Anisotropic finitesize scaling of an elastic string at the depinning threshold in a randomperiodic médium

 

S. Bustingorry A. B. Kolton¹

*Email: sbusting@cab.cnea.gov.ar
TEmail: koltona@cab.cnea.gov.ar

CONICET, Centro Atómico Bariloche, 8400 San Carlos de Bariloche, Río Negro, Argentina.

We numerically study the geometry of a driven elastic string at its sampledependent depinning threshold in randomperiodic media. We find that the anisotropic finitesize scaling of the average square width w² and of its associated probability distribution are both controlled by the ratio k = Aí/L dep, where £dep is the randommanifold depinning roughness exponent, L is the longitudinal size of the string and M the transverse periodicity of the random médium. The rescaled average square width w² /L ⁽'^dep displays a nontrivial single minimum for a finite valué of k. We show that the initial decrease for small k reflects the crossover at k ~ 1 from the randomperiodic to the randommanifold roughness. The increase for very large k implies that the increasingly rare critical configurations, accompanying the crossover to Gumbel criticalforce statistics, display anomalous roughness properties: a transverseperiodicity scaling in spite that w² <C M, and subleading corrections to the standard randommanifold longitudinalsize scaling. Our results are relevant to understanding the dimensional crossover from interface to partióle depinning.

 

I. Introduction

The study of the static and dynamic properties of ddimensional elastic interfaces in d\ 1dimensional random media is of interest in a wide range of physical systems. Some concrete experimental examples are magnetic [14] or ferroelectric [5,6] domain walls, contact lines of liquids [7], fluid invasión in porous media [8,9], and fractures [10,11]. In all these systems, the basic physics is controlled by the competition between quenched disorder (induced by the presence of impurities in the host materials) which promotes the wandering of the elastic object, against the elastic forces which tend to make the elastic object flat. One of the most dramatic and worth understanding manifestations of this competition is the response of these systems to an external drive.

The mean square width or roughness of the interface is one of the most basic quantities in the study of pinned interfaces. In the absence of an external drive, the ground state of the system is disordered but well characterized by a selfaffine rough geometry with a diverging typical width w ~ L^^eq, where L is the linear size of the elastic object and C_eq is the equilibrium roughness exponent. When the external forcé is increased from zero, the ground state becomes unstable and the interface is locked in metastable states. To overeóme the barriers separating them and reach a ñnite steadystate velocity v it is necessary to exceed a ñnite critical forcé, above which barriers disappear and no metastable states exist. For directed ddimensional elastic interfaces with convex elastic energies in a D = d + 1 dimensional space with disorder, the critical point is unique, characterized by the critical forcé F = F_c and its associated critical conñguration [12]. This critical conñguration is also rough and selfafñne such that w ~ L^dep with £d_ep the depinning roughness exponent. When approaching the threshold from above, the steadystate average velocity vanishes like v ~ (F - F_c)^ and the correlation length characterizing the cooperative avalanchelike motion diverges s& £, ~ {F - F_c)~^v for F > F_c, where ¡3 is the velocity exponent and v is the depinning correlation length exponent [1316]. At ñnite temperatura and for F <¡C F_c, the system presents an ultraslow steadystate creep motion with universal features [17,18] directly correlated with its multiaffine geometry [19,20]. At very small temperatures the absence of a divergent correlation length below F_c shows that depinning must be regarded as a nonstandard phase transition [20, 21] while exactly at F = F_c, the transition is smearedout with the velocity vanishing as » ~ T^, with tp, the socalled thermal rounding exponent [2227].

During the last years, numerical simulations have played an important role to understand the physics behind the depinning transition thanks to the development of powerful exact algorithms. In particular, the development of an exact algorithm able to target efñciently the critical conñguration and critical forcé for a given sample [28,29] has allowed to study, precisely, the selfafñne rough geometry at depinning [7,2831], the sampletosample critical forcé distribution [32], the critical exponents of the depinning transition [26, 27, 33], the renormalized disorder correlator [34], and the avalanchesize distribution in quasistatic motion [35]. Moreover, the same algorithm has allowed to study, precisely, the transient universal dynamics at depinning [36,37], and an extensión of it has allowed to study lowtemperature creep dynamics [20,21].

