Creep motion of elastic interfaces driven in a disordered landscape
Ezequiel E. Ferrero, Laura Foini, Thierry Giamarchi, Alejandro B. Kolton, Alberto Rosso
CCreep motion of elastic interfaces driven in adisordered landscape
E. E. Ferrero, L. Foini, T. Giamarchi, A. B.Kolton, and A. Rosso Instituto de Nanociencia y Nanotecnolog´ıa, Centro At´omico Bariloche,CNEA–CONICET, R8402AGP San Carlos de Bariloche, R´ıo Negro, Argentina IPhT, CNRS, CEA, Universit´e Paris-Saclay, 91191 Gif-sur-Yvette, France Department of Quantum Matter Physics, University of Geneva, 24 QuaiErnest-Ansermet, CH-1211 Geneva, Switzerland Instituto Balseiro, Centro At´omico Bariloche, CNEA–CONICET–UNCUYO,R8402AGP San Carlos de Bariloche, R´ıo Negro, Argentina LPTMS, CNRS, Univ. Paris-Sud, Universit´e Paris-Saclay, 91405 Orsay, FranceXxxx. Xxx. Xxx. Xxx. YYYY. AA:1–25https://doi.org/10.1146/((please addarticle doi))Copyright c (cid:13)
YYYY by Annual Reviews.All rights reserved
Keywords creep, domain walls, depinning, disordered elastic systems, avalanches,activated motion
Abstract
The thermally activated creep motion of an elastic interface weaklydriven on a disordered landscape is one of the best examples of glassyuniversal dynamics. Its understanding has evolved over the last 30years thanks to a fruitful interplay between elegant scaling arguments,sophisticated analytical calculations, efficient optimization algorithmsand creative experiments. In this article, starting from the pioneer ar-guments, we review the main theoretical and experimental results thatlead to the current physical picture of the creep regime. In particular,we discuss recent works unveiling the collective nature of such ultra-slow motion in terms of elementary activated events. We show thatthese events control the mean velocity of the interface and cluster into“creep avalanches” statistically similar to the deterministic avalanchesobserved at the depinning critical threshold. The associated spatio-temporal patterns of activated events have been recently observed inexperiments with magnetic domain walls. The emergent physical pic-ture is expected to be relevant for a large family of disordered systemspresenting thermally activated dynamics. a r X i v : . [ c ond - m a t . d i s - nn ] J a n ontents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Dynamical phase diagram at zero temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1. The case of the quenched Kardar-Parisi-Zhang (KPZ) depinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73. Velocity at finite temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84. Numerical methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115. Creep dynamics in the limit of vanishing temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.1. Statistics of the events and clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2. Geometry of the interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3. Optimal Paths and Barriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166. Comparison with Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.1. Creep Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.2. The Roughness puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.3. Creep avalanches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217. Conclusions and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1. Introduction
Our understanding of physics is largely based on idealized problems, the famous ‘sphericalcows’. Yet, the beauty of nature makes use of a much vast complexity. It is well knownnowadays that the presence of impurities and defects messing up with those rounded mam-mals leads to new emerging physical behavior, not observed in the idealized disorder-freeproblems. For example, the equilibration time of glasses becomes so large that it results tobe experimentally inaccessible. Such systems avoid crystallization and basically live foreverout-of-equilibrium (1, 2). Dirty metals display localization and metal insulator transitions,unseen in perfect crystals (3, 4). Systems of a broadly diverse nature show intermittentdynamics induced by the presence of disorder (5). Strained amorphous materials (6, 7, 8),fracture fronts (9, 10, 11), magnetic (12, 13) and ferroelectric domain walls (14, 15), liquidcontacts lines (16, 17), they all share a common phenomenology: when the applied drive isjust enough to induce motion, most of the system remains pinned but large regions movecollectively at high velocity. These reorganizations are called avalanches. Their locationis typically unpredictable and their size distribution display a scale free statistics. Giventhe ubiquity of this stick-slip behavior, the study avalanches has occupied a central scenein non-equilibrium statistical physics, as can be seen in the large literature of sandpilemodels (18), directed percolation and cellular automata (19).The depinning of an elastic interface moving in a disordered medium (20, 21, 22, 23, 24,25) is one of the paradigmatic examples where avalanches are well understood, thanks to theanalogy with standard equilibrium critical phenomena (22, 26). When the interface is drivenat the force f two phases are generically observed: for f < f c the interface is pinned at zerotemperature and motion is observed only during a transient time, for f > f c the line moveswith a finite steady velocity. At f c the system displays a dynamical phase transition and thediverging size of avalanches is the outcome of the presence of critical correlations. Below andabove f c the avalanches display a finite cut-off, that diverges approaching f c . We presentlyknow the statistics of avalanches sizes (27) and durations (28) and their characteristicshape (29, 30). An important observation is that subsequent depinning avalanches areuncorrelated in space and time at variance with the avalanche behavior observed in many igure 1: Left:
Sketch of the interface pulled by an external force f . The dark circles arethe impurities that contribute to the pinning energy of the interface. In the random bondcase (center) only neighboring impurities contribute while in the random field case (right) all the impurities on the left side of the interface contribute.systems where a ‘main-shock’ is at the origin of a cascade of ‘after-shocks’. The so-calledOmori law and productivity law, central in the geophysics of earthquakes (31), are notpresent at the depinning transition . Namely all the experimental observations of depinningavalanches temporally correlated were shown to be related to a finite detection threshold,created by the limited sensitivity of the measurement apparatus (34).Nonetheless, genuine aftershocks could be experimentally observed far from the depin-ning transition, in the so-called creep regime. This regime, which describes the motion ofmagnetic domain walls at finite (e.g. room) temperature and low applied fields, correspondsto an interface pulled by a small force ( f (cid:28) f c ) at finite temperature (35, 24, 25). The col-lective dynamics observed in this case is qualitatively different from the one at the criticalthreshold. In both regimes the dynamics is collective and involves large scale reorganiza-tions. But from the more recent results creep “avalanches” display complex spatio-temporalpatterns similar to the ones of observed in earthquakes.In this paper we review the main arguments and results of the last thirty years aboutcreep with particular attention to the recent progress. The paper is organized as follows.In Sect. 2 we introduce the model, present the dynamical regimes at zero temperature anddiscuss the different universality classes. In Sect. 3 we provide the scaling arguments leadingto the creep law, namely the behavior of the steady velocity as a function of the appliedforce at finite temperature. The numerical methods are discussed in Sect. 4. The morerecent results valid in the limit of vanishing temperature are presented in Sect. 5. In Sect. 6we review the creep experiments on domain wall dynamics. Conclusions and perspectivesare given in Sect. 7.
