Thermal rounding of the depinning transition

Abstract

We study thermal effects at the depinning transition by numerical simulations of driven one-dimensional elastic interfaces in a disordered medium. We find that the velocity of the interface, evaluated at the critical depinning force, can be correctly described with the power law $v\sim T^\psi$, where $\psi$ is the thermal exponent. Using the sample-dependent value of the critical force, we precisely evaluate the value of $\psi$ directly from the temperature dependence of the velocity, obtaining the value $\psi = 0.15 \pm 0.01$. By measuring the structure factor of the interface we show that both the thermally-rounded and the T=0 depinning, display the same large-scale geometry, described by an identical divergence of a characteristic length with the velocity $\xi \propto v^{-\nu/\beta}$, where $\nu$ and $\beta$ are respectively the T=0 correlation and depinning exponents. We discuss the comparison of our results with previous estimates of the thermal exponent and the direct consequences for recent experiments on magnetic domain wall motion in ferromagnetic thin films. Comment: 6 pages, 3 figures

Full text

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arXiv:0707.0345v1 [cond-mat.stat-mech] 3 Jul 2007
epl draft
Thermal rounding of the depinning transition
S. Bustingorry1, A. B. Kolton2and T. Giamarchi1
1DPMC-MaNEP, Universit´ e de Gen` eve, 24 Quai Ernest Ansermet, 1211 Gen` eve 4, Switzerland
2Dep. de F´ ısica At´ omica, Molecular y Nuclear, Universidad Complutense de Madrid, 28040 Madrid, Spain
PACS 64.60.Ht – Dynamic critical phenomena
PACS 75.60.Ch – Domain walls and domain structure
PACS 05.70.Ln – Nonequilibrium and irreversible thermodynamics
Abstract. - We study thermal effects at the depinning transition by numerical simulations of
driven one-dimensional elastic interfaces in a disordered medium. We find that the velocity of the
interface, evaluated at the critical depinning force, can be correctly described with the power law
v ∼ Tψ, where ψ is the thermal exponent. Using the sample-dependent value of the critical force,
we precisely evaluate the value of ψ directly from the temperature dependence of the velocity,
obtaining the value ψ = 0.15 ± 0.01. By measuring the structure factor of the interface we show
that both the thermally-rounded and the T = 0 depinning, display the same large-scale geometry,
described by an identical divergence of a characteristic length with the velocity ξ ∝ v−ν/β, where ν
and β are respectively the T = 0 correlation and depinning exponents. We discuss the comparison
of our results with previous estimates of the thermal exponent and the direct consequences for
recent experiments on magnetic domain wall motion in ferromagnetic thin films.
Understanding the physics of disordered elastic lines has
a direct impact on a large variety of experimental systems
in condensed matter. Indeed such systems are realized for
isolated lines by magnetic [1–4] or ferroelectric [5,6] do-
main walls, contact lines [7] and fractures [8,9] and for
periodic systems by vortex lattices [10, 11], charge den-
sity waves [12] or wigner crystals [13]. In these systems,
one particularly important question is to understand the
response of the interface to an externally applied force,
such response being directly measurable in all the above
systems.
In all these systems the competition between the disor-
der and the elasticity of the lines leads to pinned config-
urations. In order to set the system in motion it is thus
necessary to apply a force F exceeding a critical force Fc.
At zero temperature the average velocity V remains zero
for F < Fc, while the system moves for F > Fc. In addi-
tion to their experimental relevance, the understanding of
the above properties constitutes a considerable theoretical
challenge [14]. A very fruitful approach was to draw on
the analogies between such a phenomenon and a standard
critical phenomenon to predict scaling properties of the
various physical observables close to the depinning tran-
sition [15]. In that respect a very important question is
how to extend the above results to the case of finite tem-
perature. This is of course directly relevant for the exper-
imental systems. A finite temperature allows the system
to move even if F < Fcleading to a thermal rounding of
the depinning transition. Indeed, such rounding was re-
cently experimentally observed [4]. A natural way to ana-
lyze such thermal effect is to draw further on the analogy
with critical phenomena. This is however not so simple
since the depinning is after all an out of equilibrium fea-
ture and it is unclear how far the analogy with standard
critical phenomena should be carried out. Recently, seri-
ous differences between depinning and a standard critical
phenomenon were indeed pointed out [16].
