Domain walls and chaos in the disordered SOS model

arXiv:0905.4816v1 [cond-mat.dis-nn] 29 May 2009

K. Shwarz

1

, A. Karrenbauer

2,3

, G. Shehr

4

, and H. Rieger

1,4

1

TheoretishePhysik,UniversitätdesSaarlandes,66041SaarbrükenGermany

2

MaxPlankInstitute forInformatis, UniversitätdesSaarlandes,66041

SaarbrükenGermany

3

ÉolePolytehniqueFédéraledeLausanne,Lausanne,Switzerland

4

LaboratoiredePhysiqueThéorique,UniversitédeParis-Sud,91405OrsayFrane

Abstrat. Domain walls, optimal dropletsand disorderhaos at zerotemperature

arestudiednumeriallyforthesolid-on-solidmodelonarandomsubstrate. Itisshown

thattheensembleofrandomurvesrepresentedbythedomainwallsobeysShramm's

left passageformula with

κ = 4

^{whereas their fratal dimension is}

d s = 1.25

^{, and}

thereforeisnotdesribedbyStohasti-Loewner-Evolution(SLE).Optimaldroplets

with a lateral size between

L

^and

2L

^{have the same fratal dimension as domain}

walls but an energy that saturates at a value of order

O (1)

^for

L → ∞

^{suh that}

arbitrarilylargeexitationsexistwhihostonlyasmallamountofenergy. Finallyit

isdemonstratedthat thesensitivityofthegroundstateto smallhangesoforder

δ

ⁱⁿ

thedisorderissubtle: beyondaross-overlengthsale

L δ ∼ δ ^{− 1}

^theorrelationsofthe perturbedgroundstatewiththeunperturbedgroundstate,resaledbytheroughness,

aresuppressed andapproahzerologarithmially.

1. Introdution

Domainwallsindisorderedsystemsplayanimportantroleinunderstandingthestability

of the ordered phase, the energetis oflarge sale exitations, the asymptotidynamis

in and out of equilibrium as well as the sensitivity to hanges of external parameters.

They have been studied quite intensively in the reent years for Ising spin glasses

[1,2, 3,4,5, 6,7,8℄, XY spinglasses [9℄, random eldsystems [10,11,12, 13℄, random

ferromagnets[10℄,disorderedelasti manifolds[15, 16,17,18,19,20℄, andmanyothers.

Two domain wall properties are prominent: the rst onerns energy and an be

haraterized by the saling behavior of the domain wall energy with their lateral size,

whihgivesrise toa rst, sometimes universal exponent, the stiness exponent

θ

^{. The}

seond onerns geometryand gives rise toanother, sometimes universal exponent, the

fratal dimension

d s

^{,or,inase the domainisnot fratal,a roughnessexponent}

ζ

^{. The}

interplay between energetis and geometry of the domain walls (i.e. between stiness

exponentandfrataldimension)determineshowsensitivethesystemstateistohanges

ofeither externalparameterslikethe temperatureoraeld, orinternalparameters like

small disorder variations. This sensitivity is often extreme in glassy systems and goes

under the name of haos [1℄.

Domain walls of glassy systems in two spae dimensions represent fratal urves

in the plane and the question arises, whether they fall into the general lassiation

sheme for ensembles of random urves desribed by Stohasti Loewner Evolution

(SLE) [21, 22℄. Reently indiations were found that domain walls in 2d spin glasses

(at zero temperature) are indeed desribed by SLE [23, 24℄, at least for a Gaussian

distributionofthebonds,but apparentlynot forbinaryouplings[25℄. Alsothe domain

walls in the random-bond Potts model at the ritial point (i.e at nite temperature)

were found to be numerially onsistent with SLE [26℄. It appears natural to ask,

whether the domain walls in other two-dimensional disordered systems are potential

andidates for adesription by SLE.

In this paper we study domain walls and haos at zero temperature in the solid-

on-solid (SOS) model on a disordered substrate. This is a numerially onvenient

representation of a two-dimensional elasti medium, with salar displaement eld,

interating with quenhed periodi disorder. It has been studied to desribe various

physial situationsranging from vortex latties in superondutors to inommensurate

harge density wavesand rystalgrowth onadisorderedsubstrate [27,28,29℄. Here we

fous onthree questions: 1) are domainwalls inthis modeldesribed by SLE, 2)what

is the relation between size and energy of optimal exitations(droplets) inthis model,

3) doesdisorder haos exist in the ground state of this model? After a briefsummary

ofwhatisalreadyknown aboutthemodelandadesriptionofthenumerialmethodby

whihweompute the groundstateand thedomainwallsthese threeissues arestudied

in separate setions. The paper ends with adisussion of the resultsobtained.

1.1. Model

We onsider the solid-on-solid model on a disordered substrate dened by the

Hamiltonian

H = X

(ij)

(h i − h j ) ² , h i = n i + d i ,

⁽¹⁾

with

i ≡ (x i , y i ) ∈ Z ²

^{. In Eq. (1) the height variables}

n i

⁽

i = 1, . . . , N

^{) take on}

integer values

n i = 0, ± 1, ± 2, . . .

^{and the osets}

d i

^are independent quenhed random variables uniformly distributed between 0 and 1. The sum is over all nearest neighbor

pairs

(ij)

^{of aretangularlattie ofsize}

L x × L y

⁽

L x = L y = L

^{if not stated}otherwise).

The boundary onditions will be speied below in the ontext of domain walls. The

HamiltonianinEq. (1) desribes adisretemodelof atwo-dimensionalelasti medium

in a disordered environment. In the ontinuum limit, it is desribed by a sine-Gordon

model with random phase shifts (and in the absene of vorties), the so alled Cardy-

Ostlundmodel [30℄,

H CO = Z

d ² r( ∇ u(r)) ² − λ cos(2π[u(r) − d(r)]) ,

⁽²⁾

with a ontinuous salar displaement eld

u(r) ∈ ( −∞ , + ∞ )

^{and quenhed random}

variables

d(r) ∈ [0, 1]

^. Disretizing the integral and performing the innite strong ouplinglimit

λ → ∞

^{one reovers (1).}

It is well known that this model (1, 2) displays a transition between a high

temperaturephase,

T > T g = 2/π

^{wherethedisorderisirrelevantandalow}temperature phase below

T _g

^{, dominated by the disorder. The} high-temperature phase

T > T _g

^is

haraterized by a logarithmi thermalroughness

C(r) = h (h _i − h _i+r ) ² i ∼ T log r ,

⁽³⁾

where

h . . . i

^{denotes thethermalaverageand}

. . .