2. Dynamical phase diagram at zero temperature
We consider a d -dimensional interface in a d + 1 disordered medium. For simplicity weassume that the local displacement at any time t is described by a single valued function u ( x, t ) (see Figure 1 left ) and that the dynamics is overdamped. At zero temperature the Although depinning-inspired models have been adapted to produce aftershocks by adding termsof slow relaxation or memory (32, 33) ••
We consider a d -dimensional interface in a d + 1 disordered medium. For simplicity weassume that the local displacement at any time t is described by a single valued function u ( x, t ) (see Figure 1 left ) and that the dynamics is overdamped. At zero temperature the Although depinning-inspired models have been adapted to produce aftershocks by adding termsof slow relaxation or memory (32, 33) •• Creep motion 3 quation of motion of the elastic manifold writes: γ∂ t u ( x, t ) = c ∇ u ( x, t ) + f + F p ( x, u ) 1.where c ∇ u ( x, t ) describes the elastic force due to the surface tension, f is the externalpulling force and γ the microscopic friction. The fluctuations induced by impurities areencoded in the quenched stochastic term F p = − ∂ u V p ( x, u ), where the energy potential V p ( x, u ) describes the coupling between the manifold and the impurities.For simplicity we assume the absence of correlations along the x direction , whilethe correlations of V p ( x, u ) along the u direction usually belong to one of two universalityclasses: (i) In the Random Bond class (RB) the impurities affect in a symetric way the thephases on each side of the interface. They thus simply locally attract or repel the interface(see Figure 1 center ). In this case the pinning potential and the pinning force are bothshort-ranged correlated. (ii) The Random Field class (RF) describes a disorder couplingin a different way in the two phases around the interface. Thus the pinning energies areaffected by the impurities inside the entire region delimited by the interface (see
Figure1 right ). Then F p displays short range correlations while the pinning potential V p ( x, u )displays long-range correlations [ V p ( x, u ) − V p ( x (cid:48) , u (cid:48) )] ∝ δ ( x − x (cid:48) ) | u − u (cid:48) | . Here, the overlinedenotes average over disorder realizations.Equation Eq.1, so called quenched Edwards-Wilkinson equation, is a coarse-grainedminimal model governing the dynamics of the interface, at zero temperature for the moment,at large scales (22, 26, 25). It is a non-linear equation in u that has been extensively studiedby numerical simulation (37), functional renormalization group techniques (FRG) (38, 21,39) and exact mean-field solutions (40, 41, 42). For the case of a contact line of a liquidmeniscus (43) as well as the crack front of a brittle material (44) the local elastic force isreplaced by a long range one: c ∇ u → c (cid:90) ( u ( x (cid:48) , t ) − u ( x, t )) | x (cid:48) − x | α + d d d x (cid:48) α = 1 and d = 1. The qualitative phenomenology of this generalized long range modelis similar to the quenched Edwards-Wilkinson, but the universal properties (as criticalexponents and scaling functions) are different. However, for α ≥ depinning threshold f c thesteady velocity is zero, and it acquires a finite value above only above that threshold. Thevelocity vanishes continuously at the critical force as v (cid:39) ( f − f c ) β . At the depinning theinterface appears rough with a width w ( L ) = 1 L (cid:90) L u ( x )d x − (cid:18) L (cid:90) L u ( x )d x (cid:19) L ζ dep , with L being the size of the system and ζ the roughness exponent.Both β and ζ dep are universal depinning exponents depending on the dimension d of theinterface and on the range α of the elastic force; but interestingly, not on the disorder See Ref.(36) for a discussion of the correlated disorder case. igure 2:
Left:
Sketch of an avalanche below f c : the applied force f is increased infinites-imally and a finite portion of the interface is destabilized. The size S of the avalanchecorresponds to the spanned area. Right:
Dynamical phase diagram at zero temperature.At f = f c the velocity and the shape of the interface have a universal scaling behavior, thedynamics is characterized by large and scale free avalanches. At f = 0 the interface is inthe ground state with a different roughness exponent which depends on the correlation ofthe disorder (RB or RF). At very large force the interface flows with a velocity that growslinearly with the force and the quenched disorder acts as a thermal noise.type (20, 46). Slightly above f c the dynamics of a point of the interface is highly inter-mittent: for long times the point is stuck with a vanishing velocity (much smaller than theaverage value v ) and suddenly starts to move with a high velocity. In equilibrium secondorder phase transition the universality arises from the existence of a correlation length thatdiverges approaching the critical threshold. For depinning the system is out-of-equilibriumbut the presence of large spatial correlations is manifested by the collective nature of thisintermittent dynamics: at a given time, while many pieces of the interface are at rest, largeand spatially connected portions move fast and coherently.The presence of large correlations can be detected using a quasistatic protocol below (butclose to) f c . This is shown in Figure 2 left where an interface is at rest at a force f . Uponincreasing infinitesimally the force f → f + δf , an avalanche takes place: a large portion ofthe interface advances a finite amount while elsewhere only readjusts infinitesimally ( ∝ δf ).The avalanches locations cannot be predicted and their sizes (the areas spanned betweentwo consecutive metastable states) present scale free statistics P ( S ) = S − τ dep g ( S/S c ) . τ is universal as are β and ζ dep , g ( x ) is a function thatdecays fast for x ≥ x <
1. The characteristic size of the maximalavalanche increases when f → f − c . In practice, S c is the clear manifestation of the divergentcorrelation length ξ (cid:39) | f − f c | − ν dep and one expects S c (cid:39) ξ d + ζ dep (cid:39) | f − f c | − ν dep ( d + ζ dep ) .Many works have been devoted to describe the dynamics inside an avalanche (47, 33, 34,28, 48): typically the instability starts well localized at a given point and speads in spaceover a distance x ( t ) (cid:39) t /z up to a time t c (cid:39) ξ z . For the qEW equation 1 it has beenproven that there are only two independent exponents, e.g. ζ dep and z , and the other can ••
1. The characteristic size of the maximalavalanche increases when f → f − c . In practice, S c is the clear manifestation of the divergentcorrelation length ξ (cid:39) | f − f c | − ν dep and one expects S c (cid:39) ξ d + ζ dep (cid:39) | f − f c | − ν dep ( d + ζ dep ) .Many works have been devoted to describe the dynamics inside an avalanche (47, 33, 34,28, 48): typically the instability starts well localized at a given point and speads in spaceover a distance x ( t ) (cid:39) t /z up to a time t c (cid:39) ξ z . For the qEW equation 1 it has beenproven that there are only two independent exponents, e.g. ζ dep and z , and the other can •• Creep motion 5 able 1: Depinning exponents are known numerically with good precision and saturate totheir mean field values for d ≥ α . At the depinning RB and RF disorder are in the sameuniversality class. The numerical values of the roughness exponents ζ dep are taken from(49) for α = 1 and from (50) for α = 2. Those of the dynamical exponent z are taken from(51) for α = 1 d = 1, from (52) for α = 2 d = 2 and from (37) for α = 1 d = 1. Depinning Observable d = 1 d = 1 d = 2 Mean Field exponent α = 2 α = 1 α = 2 d ≥ αz t ( L ) ∼ L z αζ dep u ( x ) ∼ x ζ dep τ dep P ( S ) ∼ S − τ dep τ dep = 2 − α/ ( d + ζ dep ) 3/2 ν dep ξ ∼ | f − f c | − ν dep ν dep = 1 / ( α − ζ dep ) α − β v ∼ | f − f c | β β = ν dep ( z − ζ dep ) 1 be computed by non trivial scaling relations (see Table 1 ). Note that these relations arevalid in low dimensions, because for d ≥ α the value of the exponents saturates at theirmean field value.The physics is very different in the limits of very small or very high forces. At f = 0the interface is at equilibrium in the ground state, its roughness is characterized by a verydifferent (smaller) roughness exponent and the nature of the disorder matters: RF interfacesare rougher than RB. The ground state energy is an extensive quantity (grows as L d ) but itssample to sample fluctuations scale as L θ . The energy exponent θ obeys the scaling relation θ = 2 ζ eq + d − α (see Table 2 ). This relation is a consequence of the statistical tilt symmetryof the model which assures that the elastic constant c is not renormalized. On the otherhand, assuming that in equilibrium elastic and disorder energy scale in the same way, one hasfrom E el [ u ] = c (cid:82) ( u ( x (cid:48) ,t ) − u ( x,t )) | x (cid:48) − x | α + d d d x d d x (cid:48) the relation E eq ∝ L ζ eq L − ( α + d ) L ∼ L ζ eq + d − α .Note that for α > d/