It is thus crucial to perform a detailed analysis of the
thermal rounding of the depinning transition. Unfortu-
nately, the analytical methods able to tackle such an out of
equilibrium situation are rare. A very successful method
to describe the zero temperature depinning is the func-
tional renormalization group (FRG) [17–19]. It allowed
to obtain an expansion in 4 − d, d being the dimension
of the interface, of the depinning exponents for T = 0.
Recently, FRG equations describing the full dynamics of
the interface at finite temperature have been derived [19].
Such equations were proven to be very efficient to obtain
the dynamics at very small force (the so called creep be-
havior [10]). Unfortunately, solving them around depin-
ning at finite temperature is considerably more compli-
cated, and no complete solution exists to date, even if

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S. Bustingorry et al.
some crude prediction for the rounding exponent can be
made [19,20]. In addition to the intrinsic problem of solv-
ing the equations, the fact that they are obtained in a
4 − d expansion makes them poorly suited to quantita-
tively describe the important case of the one dimensional
interface. Alternative methods, such as numerical stud-
ies are thus clearly needed. Indeed, there were various
attempts to obtain the thermal exponent from numerical
simulations. In ref. [21] the authors used a dynamically
generated disorder environment, corresponding to an in-
finite size system in the dimension where the interface is
allowed to fluctuate. On the other hand, the random field
Ising model at finite temperature was used to compute the
thermal exponent in various dimensions [22–24]. However
in these works, in addition to the intrinsic numerical limi-
tations, the authors were forced to use the average critical
values for the pinning force, obtained by fitting the zero-
temperature velocity-force characteristics. Unfortunately,
this force has severe sample-to-sample fluctuations, which
indeed strongly depend on the size of the system [25,26].
This made the determination of the thermal rounding ex-
ponent obtained by such methods relatively imprecise, and
more importantly made it difficult to check whether one
had the proper scaling to even define such an exponent.
In view of the existing differences between depinning and
a standard critical phenomena it is indeed important to
check whether such scaling exists.
In this paper we analyze the finite-temperature dy-
namics of driven elastic interfaces evolving in a two-
dimensional random media. We use a conceptually dif-
ferent approach, where we use the pinning force for each
disorder realization as it was done in [27] for analyzing the
zero-temperature case. Such force can be now determined
with great accuracy [25]. We obtain a thermal exponent
ψ = 0.15 ± 0.01. Although we obtain the thermal expo-
nent directly from the velocity versus temperature rela-
tion at the critical force, we show that this value can also
account for finite size effects by assuming that the large-
scale behavior is controlled by a temperature-dependent
correlation length, growing with decreasing velocity in the
same way as for the zero temperature case. The existence
of such velocity-dependent correlation-length is directly
proved by the analysis of the finite-temperature structure
factor. This shows that the large-scale geometry of the
interface is only controlled by the velocity regardless on
whether one approaches the critical point from positive
forces at zero temperature or from the temperature axis
at the critical force.
We consider an elastic interface described by a single
valued function u(z,t), giving its transverse position u in
the z axis. The interface evolves with time t obeying the
equation
γ ∂tu(z,t) = c∂2
zu(z,t) + Fp(u,z) + F + η(z,t),(1)
where γ is the friction coefficient and c the elastic constant.
