Statics and dynamics of elastic manifolds in media with long-range correlated disorder

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by  Andrei A. Fedorenko

Statics and dynamics of elastic manifolds in media with long-range correlated disorder
Andrei A. Fedorenko, Pierre Le Doussal, and Kay Jo¨rg Wiese
CNRS-LaboratoiredePhysiqueThe´oriquedel’EcoleNormaleSupe´rieure,24rueLhomond,75231Paris,France
(Dated:January20,2007)
We study the statics and dynamics of an elastic manifold in a disordered medium with quenched defects
7 correlatedas∼ r−a forlargeseparationr. Wederivethefunctionalrenormalization-groupequationstoone-
0 looporder,whichallowustodescribetheuniversalpropertiesofthesysteminequilibriumandatthedepinning
0
transition. Usingadoubleε = 4−dandδ = 4−aexpansion, wecomputethefixedpointscharacterizing
2
differentuniversalityclassesandanalyzetheirregionsofstability. Thelong-rangedisorder-correlatorremains

n analyticbutgeneratesshort-rangedisorderwhosecorrelatorexhibitstheusualcusp. Thecriticalexponentsand
a universalamplitudesarecomputedtofirstorderinεandδatthefixedpoints. Atdepinning,avelocity-versus-
J forceexponentβlargerthanunitycanoccur.Wediscusspossiblerealizationsusingextendeddefects.

0
2
I. INTRODUCTION determinedatthedepinningtransitionareingeneraldifferent

] fromtheexponentandamplitudesmeasuredinequilibrium.
n
n Elastic objects in random media are the simplest example Twomethodsweredevelopedtostudythestaticsofanelas-
- ofadisorderedsystemexhibitingmetastability,glassybehav- ticmanifoldinadisorderedmedium.OneofthemistheGaus-
s
i ior,anddimensionalreduction,whicharedifficultiespresent sian variational approximation (GVA) performed in replica
d in a broader class of disordered systems [1, 2, 3]. They can space, which can be applied to both classes of elastic man-
.
t be used to model a remarkable set of experimental systems. ifolds, i.e., to interfaces [10] without overhangsand to peri-
a
Domainwallsinmagnetsbehaveaselasticinterfacesandcan odicsystems[9,11]. Withinthisapproach,whichisbelieved
m
experienceeitherrandombonddisorder(RB)asinferromag- to be exact in the mean-field limit, i.e., when the manifold
-
d nets with nonmagnetic impurities, or random field disorder lives in a space of infinite dimensions, metastability is de-
n (RF) as in disordered antiferromagnets in an external mag- scribed by breaking of replica symmetry, which allows one
o neticfield[4]. Theinterfacebetweentwoimmiscibleliquids to computethe static correlationfunctionsand to obtaindif-
c in a porous medium exhibits the same behavior and under- ferent thermodynamic properties. Another method that can
[
goes a depinning transition as the pressure difference is in- be appliedto dynamicsas well as to statics is the functional

2 creased [5]. Charge-density waves (CDW) in solids show a renormalizationgroup(FRG)[12]. Simplescalingarguments
v similar conduction threshold [6]. Another example of peri- show that large-scale properties of elastic systems are gov-
4 odic systems are vortex lines in superconductors which can erned by disorder for d < d = 4 and that perturbation
uc
3 form differentglass phases in the presenceof weak disorder theory in the disorder breaks down on scales larger than the
2
[7, 8, 9]. In all these systems, the interplay between elastic so-called Larkinscale [13]. To overcomethis difficulty, one
9
forces that tend to keep the system ordered, i.e., flat or pe- performsarenormalization-groupanalysis. Itwasshownthat
0
6 riodic,andquencheddisorder,whichpromotesdeformations inthiscase onehasto renormalizethewholedisordercorre-
0 of the local structure, formsa complicatedenergylandscape lator that becomesa nonanalyticfunctionbeyondthe Larkin
/ withnumerousmetastablestates. Thisresultsinglassyprop- scale [12, 14, 15, 16]. The appearance of a non-analyticity
t
a ertiesandanontrivialresponseofthesystemtoexternalper- in the formof a cusp atthe originis related to metastability,
m turbations. In particular, the interface becomes rough with andnicelyaccountsforthegenerationofathresholdforceat
- displacementsgrowingwiththedistancexas thedepinningtransition. ItwasrecentlyshownthattheFRG
d
can unambiguouslybe extendedto higher loop order so that
n C(x) x2ζ, (1) theunderliningnonanalyticfieldtheoryisprobablyrenormal-
o ∼
c izable to all orders[17, 18, 19]. Althoughthe two methods,
whereζistheroughnessexponent.Elasticperiodicstructures
: GVA and FRG, are verydifferent, they providea fairly con-
v in the presence of disorder lose their strict translational or-
sistent picture of the statics, and recently a relation between
i
X derandformquasi-long-rangeordercharacterizedbya slow them was found [20]. There is also good agreement with
growthofdisplacements,
r results of numerical simulations, not only for critical expo-
a
nents[21,22,23]butalsoforthewholerenormalizeddisorder
C(x)= ln x, (2)
Ad | | correlator[24]. However, many questions remain open. Al-
thoughthedynamicsinthevicinityofthedepinningtransition
wheretheamplitude isuniversalinthesimplestcase. At
d
A andatzero temperatureis wellunderstood,there isno satis-
zero temperature, a driving force f exceeding the threshold
factorytheoryforfinitetemperature,andinparticularforthe
valuef isrequiredtosettheelasticmanifoldintosteadymo-
c
tion with a velocityv thatvanishesas v (f f )β at the thermalroundingofthedepinningtransition[25].Itisalsore-
c
∼ − markablethattheexponentβ inexperimentsondepinningis
transition point. The correlationlength divergesclose to the
transition f = f as ξ (f f )−ν and the characteris- usuallylargerthan1,whileFRGandnumericalsimulationsof
c c
tictimeasτ ξz,where∼z ist−hedynamiccriticalexponent. elasticsystemswithweakdisordergivevaluessmallerthan1.
∼
Notethattheroughnessexponentandtheuniversalamplitudes Most studies of elastic manifoldsin a disordered medium
2
treatuncorrelatedpointlikedisorder. Realsystems, however, modelwithLRcorrelateddefects[42]. Thisiscloselyrelated
often contain extendeddefects in the form of linear disloca- tothebehaviorofnematicliquidcrystalsenclosedinasingle
tions,planargrainboundaries,three-dimensionalcavities,etc. pore of aerosil gel, which was recently studied in Ref. [43],
It is known that such extended defects, or pointlike defects usingtheapproximationinwhichtheporehullisconsidered
withsufficientlylong-rangecorrelations,canchangethebulk a disconnected fractal. Finally, studies of the Kardar-Parisi-
criticalbehavior[26,27,28,29,30,31,32]. Fluxlinesinsu- Zhang (KPZ) equation with power-law correlations in time
perconductorsarethemostprominentexample. Thepinning [44] bear connections to the case d = 1 considered here.
ofthefluxlinesbydisorderpreventsthedissipationofenergy However, the perturbativemethod used there cannotaddress
anddeterminesthecriticalcurrentJ ,whichisofgreatimpor- directly the zero-temperature (strong KPZ coupling) phase,
c
tanceforapplications.Itwasfoundthatextendeddefectspro- contrarytoourpresentstudy.
