Freezing into Stripe States in Two-Dimensional Ferromagnets and Crossing Probabilities in Critical Percolation

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by  Kipton Barros

Freezing into Stripe States in Two-Dimensional Ferromagnets and Crossing
Probabilities in Critical Percolation
Kipton Barros, P. L. Krapivsky, and S. Redner
Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215
When a two-dimensional Ising ferromagnet is quenched from above the critical temperature to
zero temperature, the system eventually converges to either a ground state (all spins aligned) or
9
an infinitely long-lived metastable stripe state. By applying results from percolation theory, we
0
analyticallydeterminetheprobabilitytoreachthestripestateasafunctionoftheaspectratioand
0
theformoftheboundaryconditions. Thesepredictionsagreewithsimulationresults. Ourapproach
2
generally applies to coarsening dynamicsof non-conservedscalar fieldsin two dimensions.
y
a
PACSnumbers: 64.60.My,05.40.-a,05.50.+q,75.40.Gb
M

1 We investigate the fate of a kinetic two-dimensional mately coarsens into a vertical stripe; conversely, a do-
2
Isingferromagnetfollowingaquenchfromabovethecrit- main mosaic that spans in both horizontally and verti-

ical temperature to zero temperature. Intriguingly, this cally coarsens into a ground state.
]
h evolving system can sometimes get trapped in an in-
While individually the observations (i) and (ii) are
c finitely long-lived metastable stripe state that spans the
e known,theircombineduseleadstoanalyticalpredictions
m system vertically or horizontally [1, 2, 3]. The case of for the fate of two-dimensional Ising ferromagnets. The
two dimensions is special; in one dimension the system
connection with the critical state of continuum percola-
-
t always reaches the ground state, while in three or more tion is central as it allows us to use exact results about
a
dimensions the ground state is never reached [1].
t crossing probabilities to determine the final state of the
s InthisLetterweproposeexactprobabilitiesforatwo-
. Ising system. These quantities are defined as the prob-
t dimensional Ising system to reach this metastable stripe
a abilities for the existence of a spanning cluster with a
m state. Our argument should apply to any curvature- specified topology. We denote these crossing probabili-
driven coarsening process with a non-conserved scalar
- ties for free and periodic boundary conditions as and
d order parameter, such as the time-dependent Ginzburg- F
, respectively. For a critical rectangular system with
n Landau equation [4, 5, 6]. For a square system, the P
aspectratior (ratioofheighttowidth)[10],thecrossing
o
c probability to get trapped in a vertical or a horizontal probabilities are non-trivial (i.e., strictly between 0 and
[ stripe state waspreviouslyestimated[1, 2]to be close to 1), universal functions of r [11, 12]. Beautiful exact ex-
1 13. Here we exploit the relation between coarsening and pressionsforvariouscrossingprobabilitieswereoriginally
critical percolation to argue that this probability equals
v calculatedviaconformalfieldtheory[13,14,15,16],and
1 1 √3ln27 = 0.3558... for a system with free bound- some of them have been proved in Refs. [17, 18, 19].
2 − 2π 16
2 aryconditions;thecorrespondingfreezingprobabilityfor
The connection (i) with critical percolation may seem
5 periodic boundary conditions is found to be 0.3390....
3 surprising since the initial state (with equal numbers of
. Our approach is based on two key ingredients: up and down spins) is not critical; on the square lat-
5
0 (i) Soon after the quench, the characteristic domain tice, the up spins percolate when their density exceeds
9 length scale ℓ becomes substantially larger than the lat- pc 0.5927. Hence initially neither phase percolates.
