Universal high-temperature regime of pinned elastic objects

Sebastian Bustingorry,1 Pierre Le Doussal,2 and Alberto Rosso3

1CONICET, Centro Atómico Bariloche, 8400 San Carlos de Bariloche, Río Negro, Argentina2CNRS-Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France

3CNRS–Université Paris-Sud, LPTMS, UMR 8626-Bât 100, 91405 Orsay Cedex, France�Received 24 August 2010; published 19 October 2010�

We study the high-temperature regime within the glass phase of an elastic object with short-ranged disorder.In that regime we argue that the scaling functions of any observable are described by a continuum model witha �-correlated disorder and that they are universal up to only two parameters that can be explicitly computed.This is shown numerically on the roughness of directed polymer models and by dimensional and functionalrenormalization-group arguments. We discuss experimental consequences such as nonmonotonous behaviorwith temperature.

DOI: 10.1103/PhysRevB.82.140201 PACS number�s�: 75.60.Ch, 05.70.Ln

Elastic objects pinned by quenched disorder are ubiqui-tous in nature, e.g., vortex lattices in superconductors1 andmagnetic domain walls.2 They are modeled by elastic mani-folds, of internal dimension d, parametrized by aN-component displacement field u�x�, submitted to randompotentials. These systems have been described using the col-lective pinning theory,3 in terms of the characteristic Larkinlength Rc, and more recently using functional-renormalization group �FRG� �Refs. 4 and 5� in terms of aRG fixed point at zero temperature with only few universal-ity classes depending on N ,d and the nature of elasticity anddisorder—short range �SR� or long range �LR�. For scaleslarger than Rc these objects exhibit glass phases with statis-tically self-similar ground states of roughness scaling as u�x� and self-similar energy landscape with free-energy ex-ponent �. Whenever ��0 the T=0 fixed point is attractiveand thermal fluctuations are irrelevant at low temperature forlarge systems. In some cases ��F�0 see below� the glassphase extends to all temperatures, with, however, a crossoverat T=Tdep. At high temperature T�Tdep, the system unbindsfrom individual pins but remains collectively pinned and theLarkin length increases with T.1,6,7 This has interesting con-sequences, e.g., a reentrant region in the phase diagram ofhigh Tc superconductors.8 It was generally argued that due tothis effect, the amplitude of the roughness decays with tem-perature for SR disorder.6,9,10

Besides its interest to experiments, the high-temperatureregime is also at the center of recent works on the directedpolymer �DP� d=1, a problem in close connection to Kardar-Parisi-Zhang �KPZ� growth and Burgers turbulence.11–14 Intwo dimension �N=1� it is amenable to the Bethe ansatzmethod using a continuum model with �-function correlationin the random potential.15 Since the collective pinning theoryand the FRG show that a finite correlation range in the dis-order is an important ingredient to describe pinning, an out-standing question has been the domain of validity of thismodel. Recently the distribution of the free energy F of poly-mers of length x was computed within the �-correlatedmodel.9,13,16,17 At large scale x one recovers the Tracy Wi-dom distribution,18 previously proved to hold19 for a discreteDP model but at T=0. At first sight, it suggests that the�-function model also captures the universality at low tem-

perature, in agreement with the idea of an effective scale-dependent temperature flowing to zero at large scale. How-ever, as emphasized in Ref. 9, the universal results obtainedfrom the �-function model are a priori correct only in theinfinite-temperature limit T→+� with u=u /T3 and x=x /T5

being kept fixed. Although the scaled distribution of F in thelarge size limit does not appear to depend on T, its varianceF2c�A�T�x2� exhibits an amplitude A�T� which could becomputed only at high T and it is found to decay withtemperature.9

In this Rapid Communication, we reexamine the universalhigh-temperature regime for the DP and for general pinnedelastic objects. We first argue from dimensional analysis that,in all cases where the disorder is SR and the glass phaseextends to all temperatures, the large T limit is described bya continuum model with Gaussian and �-correlated disorder.It contains only two parameters �, the elasticity, and g, theamplitude of the disorder. We then demonstrate numericallythat at high temperature the x-dependent roughness of the DPfor various discrete models falls onto a universal curve whenexpressed in the rescaled variables. The fit involves no otheradjustable parameters than g and � which, furthermore, canbe computed explicitly for each discrete model, leading to avery predictive theory. Next, we show within the FRG howto account for this high T universality and we recover thetemperature dependence of the rescaled variables which, asanticipated in Ref. 6 involves the Flory exponents.