^{theaverageoverthe quenhed disorder.}

The low-temperature or glassy phase is instead superrough, haraterized by an

asymptotiallystronger (log-square) inrease of

C(r)

^:

C(r) ∼ c(T ) · log ² r + O (log(r)) ,

⁽⁴⁾

whihmeans

ζ = 0

^{, asexpeted for arandom periodisystem. Closeto}

T g

^{, aCoulomb}

Gasrenormalizationgroup(RG)analysis tolowest order gives

c(T ) ≃ (1 − T /T g ) ² /2π ²

[29℄, in rather good agreement with numerial simulations [31℄. At

T = 0

^{, numerial}

simulations give the estimate

c(T = 0) ≈ 0.5/(2π) ² ≈ 0.012

^{[16, 32℄. While earlier}

studies,based onnearly onformal eld theory [33℄,laimed anexat resultfor

A(T )

^,

prediting

A(T = 0) = 0

^{,in lear} ontradition with numeris, amore reent approah basedonFRG,inorporatingnonanalytioperatorspreditsanon-zero

A(T = 0)

^whih

ompares reasonably withnumeris[34℄.

For free or periodi boundary onditions, the Hamiltonians (1) and (2) have a

disrete symmetry, the energy is invariant under a global height (displaement) shift

n i → n i + ∆n

⁽

u(r) → u(r) + ∆n

^{), where}

∆n

^{is an arbitrary integer. This symmetry}

Figure 1. Left: Ground state

n ⁰

^{of a}

200 × 200

^{system with the boundary sites}

(indiate in green) xed to

n i = 0

^{. The dierent height values}

n i

^{are grey-oded}

(dark=lowvalues,bright=highvalues). Middle: Groundstateongurationofthe

samesystemastotheleftwiththeupperhalfoftheboundarysites(indiatedinred)

xedto

n i = 1

^{and thelowerhalf (indiatedin green)to}

n i = 0

^{. Right: Dierene}

plot betweenthe Left and Middle plots: in the lowerwhite region the groundstate

ongurationis idential to theorrespondingsites in theleft gure, whereasin the

upper greyregion they dier by exatly

∆n = 1

^{from the} orresponding site in the middle panel. The border between the whiteand the grey region is adomain wall,

representingastepintheheightproleofthegroundstate.

will not be broken in the low temperature phase of the innite system and true long-

range order at

T < T g

^{is absent, i.e.}

h h i i = 0

^. Conomitantly the model (1), with free or periodi boundary onditions, has innitely many ground states, whih dier by a

global shift

∆n ∈ {± 1, ± 2, . . . }

^.

1.2. Domain walls

By an appropriate hoie of boundary onditions one an fore a domain wall into the

system, whih is most easily visualized at

T = 0

^{(.f. Fig. 1): onsider the square}

geometry and x the values of the boundary variables to

n i = 0

^{. This yields a unique}

ground state onguration

n ⁰ _i

^{. If one xes the boundary variables to}

n i = +1

^{, the}

orresponding ground state would be

n ^′0 _i = n ⁰ _i + 1

^{. A domain wall induing boundary}

ondition isone, in whih the lowerhalf of the boundaryvalues are xed to

n _i = 0

^and

the upper half to

n ⁰ _i = 1

^{. The ground state of this set-up is then}

n ˜ ⁰ _i = n ⁰ _i

^{in some,}

mainlythe lowerregion of the system, and

n ˜ ⁰ _i = n ^{′ 0} _i

^{in the rest - both region separated}

by a domainwall of non-trivialshape.

It turns out that these domain walls are fratal [16, 18℄, whih means that their

lengths

l dw

^{sales with linearsystem size as}

l path ∼ L ^{d s} ,

⁽⁵⁾

with

d s > d − 1 = 1

^{. The numerial estimate for}

d s

^is

d s = 1.27 ± 0.02

^{[18℄. Suh}

a fratal saling of zero-

T

^{domain walls is also found for spin glasses, in the 2d EA}

model with Gaussian ouplings it is

d s,SG = 1.27 ± 0.01

^{, and with binary ouplings it}

is

d s,SGB = 1.33 ± 0.01

^{. On the other hand zero-}

T

^{domain walls indisordered Isingor}

Potts ferromagnets are rough (i.e. are haraterized by algebrai orrelations) but not

fratal.

The energy for suh a domain wall, given by the dierene between the energy of

the groundstateof the systemwith the domainwallinduingboundaryonditions and

the one with homogeneous boundary onditions, inreases with

L

logarithmially

∆E ∼ log L .

⁽⁶⁾

This result, whih was obtained by numerial simulations [16℄, is onsistent with the

usual saling relation

∆E ∼ L ^θ

^{together with the exat result}

θ = d − 2 + 2ζ = 0

(thanks tostatistialtilt symmetry[35℄). This logarithmi behavior isharateristi of

amarginalglassphase,desribed byalineofxedpointindexedbytemperature(whih

isheremarginalintheRGsense). Foromparisonthestinessexponentin2d(3d)spin

glassesis

θ _SG2d = − 0.28 ± 0.01

^and

θ _SG2d = 0.3 ± 0.1

^(andthusharaterizedbya

T = 0

xed point),whereas for disordered Ising orPotts ferromagnets

∆E ∼ L ^d−1

^.

1.3. Method

The ground states of (1), i.e. the onguration

n ⁰ = (n ⁰ ₁ , . . . , n ⁰ _N )

^{with the lowest}

value for the energy

H[n ⁰ ]

^{for a given disorder onguration}

d = (d ₁ , . . . , d _N )

^{, an}

be omputed veryeiently using aminimum-ost-ow-algorithm[36, 16, 37℄. For the

spei details in whih domain walls are indued in the ground state it is useful to

reapitulatethe mappingontoa minimum-ost-owproblem.

After introduing the height-dierenes

n ^∗ _ij = n i − n j

^{(integer) and}

d ^∗ _ij = d j − d i

(

∈ [ − 1, +1]

^{) along the links}

k = (i, j)

^{on the dual lattie}

G ^∗

^{one obtains a ost (or}

energy) funtion that lives onthe dual lattie

H[n ^∗ ] = X

k

(n ^∗ _k − d ^∗ _k ) ² .