duced,forinstance,byheavy-ionirradiation,canincreaseJ Inthe presentpaper,we study thestatics anddynamicsof
c
byseveralordersofmagnitude[33].Systemswithanisotropic elasticmanifoldsinthepresenceof(power-law)LRcorrelated
orientationof extended defectscan be describedby a model disorderusingtheFRGapproachtoone-looporder.Thepaper
in which all defectsare stronglycorrelatedin ε dimensions isorganizedasfollows. SectionIIintroducesthemodel. Pos-
d
and randomly distributed over the remaining d ε dimen- siblephysicalrealizationsareconsideredinSec. III. Section
d
−
sions. Thecaseε = 0isassociatedwithuncorrelatedpoint- IV describes the dynamical formalism and perturbation the-
d
like defects, while extended columnar or planar defects are ory.InSec. V,werenormalizethetheoryandderivetheFRG
related to the cases ε = 1 and 2, respectively. The bulk- equations to one-loop order. In Sec. VI, we study random
d
critical behaviorin the presence of this type of disorderwas bond, in Sec. VII, random field, and in Sec. VIII, periodic
studiedinRefs.[26,27,28,29]usingaperturbativeRGanal- disorder. In Sec. IX we discuss fully isotropicextendedde-
ysisinconjunctionwithadoubleexpansioninε=4 dand fects. Inthefinalsection,wesummarizetheobtainedresults
−
ε . Thepinningoffluxlinesbycolumnardisorderwasstud- andourconclusions.
d
ied in Ref. [34], where it was shown that the system forms
a Bose glass phase with flux lines strongly localized on the
columnar defects, resulting in a zero dc linear resistivity. It II. THEMODEL
wasarguedrecentlythatthetopologicallyorderedglassphase
(Bragg glass) formed by flux lines can be destroyed in the We consider a d-dimensional elastic manifold embedded
vicinityofasingleplanardefect[35]. Ithasbeenshownthat in a D-dimensionalspace with quencheddisorder. The con-
thesmalldispersioninorientationofcolumnardefectsformsa figuration of the manifold is described by an N-component
newthermodynamicphasecalled“splayedglass”[36].Inthis displacement field denoted below u(x), or equivalently u ,
x
phase, the entanglement of flux lines enhances significantly where x denotes the d-dimensional internal coordinate of
the transportof superconductors[37]. Competition between the manifold. For example, a domain wall corresponds to
varioustypes of disorder, pointand columnar, has also been d = D 1 and N = 1, a CDW to d = D and N = 1,
−
studied,atequilibrium[38,39]andinthemovingphases[40]. andafluxlatticetod = D andN = 2. Inwhatfollows,we
focusforsimplicityonthecaseN =1andelasticobjectswith
In the case of an isotropicdistribution of disorder, power-
short-rangeelasticity. ExtensionstoN > 1andLRelasticity
lawcorrelationsarethesimplestexamplewiththepossibility
arestraightforwardforthestatics. Theenergyofthemanifold
for a scaling behavior with new fixed points (FPs) and new
inthepresenceofdisorderisdefinedbytheHamiltonian
critical exponents. The bulk-critical behavior of systems in
which defects are correlated according to a power law r−a c
= ddx [ u(x)]2+V x,u(x) , (3)
forlargeseparationr was studiedin Refs. [30, 31, 32]. The
H 2 ∇
power-lawcorrelationofdefectsind-dimensionalspacewith Z h (cid:0) (cid:1)i
exponenta= d ε canbeascribedtorandomlydistributed wherec istheelasticity andV is arandompotential. Inthis
d
extendeddefects−ofinternaldimensionε withrandomorien- paper,westudythemodelwherethesecondcumulantofthe
d
tation.Forexample,a=dcorrespondstouncorrelatedpoint- randompotentialhastheform
like defects, a = d 1 (a = d 2) describesinfinite lines
(planes) of defects w−ith random−orientation. In general, one V(x,u)V(x′,u′) = R1(u−u′)δd(x−x′)
wouldprobablynotexpectapurepower-lawdecayofcorrela- +R2(u u′)g(x x′). (4)
− −
tions.However,ifthecorrelationsofdefectsarisefromdiffer-
The first part corresponds to pointlike disorder with short-
ent sources with a broad distribution of characteristic length
range(SR)correlationsininternalspace.Thesecondpartcor-
scales, one can expect that the resulting correlations will
respondstolong-range(LR)disorderininternalspaceandthe
over several decades be approximatedby an effective power
functiong(x) x−a atlargexwitha>0. Forconvenience,
law [30]. If the correlation function of disorder can be ex- wenormalizei∼tsothatitsFouriertransformisg(q)= q a−d
pressedasafinitesumofpower-lawcontributions icir−ai, atsmallqwithunitamplitude.Aprioriweareinterested|in| the
onecanexpectthatthescaling behaviorisdominatedbythe
P case a < d, wherethe correlationsdecaysufficiently slowly
termwiththesmallesta [30]. Power-lawcorrelationswitha
nonintegervaluea=d i df canbefoundinsystemscontain- ininternalspace. Wedenoteeverywherebelow q = (2dπdq)d
ingdefectswithfractal−dimensiond [41]. Forexample,the and = ddx. Theshort-scaleuvcutoffisimpliedatq Λ
behaviorof4HeinaerogelsisarguedftobedescribedbyanXY andthxesizeofthesystemisL. R R ∼
R R
3
One couldstartwith model(4), setting R = 0; however,
1
as we show below, a nonzero R is generated under coarse
1
graining. NotethatthefunctionsR (u)canthemselvesapri-
i
oribeSR,LR,orperiodicinthedirectionofthedisplacement
fieldu. ForSR disorderin internalspaceonly,i.e., R = 0,
2
thesecasesareusuallyreferredtoasrandombond(RB),ran-
domfield[R (u) u atlargeu](RF),andrandomperiodic
1
∼| |
(RP)universalityclasses. Belowwediscusshowtheseclasses
extendtothecaseofLRinternaldisorder(R nonzero).
2
The model (3) and (4) could easily be studied using
presently available numerical algorithms for directed mani-
folds, in its statics (e.g., exact ground-state determinations)
anditsdynamics(e.g.,criticalconfigurationatdepinning),by FIG.1:(Coloronline)Lineardefectsrandomlyandisotropicallydis-
directly implementing a random potential correlated as de- tributedonparallelplaneswithrandomdistancesbetweenthem.This
scribed by Eq. (4). It is also interesting to examine which geometrymimicsdistribution(6).