2, the interface is flat ( ζ eq = 0) and the energy exponent saturates tothe central limit value θ = d/ f → ∞ the quenched pinning reduces to an annealed stochastic noise because in thecomoving frame one has F p ( x, u ) = F p ( x, δu + vt ) ∼ F p ( x, vt ). For short-range correlatedpinning force, the strength of the disorder plays the role of and effective temperature T eff . Inthis so-called fast-flow regime the motion is not intermittent, and one recovers the standardEdwards Wilkinson dynamics with the generalized fractional laplacian of Eq.2 (53). Inparticular the dynamical exponent is z = α and the roughness exponent is ζ flow = ( α − d ) / d ≤ α . For larger dimension, the Edwards Wilkinson interface is flat.For intermediate forces the physics is not fully governed by any of the three characteristicpoints described above ( f = f c , f = 0 and f → ∞ ). Therefore, one could wonder if acompletely new scaling description should be introduced. It turns out that it is not thecase, at least for f > f c . The physics of the interface can be described by a crossoverbetween short length scales, governed by the critical behaviour at f = f c , and large lengthscales, governed by the fixed point of f = ∞ . Below the depinning threshold, f < f c ,no steady-state can be defined at zero temperature rather than the complete arrest of theinterface. The presence of a finite temperature, discussed in the next section, allows toinvestigate a non-trivial stationary dynamical regime (the creep) with finite velocity atforces in between the equilibrium and the depinning fixed point, and to analyze how thistwo fixed points affect the dynamics at different scales. able 2: Equilibrium exponents for elastic manifold with random bond disorder (RB). For α = 2 the results in d = 1 are exact. In d = 2 we used the numerical results from (54)obtained using a maximal flow algorithm. For α = 1 the results are known from FRGcalculations, for RF disorder one expects ζ eq = θ = 1 /
3. Note that θ and ζ eq are notindependent, but obey to the following scaling relation θ = 2 ζ eq + d − α . Equilibrium Observable d = 1 d = 1 d = 2 Mean Field exponent α = 2 α = 1 α = 2 d ≥ αθ E ( L ) ∼ L θ (cid:39) (cid:39) d/ ζ eq u ( x ) ∼ x ζ eq (cid:39) (cid:39) τ eq P ( S ) ∼ S − τ eq τ eq = 2 − α/ ( d + ζ eq ) 3/2 ν eq ξ ∼ f − ν eq ν eq = 1 / ( α − ζ eq ) α − The quenched Edwards Wilkinson equation and its generalization to long range elasticityare well studied and understood. In all these models the non-stochastic part of the equationis linear in the displacement u and one can derive the scaling relation of table 1. However,in presence of anisotropies in the disorder (55) or in the elastic interaction (57), a non-linearity becomes relevant for short range elasticity. In this case the equation of motion ofthe interface writes: γ∂ t u ( x, t ) = c ∇ u ( x, t ) + λ ( ∇ u ( x, t )) + f + F p ( x, u ) . f → ∞ leading tothe standard Kardar-Parisi-Zhang (KPZ) (58) dynamics rather than the Edwards Wilkin-son. At depinning, if λf ≥ Ta-ble 3 . When λf < ζ dep = 1 (60). This regime has been recently observed in (61).Table 3: Exponents of the qKPZ depinning universality class. The numerical values of theroughness exponent ζ dep are taken from (50). For d = 1 the exponents z and ν dep are takenfrom (55), while for d = 2 from (56). The existence of an upper critical dimension is underdebate. qKPZ Observable d = 1 d = 2 exponent α = 2 α = 2 z t ( L ) ∼ L z ζ dep u ( x ) ∼ x ζ dep ν dep ξ ∼ | f − f c | − ν dep τ dep P ( S ) ∼ S − τ dep τ dep = 2 − ( ζ dep + 1 /ν dep ) / ( d + ζ dep ) β v ∼ | f − f c | β β = ν dep ( z − ζ dep ) ••
3. Note that θ and ζ eq are notindependent, but obey to the following scaling relation θ = 2 ζ eq + d − α . Equilibrium Observable d = 1 d = 1 d = 2 Mean Field exponent α = 2 α = 1 α = 2 d ≥ αθ E ( L ) ∼ L θ (cid:39) (cid:39) d/ ζ eq u ( x ) ∼ x ζ eq (cid:39) (cid:39) τ eq P ( S ) ∼ S − τ eq τ eq = 2 − α/ ( d + ζ eq ) 3/2 ν eq ξ ∼ f − ν eq ν eq = 1 / ( α − ζ eq ) α − The quenched Edwards Wilkinson equation and its generalization to long range elasticityare well studied and understood. In all these models the non-stochastic part of the equationis linear in the displacement u and one can derive the scaling relation of table 1. However,in presence of anisotropies in the disorder (55) or in the elastic interaction (57), a non-linearity becomes relevant for short range elasticity. In this case the equation of motion ofthe interface writes: γ∂ t u ( x, t ) = c ∇ u ( x, t ) + λ ( ∇ u ( x, t )) + f + F p ( x, u ) . f → ∞ leading tothe standard Kardar-Parisi-Zhang (KPZ) (58) dynamics rather than the Edwards Wilkin-son. At depinning, if λf ≥ Ta-ble 3 . When λf < ζ dep = 1 (60). This regime has been recently observed in (61).Table 3: Exponents of the qKPZ depinning universality class. The numerical values of theroughness exponent ζ dep are taken from (50). For d = 1 the exponents z and ν dep are takenfrom (55), while for d = 2 from (56). The existence of an upper critical dimension is underdebate. qKPZ Observable d = 1 d = 2 exponent α = 2 α = 2 z t ( L ) ∼ L z ζ dep u ( x ) ∼ x ζ dep ν dep ξ ∼ | f − f c | − ν dep τ dep P ( S ) ∼ S − τ dep τ dep = 2 − ( ζ dep + 1 /ν dep ) / ( d + ζ dep ) β v ∼ | f − f c | β β = ν dep ( z − ζ dep ) •• Creep motion 7 igure 3:
Left:
Velocity force characteristics at finite temperature. When f is very smallcompared to f c and at very small temperature, one observes the creep law ln v ∼ f − µ .Adapted from (25). Right:
First experimental verification of a creep law consistent with µ = 1 /
3. Velocity at finite temperature
At finite temperature the interface has a finite steady velocity v , even below f c . The energyof the interface can be written as the sum of three contributions: E [ u ] = (cid:90) L d d x (cid:104) c ∇ u ( x )) + V p ( x, u ( x )) − fu ( x ) (cid:105) , RHS being the elastic energy of the interface, the second, the pinningpotential, and the third, the energy associated to the driving force f . We note that theequation of motion (1) is obtained from γ∂ t u ( x, t ) = − δE [ u ] /δu ( x, t ). At finite temperatureone can write the associated Langevin equation: γ∂ t u ( x, t ) = c ∇ u ( x, t ) + f + F p ( x, u ) + η ( x, t ) , (cid:104) η ( x, t ) η ( x (cid:48) , t (cid:48) ) (cid:105) = 2 γT δ ( t − t (cid:48) ) δ ( x − x (cid:48) ) where the average is over different realizationsof the thermal noise, while the disordered landscape remains fixed.In presence of a finite drive, the energy Eq. 6 has no lower bound as it is tilted by the forceand in average decreases linearly by increasing u . Yet, the presence of pinning generatesmetastable states and barriers up to f c . The activated motion at finite temperature allowsto overcome these barriers yielding a finite steady-state velocity.The velocity force characteristics is represented in Figure 3 left . At very small forceand finite temperature a creep regime is observed, where the velocity displays a stretchedexponential behavior: v ( f, T ) = v e − (cid:16) fTf (cid:17) µ , v and f T depending on the temperature and the microscopic parameters, while µ is auniversal exponent. This creep law was verified experimentally in ferromagnetic ultrathinfilms with µ (cid:39) / et al. (62) (see Figure 3 right ). Rather strikingly, thislaw can span several decades of velocity (from almost walking speed to nails growth speed)by just varying one decade of the externally applied magnetic field at ambient temperature.The creep law was subsequently found by many other experiments(63, 64) (see Section 6 fora brief review), confirming the universality and robustness of several creep properties. Suchuniversality naturally calls for minimal statistical-physics models on which we will focus. igure 4:
Left:
Thermally assisted flux flow. The activated velocity of a single degreeof freedom in a short range disordered potential is linear in the force and exponentiallysuppressed by the size of the typical barrier E p . Right:
Creep behavior. The energeticbarrier encountered by an interface diverges when the applied force vanishes. Indeed inorder to find a new metastable state characterized by smaller energy a large portion ofthe interface has to reorganize. Scaling arguments predict that the linear size of suchreorganization scales as (cid:96) opt ∼ f − α − ζ eq .Eq. 8 has been predicted in (65, 66, 67) and derived within the functional renormal-ization group technique in (46). The stretched exponential behavior originates from thecollective nature of the low temperature dynamics of these extended objects. For a point-like system embedded in a short-range disorder potential the response to a small force willbe linear in f . The idea is to consider that the energy landscape is characterized by valleysat distance ∆ u separated by an energetic barrier of typical size E p . In presence of thetilt introduced by a finite force f , the energy gap between two consecutive valleys becomes ∼ f ∆ u (see Figure 4 ). According to the Arrhenius law, the time to jump from left to rightwill be e β ( E p − f ∆ u/ , while the time for doing it from right to left would be e β ( E p + f ∆ u/ .Therefore, the velocity can be computed as the thermally assisted flux flow (TAFF (68))across the barrier: v ∝ e − β ( E p − f ∆ u/ − e − β ( E p + f ∆ u/ (cid:39) e − βE p ∆ uf . Figure5 we show different configurations obtained at different times from the direct integration ofEq. 7. At short times one observes incoherent oscillations and the configurations differ onlyat short length scales. At much larger times the line advances in the direction of the forcewith a coherent excitation that involves a large reorganization. This collective motion leadsthe system to a local minimum characterized by a lower energy due to the presence of theforce. It is very unlikely that the interface will climb back to the previous configurationscharacterized by a higher energy. This new and deeper valley is the starting point of a newsearch in the forward direction. At these time scales the dynamics of the line can be seenas a sequence of metastable states α → α → α → . . . ••
Creep behavior. The energeticbarrier encountered by an interface diverges when the applied force vanishes. Indeed inorder to find a new metastable state characterized by smaller energy a large portion ofthe interface has to reorganize. Scaling arguments predict that the linear size of suchreorganization scales as (cid:96) opt ∼ f − α − ζ eq .Eq. 8 has been predicted in (65, 66, 67) and derived within the functional renormal-ization group technique in (46). The stretched exponential behavior originates from thecollective nature of the low temperature dynamics of these extended objects. For a point-like system embedded in a short-range disorder potential the response to a small force willbe linear in f . The idea is to consider that the energy landscape is characterized by valleysat distance ∆ u separated by an energetic barrier of typical size E p . In presence of thetilt introduced by a finite force f , the energy gap between two consecutive valleys becomes ∼ f ∆ u (see Figure 4 ). According to the Arrhenius law, the time to jump from left to rightwill be e β ( E p − f ∆ u/ , while the time for doing it from right to left would be e β ( E p + f ∆ u/ .Therefore, the velocity can be computed as the thermally assisted flux flow (TAFF (68))across the barrier: v ∝ e − β ( E p − f ∆ u/ − e − β ( E p + f ∆ u/ (cid:39) e − βE p ∆ uf . Figure5 we show different configurations obtained at different times from the direct integration ofEq. 7. At short times one observes incoherent oscillations and the configurations differ onlyat short length scales. At much larger times the line advances in the direction of the forcewith a coherent excitation that involves a large reorganization. This collective motion leadsthe system to a local minimum characterized by a lower energy due to the presence of theforce. It is very unlikely that the interface will climb back to the previous configurationscharacterized by a higher energy. This new and deeper valley is the starting point of a newsearch in the forward direction. At these time scales the dynamics of the line can be seenas a sequence of metastable states α → α → α → . . . •• Creep motion 9 haracterized by decreasing energies E α > E α > E α > . . . α , α is the metastable state with lower energy that can bereached crossing the minimal barrier. It is possible to show that for an interface of internaldimension d embedded in a d + 1 dimension the pathway obtained with such a rule is theoptimal one (and thus the one that dominates the statistics of the dynamics) in the lowtemperature limit (69). Figure 5: Configurations atdifferent times obtained by di-rect integration of Eq. 7. Atshort times one observes in-coherent oscillations and theconfigurations differ only atshort length scales. At muchlarger times the line advancesin the direction of the forcewith a coherent excitationthat involves a large reorgani-zation.The first attempts to evaluate the barriers and thelength scales associated to this coarse grained dynamicshave been done in (65, 66) and in (46) via FRG. The mainassumption in their original derivation is that, during thedynamical evolution, the energy barriers scale as the en-ergy fluctuations of the ground state at f = 0. At equi-librium the fluctuations of the free energy are known togrow with the system size with a characteristic exponent θ that depends on the equilibrium roughness exponent viaan exact scaling relation θ = 2 ζ eq + d − α . Numerical simu-lations in (70) have shown that the barriers separating twoequilibrium metastable states, that differ on a portion (cid:96) ,grow as (cid:96) ψ with an exponent consistent with ψ (cid:39) θ . Usingthese ideas one can assume that the energy barriers dueto the pinning centers and in absence of tilt grow with thesize of the reorganization E p ( (cid:96) ) ∼ (cid:96) θ = (cid:96) ζ eq + d − α E f ( (cid:96) ) ∼ f u ( (cid:96) ) (cid:96) d = f(cid:96) ζ eq + d Figure 4 right we show that the competition betweenthese two terms (Eqs.12 and 13) yields the characteristiclength scale (cid:96) opt of the optimal reorganization (and the op-timal barrier E p ( (cid:96) opt )) allowing to reach a new metastablestate with a lower energy: (cid:96) opt ∼ f − α − ζ eq E p ( (cid:96) opt ) ∼ f − θα − ζ eq . E p in Eq. 9 one recovers the creeplaw, Eq. 8, and identifies the creep exponent µ = θα − ζ eq = 2 ζ eq + d − αα − ζ eq d = 1, for RBdisorder and short range elasticity one recovers µ = 1 /
10 Ferrero et al. lthough for the average velocity there is an excellent agreement between the simplescaling arguments (65, 66) and the more sophisticated FRG analysis (46), the FRG showedclearly that other lengthscales besides (cid:96) opt (see Figure 4 right) were necessary to describethe motion, pointing to a rich dynamics in the creep regime. In particular the FRG showedthat the thermal nucleus led in the dynamics to avalanches at a larger lengthscales than (cid:96) opt itself. In order to make a full analysis of the creep regime, a numerical investigation was thuseminently suitable. This is however a highly non-trivial task considering the exponentiallylarge time and length scales. We discuss on how to undertake such a study in the nextsection.
4. Numerical methods
The direct simulation of the Langevin equation 7 has been performed in (67) and later in(71). This approach confirms a non-linear behavior for the velocity-force characteristicsbut fails in probing the specific scaling of the creep law. In fact, at low temperaturethese methods can focus only on the microscopic dynamics describing incoherent and futileoscillations around local minima (see
Figure 5 ). The forward motion that allows to escapefrom these minima occurs at very long time scales that are difficult to reach. In practiceone has to increase the temperature or the force bringing the system beyond the validity ofthe creep scaling hypothesis.A completely different strategy focus on the coarse grained dynamics at the time scales ofthe coherent reorganizations that are able to lower the energy. In practice one has to modelthe interface as a directed polymer of L monomers at integer positions u ( i ), i = 1 , . . . , L and with periodic boundary conditions ( u ( L + 1) = u (1)). The energy of the polymer isgiven by: E = (cid:88) i (cid:2) ( u ( i + 1) − u ( i )) − fu ( i ) + V ( i, u ( i )) (cid:3) . | u ( i + 1) − u ( i ) | ≤ κ, κ ∼ O (1) an integer.To model RB disorder one can define V RB ( i, u ) = R i,u with R i,u Gaussian randomnumbers with zero mean and unit variance, while for RF disorder V RF ( i, u ) = (cid:80) uk =0 R i,k ,such that [ V RF ( i, j ) − V RF ( i (cid:48) , j (cid:48) )] = δ i,i (cid:48) | j − j (cid:48) | .At the coarse grained level the dynamics corresponds to a sequence of polymer positionsdetermined using a two step algorithm. • Thermal activation.
Starting from any metastable state one has to find the compactrearrangement that decreases the energy by crossing the minimal barrier among allpossible pathways. • Deterministic relaxation.
