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arXiv:cond-mat/9809081v1 [cond-mat.stat-mech] 3 Sep 1998
Interacting Monomer-Dimer Model with Infinitely Many Absorbing States
WonMuk Hwang
Department of Physics, Boston University, Boston, MA 02215.
Hyunggyu Park
Department of Physics, Inha University, Inchon 402-751, Korea
(February 7, 2008)
We study a modified version of the interacting monomer-dimer (IMD) model that has infinitely
many absorbing (IMA) states. Unlike all other previously studied models with IMA states, the
absorbing states can be divided into two equivalent groups which are dynamically separated infinitely
far apart. Monte Carlo simulations show that this model belongs to the directed Ising universality
class like the ordinary IMD model with two equivalent absorbing states. This model is the first
model with IMA states which does not belong to the directed percolation (DP) universality class.
The DP universality class can be restored in two ways, i.e., by connecting the two equivalent groups
dynamically or by introducing a symmetry-breaking field between the two groups.
PACS numbers: 64.60.-i, 64.60.Ht, 02.50.-r, 05.70.Fh
A wide variety of nonequilibrium systems with a sin-
gle trapped (absorbing) state display a continuous phase
transition from an active phase into an absorbing phase,
which belongs to the directed percolation (DP) universal-
ity class [1–4]. Recently, systems with multiple absorbing
states have been investigated extensively. The interact-
ing monomer-dimer(IMD) model introduced by one of
us [5] is one of many models that have two equivalent
absorbing states [6–9]. These models belong to a differ-
ent universality class from DP. By the analogy to the
equilibrium Ising model that has two equivalent ground
states, this new class is called as the directed Ising (DI)
universality class [10]. When the (Ising) symmetry be-
tween the absorbing states is broken in the sense that one
of the absorbing states is probabilistically preferable, the
system goes back to the DP class [11]. Hence, the sym-
metry between the absorbing states is the key factor in
determining the universality class of models with several
absorbing states. Unfortunately, no models with higher
symmetries than the Ising symmetry (like the three-state
Potts symmetry) are found to have a stable absorbing
phase as yet.
In contrast, systems with infinitely many absorbing
(IMA) states are far less understood. All IMA systems
studied so far belong to the DP universality class [12,13].
The number of absorbing states of these IMA systems
grows exponentially with system size but there is no
clear-cut symmetry among absorbing states. Recently,
it was argued that the IMA models should belong to the
DP class unless they possess any extra symmetry among
absorbing states [14,15]. However, no IMA model with
an additional symmetry has been studied up to date and
the role of the symmetry in the IMA systems is still un-
clear.
In this Letter, we introduce an IMA model with the
Ising symmetry between two groups of absorbing states.
These two groups of absorbing states are equivalent and
dynamically separated infinitely far apart.
words, an absorbing state in one group can not be reached
from any absorbing state in the other group by a finite
number of successive local changes.
nite dynamic barrier among absorbing states inside each
group. This dynamic barrier is similar to the free energy
barrierbetween ground states of equilibrium systems that
exhibit spontaneous symmetry breaking in the ordered
phase. Our numerical simulations show that this model
belongs to the DI universality class. Furthermore, we find
that this model crosses over to the DP class by allowing
that the two absorbing groups are connected dynami-
cally and/or by introducing a symmetry-breaking field
to make one absorbing group probabilistically preferable
to the other.
Our model is a modified version of the ordinary IMD
model that we call the IMA-IMD model. Dynamic rules
of the IMA-IMD model are almost the same as those of
the IMD model with infinitely strong repulsion between
the same species in one dimension [5]. A monomer (A)
cannot adsorb at a nearest-neighbor site of an already-
occupied monomer (restricted vacancy) but adsorbs at
a free vacant site with no 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. There are no nearest-
neighbor restrictions in adsorbing particles of different
species. Only the adsorption-limited reactions are con-
sidered. Adsorbed dimers dissociate and a nearest neigh-
bor of the adsorbed A and B particles reacts, forms the
AB product, and desorbs the catalytic surface immedi-
ately. Differentiation between the IMA-IMD model and
the IMD model comes in when there is an A adsorption
attempt at a vacant site between an adsorbed A and an
adsorbed B. In the IMD model, we allow the A to ad-
sorb and react with the neighboring B, so there are two
equivalent absorbing states comprised of only monomers
In other
There is no infi-
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at alternating sites, i.e., (A0A0···) and (0A0A···) where
“0” represents a vacancy. In the IMA-IMD model, this
process is disallowed. Then, any configuration can be
an absorbing state if there are no nearest neighbor pair
of vacancies and no single vacany between two B parti-
cles, e.g., (···B0A0BB0A0A···). To impose the Ising
symmetry between the absorbing states, we introduce
the probability s of spontaneous desorption of a near-
est neighbor pair of adsorbed B particles. At finite s,
