Functional Renormalization for pinned elastic systems away from their steady states

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by  Gregory Schehr

Functional Renormalization for pinned elastic systems away from their steady states
Gregory Schehr1 and Pierre Le Doussal2
1Theoretische Physik Universit¨at des Saarlandes 66041 Saarbru¨cken Germany
2LPTENS CNRS UMR 8549 24, Rue Lhomond 75231 Paris Cedex 05, France
(Dated: February 2, 2008)
5 Using one loop functional RG we study two problems of pinned elastic systems away from their
0
equilibriumorsteadystates. Thecritical regimeofthedepinningtransition isinvestigated starting
0
from a flat initial condition. It exhibits non trivial two-time dynamical regimes with exponents
2
andscaling functionsobtained in adimensional expansion. Theaging andequilibrium dynamicsof

n the super-rough glass phase of the random Sine-Gordon model at low temperature is found to be
a characterizedbyasingledynamicalexponentz≈c/T,whereccompareswellwithrecentnumerical
J work. This agrees with thethermal boundary layer pictureof pinned systems.

0
1 PACSnumbers:

]
n Disorderedelasticsystemsoffermanyexperimentalre- addition has explored the full temperature regime [14].
n
alizationsandarealsooftheoreticalinterestasprototype Thedynamicalexponentwasfoundtodivergeasz ∼1/T
-
s models for glasses induced by quenched disorder. The at low T. One aim of this paper is to show this result
di competition between the structural order and substrate within a simple one loop FRG, and to obtain detailed
. impuritiesresultsinpinning,complexgroundstates,bar- predictions for the aging regimes at low T. This study,
t
a riersandultraslowglassydynamics. Whilegroundstate, together with the rather good agreement with numerics,
m
equilibriumdynamics,anddrivensteadystateproperties isalsoimportantasindirectevidenceforarecenthypoth-
- havebeenmuchstudiedtheoretically,muchlessisknown esis that a ”thermal boundary layer” (TBL) in the field
d
about the dynamics before the steady state is reached, theorycontrolstheactivateddynamicsof(morestrongly)
n
o or about the aging dynamics. If universality is shown pinned manifolds [15].
c there, it would be of high interest for numerous exper-
[ imental systems, e.g. magnetic domain wall relaxation

1 [1], superconductors [2], contact line depinning [3], den- t(cid:0)t0
v sity waves [4].
9
Numericalstudiesofglassydynamicsarehamperedby
9 01=z global
highbarriersinconfigurationspaceresultinginultra-long t
1 steadystate
1 timescalesmakingcomparisonwiththeoryuncertain. In
0 some cases however faster, but still interesting dynamics q(cid:0)z
5 occurs. One is zero temperature T = 0 driven dynam- (t(cid:0)t0)1=z dvoamlidaiitnyof
0
ics near the depinning transition where barriers disap-
/
at pear [5]. Recent theoretical progress has been achieved t0
m there. Functional renormalization group (FRG) studies az q(cid:0)z Lz,(cid:24)z
givemorepreciseandconsistentpredictions[6],corrobo-
-
d rated by powerful new algorithms which allow for excel-
FIG. 1: (a) solid line: initial flat configuration, dashed and
n lent determination of steady state exponents [7, 8]. One dotted lines: configurations of the string at time t′ and t re-
o
aimofthispaperistoextendtheFRGtodynamicsaway spectively. (b)Domainofvalidity,inlogarithmicscale,ofour
c
from steady states, and to show that interesting univer- approach near the depinning transition. The grey circle rep-
:
v saltwo-timedynamics (analogousto the agingdynamics resentsthesmall timeregion sensitivetomicroscopic details.
i
X innondrivensituations)alsooccursneardepinning. We
r predict new exponents and scaling functions in the criti- Theoverdampeddynamicsofasinglecomponentelas-
a cal regime. tic manifold of internal dimension d, parametrized by a
height field u(x,t)≡u is described by
A second case where T > 0 dynamics has been inves- xt
tigated is the ”marginal glass” phase exhibited by topo- η∂ u =c∇2u +F(x,u +vt)+f−ηv+ξ(x,t) (1)
t x,t x,t x,t
logicallyordered2Dperiodic systems ascapturedby the
CardyOstlund(CO)model[9]. Barrierstheregrowonly where u = 0, hξ(x,t)ξ(x′,t′)i = 2Tηδd(x − x′)δ(t −
xt
logarithmically with size allowing for precise numerics t′) is the correlator of the thermal gaussian noise,
[10]. The equilibrium [11] and aging [12] dynamics were F(x,u)F(x′,u′) = ∆(u − u′)δd(x − x′) the second cu-
studied using Coulomb gas RG methods, but only near mulant of the quenched random pinning force and v is
the glass transition temperature T [13]. A numerical the average velocity. We denote the small scale cutoff
g
study has confirmed some of the RG predictions, and in sizea∼Λ−1. Inthispaperweconsideraflatinitialcon-
0
2
figuration u = 0. Denoting u the spatial Fourier deviations were found to occur at lower T. In addition,
xt=0 qt
transform of u , we focus on the correlation Cq and the (connected) structure factorC˜q is obtained fromEq.