After the above activated move, the polymer is not nec-essarily in a new metastable state and relaxes deterministically with the non localMonte Carlo elementary moves introduced in (72).From the computational point of view the most difficult task is in the first step. In prin-ciple, one fixes a maximal barrier and enumerates all possible pathways that stay belowthe maximal allowed energy. If one of them reaches a state with a lower energy the ther-mal activation step is over, otherwise the maximal barrier is increased and the process is ••
After the above activated move, the polymer is not nec-essarily in a new metastable state and relaxes deterministically with the non localMonte Carlo elementary moves introduced in (72).From the computational point of view the most difficult task is in the first step. In prin-ciple, one fixes a maximal barrier and enumerates all possible pathways that stay belowthe maximal allowed energy. If one of them reaches a state with a lower energy the ther-mal activation step is over, otherwise the maximal barrier is increased and the process is •• Creep motion 11 epeated. This protocol is exact, it has been implemented in (69), but it has severe compu-tation limitations at low forces as the minimal barrier is expected to diverge for vanishingforces. In order to explore the low force regime, a different strategy has been adopted in(73). Instead of looking to the pathway with the minimal barrier one selects the smallestrearrangement that decreases the energy. This is done by fixing a window w and computingthe optimal path between two generic points i, i + w of the polymer using the Dijktra’s algo-rithm adapted to find the minimal energy polymer between two fixed points. The minimalfavorable rearrangement corresponds to the minimal window for which the best path differsfrom the polymer configuration. Using this strategy, it was possible not only to increaseof a factor 30 the system size, but, and more importantly, to decrease of a factor 100 theexternal drive f , unveiling the genuine creep dynamics.
5. Creep dynamics in the limit of vanishing temperature
Figure 6: Sketch of the selected pathway starting from the metastable state α k . During‘step one’ of the algorithm one searches for a polymer configuration with an energy smallerthan the one associated to α k by crossing a minimal barrier E p . During ‘step two’ thepolymer relaxes deterministically to a metastable configuration, no barriers are overcomeat this stage. Adapted from (69).Here we give a summary of the main results obtained using the coarse grained dynamicsintroduced in (69, 73). The output of the algorithm is a sequence of metastable states α k ( k = 1 , . . . , n ), as shown in Figure 6 . In (69) the barrier E p is the minimal between allpossible pathways, while in (73) the criterium of the minimal barrier has been approximatedwith the criterium of the minimal rearrangement which allows to reach much smaller forcesand much larger sizes. The area between two subsequent metastable states (see Figure 6 )defines the size of an activated event. Below this size the dynamics is futile characterized byincoherent vibrations, while once the new metastable state is reached the backward moveis suppressed.
12 Ferrero et al. .1. Statistics of the events and clusters
From the scaling arguments of Section 3 one expects that the area of the activated eventsis of the order (cid:96) d + ζ eq opt with (cid:96) opt that grows when the force decreases (see Eq. 14). Howeverthe distribution shown in Figure 7 displays a power law scaling analogous to the depinningone P ( S eve ) ∼ S − τ eve g ( S eve /S c ) . S c ( f ) grows and displays the scaling predicted inSection 3: S c ∼ (cid:96) d + ζ eq opt ∼ f − ν eq ( d + ζ eq ) . d = 1 and ζ eq depends on the nature of the disorder: for RB S c ( f ) ∼ f − / while forRF S c ( f ) ∼ f − . -2 S eve / f - ν eq (1+ζ eq ) -6 -3 P ( S e v e ) / P ( f − ν e q ( + ζ e q ) ) S eve -6 -3 P ( S e v e ) ~S eve -1.17 ~S eve −τ eq force -4 -2 S eve / f −ν eq RF (1+ζ eq RF ) -9 -6 -3 P ( S e v e ) / P ( f − ν e q R F ( + ζ e q R F ) ) S eve -9 -6 -3 P ( S e v e ) ~S eve -1.59 force ~S eve - τ eq RF Figure 7: Events size distributions P ( S eve ) for RB (left) and RF (right) at different forces.Main pannels show collapses by plotting S eve /S c with S c ( f ) = f − ν eq (1+ ζ eq ) . Insets showthe unscaled distributions. Note that for RB disorder S c ( f ) = f − / while for RF disorder S c ( f ) = f − . The perfect collapse validates the expected creep scaling (cid:96) opt ∼ f − ν eq , given S c ∼ (cid:96) (1+ ζ eq )opt . Adapted from (73).Eq. 18 implies that the typical activated events are much smaller than the one predictedby scaling arguments. However few very large events dominate the characteristic time scalesof the forward motion. The behavior of the velocity in the creep formula is then determinedby the occurrence of such large reorganizations. Indeed, the barriers associated to thelargest elementary events are expected to scale as U o pt ( f ) ∼ (cid:96) θ opt ≈ S c ( f ) θ/ ( d + ζ eq ) . Thenthe mean velocity in the Arrhenius limit writes as v ∼ exp[ − U o pt /T ] ∼ exp[ − ( f T /f ) µ /T ],with µ = θ/ (2 − ζ eq ), recovering the celebrated creep law of Eq. 8. The main differencewith the previous scaling approaches (65, 66) is that the creep law is not determined by the‘typical’ events but by the largest ones instead.To get further inside on the sequence of these events one notes that the exponent τ of P ( S eve ) is larger than the one expected in equilibrium (in particular in Figure 7 for RB τ = 1 .
17 instead of τ eq = 4 / τ = 1 .
59 instead of τ eq = 1). The anomalyobserved in the exponent τ is the first fingerprint of a discrepancy between creep eventsdistributions and other type of avalanches, as the depinning ones, going well beyond the ••
59 instead of τ eq = 1). The anomalyobserved in the exponent τ is the first fingerprint of a discrepancy between creep eventsdistributions and other type of avalanches, as the depinning ones, going well beyond the •• Creep motion 13 igure 8:
Left:
Sequence of activated events events in the creep regime. First, in theactivity map, each segment corresponds to an event and displays its longitudinal length.The full configurations of 300 consecutive metastable states are shown immediately after.An individual event of size S eve and a cluster of size S clust are exemplified. Right:
Sequenceof deterministic avalanches close to the depinning that appear randomly distributed in space.Again, both activity map and sequence of configurations are shown. Adapted from (73).anticipated differences of critical exponents. In
Figure 8 it is shown that the typicalsequence of avalanches is randomly located in space while the creep events are organizedin spatio temporal patterns very similar to earthquakes: the large events are the mainshocks that are followed by a cascade of small activated events. The events in the cascadeare the analogous of the aftershocks which are responsible of an excess of small events inthe Gutenberg-Richter exponent as reported also in the analysis of the real earthquakes(31, 74, 33) . Similar patterns for the elementary activated were observed below but nearthe depinning threshold (75).In order to analyze the spatio-temporal patterns one can study the clusters of correlatedevents, defined by the activated events enclosed by a circle in Figure 8 . All details in thedefinition of the clusters are found in (73).Surprisingly, for both RB and RF disorder, the statistics of the clusters appear as theone of the depinning avalanches with τ dep = 1 .
11 and the cut-off controlled by the systemsize and diverging in the thermodynamic limit (76) (see
Figure 9 ). An independent and complementary confirmation of these results comes from the studyof the roughness of the interface at different scales as introduced in (69). In practice onemeasures the structure factor S ( q ) = u ( q ) u ( − q ) ∼ q − ( d +2 ζ ) where u ( q ) is the Fourier The Gutenberg-Richter exponent b = ( τ −
1) for the earthquake magnitude distribution shouldbe smaller than the mean field prediction 3 /
4, but from seismic records one gets (33, 31) b (cid:39)
14 Ferrero et al. ransform of the position of the interface and the overline represents the average over manyconfigurations. The insets of
Figure 9 shows that there exists a crossover 1 /q c ∼ (cid:96) opt between two different behavior of the roughness: at small length scales the interface seemsto be at equilibrium, while at large length scales it appears at depinning. This observationsupports the idea that the clusters are depinning-like above a scale (cid:96) opt . Although such aresult is consistent with the predictions obtained by FRG in (46), it should be stressed thatthese clusters with depinning statistics above (cid:96) opt are formed by several activated eventsrather than generated by a single deterministic move.The coarse grained dynamics studied here is in the limit of vanishing temperature. Atfinite temperature the velocity is non-zero and this induces that the fast flow roughnessbecomes relevant at the large length scales (see Figure 10 ). The crossover occurs at ascale ξ that diverges at vanishing temperature. The FRG proposes a scaling form for ξ atlow temperature and force which depends on f and T (46), but this form was never testedin numerical simulation or experiments. Quenched Edwards-Wilkinson (qEW) to quenched KPZ (qKPZ) crossover.
