Strong anisotropy in two-dimensional surfaces with generic scale invariance: Gaussian and related models

TL;DR

This paper investigates strong anisotropy in two-dimensional surfaces with generic scale invariance, proposing a new anisotropic scaling Ansatz and testing it on nonlinear stochastic models relevant to thin film growth.

Contribution

It introduces a naturally adapted SA Ansatz for experimental observables and validates it using a Gaussian approximation on a nonlinear anisotropic stochastic equation.

Findings

The SA Ansatz effectively describes anisotropic surface properties.

Gaussian approximation captures key asymptotic behaviors.

The approach aligns with experimental measurements of thin film surfaces.

Abstract

Among systems that display generic scale invariance, those whose asymptotic properties are anisotropic in space (strong anisotropy, SA) have received a relatively smaller attention, specially in the context of kinetic roughening for two-dimensional surfaces. This is in contrast with their experimental ubiquity, e.g. in the context of thin film production by diverse techniques. Based on exact results for integrable (linear) cases, here we formulate a SA Ansatz that, albeit equivalent to existing ones borrowed from equilibrium critical phenomena, is more naturally adapted to the type of observables that are measured in experiments on the dynamics of thin films, such as one and two-dimensional height structure factors. We test our Ansatz on a paradigmatic nonlinear stochastic equation displaying strong anisotropy like the Hwa-Kardar equation [Phys. Rev. Lett. 62, 1813 (1989)], that was initially proposed to describe the interface dynamics of running sand piles. A very important role to elucidate its SA properties is played by an accurate (Gaussian) approximation through a non-local linear equation that shares the same asymptotic properties.