⁽⁷⁾

The ongurations

n ^∗ _k = (n ^∗ ₁ , . . . , n ^∗ _M )

^{, where}

M

^{is the number of links (or bonds) of}

the original lattie, onstitute a ow on the graph

G ^∗

^{. Suppose the original model}

(1) has free boundary onditions. Then the sum of the height dierenes along any

direted yle in the original lattie vanishes. Therefore the divergene of

n ^∗

^vanishes

atall sites

i

^:

( ∇ · n ^∗ ) i = 0 ,

⁽⁸⁾

whih means that the ow

n ^∗

^on

G ^∗

^{, in order to give rise to a height eld}

n

^{on the}

original lattie

G

^{has to be} divergene-less, i.e. without soures orsinks. The problem of determining the groundstate

n

^{of (1) isthus equivalentto nd the ow}

n ^∗

^{with the}

minimum ost (7) under the mass-balane onstraint (8) - i.e. a minimum ost ow

problem,for whih there exist very powerfulalgorithms[36,16, 37℄.

Enforing one domain wall, or step of height one, into the ground state of (1)

by appropriate boundary ondition is then equivalent to modify the onstraint (8) at

exatly two sites, the start and end point of the domain wall (see Fig. 2a-d). As an

d) a) b)

c)

Figure 2. Dierentgeometries andonstraintsondomain wallsonsideredhere: a)

boundaryonditionsinduingastep/domainwallasinFig.1.b)Boundaryonditions

induingastep/domain wall runningdiagonallyfrom oneorner oftheretangular

lattieto theopposite one. ) Boundary onditionsforairular domaininduing a

boundary alongtheequator withtwodierent orientationsof theunderlying lattie.

d)Boundaryonditionsforahalf irledomainandadomainwallwithonexedend

attheoriginandafreeendontheouterhalfirle.

example onsider the ase in whih one wants the domain wall to start at the point

(x, y) = (1, L/2)

^{and end at}

(x, y) = (L, L/2)

^{of asquare lattie. Then one hooses the}

boundary onditions for

n i

^{as follows (.f. Fig. 2 a): one xes the values for}

n i

^{at the}

lower half of the boundary (i.e. at

i = (x, 1)

^for

x = 1, . . . , L

^and

i = (1, y )

^and

(L, y)

for

y = 1, . . . , L/2

^{) to}

n i = 0

^{, and the values for}

n i

^{at the upper half of the boundary}

(i.e. at

i = (x, L)

^for

x = 1, . . . , L

^andat

i = (1, y)

^and

(L, y )

^for

y = L/2 + 1, . . . , L

^)to

n i = 1

^{. Translatingtheseboundaryonditionsfortheheightvariables}

n

^{intoonstraints}

for the ow variables

n ^∗

^one immediately sees that at the point

(x, y) = (1, L/2)

^and

(x, y) = (L, L/2)

^{,wherethestepintheheightprolestartsand}terminates,respetively, the onstraint (8)is modiedinto

( ∇ · n ^∗ ) (1,L/2) = +1 , ( ∇ · n ^∗ ) (L,L/2) = − 1 .

⁽⁹⁾

In other words: the indued step sends a unit of ow from the starting point of the

domainwall,whihisthe asoureof unit strength, arossthe sampletothe end point,

the sink, along an optimal (minimum ost/energy) path. In what follows we identify

domainwallsimmediatelywith the optimal path forthe extra ow unit dened by the

modied mass balaneonstraints (9).

With the help of this onept one an then also onsider situations in whih the

startingpointofthedomainwallisxed buttheendingisonlyforedtobeonaspei

region of the boundary, opposing the starting point (see Fig. 2d). Suppose one wants

he domain wall to start at

i s = (1, L/2)

^{, and terminate somewhere on the opposing}

boundary

i t = (L, y)

^with

y ∈ { 1, . . . , L }

^{. Then one introdues an extra node into the}

dual graph

G ^∗

^{, denoted as the target node, onnets it with bonds of zero ost to all}

sites onthe terminalboundary,andassignsasinkstrength

− 1

^{toit. The sourenodeis}

the one losest to

i s

^{in the dual graph and has sourestrength}

+1

^{. The minimum ost}

ow of this arrangement is then the desired groundstate onguration with a domain

wall startingat

i s

^{and ending somewhereon the opposite boundary.}

2. Shramm-Loewner evolution (SLE)

Sine the domain walls as dened above represent fratal urves embedded in a two-

dimensional spae the question arises whether they fall into the lassiation sheme

of Shramm-Loewnerevolution (SLE)likeloop-erasedrandomwalks, perolation hulls,

and domainwalls atphase transitions in2din the salinglimit[22,21℄. The neessary

(and suient) ondition for a set of random urves onneting two points on the

boundary

D

^{of a domainto be desribed by SLE are 1) the measure for these random}

urves has to fulll a Markov property, 2) the measure has to be invariant under

onformalmappingsofthe domain. Reentlyitwassuggestedthat alsodomainwallsin

2dspin glasses an bedesribed by SLE [23, 24℄althoughneither onformalinvariane

isfullledforeahindividualdisorderrealizationnortheMarkovpropertyafterdisorder

averaging. However, onformal invariane might hold for the disorder averaged model

and the Markov property mightbe fullled almost always ina statistialsense.

A single parameter

κ

parameterizes all SLEs and it is related to the fratal dimension of the urve via

d s = 1 + κ/8 .

⁽¹⁰⁾

The parameter

κ

^{is the diusion oeient of the Brownian motion that underlies the}

SLEand generates viaa randomsequene ofsimple onformalmaps thefratalurves.

Inadditiontothefrataldimensionitdeterminesvariousothergeometriandstatistial

propertiesoftheSLEurves. Oneofthemisforinstanethe probabilitythataurvein

the upperhalfplane

H

^{generatedbySLEwillpass totheleftofagiven point}

z = x + iy

is given by Shramm'sleft passageformula [38℄

P κ (z) = 1

2 + Γ _κ ⁴

√ π Γ ^8−κ _2κ 2 F 1

1 2 , 4

κ ; 3 2 ; −

x y

2 ! x

y ,

⁽¹¹⁾

where

2 F 1

^{is the} hyper-geometri funtion

2 F 1 (a 1 , a 2 ; b; z) =

∞

P

k=0

Γ(k+a 1 ) Γ(a 1 )

Γ(k+a 2 ) Γ(a 2 )

Γ(b) Γ(k+b)

z ^k k!

^.

Sinetheprobabilitydependsjustontheratiobetween

Re(z)

^and

Im(z)

^{,itissometimes}

useful toreplaethis ratioby afuntion ofanangle. Wedeided touse:

tan (φ) = x/y

^.

So

φ ∈ ] − ^π ₂ ; ^π ₂ [

^{isthe angleattheoriginbetween theimaginaryaxisand}

z

^{. Thisformula}

holds for the domain of the SLE being the upper halfplane with the start point of the

urvebeingidentialtotheoriginofthe oordinatesystemandtheendpointatinnity.