typeofcorrelationsinarandommediumcannaturallyleadto
Eq. (4) and how such disorder could be realized from, e.g.,
distributionsofextendeddefects,sincesomeofthemmaybe Although we mostly discuss extended defects, other
experimentallyfeasible. sources of long-range correlations are possible, such as de-
fects where each single one creates a long-ranged disorder
potential, or a substrate matrix itself quenched at a critical
III. REALIZATIONSANDUNIVERSALITYCLASSES point.
A. Defectpotential
B. Couplingtothemanifold
Letus first recallhow long-rangecorrelationscan arise in
Wenowexaminehowthelong-rangecorrelateddefectpo-
thepotentialcreatedbydefects. Tothispurpose,callv(r)the
tential couples to the elastic manifold and what type of LR
defectpotential,inthesimplestcasetakentobeproportional
model results. A general formulation of this coupling (see,
to defect density. Consider for simplicity a large number of
e.g.,[3])hastheform
weakdefectlineswithauniformandisotropicdistributionin
a space of dimension D. These create an almost Gaussian
randompotentialv(r)with V(x,u)= dD−dzv(x,z)ρ(x,z,u), (7)
Z
v2
v(r)v(r′) LR forr (5) where the defect potential lives in the D-dimensional space
∼ r r′ a →∞ parametrizedby(x,u)andx Rd istheinternalcoordinate
| − | ∈
ofthemanifold. ρ(x,z,u)isthemanifolddensity. Eachtype
and a = D 1. To derive this, consider defects of finite
− ofcouplingtothedisordercorrespondstoadifferentfunction
radius a . The probability that point r′ is contained in the
d ρ(x,z,u),andwenowindicatethemaincases.
defectgoingthroughr is (a /r r′ )D−1,i.e., inversely
d
∼ | − |
proportional to the sphere of radius r r′ . This is easily
| − |
generalized to isotropic distributions of extended defects of
1. Elasticinterfacesinrandombonddisorder
internaldimensionε ,witha=D ε .Notethatbyextended
d d
−
defects we mean defects that are perfectly correlated along
Letusfirstdiscusselasticinterfacesintheso-calledrandom
their internal dimension. Generalizations where defects are
bond(RB)case,wherethecouplingbetweendisorderandin-
themselves(anisotropic)fractalscanalsobeconsidered.
terfaceoccursonlyinthevicinityoftheinterfaceas,e.g.,for
Animportantcaseisauniformdistributionofextendedde-
domain walls in magnets with random bond disorder. This
fectsinD-dimensionalspace,butisotropiconlywithinalin-
earsubspaceofdimensionD′. Forinstance,onecanirradiate correspondstothechoice
amaterialinthebulkwhilesimultaneouslyrotatingitalongan
ρ(x,z,u) δ(z u), (8)
axis. Thisproducesa distributionoflineardefects(εd = 1), ∼ −
isotropic within the plane (D′ = 2), and normal to the axis hencetheadditionalvariablezintroducedinEq.(7)isidenti-
(seeFig.1).Moregenerally,thisyieldsadefectpotentialwith caltou,thedisplacementfield(withingeneralD d=N ).
secondcumulant, Inthatcase, −
v(r,z)v(r′,z)=g(r r′)f(z z′), (6)
V (x,u) v(x,u). (9)
− − RB
g(r) r−a , ∼
∼ Consider now a uniform distribution of defects in the D-
whilef(z)isshort-ranged(herer RD′,z RD−D′,a = dimensionalplanebutisotropicallydistributedwithinthe(av-
D′ ε ). ∈ ∈ erageddirection)oftheinternalspaceofthe manifold. This
d
−
4
isgivenbyEq. (6)abovewithD′ =d, thatsubspace.TherandompotentialexperiencedbytheCDW
isgivenby
V (x,u)V (0,0)=g(x)R (u), (10)
RB RB 2
V(x,φ)=h (x)cosφ(x)+h (x)sinφ(x), (13)
1 2
whichismodel(4)withaSRfunctionR (u)and,infullgen-
2
with Gaussian distributed h (x) = v(x)cos(2k x ) and
erality, a = d ε . Thephysicalrealizationin termsof ex- 1 F ⊥
d
− h (x) = v(x)sin(2k x ). Onlargescalesk x 1, and
tendeddefectsisthusaninterface(d=2)inD =3withline 2 F ⊥ F ⊥
≫
theircumulantcanbeapproximatedby[fromEq. (6)]
defects all orthogonal to the u directions, isotropically dis-
tributedwithinthe(average)planeoftheinterface,anda=1.
1 1v2
ThisisillustratedinFig. 1. h (x)h (0)= v2 δ δd(x)+ LRδ δ(x ), (14)
i j 2 SR ij 2 xa ij ⊥
Another physical realization consists of extended defects k
withfiniterandomlengthssuchthatthedistributionoflengths
where we have omitted all rapidly fluctuating contributions.
has a power-law tail for large lengths. For instance, needles
Equations(13)and(14)givethepotentialcorrelatorinaform
ofvariablelengthsalignedalongonedirectioncouldactona
thatcanbegeneralizedto
directed polymer d = 1 as power-law correlated disorder in
internalspace.
V(x,u)V(x′,u′)=R (u u′)δd(x x′)
An interesting, though qualitatively different, case occurs 1 − −
whentheextendeddefectsaredistributedisotropicallyinthe +R2(u−u′)g(xk−x′k)δd⊥(x⊥−x′⊥), (15)
whole(x,u)space.ThiswillbediscussedinSec. IX. Finally,
note that we considerweak Gaussian disorder. It is possible withd⊥ =1andbarefunctionsRi(φ)= 12vi2cos(φ),u≡φ.
thatatstrongdisorderanotherphase existswherethe line or Thus periodic systems are described by periodic functions
manifoldgetslocalizedalongthestrongestextendeddefect. Ri(u). Hered⊥ isthedimensionofthetransversesubspace.
NotethattheHamiltonian = ddx[1( φ)2+V(x,φ)]
HXY 2 ∇
withV(x,φ)givenbyEq.(13)andaGaussiandistributionof
R
2. Elasticinterfacesinrandomfielddisorder fieldshi(x)hj(x′) g(x x′)describestheXY modelwith
∼ −
long-rangecorrelatedrandomfields. Therefore,thelattercan
be mapped onto periodic manifolds with correlator (15) and
Randomfield(RF)disorderisdescribedbythefunction
d = 0, i.e., tomodel(4) withperiodicfunctionsR (u). In
⊥ i
thenextsection,wewillshowhowtheFRGpictureofmodel
ρ(x,z,u) Θ(u z), (11)
∼ − (15) can be obtainedfrom the FRG results for model(4). It
where Θ(z) is the Heaviside step function. This means that isworthwhiletonotethatinthecaseofperiodicsystems,the
the changein energywhen the interfacemovesbetween two integration in Fourier space is supposed to be over the first
configurationsis proportionalto the sum ofall defectpoten- Brillouin zone. Note also that we have neglected the cou-
tialsinthevolume(inRD)spannedbythischange. Thedis- pling of disorder to the long wavelength part of the density
cussionofthegeometryofdefectsneededtoproduceLRdis- ρ0 ddxv(x) u(x)asitisusuallyirrelevantneartheupper
− ∇
criticaldimension. Indeed,inthereplicatedHamiltonian(see
orderininternalspaceisidenticaltothelastsection.Substitu-
R
below), this coupling generates additionally to the SR term
tionofEq.(11)intoEq.(7)yieldstheRFdisordercorrelator,
whichcanbeapproximatedbyEq.(4)withR (u) ufor 1/T ddxσ1 ua(x) ub(x)theLRterm
i − ∇ ∇
∼ −
largeu.
1 R
ddxddx′σ g(x x′)δd⊥(x x′ ) u (x) u (x′).