The roughnessexponent measured at large scales ζ dep ≈ .
25 (see the inset of
Figure 9 ) is in agreementwith the depinning exponent of the quenched Edwards-Wilkinson universality class.The qEW depininning exponents are expected when the elastic interactions are harmonicand short range as in Eq. 6. When the interactions are anharmonic (57, 77) or a metricconstraint as Eq. 17 is present, the depinning is in the quenched KPZ universality class. Inparticular the roughness exponent is expected to be ζ qKPZdep ≈ .
63 (57, 69). The reasons ofwhy simulations deep in the creep regime (but with the metric constraint of (17)) apparentlydisplay a crossover from ζ eq to ζ dep instead of a crossover from ζ eq to ζ qKPZdep are analyzed in(78). The exponents of the qEW universality class show up at an intermediate regime, but -3 S clust / f - ν eq (1+ζ eq ) -9 -6 -3 P ( S c l u s t ) f - τ d e p ν e q ( + ζ e q ) -2 -1 q f - ν eq -3 S ( q ) / S ( f ν e q ) ~S - τ dep ~S - τ eq force ~q - (1+2 ζ dep ) ~q - (1+2 ζ eq ) -4 -2 S clust / f - ν eq RF (1+ζ eq RF ) -9 -6 -3 P ( S c l u s t ) f - τ d e p ν e q R F ( + ζ e q R F ) -2 -1 q f - ν eq -6 -3 S ( q ) / S ( f ν e q ) ~S - τ eq RF ~S - τ dep force ~q - (1+ζ eq RF ) ~q - (1+ζ dep ) Figure 9: Cluster area distribution P ( S clus ) for different forces for RB (left) and RF (right) disorder. A characteristic size S c ( f ) separates small clusters that follow equilibrium -likestatistics from big clusters that follow a depinning -like one. This result is confirmed bythe study of the rescaled structure factor S ( q ) for the same forces (insets): a geometricalcrossover is observed from equilibrium -like roughness at small scales to a depinning -likeroughness at large scales. Adapted from (73). ••
63 (57, 69). The reasons ofwhy simulations deep in the creep regime (but with the metric constraint of (17)) apparentlydisplay a crossover from ζ eq to ζ dep instead of a crossover from ζ eq to ζ qKPZdep are analyzed in(78). The exponents of the qEW universality class show up at an intermediate regime, but -3 S clust / f - ν eq (1+ζ eq ) -9 -6 -3 P ( S c l u s t ) f - τ d e p ν e q ( + ζ e q ) -2 -1 q f - ν eq -3 S ( q ) / S ( f ν e q ) ~S - τ dep ~S - τ eq force ~q - (1+2 ζ dep ) ~q - (1+2 ζ eq ) -4 -2 S clust / f - ν eq RF (1+ζ eq RF ) -9 -6 -3 P ( S c l u s t ) f - τ d e p ν e q R F ( + ζ e q R F ) -2 -1 q f - ν eq -6 -3 S ( q ) / S ( f ν e q ) ~S - τ eq RF ~S - τ dep force ~q - (1+ζ eq RF ) ~q - (1+ζ dep ) Figure 9: Cluster area distribution P ( S clus ) for different forces for RB (left) and RF (right) disorder. A characteristic size S c ( f ) separates small clusters that follow equilibrium -likestatistics from big clusters that follow a depinning -like one. This result is confirmed bythe study of the rescaled structure factor S ( q ) for the same forces (insets): a geometricalcrossover is observed from equilibrium -like roughness at small scales to a depinning -likeroughness at large scales. Adapted from (73). •• Creep motion 15 t very large scales the qKPZ exponents are recovered, as expected. The crossover betweenthe two depinning regimes is estimated to be L anh ∝ (cid:96) ζ dep − ζ eq ζ dep − opt . Figure 11 left for the structure factor and in
Figure 11 right for the cluster size statistics.
The exact algorithm for simulating the coarse-grained dynamics below the depinning thresh-old is computationally expensive but has the advantage that gives access to the energybarriers of the activated motion (69). If the interface moves on a torus (namely, periodicboudary conditions are assumed both in x and in u ) the dynamics reaches a stationarystate independent on the initial condition, with a finite sequence of metastable states α k separated by barriers E p ( α k → α k +1 ) that can be computed exactly.Barriers are important, since the Arrhenius activation formula tell us that at van-ishing temperatures the steady state forward motion of the elastic interface is fully con-trolled in a finite sample by the largest barrier U = max k E p ( α k → α k +1 ) encountered inthe stationary sequence of metastable states. The dominant configuration α k ∗ such that U = E p ( α k ∗ → α k ∗ +1 ) is the largest barrier in a given sample plays a role similar to aground state configuration in an equilibrium system; in the sense that its attributes tendto dominate the average properties at low enough temperatures (compared with the gapbetween the first and second largest energy barriers).In Figure 12 left we show the mean value U as a function of the force. As expectedfrom the creep formula U grows with decreasing the force. Unfortunately, the computationalcost of applying the exact algorithm is too high to verify the asymptotic scaling U ∼ f − µ when f →
0. When f → f c , the barrier vanishes and the size of the activated eventFigure 10: Left:
Dynamical phase diagram proposed in (46) at finite temperature. Below f c the crossover between equilibrium and depinning occurs at the scale (cid:96) opt . At finitetemperature there is also a crossover at a length scale ξ between depinning and fast flow.However ξ diverges in the limit of small temperature. Right:
Behavior of the roughnessmeasured from the structure factor consistent with the dynamical phase diagram. Adaptedfrom (69).
16 Ferrero et al. -2 -1 q S q [ s h i f t e d ] -10 -8 -6 q L anh2.33 -10 -5 S q / S q ( - ) ~q -(1+2 ζ depqKPZ ) ~q -(1+2 ζ dep ) S clust / f −ν eq (1+ζ eq ) -8 -6 -4 -2 P ( S c l u s t ) f − ν e q ( + ζ e q ) τ d e p q K P Z ~S - τ dep ~S - τ eq f ~S - τ dep qKPZ Figure 11:
Left:
Structure factor for the Random Bond case showing the characteristiclengthscale L anh which separate the harmonic depinning regime with roughness exponent ζ dep from the anharmonic depinning regime with exponent ζ qKPZdep , for different high forces f ∈ { . , . , . , . , . , . } , L = 3360. The bottom-left inset shows the raw structurefactor arbitrarily shifted in the vertical direction for different forces for a better display. Themain panel shows the structure factor rescaled with L anh ∝ ( (cid:96) opt /L c ) / , as proposed in Eq.20 for RB disorder. Straight gray lines are a guide to the eye, showing slopes correspondingto ζ dep (cid:39) .
25 (full line) and ζ qKPZdep (cid:39) .
65 (dash line).
Right:
Cluster size distributionsfor L = 3360 and f ∈ { . , . , . } . The anharmonic crossover has consequences in thecluster distribution for large cluster sizes. In the depinning regime the power law decay hasa crossover from a regime described by τ dep ≈ .
11 to a regime described by τ qKPZdep ≈ . L eft: Average over disorder realizations of the dominant barrier, as obtained byusing the exact transition pathways algorithm. Adapted from (69). R ight: Rescaled energybarrier as a function of H/H dep for different materials and temperatures ranging from 10to 315 K (25 curves in total), from (79). Black circles correspond to the barriers shown onthe left.becomes of the order of the Larkin length, the length for which the relative displacements ••
11 to a regime described by τ qKPZdep ≈ . L eft: Average over disorder realizations of the dominant barrier, as obtained byusing the exact transition pathways algorithm. Adapted from (69). R ight: Rescaled energybarrier as a function of H/H dep for different materials and temperatures ranging from 10to 315 K (25 curves in total), from (79). Black circles correspond to the barriers shown onthe left.becomes of the order of the Larkin length, the length for which the relative displacements •• Creep motion 17 re of the order of the interface thickness (or the correlation length of the disorder) (24).This matches nicely with the behavior expected for the critical configuration at f = f c .There, the barrier is zero as the configuration is marginally stable and the soft mode islocalized (Anderson-like) with a localization length that can be identified with the Larkinlength (80). In Figure 12 right we show the same quantity obtained in experiments fordifferent ferromagnetic domain walls.