We consider the standard model for pinning of partitionfunction Z=�D�u�e−H/T with

H = �x��

2��xu�x��2 + V�u�x�,x� , �1�

where �x=�ddx. The second cumulant of the random poten-tial is given by V�u ,x�V�u� ,x��c=g�d�x−x��R�u−u��, whereR�u� is a short-range function with �dNuR�u�=1. A smallscale cutoff x0 is implicit whenever needed. For zero averageGaussian-random potential one finds using replica that Zp

=�D�u�e−Srepwith

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Srep =�

2T�=1

p �x

��xu��x��2 −g

2T2 ��=1

p �x

R�u�,�� , �2�

where u�,�=u��x�−u��x�. To study the high-T regime wefirst substitute the rescaled coordinates and fields, x=bx, u=au in the replicated action, Eq. �2�. For d2 the smallscale cutoff x0 can be set equal to zero and the temperaturedependence is removed by the choice,

�bd−2a2

T= 1

bdg

T2aN = 1, �3�

which is solved as

a = ��2

g���F−2�F�/��F�4+N���T

����F/�F�

,

b = ��2

g�2/��F�4+N���T

���1/�F�

, �4�

where the first condition in Eq. �3� requires �F=d−2+2�F,and the second �F= �4−d� / �4+N�, i.e., the Flory roughnessand free-energy exponents. For d2, as discussed later, thepresence of a small but finite cutoff x0 slightly modifies Eq.�3�, and Eq. �4� remains valid if we replace �F by 2�F leadingto a= �T /��1/2 and b=Rg�T /��1/�2�F� where Rg= � �2

g �1/�4−d� issimilar to the T=0 Larkin length. In the glass phase, where�F ,�F�0, the high-temperature regime corresponds to largea. Using lima→� aNR�au�=�N�u�, the model becomesequivalent to

Srep = �x

1

2 �=1

p

��xu��x��2 −1

2 ��=1

p �x

�N�u�,�� , �5�

i.e., a continuum model where �, T, and g have been set tounity and the disorder is � correlated. Note that at the ther-mal fixed point T=� �and u�x1−d/2�, the disorder termscales as x �2d+N�d−2��/2�=x�1/2���F�4+N��. This means that for�F�0, the glass phase exists at all temperature.

Let us consider the high-temperature regime for muchmore general models. First one can break the statistical-tiltsymmetry �STS� u��x�→u��x�+��x�, which implies thenonrenormalization of �, by considering, e.g.,V�u ,x�V�u� ,x��c=R�u−u� ,x−x��. Then one easily sees thatthe same rescaling yields lima→�,b→� aNbdR�au ,bx�=�d�x��N�u�. Similarly a third cumulant of disorder yields ag3��u�,����u�,�� interaction in the replicated action but itsstrength decays as g3 /g�1 / �TaN� at high T and similarly forhigher cumulants. It is thus quite reasonable to expect thatprovided a model is SR it will fall at high T in the sameuniversal behavior as model �5� �corresponding to Gaussiandisorder� hence will be also described by universal scalingfunctions up to two parameters � and g.

We now explore the consequences of Eq. �5� on the DP�d=N=1� using numerics. We recall that, for this model, �=2 /3, �=1 /3, and the Flory exponents are �F=3 /5 and �F=1 /5. For clarity, we focus on the behavior of the roughnessin the so-called droplet geometry where a line of size x hasone end pinned at the origin �u�0�=0� and the other one isfree. Similar conclusions are expected also for other bound-

ary conditions and for other observables, such as the fluctua-tion of the free energy. A directed polymer is an elastic lineliving on a square lattice with a solid-on-solid restriction�u�x+1�−u�x��=1. Impurities of energy V�u ,x� are drawn oneach site of the lattice. The sum of the energies associated tothe sites linked by a polymer defines the energy of the poly-mer. Here we release the solid-on-solid restriction allowinglonger elongations �u�x+1�−u�x��=2j−1 �j=1, . . . ,n� withan elastic cost of j2. Figure 1�a� shows the geometry of themodel for n=2. For n=1 we recover the standard-directedpolymer. These lattice models violate STS because the re-striction on the local elongation generates nonharmonicterms in the elastic energy. For this reason, at low tempera-ture, we expect a renormalization of both � and g taking adifferent value respect to the bare ones. A major simplifica-tion occur in the high-temperature regime where one canexplicitly compute � and g. In practice, � can be extractedfrom the model without disorder for which the polymer be-haves like a particle diffusing on a one dimensional lattice�u�x� being the particle “position” at “time” x�. The mean-square displacement of the particle is given by �u2�x��T,n=Tx /�n. The ratio T /�n coincides with the mean-squarejump of the particle, i.e., with the mean-square local elonga-tion of the polymer, hence

