection.

We choose the initial configuration with a defect at tcenter in the preferred absorbing state. We seth50.5 forconvenience. Our estimates for the critical probability athe dynamic scaling exponents arepc50.4138(3),d50.163(5), h50.315(5), andz51.265(5) ~see Fig. 11!.These values are also improved compared with our prevresults@29# and are in excellent accord with the standard Dvalues;d50.1596(4), h50.3137(10), andz51.2660(14).

C. Interface dynamics

As the two absorbing states are not probabilisticaequivalent, the dynamics of the interface between the pferred and unpreferred absorbing states shows a comple

FIG. 9. Time dependence of the average concentration of dimat criticality for various system sizesL5322512 for the asymmet-ric IMD model with h50.5. The solid line is of slope20.165.

FIG. 10. Size dependence of the characteristic time at criticafor various system sizesL5322512 for the asymmetric IMDmodel withh50.5. The solid line is of slope 1.45.

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different behavior from the symmetric case. In Fig. 12, evlutions of the interface below, at, and above criticality ashown. Forp,pc , the active region grows linearly in timin both directions. Of course, the left interface front near

FIG. 11. Plots of the effective exponents against 10/t for thedefect dynamics of the asymmetric IMD model withh50.5. Threecurves from top to bottom in each panel correspond top50.4132,0.4138, and 0.4144.

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preferred absorbing state~we call it P-interface front! movesslower than the right interface front~U-interface front! nearthe unpreferred absorbing state due to the symmebreaking field. At the critical point, theP-interface front be-haves like a critical interface~zero velocity with nontrivialtemporal scaling! as in the critical defect dynamics, but thU-interface front still moves with a finite velocity. As wincreasep further into the absorbing phase, the active regcannot grow and the preferred absorbing state dominaThe P-interface front moves with the same velocity and tsame direction as theU-interface front. The width of theactive region is finite, in contrast to its diffusive behaviorthe absorbing phase for the symmetric case. Here, wesider the critical case only.

In the symmetric case, four different interface dynamare introduced. In the ordinary~type-A) interface dynamics,the exponentz/2 converges to a trivial value of unity, due tthe ballistic nature of theU-interface front even at criticality.The type-B interface dynamics also involve the dynamicstheU-interface front, so it does not yield nontrival valuesthe exponents. The type-C and type-D interface dynamicsfocus only on the profile between the initial position of thinterface ~the center! and theP-interface front which be-haves in a critical fashion.

For the type-C dynamics, we run 23104 independentsamples up to 33104 time steps atpc50.4138 withh50.5.Our estimates ared50.37(1), h50.11(1), andz51.27(1)~Fig. 13!. As expected, the value ofz agrees well with theDP value and these exponents satisfiy the generalized hyscaling relation for the DP universality class, i.e.,d1h50.48(2) is in excellent accord with the standard DP vaof 0.473~1!.

Similarly, we find d50.02(2), h50.45(1), andz51.23(2) for the type-D dynamics~Fig. 14!. Again thesevalues agree well with the DP values. The concept of ccentrating only on the relevant interface front may be applto other types of models in which many types of interfaccoexist and some of them are not equivalent.

IV. OTHER MODELS

A. Probabilistic celluar automata

Grassberger, Krause, and von der Twer@13,14# studiedtwo models of probabilistic cellular automata~PCA!,

FIG. 12. Evolutions of the asymmetric IMD interface dynamics for~a! p,pc , ~b! p5pc , and~c! p.pc . The region of the preferred~unpreferred! absorbing state is shown in black~grey!. The active sites are represented by white pixels.

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namely, modelsA andB about ten years ago. TheA modelevolves with rule number 94 in the notation of Wolfram@37#except 110 and 011 configurations, where the central spflips to 0 with probabilityp and remains unflipped with probability 12p. TheB model evolves with rule number 50 except 110 and 011 configurations, where the central spiflips to 0 with probability 12p and remains unflipped withprobability p. These are the first models investigated, whare not in the DP universality class. Both models have tequivalent absorbing states, i.e., (1010•••) and (0101•••),and exhibit an absorbing phase transition that belongs toDI universality class. But these models behave differentlythe absorbing phase. Once the system enters into one otwo absorbing states, it remains in that state forever in moA but oscillates from one state to the other in modelB. Inspite of the discriminating behavior, these models belongthe same DI universality class. In the kink representation,total number of kinks are conserved modulo 2 in the dynaics.

First, consider theA model. We can introduce thsymmetry-breaking field which makes the system to pre(1010•••) to (0101•••). The system must go through a 11

FIG. 13. Plots of the effective exponents against 10/t for thetype-C interface dynamics of the asymmetric IMD model forh50.5 at criticality.

