Mapping of 2+1-dimensional Kardar-Parisi-Zhang growth onto a driven lattice gas model of dimer

0.31 MB·English

by  Geza Odor

Mapping of 2+1-dimensional Kardar-Parisi-Zhang growth onto a driven lattice gas
model of dimers
G´eza O´dor (1), Bartosz Liedke (2) and Karl-Heinz Heinig (2)
(1) Research Institute for Technical Physics and Materials Science,
P.O.Box 49, H-1525 Budapest, Hungary
(2) Institute of Ion Beam Physics and Materials Research
Forschungszentrum Dresden - Rossendorf
P.O.Box 51 01 19, 01314 Dresden, Germany
9
0
Weshowthata2+1dimensionaldiscretesurfacegrowthmodelexhibitingKPZclassscalingcan
0
be mapped onto a two dimensional conserved lattice gas model of directed dimers. In case of KPZ
2
height anisotropy the dimers follow driven diffusive motion. We confirm by numerical simulations

n thatthescalingexponentsofthedimermodelareinagreementwiththoseofthe2+1dimensional
a KPZ class. This opens upthepossibility of analyzing growth models viareaction-diffusion models,
J
which allow much more efficient computer simulations.

8
2 PACSnumbers: 05.70.Ln,05.70.Np,82.20.Wt

]
h
I. INTRODUCTION but an earlier numerical work [12] does not support this
c
e claim.
m The Kardar-Parisi-Zhang (KPZ) equation [1] moti- In one dimension a discrete, restricted solid on solid
- vatedbyexperimentallyobservedkineticrougheninghas realizationof the KPZgrowthis equivalent to the asym-
at been the subject of large number of theoretical studies metric simple exclusion process (ASEP) of particles
t [2,3]. Lateritwasfoundtomodelotherimportantphys- [13, 14] (see Fig. 1).
s
. ical phenomena such as randomly stirred fluid, [4], dissi-
t
a pativetransport[5,6],directedpolymers[7]andthemag-
m netic flux lines in superconductors [8]. It is a non-linear
- stochastic differential equation, which describes the dy-
d
namics of growth processes in the thermodynamic limit
n
specified by the height function h(x,t)
o
c
[ ∂th(x,t)=v+σ∇2h(x,t)+λ(∇h(x,t))2+η(x,t) . (1) FfaIcGe.g1r:ow(Ctholoorntoonltinhee)1MdaAppSiEnPgomfothdeel.1+Su1rdfaimceenastitoancahlmseunrt-

3 (withprobabilityp)anddetachment(withprobabilityq)cor-
Here v and λ are the amplitudes of the mean and local
v responds to anisotropic diffusion of particles (bullets) along
growth velocity, σ is a smoothing surface tension coeffi-
9 the1d base space.
cientandη roughensthesurfacebyazero-averageGaus-
0
9 sian noise field exhibiting the variance
1
. η(x,t)η(x′,t′) =2Dδd(x x′)(t t′) . (2) In this discrete so-called ’roof-top’ model the heights
0 h i − − are quantized and the local derivatives can take the
1
8 Heredisusedforthedimensionofthesurface,D forthe values ∆h = 1. By considering the up derivatives
±
0 noiseamplitudeand denotesaverageoverthenoisedis- (∆h=1) as particles and down ones as holes the rough-
hi
: tribution. In 1+1 dimensions it is exactly solvable [7], ening dynamics can be mapped onto a driven diffusive
v
but in higher dimensions only approximations are avail- systemofparticleswithsinglesiteoccupancy. TheASEP
i
X able(see[9]). Ind>1spatialdimensionsduetothecom- model on the other hand is well known and its scaling
r petition of roughening and smoothing terms, models de- properties are explored (for a recent review see [15]).