The pinning force Fp(u,z) = −∂uU(u,z) represents the ef-
fects of a random-bond disorder described by the potential
012
F
34
0
1
2
3
4
5
V
T = 0.1
T = 0.05
T = 0.02
T = 0.01
T = 0
1.8 2 2.2
F
0
1
2
V
-0.200.2
f
0
0.5
1
v
Fc = 1.90415
Fc,1
Fc,2
Fig. 1: Velocity V versus force F for a given disorder realiza-
tion and different temperatures. The system size is L = 1024,
and the critical force Fc is also indicated. The left-inset shows
the velocity-force curves for two different disorder realizations,
with critical forces Fc,1 = 1.82395 and Fc,2 = 1.90555 respec-
tively, at a fixed small temperature T = 0.001 and for a small
size L = 128. The right-inset shows the same data as in the left-
inset but in reduced variables v = V/Fc and f = (F − Fc)/Fc.
U(u,z), whose sample to sample fluctuations are given by
[U(u,z) − U(u′,z′)]2= δ(z − z′)R(u − u′), where R(u)
stands for a correlator of range rf [19], and the overline
indicates average over disorder realizations. The thermal
noise η(z,t) satisfies ?η(z,t)? = 0 and ?η(z,t)η(z′,t′)? =
2γTδ(t− t′)δ(u − u′). Finally, the force F corresponds to
an uniform and constant external field.
In order to numerically solve eq. (1) we discretize the
z direction in L segments of size δz = 1, i.e.
j = 0,...,L − 1, while keeping uj(t) as a continuous vari-
able.Then the equation is integrated using the Euler
method with a time step δt = 0.01.
random potential is modeled by a cubic spline passing
through M regularly spaced uncorrelated Gaussian num-
bers points [25,28]. The numerical simulations are per-
formed using γ = 1, c = 1, rf = 1, and with R(0) = 1
giving the strength of the disorder. Periodic boundary
conditions are used in both spatial directions, thus defin-
ing a L × M system.
Figure 1 shows a typical velocity-force characteristics for
a single sample of size L = 1024, obtained by computing
the steady-state velocity V = ?∂tu(z,t)? through numeri-
cal integration of Eq. (1). The sample-dependent critical
force is also quoted, which can be obtained for each realiza-
tion of the disorder configuration using a fastly-convergent
algorithm [25]. We can observe that the sharp depinning
transition is rounded by temperature as expected. The
straight dashed-line indicates the fast-flow limit V = F,
which is reached at high force. The left-inset shows the
velocity-force curve for T = 0.001, for a smaller system
size L = 128, and for two different values of the critical
z →
The continuous

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Thermal rounding of the depinning transition
force, as indicated. At large forces both curves asymptot-
ically approach the fast-flow limit V = F. However, as
it is shown in the left-inset, near the depinning transition
the velocity strongly depends on the precise value of the
critical force. Therefore, in order to properly analyze the
rounding of the depinning transition, averages of small ve-
locities must be carefully defined. To avoid this problem
we exploit the access we have to a high-precision value of
Fc for each sample [25], as was previously done for the
T = 0 case [27]. We thus define the reduced velocity and
force variables, v = ?∂tu(z,t)/Fc? and f = (F − Fc)/Fc,
respectively. As an example, in the right-inset of Fig. 1
we show the same data than in the left-inset but in the
reduced variables. The improvement of the scaling is ob-
vious. This allows us to define our systematic procedure
to extract the physical properties.
For a given disorder realization we first compute the
critical-force of the sample and then average the reduced
velocity v for a fixed value of the control parameters f
and T, both set with high precision. Since thermal and
disorder averages must be taken in the steady state, we
start with a flat initial condition and let the system evolve
until the steady state is reached.
during a sweep over the lateral size M.
one sweep is also good enough to properly account for
the thermal fluctuations of v. Disorder average is finally
performed by repeating this procedure for N samples of a
given system size. In the following v represents the final,
thermal and disorder, averaged value of the steady-state
velocity.