x,t tt′ tt
the connected (w.r.t. the thermal fluctuations) correla- (6) in the limit w →0, u→1 keeping w/(u−1)=qzCOt
tion C˜q : fixed. Therefore the dynamical exponent z associated
tt′ CO
to equilibrium fluctuations coincides with the one asso-
Ctqt′ =huqtu−qt′i , C˜tqt′ =huqtu−qt′i−huqtihu−qt′i ciated to non-equilibrium relaxation, which was actually
(2) numerically computed in Ref. [14].
and the response Rqtt′ to a small external field h−qt′ InbothcasesEq. (1)isstudiedusingthestandarddy-
namical (disorder averaged) MSR action. Correlations
Rqtt′ =δhuqti/δh−qt′ , t>t′ (3) (2) and response (3) are obtained as functional deriva-
tives of the dynamical effective action Γ[u,uˆ]. It is per-
where .. and h..i denote respectively averages w.r.t. dis-
order and thermal fluctuations. In the following Rq = turbatively computed [12] using the Exact RG equation
η−1e−q2(t−t′)/η (respectively Cq ) denote the barett′re- (ERG) associatedto the multi-localoperatorsexpansion
tt′ introduced in [16, 17] and extended to non-equilibrium
sponse function (respectively correlation),i.e. in the ab-
dynamics in [12]. The ERG equations are obtained by
sence of disorder. We now specify the time regime t,t′.
varying an infrared (large scale) cutoff Λ = Λ e−l in-
In the first situation studied below, i.e. an interface l 0
troduced in the bare response and correlation functions
very near the depinning threshold, we set T = 0 in (1).
(R →R, C →C). The information about non equilib-
NumericalsimulationsofamanifoldofinternalsizeLare l l
rium dynamics is contained in the interacting part of Γ:
typically performed on a cylinder, i.e. with periodicity
in u∼u+W with W ∼Lζ. After a time empirically of
order fixed number of turns, i.e. τ ∼ Lz according to 1
L Γint = iuˆxtFlt[ux]− iuˆxtiuˆxt′∆ltt′(uxt−uxt′)
scaling(z thedynamicalexponent),thesystemreachesa Zxt 2Zxtt′
unique global time-periodic steady state which has been (7)
extensively studied [7, 8]. Here we are interested instead where only the solution of the ERG equation to lowest
inthe dynamicalregimebefore thissteadystate isestab- orderin∆≡∆l andtooneloopisneededhere. Itreads:
lished, i.e. in the limit of times t, t′ very large compared
δF [u ]
to microscopic time scales az, and such that: δultxt′x =Rlxt=t′0∆′′(uxt−uxt′) (8)
t,t′ ≪Lz,ξz (4) 1 1
∆ltt′(u)=∆(u)+∆′′(u)(Clxt=t′0− 2Clxt=t0− 2Clxt=′t′0)
where ξ ∼ |f −f |−ν is the scale above which interface
c
motion becomes uncorrelated, i.e. we mainly focus on As in [12] we need however the FRG equation for the
the criticalregime of highly correlatedavalanche motion ”statics” part to one loop and next order. It has the
(Fig. 1). In the computation of the correlations (2) and standard form:
response (3), we will be interested in the scaling limit
∂∆˜(u)=(ǫ−2ζ)∆˜(u)+ζu∆˜′(u)+T˜∆˜′′(u)
q/Λ ≪1, keeping the scaling variables l
0
1
− [(∆˜(u)−∆˜(0))2]′′ (9)
w =qz(t−t′) u=t/t′ (5) 2
where one defined the rescaled, dimensionless disorder
fixed. In the context of the depining transition, the do-
∆˜ (u) = S Λ−ǫ∆ (u) and temperature T˜ = S Λd−2T/c
main of validity of our approach is depicted in Fig. 1. l d l l l d l
withǫ=4−d. TheresponseRq andcorrelationfunction
To study the second situation, i.e. the relaxational tt′
Cq atthe fixedpoint (l→∞)are given,to one loop,by
dynamics of the d=2 Cardy-Ostlundmodel, we set f = tt′
the equations:
v = 0 in (1), ∆(u) being a periodic function of period
unity. The super-rough glass phase [11] for T < T is,
g t t
withinRG,describedbyalineoffinitetemperaturefixed (η∂ +q2)Rq = dt Σ Rq − dt Σ Rq (10)
t tt′ Z 1 tt1 tt′ Z 1 tt1 t1t′
points(FP).Recently,wehaveobtainedanalytically[12] 0 0
C˜q (2) under the scaling form:
tt′
t′ t t′
C˜q = T t −λ−zd+2 F (qzCO(t−t′),t/t′) (6) Ctqt′ =2TZ dt1Rqtt1Rqt′t1+Z dt1Z dt2Rqtt1Dt1t2Rqt′t2
tt′ q2 (cid:18)t′(cid:19) C˜ ti ti ti
(11)
whereFC˜(w,u)isauniversal[18]scalingfunction,which where the fixed point self-energy is Σtt′ = δFδultx[ut′x]|u=0
wasrecentlyconfirmedbynumerics[14]inawider range and the disorder-noise kernel Dtt′ = ∆ltt′(u = 0) are
oftemperature. Andalthoughthenumericallymeasured local in space to this order of computation.
