# Microscopic interplay of temperature and disorder of a one-dimensional
elastic interface

Nirvana Caballero<EMAIL_ADDRESS>Department of Quantum Matter
Physics, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva,
Switzerland  Thierry Giamarchi  Department of Quantum Matter Physics,
University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva, Switzerland
Vivien Lecomte  Université Grenoble Alpes, CNRS, LIPhy, FR-38000 Grenoble,
France  Elisabeth Agoritsas  Institute of Physics, Ecole Polytechnique
Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

###### Abstract

Elastic interfaces display scale-invariant geometrical fluctuations at
sufficiently large lengthscales. Their asymptotic static roughness then
follows a power-law behavior, whose associated exponent provides a robust
signature of the universality class to which they belong. The associated
prefactor has instead a non-universal amplitude fixed by the microscopic
interplay between thermal fluctuations and disorder, usually hidden below
experimental resolution. Here we compute numerically the roughness of a one-
dimensional elastic interface subject to both thermal fluctuations and a
quenched disorder with a finite correlation length. We evidence the existence
of a novel power-law regime at short lengthscales. We determine the
corresponding exponent $\zeta_{\text{dis}}$ and find compelling numerical
evidence that, contrarily to available analytic predictions, one has
$\zeta_{\text{dis}}<1$. We discuss the consequences on the temperature
dependence of the roughness and the connection with the asymptotic random-
manifold regime at large lengthscales. We also discuss the implications of our
findings for other systems such as the Kardar–Parisi–Zhang equation and the
Burgers turbulence.

††preprint: APS/123-QED

##  I Introduction

Interfaces are ubiquitous in Nature and provide remarkable challenges both
from theoretical and experimental fronts [1]. Experimentally they span very
different underlying physics and characteristic scales, with examples ranging
from domain walls in ferromagnetic or ferroelectric thin films [2, 3, 4, 5, 6,
7, 8, 9] to imbibition fronts in porous media [10], fracture surfaces [11, 12,
13] and growing fronts of cell colonies [14, 15, 16]. On the theory side,
describing the competition between the elastic interactions that tend to order
them, and the temperature and system heterogeneity that hinder this tendency,
is a considerable theoretical challenge, with a resulting out-of-equilibrium
physics akin to the one of glasses [17].

A successful theoretical tool to study interfaces is provided by the
_disordered elastic systems_ framework [18, 19, 20, 21, 22, 23]: interfaces
are modeled as elastic objects evolving in a quenched disordered landscape and
subject to thermal noise. Remarkably, this minimal description is enough to
account for many key statistical features of the geometry and dynamics of
static or driven interfaces [24]. In particular, geometrical fluctuations are
scale-invariant at sufficiently large lengthscales, evidenced by a power-law
behavior of the roughness, defined as the variance of the relative
displacement between two points in the interface. The associated roughness
exponent provides a robust signature of the universality class to which an
interface belongs [25], depending solely on its dimensionality and on the
nature of its elasticity and underlying disorder [22].

Beyond this hallmark of universality, the roughness prefactor itself encodes
quantitative information about the microphysics of a given system. Labelled as
non-universal, this roughness feature has been poorly exploited up to now,
despite of its crucial experimental relevance for the quantitative
determination of characteristic scales and the validity range of theoretical
predictions in a given experimental setup [26, 27, 28]. At equilibrium its
amplitude is fixed at short lengthscales by the microscopic interplay between
thermal fluctuations and a spatially-correlated disorder. The resulting
temperature crossover below a characteristic energy scale ${T_{c}(\xi)}$,
associated to the disorder strength and finite correlation length $\xi$, is
thus a macroscopic ‘smoking gun’ of the spatial structure of microscopic
disorder [29, 30, 31]. Furthermore, studies using perturbative [32] or
variational [29] methods suggest at short-lengthscales a power-law excess
roughness –in addition to thermal fluctuations– with a characteristic exponent
${\zeta_{\text{dis}}=1}$. Unfortunately a quantitative characterization of
this microscopic interplay, usually hidden below experimental resolution, has
been difficult to access up to now.

For static one-dimensional (1D) interfaces, understanding this interplay is
important to characterize the finite-time and steady-state fluctuations of the
1D Kardar–Parisi–Zhang (KPZ) equation [33, 34]. Considerable advances were
achieved recently [35, 36, 37, 38, 39] allowing for the computation of
universal exponents and distributions [38, 39] for spatially-uncorrelated
noises. Nevertheless, a regime which resists an exact analytical treatment is
the low-temperature limit in a spatially-correlated disorder [40, 41, 42, 30,
31, 43, 44, 45, 46], despite its relevance to analyze experimental
realizations of KPZ [47, 48, 38, 49], or of Burgers turbulence [50, 51].

In this paper, we address these issues by numerically computing the roughness
of a 1D interface with short-range correlated disorder and thermal
fluctuations. We unveil the key regime where the interplay between temperature
and disorder leads to a roughness excess – compared to bare thermal
fluctuations – and show that it relaxes towards a power-law behavior
${B_{\text{dis}}(r)\approx A_{\text{dis}}(T)\,r^{2\zeta_{\text{dis}}}}$ at
short lengthscale $r$. For the specific model we consider, previous analytical
but approximate predictions propose $\zeta_{\text{dis}}=1$; yet, we find
compelling numerical evidence showing that $\zeta_{\text{dis}}<1$, with
important consequences on the temperature dependence of the roughness and the
connection to the asymptotic random-manifold regime.

The outline of the paper is the following. After describing the model in Sec.
II, we focus on the roughness and disentangle its different contributions in
Sec. III. We then present a scaling description for the crossovers between
different roughness regimes in Sec. IV. We further discuss the implications of
our results for KPZ-related systems in Sec. V, and conclude in Sec. VI.

Figure 1:  a) Snapshot of an initially flat interface $u(y,t)$ evolved under
the qEW dynamics (1) up to time $t=10^{9}$. b) Roughness ${B(r,t)}$ obtained
for interfaces that evolved during different times averaged over 50
realizations (10 or 5 realizations for the larger studied times – indicated in
the color scale) with and without disorder (continuous and dotted lines,
respectively) at $T=0.01$ and $\epsilon=0.1$. Shaded fluctuations are shown in
the same color scale. Dashed lines correspond to the analytical prediction (5)
in a clean system. The expected power-law at large distances in the disordered
case (with $\zeta_{\text{KPZ}}=2/3$) is indicated in a brown dotted line.

##  II Model

We consider a 1D interface parametrized at time $t$ by
${(y,u(y,t))\in[0,L]\times\mathbb{R}\subset\mathbb{R}^{2}}$, as shown in Fig.
1 a). The displacement field ${u(y,t)}$ is univalued with periodic boundary
conditions ${u(y=0,t)=u(y=L,t)}$. Starting from a flat initial condition
${u(y,t=0)=0}$ the interface evolves according to the quenched
Edwards–Wilkinson (qEW) equation [52, 53, 54, 55] in absence of an external
force:

$\eta\partial_{t}u(y,t)=c\partial^{2}_{y}u(y,t)+F_{p}\left(y,u(y,t)\right)+\chi(y,t)\,.$
(1)

Thermal fluctuations are described by a centered Gaussian white noise
${\chi(y,t)}$ of two-point correlator
${\left\langle\chi(y_{1},t_{1})\chi(y_{2},t_{2})\right\rangle=2\eta
T\delta(y_{2}-y_{1})\delta(t_{2}-t_{1})}$, where
${\left\langle\dots\right\rangle}$ denotes the thermal average, $\eta$ is the
friction coefficient and $T$ the temperature (the Boltzmann constant is fixed
to ${k_{B}=1}$). We fix the units of time and energy by setting $\eta=1$ and
the elastic constant $c=1$. The disorder is a quenched random potential
${V_{p}(y,u)}$ and its associated pinning force
${F_{p}(y,u)=-\partial_{u}V_{p}(y,u)}$, both Gaussian with zero mean and
correlators:

$\begin{split}&\overline{V_{p}(y_{1},u_{1})V_{p}(y_{2},u_{2})}=D\,R_{\xi}(u_{2}-u_{1})\,\delta(y_{2}-y_{1})\,,\\\
&\overline{F_{p}(y_{1},u_{1})F_{p}(y_{2},u_{2})}=\Delta_{\xi}(u_{2}-u_{1})\,\delta(y_{2}-y_{1})\,,\end{split}$
(2)

where $\overline{\vphantom{|}\cdots\vphantom{|}}$ denotes the average over
disorder realizations. We consider the case of ‘random-bond’ disorder with
finite correlation length ${\xi}$, i.e. with a short-range correlator
${R_{\xi}(u)=\xi^{-1}R_{1}(u/\xi)}$ normalized as
${\int_{\mathbb{R}}\mathrm{d}u\,R_{\xi}(u)=1}$ and $D$ the disorder strength.
Both correlators are even and related through
${\Delta_{\xi}(u)=-D\,R_{\xi}^{\prime\prime}(u)}$ [21], so that
${\int_{\mathbb{R}}\mathrm{d}u\,\Delta_{\xi}(u)=0}$.

We numerically integrate (1) keeping $u$ as a continuous variable while
discretizing the $y$ direction in $L=512$ segments of unit length. These qEW
settings allow us to advantageously replace the Dirac ${\delta(y-y^{\prime})}$
by the Kronecker ${\delta_{yy^{\prime}}}$ in (2) and thus to implement an
uncorrelated disorder along the internal direction $y$. In the transverse
direction, the disorder potential is dynamically generated [56, 55] with
random numbers taken from a uniform distribution in the range
$[-\frac{\epsilon}{2},\frac{\epsilon}{2}]$ at equidistant positions, spaced by
${\Delta u=1}$. To obtain the random-bond pinning force at a point $u$, we
interpolate the two nearest random numbers with a linear spline and take its
derivative with respect to $u$. With these settings, the correlator
${\Delta_{\xi}(u=u_{2}-u_{1})=\frac{2}{3}\frac{D}{\xi^{3}}\Delta_{\text{adim}}(\hat{u}=u/\xi)}$
is fully given by a piecewise linear continuous function,
${\Delta_{\text{adim}}(\hat{u})}$, that connects the values

$\left\\{\begin{array}[]{l}\Delta_{\text{adim}}(\hat{u}=0)=2\\\
\Delta_{\text{adim}}(|\hat{u}|=1)=-1\\\ \Delta_{\text{adim}}(|\hat{u}|\geq
2)=0\end{array}\right.$ (3)

(see Appendix A). Our effective parameters are then
$\\{\xi=1,D=\frac{\epsilon^{2}}{12}\\}$, where we took $\epsilon=0.1$, and we
explore the temperature range ${T\in\\{0.005,\dots,0.074\\}}$. Further
technical details on the numerical integration are reported in Appendix B.

We emphasize that we explicitly consider a spatially-correlated disorder in
the transverse direction, i.e. allowing for a resolution at displacements
smaller than ${\xi}$. This is a key difference compared to other numerical
studies of the qEW dynamics, which usually assume a discrete description of
the interface on a lattice, with a disorder completely uncorrelated from site
to site. The latter allows for highly efficient computations to access
roughness scalings at asymptotically large lengthscale [57, 58, 54], but by
definition it prevents any study of the microscopic interplay between thermal
fluctuations and disorder. For equilibrated 1D interfaces, an alternative
approach is provided by the ‘1+1 directed polymer’ in a correlated disorder
with ${\xi>0}$, studied through the KPZ equation that its free energy
satisfies [31]. The latter requires, however, the disorder to have a finite
correlation length along the internal direction $z$ and bears by construction
an artifact at short lengthscales –these two aspects becoming increasingly
problematic at lower temperatures. Our numerical approach bypasses these
issues in order to provide a reliable characterization, for the qEW dynamics
(1), of the temperature-dependent microscopic interplay we are interested in.

##  III Disentangling roughness contributions

To characterize the geometrical fluctuations of the interface we focus on the
roughness function (also called ‘height-height correlation function’ in the
context of growing surfaces [25]):

$B\left(r=|y_{2}-y_{1}|,t\right)=\overline{\left\langle[u(y_{2},t)-u(y_{1},t)]^{2}\right\rangle}\,.$
(4)

It quantifies the variance of the relative displacements of the interface, as
a function of the lengthscale $r$, and inherits the translation invariance in
$y$ of the microscopic disorder. For a clean system ($F_{p}(y)=0$) we can
compute analytically the time dependence of this correlation for an infinite
interface [52, 59]:

$B_{\text{th}}(r,t)=\frac{Tr}{c}\bigg{[}1-\frac{1}{\sqrt{\pi}zr}\Big{(}e^{-z^{2}r^{2}}-1\Big{)}-\frac{2}{\sqrt{\pi}}\int_{0}^{zr}\\!\\!\\!\\!\\!\\!\mathrm{d}t\,e^{-t^{2}}\bigg{]}$
(5)

where ${z=\sqrt{\frac{\eta}{8ct}}}$. At large times, (5) converges to the
static thermal roughness ${B_{\text{th}}(r)\equiv\frac{Tr}{c}}$. To
disentangle different contributions on $B\left(r,t\right)$, we introduce the
excess roughness ${B_{\text{dis}}=B-B_{\text{th}}}$, defined as the difference
between the total roughness and its analytical value in the clean case. Note
that for a finite size $L$ with periodic boundary conditions, we have
${B_{\text{th}}(r,L)=\frac{Tr}{c}(1-r/L)}$, thus predicting a saturation for
lengthscales ${r\lesssim L/2}$ and a decrease of the roughness beyond (also
expected for the whole roughness function).

In Fig. 1, we show the evolution of $B(r,t)$ at a fixed temperature in clean
and disordered systems. For the clean case, results match the analytical
prediction (5). For the disordered case, at very large times and large
lengthscales, the power-law ${\sim r^{2\zeta_{\text{KPZ}}}}$ with
$\zeta_{\text{KPZ}}=\frac{2}{3}$, characteristic of a random-bond disorder is
reached. Note, however, that already after $t=10^{3}$ the roughness has
converged at short lengthscales ($r\lesssim 20$).

Figure 2:  a) Roughness of interfaces evolving in a spatially-correlated
disorder, for different temperatures $T$. Each curve corresponds to an average
over realizations that evolved with the qEW dynamics (1) for a time $t=10^{6}$
starting from a flat initial condition. Dashed lines represent the roughness
(5) in absence of disorder. b) Corresponding excess roughness
$B_{\text{dis}}(r)$ due to disorder, obtained as the difference between the
total roughness and the thermal roughness (5). The decrease beyond ${r=L/2}$
is an artifact of the periodic boundary conditions. Our numerical study shows
the existence of a power-law regime for the excess roughness at small scales,
characterized by an exponent $\zeta_{\text{dis}}$.

In Fig. 2 a), we show $B(r,T)$ at fixed large time $t=10^{6}$ averaged over 50
realizations [60]. To increase the statistics, for each realization that
evolved for a time $10^{6}$, we included 100 more configurations equally
spaced in time intervals of $10^{3}$. This procedure allows us to increase the
statistics for lengthscales at which the interface has already equilibrated.
In Fig. 2 b), we report the corresponding excess roughness
${B_{\text{dis}}(r,T)}$ at different temperatures. The numerical data shows
the existence of a power-law regime of ${B_{\text{dis}}(r,T)}$ characterized
by an exponent $\zeta_{\text{dis}}$. This regime, which can be obscured by the
existence of the large thermal component of the roughness, is nevertheless
present and universal 111In Appendix D, we compare our results for two very
different disorder distribution, generated from potentials which have bounded
and unbounded support. We show that ${B_{\text{dis}}(r,T)}$ presents the same
exponent $\zeta_{\text{dis}}$, providing compelling evidence for the
universality of the power-law regime characterized by this exponent. , and
results from the interplay between the finite correlation of the disorder and
the thermal fluctuations. As we discuss in Sec. IV and Sec. V, besides the
existence of the regime itself, the value of the exponent $\zeta_{\text{dis}}$
has important consequences for the temperature dependence of the roughness.