a
scribedby the KPZequationexhibita rougheningphase The extension of this kind of lattice-gas analogy to
transitionbetweenaweak-couplingregime(λ<λ ),gov- higherdimensionshasnotbeenconsideredtoourknowl-
c
erned by λ=0 Edwards-Wilkinson fixed point [10], and edge. Instead hypercube stacking models were con-
astrongcouplingphase. The strongcouplingfixedpoint structed [14, 16, 17] and surface configurations were
is inaccessible by perturbative renormalization method. mapped onto the d-state Potts spins defined on the sub-
Therefore the KPZ phase space has been the subject of strate lattice itself. Especially 2 + 1 dimensional sur-
controversies and the value of the upper critical dimen- faces were shown to be related to the six-vertex model
sionhasbeendebatedforalongtime. Veryrecentlynon- with equal vertex energies [18] and to the ground-state
perturbative renormalization and mode coupling theory configurations of the anisotropic Ising model defined on
has revealed a rich phase diagram, with more than one the triangular lattice [19]. As a consequence the height-
lines offixed point solutions in the d - λ space [11]. This height correlation functions can be related to four-spin-
suggests an upper critical dimension d = 4 for KPZ, correlation functions of the spin system. Very recently
c
2
the conformal invariance of the isoheight lines has also the generalized Kawasakiupdating rules
been pointed out [20].
1 1 1 1
Here we show that a 2+1 dimensional growth model − ⇀↽ − (3)
exhibitingKPZscalingcanalsobemappedontoadriven (cid:18)−1 1(cid:19) (cid:18)1 −1(cid:19)
lattice gas. This is important from theoretical point of
with probability p for attachment and probability q for
view,becausethescalingbehaviorofdrivendiffusivesys-
detachment. We can also call the ’+1’-s as particles and
tem (DDS) has been studied intensively for a long time
the ’ 1-s as holes of the base square lattice. In this
(for a review see ref. [21]), thus results for DDS may be −
way an attachment/detachment update can be mapped
exploited to understand KPZ better and vica versa [22].
onto a single step motion of an oriented dimer in the
Furthermorethisconservedlatticegasanditsgeneraliza-
bisectrixdirectionofthexandyaxes. Tomakeaone-to-
tions with anisotropies, disorder, or higher order terms
one mapping we update the neighborhood of sub-lattice
canbe studiedeffectivelybybit-codedalgorithmsforex-
points, which are denoted by the crossing-points of the
ample.
dashed lines only.
Since the three dimensional space can’t be filled fully
by octahedra, holes can occur among them, below the
II. MAPPING ONTO LATTICE GAS IN TWO surface. Thereforethisapproximationofasurfacegrowth
DIMENSIONS may not sound to be faithful and the validity of KPZ
growth rules requires confirmation. Note, however that
in reality atoms are not cubes either and do not tile the
As a generalization of the 1+1 dimensional roof-top
three dimensional space completely. Furthermore very
model,wherethebuildingblocksaresquareslet’sputoc-
recentlyinbi-disperseballisticdepositionmodels[24,25],
tahedra on the square lattice, such that we get back the
inwhichunder-surfacevacanciesmayoccurKPZscaling
1+1dimensionalmodelinthe x ory directionasshown
has been reported as well.
onFigure2. Surfaceadsorptionordesorptioneventscor-
ThedeterministicpartoftheKPZequation(1),which
respond to attachment or detachment of octahedra, re-
can be obtained by averaging over the noise can be de-
spectively. The surface built up from the octahedra can
rived from the surface/dimer model similarly as it was
done in 1+1 dimension [13]. If we apply the transfor-
mation
v(x,t)= h(x,t) (4)
∇
we get the Burgers equation for the height profile
∂ v(x,t)=σ 2v(x,t)+λv(x,t) v(x,t) . (5)
t
∇ ∇
Oursystemis representedby twomatrices∆ and∆
x y
of sizes L L, which contain discrete derivatives +1 or
×
1inxandy direction,respectively(seeEqs.(10),(11)).
