Roughening of the anharmonic Larkin model

PURRELLO, VÍCTOR H.; IGUAIN, JOSÉ L; KOLTON, ALEJANDRO B.

Bariloche

Workshop; Yielding Phenomena in Disordered Systems 2019; 2019

Centro Atómico Bariloche

We study the roughening of d-dimensional directed elastic interfaces subject to quenchedrandom forces. As in the Larkin model, random forces are considered constant in the dis-placement direction and uncorrelated in the perpendicular direction. The elastic energydensity contains an harmonic part, proportional to (∇u)2 , and an anharmonic part, pro-portional to (∇u)?2n, where u is the displacement field and n> 1 an integer. By heuristicscaling arguments, we obtain the global roughness exponent ζ, the dynamic exponent z,and the harmonic to anharmonic crossover length scale, for arbitrary d and n, yielding anupper critical dimension dc(n)=4n. We find a precise agreement with numerical calculationsin d=1. For the d=1 case we observe, however, an anomalous ?faceted? scaling, with thespectral roughness exponent ζs satisfying ζs > ζ > 1 for any finite n> 1, hence invalidatingthe usual single-exponent scaling for two-point correlation functions, and the small gradientapproximation of the elastic energy density in the thermodynamic limit. We show that suchd=1 case is directly related to a family of Brownian functionals parameterized by n, rang-ing from the random-acceleration model for n=1, to the Lévy arcsine-law problem for n=∞.Our results may be experimentally relevant for describing the roughening of non-linear elasticinterfaces in a Matheron-de Marsilly type of random flow.