−T 2 k− k ⊥− ⊥ ∇ a ∇ b
Z
3. Periodicsystems Forsmalldisorderinthevicinityoftheuppercriticaldimen-
sion,bothofthemrenormalizetozeroaccordingto
Asanexampleofperiodicsystems,weconsiderincommen-
surate single-QCDWs. In thatcase D = d, hencethe func- dℓσ1 = (2 d 2ζ)σ1+..., (16)
− −
tionρ(x,z,u) = ρ(x,u) in Eq. (7). Theelectrondensityof d σ = (2 a d 2ζ)σ +... (17)
ℓ 2 ⊥ 2
CDWs neglectingeffectscausedbyan appliedstrain hasthe − − −
form[3,6]
IV. DYNAMICALFORMALISM
ρ(x,φ) ρ +ρ cos 2k [x u(x)] , (12)
0 1 F ⊥
∼ { − }
Theoverdampeddynamicsoftheelasticmanifoldinadis-
wherethedisplacementu(x)ofthemaximumofthedensity
orderedmediumcanbedescribedbytheequationofmotion
is related to the standardphase field via φ(x) = 2k u(x),
F
−
wherekF istheFermiwavevector. Thed-dimensionalspace η∂tuxt =c 2uxt+F(x,uxt)+fxt, (18)
is split intox = (x ,x ), with x denotingthe modulation ∇
k ⊥ ⊥
directionoftheCDWandk theFermiwavevector. where η is the friction coefficient. In the presence of
F
We again consider the situation of extended defects all an applied force f, the center-of-mass velocity is v =
aligned with the direction x and isotropically distributed in L−d ∂ u . The pinning force reads F = ∂ V(x,u),
k x t xt − u
R
5
andthus,forcorrelator(4),thesecondcumulantoftheforce
isgivenby
F(x,u)F(x′,u′) = ∆ (u u′)δd(x x′)
1
− −
+∆ (u u′)g(x x′), (19)
2
− −
with ∆ = R′′(u) in thebaremodel. Inthe following,we
willalwiays−useig(q)= q a−dandg(x)= eiqxg(q).
| | q
Themostimportantquantityofinterestistheroughnessex-
R
ponentζ measuredin equilibriumorat the depinningtransi- FIG.2: Diagrammatic rules: a, propogator; b, SRdisorder vertex;
tionf =fcdefinedby c, LR disorder vertex; d and e, one-loop diagrams generating the
criticalforceatthedepinningandgivingcorrectiontothemobility
C(x x′)= ux ux′ 2 x x′ 2ζ. (20) andelasticity.
− | − | ∼| − |
Thevelocityvanishesatthedepinningtransitionasv f
f β,whilethecorrelationlengthdivergesatthetrans∼itio|na−s elasticity. Thequadraticpartoftheaction(21)yieldsthefree
c
ξ | f f −ν. Onecanalsointroducethedynamiccritical responsefunction
c
∼ | − |
exponentz,whichrelatesspatialandtemporalcorrelationsvia
t xz. uqtiuˆ−q0 0 =Rqt = Θ(t)e−(cq2+m2)t/η, (23)
∼ h i η
Let us briefly sketch how one can construct the perturba-
tiontheoryin disorder. We adoptthedynamicformalism. It whichcanbeusedtogeneratetheperturbationtheoryindis-
alsoallowsustoobtainthestaticsequations(tooneloopand order.Thetheoryhastwodisorderinteractionvertices∆ (u)
1
N = 1thesecaneasilybededuced,ascanbecheckedusing and ∆ (u). At each vertex ∆ (u) there is one conservation
2 i
replica).Insteadofadirectsolutionoftheequationofmotion ruleformomentumandtwo forfrequencywhile eachvertex
(18) with consequent averaging over different initial condi- ∆ (u)carriesadditionalmomentumdependence.Inwhatfol-
2
tions and disorder configurations, we employ the formalism lows, wegeneralizethesplitteddiagrammaticmethoddevel-
of generating functional. Introducing the response field uˆ oped in Ref. [18], shown in Fig. 2. As is the case for the
xt
wederivetheeffectiveaction,whichreads modelwithSRdisorder,ourmodelexhibitsthe so-calleddi-
mension reduction, both in the statics and in the dynamics.
S = iuˆ (η∂ c 2+m2)u iuˆ f The naive perturbation theory obtained taking the functions
xt t xt xt xt
− ∇ − ∆ (u)analyticatu = 0leadstothesameresultasthatcom-
Zxt Zxt i
1 putedfromtheGaussiantheorysetting∆i(u)=∆i(0). Inthe
−2Zxtt′ iuˆxtiuˆxt′∆1(uxt−uxt′) limitm→0,thetwo-pointfunctionthenreadstoallorders:
−21Zxx′tt′ iuˆxtiuˆx′t′g(x−x′)∆2(uxt−ux′t′),(21) uqtu−qt′ = ∆c21q(04) + c2∆q42+(0d−)a. (24)
where we have added a small mass m, which plays the role The first term in Eq. (24) dominates in the limit q 0 for
→
ofanIRcutoff. Tostudythecriticaldomain,onehastotake a d, and LR disorder is irrelevant in this case, while the
≥
thelimitm 0. TheaverageoftheobservableA[u ]over last term dominates for a < d. Equation (24) results in
xt
dynamictraj→ectorieswithdifferentinitialconditionsandover ζ =(4 d)/2fora dandζ =(4 a)/2fora<d,which
− ≥ −
differentdisorderconfigurationscanbewrittenasfollows: areknowntobeincorrect. Thephysicalreasonforthisisthe
existenceofalargenumberofmetastablestates. Therough-
nessexponentcanbeestimatedusingFloryargumentssetting
A[u ] = [u] [uˆ]A[u ]e−S[u,uˆ]. (22)
h xt i D D xt u xζ. Thenthegradienttermscalesas 2ux xζ−2. The
Z pin∼ningforceforSRdisorderscalesasF(x∇,u )∼ x−(d+ζ)/2
x
Furthermore, the response to the external perturbation fxt, andforLRdisorderasF(x,ux) x−(a+ζ)/2. T∼herefore,in
which is local in time and in space, can be computed using the regime where the behavior is∼governed by SR disorder,
hA[uxt]iuˆxti = δfδxthA[uxt]i. Notethatcausalityisfulfilled, the Flory estimate gives for RF disorder the Imry-Ma value
andhereweadopttheItoconvention,whichresultsingetting ζF = (4 d)/3 while for LR RF disorder we get ζF =
SR − LR
ridofallclosedloopscomposedofresponsefunctions. (4 a)/3. Asimilarargumentconstructedfromthepotential
−
In the absence of LR correlated disorder action, Eq. (21) correlatorsR (u)yieldstheFloryestimatesζF =(4 d)/5
exhibitsthe so-calledstatisticaltiltsymmetry(STS),i.e., the and ζF = (i4 a)/5, respectively, for theScRase of r−andom
LR −
invarianceofthedisordertermsunderthetiltu u +h bonddisorder. To obtaincorrectionsto the Flory values, the
xt xt x
→
withanarbitraryfunctionh .TheSTSgivestheexactidentity FRG developedin Refs. [12, 14, 15, 16, 17, 18] willbe em-
x
=1/cq2fortheresponsefunction = u iuˆ , ployed. The solution is nontrivial because the renormalized
tRqt Rqt h qt −q0i
whichimpliesthattheelasticityisuncorrectedbydisorderto disorderbecomesnonanalyticabovetheLarkinscale,andone
R
allorders. LRcorrelateddisorderdestroystheSTSofaction hastodealwithanonanalyticfieldtheory.Herewegeneralize
(21),andthus,inprinciple,allowsforarenormalizationofthe thisapproachtothecaseofLRcorrelateddisorder.