an absorbing configuration cannot have this BB pair.
Hence only those configurations that have particles at
alternating sites and no two B’s at consecutive alternat-
ing sites become absorbing states, e.g., (A0A0B0A0···)
and (0A0A0B0A···). The absorbing states are divided
into two groups with particles occupied at odd- and even-
numbered sites (O group and E group). The number of
absorbing states in each group grows exponentially with
system size and there is a one-to-one mapping between
absorbing states in two groups. It is clear that one can
not reach from an absorbing state in one group to an ab-
sorbing state in the other group by a finite number of suc-
cessive local changes. Any interface (active region) be-
tween two absorbing states in the different groups never
disappears by itself in a finite amount of time, so there
is an infinite dynamic barrier between the two groups.
These interfaces annihilate pairwise only.
The order parameter characterizing the absorbing
phase transition is the density of active sites or kinks
(domain walls). In the IMD model, the dimer density
served well as the order parameter, but it cannot do in
this model. We use the kink density as the order param-
eter. Kinks are defined such that all absorbing configura-
tions have no kinks but any local change of the absorbing
configurations should produce kinks. In this model, one
should examine, at least, three adjacent sites to check
the existence of kinks. There are 13 possible configura-
tions for three adjacent sites. We assign a kink to eight
different configurations; 000, 00A, A00, B00, 00B, B0B,
BB0, and 0BB. Five others, A0A, A0B, B0A, 0A0, and
0B0, do not have a kink. In this kink representation,
there is no mod(2) conservation of the total number of
kinks.
Three independent critical exponents characterize the
critical behavior near the absorbing transition: the order
parameter exponent β, correlation length exponent ν⊥,
and relaxation time exponent ν?[2]. Elementary scaling
theory combined with the finite size scaling theory [16]
predicts that the kink density ρ(pc,L) at criticality in the
(quasi)steady state scales with system size L as
ρ(pc,L) ∼ L−β/ν⊥. (1)
One can also expect the short time behavior of the kink
density as ρ(pc,t) ∼ t−β/ν?and the characteristic time
scales with system size as τ(pc,L) ∼ Lν?/ν⊥.
In Monte Carlo simulations, a monomer is attempted
to adsorb at a randomly chosen site with probability
(1 − s)p and a dimer with probability (1 − s)(1 − p).
With probability s, a randomly chosen nearest neighbor
pair of adsorbed B’s (if there is any) is desorbed from
the lattice. We choose the dimer desorption probabil-
ity s = 0.5 and run stationary Monte Carlo simulations
starting with an empty lattice with size L = 25up to
211. The system reaches a quasisteady state first and
stays for a reasonably long time before finally entering
into an absorbing state. We measure the kink density
in the quasisteady state and average over many survived
samples. The number of samples varies from 2 × 105for
L = 25to 2×103for L = 211. The number of time steps
ranges from 103to 2 × 105.
From Eq. (1), we expect the ratio of the critical kink
densities for two successive system sizes ρ(L/2)/ρ(L) =
2β/ν⊥, ignoring corrections to scaling. This ratio con-
verges to unity in the active phase (p < pc) and to 2
in the absorbing phase (p > pc) in the limit L → ∞.
We plot the logarithm of this ratio divided by ln2 as a
function of p for L = 26,···,211in Fig. 1. The cross-
ing points between lines for two successive sizes converge
slowly due to strong corrections to scaling. In the limit
L → ∞, we estimate the crossing points converge to the
point at pc= 0.425(4) and β/ν⊥= 0.49(3). The value of
β/ν⊥agrees well with the standard DI value 0.50.
In Fig. 2, we show the time dependence of the criti-
cal kink densities ρ(pc,t) for various system sizes with
pc = 0.425. From the slope of ρ(pc,t) we estimate
β/ν?= 0.275(5). Insets show the size dependence of the
relaxation time τ(pc,L) and the steady-state kink den-
sity ρ(pc,L) at criticality. We estimate ν?/ν⊥= 1.74(4)
and β/ν⊥ = 0.494(6), respectively. All of these results
are in excellent agreement with the DI values.
We run dynamic Monte Carlo simulations with various
initial configurations and get a more precise estimate of
the critical probability pc= 0.425(1). Our estimates for
the dynamic scaling exponents are δ + η = 0.28(1) and
z = 1.14(1) [17], where δ +η characterizes the growth of
the number of kinks averaged over survived samples and
z the spreading of the active region [2]. These values are
also in excellent agreement with the DI values.
To check the importance of the Ising symmetry among
the absorbing states, we introduce a symmetry breaking
field such that the monomer adsorption attempt at an
even-numbered site is rejected with probability h [11].
For finite h, the O group of absorbing states is proba-
bilistically preferable to the E group. We set h = 0.1 and
run stationary Monte Carlo simulations for lattice sizes
L = 25up to 29. In Fig. 3, we plot ln[ρ(L/2)/ρ(L)]/ln2
versus p, from which we estimate pc = 0.304(2) and
β/ν⊥ = 0.24(1). The value of β/ν⊥ is clearly different
from the DI value but agrees well with the standard DP
value 0.2524(5). More detailed study including dynamic
Monte Carlo simulations confirms that the systems with
finite h belong to the DP universality class [17].
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Similar to the case of the ordinary IMD model,