Figure 3:  Collapse of the excess roughness $B_{\text{dis}}(a,T)$ for
different fixed lengthscales $a$ as a function of temperature. The collapse is
obtained with a power-law $a^{2\zeta_{\text{dis}}}$, with
$\zeta_{\text{dis}}=0.91\pm 0.05$. The insert shows that the collapse is lost
for ${\zeta_{\text{dis}}^{\text{th}}=1}$.

A fit of $B_{\text{dis}}(r)$ with a power-law ${\sim r^{2\zeta_{\text{dis}}}}$
in the regime of small values of $r$ gives an exponent
$\zeta_{\text{dis}}=0.91$ with a temperature-dependent amplitude. Given the
importance of estimating if $\zeta_{\text{dis}}=1$, we now explore this issue.
This result is further detailed in Fig. 3 where the rescaling of
$B_{\text{dis}}$ with the power-law $r^{2\zeta_{\text{dis}}}$ is shown. The
collapse of the curves is very sensitive to the precise value of the exponent
and allows us to rule out with a high confidence a value of
$\zeta_{\text{dis}}=1$ (see Appendix C). Interestingly, the only currently
available analytical predictions gives
${\zeta_{\text{dis}}=\zeta_{\text{dis}}^{\text{th}}\equiv 1}$, in distinct
methods: a finite-temperature perturbative approach [32] and two different
computations based on a Gaussian-Variational-Method (GVM) [29]. Our numerical
findings thus provide evidence that non-perturbative approaches, beyond the
usual variational alternatives, are required to determine
$\zeta_{\text{dis}}$. We also checked that a different disorder distribution
leads to the same value of $\zeta_{\text{dis}}$ (see Appendix D), supporting
the robustness of our finding that ${\zeta_{\text{dis}}<1}$.

##  IV Crossovers between power-law roughness regimes

To analyze the implications of this result, we determine the relevant
lengthscales. The static roughness ${B(r)=B_{\text{th}}(r)+B_{\text{dis}}(r)}$
crosses over from a thermal regime ${B(r)\approx
B_{\text{th}}(r)=Tr^{2\zeta_{\text{th}}}/c}$ at very short lengthscales to the
random-manifold regime ${B(r)\approx
B_{\text{dis}}(r)\approx\xi_{\text{eff}}^{2}\,(r/L_{c})^{2\zeta_{\text{KPZ}}}}$
at lengthscales larger than a characteristic length $L_{c}$, closely related
to the ‘Larkin length’ in higher-dimensional interfaces [62, 19]. In our
system, ${\zeta_{\text{th}}=\frac{1}{2}}$ and
${\zeta_{\text{KPZ}}=\frac{2}{3}}$. Previous analytical studies [29, 30, 44]
have shown that ${L_{c}=\frac{(T/f)^{5}}{cD^{2}}}$ and
${\xi_{\text{eff}}=\frac{(T/f)^{3}}{cD}}$, with a temperature-dependent
parameter $f$. Defining the energy scale ${T_{c}=(\xi cD)^{1/3}}$ we expect a
monotonous crossover from ${f\approx 1}$ at ${T\gg T_{c}}$ to ${f\sim
T/T_{c}}$ at ${T\ll T_{c}}$ 222See Eqs. (49), (97) in [30]. The _effective_
characteristic energy scale ${\widetilde{E}=T/f}$ controls all the relevant
scales for the geometrical fluctuations and in the two previous limits, is
fixed by thermal fluctuations (${\widetilde{E}\approx T}$) or by disorder
(${\widetilde{E}\approx T_{c}}$). From our numerical findings
${B_{\text{dis}}(r)}$ has two power-law regimes with a crossover lengthscale
$r_{0}$:

$B_{\text{dis}}(r)=\left\\{\begin{array}[]{ll}A_{1}(r/r_{0})^{2\zeta_{\text{dis}}}:&r\ll
r_{0}\,,\\\ A_{2}(r/r_{0})^{4/3}:&r\gg r_{0}\,.\end{array}\right.$ (6)

Let us fix $r_{0}$ with a scaling argument to determine
${\\{L_{c},\xi_{\text{eff}}\\}}$ [44]. We rescale the spatial coordinates
${(y=b\bar{y},u=a\bar{u})}$ and parameters
${\\{c=c^{\prime},D=D_{0}D^{\prime},T=\widetilde{E}T^{\prime},\xi=a\xi^{\prime}\\}}$
while leaving the Boltzmann weight invariant so that
${B(r;c,D,T,\xi)=a^{2}B(r/b;c^{\prime},D^{\prime},T^{\prime},\xi^{\prime})}$.
To focus on the microscopic interplay of temperature and disorder, we fix the
scales ${a=\xi}$ and ${\widetilde{E}=T/f}$ (i.e.
${\\{\xi^{\prime}=1,T^{\prime}=f\\}}$) which implies
${D_{0}=\frac{(T/f)^{3}}{c\xi}}$ and ${b=\frac{c\xi^{2}}{T/f}}$ 333The high-
temperature version of this scaling choice was reported in Table (2a) of [44],
but had an unpleasant ill-defined limit at asymptotically _large_ lengthscales
for both ${T\to 0}$ and ${\xi\to 0}$. We argue here that it is instead
relevant for _short_ lengthscales.. The short-lengthscale regime where the
disorder correlation length and the effective energy scale $\widetilde{E}$ are
equally relevant lies at ${r\leq b}$, so we can identify ${r_{0}=b}$. Note
that the effective disorder strength ${D^{\prime}=\frac{T_{c}}{T}f}$ crosses
over from ${D^{\prime}\approx 1}$ at ${T\ll T_{c}}$ to ${D^{\prime}\sim
T_{c}/T}$ at ${T\gg T_{c}}$. As physically expected, the latter case supports
a small-disorder perturbative expansion at high temperature.

We thus can determine the prefactors for ${B_{\text{dis}}(r\leq r_{0})}$ in
(6): ${A_{1}=B_{\text{dis}}(r_{0})=A_{2}}$ with $A_{2}$ fixed from the large
scales from ${A_{2}/r_{0}^{4/3}=\xi_{\text{eff}}^{2}/L_{c}^{4/3}}$. Using
${r_{0}=\frac{c\xi^{2}}{T/f}}$ we get

$A_{\text{dis}}(T)=\frac{B_{\text{dis}}(r\leqslant
r_{0})}{r^{2\zeta_{\text{dis}}}}\propto\frac{\xi^{2}T_{c}^{2}}{\left(c\xi^{2}\right)^{2\zeta_{\text{dis}}}}\left(\frac{f}{T}\right)^{2(1-\zeta_{\text{dis}})}$
(7)

Note that this implies that
${\zeta_{\text{dis}}=\zeta_{\text{dis}}^{\text{th}}=1}$ leads to a
temperature-independent roughness prefactor. On the contrary, a value
${\zeta_{\text{dis}}\neq 1}$, as we find numerically, implies that
${A_{\text{dis}}(T)}$ _does_ depend on temperature. We report this feature in
Fig. 3, thus further supporting $\zeta_{\text{dis}}<1$.