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configuration right before entering into an absorbing stai.e., (•••010111010•••)→(•••010101010•••). If the flip-ping probability of the central spin in the 111 configuratiodepends on the evenness or oddness of the position ocentral spin, the symmetry between two absorbing statesbe broken. With probabilityh, we reject the flipping attempof the central spin in the 111 configuration when it is ateven-numbered site. Then the absorbing state with 1’s atodd-numbered sites is probabilistically preferable toother one. Again the parity in the total number of kinksconserved even with the symmetry-breaking fieldh.

We run the defect and type-C interface dynamics withh50.1. Our estimates for the defect dynamics arepc50.2435(4), d50.1625(25), h50.310(5), and z51.245(5)~Fig. 15!. These values agree very well with thstandard DP values. For the type-C interface dynamics acriticality, we find d50.372(2), h50.115(5), and z51.275(5) ~Fig. 16!, which agree with the results for thIMD model with the symmetry-breaking field~see Sec.III C !. So the value ofd in the type-C dynamics seems to buniversal. In order to check whether the value ofd in thetype-C dynamics is universal in the symmetric case, we r

FIG. 14. Plots of the effective exponents against 10/t for thetype-D interface dynamics of the asymmetric IMD model forh50.5 at criticality.

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the type-C interface dynamics whenh50 and findd50.395(5), h520.08(2), and z51.17(2) ~Fig. 17!,which is consistent with those found in the IMD model witout the symmetry-breaking field~see Sec. II B!.

We also run stationary simulations at criticality with siL532 up to 512 withh50.1. Our estimates for the steadstate exponents areb/n'50.245(5), b/n i50.155(5), andn i /n'51.63(5) ~Fig. 18!, which also agree reasonably wewith the DP values. We measure the number of stationruns falling into the unpreferred and preferred absorbstates respectively, i.e.,Nu and Np at criticality in the longtime limit. The ratioR5Nu /Np versus system sizeL is plot-ted in Fig. 19. This ratio vanishes exponentially in systsize;R;exp(2L/L0) with L0.13. It means that the chancof entering into the unpreferred absorbing state is negligso the system behaves like having a single absorbing stathe preferred one.

For the B model, the situation is quite differenThe system oscillates between the two absorbstates; (1010•••)↔(0101•••). In this model, the systemmust go through a 000 configuration right before enting into an absorbing state, i.e., (•••101000101•••)

FIG. 15. Plots of the effective exponents against 10/t for thedefect dynamics of the asymmetric PCAA model with h50.1.Three curves from top to bottom in each panel correspond tp50.2443, 0.2435, and 0.2427.

ryg

leof

g

-

→(•••010101010•••). We can introduce a rejection probability discriminating the even- and odd-numbered sisimilar to that in theA model, but it cannot make the twabsorbing states probabilistically unequivalent due to thecillatory nature in the dynamics. In fact, it is meaninglessdistinguish the two absorbing states without any dynambarrier. In theA model, there exists a dynamic barrier btween the two absorbing states, which makes the systeman infinitely long time to hop from near one absorbing stto near the other absorbing state. This dynamic barriesimilar to the free energy barrier between two ground stain the equilibrium Ising model. Without this barrier, thereno way to distinguish the two absorbing states. Thereforseems impossible to find the crossover from the DI to theuniversality class in the B model.

B. Nonequilibrium kinetic Ising model

Nonequilibrium kinetic Ising model~NKI ! recently intro-duced by Menyha´rd @15–17# evolves with the competing effect of spin flips at zero temperature (T50) and nearest-neighbor spin exchanges atT5`. The spin-flip dynamics

FIG. 16. Plots of the effective exponents against 10/t for thetype-C interface dynamics of the asymmetric PCAA model withh50.1 at criticality.

thicsquiti

gpWgy

ersterup

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al

the

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occurs with probabilityp and spin-exchange dynamics wi12p. The competition between the two different dynamat different temperatures drives the system into an nonelibrium steady state and there is a continuous phase transas the competition parameterp varies.

In theT50 spin-flip dynamics, the system evolves tryinto lower the energy. A spin is allowed to flip only if the flilowers the energy of the system or leaves it unchanged.use the parameterr to distinguish the cases when the eneris lowered or unchanged. We flip a spin with probabilityr inthe former case and flip a spin freely in the latter case. Hwe setr 50.5. In theT5` spin-exchange dynamics, neareneighbor spins are freely exchanged regardless of the enchange. Any up-down pair of spins can flip to the down-pair of spins if they are in the nearest neighbor.