6
V. FUNCTIONALRENORMALIZATION
a b
We now consider the renormalizationof model (21). The
subtleties arising for the correlator (15) will be discussed
briefly at the end. We carry out perturbation theory in the
baredisordercorrelators∆ (u)andthenintroducetherenor-
i0
malizedcorrelators∆ (u). Wewillsuppressthesubscript”0”
i
to avoid an overly complicated notation. According to the
c d
standard renormalizationprogram, we compute the effective
actiontoone-looporder. Hereweadoptthedimensionalreg-
ularization of integrals and employ the minimal subtraction
schemetocomputetherenormalizedquantitiesandabsorbthe
polesinε=4 dandδ =4 aintomultiplicativeZfactors. FIG.3: One-loopdiagramscorrectingdisorder. Thedottedlinecor-
− −
When derivativesof the ∆ at u = 0 occur, in the dynamics respondstoeitherSRdisordervertex(dashedline)ortoLRdisorder
i
(i.e.,atthedepinningtransitionfordynamicalquantities)they vertex (wavy line). Diagrams of type a, b, and c contribute to SR
aretakenatu = 0+ ascanbejustifiedexactlyforN = 1. In disorder.OnlydiagramsoftypedcorrecttheLRdisorder.
thestatics, thetreatmentismoresubtle(asdiscussedintwo-
loop studies [19]) but is not needed in the present one-loop
study. read
Letusfirstlyconsiderthefirst-ordertermsgeneratedbyex-
1 mˆ−ε
pansionofe−Sindisorder.Thesetermsaregivenbydiagrams I = =K + (1), (29)
1 (q2+mˆ2)2 4 ε O
dandeshowninFig.2. Westartfrom Zq
qa−d mˆ−δ
I = =K + (1), (30)
2 (q2+mˆ2)2 4 δ O
iuˆxt∆1(uxt uxt′)iuˆxt′ Zq
Zt>t′,x −
where we set mˆ = m/√c and K is the area of a d-
+ iuˆxt∆2(uxt ux′t′)g(x x′)iuˆx′t′.(25) dimensional sphere divided by (2π)d.d To remove the poles
Zt>t′,x,x′ − − in the mobility, we introduce the corresponding Z factor
η =Z−1[∆ ]η,whichtoone-looporderisgivenby
Expanding∆i(u)inaTaylorseriesandcontractingoneiuˆ,we R η i
obtain the leadingcorrectionsto the thresholdforce, friction
andelasticity. Thetermsgivingthethresholdforcetoleading Zη−1 =1−∆ˆ′1′(0+)I1−∆ˆ′2′(0+)I2. (31)
orderare
In the absenceof LR correlateddisorder, the elasticity re-
mainsuncorrectedtoallordersduetotheSTS,whileherethe
iuˆxt∆′1(0+)Rx=0,t−t′ correctionreads
Zt>t′,x
+Zt>t′,x,x′iuˆxt∆′2(0+)g(x−x′)Rx−x′,t−t′. (26) δc = 21d∆′2′(0)Zxtx2Rx,tg(x)
1 1
Tarheeynoanruensitvreornsgall.yTuvheditveremrgsinpgro(p∼orΛtiod−na2l+toΛ∆a−′′(20)+, a)ncdanthbues = −2d∆′2′(0)Zqg(q)∇2qcq2+m2
i K ε
rewrittenascorrectionstofrictionandelasticityusingtheex- = c d∆ˆ′′(0) mˆ−δ. (32)
pansion − d 2 δ
We have not set the second derivative at 0+ as ∆ remains
∂ 2
uxt ux′t′ =(t t′)∂tuxt+(x x′)i uxt analyticasisdiscussedbelow. Furthermore,thecorrectionto
− − − ∂x
i elasticity(32)isfiniteforε,δ 0,andthuscdoesnotacquire
1 ∂2u →
+(x x′) (x x′) xt + (∆t2,∆x3).(27) ananomalousdimension. However,weexpectcorrectionsat
i j
− − 2∂xi∂xj O two-looporder. If thisis the case, one hasto introducea Z-
factorthatrenormalizeselasticity:c =Z−1[∆ ]cwithZ =
R c i c
ThefirstterminEq.(27)givesthecorrectiontofriction, 1+ (∆2).
O i
In principle, due to the lack of STS, the KPZ term
δη = −∆′1′(0+)ZttRx=0,t−∆′2′(0+)ZxttRx,tg(x) iλn(∇thuexetq)u2abtiroenakoifnmgothtieosny(m18m)eattrythued→ep−inunicnagntrbaensgietinoenr.atIend-
= η ∆ˆ′′(0+)I +∆ˆ′′(0+)I , (28) deed, diagram e in Fig. 2, when expanding ∆(u) to second
− 1 1 2 2 orderinu,usingEq.(27),gives
h i
wherewehaveintroduced∆ˆ (u) = ∆ (u)/c2. Theone-loop 1
i i δλ= ∆′′′(0+) x2R g(x). (33)
integralsI1 and I2 divergelogarithmicallyand for ε,δ → 0 2d 2 Zxt x,t
7
Moreover,thetermwithcubicsymmetry(M = 2)andterms NotethatinthepresenceofSTS,ψ = 0,andwerecoverthe
with higher-order symmetries (M > 2) λ (∂ u )2M conditions obtained at the end of Sec. III. The actual value
M i i xt
canbegeneratedbydiagrame, of ζ will be fixed by the disorder correlators at the FP. The
P
elasticityexponentψandthedynamicexponentzread
1
δλ = ∆(2M+1)(0+) R g(x) x2M. (34)
M d(2M)! 2 Zxt x,t Xi i ψ = −mddmlnZc(∆˜i) 0, (42)
However,aswewillshowlater,ifwestartfrombareanalytic d (cid:12)
disorderdistribution, the LRdisorderremainsanalyticalong z = 2+ψ+mdmlnZ(cid:12)(cid:12)η(∆˜i) 0, (43)
theFRGflowandthecorrespondingFPvalue∆∗(u)isalsoan (cid:12)
2 where subscript “0” means a derivative at co(cid:12)nstant bare pa-
analyticfunction.Thusterms(33)and(34)arezero,provided (cid:12)
rameters.Toone-looporderthisyields
thattheyareabsentinthebeginning.Moreover,theterms(34)
are irrelevant in the RG sense for M > 2 [but not the KPZ
ψ = (ε2,εδ,δ2), (44)
term(33);seeRef.[46]].Thisprovesthatourbaremodel(21) O
isaminimalmodelforthedescriptionofelasticmanifoldsin z = 2 ∆˜∗′′(0) ∆˜∗′′(0). (45)
− 1 − 2
arandommediawithLRcorrelateddisorder.