From the behavior of the interpolating parameter ${f\approx 1}$ at ${T\gg
T_{c}}$ and ${f\sim T/T_{c}}$ at ${T\ll T_{c}}$, Eq. (7) implies that
$A_{\text{dis}}(T)$ decreases with increasing $T$ at ${T\gg T_{c}}$ and
saturates at ${T\to 0}$. This scenario is at variance with our numerical
findings in Fig. 3: $A_{\text{dis}}(T)$ does indeed present a regime of
temperature $\sim T^{-2(1-\zeta_{\text{dis}})}$ compatible with (7). However,
such power-law behavior is expected to hold at high $T$ and to provide an
upper bound to the low-$T$ saturation. The reasons for this discrepancy are
unclear but could stem from: (i) numerical prefactors in the estimation of
$T_{c}(\xi)$, so that the regime ${T\ll T_{c}}$ would in fact be reached below
our temperature range; (ii) the discretization along the internal direction
$y$, which could add one lengthscale to take into account in the scaling
analysis. A breakdown of the above scaling argument cannot be excluded, but
${r_{0}={c\xi^{2}}f/{T}}$ does coincide with a more involved prediction
obtained in the KPZ language 444See Eq. (99) in [30], where the DP ‘time’
${t_{\text{sat}}}$ can be identified with our crossover $r_{0}$.. This
discrepancy, which does not question the $\zeta_{\text{dis}}$ roughness regime
existence requires further investigations that are beyond our present scope.

Let us now turn to the case of very small thermal fluctuations at $T\to
0^{+}$, where the crossover between the thermal and $\zeta_{\text{dis}}$
power-law regimes can be further examined analytically and numerically. The
crossover lengthscale $r_{0}$ is always bounded by the Larkin length $L_{c}$:

$\left\\{\begin{array}[]{ll}T\gg T_{c}:&r_{0}\ll L_{c}\,,\text{ with
}r_{0}\sim\frac{c\xi^{2}}{T}\,,\,L_{c}\sim\frac{T^{5}}{cD^{2}}\,,\\\ T\ll
T_{c}:&r_{0}\lesssim
L_{c}\sim\frac{T_{c}^{5}}{cD^{2}}=\frac{c^{2/3}\xi^{5/3}}{D^{1/3}}\,.\end{array}\right.$
(8)

At high temperature, the regime ${r\lesssim r_{0}}$ where
${B_{\text{dis}}(r)\sim r^{2\zeta_{\text{dis}}}}$ has a small extension and is
screened by the thermal contribution $Tr/c$ in $B(r)$; hence, as $r$
increases, the total roughness $B(r)$ crosses over directly from the thermal
to the random-manifold regime, with a single crossover at the high-$T$ Larkin
length $L_{c}$ such that ${B_{\text{th}}(L_{c})\approx
B_{\text{dis}}(L_{c})}$. On the contrary, in the limit ${T\to 0^{+}}$ we have
${B(r)\approx B_{\text{dis}}(r)}$, which crosses over from the new power-law
regime that we characterized ($\zeta_{\text{dis}}$) to the well-studied
random-manifold regime (${\zeta_{\text{KPZ}}}$), and a vanishing thermal
regime at ${r\leq r_{1}\sim T^{1/[1-2(1-\zeta_{\text{dis}})]}}$ 555At low $T$,
${B_{\text{th}}(r)}$ intersects ${B_{\text{dis}}(r)}$ at a crossover
${r_{1}<r_{0}}$. We can evaluate $r_{1}$ at ${T\ll T_{c}}$ from
${Tr_{1}/c\equiv A_{1}(r_{1}/r_{0})^{2\zeta_{\text{dis}}}}$, which yields
${r_{1}^{1-2(1-\zeta_{\text{dis}})}=\frac{T\,(T/f)^{4/3}}{c\xi^{4/3}D^{2/3}}r_{0}^{2(1-\zeta_{\text{dis}})}}$
with ${T/f\sim T_{c}}$. By studying the lengthscale $r_{1}$ (see Appendix E),
we find our numerical results to be compatible with a scaling
$\zeta_{\text{dis}}\approx 0.91$, but not with
$\zeta_{\text{dis}}=\zeta_{\text{dis}}^{\text{th}}=1$.

Figure 4: Summary of the crossover scales at high versus low temperature for
the equilibrated 1D qEW interface. Red (respectively blue) arrows indicate how
crossover scales change when the temperature increases (respectively
decreases), in both temperature regimes. We always have ${r_{0}\lesssim
L_{c}}$. Note that assuming ${\xi=0}$ corresponds to setting ${T_{c}=0}$, in
which case we could account only for the high-temperature regime.

Physically, the characteristic lengthscales ${\\{L_{c},r_{0},r_{1}\\}}$ and
the energy scale ${\widetilde{E}=T/f}$ are related to the microscopic
interplay between temperature and disorder. The Larkin length can be deduced
from asymptotically large lengthscales, easier to access analytically and
experimentally, hence its crucial role in previous studies [62, 19, 21, 29,
30, 44]. In this paper, we argue instead that it stems as a consequence of
short-lengthscale properties, usually hidden below experimental resolution,
that one can nevertheless investigate via the excess roughness due to
disorder. We expect this scenario –relating the microscopic interplay to the
quantitative amplitude of the macroscopic fluctuations– to be valid beyond the
specific interface (1D qEW) we considered.

##  V Implications for other KPZ-related problems

The relevance of these results extends well beyond the sole equilibrium 1D
interface, thanks to the exact mapping onto the so-called 1+1 directed polymer
(DP) and the KPZ settings [33, 67]. An interface segment can be seen as a
polymer growing along a DP ‘time’: fixing an extremity of the DP at the origin
(i.e. ${u(0)=0}$), its endpoint distribution encodes, at a fixed ‘time’ $r$,
the geometrical fluctuations of an equilibrated interface at a lengthscale $r$
and _in a given disorder realization_. The DP free-energy, i.e. the logarithm
of this distribution, then obeys a KPZ equation with a ‘time’ $r$ and a
spatially-correlated noise $V_{p}$ (our random potential), with a ‘sharp-
wedge’ initial condition [67]. In this language, the asymptotic random-
manifold scaling at ${r\to\infty}$ describes the approach to the KPZ steady-
state regime, and the microscopic interplay at short lengthscales corresponds
to the KPZ evolution at short ‘times’.

The DP formulation allows one to disentangle the thermal and sample-to-sample
fluctuations (thermal and disorder averages, respectively). It provides in
particular a natural decomposition of the roughness as
${B(r)=B_{\text{th}}(r)+B_{\text{dis}}(r)}$, with a direct definition of the
excess roughness as the second _disorder_ cumulant
${B_{\text{dis}}(r)=\overline{\langle u(r)-u(0)\rangle^{2}}^{c}}$ 666See Eqs.
(B.12)-(B.13) in [30], derived from the statistical tilt symmetry. Note that
this relation is valid under the condition that ${u(0)=0}$ for any given
disorder realization $V_{p}$; since our numerical approach does not impose
such a boundary condition, we cannot directly evaluate $B_{\text{dis}}(r)$
from such an identity.. Such writing in fact encodes a more general
decomposition: in a given disorder realization, the DP free-energy can be
decomposed as the sum of a purely thermal contribution (that remembers the
initial condition $u(0)=0$) and a disorder contribution (that is invariant by
translation in distribution and thus forgets –statistically– the initial
condition). Such a manifestation of the so-called ‘statistical tilt symmetry’
(STS) (valid at $\xi=0$ and $\xi>0$ 777See Appendix B in Ref. [30].) means
that $B_{\text{dis}}(r)$ directly encodes the two-point correlations of the DP
endpoint when the thermal fluctuations (that fully capture the initial
condition) have been subtracted. This explains why $B_{\text{dis}}(r)$ gives
access to a regime of fluctuations that is inherent to disorder and that, in
spite of its seemingly artificial definition, it bears a _bona fide_ physical
content. We expect that the excess roughness $B_{\text{dis}}(r)$ can be
generalized to other systems presenting STS (e.g. for interfaces in higher
dimensions or that are discretized in the longitudinal direction).