≈
0 tice spacing a while remaining much less than the sys- Howeverthe systemquickly approachesthe criticalstate
v: tem size: L, a ℓ L. At this stage, the domain ofcontinuum percolationin which the crossingprobabil-
≪ ≪
i mosaic has reached the critical point of continuum per- ities are non-trivial. To appreciate this property, con-
X
colation (Fig. 1(a)), as previously observed in various sider for concreteness domain coarsening in the square
r two-dimensional systems [7, 8]. geometry with free boundary conditions. Let be the
a v
F
probabilitythataspinupdomaincrossesvertically(with
(ii) During the coarsening regime, ℓ(t) a, the dynam-
≫ no constraint on horizontal crossing); also equals the
ics becomes deterministic and domain wall evolution is v
F
horizontal crossing probability (because the system is
driven only by local curvature [4, 5, 6]. Thus the global
square),andfurtherequalsthe crossingofspindowndo-
domain topology does not change once the coarsening
mains (due to up/down symmetry). A vertical spanning
regime is reached [9].
spin up domain exists if and only if a horizontal span-
These features imply that the ultimate fate ofthe sys-
ning spin down domain does not. (This last statement
tem is pre-determined once the critical percolation state
is predicated on the non existence of singular points in
is reached. For instance, if a domain exists that crosses
which more than two domains meet, see Fig. 2.) Hence
the system only horizontally or only vertically, a stripe
=1 ,sothat = 1,whichisindeednon-trivial.
state is reached. Figure 1 shows an example of a verti- Fv −Fv Fv 2
cal spanning domain shortly after the quench that ulti- We begin with the analytically simpler case of free
2
(a) (b) (c) (d)
FIG. 1: Coarsening in the kinetic Ising model on a 1024×1024 square lattice with periodic boundary conditions at times (a)
200,(b)1000,(c)5000,and(d)2.5×104 MonteCarlostepsfollowingaquenchfromT =∞to0. Domainsareregionsofeither
spinup(gray)orspindown(black). Aspanningspin-updomain,thateventuallycoarsensintoaverticalstripe,ishighlighted.
while the crossing probability for dual spanning satisfies
1
(r)= (1 (r) (r))= (1/r). (4)
Fhv 2 −Fhv −Fhv Fhv
The basic relation between Ising domains and critical
percolation implies that, e.g., the crossing probability
(a) (b)
coincides with the probability for the Ising system
Fhv
to freeze into a vertical stripe state as a function of r.
FIG. 2: (a) A state where 4 Ising domains meet at a single
To test this basic prediction for the freezing probabil-
point evolves into a state (b) where only two domains can
ity, we simulate the kinetic Ising model at zero tempera-
meet underzero-temperature single-spin flip dynamics.
ture using single-spin flip dynamics with the Metropolis
acceptance criterion — a spin is flipped if its energy de-
creases or remains the same as a result of the flip. To
boundary conditions. In this setting, every domain mo-
make this simulation more efficient, a list of “active”
saic either spans only vertically, only horizontally, both
spins — those whose energy will not increase upon be-
horizontally and vertically (dual spanning), or the mo-
ingflipped—ismaintainedandconstantlyupdateddur-
saicdoesnotcontainaspanningcluster. Theirrespective
ing the dynamics. In each update step an active spin
probabilities, , , ,and (thedual-spanning
Fhv Fhv Fhv Fhv is chosen at random and flipped. One Monte Carlo step
and non-probabilities are identical by up/down symme-
correspondstoeachactivespinflippingonce,onaverage.
try) therefore satisfy the normalization condition
We simulate many quenches from T = to 0 on lat-
∞
+ +2 =1. (1) tices of dimension (256/r) 256. For each value of r,
Fhv Fhv Fhv weperformed2 104 simula×tionruns,witheachstarting
×
Moreover,the exact form of is known to be [15, 16] fromadifferentrandominitialcondition. Wedefineado-
Fhv main as a connected cluster of nearest-neighbor aligned
√3 4 5 spins. Clustersareidentifiedusingaclustermultilabeling
Fhv(r)= 2π λ 3F2(cid:18)1,1,3;3,2;λ(cid:19), (2) method [21]. To determine whether a quenched system
ultimatelyfreezesintoastripestate,oneshould,inprin-
where F (a ,...a ;b ,...b ;λ)isthegeneralizedhyper- ciple, simulate until the system ceases to evolve. The fi-
p q 1 p 1 q
geometric function [20], λ=λ(r) is defined implicitly by nalstagesoftheevolutiontakeadisproportionatelylarge
amountofCPUtime,however,anditisadvantageousto
1 k 2 2K(k2) stopthesimulationwhenthedomainmosaicfirstreaches
λ= − , with r = , itsfinalstatetopology. ForthisIsingsystem,oursimula-
(cid:18)1+k(cid:19) K(1 k2)
− tions indicate that after 200 Monte Carlo steps, domain
mosaicshavereachedthetopologyofthe finalstate with
and K(u) is the complete elliptic integral of the first a probability that exceeds 0.998. Thus we may identify
kind [20]. thestateofthesystematthisearlytimeasthepredictor
The corresponding horizontal crossing probability fol- of the topology in the final state.