In practice, the algorithm for targeting the critical conñguration [28, 29] has been numerically applied to directed interfaces of linear size L displacing in a disordered potential of transverse dimensión M, applying periodic boundary conditions in both directions in order to avoid border effects. This is thus equivalent to an elastic string displacing in a disordered cylinder. The aspect ratio between longitudinal L and transverse M periodicities must be carefully chosen, in order to have the desired thermodynamic limit corresponding to a given experimental realization. In Ref. [32] it

was indeed shown that the critical forcé distribution P(F_C) displays three regimes associated with M: (i) At very small M compared with the typical width L^^dep of the interface, the interface wraps the computational box several times in the transverse direction, as shown schematically in Fig. l(b), and therefore the periodicity of the random médium is relevant and P(F_C) is Gaussian; (ii) At very large M compared with L^^dep, as shown schematically in Fig. l(c), periodicity effects are absent but then the critical forcé, being the máximum among many independent subcritical forces, obeys extreme valué statistics and P(F_C) becomes a Gumbel distribution; (iii) In the intermedíate regime, where M « ¿Cdep an_c[ periodicity effects are still irrelevant, as shown schematically in Fig. l(a), the distribution function is in between the Gaussian and the Gumbel distribution. It has been argued that only the last case, where M « L^^dep, corresponds to the randommanifold depinning universality class (periodicity effects absent) with a ñnite critical forcé in the thermodynamic limit L, M -> oo. This criterion does not give, however, the optimal valué of the proportionality factor between M and L^^dep, and must be modiñed at ñnite velocity since the crossover to the randomperiodic universality class at large lengthscales depends also on the velocity [38]. To avoid this problem, it has been therefore proposed to deñne the critical scaling in the ñxed center of mass ensemble [39]. The crossover from the randommanifold to the randomperiodic universality class is, however, physically interesting, as it can occur in periodic elastic systems such as elastic chains. Remarkably, although the mapping from a periodic elastic system (with given lattice parameter) in a random potential to a nonperiodic elastic system (such as an interface) in a random potential with periodic boundary conditions is not exact, it was recently shown that the lattice parameter does play the role of M for elastic interfaces with regard to the geometrical or roughness properties [38]. Since the periodicity can often be experimentally tuned in such periodic systems it is thus worth studying in detail the geometry of critical interfaces of size L as a function of M with periodic boundary conditions, and thus complement the study of the critical forcé in such systems [32].

In this paper, we study in detail, using numerical simulations, the geometrical properties of the one dimensional interface or elastic string critical configuration in a randomperiodic pinning potential as a function of the aspect ratio parameter /c, conveniently defined as k = M/L^^dep. We show that k is indeed the only parameter controlling the finitesize scaling (i.e. the dependence of observables with the dimensions L and M) of the average square width and its sampletosample probability distribución. The scaled average square width w²L~²⁽*^dep is described by a universal function of k displaying a nontrivial single minimum at a finite valué of k. We show that while for small k this reflects the crossover at k ~ 1 from the randomperiodic to the randommanifold depinning universality class, for large k it implies that in the regime where the depinning threshold is controlled by extreme valué (Gumbel) statistics, critical configurations also become rougher, and display an anomalous roughness scaling.

 

Figure 1: (a) Elastic string driven by a forcé F in a randomperiodic médium with periodic boundary conditions. It is described by a displacement field u(z) and has a mean width w. Trie anisotropic finitesize scaling of width fluctuations are controlled by the aspectratio parameter k = M/L^^dep, with £d_ep the randommanifold roughness exponent at depinning. In the case fc<l (b) periodicity effects are important, while when k ^ 1 (c) they are not important but the roughness scaling of the critical configuration is anomalous.