The DP free-energy steady-state properties at ${\xi>0}$ have been studied
through non-perturbative functional-renormalization-group [45], but the
understanding of the short-DP-‘time’ regime is incomplete, either through
variational 888See Eqs. (56), (126) in [29], or Eqs. (6.21), (6.26) in [80].,
perturbative 999See Eq. (20) in [32] for the second-order correction at finite
temperature, leading to the conjecture stated in Eq. (21)., or indirect
101010See Sec. VI in [30] and the summary in Fig. 7.1 in [80]. approaches. On
the numerical side, we have previously reported in Ref. [31] an apparent
power-law behavior for $B_{\text{dis}}$ at short lengthscales, with
$\zeta_{\text{dis}}\in[2,2.5]$ depending on the temperature. These exponent
values cannot be directly compared to the one found in the present paper for
two reasons: (i) In Ref. [31], a singularity at initial KPZ ‘time’, due to the
approach used, had to be amended through a short-time regularization that can
affect $B_{\text{dis}}$ in that regime. (ii) In Ref. [31], the disorder
presented a finite correlation length $\xi_{z}$ in the internal direction,
with ${\xi_{z}\approx 1}$ so that ${r_{0}\approx\xi_{z}}$; this is at odds
with the assumption ${\xi_{z}=0}$ inherent to Eq. (2), especially because
$\xi_{z}$ is larger than the typical lengthscales of the regime studied in the
present paper. In fact, addressing these issues partly triggered the present
study, whose settings also have the advantage to apply to generic non-
stationarized interfaces (and not exclusively to the specific static 1D
interface of Ref. [31]).

The scope of the results obtained in the present paper can thus be extended as
follows. Consider the two-point correlator of the derivative of the DP free-
energy at ‘time’ $r$. It is a central quantity that encodes the scalings of
the amplitude of the KPZ field. Strictly at $\xi=0$, it contains a singular
Dirac delta contribution, that is rounded at finite $\xi>0$: its properties
thus reflect how the disorder correlation length $\xi$ affects the KPZ
scaling. In Ref. [30], it was shown that scale $r_{0}$ corresponds to the
typical DP ‘time’ at which the maximum of this correlator saturates to its
steady-state value. The short-lengthscale power-law regime of
$B_{\text{dis}}(r)$ governed by $\zeta_{\text{dis}}$ that we uncovered
describes how the DP free-energy (i.e. the KPZ field) distribution evolves at
short times. This indicates that, to access the KPZ scalings at short times,
one needs to disentangle the disorder from the thermal fluctuations, and that
this happens in the scale-invariant regime governed by $\zeta_{\text{dis}}$.

Since ${r_{0}\to 0}$ as ${T\to 0}$, the range of this regime vanishes in high-
temperature limit ${T\gg T_{c}}$ (see Fig. 4). Nevertheless, already in this
limit, the KPZ equation at short times with different initial conditions
displays distinct large deviations [73]: we thus expect that at finite $T$ and
$\xi$, the short-time regime we have uncovered will have a counterpart in such
large-deviation analysis. How these features, and more generally the exact
predictions for the KPZ equation available at ${\xi=0}$ [74], are modified for
a given disorder correlator ${R_{\xi>0}}(u)$, in the low-temperature limit,
remains currently an open issue. Although the asymptotic roughness exponent
${\zeta_{\text{KPZ}}=\frac{2}{3}}$ is known to be robust to finite $\xi$ [44],
the functional form of the KPZ steady-state distribution and correlators –as
well as their overall amplitude and characteristic scales– will be modified by
the short-‘time’ evolution. For instance, in experiments of growing interfaces
in liquid crystals which exhibits geometrical fluctuations remarkably
compatible with the KPZ predictions at ${\xi=0}$ [47, 48, 74]. In particular,
short-time fluctuations could be probed within our approach for the droplet
initial condition (through the excess roughness or more generally after
removing a thermal-like contribution to the KPZ field). Also, in the large-
time regime, their two-point correlator has a voltage-dependent amplitude
which could be a signature of our ‘low-temperature’ regime 111111See Sec.
7.2.2 in [80]..

Finally, since the noisy Burgers equation is derived from the KPZ one, our
results translate to the Burgers turbulence [51] where the small-temperature
asymptotics corresponds to the inviscid limit and large-scale forcing [50,
46]. In particular, the regime we have identified could be compared to the
recent results of Ref. [76], where, within a Fourier approach, a fluctuation
regime scaling as $\sim k^{-1}$ (with $k$ the Fourier mode) was identified
numerically and could interestingly be analysed with other scaling exponents.

##  VI Conclusion

In this work, we find that, hidden under thermal fluctuations of a 1D
interface, there exists a scale-invariant regime at short lengthscales
governed by an exponent $\zeta_{\text{dis}}$. Our numerical analysis shows
that ${\zeta_{\text{dis}}<1}$, at variance with the existing analytical
estimates for such exponent $\zeta_{\text{dis}}^{\text{th}}=1$. The
discrepancy hints towards the non-perturbative nature of the short-lengthscale
regime of ${B_{\text{dis}}(r)}$, which thus requires further analytical
investigations. In systems where thermal fluctuations are strong enough, the
scaling regime described by $\zeta_{\text{dis}}$ is hidden under thermal
fluctuations. When the temperature becomes small or when the disorder
correlation length becomes large, the power-law behavior $B(r)\sim
r^{2\zeta_{\text{dis}}}$ can be manifested on experimentally or numerically
accessible ranges. For instance, large roughness exponents were found in a
recent modelization [77] of experimental measurements [78] of magnetic domains
subjected to an AC field. A possible cause can be that the AC field
effectively increases the disorder correlation length in the sample, thus
unveiling the $B(r)\sim r^{2\zeta_{\text{dis}}}$ regime. It is worth
investigating the existence of such scaling regime in interfaces, and to test
its robustness to features such as overhangs (for instance in disordered
Ginzburg–Landau models [79], known to reduce to qEW [59]). Adding a driving
force generates a velocity in a non-linear ‘creep regime’, whose scaling is
controlled by the static geometrical exponents. It is natural to wonder if the
scaling regime we unveiled has consequences for the creep regime, for instance
in avalanches statistics. The case of higher dimensions random manifolds is
also open.

###### Acknowledgements.

We thank discussions with Jean-Pierre Eckmann and Sebastian Bustingorry. This
work was supported in part by the Swiss National Science Foundation under
Division II. N.C. acknowledges support from the Federal Commission for
Scholarships for Foreign Students for the Swiss Government Excellence
Scholarship (ESKAS No. 2018.0636). E.A. acknowledges support from the Swiss
National Science Foundation by the SNSF Ambizione Grant PZ00P2_173962. V.L.
thanks the Université de Genève (where part of this work was performed) for
its warm hospitality, and acknowledges support by the ERC Starting Grant No.
680275 MALIG, the ANR-18-CE30-0028-01 Grant LABS and the ANR-15-CE40-0020-03
Grant LSD. The simulations were performed at the University of Geneva on the
Mafalda cluster of GPUs.

##  Appendix A Pinning force correlator and disorder strength

In this appendix, we detail how we generate the quenched random potential
${V_{p}(y,u)}$ and its associated pinning force
${F_{p}(y,u)\equiv-\partial_{u}V_{p}(y,u)}$. We recall that the interface is
parametrized by the univalued displacement field ${u(y,t)}$, with the internal
coordinate $y$ taking discrete values ${y=j\Delta y}$ (where $j$ is an
integer, ${j=0,\dots L_{y}/\Delta y}$), and the transverse coordinate $u$
taking continuous values. We consider a disorder with Gaussian distribution
fully characterized by a zero mean and the two-point correlators:

$\begin{split}&\overline{V_{p}(y_{1},u_{1})V_{p}(y_{2},u_{2})}=D\,R_{\xi}(u_{2}-u_{1})\,\delta_{y_{2}y_{1}}\,,\\\
&\overline{F_{p}(y_{1},u_{1})F_{p}(y_{2},u_{2})}=\Delta_{\xi}(u_{2}-u_{1})\,\delta_{y_{2}y_{1}}\,,\end{split}$
(9)

where $\overline{\vphantom{|}\cdots\vphantom{|}}$ denotes the average over
disorder realizations. We chose the normalization
${\int_{\mathbb{R}}\mathrm{d}u\,R_{\xi}(u)=1}$, and the correlators are simply
related by ${\Delta_{\xi}(u)=-DR_{\xi}^{\prime\prime}(u)}$. In the following,
we thus make explicit the functional ${\Delta_{\xi}(u)}$ and the disorder
strength $D$.