Since the critical force distribution and the geometry of
the critical configuration depends on the relation between
L and M for a periodic system [26], it is important to prop-
erly set the aspect-ratio of the system before attempting
any finite-size analysis of the data. An efficient choice [26]
is to work with M = Lζdep, where ζdepis the roughness of
the line at T = 0 depinning. We take the value ζdep= 1.25
determined numerically [27,29]. This choice ensures on
one hand that M is large enough for periodicity effects
to be absent and decorrelation at the largest length-scales
to take place in one sweep, and on the other hand that
M is small enough to have no more that one dominant
configuration [16] controlling the small velocity regime in
each sample. Otherwise this would complicate drastically
the analysis of thermal effects at low temperatures. In the
following, we present results with L = 128, 256, 512, 1024
and 2048, with M = 430, 1024, 2436, 5792 and 13770, re-
spectively.
The inset of Fig. 2 shows v against T at Fc, i.e.
f = 0, for different system sizes. Strong finite size effects
can clearly be seen at the smallest temperatures, where
the motion becomes increasingly more correlated. More-
over, we find that finite size effects are appreciable up to
L = 512 in all the temperature range analyzed in Fig. 2.
However, for L ≥ 1024 all curves collapse very well for
T > 5.10−4, as can be seen by comparing L = 1024 and
L = 2048. This data is averaged over N = 1000 samples
We then compute v
We find that
for L = 128, 256, N = 500 samples for L = 512, 1024,
and N = 100 for L = 2048.
phenomena, one can assume that the steady-state veloc-
ity, which represents the order parameter of the depinning
transition [15], is an homogeneous function of the “state
variables”, f and T, although only for positive f [16].
This is consistent with the existence of a growing cor-
relation length controlled by the velocity. With this as-
sumption, scaling arguments lead to universal functions
allowing to describe different finite-size regimes. In the
zero-temperature case such scaling relation can be writ-
ten in terms of the system size and the force as
In analogy with critical
v ∼ L−β/νg
?
f L1/ν?
.(2)
where the scaling function g(x) is such that g(x) ∼ 1 for
x ≪ 1 and g(x) ∼ xβfor x ≫ 1. This scaling relation
accounts for finite-size effects, allowing to obtain a precise
estimate [27] of the depinning exponent β. It has also been
used successfully to analyze the transient critical dynamics
of driven elastic lines [30]. Under equivalent assumptions
we can write the following scaling relation for the system
at the critical force f = 0 at finite temperature [17,31],
v ∼ L−β/νh
?
T Lβ/(ψν)?
,(3)
with a scaling function h(x) behaving as h(x) ∼ 1 for
x ≪ 1 while h(x) ∼ xψfor x ≫ 1. We use (3) to an-
alyze the data presented in the inset of Fig. 2. The re-
sults are presented in the main panel of Fig. 2. A good
collapse is obtained only for the two largest system sizes
L = 1024,2048, while a poor collapse is obtained for sys-
tem sizes up to L = 512, due to the large finite-size effects
which reduce significantly the region for thermal scaling.
Using the values β = 0.33, ν = 1.33 [30] we extract the
thermal exponent ψ by fitting the power law v ∼ Tψfor
these two largest systems sizes. The best fit is obtained
with ψ = 0.15±0.01. The power-law behavior is indicated
by the continuous dashed-line in Fig. 2.
The scaling relation (3) strongly suggests the existence
of a growing dynamical correlation length ξTat depinning,
such that ξT ∼ T−νψ/βat the critical force.
this hypothesis by measuring the temperature-dependent
structure factor S(q) of the interface. The structure factor
is defined by
We test
S(q) =
???????
1
L
L−1
?
j=0
uj(t)e−iqj
??????
?
,(4)
where q = 2πn/L, with n = 1,...,L − 1. From simple
dimensional analysis one infers that for small q, S(q) ∼
q−(1+2ζ)for a line with a roughness exponent ζ.