A standard hek whether an ensemble of random urves is a potential andidate

for SLE therefore is to simultaneously determine their fratal dimension

d s

^{and the}

10 100 1000 10000

10 100 1000

L path

L

∝L ^1.25 L•L geometry L•2L geometry

Figure 3. Averagelength ofdomain wallsspanning thesystemfrom oneend tothe

oppositeoneasafuntionofthesystemsize

L

^{inalog-logplot. Thestraightlinesare}

leastsquaretsto(5)andyieldtheestimateforthefrataldimension

d s = 1.25 ± 0.01

^.

left passage probabilities, and to test whether the latter t (11) with

κ = 8(d s − 1)

[23, 24,25℄.

In Fig. 3 we show our data for the average length of a domain wall in the

onguration depited in Fig. 2 b, i.e. starting at the point

i s = (1, 1)

^{and ending}

at

i t = (L, L)

^{in a}

L × L

^{geometry and at}

i t = (L, 2L)

^{in a}

L × 2L

^{geometry. Least}

square ts to the saling law (5)yield the estimate

d s = 1.25 ± 0.01

^{whih agrees with}

the value found in[18℄.

This value for the fratal dimension would imply

κ = 2.00 ± 0.08

^{if the domain}

wallsare desribed bySLE. Next wedetermined fordierentpoints

(x, y )

^{of thelattie}

the frequeny that adomain wall passes tothe left of it, yielding a probability

p(x, y )

^,

whih we ompared with Shramm's left passage formula

P κ (x, y)

^{for xed}

κ

^{. For the}

irledomainand the hoieof thestart and endpointsof the domainwallasshownin

Fig. 2, the formula (11) ismodied:

Let

E

^{be the unit irle in the omplex plane. The Cayley funtion}

g : E → H , z 7→ i ^1+z _1−z

^{maps the unitirle onformallyintothe upperhalfplane}

H

^{. Furthermore}

g( − 1) = 0, g(1) = ∞

^{, thus urves in}

E

^{starting at}

z = − 1

^{and ending at}

z = 1

^are

onformallymapped onurvesin

H

^{. Forthe latter, if the urvesare desribed by SLE,}

Shramm'sformula(11) holds, sothat for the former, the modiedformula

P κ,g (z) = 1

2 + Γ _κ ⁴

√ π Γ ^8−κ _2κ 2 F 1

1 2 , 4

κ ; 3 2 ; −

Re g (z) Im g(z)

2 !

Re g(z)

Im g(z) .

⁽¹²⁾

holds. Hene we lled the unit irle with ner and ner grids

G

approximating better and better the ontinuum limitfor the urves in

E

^{that we want to hek for SLE. For}

this we dene the funtion

f G (κ) = X

(x,y)∈G

[p(x, y ) − P κ,g (x, y )] ²

⁽¹³⁾

that measures the umulative squared deviation of the probabilities

p(x, y)

^{from the}

0 20 40 60 80 100 120 140

3 3.5 4 4.5 5

f G ( κ )

κ

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

-1 -0.8 -0.6 -0.4

-0.2 0 0.2 0.4

0.6 0.8 1 -1 -0.8

-0.6 -0.4

-0.2 0

0.2 0.4

0.6 0.8

1

0.002 0 0.004 0.006 0.008 0.01

Figure 4. Left: The umulative squared deviation

f G (κ)

^{of the omputed left}

passage probabilities

p(x, y)

^{from the values}

P κ,g (x, y)

^{given by (12) as a funtion}

of

κ

^{. The underlying lattie geometry is the irle as skethed in Fig. 2., the}

orrespondingonformalmap

g(z)

^{entering(12)isgiveninthetext. Theminimumis}

at

κ = 4.00 ± 0.01

^{withasquareddierenepergridpointof about}

2 · 10 ^{− 5}

^{. Right:}

Absolutedierenebetweenthealulatedleftpassageprobability

p(x, y )

^andtheSLE

expetation

P κ,g (x, y)

^(12)for

κ = 4

^{asafuntion ofthe2dlattieoordinates}

(x, y)

^.

Forthe whole unit irle thedeviation is almost everywheresmaller than1%. Note

thatthedomainwallisxedat

( − 1, 0)

^and

(1, 0)

^{,wherethelargestdeviationsour.}

0 50 100 150 200 250 300 350 400 450 500

3 3.5 4 4.5 5

f G ( κ )

κ

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

0 50 100 150 200 250 300 350 0 50 100

150 200 250 300

350 0

0.005 0.01 0.015 0.02

Figure5. Cumulativedeviation

f G (κ)

^asfuntionof

κ

^{andabsolutedierenebetween}

p(x, y)

^and

P κ,g (x, y)

^for

κ = 4

^{asin Fig.4 but forthe squaregeometry asdepited}

in Fig.2b. Notethat the domainwallis xedat

(0, 0)

^and

(L, L)

^{, where thelargest}

deviationsour.

SLE-value for given

κ

^{. The result is shown in Fig. 4. The minimal deviation of the}

data from the expeted SLE result is at

κ = 4.00 ± 0.01

^{whih learly diers from the}

value

κ = 2.00 ± 0.08

^{that one would expet fromthe fratal dimension if the domain}

walls would be desribed by SLE.

Next we varied the geometry and the domain wall onstraints and studied the

ase depited in Fig. 2b, i.e. a quadrati domain

D

^{with orners at}

0

^,

p

^,

p + ip

^,

ip

⁽

p

^{real and positive). The funtion}

g : D → H : z 7→ − p(z; S)

^with

S =

{ 2n 1 p + i · 2n 2 p | n 1 , n 2 ∈ Z }

^{denes a onformal map from}

D

^into

H

^{, where}

p(z; S) =

-0.02 -0.01 0 0.01 0.02

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 p( φ )-P 4 ( φ )

φ/π

R=50 R=75 R=100 R=125

Figure 6. Deviation of the left passageprobability

p(φ)

^{from the SLEexpetation}

P κ (φ)

^{(11) for}

κ = 4

^{in thehalf irle geometry depitedin Fig. 2 d (}

p = 510, q = 255, R = 500

^{),i.e.onefreeendofthedomainwall.}

1

z ² + P

ω∈S\{0}

h 1

(z−ω) ² − _ω ^{1 2} i

istheWeierstraÿ

p

^{-funtion. Furthermore}

g(p+ip) = 0, g(0) =

∞

^,thusurvesin

D

^startingat

z = 0

^andendingat

z = p + ip

^{areonformallymapped}

on urves in

H

^{starting at the origin and extending to innity. Using}

g

^{in (12) we}

determined

κ

^{in the same way it was done in the ase of the unit irle. The result is}

shown in Fig. 5and yields

κ = 4.00 ± 0.01

^.

