Roughening of the anharmonic Larkin model

We study the roughening of $d$-dimensional directed elastic interfaces subject to quenched random forces. As in the Larkin model, random forces are considered constant in the displacement direction and uncorrelated in the perpendicular direction. The elastic energy density contains an harmonic part, proportional to ${\left({\partial }_{x}u\right)}^2$, and an anharmonic part, proportional to ${\left({\partial }_{x}u\right)}^{2n}$, where $u$ is the displacement field and $n>1$ an integer. By heuristic scaling arguments, we obtain the global roughness exponent $\zeta$, the dynamic exponent $z$, and the harmonic to anharmonic crossover length scale, for arbitrary $d$ and $n$, yielding an upper critical dimension $d_c\left(n\right)=4n$. We find a precise agreement with numerical calculations in $d=1$. For the $d=1$ case we observe, however, an anomalous “faceted” scaling, with the spectral roughness exponent ${\zeta }_s$ satisfying ${\zeta }_s>\zeta >1$ for any finite $n>1$, hence invalidating the usual single-exponent scaling for two-point correlation functions, and the small gradient approximation of the elastic energy density in the thermodynamic limit. We show that such $d=1$ case is directly related to a family of Brownian functionals parameterized by $n$, ranging from the random-acceleration model for $n=1$ to the Lévy arcsine-law problem for $n=\infty$. Our results may be experimentally relevant for describing the roughening of nonlinear elastic interfaces in a Matheron-de Marsilly type of random flow.

Figure 1

Schematics of the anharmonic Larkin model (ALM) for the particular $d=2$ case, corresponding to a two-dimensional interface in a three-dimensional medium. The directed interface is described by an instantaneous scalar displacement field in the vertical direction $u(x,t)$ governed by Eq. (7), with $x\equiv (x_1,x_2)$ the internal coordinate. The elasticity of the interface is nonlinear, with an elastic energy given by Eq. (6). Arrows indicate that each point $x$ of the interface is subject to a quenched uncorrelated scalar random force $f\left(x\right)$, described by Eq. (3).

Figure 2

Disorder-averaged last-end-point squared displacement of a $d=1$ ALM interface divided by the expected size scaling, as a function of the system size $L$, using ${10}^4$ samples for each value of $n$ and $L$. Constant behavior at large $L$ confirms the validity of the predicted global roughness exponent $\zeta (d=1,n)$ of Eq. (22). The error bars are estimated from the standard deviations.

Figure 3

Rescaled disorder-averaged quadratic displacement of the end-point combining both an harmonic and an anharmonic elasticity. The master curve confirms the general crossover scaling predicted in Eqs. (26) and (32), with $L_{\text{anh}}$ particularized for $d=1$ and $n=2$ [Eq. (33)]. The dashed and solid lines indicate the scalings $\overline{u_L^2}\sim L^{\zeta (d=1,n=1)}$ and $\overline{u_L^2}\sim L^{\zeta (d=1,n=2)}$, respectively, with the roughness exponents predicted by Eq. (22).

Figure 4

Rescaled structure factors of the harmonic string (i.e., $d=1, n=1$) Larkin model (LM) for different times $t$, rescaled according to Eq. (5) with $\zeta \equiv \zeta (d=1,n=1)=3/2$ and $z\equiv z(d=1,n=1)=2$. The dashed line indicates a $q^{-(1+2{\zeta }_s)}$ scaling, with ${\zeta }_s=\zeta$. Inset: raw data.

Figure 5

Structure factor time evolution for the anharmonic string $n=2, d=1$. The spectral roughness exponent ${\zeta }_s$ fits individual curves as $S(q,t)\sim q^{-(1+2{\zeta }_s)}$ for $q{t}^{1/z}\gg 1$, while the global exponent $\zeta$ fits the curves envelope as $S(q,t)\sim q^{-(1+2\zeta )}$. Scaling is anomalous, ${\zeta }_s\ne \zeta$, in contrast with the harmonic model (see Fig. 4).

Figure 6

Rescaled structure factor evolution for the $d=1$ and $n=2$ anharmonic Larkin model, using $\zeta \equiv \zeta (d=1,n=2)=7/6$ and $z\equiv z(d=1,n=2)=5/3$, as predicted by Eqs. (22) and (23). The dashed line corresponds to the power-law $1/q^{1+2{\zeta }_s}$, indicating a spectral roughness exponent ${\zeta }_s\approx 1.39$.

Figure 7

Global roughness exponent $\zeta$, spectral roughness exponent ${\zeta }_s$, and dynamical exponent $z$ obtained from simulations of the $d=1$ harmonic and anharmonic Larkin models as a function of the elasticity exponent $n$. $n=1$ corresponds to the usual harmonic Larkin Model, while $n>1$ correspond to different anharmonic Larkin models. The spectral roughness exponent equals the global exponent only for the harmonic case, $n=1$. For $n>1$, we find ${\zeta }_s(d=1,n)-\zeta (d=1,n)\approx 0.23$, as indicated by the dashed line.

Figure 8

Rescaled end-point distribution $P(u_L,n)$. The dashed and the solid lines indicate the analytical results known for the random acceleration process and for the Lévy arcsine-law process, respectively. The first maps to the harmonic Larkin model ($n=1$); the second maps to the extreme ($n=\infty$) anharmonic Larkin model.

Figure 9

Snapshots of $L=1024$ interfaces using the dynamic solution for different $n$ and the same realization of the random forces $f\left(x\right)$. The $n=1$ corresponds to the $1d$ harmonic Larkin model. The interfaces become more “faceted” with further increasing $n$.