6. Comparison with Experiments
The creep regime has been studied in different types of domain walls. Paradigmatic exam-ples are domain walls in thin film ferromagnets with out of plane anisotropy (12), driven byan external magnetic field or by an external electric current. In these systems, the domainwalls can be directly observed by microscopy techniques based on magneto-optic Kerr effect(MOKE). This allows to measure the mean velocity as a function of the applied field andthe domain wall geometry. More recently, the analysis of the images has allowed to identifythe sequence of events connecting different metastable domain wall configurations in pres-ence of a uniform weak drive. In this section we briefly review part of such experimentalliterature. For a dedicated review of the experimental literature on magnetic domain wallsup to 2013, including reports of different values of µ and strong pinning issues, see (12).As a side remark we also mention the possibility to study the creep regime of domain wallsin ferroelectric materials driven by an external electric field and observed with piezoforcemicroscopy (14, 15). The creep law Eq. 8 was first experimentally tested in thin ferromagnetic films(Pt/Co(0.5nm)/Pt) driven by a magnetic field H by Lemerle et al. (62). They observed aclear stretched exponential behavior (log v ∝ − H − µ ) of the stationary mean velocity as afunction of the applied field. Rather strikingly, such law can span several decades of velocity,from almost walking speeds to the speed of nails growth. The creep exponent µ was foundto be compatible with the prediction µ = (2 ζ eq − / (2 − ζ eq ) = 1 / ζ eq = 2 / Fe (0.3nm)/Pt ferromagneticthin film wires (63). In this paper not only Eq. 8 with µ ≈ / d : 1 →
0) in the velocity force characteristic at lowfield. Indeed, decreasing the magnetic field the length scale (cid:96) opt grows as ∼ H − ν eq with ν eq = 1 / (2 − ζ eq ) up to the size of the wire’s width where it saturates. As a consequencethe barrier E p ∼ (cid:96) θ opt saturates inducing the breakdown of the creep law of Eq. 8 when (cid:96) opt becomes of the order of the wire width. A dimensional crossover ( d : 1 →
0) then takesplace, from creep, Eq. 8, to a TAFF like regime, Eq. 9.From the creep theory perspective the experiments of Refs. (62, 63) hence providecrucial information: ( i ) Although domain walls are actually two dimensional objects inthree dimensional materials, they effectively behave as a simpler one dimensional elasticobject. In other words, the thickness of the magnetic film is smaller than (cid:96) opt and thedynamics is governed by energy barriers with θ ( d = 1). ( ii ) Dipolar interactions originatedby stray magnetic fields seem to be unimportant otherwise the nonlocal elasticity wouldchange the exponent µ . ( iii ) The disorder is of RB type as for RF disorder one expects
18 Ferrero et al. eq = 1, yielding µ = 1. This is particularly relevant, since the nature of the DW pinningis one of the less controlled properties of the hosting materials.In particular since the pioneer work by Lemerle et al. (62) there have been severalrecent works in thin magnetic systems reporting a consistent creep behavior with a meandomain wall velocity showing a stretched exponential law with µ = 1 / U = − K B T log v/v with v is a characteristic field independent velocity (64). Its behavioras a function of H was found to be universal for a large family of materials: U diverges atsmall fields as predicted by the creep law, U ∼ H − µ and vanishes at the depinning field as U ∼ ( H − H d ) (see Figure 12 right ). Both asymptotic behaviors are well described by thematching expression U ∼ (1 − ( H d /H ) µ ). Moreover, the behavior experimentally observedfor U as a function of H is in perfect agreement with the value U found in (69) and shownin Figure 12 left .Figure 13:
Left:
Roughness exponents obtained in (62) by fitting the displacement cor-relator function [ u ( x + L ) − u ( x )] ∼ L ζ with 1 µ m < L < µ m and v = 7 nm/s. Theaverage exponent is ζ ≈ . ± . Right:
Roughness exponent obtained in (84) by fittingthe detrended width. Different symbols correspond to two domain wall configurations at v ≈ ζ dep ≈ .
5, the dashed line a qKPZscaling 2 ζ qKPZdep = 1 . Another important test of the creep theory is to study the steady-state roughness of theinterface. From
Figure 10 we expect that the width of a domain wall of size L , w ( L ) (seeEq. 3) should scale as w ( L ) ∼ L ζ eq if L < (cid:96) opt L ζ dep if (cid:96) opt < L < ξL ζ flow if ξ < L . et al. (62) and various following works report ζ ≈ . ± .
1, in agreement with theequilibrium value ζ eq = 2 / ζ dep = 1 . ••
1, in agreement with theequilibrium value ζ eq = 2 / ζ dep = 1 . •• Creep motion 19 s we discuss below however, in the light of the current theory for creep and more recentexperiments, the identification of the observed ζ with ζ eq = 2 / H dep ≈ T dep ≈ K at room temperature( T = 300 K ). With these values it is possible to estimate (cid:96) opt using the assumptions ofweak pinning (88, 89, 90): (cid:96) opt = L c ( H dep /H ) ν eq L c = ( k B T dep ) / ( M s H dep w c δ ) 22.The microscopic Larkin length L c can be evaluated as a function of the domain wall width w c , the thickness of the sample δ and the saturation magnetization M s . All these mi-cromagnetic parameters are known, yielding L c ≈ . µm (see (83) for the analysis fordifferent materials). Using a spatial resolution of 1 µ m, typical for MOKE setups and themeasured H dep ≈ H (cid:46) . (cid:96) opt > µ m. Interestingly, (cid:96) opt was estimatedin Ta/Pt/Co Fe (0.3nm)/Pt wires (63) with a completely different method, observingfinite size effects as the wire width w was reduced. A good scaling (cid:96) opt ∼ H − ν eq with1- d RB exponents, compatible with ζ eq = 2 /
3, was found. For these samples a field of H = 16 Oe gives (cid:96) opt ≈ . µ m, remarkably in good agreement with the above estimatefor the Pt/Co/Pt film. Unfortunately, no direct roughness exponent measuremnts werereported in Ref (63). The above estimates suggest that the range of length scales used tofit experimentally the roughness exponent exceed the size of (cid:96) opt . This implies that thevalue ζ ∼ . − . Figure 2 )The fast flow exponent predicted for RB or RF systems is ζ flow = 1 / ζ dep (cid:39) .
25 and the quenched KPZ with ζ qKPZdep (cid:39) .