Wealsoompared

p(x, y)

^diretlywith

P κ=4 (x, y)

^{forurvesintheupperhalfplane}

H

^{starting at the origin. For this we onsidered the geometry depited in Fig. 2d, in}

whihtheurvesstartingattheoriginanendeverywhereonthehalfirle. Weheked

the predition (11) for dierent radii

R

^{and angles}

Φ = arctan

x y

∈

− ^π ₂ ; ^π ₂

. The

resultisshowninFig. 6,wherethedeviationsfrom

P κ (φ)

^(11)for

κ = 4

^{are everywhere}

less than 2% for the size

L = 500

^{and radii shown in Fig. 6. We alsoobserve that the}

deviationsystematiallydereaseswith thesystem size

L

^{forxed ratio}

R/L

^indiating

vanishing deviations

P (Φ) − P κ=4 (Φ)

^{inthe limit}

L → ∞

^.

For allthe geometries that we studied we found that the data an be niely tted

with

κ = 4

^{,butthefrataldimensionsofthe domainwallsisunhangedbythe dierent}

set-ups:

d s = 1.25 ± 0.01

^{. The onlusion is that domain wallsin the SOS model ona}

disordered substrate are not desribed by SLE. The questionarises whihondition for

SLE is atually violated. The fat that for all geometries the left passage probability

we studied is well desribed by a ommon expression, Shramm's formula ontaining

theonformalmapofthehalfplanetothe speigeometryunderonsideration,would

atuallyhintatonformalinvariane. Butobviouslythisdoesnotexludethepossibility

thatabreakingofonformalinvarianemanifestsitselfinotherquantities,orevenother

geometries. We did not attempt to hek the domain Markov property, as was tried

in [24℄.

3. Exitations

In this setion we study large sale exitationsthat ost a minimum amountof energy,

alsodenoted as droplets[2℄, and the size dependene of their energy. Aording to the

usual argumentsindropletsaling theory [2℄this exitationenergy is expeted tosale

inthesame way asa domainwallof lateralsize

L

^,i.e.like

∆E ∼ L ^θ

^with

θ

^{the stiness}

exponent,whih forthe system we onsider is

θ = 0

^{. The domain wallenergy sales as}

∆E DW ∼ log L

^{.f.(6),and ifitisorretthat allexitationsofsale}

L

^{displaythesame}

behavior the question arises: howan thermalfutuations destabilize the ground state

suhthatthe presentmodelisindeedharaterizedby alineofxed points,indexed by

temperature(andthusdierentfroma

T = 0

^{xed pointlikeinspinglasseswith}

d > 2

^?

A

log L

^{saling of exitations still implies that larger and larger} exitations, neessary to our to deorrelate the system's state from the ground state, need more and more

energy and are thereforemore and more unlikely,i.e. ourring with a probability that

deays as

exp( − ∆E/T ) ∼ L ^−c/T

^at

T > 0

^.

The solution to this problem lies in the freedom that domain walls of exitations

of sale

L

^{have to optimize their energy (.f. for a similar disussion in the ontext of}

optimized disloationsin [18℄). Findingthe optimalexitation of agiven sale requires

ahighereortthansearhing adomainwallwithgiven start andendpoints(aswe have

seen above, there are even eient ways to optimize the positions of these endpoints).

Thereisasimplewaytoindueexitationsofanarbitrarysale,asrstproposedinthe

ontextof spin glasses[4℄and used again in[8℄: we ouldompute the groundstate for

xed boundaryonditionsasdepitedintheleftpanelofFig.1,thisyields

n ⁰

^{. Thenwe}

hoose aentralsite(say

i = (x, y) = (L/2, L/2)

^),xitto

n i = n ⁰ _i + 1

^{, xthe boundary}

as before and ompute the groundstate again,giving

˜ n ⁰

^{. This willdierfrom}

n ⁰

^only

by aompat luster

C

^{that ontainsthe site}

i

^{and whihhas}

n ˜ ⁰ _i = n ⁰ _i + 1

^{. This isthen}

in fatan optimal ordroplet exitation, but its size

V = # { i ∈ C }

^{an vary from 1to}

L ²

^{. Droplets of axedsize annotbe generated inthis way.}

The SOSmodelonadisordered substrate atuallyallowsfor aneientsearhfor

optimalexitations of agiven sale, as wedesribein the following:

A droplet exitations in the SOS model is a simply onneted luster

C

^{of sites}

whoseheightvaluesareallinreased(oralldereased)byoneasomparedtotheground

state:

n _i = n ⁰ _i + 1 ∀ i ∈ C

^{, see Fig. 7. Pitorially it is, in the height} representation, an extra-mountain (valley)of height

+1

⁽

− 1

^{). Its boundaryis adireted yle inthe dual}

lattie,and rememberingthe mappingto theminimum-ost-owproblem(7-8),adding

a yle to a feasible solution maintains the divergene-free onstraint (8). Hene, the

transitionfromone state tothe other isdeterminedby suh ayle. Thus rst one has

to ompute the ground state of a given disorder realization. Then in order to nd a

dropletexitationof this ground state thathas a given lateral size one an for instane

fore this extra-yle to run within an annulus of inner radius

L/4

^{and outer radius}

3L/4

^{(i.e it's average diameter is}

L/2

^{). This an be ahieved by simply removing all}

sites / bonds outside this annulus and then omputing the optimal yle within this

t

a) b)

c)

Figure7. a)Skethofadropletexitationrepresentingasimplyonnetedluster

C

ofsites

i

^{whose height valuesare inreasedby one asompared tothe groundstate:}

n i = n ⁰ _i + 1 ∀ i ∈ C

^{. b) Direted yleinthe duallattie}orrespondingto theluster in a) representingthe step in height prole indued by raisingthe luster

C

^{by one}

height unit. ) Sketh of an

s − t

^{ut of thenodes of theoriginal lattiegraphsuh}

that those sites onneted to the external node

s

^{are foredto be in the set}

S

^and

thoseonnetedto

t

^intheset

T

^(seetext).