Figure 5:  For a fixed coordinate $y$, the random potential ${V_{p}(y,u)}$ is
generated by a linear interpolation between random numbers taken from a
uniform distribution with zero mean (orange crosses). The associated pinning
force $F_{p}$ is obtained from the associated piecewise derivative
${F_{p}(y,u)=-\partial_{u}V_{p}(y,u)}$.

###  A.1 Generating a spatially-correlated disorder

The pinning potential ${V_{p}(y,u)}$ is defined independently for each
discrete value of the coordinate $y$, so that in what follows, one fixes $y$
and considers only the coordinate $u$. The procedure to generate the
corresponding quenched ‘landscape’ ${U(u)\equiv V_{p}(y,u)}$ at fixed $y$,
spatially-correlated with a finite correlation length $\xi$, is the following
(see Fig. 5). We first discretize the direction $u$ with fixed steps ${\Delta
u=\xi}$. A random number $U_{k}$ is generated independently at each site
${u=k\Delta u}$ (with $k$ an integer) from a probability distribution function
${\mathcal{P}(U_{k})}$ of zero mean.

Then, on every interval ${u\in[k\Delta u,(k+1)\Delta u]}$ (i.e. one has
${k=\lfloor u/\Delta u\rfloor}$) the random pinning potential is defined as a
linear interpolation between $U_{k}$ and $U_{k+1}$:

$U(u)=U_{k}+\frac{U_{k+1}-U_{k}}{\Delta u}({u-k\Delta u})\,.$ (10)

One can easily check that, as required, this definition satisfies ${U(k\Delta
u)=U_{k}}$ and ${U((k+1)\Delta u)=U_{k+1}}$. The associated pinning force
${F_{p}(u)=-\partial_{u}U(u)}$ is finally given by the discrete derivative

$F_{p}(u)=-\frac{U_{k+1}-U_{k}}{\Delta u}$ (11)

with ${k=\lfloor u/\Delta u\rfloor}$ as above.

For our simulations we sort each reference value $U_{k}$ from a _uniform_
distribution on the interval ${[-\frac{\epsilon}{2},\frac{\epsilon}{2}]}$,
with ${\epsilon>0}$. It has consequently a zero mean and a variance

$D_{0}\equiv\overline{(U_{k})^{2}}=\frac{\epsilon^{2}}{12}\,.$ (12)

To keep the discussion general, thereafter we generically denote the variance
of $U_{k}$ as a control parameter $D_{0}$. In addition, we discuss in Appendix
D how choosing an alternative distribution ${\mathcal{P}(U_{k})}$ (non-uniform
but with the same variance) leads to physically consistent results.

###  A.2 Piecewise linear force correlator

We start by defining the intermediate two-point correlator of the force as

$\Delta^{(2)}_{\xi}(u_{1},u_{2})=\overline{F_{p}(u_{1})F_{p}(u_{2})}\,.$ (13)

Because it is associated to the specific set of intervals ${u\in[k\Delta
u,(k+1)\Delta u]}$ with ${k=\lfloor u/\Delta u\rfloor}$, it is important to
notice that it is _not_ invariant by translation along the $u$ direction.
Indeed, pairs of points ${(u_{1},u_{2})}$ separated by a same distance
${u=u_{2}-u_{1}}$ can either lie in the same interval $[k\Delta u,(k+1)\Delta
u]$ or not.

One has in fact three possibilities: if ${(u_{1},u_{2})}$ are

  * •

in the same interval, one has

$\Delta^{(2)}_{\xi}(u_{1},u_{2})=2\overline{(U_{i})^{2}}=\frac{2D_{0}}{\Delta
u^{2}}\,;$ (14)

  * •

in adjacent intervals $[(k-1)\Delta u,k\Delta u]$ and $[k\Delta u,(k+1)\Delta
u]$:

$\Delta^{(2)}_{\xi}(u_{1},u_{2})=-\overline{(U_{i})^{2}}=-\frac{D_{0}}{\Delta
u^{2}}\,;$ (15)

  * •

in more distant intervals $[k\Delta u,(k+1)\Delta u]$ and $[j\Delta
u,(j+1)\Delta u]$ with ${|k-j|\leq 2}$:

$\Delta^{(2)}_{\xi}(u_{1},u_{2})=0\,.$ (16)

And from now on we use that ${\xi=\Delta u}$ to emphasize the explicit
dependence on the correlation length $\xi$.

To recover a translation-invariant correlator, as required in the definitions
(9), one must average the intermediate correlator
${\Delta_{\xi}^{(2)}(u_{1},u_{2})}$ over all pairs of points ${(u_{1},u_{2})}$
separated by the same distance $u$:

$\Delta_{\xi}(u)=\int_{\mathbb{R}^{2}}\mathrm{d}u_{1}\mathrm{d}u_{2}\>\delta(u_{2}-u_{1}-u)\,\Delta^{(2)}_{\xi}(u_{1},u_{2})\,.$
(17)

One finds by an explicit computation

$\Delta_{\xi}(u)=\frac{D_{0}}{\xi^{2}}\,\Delta_{\text{adim}}\big{(}u/\xi\big{)}$
(18)

with ${\Delta_{\text{adim}}(\hat{u})}$ the piecewise linear continuous
function that connects the values:

$\displaystyle\Delta_{\text{adim}}(\hat{u})=2\quad\text{for}\quad\hat{u}=0\,,$
(19)
$\displaystyle\Delta_{\text{adim}}(\hat{u})=-1\quad\text{for}\quad|\hat{u}|=1\,,$
(20)
$\displaystyle\Delta_{\text{adim}}(\hat{u})=0\quad\text{for}\quad|\hat{u}|\geq
2\,.$ (21)

The complete function is plotted in the inset of Fig. 6.

As a self-consistent validation of our procedure, we evaluated numerically the
correlator ${\Delta_{\xi}(u)}$, and as shown in Fig. 6 we find an excellent
agreement with the expression (18).

Figure 6:  Numerical evaluation of the pinning force correlator for 100
different disorder realizations with $\epsilon=1$ (gray lines) and its average
(pink). In black dashed line we show ${\Delta_{\xi}(u)}$, the expected
correlator given by Eq. (18) with $D_{0}=1/12$. In the inset, we show the
adimensionalized force correlator ${\Delta_{\text{adim}}(\hat{u})}$ connecting
the values of Eqs. (19)-(21).

###  A.3 Correlators in Fourier space and disorder strength

One can check that
${\int_{\mathbb{R}}\mathrm{d}\hat{u}\,\Delta_{\text{adim}}(\hat{u})=0}$, as
expected for the ‘random-bond’ disorder we consider. To access the disorder
strength $D$, we switch to Fourier space where we can more easily exploit the
relation between the correlators
${\Delta_{\xi}(u)=-DR_{\xi}^{\prime\prime}(u)}$ from Eq. (9).