II. Method

The model we consider here is an elastic string in (1 + 1) dimensions described by a single valued function u(z,t), which gives the transverse displacement wasa function of the longitudinal direction z and the time t [see Fig. l(a)]. The zerotemperature dynamics of the model is given by

7 dtu(z, t) = cd_zu(z, t) + F_p(u, z) + F, (1)

where 7 is the friction coefficient and c the elastic constant. The first term in the right hand side derives from an harmonic elastic energy. The effects of a randombond type disorder is given by the pinning forcé F_p(u,z) = -d_uU(u,z). The disorder potential U(u, z) has zero average and sampletosample fluctuations given by

[U(u, z) - U(u', z')\ = 5(z - z^r) R²(u - i¿⁷), (2)

where the overline indicates average over disorder realizations and R{u) stands for a correlator of finite range Vj [18]. Finally, F represents the uniform external drive acting on the string. Physically, this model can phenomenologically describe, for instance, a magnetic domain wall in a thin film ferromagnetic material with weak and randomly located imperfections [1], being F proportional to an applied external magnetic field pushing the wall in the energetically favorable direction.

In order to numerically solve Eq. (1), the system is discretized in the ¿direction in L segments of size 5z = 1, i.e. z ->. j = 0,..., L - 1, while keeping Uj(t) as a continuous variable. To model the continuous random potential, a cubic spline is used, which passes through M regularly spaced uncorrelated Gaussian number points [30]. For the numerical simulations performed here we have used, without loss of generality, 7 = 1, c = 1 and rf = 1 and a disorder intensity R(0) = 1. In both spatial dimensions we have used periodic boundary conditions, thus defining a L x M system.

The critical configuration u_c(z) and forcé F_c are defined from the pinned (zerovelocity) configuration with the largest driving forcé F in the long time limit dynamics. They are thus the real solutions of

cd_zu(z) + F_p(u, z) + F = 0,              (3)

such that for F > F_c there are no further real solutions (pinned configurations). Middleton theorems [12] assure that for Eqs. (3) the solution exists and it is imique for both u_c(z) and F_c, and that above F_c the string trajectory in an L dimensional phasespace is trapped into a periodic attractor (for a system with periodic boundary conditions as the one we consider). In other words, the critical configuration is the marginal fixed point solution or critical state of the dynamics, being F_c the critical point control parameter of a Hopf bifurcatión. Solving the Ldimensional system of Eqs. (3) for large L directly is a formidable task, due to the nonlinearity of the pinning forcé F_p. On the other hand, solving the longtime dynamics at different driving forces F to localize F_c and u_c is very inefficient due to the critical slowing down. Fortunately, Middleton theorems, and in particular the "nonpassing rule", can be used again to devise a precise and very efficient algorithm which allows to obtain the critical forcé F_c and the critical configuration u^c for each independent disorder realization iteratively without solving the actual dynamics ñor directly inverting the system of Eqs. (3) [30]. Once the critical forcé and the critical configuration are determined with this algorithm, we can compute the different observables. In particular, the square width or roughness of the string at the critical point for a given disorder realization is defined as

Computing w² for different disorder realizations allows us to compute its disorder average w² and the sampletosample probability distribution P(w²). In addition, the average structure factor associated to the critical configuration is where q = 27rn/L, with n = 1,...,_L - 1. One can show, using a simple dimensional analysis, that given a roughness exponent £, such that w² ~ L^2c*, the structure factor behaves as S(q) ~ q~(^1+2c*' for small q, thus yielding an estimate to ( without changing L. To compute averages over disorder and sampletosample fluctuations, we consider between 10³ and 10⁴ independent disorder realizations depending on the size of the system.

 

Figure 2: The scaling of w² for the critical configuration at different M valúes as indicated. The curves for M = 64 and 16384 are shifted upwards for clarity. The dashed and dotted lines are guides to the eye showing the expected slopes corresponding to the different roughness exponents.