the symmetry-breaking field makes the system behave
like having only one (preferred) group of absorbing
states [10]. Evolutions of the critical interfaces (active
region) (a) for the symmetric case (h = 0) and (b) for
the asymmetric case (h = 0.1) are shown in Fig. 4. In
the symmetric case, the interfaces between the O and
E group of absorbing states diffuse until they meet and
form a loop to disappear, which is the essential charac-
teristic of the DI universality class. In the asymmetric
case, the absorbing region of the unpreferred (E) group
quickly vanishes and the interfaces between the differ-
ent groups become irrelvant. The interfaces inside the
preferred (O) group, which can disappear by themselves
without forming loops, become dominant and force the
system into the DP universality class.
When the desorption process of a nearest neigh-
bor BB pair is forbidden (s = 0), the system can
find many more absorbing states with BB pairs, e.g.,
(···B0A0BB0A0A···), in addition to the two groups of
the absorbing states for s ?= 0. These new extra ab-
sorbing states are generically mixtures of the O and E
group of the absorbing states. The O and E groups are
now connected dynamically via new mixture-type ab-
sorbing states. Consider a configuration with an inter-
face between two absorbing states in the different groups,
(···B0A0000A0A···), where the interface is placed in
two central vacancies 00. With nonzero s, this configu-
ration never evolves into an absorbing state. However,
in the case of s = 0, it can evolve into a mixture-type
absorbing state by adsorbing a dimer BB in the center.
Actually, any interface can disappear by itself in a finite
amount of time, so there is no infinite dynamic barrier
between absorbing states. Therefore the evolution of the
interfaces resembles the asymmetric case in Fig. 4.
Absorbing states for s = 0 no longer possess the clear-
cut global symmetry which drives the system into the DI
class. So we expect the system falls into the DP class
like the other IMA models without an extra symmetry.
We run dynamic Monte Carlo simulations starting with a
lattice occupied by monomers at alternating sites except
at the central vacant site, (···A0A000A0A···), where
0 represents a defect. In Fig. 5, we plot three effective
exponents against time; δ(t), η(t), and z(t) [2]. Off crit-
icality, these plots show some curvatures.
of the dynamic scaling exponents can be extracted by
taking the asymptotic values of the effective exponents
at criticality. From Fig. 5, we estimate pc = 0.105(1),
δ = 0.02(1), η = 0.48(5), and z = 1.33(5). The values
of δ + η and z are in good agreement with those of the
DP values [18]. Introduction of the symmetry breaking
field h only changes the location of pc. Stationary Monte
Carlo simulations also confirm our results [17].
In summary, we found the first IMA model that does
not belong to the DP class, but belong to the DI class.
This can be achieved by imposing a global Ising symme-
The values
try on the absorbing states, i.e., making the two equiv-
alent group of IMA states that are dynamically sepa-
rated infinitely far apart. When the symmetry between
these groups is broken, one group of absorbing states be-
comes completely obsolete and the evolution morphology
changes from a loop-like to a tree-like structure, which
ensures the system in the conventional DP class. We also
found that the system goes back to the DP class if the
mixture-type absorbing states between the two groups
are added. These extra absorbing states connect the two
separated groups dynamically and make the loop-forming
process of the interfaces irrelvant. The absorbing states
in all other previously studied IMA models are dynam-
ically connected in the sense as mentioned above. This
may explain why those models belong to the DP class.
HP wishes to thank M. den Nijs for his hospitality
during his stay at the University of Washington where
this work was completed. This work was supported by
Research Fund provided by Korea Research Foundation,
Support for Faculty Research Abroad (1997).
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librium phase transitions in lattice models (Cambridge
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373 (1979).
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A 40, 4820 (1989).
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H. Park, M. H. Kim and H. Park, Phys. Rev. E 52, 5664
(1995).
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A 17, L105 (1984); P. Grassberger, J. Phys. A 22, L1103
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[7] N. Menyh´ ard, J. Phys. A 27, 6139 (1994); N. Menyh´ ard
and G.´Odor, J. Phys. A 28, 4505 (1995).
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4094 (1996); Phys. Rev. E 55, 5225 (1997); K. S. Brown,
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E 57, 6438 (1998).
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27, L61 (1994); I. Jensen and R. Dickman, Phys. Rev. E
48, 1710 (1993).
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[15] M. A. Mu˜ noz, G. Grinstein, R. Dickman, and R. Livi,
Phys. Rev. Lett. 76, 451 (1996).
[16] T. Aukrust, D. A. Browne, and I. Webman, Phys. Rev.
A 41, 5294 (1990).
[17] Detailed numerical results will be published elsewhere.
[18] The values of the exponents, δ and η, depend on initial
configurations, but their sum is universal [13].
0.4050.4100.4150.4200.4250.4300.4350.4400.4450.450
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
L=64
L=128
L=256
L=512
L=1024
L=2048