−
Intwodimensionsweintroducethevectorvariableσ =
i,j
(∆ (i 1,j),∆ (i,j 1)). Thishasthevalue(1,1)incase
x y
− −
of a dimer and ( 1, 1) for a pair of holes. By setting
FIG.2: (Color online)Mappingofthe2+1dimensionalsur- up the master eq−uati−on
face growth onto the 2d particle model (bullets). Surface at-
tachment (with probability p) and detachment (with proba- ∂ P( σ ,t) = w′ ( σ )P( σ′ ,t)
bility q) corresponds to Kawasaki exchanges of particles, or t { } i,j { } { }
Xi,j
toanisotropic diffusion of dimers in thebisectrix direction of
the x and y axes. The crossing points of dashed lines show w ( σ )P( σ ,t) (6)
i,j
− { } { }
the base sub-lattice to be updated. Thick solid/dashed lines Xi,j
on the surface show the x/y cross-sections, corresponding to
the1d model (Fig. 1.) for the probability distribution P( σ ,t), where the
{ }
prime index denotes a state as a result of a generalized
Kawasakiflip (3) the transition probability is given by
be described by the edges meeting in the up/down mid- 1
dle vertexes. The up edges in the x or y directions are wi,j({σ}) = 8[2−σi+1,j+1σi,j +λ(σi+1,j+1−σi,j)
representedby’+1’-s,whilethedownonesby’ 1’inthe (1 λ)
model. In this way a single site deposition flip−s the four − (σi+1,j+1 σi,j)2] , (7)
− 2 ×
edges and means two ’+1’ ’ 1’ (Kawasaki) exchanges:
one in the x and one in the↔y−direction. This canalso be with λ=2 p 1 parametrization. This formally looks
p+q −
understood as a special 2d cellular automaton [23] with like the one-dimensional Kawasaki exchange probability
3
(shown in [13]), except the cross-product term, which is attempts correspond to one MCS. After certain time in-
necessary to avoid surface discontinuity creation. The tervalsdataevaluationrequiresthereconstructionsofthe
cross-product as a determinant cancels updates between surface heights h (t) by summing up the sequence of
x,y
configurations like (1,1) (1, 1). The nonlinear term local slopes ∆ , ∆ .
x y
→ −
vanishes for p = q (λ = 0). The sign of the coefficient λ
of the nonlinear term can be interpreted as follows. For
p>q positive nonlinearity (positive excess velocity)it is IV. RESULTS
a consequence of growth with voids.
ToobtainEq.(5)firstoneaveragesovertheslopevec- Starting from periodic, vertically striped particle dis-
tors tribution, which corresponds to a flat initial surface we
update the particle model by the rules defined in the
σ = σ P( σ ,t) . (8)
h i,ji i,j { } previous section. At certain time steps we calculate the
X
{σ}
h (t) heights from the height differences ∆ . The
x,y x,y
Then calculating its time derivative using the master morphologyofa growingsurface is usually characterized
equation the cross-product term drops out and one ob- by its width
tains
1 L 1 L 2 1/2
2∂thσi,ji =+ hλσσi−i,1j,(j−σi1+i1−,j+21hσi,jσii−+1,hjσ−i1+)1,j.+1i (9) W(L,t)=hL2 Xx,y h2x,y(t)−(cid:16)L2 Xx,y hx,y(t)(cid:17) i .
h − i (12)
Hereonecanseethediscretesecondandfirstdifferentials In the absence of any characteristic length, growth pro-
ofσ correspondingtotheoperatorsof(5). Thesediffer- cesses are expected to show power-law behavior of the
i,j
entials are one-dimensional because the dimer dynamics correlationfunctions in spaceandheightandthe surface
is also one-dimensional. Making a continuum limit in is described by the Family-Vicsek scaling [26]
both directions and taking into account the relation of
heightandslopevariables(4)we canarriveto the deter- W(L,t) tβ, for t0 <<t<<ts (13)
∝
ministic part of the KPZ equation (1). Lα, for t>>t . (14)
s
ThisagreementdoesnotprovetheequivalenceofKPZ ∝
and the dimer model since they are just the first equa- Here α is the roughness exponent and characterizes the
tions in the hierarchy of equations for correlation func- deviation from a flat surface in the stationary regime
tions. On the other hand from universal scaling point (t >> t ), in which the correlation length has exceeded
s
of view they show the equivalence of the leading order the linear system size L; and β is the surface growth
terms. We will show by numerical simulation that our exponent, which describes the time evolution for earlier
mapping is faithful and reproduces the KPZ class sur- (non-microscopic t >> t0) times. The dynamical expo-
face growth behavior. nent z can be expressed by the ratio
z =α/β . (15)
III. THE SIMULATION ALGORITHM
In case of up-down symmetry (p = 1, q = 1) the non-
In the algorithm we extend the sequence of discrete lineartermisdropped,andtheKPZequation(1)simpli-
slopes of the 1d ASEP model (Fig. 1) to local deriva- fiestotheEdwards-Wilkinson(EW)equation[10]. Since
tives at (i,j) sites in x and y directions of the surface the upper critical dimension of this equation is: dc = 2,
(see Fig. 2). The initially flat surface is presented as a mean-fieldbehavior,characterizedbyα=β =0andlog-
regular sequence of ’+1’-s and ’ 1’-s within both matri- arithmic scaling is expected by field theory. Indeed, the
ces. Periodicboundaryconditio−nsareappliedtoxandy width of the surface grows like
direction. Thesystem’sevolutionissimulatedasfollows.
