# Roughening of the anharmonic Larkin model

## TL;DR

This paper investigates the roughening behavior of d-dimensional elastic interfaces with anharmonic elasticity under quenched disorder, revealing a new anomalous scaling in one dimension and providing scaling laws for arbitrary dimensions and anharmonicity levels.

## Contribution

It introduces a heuristic scaling framework for anharmonic elastic interfaces, identifies an upper critical dimension, and uncovers anomalous faceted scaling in one dimension.

## Key findings

- Derived roughness and dynamic exponents for arbitrary d and n.
- Identified anomalous faceted scaling in 1D with spectral roughness exponent > 1.
- Established connection to Brownian functionals and potential experimental relevance.

## 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 L\'evy 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.