lows by symmetry Figure 3 shows our numerical results for the probabil-
ity for a specified ultimate fate of an Ising system with
(1/r)= (r). (3) free boundary conditions for a variety of aspect ratios
Fhv Fhv
3
y 1 y 1
g g
o o
ol ol
p p
o o
T 0.75 T 0.75
g g
n n
nni Fhv nni P0,1
Spa 0.5 2Fhv Spa 0.5 2PX
of Fhv of P1,0
y y
t 0.25 t 0.25
bili bili
a a
b b
o o
Pr 0 Pr 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Aspectratior Aspectratior
FIG. 3: Probabilities of various domain topologies in the FIG. 4: Probabilities of various domain topologies in the ki-
kinetic Ising model following a quench with free boundary netic Ising model following a quench with periodic boundary
conditions: verticalstripes(+),dual-spanningconfigurations conditions: verticalstripes(+),dual-spanningconfigurations
((cid:3)), and horizontal stripes (×). Error bars are about 1/3 ((cid:3)), and horizontal stripes (×), with error bars about 1/3
thesymbolsize. Thelatticedimensionsare(256/r)×256for of the symbol size. The lattice dimensions are (256/r)×256
variousaspectratiosbetween0and1. Alsoshownarethecor- for aspect ratios between 0 and 1. Also shown are the corre-
respondingexactpercolation crossing probabilitiesFhv,Fhv, sponding exact percolation crossing probabilities P1,0, P0,1,
and 2Fhv, respectively, from Eqs. (1)–(4). and 2PX, respectively, from Eqs. (6) and (7).
r. These agree well with the exact crossing probabilities ditions. This probability is given by [14],
that follow from Eqs. (2)–(4). For the important special
1
caseofasquaregeometry,r =1,Eq.(2)canbesimplified (r)= Z Z Z
to [22], Pa,b Xl Z(cid:20) 3al,3bl− 2al,2bl−2 (3l+1)a,(3l+1)b
∈
1
1 √3 27 Z +Z , (6)
Fhv(1)=Fhv(1)= 4 − 4π ln16 =0.1779... (5) −2 (3l+2)a,(3l+2)b (2l+1)a,(2l+1)b(cid:21)
fromwhichtheprobabilityoftheIsingsystemcoarsening where Zm,n is a shorthand for Zm,n(32;r); generally
intoastripestateequals2 (1)=0.3558.... Anearlier
numericalestimatefortheFphrovbabilityofreachingastripe Z (g;r)= √g e πg(m2/r+n2r),
m,n √rη2(e 2πr) −
state [1] is consistent with this exact result. −
For periodic boundary conditions, a parallel set of re-
and η(q) = q1/24 (1 qk) is the Dedekind η func-
sults can be constructed to again connect the ultimate k 1 −
tion [20]. AdditQion≥ally, the configuration with cross
fateoftheIsingsystemandpercolationcrossingprobabil-
topology occurs with probability [14]
ities. The nature ofthe crossingprobabilities is substan-
tiallymorecomplexforsystemswithperiodicboundaries 1 8 8 1
(r)= Z ,1;r Z , ;r , (7)
because spanning clusters can wrap around the torus PX 2(cid:20) (cid:18)3 (cid:19)− (cid:18)3 2 (cid:19)(cid:21)
multiple times in the vertical and horizontal directions.