modied graph, the ost for whih depends on the ground state onguration and the

substrate heights. One assigns osts to eah direted edge that orresponds to the

energy ost for inreasing the height dierene between its left and right side by one

unit [36, 16, 37℄. Note that in pratie one may use the redued osts emerging from

the groundstate. Thatis,if and onlyif afeasible owhas minimumenergy,then there

are non-negative redued osts suh that the osts of eah yle remainsunhanged. If

the suessive shortest path algorithmis used to solve the minimum-ostow-problem

when omputing the groundstate

n ⁰

^{, then no extra work is neessary to ompute the}

desired non-negative redued osts. Hene, Dijkstra's shortest path algorithm an be

used to nd the shortest direted yle around the annulus, i.e. the one separating the

inner and the outer ring ofthe annulus. To this end, the annulus is ut from the outer

to the inner ring, i.e.the orresponding edges are removed from the graph, to prevent

theshortestpathfromshort-utting. Foreahoftheseremoved edges,the shortestpath

Figure8.Examplesofoptimalexitationsofsale

L

^{,whoseboundary(domainwall)}

is fored to lie in the interior of the area indiated in green (an annulus in square

geometry). Fromthe toprowto thebottom itis

L = 32

^,

64

^,

128

^and

256

^{. Pitures}

aresaled tohavethesamesize.

fromitshead toitstailisomputed,where onlythethe remainingedgesare used. The

obtained shortest paths are ompleted with the orresponding direted edges to form

ylesaroundthe annulus. Thisproedurehastoberepeatedforall(orarepresentative

number of)positions ofthe annulus withinthe originallattie(inpratie one xes the

annulusandshiftsthedisorderongurations, wrappingitaroundatoroidalgeometry).

The proedure of nding the optimal yle in a given annulus an be simplied

by observing that the dropletexitation inthe heightrepresentation orresponds to an

s

^-

t

^{-ut of the underlying graph [37℄}

‡

^{in suh a way that one fores all nodes of the}

inner irle of the annulus to belong to

S

^{and all nodes belonging to the outer ring of}

the annulus to

T

^{, see Fig. 7 . The minimum}

s

^-

t

^{-ut with respet to the} non-negative redued edge osts of the groundstate

n ⁰

^{is then exatly the boundary of the optimal}

exitation (or the optimal yle) one is searhing for a given annulus arrangement.

‡

^An

s

^-

t

^{-utisapartition ofthenodesofagraph}

G

^{intotwodisjointsets}

S

^and

T = G / T

^{,suh that}

s ∈ S

^and

t ∈ T

^.

0 100 200 300 400 500 600 L

3 3.5 4 4.5 5

∆Ε

10 100 1000

L 3

3.5 4 4.5 5

∆Ε

Figure9. Disorderaveragedenergyoftheoptimalexitationsofsale

L

^asafuntion

of

L

^{-leftinlinearsale,rightinalog-logsale.}

Aording to the famous Min-Cut-Max-Flow theorem one an ompute the minimum

s

^-

t

^{ut in polynomial time by solving the assoiated maximum-ow problem [16, 37℄.}

Wehave implementedthis proedure andshowfor illustrationanumberofexamples in

Fig. 8.

Fig. 9 shows our result for the disorderaveraged energy of the optimal exitations

ofsale

L

^{thatwe obtainwith theproeduredesribed above. Thisrepresentsanupper}

bound for the optimal exitations of sale

L

^{sine the annulus} arrangement does not inludeallpossibleexitationsofsale

L

^{. Asoneansee thisboundsaturatesatanite}

energy of order

O (1)

^{in the limit}

L → ∞

^. Consequently arbitrarily large exitations exist that ost onlya smallamount of energy, whihrenders the ground state unstable

atan non-vanishing temperature.

Fig. 10 shows the distributionof optimalexitation energies for dierentvalues of

L

^{. As an be seen the} distribution is nearly Gaussian with a nite width, i.e. it does not display long, e.g. algebraially deaying, tails, whih implies that the average is

representative foralmost alldisorderongurations. It an alsobeseen that the whole

probability distribution

P _L (∆E)

^beomes independent of the length sale

L

^{for large}

L

^{. This is an important} observation sine droplet saling theory [2℄ one would expet

P L (∆E) ∼ l ^−θ p(∆E/L ˜ ^θ )

^{. For}

θ = 0

^{,asitistheasehere,itisnotapriorilearwhether}

this implies

P L (∆E) ∼ (ln L) ⁻¹ p(∆E/ ˜ ln L)

^{,i.e.asalingwith the averagedomainwall}

energy

ln L

^,or

P L (∆E) ∼ p(∆E) ˜

^{,i.e.dropletsize}independene. Fig.10shows thatthe latter is orret.

This has important impliations for the saling of the average droplet energy

within a system of lateral size

L

^{as determined in [20℄. There the average energy of}

droplets of size

l < L

^,

(∆E) ^min _L

^{, i.e. without a lower bound, was estimated to behave}

like

(∆E) ^min _L ∼ ln L

^{. In order to make ontat with our result one should note that}

this energy is the minimum among droplets of the kind we determined here, with size

l ∈ [L/2, L]

^,

l ∈ [L/4, L/2]

^,

l ∈ [L/8, L/4]

^,

. . .

^{, i.e a minmum of} approximately

ln L

randomnumbers. Onlyiftheprobabilitydistributionoftheseenergiesisa)independent

and b)identiallydistributed, theirminimum goeslikethe inverse of their number, i.e.

0 1 2 3 4 5 6 7 8 9 10

∆Ε 0

0.0005 0.001 0.0015 0.002 0.0025 0.003

p( ∆Ε)

L = 32 L = 128 L = 512

Figure10. Distributionoftheoptimalexitationenergiesofsale

L

^{forthree values}

of

L

^{. For}

L ≥ 128

^thisdistribution beomesindependentof

L

^.

(∆E) ^min _L = min { (∆E) L/2 ¹ , (∆E) L/2 ² , . . . , (∆E) _L/2 ^k } ∼ 1/k

^with

k ∼ ln L

^{. Aording to}

our result depited in Fig. 10 the assumption b, whih is impliit in the reasoning of

[20℄, is indeedfullled.

Finally we note that also determined the fratal dimension of the boundary of

the optimalexitations and found that it is idential with the frataldimension of the

domainwallswith xed start and endpoints:

d _s = 1.25 ± 0.01

^.

4. Disorder Chaos

In this setion, we study the sensibility of the ground state to a small hange of the

quenheddisorderonguration. Tothispurposewegenerateaongurationofrandom

osets

d ¹ _i

^{and ompute the assoiated ground state}

h ¹ _i

^{. Then we slightly perturb this}

ongurationofrandomosets

d ² _i = d i + δǫ i

^with

δ ≪ 1

^andwhere

ǫ i

^'sareindependent and identially distributed Gaussian variables of unit variane and we ompute the

assoiated ground state

h ² _i

^{. The question we ask is : how dierent are these two}

ongurations of the systems

h ¹ _i

^and

h ² _i

^?