ln[ ρ(L/2)/ρ(L) ] / ln2
p
FIG. 1. Plots of ln[ρ(L/2)/ρ(L)]/ln2 versus p for the sym-
metric case.
567
log2(L)
89 10 11
-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0


log2 ( ρ )
567
log2(L)
89 10 11
6
8
10
12
14
16


log2(τ)
1 10100100010000100000
0.1
1


ρ(t)
t
FIG. 2.
pc = 0.425.
Insets show the size dependence of the relaxation time τ and
the steady-state kink density ρ at criticality. The solid lines
are of slope 1.74 (= ν?/ν⊥) and −0.494 (= −β/ν⊥).
The time dependence of the kink density at
The straight line is of slope 0.275 (= β/ν?).
0.2950.3000.3050.3100.3150.320
0.1
0.2
0.3
0.4
0.5
L=64
L=128
L=256
L=512


ln[ρ(L/2)/ρ(L)] / ln2
p
FIG. 3. Plots of ln[ρ(L/2)/ρ(L)]/ln2 versus p for the
asymmetric case with h = 0.1.
FIG. 4.
symmetric case and (b) the asymmetric case. Monomers (A)
are represented by black, dimer particles (B) by grey, and
vacancies by white pixels.
Evolutions of the critical interfaces for (a) the
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0.0000.001 0.0020.003
1.1
1.2
1.3
1.4
1.5
1.6


z(t)
10 / t
0.0000.0010.002 0.003
0.2
0.3
0.4
0.5
0.6
0.7


η(t)
10 / t
0.0000.0010.0020.003
-0.06
-0.04
-0.02
0.00


-δ(t)
10 / t
FIG. 5. Plots of the effective exponents against 10/t for s = 0 and h = 0. Five curves from top to botton in each panel
correspond to p = 0.100, 0.104, 0.105. 0.107, and 0.110.
5