The absorbing phase consists of two completely fermagnetically ordered states that are equivalent. One caflip a spin in these absorbing states because it increaseenergy. These absorbing states are the same as the twgenerate ground states of the equilibrium ferromagnetic Ismodel. The absorbing transition of the NKI model belongsthe DI universality class. In terms of ordinary domain-w

FIG. 17. Plots of the effective exponents against 10/t for thetype-C interface dynamics of the symmetric PCAA model at criti-cality.

i-on

e

e,-gy

-ot

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ol

FIG. 18. The kink density in the quasisteady state againstsystem size, time dependence of the kink density atL5512, andsize dependence of the characteristic time are plotted for the asmetric PCAA model withh50.1 at criticality. The solid lines areof slope20.245,20.155, and 1.63 from top to bottom.

FIG. 19. The semilogarithmic plot for the raioR against systemsizeL. The solid line isR51.44 exp(2L/12.9).

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n

lit

al

ver

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language, a domain wall between different spins is intpreted as a walker in the BAW model. Then the NKI modcan be mapped exactly to the BAW model with two ospring with a control parameter for the two walker annihition process@27,38#.

We introduce a symmetry-breaking field that prefersspins over down spins. So this field plays like an extermagnetic field in the equilibrium Ising model. For convnience, we define the symmetry-breaking fieldh as a prob-ability of not allowing an up spin to flip. Therefore the asorbing state with all spins up becomes the preferabsorbing state. Again the parity in the total number of dmain walls is still conserved even with the symmetrbreaking field.

We run the defect dynamics and stationary simulatiowith h50.1. Our estimates arepc50.190(5), d50.17(1),h50.32(1), andz51.25(5)~Fig. 20!, which agree well withthe standard DP value. Stationary simulations at criticayield b/n'50.24(1), b/n i50.155(5), andn i /n'51.50(5)~Fig. 21!, which also agree reasonably well with the DP vues.

FIG. 20. Plots of the effective exponents against 10/t for thedefect dynamics of the asymmetric NKI model withh50.1. Threecurves from top to bottom in each panel correspond top50.18,0.19, and 0.20.

r-l

-

pl

d-

s

y

-

V. SUMMARY AND DISCUSSION

All models studied in this paper~IMD, PCA, NKI! pre-serve the parity of the total number of kinks but cross ofrom the directed Ising~DI! to directed percolation~DP! uni-versality class when the parity-conserving symmetbreaking field is introduced. As we argued in our preliminapaper@29#, the essential factor that determines the university class of a nonequilibrium absorbing phase transitionnot the conservation laws in dynamics but the symmetrytween absorbing states.

We take a careful look at various kinds of kinks in themodels. First, consider the NKI model, which is the simpleone in the domain wall~or kink! representation. In the NKImodel, a kink is assigned between two neighboring spinsthe opposite direction. Only one type of kink exists in tNKI model and there is a two-to-one mapping between sconfigurations and kink configurations. The two absorbstates correspond to the vacuum configuration in the krepresentation. The evolution dynamics conserves the t

FIG. 21. The kink density in the quasisteady state againstsystem size, time dependence of the kink density atL5512, andsize dependence of the characteristic time are plotted for the asmetric NKI model withh50.1 at criticality. The solid lines are oslope20.24,20.155, and 1.50 from top to bottom.

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number of kinks modulo 2. By identifying a kink as a walkin the BAW model, the NKI model can be exactly mappedthe BAW model with two offspring with a control parametfor the two walker annihilation process@27,38#. Numericalresults @22,26# and recent field theoretical works for thBAW models@39,40# suggest that the parity conservationresponsible for the DI universality class. The symmetbreaking field in the NKI model cannot be represented blocal kink operator in the field theory language, similar to tmagnetic field in the equilibrium Ising model in the Blocwall representation. Therefore the recent field theoreticalsults by Cardy and Ta¨uber @39,40# do not apply when thesymmetry-breaking field is introduced. In order to considthe symmetry-breaking field, one should include the lonrange string operator, i.e., the global product of the numoperators of kinks in the quantum Hamiltonian, which bcomes a highly nontrivial problem. Our numerical resusuggest that this long-range string operator is relevantmakes the system leave a DI fixed point and flow into afixed point by the renormalization-group transformations.

The PCA models are similar to the NKI models evthough there seems no trivial mapping into the BAW moels. These models contain two types of kinks, which aresigned between two neighboring 1’s and 0’s, respectivThere is no kink between 1 and 0, so the two absorbing stcorrespond to the vacuum in the kink representation. Duthe parallel updating procedure, it is not easy to examwhether these two types of kinks can be represented bykink operator in the field theory. But the parity of the totnumber of kinks is conserved during the evolution. Tsymmetry-breaking field that discriminates the evenumbered and odd-numbered sites should be also resented by a long-range string operator like that in the Nmodel. However, theB model does not have any dynambarrier between two absorbing states due to the oscillanature, so it is not affected by the symmetry-breaking fieExcept for that, we can draw the same conclusion forPCA models as in the NKI model.