FRG calculations show [19] that the large-scale motion
of an interface at T = 0 and finite velocity (i.e. f > 0) can
be described by the Edwards-Wilkinson model with an ef-
fective temperature determined by the velocity and by the

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S. Bustingorry et al.
10-610-510-410-310-210-1
T
2.10-1
5.10-1
v
L = 128
L = 256
L = 512
L = 1024
L = 2048
101
102
103
104
105
T Lβ/νψ
5.100
100
v Lβ/ν
L = 1024, 2048
Fig. 2: Finite-size scaling of the velocity. Main panel: rescaled
velocity for L = 1024 and L = 2048 from which we have ex-
tracted the thermal exponent ψ = 0.15 ± 0.01 using eq. (3).
The power law fit is represented by the continuous dashed line.
The inset shows v for all the system sizes. The other, previ-
ously known, T = 0 critical exponents used in the scaling are
β = 0.33 and ν = 1.33.
disorder strength. At T = 0 the corresponding crossover
length ξf, diverges as f → 0+ (see however Ref. [16] for
the situation below threshold) and separates two rough-
ness regimes [19,27]: at length scales lower than ξf, the
roughness scales with the depinning exponent ζdep, while
at length scales larger than ξfit scales with a purely ther-
mal roughness exponent ζT = 1/2, with ζT < ζdep. It is
thus natural to check whether the correlation length ξT
which controls the rounding of the depinning transition at
f = 0 and T > 0 has the same geometrical interpretation.
In such a case, for f = 0 and T > 0, the structure factor
would scale as
S(q) ∼ T−νψ(1+2ζdep)/βs
?
q T−νψ/β?
.(5)
where the scaling function s(x) behaves as s(x)
x−(1+2ζT)for x ≪ 1 and s(x) ∼ x−(1+2ζdep)for x ≫ 1.
∼
The main panel of Fig. 3 shows the rescaling of the
structure function S(q) according to Eq. (5) for different
temperatures (the unscaled data is shown in the inset).
The system size used is L = 1024 and the range of tem-
peratures analyzed covers well the power law regime of v
from which we have measured the thermal exponent ψ.
To rescale S(q) we have used all the previously known
exponents: ψ was extracted from the velocity in Fig. 2,
ζT = 0.5 and ζdep = 1.25 from T = 0 depinning sim-
ulations [27,29]. As it can be observed, the collapse of
curves is excellent in the three temperature decades ana-
lyzed. Fig. 2 and Fig. 3 thus give strong support to the
view that the scaling behavior at the depinning transition,
both for f > 0, T = 0 and the f = 0, T > 0 cases, is con-
trolled by the same velocity-dependent correlation length,
Fig. 3: Scaling of the finite-temperature structure factor for
L = 1024. The inset shows S(q) for different temperatures,
as indicated, and the main panel shows its rescaled form using
eq. (5). We have used the exponent ψ extracted from the veloc-
ity in fig. 2, and the known roughness exponents ζT = 0.5 and
ζdep= 1.25. Dotted and dashed lines correspond to the asymp-
totic behaviors x−(1+2ζT)and x−(1+2ζdep), respectively, with
x = q T−νψ/β. The characteristic thermally-induced length-
scale ξT ∼ T−νψ/βseparates these two roughness regimes.
having identical geometrical interpretations. This length
diverges with the velocity as ξ ∼ v−ν/βregardless of how
the v = 0 critical point is approached (f → 0+ at T = 0
or T → 0 at f = 0).