The corrections to disorder are given by the diagrams Thescalingrelationsthenread[46]
showninFig.3. Thecorrespondingexpressionsread
1
ν = , (46)
δ1∆ˆ (u)= ∆ˆ′(u)2+[∆ˆ (u) ∆ˆ (0)]∆ˆ′′(u) I 2 ζ+ψ
1 −{ 1 1 − 1 1 } 1 − z ζ
−{2∆ˆ′1(u)∆ˆ′2(u)2+[∆ˆ2(u)−∆ˆ2(0)]∆ˆ′1′(u) β =ν(z−ζ)= 2 −ζ+ψ. (47)
+∆ˆ (u)∆ˆ′′(u) I [∆ˆ′(u)2+∆ˆ (u)∆ˆ′′(u)]I , (35) −
1 2 } 2− 2 2 2 3
δ1∆ˆ (u)= ∆ˆ (0)∆ˆ′′(u)I ∆ˆ (0)∆ˆ′′(u)I . (36) At zero velocity, the above calculation can be consid-
2 − 1 2 1− 2 2 2 eredasa dynamicalformulationoftheequilibriumproblem.
The one-loop integrals I and I have been defined in However, one has to be careful with mapping the dynamic
1 2
Eqs.(29)and(30),whereasI isgivenby FRG equations to the static equations, because as shown in
3
Ref. [19] the bare relation ∆ = R′′(u) breaks down for
i − i
q2(a−d) K mˆ−2δ+ε theSRcaseattwo-looporder. Thestandardderivationofthe
I = = 4 + (1). (37)
3 (q2+mˆ2)2 2δ ε O FRG equationsin the statics is based on the renormalization
Zq − of the replicatedHamiltonian. We have checkedthatsimilar
Letusdefinetherenormalizeddimensionlessdisorder∆Ras toothersystemswithonlySRdisorder,thestaticFRGequa-
i
tionsforsystemswith LRdisordercanbe obtainedfromthe
mε∆R =∆ˆ (u)+δ1∆ˆ (u), (38) dynamicflow equationsto one-looporderusing the identifi-
1 1 1 cation∆˜ = R′′(u). Theyread
mδ∆R =∆ˆ (u)+δ1∆ˆ (u). (39) i − i
2 2 2
∂ R (u) = (ε 4ζ)R (u)+ζuR′(u)
Theβ functionsaredefinedasthederivativeof∆R(u)with ℓ 1 − 1 1
i i 1
dreesrpteoctatttoaitnheamfixaesds pmoiantt,fiixteisdnbeacreessdairsyortdoerre∆scia(lue)t.heInfieolrd- +2[R1′′(u)+R2′′(u)]2+AR1′′(u), (48)
u by mζ and write the β functions for the functions ∆˜i =: ∂ℓR2(u) = (δ−4ζ)R2(u)+ζuR2′(u)+AR2′′(u), (49)
K m−2ζ∆R(umζ),
4 i whereA= [R′′(0)+R′′(0)].
− 1 2
∂ ∆˜ (u) = (ε 2ζ)∆˜ (u)+ζu∆˜′(u) InthecaseofthemodelwithcorrelatorgivenbyEq.(15),
ℓ 1 − 1 1 onehastodistinguishbetweenthetransverseandparalleldi-
1 d2
[∆˜ (u)+∆˜ (u)]2+A∆˜′′(u), (40) rections,andthereforeintroducecorrespondingelasticcoeffi-
− 2du2 1 2 1 cientsc andc . Inthetransversedirection,disorderisonly
⊥ k
∂ℓ∆˜2(u) = (δ−2ζ)∆˜2(u)+ζu∆˜′2(u)+A∆˜′2′(u), (41) δ-correlatedandasaresultthetransverseelasticityisnotcor-
rectedandcanbesetto1.Thepowercountingshowsthatthe
whTehreeAsca=lin[∆˜g1b(e0h)a+vio∆˜r2o(f0)th]eansdys∂teℓm:=is−cmon∂t∂mro.lled by a sta- LTRhedoinsoer-dloeorpisinntaeigvreallysraerleevloagnatrfiothrmδ1ica=lly4d−ivedr⊥ge−ntaand<fo0r.
ble fixed point [∆˜∗(u),∆˜∗(u)] of flow equations (40) and ε,δ 0aregivenbyEqs.(29),(30),and(37)withδ δ .
1 2 1 1
(41). To determine the critical exponents, let us start from Thus→the aboverenormalizationcan be generalizedto m→odel
powercountingfollowingRef.[46]. Thequadraticpartofac- (15)ifoneformallyreplacesδ δ .
1
tion (21) is invariant under x xb, t tbz, u ubζ, Letusshowhowanonanalyt→icityofthedisorderappearsin
uˆ uˆb2−z−ζ−d+ψ. Underthis→transform→ation,them→obility, theproblem. Westartfromthebareanalyticcorrelatorswith
ela→sticity, and disorder scale at the Gaussian FP as c bψ, ∆˜′′(0) <0. Theflowequationfory := ∆˜′′(0) ∆˜′′(0)
η ∼ b2−z+ψ, ∆˜1 ∼ b4−d−2ζ+2ψ, and ∆˜2 ∼ b4−a−2∼ζ+2ψ. R1′i′′′(0)+R2′′′′(0)>0reads − 1 − 2 ≡
Thus SR disorder becomes relevant for ζ ψ < (4 d)/2
whileLRdisorderisnaivelyrelevantforζ− ψ <(4 −a)/2. ∂ y =εy+3y2+γ(m), (50)
ℓ
− −
8
whereγ(m) = (ε δ)∆˜′′(0). Aswe showbelow,thefunc- Usingourfreedomto rescaleR (u), we introduceδˆ:= δ/ε,
tion∆˜2(u)remains−analyt2icalongthewholeFRGflowandat Ri(u) =: εri(u) and fix r1′′(0)i =: −x and r2′′(0) = −1,
thefixedpoint(FP).ThesolutionofEq.(50)foranyfunction where x is the parameter to be determined. The stationarity
γ(m)boundedfrombelowblowsupatsomefinitescalem∗, conditionofEqs.(48)and(49)reads
whichcanbeassociatedwiththeinverseLarkinlength. This
blowupofycorrespondstothegenerationofacuspsingular- 1 4δˆ r (u)+ δˆur′(u)
ity:∆˜ (u)becomesnonanalyticattheoriginandacquiresfor − 5 1 5 1
1
m < m∗ a nonzero ∆˜′1(0+). The precise estimation of the (cid:16) +(cid:17)1[r′′(u)+r′′(u)]2+(1+x)r′′(u)=0, (55)
Larkinscalerequiresthesolutionofthepairofflowequations 2 1 2 1
forboth∆˜i(u). δˆ δˆ
Before studying different FPs, let us note an important 5r2(u)+ 5ur2′(u)+(1+x)r2′′(u)=0. (56)
propertythatisvalidunderallconditions:if∆˜ (u)(i=1,2)
i
is a solution of Eqs. (40) and (41), then κ2∆˜ (u/κ) is also SinceEq.(56)islinearinr2,itcanbesolvedforfixedxby
i
a solution. Analogously, if R (u) is a solution of Eqs. (48)
and(49),thenκ4R (u/κ)isalisoasolution. We canusethis 5(1+x) δˆu2
i r (u)= exp . (57)
property to fix the amplitude of the function in the nonperi- 2 δˆ −10(1+x)!