There are two types of spanning clusters [14, 23, 24]. where Z(g,f;r) = f m,n ZZfm,fn(g;r). By symme-
“Winding”clusters arelabeledbytheir verticalandhor- try, X(r) alsorepresPentsth∈e probabilitythatno cluster
P
izontal winding numbers, (a,b). For example, winding spans the system.
numbers (0,1) and (1,0) correspond to a vertical and a To compute the crossing probabilities numerically, it
horizontal stripe, respectively. A spanning cluster that is preferable to employ the asymptotic expansions [23]
wrapsaroundthetorusonceintheverticaldirectionand
(r) = 1 2ρ5/4+ρ3+2ρ12 4ρ53/4+3ρ15...
onceinthehorizontaldirectioncanhaveoneoftwowind- P0,1 − −
ing number pairs, (1,1) or (1, 1), and gives a diagonal 2r
stripe configurationwhen the t−orus is unwound onto the P1,0(r) = r 3 ρ3−ρ15−ρ27+4ρ35...
(cid:0) (cid:1)
square. The other cluster type is the “crosstopology”in (r) = ρ5/4 ρ3 2ρ12+2ρ53/6 2ρ15...,
X
whichaspanningclusterisformedbytheunionoftwoor P − − −
more spanning clusters with distinct winding numbers. where ρ e π/6r, rather than evaluating the special
−
≡
Let (r) denote the crossing probability for a span- functions directly. These expansions provide an excel-
a,b
P
ning cluster with winding numbers (a,b) to exist on a lent approximation for the entire range of aspect ratio
rectanglewithaspectratior andperiodicboundarycon- 0<r <1 [23].
4
Again, a domain mosaic characterized by winding case, the coarsening regime requires that ξ ℓ(t) L
≪ ≪
numbers (0,1) [or (1,0)] occurs with probability P (or (where the equilibrium correlation length ξ may be ar-
0,1
P ),asgivenbyEq.(6),andcoarsensintoavertical(or bitrarily large), and metastable stripe states last a finite
1,0
a horizontal) stripe state. Similarly, a mosaic with cross but large time compared to the coarsening timescale.
topology occurs with probability 2 given by Eq. (7),
X
P
and coarsens directly into the ground state (either all
We are grateful to R. Ball and W. Klein for very
spins pointing up or all pointing down). However, a do-
useful conversations. This work is supported by
main mosaic can also reach the ground state by the in-
DOE grant DEFG-0295ER14498 and NSF Grant DGE-
direct route of first forming a diagonal stripe state with
0221680(KB),NSFgrantCCF-0829541(PLK),andNSF
non-zero winding numbers in both directions. As found
grant DMR0535503 (SR).
previously for the specific case of the (1,1) stripe, such
states are long lived [1]; namely, they reach the ground
state at a time scale that is muchlargerthan the typical
coarsening time (L2).
O
InFig.4weplottherealizationsthatevolvetoatopol- [1] V.Spirin,P.L.Krapivsky,and S.Redner,Phys.Rev.E
ogy with winding number (0,1) or (1,0), or to the cross 63, 036118 (2001); 65, 016119 (2001).
topology for a variety of aspect ratios r. These again [2] P. Sundaramurthy and D. L. Stein, J. Phys. A 38, 349
agree well with the exact percolation crossing probabil- (2005).
ities that follow from Eqs. (6) and (7). In the specific [3] E.E.FerreroandS.A.Cannas,Phys.Rev.E76,031108
(2007).
case of the square system (aspect ratio r =1), the prob-
[4] I. M. Lifshitz, Sov. Phys.JETP 15, 939 (1962).
ability of reaching an infinitely along lived stripe state
[5] J. D. Gunton,M. San Miguel, and P. S.Sahni, in Phase
is 0.3390.... Because the kinetic Ising model can also Transitions andCritical Phenomena,editedbyC.Domb
evolveto diagonalstripe topologies, 0,1+ 1,0+2 X is and J. L. Lebowitz (Academic, New York,1983), vol. 8.
less than 1. P P P [6] A. J. Bray, Adv.Phys. 43, 357 (1994).