Suh questions rst arose inthe ontext of spin glasses [1℄, where it was proposed

that disorder indued glass phases may exhibit stati haos, i.e. extreme sensitivity

to suh small modiations of external parameters (like disorder onsidered here or

temperature). Suh small perturbations are argued to deorrelate the system beyond

the so alled overlap length

L δ

^{whih diverges for small}

δ

^as

L δ ∼ δ ^−1/α

^{, with}

α

^the

haos exponent. As an example let us rst onsider the ase of an Ising spin-glass

with a ontinuous Gaussian distribution of random exhange interations, of width

J

^{. The system has two ground states related by a global spin reversal. Within the}

phenomenologialdroplettheory[1,14,41℄,alow-lyingenergyexitationsofthesystem

involves anoverturned droplet of linearsize

L

^{and osts anenergy}

JL ^θ

^{. If wenow add}

a small random bond perturbation, say Gaussian of width

δJ

^{, the exess energy of}

a droplet is modied. For suh spin-glass system, this energy omes only from the

bonds whih are at the surfae of the droplet. Their ontribution is thus the sum of

L ^{d s}

independent random variables of width

δJ

^{, where}

d s

^{is the fratal dimension of}

the droplet : it is thus of order

± δℓ ^{d s /2}

^{. Therefore the ground state is unstable to}

the perturbation on length sales

L

^{suh that}

δJL ^{d s /2} > JL ^θ

^{, i.e.}

L > L δ

^where

L δ ∼ δ ^{−1/α SG}

^with

α SG = d s /2 − θ

^{. One thussees that, for spin glasses, disorder haos}

is losely relatedto the (geometrial)properties of the domain walls.

The situation is rather dierent for elasti systems in a random potential as

onsidered here. Here we onsider the SOS model on a disordered substrate dened

in Eq. (1) as

H = X

(ij)

(h i − h j ) ² , h i = n i + d i ,

⁽¹⁴⁾

with

i ≡ (x i , y i ) ∈ Z ²

^{. In Eq. (14) the height variables}

n i

⁽

i = 1, . . . , N

^{) take on}

integer values

n i = 0, ± 1, ± 2, . . .

^{and the osets}

d i

^are independent quenhed random variables uniformly distributed between 0 and 1. This system (with free of periodi

boundary onditions) has innitely many ground states whih dier by a global shift

∆n ∈ {± 1, ± 2, ... }

^{. Again, within the dropletargument [14, 42, 43℄ a low-lying energy}

exitationof the system involves adroplet of size

L

^{where the height eld isshifted by}

unity, say

∆n = 1

^{and it osts an energy}

L ^θ

^{. But now if we had a small} perturbation

d ² _i = d ¹ _i + δǫ i

^{,theexess energyofthis dropletomesfromthebulkof thedroplet,whih}

experienes a random fore eld. This ontribution is thus the sum of

L ^d

^random

variables of width

δ

^{and therefore in this ase the ground state is unstable to the}

perturbation on length sales

L > L δ

^where

L δ ∼ δ ^−1/α

^with

α = d/2 − θ

^{. Here}

d = 2

^and

θ = 0

^{and thusoneexpets}

L δ ∝ δ ⁻¹

^{. Atvarianewith}spin-glasses disussed above, one thus sees that for disordered elasti systems, disorder haos is not diretly

related tothe properties of domainwalls.

For the present model(14), disorderhaos wasdemonstrated analytially atnite

T

^{near the glass transition}

T g

^{using a Coulomb Gas} Renormalization Group [44℄. At

T = 0

^{,someindiationsofdisorderhaoswere alsofoundnumeriallyinRef. [16℄where}

global orrelations between

h ¹ _i

^and

h ² _i

^{where studied through}

χ(δ) = P

i (h ¹ _i − h ² _i ) ²

^{. In}

this paper, followingRef. [43℄, weharaterizethe loalorrelationsbetween these two

ongurations by the orrelation funtion

C ij (r)

^with

r ≡ (x, y)

^(with

i, j = 1, 2

⁾

C _ij (r) = (h ⁱ _k − h ⁱ _k+r )(h ^j _k − h ^j _k+r ) ,

⁽¹⁵⁾

where

k + r ≡ (x k + x, y k + y)

^{and for a} rotationnaly invariantsystem onsidered here one has

C ij (r) ≡ C ij (r)

^with

r = | r |

^{. In the following we will talk about intralayer}

orrelationsfor

C _ii (r)

^{and interlayer}orrelationsfor

C _i6=j (r)

^. Equivalently,onean also study suh orrelations (15) inFourier spae and dene

S _ij (q)

^with

q ≡ (q _x , q _y )

^as:

S ij (q) = ˆ h ⁱ _q ˆ h ^j _−q , ˆ h ^j _q = 1 L ²

X

k

h ^j _k e ^iq·k

⁽¹⁶⁾

where

q · k = q x x k + q y y k

^{. Fora}rotationnalyinvariantsystem onsidered here, one has

S ij (q) ≡ S ij (q)

^where

q = | q |

^.

Disorder haos at

T = 0

^{in generi disordered elasti systems in dimension}

d

^was

reently studied analytially using the Funtional Renormalization Group (FRG) [43℄.

At one loop order in a dimensional expansion in

d = 4 − ǫ

^{it was found that for short}

rangedisorder(likerandombond problems)and randomperiodisystems indimension

d > 2

^{(inluding one omponent}Bragg-Glass), one has [43℄

C 12 (r) = r ^2ζ Φ(δr ^α ) with Φ(x) ∼

( c ^st , x ≪ 1 , x ^−µ , x ≫ 1 ,

(17)

with

c ^st

^aonstant,

ζ

^{theroughnessexponentandwhere}

µ

^{isthedeorrelationexponent.}

Translated intoFourier spae,this yields

S 12 (q) = L ^(d+2ζ) _δ ϕ(qL δ ) with ϕ(x) ∼

( x ^{−d−2ζ+µ} , x ≪ 1 , x ^−d−2ζ , x ≫ 1 ,

(18)

with

L δ ∼ δ ^−1/α

^{and where the behavior for large}

x

^{is then suh that the dependene}

on

L _δ

^{anels inthis limit,as itshould.}

The two-dimensional disordered SOS model we are onsidering here (14)

orresponds preisely to the marginal ase

d = 2

^(with

ζ = 0

^{and thus}

θ = 0

^{) where,}

as disussed in Ref. [43, 45℄, the analysis yielding the result in Eq. (17) eases to be

valid. Indeed in that ase non loal terms, irrelevant in

d > 2

^{are generated under}

oarse-grainingandthese additionaltermshavetobehandled withare at

T = 0

^{. One}

thusonsiders theHamiltonianassoiated tothe twoopiesofthe system parametrized

by the salarelds

u ⁱ ≡ u ⁱ (r)