exponents z and λ were found to be in good agree- We first apply the above equations to the case of the
CO
ment with one loop RG predictions near T , significant depinning transition, just above threshold f = f+. In
g c
3
that case one further introduces a small finite velocity We nowfocus onthe relaxationaldynamics (1) ofran-
v → 0+ in the above equations (shifting u → u +vt dom periodic system, at low temperature. We first start
xt xt
evrywhere). Eq (9), setting T = 0 reaches the standard by deriving the low T expression of the equilibrium dy-
one loop FP for depinning transition with ζ = ǫ/3. namical exponent z . It is given by the one loop FRG
dep CO
ThisFPfunctionbeingnon-analyticatu=0,thisresults equation (9) together with ∂ lnη =−∆˜′′(0)=z −2.
l l CO
in Σtt′ = Rlxt=t′0∆′′(0+) using the limit v = 0+. We can Wewillspecializetod=2(weremindS2 =1/(2π))thus
now solve the equation for Rq (10), perturbatively in ǫ= 2 and therefore T˜ = T˜ =T/(2πc) is not flowing. In
tt′ l
the disorder, as in [12], and obtain a solution consistent that case, there is a line of fixed points ∆˜∗(u) indexed
T
with the scaling form by temperature, analyzed in Ref. [23], for T < T . The
g
transition at T =T can be analyzed in the Fourier rep-
g
Rq = t θRqz−2F (qz(t−t′),t/t′) (12) resentation ∆˜(u) = n6=0∆˜ncos(2πnu), the linear part
tt′ (cid:18)t′(cid:19) R of Eq. (9) being ∂∆˜P=2(1− T n2)∆˜ +O(∆2), where
l n Tg n
with z−2=−∆˜∗′′(0+) and the novelexponent θ asso- Tg = c/π (i.e. 2− (2π)2T˜g = 0). The transition cor-
R
responds to the lowest harmonic becoming relevant. At
ciated to large time off equilibrium relaxation:
any temperature the fixed point solution of (9) can be
1 ǫ written in that case:
θ =− ∆˜∗′′(0+)=− (13)
R
2 9
u=(3/ǫ)G(T˜,T˜+∆˜∗(0)−∆˜∗(u)) (17)
F (w,u) ≡ F (w) is a universal [18] scaling function,
R R 4y =3∆˜∗(0)+T˜± 3(3∆˜∗(0)+T˜)(∆˜∗(0)+3T˜)
whose expression is given at one loop order by: ±
q
FR(w)=e−w+z−2 2((w−1)Ei(w)e−w+e−w−1) (14) where G(a,b) = abydy/q(y−T˜)(y−y−)(y−y+), and
y < 0 < T˜ < yR . The condition 1 = G(T˜,y ) yields
− + 2 +
whereEi(w)istheexponentialintegralfunction,withthe ∆˜∗(0) as a function of T. As T → 0 this solution
largewpowerlawbehaviorFR(w)∝w−2. Thisoneloop converges to the zero temperature solution ∆˜∗T=0(u) =
scalingformqz−2FR(qz(t−t′))(12)canbewrittenasthe ∆˜∗T=0(0)− 6ǫu(1−u), and ∆˜∗T=0(0) = ǫ/36. From the
Fourier transform, w.r.t time variable of 1/(q2+Σ(iω)) FRG equation at u = 0 one has the exact relation for
with Σ(s) = s2/z +s+O(s2). Such scaling forms (12) all T:
arise in the context of critical points [19]. Solving Eq.