�n�T� =

j=1

n

e−�j2

j=1

n

��2j − 1�2e−�j2

. �6�

The temperature behavior of �n is shown in Fig. 1�b�. At lowtemperatures, only the smallest elongation is activated and�n�T→0�=�1=T. At high T all allowed elongations areequally activated and �n�T→��=3T / �4n2−1�. The con-tinuum elastic string limit is achieved for large n and T withn2 T and �n→�=1 /2. Two impurity distributions wereimplemented: �i� V�u ,x� are uncorrelated Gaussian numbersof variance g �ii� “Poissonian” disorder where each site con-tains zero or one impurity �with probability p� of fixedstrength d. The second parameter is defined as g=�u,x�,�u�,x��V�u ,x�V�u� ,x��c. In the latter case although allthe disorder cumulants are nonzero, only the second one withg=d2p�1− p�, remains relevant in the large temperature limit.

The weight Zu,x of all polymers starting in �0,0� and end-ing in �u ,x� is given by the following recursion:

u

x

1

(a)

10-1

100

101

102

T

10-1

100

101

102

κ n(T)

n = 1

2345678

(b)

FIG. 1. �Color online� �a� Sketch of the model. Solid line cor-responds to a polymer with elongations up to n=2. �b� Temperaturedependence of �n for different n.

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Zu,x = e−�Vu,xj=1

n

e−�j2�Zu−2j+1,x−1 + Zu+2j−1,x−1�

with Zu,0=�u,0. The probability, for a given disorder realiza-tion, to observe a polymer ending in �u ,x� is Zu,x /uZu,x.Since Zu,x grows exponentially we divided all weights atfixed x by the biggest one11 which does not change the end-ing probability. The roughness is defined as the mean-squaredisplacement of the free end

B�x� = �u2�x��n,T = u

u2Zu,x/u

Zu,x. �7�

In the rescaled variable the roughness takes the form B�x�=H�x�, where H is universal once the boundary conditionsare specified. Here we have H�x→0�= x and H�x→��=cx4/3 where c is a universal constant. Hence in the originalvariables one has

B�x,T� = a2H�x/b� =T6

�n2g2H�x

�ng2

T5 � . �8�

This function has a crossover between a pure thermal and astrong-disorder regime for x T5 /�ng2, where

B�x,T� → c� g

�nT�2/3

x4/3. �9�

A decay of the roughness amplitude is thus expected at hightemperature. This nonintuitive behavior in T was pointed outin the past6,9 but never observed in numerical simulationssince high enough temperatures where not reached �see Ref.20 and references therein�. Here we further check the depen-dence in �n and g. In Fig. 2 we show B�x ,T� as a function ofthe temperature for a fixed large value of x. Three regimesare observed: �i� low temperature T�Tdep �here we estimateTdep�1�: B�x ,T�, at this large scale, is temperature indepen-dent. �ii� Roughness decay: for larger temperature B�x ,T� isa decreasing function of T well fitted by the asymptotic be-havior of Eq. �9�. �iii� Very high temperature T5 �ng2x: thepure-thermal regime B�x ,T�=Tx /�n. For n=1, the roughnessachieves a flat plateau while for n�1 as well as for thecontinuum elastic string a nonmonotonic behavior is ex-pected as shown here for n=8 �for any finite n the increasingroughness finally reaches a plateau not shown in the figure�.

Let us stress that if x is not large enough the roughness decayis not observed and the polymer crosses directly from a lowtemperature to a pure-thermal regime.

Finally, we test the full scaling relation Eq. �8�. The insetof Fig. 3 shows the raw data for different parameters. Asobserved in the main panel of Fig. 3, using Eq. �8� and nofree parameter, all the data are perfectly collapsed into asingle universal function, H�x�, with the fitted value c=0.54�0.01. This amazing scaling rely on the high-temperature properties. For the same models, at T=0, there isnot a universal function H0 and two model-dependent con-stants a, b such that B�x�=a2H0�x /b�.

We now study the high-T regime using the FRG. The

rescaled disorder correlator R�u�, defined through R�u�=Ad

−1x0d−4�2e−��−4���R�ue−��� �Ad a constant�, satisfies, under

coarse graining over short scales xx0e−�, the flowequation,4,5

��R�u� = �� − 4��R�u� + �uiRi��u� + Te−�lRii��u� + ��R� ,

�1loop�R� =1

2Rij� �u�Rij� �u� − Rij� �u�Rij� �0� , �10�

where Ri���uiR, T=Bdx0

2−dT /�, and we indicated the betafunction to one loop. The key point is that at high tempera-ture T, there is a first regime of the flow, 0��T, wherethe nonlinear terms are negligible. At ���T they match therescaling ones, leading to the approach to the T=0 fixedpoint as � �T. Although noticed in Ref. 7 all consequenceshave not been discussed.