63. The first value is consistent withthe roughness exponent obtained in (84) at low velocity, while the last value is remarkablyclose to the values at higher velocity reported in (62). A possible way to solve this puzzle isto invoke a crossover qEW / quenched KPZ already observed in the numerical simulationsin Section 5.2. There, at low drive, the crossover occurs at very large length scales, and theqEW exponents are measured. At higher drive the quenched KPZ is recovered already atshort distances. To invoke such an identification however, we have to justify the presence ofa KPZ term in the effective DW equation of motion. At least two mechanisms can justify thepresence of a non-linear KPZ term: (i)
A kinetic mechanism yields λ ∼ v (58) for interfacesdriven by a pressure (i.e. driven by a force locally normal to the interface). (ii) A quencheddisorder mechanism induced by the anisotropy of the disorder (55) or anharmonicities inthe elasticity (57, 50, 77) yields a velocity independent λ . At the depinning transition onlythe second mechanism is relevant but at the moment we lack a microscopic derivation andthe presence of crossovers between qEW and qKPZ is still under debate.To shed light on this puzzle another important ingredient that should potentially be
20 Ferrero et al. aken into account is the presence of defects such as bubbles and overhangs, at short length-scales. The effects of these defects on the large scale properties of the domain wall are notyet well understood. Large scale simulations on the 3- d random field Ising model showed ananomalous behavior of the roughness of the interface which doesn’t match with the qEWprediction (94) (see also (95)). Figure 14:
Left:
Large reorganizations as obtained by Repain et al. (96) in irradiatedPt/Co/Pt thin films. The inset shows the successive domain wall configurations in a 92 × µ m field of view. Time interval between two images is ∆ t = 200 s. Right:
Sequences ofmagnetization reversal areas detected deep in the creep regime of Pt/Co/Pt thin films, asobtained by Grassi et al. (84). In this image time windows of ∆ t = 15 s were used.A direct experimental access to the thermal activated events and clusters would consti-tute a strong test for the current theoretical picture.Repain et al. in (96) observed reorganizations in the creep regime whose characteristicsize qualitatively increases when lowering the field. It is not clear if these reorganizationscan be identified with the thermal activated events as they look like chains of concatenatedarcs (see inset in Fig.14) suggesting the presence of strong diluted pinning. More recently,Grassi et al. (84) performed a detailed and more quantitative analysis in non-irradiatedPt/Co/Pt films, focusing on regions of the sample where strong pinning was not present.They observed almost independent thermally activated reorganizations. Their observationsare consistent with the existence of “creep avalanches” with broad size and waiting-time dis-tributions. It is tempting to identify them with the clusters found in numerical simulationsdiscussed in Section 5.1.The quantitative experimental study of creep events remains a big experimental chal-lenge. The single thermally activated event or “elementary creep event” of Sec 5 appearsto be systematically too small to be resolved by Kerr microscopy, even for velocities oforder of v ∼ (cid:96) opt or to activated depinningfrom strong centers. ••
Sequences ofmagnetization reversal areas detected deep in the creep regime of Pt/Co/Pt thin films, asobtained by Grassi et al. (84). In this image time windows of ∆ t = 15 s were used.A direct experimental access to the thermal activated events and clusters would consti-tute a strong test for the current theoretical picture.Repain et al. in (96) observed reorganizations in the creep regime whose characteristicsize qualitatively increases when lowering the field. It is not clear if these reorganizationscan be identified with the thermal activated events as they look like chains of concatenatedarcs (see inset in Fig.14) suggesting the presence of strong diluted pinning. More recently,Grassi et al. (84) performed a detailed and more quantitative analysis in non-irradiatedPt/Co/Pt films, focusing on regions of the sample where strong pinning was not present.They observed almost independent thermally activated reorganizations. Their observationsare consistent with the existence of “creep avalanches” with broad size and waiting-time dis-tributions. It is tempting to identify them with the clusters found in numerical simulationsdiscussed in Section 5.1.The quantitative experimental study of creep events remains a big experimental chal-lenge. The single thermally activated event or “elementary creep event” of Sec 5 appearsto be systematically too small to be resolved by Kerr microscopy, even for velocities oforder of v ∼ (cid:96) opt or to activated depinningfrom strong centers. •• Creep motion 21 . Conclusions and Perspectives
Elastic interfaces driven in disordered media represent a dramatic simplification of physicalsystems, such as magnetic domain walls in disordered ferromagnets. However, by encom-passing the key interplay between elasticity and disorder, these models are able to predictwith extraordinary precision some properties which are practically impossible to infer frommore realistic microscopic approaches. An important example is provided by the creepregime. The theoretical picture is now well understood: • The velocity versus the force characteristics displays a stretched exponential behavior. • The geometrical properties of the interface show a crossover from an equilibrium-likebehavior at short length scales to a depinning-like behavior at large length scales. • The dynamics displays spatio-temporal patterns (“creep avalanches”) made of manycorrelated activated events. The statistical properties of these avalanches are de-scribed by the depinning critical point.The creep regime is relevant for many physical systems, ranging from fracture fronts, contactlines or ferroelectric domain walls. The most striking confirmation comes however fromthe experiments in ferromagnetic films. There, the stretched exponential behavior of thevelocity is today well established. More recently, the analysis of the MOKE images showedthe fingerprints of an avalanche creep dynamics.Despite of the success of the elastic interface model many important questions remainopen. First, the statistical properties of the creep avalanches are still an experimentalchallenge: the elementary events are too small to be resolved with MOKE microscopy andthe spatio-temporal correlations have not been characterized. Second, there is a mismatchbetween the roughness exponents observed in numerical simulations and the ones observedexperimentally. To find a solution for this puzzle is probably one of the biggest currentchallenges in the field. We hope these questions will motivate further research on theuniversal collective dynamics of elastic interfaces in random media.
DISCLOSURE STATEMENT
The authors are not aware of any affiliations, memberships, funding, or financial holdingsthat might be perceived as affecting the objectivity of this review.
ACKNOWLEDGMENTS
We warmly acknowledge collaborations and uncountable vivid discussions with E. Agorit-sas, S. Bustingorry, J. Curiale, G. Durin, E .A. Jagla, V. Jeudy, W. Krauth, V. Lecomte,P. Le Doussal, P. Paruch and K. Wiese. We acknowledge the France-Argentina projectECOS-Sud No. A16E01. ABK acknowledges partial support from grants PICT2016-0069/FONCyT from Argentina. EEF acknowledges support from grant PICT 2017-1202,ANPCyT (Argentina). TG support from the Swiss National Science foundation underDivision II. This work is supported by “Investissements d’Avenir” LabEx PALM (ANR-10-LABX-0039-PALM) (EquiDystant project, L. Foini).
22 Ferrero et al.
ITERATURE CITED
1. Eds: Barrat JL, Dalibard J, Feigelman M, Kurchan J. 2003. Slow relaxations and nonequilibriumdynamics in condensed matter. Springer, Berlin2. Berthier L, Biroli G. 2011.
Reviews of Modern Physics
Physical review
Reviews of Modern Physics
Nature
Physical review letters
Proceedings of the National Academy of Sciences
Reviews of Modern Physics
Physics Reports
Physical Review Letters
Physical review letters
Comptes RendusPhysique
Physical Review B
Comptes Rendus Physique
Annu. Rev. Mater. Res.
Physical Review E
EPL (Europhysics Letters)
Physica A: Statistical Mechanics and its Applications
Phys. Rev. B
J. Phys. IIFrance
Physics Reports
Phys. Rev. B
Physica B: Condensed Matter
Comptes Rendus Physique
Physics Reports
Physical Review B
EPL (Europhysics Letters)
Nature Physics
Nature communica-tions
Journal of Geophysical Research: Solid Earth
Phys. Rev. Lett.
Physical review letters
Phys. Rev. Lett.
Phys. Rev. E
Phys. Rev. E
Phys. Rev. B
Phys. Rev. B
Phys. Rev. B ••
Phys. Rev. B •• Creep motion 23
1. Alessandro B, Beatrice C, Bertotti G, Montorsi A. 1990.
Journal of Applied Physics
Physica C: Superconductivity
The journal of chemical physics
Journal of applied mechanics
Phys. Rev. E
Phys. Rev. B
Phys. Rev. E arXiv preprint arXiv:1909.09075
49. Rosso A, Krauth W. 2002.
Phys. Rev. E
Phys. Rev. E
Phys. Rev. B
Physica A: Statistical Mechanics and its Applications
Phys. Rev. E
Phys. Rev. E
Phys. Rev. Lett.
Physica A: Statistical Mechanics and its Applications
Phys. Rev. Lett.
Physical Review Letters
Phys. Rev. A
Phys. Rev. Lett.
Phys. Rev. Lett.
Phys. Rev. Lett.
Nature
Phys. Rev. Lett.
Journal of Physics C: Solid State Physics
Europhysics Letters (EPL)
Phys. Rev. Lett.
Rev. Mod. Phys.
Phys. Rev. B
Phys. Rev. E
Phys. Rev. Lett.
Physical Review B
Phys. Rev. Lett. to be published in Physics Report
75. Purrello VH, Iguain JL, Kolton AB, Jagla EA. 2017.
Phys. Rev. E
Physical Review B
Phys. Rev. E
Phys. Rev. Lett.
Phys. Rev. E
Phys. Rev. B
Phys. Rev. B
Phys. Rev. B
24 Ferrero et al.
Phys. Rev. B
Phys. Rev. B
Phys. Rev. B
Phys. Rev. B
Journal of Low Temperature Physics
Phys. Rev. B
Journal of Statistical Mechanics: Theory and Experi-ment
Phys. Rev.Lett.
Phys. Rev. Lett.
Phys. Rev. E
Phys. Rev. E
Europhys. Lett. ••