We first rewrite, similarly to Eq. (18), the random potential correlator in
terms of its adimensionalized version, starting from its very definition

$\overline{U(u)U(0)}\equiv
DR_{\xi}(u)\equiv\frac{D}{\xi}R_{\text{adim}}(u/\xi)\,.$ (22)

First, by direct comparison with the definition in Eq. (12) we can establish
its relation to the variance $D_{0}$:

$D_{0}\equiv\overline{U(0)^{2}}=DR_{\xi}(u=0)=\frac{D}{\xi}R_{\text{adim}}(\hat{u}=0)\,,$
(23)

and secondly the relation between the correlators
${\Delta_{\xi}(u)=-DR_{\xi}^{\prime\prime}(u)}$ becomes

$\frac{D_{0}}{\xi^{2}}\Delta_{\text{adim}}(\hat{u})=-\frac{D}{\xi^{3}}R_{\text{adim}}^{\prime\prime}(\hat{u})\,.$
(24)

Defining the Fourier transform along the transverse direction as
${\hat{\Delta}_{\text{adim}}(\hat{q})=\int_{\mathbb{R}}\mathrm{d}\hat{u}\,e^{i\hat{q}\hat{u}}\Delta_{\text{adim}}(\hat{u})}$,
one finds by direct computation that it takes the simple form:

$\hat{\Delta}_{\text{adim}}(\hat{q})=16\frac{\big{[}\sin\frac{\hat{q}}{2}\big{]}^{4}}{\hat{q}^{2}}\,.$
(25)

Defining similarly the Fourier transform
${\hat{R}_{\text{adim}}(\hat{q})=\int_{\mathbb{R}}\mathrm{d}\hat{u}\,e^{i\hat{q}\hat{u}}R_{\text{adim}}(\hat{u})}$,
Eq. (24) rewrites

$\hat{\Delta}_{\text{adim}}(\hat{q})=\frac{D}{D_{0}\xi}\hat{q}^{2}\hat{R}_{\text{adim}}(\hat{q})\,.$
(26)

At this point, we might think that we have some freedom to define
${\hat{R}_{\text{adim}}}$ up to an arbitrary constant, but we have in fact to
enforce the imposed normalization
${\int_{\mathbb{R}}\mathrm{d}u\,R_{\text{adim}}(\hat{u})=1}$ or equivalently
${\hat{R}_{\text{adim}}(\hat{q}=0)=1}$. This is achieved by imposing
${\frac{D}{D_{0}\xi}=1}$ and thus

$\hat{R}_{\text{adim}}(\hat{q})=16\frac{\big{[}\sin\frac{\hat{q}}{2}\big{]}^{4}}{\hat{q}^{4}}\quad\Rightarrow\quad\lim_{\hat{q}\to
0}\hat{R}_{\text{adim}}(\hat{q})=1\,.$ (27)

Furthermore, we have by inverse Fourier transform:

$R_{\text{adim}}(\hat{u}=0)=\int_{\mathbb{R}}\frac{\mathrm{d}\hat{q}}{2\pi}\,\hat{R}_{\text{adim}}(\hat{q})=\frac{2}{3}\,.$
(28)

We have, at last, directly access to the disorder strength:

$D\stackrel{{\scriptstyle\eqref{eq:def-D0-D-Rhat}}}{{=}}\frac{D_{0}\xi}{R_{\text{adim}}(\hat{u}=0)}\stackrel{{\scriptstyle\eqref{eq-
linear-Rhat-peak}}}{{=}}\frac{3}{2}D_{0}\xi\,.$ (29)

The last expression is valid for any random potential generated by a linear
interpolation between uncorrelated random points, drawn from an arbitrary
distribution ${\mathcal{P}(U_{k})}$ with zero mean and variance $D_{0}$. For
the uniform distribution we consider, Eq. (12) implies

$D^{\text{uniform}}=\frac{3}{2}D_{0}\xi=\frac{1}{8}\epsilon^{2}\,.$ (30)

This is the expression that we used at the end of Sec. II in the main text in
order to fix the disorder strength.

##  Appendix B Parameters of numerical simulations

To solve the quenched Edwards-Wilkinson equation (Eq. (1) of the main text)
and compute the interface roughness, we take advantage of massively parallel
accelerated computing with a CUDA C++ code running in NVIDIA GPUs with a Volta
architecture, in double precision. At each simulation step we approximate the
second derivative of $u$ along the $y$-direction by a two-point central finite
difference scheme and integrate in time with a first-order Euler step. Pseudo-
random numbers are generated with a counter-based RNG (Philox, allowing
$2^{64}$ parallel and distinct streams with a period of $2^{128}$ [56]). For
the thermal noise, we use Gaussian distributed numbers while for the quenched
disorder, we use the method described in Appendix A (and in Appendix D in the
consistency check described in the same section): the method consists in a
linear interpolation of uniformly distributed random numbers with the
implementation proposed in [55]. These random numbers, uncorrelated from site
to site, are dynamically generated along the evolution of the interface, i.e.
we build the disorder at larger $u$ only if the interface has locally wandered
further away.

Space discretization along the $y$-direction is set to $1$ and time
discretization to $10^{-2}$. We can simulate 4 realizations of systems of 512
sites for $10^{8}$ steps in approximately 8 hours. These scheme and parameters
give roughness functions in the clean case which are in very good agreement
with the theoretical prediction (Eq. (5) in the main text). The corresponding
roughness of the simulated systems differs from the theoretically predicted
values in less than $10^{-4}$ for $r\leq 50$ and less than $10^{-3}$ for
larger values of $r$ (both, simulated and predicted roughness functions are
shown in Fig. 1 in the main text). This excellent agreement is a strong
consistency check that provides a good support for the validity of the
numerical procedure in the disordered case.

##  Appendix C Determination of the best value of $\zeta_{\text{dis}}$

To obtain $\zeta_{\text{dis}}$ from the numerical data presented in the main
text, we fit $B_{\text{dis}}(r,T)$ independently for each value of $T$ in the
range $r=[1,r_{f}]$. For the nine lowest temperatures we have studied, we find
values of $\zeta_{\text{dis}}$ between 0.9 and 0.92. To determine the best
value of this exponent that is compatible with every temperature we
considered, we rely on the following scaling argument. We predict that
$B_{\text{dis}}(r_{0},T)/r_{0}^{2\zeta_{\text{dis}}}$ should be independent of
$r_{0}$ for all temperatures $T$. In Fig. 7, we illustrate that the best
choice of $\zeta_{\text{dis}}$ that ensures this collapse is
$\zeta_{\text{dis}}=0.91\pm 0.01$. We quantify the spread of the functions
$B_{\text{dis}}(r,T)/r^{2\zeta_{\text{test}}}$ around their mean value for
different values of $\zeta_{\text{test}}$ by computing the function
$F_{T,r}(\zeta)=\Sigma_{T,r}\Big{(}\frac{B_{\text{dis}}(r,T)/r^{2\zeta}-\Sigma_{r_{i}}B_{\text{dis}}(r_{i},T)/r_{i}^{2\zeta}}{\Sigma_{r_{i}}B_{\text{dis}}(r_{i},T)/r_{i}^{2\zeta}}\Big{)}^{2}$
with $r_{i}=1,\dots,5$, for the 5 lowest studied temperatures. The resulting
function has a minimum in $\zeta=0.91$, as shown in the inset of Fig. 7.

Figure 7: Numerical collapse of the excess roughness, described in the text,
for values $\zeta_{\text{test}}$ of $\zeta_{\text{dis}}$ ranging from $0.89$
to $0.93$. The value achieving the best collapse is $\zeta_{\text{dis}}=0.91$.
At 0.91 the function $F_{T,a}(\zeta)$ (see text) has a minimum, as shown in
the inset. The different dashed lines correspond to the value of $a$ indicated
in the caption. Figure 8: Roughness excess obtained for interfaces that
evolved for $10^{5}$ steps under the same conditions at $T=0.01$ for different
system sizes $L$ averaged over 50 realizations. In the inset we show the
difference between the exponents obtained from fitting the curves with a
power-law $\sim\zeta_{\text{dis}}$ in the range $[1,10]$.
$\zeta_{\text{dis}}^{\infty}$ corresponds to the exponent obtained for the
largest system $L=2^{20}$.