odiccase,whilefortheperiodiccasethesolutionisuniqueas
theperiodisfixed. FromtheTaylorexpansionofEq.(55)aroundu=0,wefind
5(1 x2)
r (0) = − ,
VI. NONPERIODICSYSTEMS:RANDOMBOND 1 8δˆ 10
−
DISORDER r′(0) = 0, (58)
1
Inthissection, we studythescalingbehaviorof anelastic wherethesecondconditionexcludesthepossibilityofasuper-
interfaceinadisorderedenvironmentwithLRcorrelatedRB cusp(thefirstlinedoesnotdivergesincex=1forδ =5/4).
disorder. To this aim, we have to find a stationary solution Thusforfixedδˆthesimultaneousequations(55)and(56)have
(FP) of Eqs. (48) and (49) that decays exponentially fast at auniquesolutionforanyx,butonlyforaspecificxdoesthe
infinity as expected for RB disorder. The SR RB FP with solutionr1(u)decayexponentiallyfastto0forlargeu.Tode-
R (u)=0,whichdescribessystemswithonlySRcorrelated terminethisvalue,weemploytheshootingmethodchoosing
2
disorder,wascomputednumericallyinRefs.[12,17,18].The xasourshootingparameter.Forfixedx,weintegratenumer-
correspondingroughnessexponenttoone-looporderisgiven ically Eq. (55) with r2(u) givenby Eq. (57) from0 to some
byζSR =0.208298ε+ (ε2). WenowlookforaLRRBFP largeumaxwithinitialconditions(58). Thentheshootingpa-
withR (u)=0. IntegraOtingEq.(49),weobtain rameterx canbefoundbysolvingnumericallythe algebraic
2
6 equation r (u ;x) = 0. Increasing u , we acquire the
1 max max
∞ ∞ desiredaccuracyforxandr (x). Wewereabletofindthenu-
1
∂ℓ R2(u) = (δ 5ζ) R2(u). (51) mericalsolution with reasonableaccuracyonly for δˆ 1.1.
Z − Z ThetypicalFPfunctionsr∗(u)andr∗(u)areshownin≥Fig.4.
0 0 1 2
Theactualvaluesofxobtainedbyshootingfordifferentδˆare
Therefore,thenewLRRBFP,ifitexists,has summarizedinTableI.
Let us now check the stability of SR and LR FPs. To
δ
ζ = + (ε2,δ2,εδ). (52) that end we linearize the FRG equations about each FP. In
LRRB
5 O
the vicinity of a FP, the linearized flow equationshave solu-
tionsthatarepurepowerlawsinm,i.e.,scaleasm−λ witha
The direct inspection of diagrams contributing to the FRG
discrete spectrum of eigenvaluesλ. A stable fixed pointhas
equationforR showsthatthehigherorderscanonlybelin-
2 all eigenvalues λ < 0. Substituting r r∗(u) + z (u)
earinevenderivativesofR2(u). Theonlytermthatislinear i → i i
into the flow equations and keeping only terms that are lin-
inR (u)comesfromtherenormalizationoftheelasticityand
2
earinz (u),wederivethelinearizedflowequationsattheFP
canberewrittenas2ψR (u)toallorders.Therefore,inhigher i
2 r∗(u),r∗(u) ,
orderswehave { 1 2 }
∞ ∞ (1 4ζ )z (u)+ζ uz′(u)+[r∗′′(u)+r∗′′(u)]
− 1 1 1 1 1 2
∂ R (u) = (δ 5ζ+2ψ) R (u), (53) [z′′(u)+z′′(u)]+(1+x)z′′(u)
ℓ 2 − 2 × 1 2 1
Z0 Z0 + A0r1∗′′(u)=λz1(u), (59)
athnedFaRsaGcroenssuelqtiunegnicnet,he0∞exdaucRti2d(eun)tiitsyexactlypreservedalong (δˆ +− 4(1ζ1+)zx2)(zu2′)′(+u)ζ+1uAz2′0(ru2∗)′′(u)=λz2(u), (60)
R
ζLRRB = δ+2ψ. (54) wanhderλewiseahlasvoeminetarosudruecdedinζu=niεtsζ1o,fAε0 .=N−o[zte1′′(t0h)at+bze2′c′(a0u)s]e,
5
9
TABLEI:Long-rangecorrelatedrandombondfixedpoint.Shooting 1.0
parameter x = −r′′(0), themaximal eigenvalue and theuniversal
1
amplitudefordifferentvaluesofδˆ. 0.8
-r1* ’’
δˆ x(δˆ) λ1 B(δˆ) 0.6 -r2* ’’
1.1 1.931986 33.89
0.4
1.2 1.121722 −0.160 31.37
1.3 0.922046 −0.262 31.64 0.2
1.4 0.825747 −0.365 32.41
1.5a 0.766976 −0.469 33.34 0.0
0 1 2 3 4 5 6 7 8
2.0b 0.639151 −1 38.44
-0.2
3.0c 0.562357 −2.120 48.10 u
∞ 0.463619 ∞ -0.4
aRandomlinesinaplanarinterface(d=2,a=1). FIG.4: Fixedpointdescribingtheinterfaceinamediumwithlong-
bcRRaannddoommlpilnaenseisninaa3d3dmmanainfoiflodld(d(d==3,3a,a==2)1.). rangecorrelatedrandombonddisorder(LRRBFP)forδˆ= 2. The
SRpartr∗(u)isanonanalyticfunctionwithr∗′′(0+)6=0. TheLR
1 1
partr∗(u)isananalyticfunction.Herewereportminustheirsecond
2
derivative.