In conclusion,the probabilities with which Ising ferro- [7] R. Zallen and H. Sher, Phys. Rev. B 4, 4471 (1971);
magnets freeze into metastable stripe states correspond A. Weinrib, Phys.Rev.B 26, 1352 (1982).
[8] J. J. Arenzon, A. J. Bray, L. F. Cugliandolo, and A. Si-
exactly to crossing probabilities in critical percolation.
cilia, Phys.Rev.Lett.98,145701(2007); A.Sicilia,J.J.
Our simulation results for the probabilities to reach a
Arenzon, A. J. Bray, and L. F. Cugliandolo, Phys. Rev.
specifiedultimatefate(Figs.3and4)agreewiththeoret- E 76, 061116 (2007).
icalpredictions. Ourapproachalsoappliestoarbitrarily- [9] N.A.Gross,W.Klein,andK.Ludwig,Phys.Rev.E56,
shaped domains and boundary conditions, and also can 5160 (1997).
beusedtodeterminemoresubtlecharacteristics,suchas [10] We deal only with rectangular systems and the thermo-
the distribution of the number of stripes. dynamiclimitistakenwithafixedaspectratior.Taking
thelimit inthisspecificwayisnecessary sincethecross-
Thecorrespondencewithcontinuumpercolationatthe
ing probabilities dependon r.
critical point depends crucially on the initial state con-
[11] R. P. Langlands, C. Pichet, P. Pouliot, and Y. Saint-
sisting of an equal number of up and down spins. This Aubin,J. Stat.Phys. 67, 553 (1992).
equalityalwaysholdsifthesystemisquenchedfromequi- [12] J.-P.HoviandA.Aharony,Phys.Rev.E53,235(1996).
libriumataninitialtemperaturethatisabovethecritical [13] J. L. Cardy, J. Phys. A 25, L201 (1992).
temperature T . If, however,the systemis in a magnetic [14] H. T. Pinson, J. Stat. Phys. 75, 1167 (1994).
c
fieldthatisswitchedtozerosimultaneouslywithquench- [15] G. M. T. Watts, J. Phys.A 29, 363 (1996).
[16] J. J. H.Simmons, P.Kleban, and R. M. Ziff, J. Phys. A
ing, or if the initial temperature is below T , then the
c 40, F771 (2007).
Ising ferromagnet always freezes into the ground state [17] S. Smirnov,C. R.Acad. Sci. Paris 333, 239 (2001).
corresponding to the initial majority phase [1]. This be- [18] O. Schramm, Elect. Comm. Probab. 6, 115 (2001).
havior immediately follows from the percolation view- [19] J. Dub´edat, Probab. Theory Relat. Fields 134, 453
point. Such a system is above the percolation threshold (2006).
and there always exists a spanning domain of the ma- [20] M.AbramowitzandI.A.StegunHandbookofMathemat-
jority phase that percolates in both directions. Conse- ical Functions (Dover,New York,1972).
[21] M. E. J. Newman and R. M. Ziff, Phys. Rev. Lett. 85,
quently, the system coarsens into the ground state.
4104 (2000).
WhilewefocusedontheIsingmodelwithzerotemper- [22] R. S.Maier, J. Stat. Phys. 111, 1027 (2003).
ature spin-flip dynamics, our theory applies more gener- [23] G.PruessnerandN.R.Moloney,J.Stat.Phys.115,839
ally to phase ordering kinetics in two-dimensional sys- (2004).
tems with non-conservedscalar order parameter, as well [24] P.diFrancesco,H.Saleur,andJ.B.Zuber,J.Stat.Phys.
astoquenchestosub-criticaltemperatures. Inthe latter 49, 57 (1987).