(10) for any finite Fourier mode q, one obtains the local T˜∆˜∗′′(0)=−ǫ∆˜∗(0) (18)
response function Rx=0
tt′ It implies that as T →0, ∆˜∗′′(0)∼−ǫ∆˜∗ (0)/T˜ which
T=0
A0 +A1 ln(t−t′) t θR gives, using T˜ =T/(2π2Tg) the one loop estimate:
Rx=0 = R R (15)
tt′ (t−t′)1+(d−2)/z (cid:18)t′(cid:19) 2π2T T
g g
z −2∼ ≃2.19 (19)
CO
9 T T
where A0 ,A1 are non universal, Λ -dependent, ampli-
R R 0
tudes (the logarithmiccorrectionscoming fromthe large We now compute the response Rq (3) and the corre-
tt′
w behavior of F (w,u)). At this order, (15) is compati- lation Cq (2) for the relaxational dynamics defined by
R tt′
blewithlocalscaleinvariancearguments[20]. Notehow- Eq. (1). By solving to first order in the disorder the
ever that at this order (15) could also be written as [21] equations (10,11) for the present case, one obtains, in
Rxtt=′0 =AR(t/t′)θR(t−t′)−(1+a) witha6=1+(d−2)/z: thelimitq/Λ0 ≪1keepingqz(t−t′),t/t′ fixed,thesame
clarifying this point requires higher order calculations, result as for the the depinning (12, 16) with the substi-
left for future investigations. tution of the exponents z and θR by zCO given by Eq.
The correlation function is obtained by solving per- (19) and θCO given to one loop, as T →0 by
turbatively the equation (11), at T = 0 (thus the bare
correlationCtxt=′ 0andtheconnectedoneC˜tqt′ vanish). One θCO ∼ π2Tg ≃1.09Tg (20)
finds in perturbation to one loop and lowest order that 9 T T
it is consistent with the scaling form: ForthepresentcasewhereT 6=0,theconnectedcorre-
lationfunctionC˜q isnonzero. Itisgivenbyanequation
t θC tt′
Cq =q−(d+2ζdep) F (qz(t−t′),t/t′) (16) exactly similar to Eq. (11) with the substitution of Dtt′
tt′ (cid:18)t′(cid:19) C byDc giventooneloopbyDc =∆′′(0)Cx=0. Bysolv-
tt′ tt′ ltt′
FC(w,u)=∆˜∗(0)(1−e−wu/(u−1))(1−e−w/(u−1)) ingperturbativelytheequationforC˜tqt′,wefindasolution
consistent with the scaling form given in Eq. (6) where
where a priori θ = O(ǫ) which, as F (w,u) is already z is given by (19) and λ = d to the order of our cal-
C C CO
of order O(ǫ), requires a second order calculation [22]. culation, in good agreement with the numerics [14] and
4
0.5
1/zCO independently of w, a consequence of (22). Notice also
0.45 RG
FRG1 the identity, given the value of zCO (19), obtained here
0.4 FRG2 up to one loop, z = 1 . This relation was also found
CO X∞
0.35 in the vicinity of Tg [12] and is consitent with numerical
simulations at low T [14].
0.3
zCO 0.25 In conclusion we have defined and computed new uni-
1/ versal exponents and scaling form for driven interfaces
0.2
near the depinning transition, We also showed that the
0.15
onelooptruncationoftheFRGyieldsagoodapproxima-
0.1 tion to numerics for low T aging dynamics of the pinned
0.05 periodicmanifoldind=2. Furthernumericsneardepin-
0 ning and investigations of other predictions of the TBL
0 0.2 0.4 0.6 0.8 1
picture (e.g. barrier fluctuations), together with more
T/Tg
precise RG calculations, would be of high interest.
FIG.2: 1/zCO asafunctionofT/Tg. Thesquaresymbolsare GS acknowledges the financial support provided
theresultsofthenumericalsimulationofRef.[14]. Thecurves
through the European Community’s Human Poten-
FRG1 and FRG2 correspond to our one loop FRG estimate
tial Program under contract HPRN-CT-2002-00307,
as explained in the text. For comparison, we have plotted in
dashed line the result of the one loop RG calculation in the DYGLAGEMEM.
vincinity of Tg [11].
where F (w,u) is a universal [18] scaling function:
C˜
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4π2T
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[7] A. Rosso and W. Krauth Phys. Rev. Lett. 87,187002
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CO
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5
[19] For a recent review see P. Calabrese and A.Gambassi, qθdep∂t′Ctqt′/Rtqt′ = (t/t′)θC−θRFTeff(qz(t − t′),t/t′)
cond-mat/0410357. where,tohigherloopθdep =d−2+2ζdepwillpresumably
[20] M. Henkelet al.,Phys. Rev.Lett. 87, 265701 (2001). turnout tobedifferent from theequilibrium freeenergy
[21] G.Schehrand R.Paul, cond-mat 0412447. fluctuationexponentθ=d−2+2ζ,giventhatζdep 6=ζ.
[22] Note the interesting dimensionless ratio, superficially [23] P. Chauve et al. Phys.Rev.B 62, 6241, (2000).
analogous, but physically different from a FDR