We note that the regime where we observe the power-law behavior characterized
by $\zeta_{\text{dis}}$ emerges at relatively short scales. We obtained the
roughness of interfaces that evolved from a flat initial condition according
to Eq. 1 under the same parameters for different system sizes. As shown in
Fig. 8 at the scales we are interested in the difference in the obtained
exponents characterizing the excess power-law regime is of the order
$10^{-3}$. For our study we then fix $L=2^{9}=512$ in this work.

Figure 9: Roughness excess obtained for interfaces that evolved for different
times (as indicated by the color scale) at $T=0.01$ for a fixed system size
$L=512$ averaged over 50 realizations in all cases, except for $t=2\times
10^{8}$, where the average was taken over 10 realizations and $t=4\times
10^{8}$, where the average was taken over 5 realizations. The curves were
shifted for clarity. See Fig. 10 for the same analysis in the case $\xi=2$.

As shown in Fig. 1, the roughness of an initially flat interface will converge
to a steady state at sufficiently long times. To test how this influences the
regime we are interested in, we follow the roughness of interfaces that
evolved for long times at a fixed temperature $T=0.01$. As shown in Fig. 9,
the exponent characterizing the excess regime fluctuates around values which
are lower than one.

Figure 10: Roughness excess obtained for interfaces that evolved for different
times (as indicated by the color scale) with a disorder correlation lenght
$\xi=2$ at $T=0.01$ for a fixed system size $L=512$ averaged over 50
realizations in all cases, except for the largest studied time, where the
average was taken over 7 realizations. The curves were shifted for clarity. In
the inset we show the exponents obtained from fitting the curves with a power-
law $\sim\zeta_{\text{dis}}$, shown in dotted black lines.

The region in which the excess regime emerges depends on the disorder
correlation length $\xi$. We can further verify our numerical findings by
studying the roughness excess for interfaces evolving under the same
conditions as before, but with a larger disorder correlation length $\xi=2$.
We observe that also in this case the exponent $\zeta_{\text{dis}}$ is
compatible with the value 0.91, as shown in Fig. 10.

##  Appendix D Disorder generated from a Gaussian distribution

Figure 11: Roughness (dotted lines) and excess roughness (continuous lines) at
$T=0.01$ averaged over 50 realizations with increased statistics, obtained for
disorders generated from a uniform and a Gaussian distribution. In the inset
we show the difference between both excess roughness.

For the temperature $T=0.01$, we determined the excess roughness
$B_{\text{dis}}^{\text{Gauss}}(r)$ for a disorder potential where the
$U_{i}$’s are distributed with a Gaussian distribution with the same variance
$\overline{(U_{i})^{2}}=1/12$ as the uniform one, see Eq. (12). On Fig. 11,
$B_{\text{dis}}^{\text{Gauss}}(r)$ is compared to the excess roughness
$B_{\text{dis}}(r)$ computed –as in the rest of the paper– for a disordered
potential with the $U_{i}$’s drawn from a uniform distribution, as described
in Appendix A.

The results show that $B_{\text{dis}}^{\text{Gauss}}(r)$ is very close to
$B_{\text{dis}}(r)$, as mentioned at the end of Sec. III. This provides a
strong evidence supporting the following points: in the asymptotic regime
$r\to 0$, the power-law behavior $B_{\text{dis}}(r)\sim
r^{2\zeta_{\text{dis}}}$ is universal, _i.e._ presents an exponent
$\zeta_{\text{dis}}$ which does not depend on the specific random-potential
disorder distribution. This is an important aspect, as the asymptotic regime
$r\to 0$ where $B_{\text{dis}}(r)$ presents the power-law behavior $\sim
r^{2\zeta_{\text{dis}}}$ could have been sensitive to the details of the
disorder correlator at small scales. Also, the prefactor $A$ in
$B_{\text{dis}}(r)\sim A\,r^{2\zeta_{\text{dis}}}$ is mainly governed by the
variance $D_{0}$ of the disorder distribution (which is the same in the
uniform and in the Gaussian distribution we have used for $\mathcal{P}(u)$).
Last, we expect also that this prefactor $A$ depends, through a numerical
constant, on the rescaled shape $\Delta_{\text{adim}}(\hat{u})$ of the
disorder correlator. This is seen in the small difference
$|B_{\text{dis}}^{\text{Gauss}}(r)-B_{\text{dis}}(r)|\ll B_{\text{dis}}(r)$
shown in the inset of Fig. 11. Such difference also scales as
$r^{2\zeta_{\text{dis}}}$ in the asymptotic regime $r\to 0$, indicating that
indeed only the prefactor $A$ is affected by the adimensionalized shape of the
disorder correlator.

##  Appendix E Scaling of the crossover scale $r_{1}(T)$

At the end of Sec. IV, we mention that the crossover scale $r_{1}$ is
determined numerically by the intersection between the two power-law scalings
${B_{\text{dis}}(r)\sim r^{2\zeta_{\text{dis}}}}$ and the thermal regime
${Tr/c}$. The scaling arguments presented in the main text indicate that
$r_{1}$ scales with temperature as

$r_{1}(T)\sim\left\\{\begin{array}[]{ll}T^{1/[1-2(1-\zeta_{\text{dis}})]}&\leavevmode\nobreak\
\text{for\leavevmode\nobreak\ }T\ll T_{c}\,,\\\ T^{5}&\leavevmode\nobreak\
\text{for\leavevmode\nobreak\ }T\gg T_{c}\,.\end{array}\right.$ (31)

To determine how our numerical results are compatible with such predictions,
we have used a fitting function

$r_{1}^{\text{fit}}(T)=C\,T^{\alpha_{-}}\big{[}1+(T/T^{\star})^{\alpha_{+}-\alpha_{-}}\big{]}$
(32)

(where $C$ and $T^{\star}$ are constants and $\alpha_{-}<\alpha_{+}$) that
interpolates between the regimes $r_{1}^{\text{fit}}(T)\sim T^{\alpha_{-}}$
for $T\ll T^{\star}$ and $r_{1}^{\text{fit}}(T)\sim T^{\alpha_{+}}$ for $T\gg
T^{\star}$. The prediction of Eq. (31) corresponds to

$\alpha_{+}=5\,,\quad\alpha_{-}=1/[1-2(1-\zeta_{\text{dis}})]\,.$ (33)

A first scenario where $\zeta_{\text{dis}}=\zeta_{\text{dis}}^{\text{th}}=1$
(according to the perturbative analysis of [32], for instance) corresponds to
$\alpha_{-}=1$. A second scenario where $\zeta_{\text{dis}}\approx 0.9<1$
corresponds to $\alpha_{-}\approx 1.25>1$. To distinguish between these two
possibilities, we have fitted the values of $r_{1}(T)$ obtained numerically
with the function (32), where we fixed the exponent $\alpha_{\pm}$ and we left
the constants $C$ and $T^{\star}$ as free parameters. As shown on Fig. 12, the
data provide a strong evidence supporting the second scenario
$\zeta_{\text{dis}}\approx 0.9<1$. Also, leaving $\alpha_{-}$ as a free
parameter for the fit, one finds $\alpha_{-}\approx 1.33$ which corresponds to
the value $\zeta_{\text{dis}}\approx 0.88$: it is compatible with the value
$\zeta_{\text{dis}}\approx 0.91$ obtained in the main text with a completely
different method.

Figure 12: Behavior of the crossover scale $r_{1}(T)$. The points are the
result of numerical simulations, while the dashed and full line curves
correspond to the two scenarios discussed in the text. The exponent
$\alpha_{-}\approx 1.25>1$ is more compatible with the numerical data, and
corresponds to $\zeta_{\text{dis}}\approx 0.9<1$.

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