Roughening of the anharmonic Larkin model

Abstract

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 $({\partial }_{x}u{)}^2$, and an anharmonic part, proportional to $({\partial }_{x}u{)}^{2n}$, 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 $d_c(n)=4n$. 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 non-linear elastic interfaces in a Matheron-de Marsilly type of random flow.

12 pages, 9 figures, published version

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• Nonequilibrium criticality driven by Kardar-Parisi-Zhang fluctuations in the synchronization of oscillator lattices
• Kardar-Parisi-Zhang fluctuations in the synchronization dynamics of limit-cycle oscillators
• Kardar-Parisi-Zhang universality class in the synchronization of oscillator lattices with time-dependent noise
• Dynamic response and roughening of ferroelectric domain walls driven at planar electrode edges
• Influence of coupling symmetries and noise on the critical dynamics of synchronizing oscillator lattices
• Roughening of the Anharmonic Elastic Interface in Correlated Random Media