 

III. Results

i. Roughness at the critical point

Figure 2 shows the scaling of the square width of the critical configuration w² with the longitudinal size of the system L for L = 32, 64,128, 256, 512 and different valúes of M. When M is small, M = 8, for all the L valúes shown we observe w² ~ L²^^h with £l = 15, corresponding to the Larkin exponent in (1 + 1) dimensions. This valué is different from the valué (dep = 125 [33,40] expected for the randommanifold universality class, and is thus indicating that the periodicity effects are important for this joint valúes of M and L. This situation is schematically represented in Fig. l(b). This result is a numerical confirmation of the twoloop functional renormalization group result of Ref. [16] which shows that the ( = 0 fixed point, leading to a universal logarithmic growth of displacements at equilibrium is unstable. The fluctuations are governed, instead, by a coarsegrained generated randomforce as in the Larkin model, yielding a roughness exponent (l = (4 - d)/2 in d dimensions [16], which agrees with our result for d = 1. We can thus say that for small enough M (compared to L) the system belongs to the same randomperiodic depinning universality class as charge density wave systems [14,41], which strictly correspond to M = 1.

 

Figure 3: Structure factor of the critical configuration for L = 256 and different M values, as indicated. The curves for M = 64 and 16384 are shifted upwards for clarity. The dashed and dotted lines are guides to the eye showing the expected slopes corresponding to the different roughness exponents.

Figure 4: Scaling of the structure factor of the critical configuration for L = 256 and different valúes of the transverse size M = 2^P with p = 3,4,..., 14 M. Although the valúes of the two exponents are very cióse, the change in the slope of the scaling function against the scaling variable x = qM¹'^<*^dep is clearly observed.

 

When M is large, on the other hand, M = 16384 in Fig. 2, for all the L valúes considered the exponent is consistent with Cdep? of the randommanifold universality class. This situation is schematically represented in Fig. l(c), and we will show later that, for this elongated samples, the effects of extreme valué statistics are already visible.

For intermedíate valúes of M, such as M = 64 in Fig. 2, we can observe the crossover in the scaledependent roughness exponent C(^) 2 d\o L cⁿang^mg irom Qep to (,l as L ulereases, as indicated by the dashed and dotted lines. This crossover, from the randommanifold to the randomperiodic depinning geometry, oceurs at a characteristic distance /* ~ ^VCdep^ when the width in the randommanifold regime reaches the transverse dimensión or periodicity M. At finite velocity, this crossover length remains constant up to a nontrivial characteristic velocity and then decreases with increasing velocity [38].

The above mentioned geometrical crossover can be studied in more details through the analysis of the structure factor S(q), for a line of fixed size L. In Fig. 3 we show S(q) for L = 256 and M = 8, 64,16384. For the intermediate value M = 64 a crossover between the two regimes is visible, and can be described by

The collapse of Fig. 4 for L = 256 and different valúes of M = 2^P with p = 3,4,..., 14 shows that this scaling form is a very good approximation. However, as we show below, small corrections can be expected fully in the randommanifold regime in the large ML~^^dep limit of very elongated samples. In Fig. 5(a), we show w² as a function of the transverse periodicity M for different valúes of the longitudinal periodicity L. Remarkably, w² is a nonmonotonic function of M. For small M it decreases towards an L dependent minimum m*, and then increases with increasing M, in the regime where the extreme valué statistics starts to affect the distribution of the critical forcé [32]. Since

Figure 5: (a) Squared width of the critical configuration as a function of M for different system sizes L as indicated. (b) Scaling of the width in (a), showing that the relevant control parameter is M/L^ζdep. The dashed line in (a) and (b) corresponds to w² = M², which is always to the left of the minimum of w²

The solid line indicates k²^¹~^'^L'^^áep^ which is the behavior expected purely from the randomperiodic to randommanifold crossover at the characteristic distance l* ~ M¹/^⁶?.