of the freedom to rescale r (u), we always have the eigen-
i
mode z(0) with marginal eigenvalue λ = 0. As shown in
i 0
Ref. [45] for SR RB FP the corresponding eigenfunction is eigenvalueλ= 1andreads
−
givenby z(0) = ur∗′(u) 4r∗(u),z(0) = 0, while the next
eigenvalue1λ1 = −11 cor−respo1nds to2z1(1) = ζ1SRur1∗′(u) + z1(u) = −13ur1∗′(u)+ 21r1∗(u)+ 65r2∗(u), (63)
(1 4ζSR)r∗(u),z(0) = 0. Here r∗,r∗ = 0 is the SR 1
RB−FP a1nd t1he Tay2lor expansion of{th1e f2unction}r1∗ can be z2(u) = −3[ur2∗′(u)+r2∗(u)]. (64)
found in Ref. [45]. Thus the SR RB FP is stable in the
SR disorder subspace (r = z = 0). Let us check its ForothervaluesofδˆwesolveEq.(59)numericallyusingλas
2 2
stability with respect to introduction of LR correlated disor- ashootingparameterandrequireanexponentiallyfastdecay
der. From Eq. (60) it follows that the maximal eigenvalue ofz (u)forlargeu. Tocomputethe numericalsolution,we
1
λ = δˆ 5ζSR correspondstotheexponentialeigenfunc- need the initial conditions. Expanding Eq. (59) in a Taylor
tiomnaxz (u)−=ex1p( ζSRu2/2r∗′′(0))withr∗′′(0)= 0.577 series,weobtain
2 − 1 | 1 | 1 −
forSRRBFP.Asaconsequence,theLRcorrelateddisorder
destabilizesthe SR RB FP if δˆ > 5ζSR 1.041, or equiva- z (0) = 5[x2(2δˆ+5λ)+5x(δˆ+λ)+3δˆ], (65)
lently, using Eq. (52), if ζSR < ζLR1. T≈his criterion was of 1 3δˆ(5 4δˆ 5λ)
− −
courseexpected. z′(0) = 0. (66)
1
We now check the stability of the LR RB FP
{wri1∗th(ue)i,gre2∗n(fuu)nc6=tio0n}s.gItivaelsnobhyasza(0m) a=rgiunra∗l′e(uig)enva4lru∗e(λu0)=tha0t vAapluaretifsroλm1.thIteimsashrgoiwnanlfeoigrednifvfaelrueenλtδ0ˆ=>01,.1thienlaTragbelsetIe.igTehne-
canbecheckedbydirectsubstitiutionintoiEqs.−(59)iand(60). negativesignofλ1 reflectsthestabilityoftheLRRBFP.For
Equation(60)allowsforananalyticalsolutionthatreads δˆ 1.1wefailedtocomputethenumericalsolutionwithrea-
≤
sonableaccuracy. However,thelargesteigenvaluecomputed
5A δˆu2 δˆu2 atLRRBFPλ1tendsto0forδˆ 1.1andtheSRRBFPbe-
z2(u)= 2δˆ+05λ"5(1+x) −1#exp −10(1+x)!.(61) cdoismoredseur.nsTtahbulesfwoer δˆex>pe1ct.0t4h1atw→tihtehLreRspRecBt tFoPLRis-csotarbrelelatfeodr
δˆ>1.041.Moreover,thelargesteigenvaluewithinaccessible
Wearefreetofixthelengthoftheeigenvectors,forinstance,
accuracyiswellapproximatedbyλ = 0.1917(ζSR ζLR),
bytheconditionz′′(0)=1,whichgives 1 1 − 1
2 whichgivesλ = 0.06forδˆ=1.1.
1
−
Besidestheroughnessexponent,thereisanotheruniversal
1
A = (1+x)[2δˆ+5λ]. (62) quantitythatisofinterest.Thisisthedisplacementcorrelation
0 3δˆ function,whichbehaveslike
Thus to find the eigenvalue λ and the eigenfunction z1, we u u = q−(d+2ζ). (67)
havetosolveEq.(59)withconditionz′′(0) = 1 A and q −q Ad
1 − − 0
requireanexponentiallyfastdecayofthesolutionatlargeu. Let us show that in contrast to systems with only SR-
Theonlycase forwhichwesucceededto constructthesolu- correlateddisorder,thissystem,whosebehavioriscontrolled
tionanalyticallyisδˆ= 2,whichisdepictedinFig.5. Ithas by the LR RB FP, has a universal amplitude . Indeed,
d
A
10
LetuslookforasolutionofEqs.(40)and(41),whichde-
0.8 caysexponentiallyfastatinfinityasexpectedforRFdisorder.
-z1’’ From Eq. (41) it followsthat (hereafterwe dropthe tilde on
∆ )
i
0.4 -z2’’
∞ ∞
∂ ∆ (u) = (δ 3ζ) ∆ (u). (71)
ℓ 2 2
0.0 −
0 1 2 3 4 5 6 7 8 Z0 Z0

-0.4 u Therefore, 0∞∆2(u)ispreservedalongtheFRGflowfixing
theroughnessexponentto
R
-0.8 ζ = δ + (ε2,δ2,εδ), (72)
LRRF
3 O
FIG.5: Secondderivativeofeigenfunctions z (u)andz (u)com- which coincides with the Flory estimate. Introducing
1 2
putedattheLRRBFPforδˆ=2. ∆i(u) = εyi(u), ζ = εζ1 andfixingy1(0) = x, y2(0) = 1,
we can rewrite the stationary form of Eqs. (40) and (41) as
follows(ζ =δˆ/3):
1
according to Eq. (51), the integral duRtr(u) is preserved
2
along the flow and is fixed to its bare value Q, where we 1 d2
haveintroducedtheactualrenormaliRzedcorrelatorR2tr which (1−2ζ1)y1(u) + ζ1uy1′(u)− 2du2[y1(u)+y2(u)]2
is connected to R2 given by Eq. (57) by the relation Ritr = + [1+x]y′′(u)=0, (73)
κ4R (u/κ)withκgivenby 1
i (δˆ 2ζ )y (u) + ζ uy′(u)+[1+x]y′′(u)=0. (74)
− 1 2 1 2 2
3/10
Q1/5 δˆ Equation(74)canbesolvedanalyticallygiving
κ= , (68)
(2π)1/10 5(1+x)!
δˆu2
whereweused duRtr(u) = Q. Thentheamplitudecanbe y2(u)=exp . (75)
2 −6(1+x)!
writtentoone-looporderasfollows[19]:
R
Substituting the FP function (75) in Eq. (73), we obtain a
1
Ad = K [−R1tr′′(0)−R2tr′′(0)] closed differential equation for the function y1(u). Expand-
4 ingaroundu=0,wefind
κ2
= (1+x)=Q2/5B(δˆ), (69)
K4 y′(0) = 1 9x+3δˆ 6xδˆ, (76)
1 −3 −
wherewehaveintroducedtheuniversalfunction q
1 δˆx 2
8π2 δˆ 3/5 y1′′(0) = 3 − 9x−+1, (77)
B(δˆ)= (2π)1/5(1+x(δˆ))2/5 5! . (70) y2′(0) = 0, (78)
δˆ
y′′(0) = . (79)
Valuesforx(δˆ)andforB(δˆ)fordifferentδˆareshowninTa- 2 −3(x+1)
bleI.
Thuswecancomputenumericallythesolutiony (u)forany
1
fixedδˆand y (0) x, howeveronlyfor specialvaluesof x
1
≡
VII. NONPERIODICSYSTEMS:RANDOMFIELD doesthissolutiondecayexponentiallyatinfinity. Thecorre-
DISORDER spondingsolutioncanbecomputedusingtheshootingmethod
asdescribedabove,usingxasa shootingparameter(see Ta-
We now address the problem of an elastic interface in a bleII).
mediumwithLR-correlatedRFdisorder.Weexpectthatsim- A pair of typical FP functions is shown in Fig. 6. Sur-
ilar to systems with uncorrelated disorder, this universality prisingly, the function y (u) obtained by shooting satisfies
1
∞
classalsodescribesthedepinningtransition. Toseethatsys- du y (u) = 0, characteristic for RB-type correlations
0 1
temswithRBdisorderflowinthedynamicstotheRFFP,one along the u direction. In other words, the LR RF FP is in
R
has to include either effects of a finite velocity or consider factofmixedtype:RBfortheSRpartandRFfortheLRpart
two-loopcontributions,whichgo beyondofthe scopeofthe ofthedisordercorrelator. Thiscanbeunderstoodasfollows:
∞
presentwork;butweexpectthemechanismtobethesameas Considertheflowof duy (u). Itisobtainedbyintegrat-
0 1
inRef.[18]. ing the l.h.s. of Eq. (73) from 0 to infinity, and by inserting
R