Let us now compare our results with previous analysis
of the thermal rounding. In ref. [21], the authors gave
a first estimate for the thermal exponent ψ of an elastic
string in a random medium. Instead of dealing with a
finite-size system, they simulated the driven interface in a
semi-infinite medium with M → ∞ by dynamically gen-
erating disorder in a small region of the sample around
the moving system. By fitting the resulting time-averaged
velocity-force characteristics they have calculated the crit-
ical force, and the critical exponents. The value they es-
timated for the thermal exponent, ψ = 0.16 [21], is very
close to our present estimate, ψ = 0.15. However, their
numerical procedure also gives β ≈ 0.24, which is much
lower than the one obtained in more recent numerical sim-
ulations, β ≈ 0.33 [27,30] and analytical calculations [19],
prompting for questions on the reliability of such a semi-
infinite procedure to obtain the critical exponents. The
thermal exponent ψ has been also measured in numeri-
cal simulations of domain wall motion in the random field
Ising model, by fitting various parameters of the V (F,T)
curves in order to obtain universal functions. The reported
value for this indirect measure is ψ ≈ 0.2 in (1+1) di-
mensions [22], higher than our value. Although our study
rests on random-bond disorder, it has been shown [19,32]
that for the T = 0 dynamics random-bond disorder and
random-field are in the same universality class, contrarily
to the statics. One would thus naively expect the same

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Thermal rounding of the depinning transition
thermal exponent as the one found in Ref. [22], assum-
ing that anharmonic corrections to the elasticity, present
in the random fied Ising model but absent in our model,
can not change the value of ψ. If that is the case for the
random field Ising model, the difference in the thermal
exponent can be attributed to the numerical limitations
in [22] compared to our method, where the control pa-
rameter F −Fccan be determined with high accuracy, for
obtaining this exponent.
More recently, a finite temperature study of the depin-
ning transition in a model of extremal activated dynam-
ics [33] was reported. Analyzing the geometry of the line
at depinning with this artificial dynamics gives a charac-
teristic length l ∼ T−0.95which separates the ζdep= 1.25
and ζT = 0.5 regimes of roughness and which can be also
associated with the distribution of subcritical forces along
the front. If l was the velocity-dependent dynamical cor-
relation length of the depinning transition ξ, this result
would imply v ∼ T0.24, yielding a value of ψ = 0.24, which
is much higher than ours and previous reported values. So
most likely this artificial dynamics does not allow to infer
the velocity for the depinning problem.
One of the interests in trying to obtain an accurate
determination of the rounding exponent is to determine
which scaling law governs it and whether it is an inde-
pendent exponent from the T = 0 depinning exponents
or not. Indeed, in the simpler case of a particle in a one
dimensional potential, the value of the thermal exponent
is related to the first-passage-time problem of overcoming,
by thermal fluctuations, the vanishing barrier at the de-
pinning or saddle-node bifurcation [34,35]. This leads to
the relation ψ = β/(2 − β) = 1/3 with β = 1/2 for any
analytical potential in one dimension with non-vanishing
second derivative at the critical point. For the case of
the interface no solid determination of the rounding ex-
ponent exists. One proposal [20] for the rounding expo-
nent is ψ = β/(1 + 2β). This relation, with β = 1/3
would lead to ψ = 1/5 which would be much higher
than our numerical estimate, although it compares well
with the random field Ising model value. It would also
work poorly for the d = 2 case of the random field Ising
model (using β = 2/3 it would give ψ = 2/7 instead of
the measured [23] ψ = 0.42) and seems thus to be ruled
out. For charge-density-waves it was proposed [17] using
mean-field theory that ψ = 3β/2. This law agreed with
numerical simulations for finite-dimension charge-density-
waves models [31]. Using the analogy with standard crit-
ical phenomena (viewing F − Fcas T − Tc, V as a mag-
netization, and T as an external magnetic field), a scal-
ing relation for the thermal exponent can be obtained by
using the standard hyperscaling relation [23], leading to
ψ = β/[(d+1)ν −β], where d is the internal dimension of
the interface (d = 1 in the present case). Such an estimate
gives good results for the d = 2 rounding exponents for the
random field Ising model [23]: using β = 2/3 and ν = 3/4
it predicts ψ = 8/19 ≈ 0.421 which compares well with the
numerical value ψ = 0.42. For our one-dimensional case, it
would predicts the value ψ = 1/7 ≈ 0.143, lower but very
close to our numerical value. This relation seems thus
empirically quite good. Of course, this approach is purely
phenomenological as there is not any clear justification for
using the equilibrium scaling relations for non-equilibrium
dynamical transitions.