^:

H 2 copies = 1

T X

i=1,2

Z d ² r

1

2 ( ∇ ^r u ⁱ ) ² + V i (u ⁱ , r) − µ ⁱ (r) · ∇ ^r u ⁱ

,

⁽¹⁹⁾

where

µ ⁱ ≡ (µ ⁱ _x , µ ⁱ _y )

^are two-omponent random tilt elds, whih are generated upon renormalization. Whilethey areirrelevantin

d > 2

^{they beomerelevantin}

d = 2

^where

they playaruial role. Thesetwoopies

u ¹ , u ²

^arethusindependent(19)butthey feel two mutually orrelated random potentials

V _i (u, r)V _j (u ^′ , r ^′ ) = R _ij (u − u ^′ )δ ² (r − r ^′ ) ,

⁽²⁰⁾

µ ⁱ _ρ (r)µ ^j _ρ ′ (r ^′ ) = σ ij δ ρρ ^′ δ ² (r − r ^′ ) ,

⁽²¹⁾

where

i = 1, 2

^{is the index of theopy and}

ρ = x, y

^{is aspatial index. This leads tothe}

repliated Hamiltonian,

Z ⁿ = exp ( − H _rep )

^:

H rep = 1 2T

Z d ² r

2

X

i=1 n

X

a=1

1

2 ( ∇ ^r u ⁱ _a ) ² − 1 2T ²

2

X

i,j=1 n

X

a,b=1

Z

d ² r[R ij (u ⁱ _a − u ^j _b )

⁽²²⁾

− 1

2 ∇ ^r u ⁱ _ab ∇ ^r u ^j _ab G ij (u ⁱ _a − u ^j _b )] ,

where weused the notation

u ⁱ _ab = u ⁱ _a − u ⁱ _b

^{. In the bare model, one has}

G ij (u) = σ ij

^.

Close to the transition

T . T g

^{, this model (22) was studied using Wilson RG}

analysis by varying the short sale momentum uto

Λ ℓ = Λe ^−ℓ

^{, with}

ℓ

^{the log-sale.}

It was shown, using Coulomb Gas tehnique at lowest order, that

G ii ∝ σℓ

^{. This}

leads to the orrelation funtion in Fourier spae

S ii (q) ∝ σℓ/q ²

^{, whih yields, setting}

ℓ = log(1/q)

^{, tothe}

log ² (r)

^{behaviorofthe intralayer}orrelations(thisresult for

C ii (r)

an be derived ina more ontrolled way using the Exat RenormalizationGroup [46℄).

Conerning haos properties, it was shown in Ref. [44℄ that

G 12 (0)

^{grows linearly with}

ℓ

^{for small}

ℓ

^{beforeit saturates to aonstant for large}

ℓ

^,

G 12 (0) ∼ σ > ˆ 0

^{, whih yields}

S 12 (q) ∝ σ/q ˆ ²

^{. These resultslose to}

T g

^{an besummarized as}

S 12 (q) ∼



 



 



σ log 1/q

q ² , q ≫ L ⁻¹ _δ ˆ

σ

q ² , q ≪ L ⁻¹ _δ

(23)

whileinreal spae,the behaviorof theinterlayerorrelationfuntion

C 12 (r)

^for

T . T g

is thus

C 12 (r) ∼

( σ log ² (r) , r ≪ L δ

ˆ

σ log(r) , r ≫ L _δ .

(24)

At

T = 0

^{, it was reently shown [34℄, using FRG to one loop inluding the term}

G 11 (u)

^{thattheintralayerorrelationfuntionalsobehaveslike}

C(r) ∝ log ² (r)

^,inrather

good agreement with numeris [16, 32℄. One thus also expets that

C 12 (r) ∝ log ² (r)

for

r ≪ L δ

^{[16, 32, 34℄. For}

r ≫ L δ

^{, a behavior of}

C 12 (r) ∼ σ ˆ log r

^{as in Eq. (23)}

was disussed inRef. [43℄. To determine analytially whether

σ > ˆ 0

^at

T = 0

^requires

a detailed and diult analysis of the oupled FRG equations for

R ij (u), G ij (u)

^(22),

whihgoesbeyondthe previousstudiesdoneinthat diretioninRef. [34,43,45℄. Here,

we willanswer this question using numerial simulations.

The purpose of our study is atually to answer the two main questions : (i) what

are the residualorrelations beyond

L _δ

^{and in partiular is}

σ ˆ

^{alsonite at}

T = 0

^{? (ii)}

what is the saling form of this orrelation funtion

C ₁₂ (r)

^{, i.e. the analogous of Eq.}

(17) from whih one an extrat the overlap length

L δ

^{and hek the value of haos}

exponent

α = 1

^{as expeted fromdroplet saling?}

Here will use the minimum-ost-ow algorithm desribed above to ompute the

two ground states

h ¹ _i

^and

h ² _i

^{with free boundary onditions. Instead of the orrelation}

funtion

C 12 (r)

^{we ompute numerially the Fourier transform}

S 12 (q)

^{of the overlap}

between the ongurations (16). Our simulations have been performed on a square

lattie of linear size

L = 256

^{and we have hosen}

q = (q, 0)

^with

q = 2πn/L

^with

n = 0, 1, 2, · · · , L − 1

^{. The disorder average has been performed over}

10 ⁶

independent

samples. In Fig. 11 (left) we show our numerial data for

S ₁₂ (q)

^{. Its behavior lose}

to

T _g

^{(23) suggests strongly to plot}

q ² S ₁₂ (q)

^{as a funtion of}

q

^{. Given that we are}

working on a disrete lattie of nite size, it is more onvenient to work with the

variable

y = sin (q/2) = [(1 − cos (q))/2] ^1/2

^insteadof

q

^{(of ourse for small}

q

^{it makes}

no dierene). In Fig. 11(left) we atually show a plot of

S 12 (q)[sin (q/2)] ² /A(δ)

^{as a}

funtion of

sin (q/2)

^{on a log-linear plot for dierent values of}

δ = 0, 0.1, 0.2, 0.3

^{. On}

this plot the amplitude

A(δ)

^{is hosen suh that the urves for dierent values of}

δ

^do

oinide for

sin (q/2) ∼ 1

^{, with}

A(0) = 1

^{. For}

δ = 0

^{, where}

S 12 (q) = S 11 (q)

^{, this plot}

is almost a straight line, whih suggests indeed that for small

q

^where

sin (q/2) ∼ q/2

^,