the only typical transverse scale in Fig. 5(a) is set by the mínimum m*, we can expect w² ~ m*²G(M/m*) with G(x) some universal function. On the other hand, since the only relevant characteristic lengthscale of the problem is set by the crossover between the randomperiodic regime and the randommanifold regime, we can simply write m* ~ Z>^dep and therefore

where k* = m*L~^^áep. The fact that the randomperiodic roughness exponent (l = 3/2 is larger than the randommanifold one (dep ~ 5/4 consequently implies an initial decrease oí G(k) as G(k) ~ fc²'⁵, as shown in Fig. 5(b) by the solid line. Periodicity effects, or the crossover from randomperiodic to randommanifold, thus explain the initial decrease of G{k) seen in Fig. 5(b), or the initial decrease of w² against M for fixed L, seen in Fig. 5(a). At this respect, it is then worth noting that the line w² = M², shown by a dashed line, lies completely in the regime k < k* implying that the naive criterion w² < M² is not enough to avoid periodicity effects, and to have the system fully in the randommanifold regime. As we show later, this is related with the shape of the probability distribution of P(w²) which displays sampletosample fluctuations of the order of the average w².

The presence of a minimum at k* in the function G{k) and in particular its slower than powerlaw increase for k > k* is nontrivial and constitutes one of the main results of the present work. This result shows that corrections to the standard scaling w² ~ L^^dep may arise from the aspectratio dependence of the prefactor G{k). On the one hand, w² now grows with M for L fixed, in spite that w² ^ M², i.e. transversesize/periodicity scaling is present. On the other hand, the scaling of w² with L is slower in this regime, due to subleading scaling corrections coming from G{k). The precise origin of these interesting leading and subleading corrections in the finitesize anisotropic scaling are highly nontrivial. Since the critical configurations in this regime have the constant roughness exponent £dep of the randommanifold universality class, the slow increase of G{k) cannot be attributed to a geometrical crossover effect, as for the case k < k*. However, we might relate this effect to the crossover in the critical forcé statistics, from Gaussian to Gumbel, in the k ^ k* limit [32]. In the Gumbel regime, the average critical forcé is expected to mcrease as t_c ~ log(M/L¿ ) = iog/c [39], smce the sample critical forcé can be roughly regarded as the máximum among M/L_á mdependent subcritical forces and configurations [32]. The increase in the critical forcé might be therefore correlated with the slow increase of roughness. The physical connection between the two is subtle though, since a large critical forcé in a very elongated sample could be achieved both by profiting very rare correlated pinning forces such as accidental columnar defects, or by profiting very rare noncorrelated strong pinning forces. Since in the first case the critical configuration would be more correlated and in general less rough than for less elongated samples (smaller k), contrary to our numerical data of Fig. 5(b), we think that the second cause is more plausible. We can thus think that in the k ^ k* limit of extreme valué statistics of F_c, the effective disorder strength on the critical configuration increases with k. This might be translated into the universal function G(k), such that w² ~ L^2<*^depG(k) can increase for increasing valúes of k at fixed L in such regime. A quantitative description of these scaling corrections remains an open challenge.

ii. Distribution function

We now analyze sampletosample fluctuations of the square width w² by computing its probability distribution P(w²). This property is relevant as w² fluctuates even in the thermodynamic limit for critical interfaces with a positive roughness exponent [42]. It has been computed for models with dynamical disorder such as randomwalk [43] or EdwardsWilkinson interfaces [44,45], the Mullins Herrings model [46] and for nonMarkovian Gaussian signáis in general [47, 48]. It has also been calculated for nonlinear models such as the onedimensional KardarParisiZhang model [49, 50] and for the quenched EdwardsWilkinson model at equilibrium [51].

Figure 6: Scaling function $(x) for L = 256 and different valúes of M = 8,128,2048,16384, which shows the change with the transverse size M.