W2(t)=aln(t)+b (16)
Asite(i,j)onthesubstrateplaneisselectedrandomly.
Then, we choose an attachment or detachment attempt
as shown in Fig. 3. The prefactor a obtained by fitting
according to their probabilities p and q. Generalized
the L=1024 curve in the 20<t<1000 regionwith the
Kawasakiexchanges(3)ofattachmentordetachmentare
form (16) is a=0.152(8). This is in agreement with the
realized if
theoreticalestimatefortheEWequationD/(4πσ)[27]if
∆x(i−1,j) ∆x(i,j) = −1 1 (10) takeintoaccounttheexactvalueforthestiffnessconstant
(cid:18) ∆y(i,j 1) ∆y(i,j) (cid:19) (cid:18) 1 1 (cid:19) (or surface tension): σ¯/D = π/9. This constant was
− −
identified by [19] through the correspondence between
∆ (i 1,j) ∆ (i,j) 1 1
or x − x = − , (11) theexactcalculationofthefour-spincorrelationfunction
(cid:18) ∆y(i,j 1) ∆y(i,j) (cid:19) (cid:18)1 1(cid:19)
− − ofthe zero-temperaturetriangularIsing antiferromagnet
respectively. Throughout this paper the time is mea- [28]andthediscreteheight-heightcorrelationfunctionin
sured by Monte Carlo steps (MCS), i.e. L L jump real space in the interface model. A factor σ¯ = 2/3σ is
×
4
3.0 of [30]. Using these surface exponents and the scaling
2.5
100
2.5 nf.) 2.0
2w(i 1.5
2.0
1.0
2 101 102 L103 104
w
8
3
1.5 0.
L
w/
1.0
10−1
0.5
100 101 102 103 104 105
t
10−6 10−5 10−4 10−3 10−2 10−1 100 101 102
t/L1.67
FIG. 3: (Color online) Logarithmic surface growth in
case of up-down symmetry for different sizes L =
64,128,256,512,1024(bottomtotop). Thedashedlineshows FIG. 4: (Color online) Scaling collapse for p=1, q =0 with
thefittingwith theform (16). Inset: widthsaturation values 2 + 1 dimensional KPZ class exponents for different sizes:
for different system sizes L in thelong time limit. L=64,128,256,512,1024 (bottom to top).
law (15) we estimated the dynamical exponent to be:
coming from the 2/√3 triangular lattice site per surface z = 1.64(1), which is somewhat greater than what one
element and the 1/√3 of the octahedron/cube surface findsforthe2+1dimensionalKPZclassin[9](z =1.58).
fraction, thus the theoretical estimate is: a 0.151981. We think thatthis is due to the correctionto scaling ob-
≃
Thesaturationvaluesareexpectedtoexhibitlogarith- served in the time dependence discussed above. If we
mic growth scale the time with the dynamical exponent z = 1.64
we obtain a good scaling collapse of the growth data for
W2(inf.)= lim W2(t)=cln(L)+d (17) different sizes (Fig.4) in agreement with the (13,14) law
t→∞
again. Ourexponent estimates alsosatisfy the α+z =2
with the system size [27]. As can be seen in the inset of scaling relation within error margin. This implies that
Fig.3this reallyhappens withthe prefactorc=0.30(1), the Galilean invariance holds and the lattice model in-
which agrees with the theoretical value c = 2a 0.304 deedliesinthe 2+1dimensionalKPZuniversalityclass.
≃
again.