A rigorous derivation is thus clearly needed. Although
in principle the finite temperature FRG calculation [19]
allows to reach the thermal exponent, the equations are
quite complicated and the involved analytical approach
has not been accomplished so far. In that respect our re-
sults provide an important clue by showing that the scal-
ing properties near the critical point f = T = 0 are con-
trolled by the same velocity-dependent correlation length
as for the T = 0, f > 0 case. They show that the large-
scale geometry is also the same in both cases, regardless
the origin of the steady-state velocity. In the FRG this
implies that the flow of the friction/velocity (the param-
eter λ in [19]) is identical to the zero-temperature case,
providing an independent confirmation of the hypothesis
used to study the T = 0 flow, that the rounding of the
cusp occurs at a scale ρλwhich is negligible compared to
λ close to the depinning [19]. Note that since v ∼ Tψand
ψ < 1 the velocity is indeed large compared to the temper-
ature. This makes it likely that the whole rounding of the
cusp is still controlled by the velocity, although in principle
it could be also possible that, even if the temperature is
smaller than λ, it is larger that the velocity cusp-rounding
scale, ρλ. Our result thus urge for a reexamination and a
further analysis, either analytical or numerical, to check
these possible scenarios and to extract the rounding expo-
nent directly from the FRG calculation [36]. Finally, let us
point out that, although the results here presented corre-
spond strictly to the steady-state evolution of the driven
interface, we have also found that the value of ψ agrees
well with the ones obtained from the analysis of the non-
steady short-time relaxation of the interface at f = 0 and
T > 0 [36].
Our results are directly relevant for recent experiments
on the domain wall motion in ferromagnetic thin films [4].
Indeed, in these experiments both the thermal exponent
ψ and the geometrical properties such as the dynamical
correlation length ξ and the roughness exponents ζT and
ζdep could be obtained by imaging the structure of the
moving domain walls. This would allow to check the cur-
rent picture of the depinning transition as a collective phe-
nomenon. It is also worth stressing that our study, and
in particular our finite-size effects analysis, is relevant for
other model systems with system-size limitations, such as
the recently reported simulations on the temperature de-
pendence of the flux lines dynamics in high-temperature
superconductors [37].
In conclusion we have studied the thermal rounding of
the depinning transition by analyzing both the steady-
state velocity and geometry of a one-dimensional inter-
face moving in a two dimensional random medium. We
have obtained the thermal exponent ψ = 0.15 ± 0.01 by

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S. Bustingorry et al.
directly extracting the temperature dependence of the ve-
locity at the critical force, v ∼ Tψ. As it was recently
done in Ref [27] for the T = 0 case, we have exploited
the easy access to the exact critical force for each disor-
der realization by using the powerful Rosso-Krauth algo-
rithm [25,29]. This allowed us to eliminate the statistical
uncertainty in the control parameter induced by the sam-
ple to sample fluctuations of the critical force, which is
present in all previous Langevin-dynamics numerical ap-
proaches to the thermal-rounding problem. Moreover, we
have shown explicitly that the value of ψ is consistent with
the existence of a velocity-dependent correlation length ξ
separating two regimes of roughness, and we find that ξ
diverges as v → 0 the same way, regardless we approach
the depinning threshold f = T = 0 from positive forces, or
from the temperature axis. Our results are relevant for re-
cent experiments on domain wall motion in ferromagnetic
thin films [4].
∗ ∗ ∗
We thank A. Rosso for illuminating discussions. This
work was supported in part by the Swiss National Science
Foundation under MaNEP and Division II.
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