For ��T it is simpler to integrate the flow equation of

the unrescaled R�u�, ��R�u�= Te�2−d��Rii��u�, hence

R��u� = �4�D��−N/2� du�e−�u − u��2/4D�R0�u��

with D�= T�0�e�2−d��. We first discuss d2. Then D�

� Te�2−d�� / �2−d� grows hence R��u�→g e−u2/4D�

�4�D��N/2 , where herewe define g=�du�R0�u��, which is the only memory of theinitial condition. This initial regime involves Flory expo-

10-1

100

101

102

T

104

105

106

B(x,T)

n=8

n=1

FIG. 2. �Color online� Temperature behavior of B�x ,T� for fixedx=16384. Circles �squares� corresponds to n=8�n=1�. Continuumlines correspond to the pure thermal regime. Dashed lines corre-spond to Eq. �9� with c=0.54.

10-3

10-2

10-1

100

101

102

103

104

~x

10-3

10-2

10-1

100

101

102

103

104

105

H(~ x)

n=1n=2n=8g=8, n=1P, n=1n=1, T=2

100

102

104x

100

106

B(x

,T)

FIG. 3. �Color online� Inset: raw data for B�x ,T�. Main: collapseusing Eq. �8�. Dashed lines emphasize the two limit behaviors. T=4 except for the last data set with T=2. P stands for Poissoniandisorder �with d=2 and p=0.2�.

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nents, i.e., if we choose �=�F, we see that g=�du�R�u�� ispreserved and

R��u� → g��−Nf� u

���,�� = �D�e−�F� � �Te−��F/2�� �11�

with the “fixed point” shape f�x�= �4��−N/2e−x2/4 and g=Adx0

4−dg�−2. Inserting Eq. �11� into Eq. �10� the rescaling

and temperature terms are of the same order while the �R��2

is smaller by a factor g /��4+N.21 The length �T is thus deter-

mined as g /��4+N�1. Defining b=x0e�T one recovers for b

formula �4� up to universal constants �x0 cancels�. For ���T one can choose the asymptotic value for � and theasymptotic form for the flow is then R���u�=e−��+2���−�T�+2�F�TR��ue−���−�T�−�F�T�. This gives the pre-dicted roughness decay as

u�− q�u�q� �− R�=ln�a/q�� �0�

q4 →T−2��−�F/�F�

qd+2� R���0� . �12�

Furthermore, the flow enters the low-temperature regionwhere nonlinear terms are important with a T-independent

initial condition since the shape of R�u� has converged tof�x�. The unique RG trajectory explains the universality inthe function H�x� and others.

For d�2, D�→ T / �d−2� which is large at high T. Uni-

versality still holds, with Eq. �11� and ��� 1d−2 Te−�F�. Insert-

ing Eq. �11� into Eq. �10� the temperature term is negligiblecompared to the rescaling term but �R��2 is again exactly

down by a factor g /��4+N. This leads to the result for b given

in Eq. �4� with �F→2�F, provided that in the initial model�2� one replaces � /T→x0

2−d� /T and g /T2→x02�2−d�g /T2 to

eliminate the x0 dependence in b. This implies that for d�2, at high T, the continuum model is the small x0 �rescaledcutoff� limit of

Srep =x0

2−d

2�

x�

�=1

p

��xu��x��2 − x0�2−d�

��=1

p

�N�u��� .

Moreover the roughness decay Eq. �12� holds if �F→2�F.Let us discuss LR disorder R�u��1 /u� at large u. The

roughness is known4 to be given by the LR Flory exponent�LR=�LR

F = 4−d4+� for ��c�N� such that �LR��SR. This means

that the roughness decay Eq. �12� is not present for LR dis-order, e.g., random-field disorder. The condition g finite im-plies ��N hence �LR�SR

F . Since �SRF �SR, a finite g guar-

antees that at high T the system is described by the SR fixedpoint. Similar conclusions hold for LR correlated disorder inx.22

We have demonstrated numerically on the DP that a uni-versal high-T regime exists, described by a two parameter�-correlated model. The dimensional and FRG argumentsshow that this universality within the glass phase, e.g.,

B�x�=a2H�x /b� for the roughness and F2c=T2F�x /b� for the

free-energy variance, extends to general d ��F�0� andboundary conditions.

This work was supported by ANR under Grant No. 09-BLAN-0097-01/2 and by MINCYT-ECOS under ContractNo. A08E03.

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