For pure deposition p = 1, q = 0, or in case of other
V. CONCLUSIONS AND OUTLOOK
generalup/downasymmetriccases,wesawpower-lawin-
creaseofthesurfacewidth,inagreementwiththescaling
hypothesis (13) (see Fig. 4). For the the largest system Wehavepointedoutthepossibilityofmappingofadis-
that we have investigated (L = 1024) we fitted W(t) in crete surface growth processes onto a conserved, driven
the 100<t <10000 time window with a power-law and latticegasmodeloforienteddimers,whichmoveperpen-
obtained β = 0.23(1). This value agrees quite well with dicularly in two dimensions. The straight line motion of
the numerical estimate for the 2+1 dimensional KPZ dimersinthetwodimensionalspaceisverysimilartothe
class (β = 0.24) provided in ref. [9]. Note however that motion of particles of the ASEP process. The difference
for smaller system sizes the exponent estimate is some- is that since the dimers are extended objects, their mo-
what smaller, due to corrections to scaling, but one can tion is slowed down by the dimer particle exclusion and
clearly see a convergence towards higher values and a thesub-latticeupdateascomparedtothesingleparticles
better collapse as L . Large scale simulations with ofthe ASEP.As aconsequencetheir motionisdescribed
→ ∞
an effective, bit-coded versionof our algorithmcould re- by somewhatlargerdynamicalexponent(z 1.64)than
≃
sult in very precise estimates. The systematic tendency that of the ASEP (z =3/2),so the change of z(d) seems
towardsanasymptoticbehaviorhasbeenfoundinFig.4. to be a purely topologicalphenomena in KPZ.This pro-
The saturation values W(inf.) for different system vides a better understanding of the relation of univer-
sizes also scale well with (14) and with the exponent sality classes of surface classes to those of the reaction-
α=0.38(1)ofthe2+1dimensionalKPZclass[9,29]. As- diffusionmodels. [31,32,33]. Interestinglythex/y sym-
sumingcorrectionstoscalingoftheformW A2Lα(1+ metricsurfacedynamicsmapsontoastronglyanisotropic
B2L−ω the fitting to our data resulted in ve≃ry small ef- reaction-diffusion model.
fect: α = 0.377(15), which marginally overlaps with the We have found KPZ or EW scaling by numerical sim-
value of [29] but does not support the proposal α= 2/5 ulations, hence we showed that lattice anisotropy and
5
under-surface vacancies are irrelevant. Our simulation of our code, which manipulates the two-dimensionalbit-
resultsforthe2+1dimensionalEWcasereproducedthe field by logical operations runs roughly 10 times faster
theoretically expected logarithmic scaling, with the cor- thanthecurrentversionandwillbepublishedelsewhere.
rect leading order coefficients. For the KPZ scaling our For example the Bradley-Harper [39] and the debated
roughness exponent result is in the middle of the range Kuramoto-Sivashinsky [40] models with their modifica-
obtained by various numerical exponent estimates: i.e. tionscanbeinvestigatednumericallyandarethesubject
between α=0.36 [34, 35] and the field theoretical value of a forthcoming publications.
α = 0.4 [30]. Our α = 0.377(15) coincides with that of
thenumericalstudy[36]andagreeswiththerenormaliza-
tionresultsα=0.38[37]. Itoverlapsmarginallywiththe
Acknowledgments:
simulation results α=0.393(3) [29] as well. Our growth
exponentestimateβ =0.23(1)matchestheresultsof[34]
(β = 0.221(2)) and [36] (β = 0.229(5)), obtained by in- We thank Zoltan Ra´cz and Uwe T¨auber for the useful
dependent numerical fitting procedures. The dynamical comments. Support from the Hungarian research fund
exponent of this study is also in the range provided in OTKA (Grant No. T046129), the bilateral German-
[36]. Hungarian exchange program DAAD-MO¨B (Grant Nos.
Ourmodelprovidesanefficientwayofsimulationsand D/07/00302, 37-3/2008) and from the German Science
opens us the possibility to study more complex growth Foundation(DFGresearchgroup845,projectHE2137/4-
models relevant in recent interest of self-organizing sur- 1) is acknowledged. The authors thank for the access to
face nanosystem [38]. An optimized, bit-coded version the HUNGRID.
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