The IMD model has a more complex kink representatdue to its multicomponent nature. Three different typeskinks are found betweenB andB, B and 0, 0 and 0, whereBis a site occupied by a dimer atom and 0 is a vacancy.parity of the total number of kinks is conserved and tsymmetry-breaking field is similar to that in the PCA moels. Our numerical results show that the IMD model exhibthe same critical behavior as in the NKI and PCA modelmay imply that the differences between various kinks inIMD and PCA models are just irrelevant details that do naffect the universal behavior. It may be interesting to stuthese differences in the field theoretical models.

Recently, a few other models have been introduced wequivalent absorbing states. Those are generalized monomonomer models studied by Bassler and Browne@30–32#,and generalized Domany-Kinzel models and generalicontact processes studied by Hinrichsen@33#. These modelsare multicomponent models so there are many typeskinks. Unlike the IMD model, there appears no explicit paity conservation law in the kink representation of these mels, even though they all have two equivalent absorbstates. Numerical simulations showed that these modelslong to the DI universality class. By introducing a symmetr

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breaking field, these models cross over from the DI touniversality class as in the models studied in this papThese results strongly support our conclusion that the smetry between absorbing states, not the conservation lawthe essential property to determine the universality classthe absorbing phase transitions. However, we do not excthe possibility of the hidden conservation law in the kindynamics of these models. In their absorbing phase, numcal simulations show that large domains of two different asorbing states are formed and active regions betweendifferent domains~domain walls with finite width! surviveand diffuse until they annihilate pairwise@33#. It implies thatthere may be an effective parity conservation law in domwalls, even though there is no parity conservation in micscopic kinks. With the symmetry-breaking field, large dmains of the unpreferred absorbing state completely dispear and domain walls~active regions! annihilate bythemselves.

In this paper, we also introduce and investigate variointerface dynamics. Without the symmetry-breaking fiewe find new universal exponents ford in the type-B andtype-C interface dynamics, but the hyperscaling relationthe DI universality class is always intact. With thsymmetry-breaking field, the interface moves, in averagwith a constant velocity in the direction of the unpreferrabsorbing state. By focusing only on the side of the preferabsorbing state, we find new exponents ford for the type-Cand type-D dynamics and the hyperscaling relation for tDP universality class is obtained. These new exponentsalso shown to be universal. These types of interface dynics should be useful in studying some other models wmany but inequivalent absorbing states.

It is interesting to compare the two-dimensional equilrium Ising model with the one-dimensional NKI model. Itwell known that two-dimensional equilibrium models are rlated to one-dimensional kinetic models via transfer maformalism @41#. An extra space dimension is interpretedthe time dimension in one-dimensional kinetic models. Ocan write down the evolution operator of the kinetic modcorresponding to a given two-dimensional equilibriumodel. This evolution operator is Hermitian for the equilirium model. General nonequilibrium kinetic models in odimension can be obtained by modifying the above Hermian evolution operator in a non-Hermitian form, i.e., breaing the detailed balance. Then equilibrium models ind di-mensions and nonequilibrium kinetic models ind21dimensions can be directly compared.

The NKI model is a special case of general nonequilrium kinetic Ising models. In the NKI model, the time revesal symmetry is broken completely and its dynamics favone time direction over the other. Therefore, by addingdirectional sense in the time direction to the equilibriuIsing model, one can see the crossover from the equilibrIsing universality class to the nonequilibrium DI universaliclass.

Similar things happen for the models with a single asorbing state that belong to the DP universality class. Tpercolation problem is equivalent to theq→1 limit of theq-state Potts model@42#. The DP problem is defined by adding a directional sense to the percolation problem. Similaone can define nonequilibrium models withq equivalent ab-

et-

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sorbing states by adding a directional sense to theq-statePotts model. Both models have the permutation symmbetweenq ground~or absorbing! states, so these nonequilibrium models may be called as theq-state directed Pottsmodel. In this sense, the NKI model may be called therected Ising model and the NKI universality class as therected Ising~DI! universality class. It will be interesting toset up and investigate the directed Potts models withq>3@33#. Of course, other types of generalized models withrotational or cyclic symmetry are also interesting.

There is a difference in connecting models with differevalues ofq. Crossover from the DI to the DP universaliclass is obtained by introducing a symmetry breaking fieBut the Ising phase transition disappears when the magn

r,

r-

ry

i-i-

e

t

.tic

field is applied to the two-dimensional equilibrium Isinmodel. The percolation universality class appears othrough the random cluster formulation of the Potts mo@43#. Therefore the analogy between theq-state Potts modeand theq-state directed Potts model is not complete. Thesimilarities and differences should be further investigatedthe future.

ACKNOWLEDGMENTS

This work was supported in part by the Korea Science aEngineering Foundation through the SRC program of SNCTP, and by the academic research fund of the MinistryEducation, Republic of Korea~Grant No. 97-2409!.

E

es,

a

0

v.

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