 

In particular, the probability distribution P(w²) of critical interfaces at the depinning transition was studied analytically [52], numerically [31] and also experimentally for contact lines in partial wetting [7]. Remarkably, nonGaussian effects in depinning models are found to be smaller than 0.1% [31,52], thus showing that P{w²) is strongly determined by the selfaffine (critical) geometry itself, rather than by the particular mechanism producing it. As in all the above mentioned systems the width distribution P(w²) at different universality classes of the depinning transition was found to scale as

with <&£ an universal function, which only depends on the roughness exponent £ and on boundary conditions when the global width is considered [47,48]. In this way, w² is the only characteristic lengthscale of the system, absorbing the system longitudinal size L, and all the nonuniversal parameters of the model such as the elastic constant of the interface, the strength of the disorder and/or the temperature. Since <£>£ can be easily generated using nonMarkovian Gaussian signáis [53], the quantity w²P(w²) is a good observable to extract the

 

Figure 7: Scaling function $(x) for different valúes of L = 32,64,128,256 while keeping (a) k = M/L^^dep p=¡ 1 and (b) k = M/L^^dep p=¡ 0.025. The dotted line corresponds to the scaling function of the nondisordered EdwardsWilkinson equation [43], while the continuous and dashed lines correspond to the scaling functions of Gaussian signáis with £ = 1.25 and £ = 1.5, respectively [31,53].

Fig. 7(a) and k ~ 0.025 <C k* in Fig. 7(b), with k* the minimum of w². Since data for the same k practically collapses into the same curve, we can write for our case:

as expected from the existence of the geometric crossover between the roughness exponents £l for k ->. 0 and £dep for k > k*. For intermedíate valúes k < k*, however, <& í =, k) does not necessarily coincide with the one of a Gaussian signal function <£>£ for a given £, since the critical configuration includes a crossover length /* < L. Whether multiafñne or effective exponent selfafñne nonMarkovian Gaussian signáis can be used to describe satisfactorily these intermedíate cases is an interesting open issue.

In Fig. 6, we show how the scaled distribution function $(x) = w² P{x w²) looks like for the depinning transition in a randomperiodic médium for a fixed valué L = 256 and different valúes of M. We see that $(x) depends on M for small M but converges to a fixed shape for large M. We also note that for all M $(x) extends appreciably beyond x = 1 explaining why the criterion w² < M² is not enough to be fully in the randommanifold regime, as noted in Fig. 5.

In Fig. 7, we show the scaling function $(x) for different valúes of L and M but fixing the aspectratio parameter k = M/L^^dep, k ~ 1 > k* in

IV. Conclusions

We have numerically studied the anisotropic finitesize scaling of the roughness of a driven elastic string at its sampledependent depinning threshold in a random médium with periodic boundary conditions in both the longitudinal and transverse directions. The average square width w² and its probability distribution are both controlled by the parameter k = M/L^^dep. A nontrivial single minimum for a finite valué of k was found in w²/L²⁽*^dep. For small fc, the initial decrease of w² reflects the crossover from the randomperiodic to the randommanifold roughness. For very large k₁ the growth with k implies that the crossover to Gumbel statistics in the critical forces induces corrections to G(k), that grow with k, to the string roughness scaling w¹ k, G(£;)L²^^dep. These increasingly rare critical conñgurations thus liave an anomalous roughness scaling: they have a transversesize/periodicity scaling in spite that its width is w¹ <C M , and subleading (negative) corrections to the standard randommanifold longitudinalsize scaling.

Our results could be useful for understanding roughness fluctuations and scaling in ñnite experimental systems. The crossover from randomperiodic to randommanifold roughness could be studied in periodic elastic systems with variable periodicity such as conñned vortex rows [54] and singlefiles of macroscopically charged partióles [55] or colloids [56], with additional quenched disorder. The rareevent dominated scaling corrections to the interface roughness scaling could be studied in systems with a large transverse dimensión, such as domain walls in ferromagnetic nanowires [57]. For the later case, it would be interesting to have a quantitative theory, making the connection between the extreme valué statistics of the depinning threshold and the anomalous scaling corrections to the roughness of such rare critical conñgurations. This would allow to understand the dimensional crossover, from interface to partióle depinning.

 

Acknowledgements This work was supported by CNEA, CONICET under Grant No. PIP11220090100051, and ANPCYT under Grant No. PICT2007886. A. B. K. acknowledges Universidad de Barcelona, Ministerio de Ciencia e Innovaci´on (Spain) and Generalitat de Catalunya for partial funding through I3 program.

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