Depinning in the quenched Kardar-Parisi-Zhang class I: Mappings, simulations and algorithm

TL;DR

This paper investigates the depinning transition in the quenched KPZ universality class through mappings, simulations, and algorithms, revealing universal behaviors and providing new tools for analyzing critical exponents and effective parameters.

Contribution

It introduces a comprehensive analysis of the quenched KPZ class, including mappings to cellular automata, scaling arguments, and a novel algorithm for estimating key parameters, establishing universality.

Findings

The qKPZ class encompasses anharmonic depinning and certain cellular automata.

The universal KPZ amplitude is approximately 1.10 in one dimension.

The new algorithm effectively estimates the elasticity and non-linearity parameters.

Abstract

Depinning of elastic systems advancing on disordered media can usually be described by the quenched Edwards-Wilkinson equation (qEW). However, additional ingredients such as anharmonicity and forces that can not be derived from a potential energy may generate a different scaling behavior at depinning. The most experimentally relevant is the Kardar-Parisi-Zhang (KPZ) term, proportional to the square of the slope at each site, which drives the critical behavior into the so-called quenched KPZ (qKPZ) universality class. We study this universality class both numerically and analytically: by using exact mappings we show that at least for $d=1,2$ this class encompasses not only the qKPZ equation itself, but also anharmonic depinning and a well-known class of cellular automata introduced by Tang and Leschhorn. We develop scaling arguments for all critical exponents, including size and duration of avalanches. The scale is set by the confining potential strength $m^2$. This allows us to estimate numerically these exponents as well as the $m$-dependent effective force correlator $\Delta(w)$, and its correlation length $\rho:= \Delta(0)/|\Delta'(0^+)|$. Finally we present a new algorithm to numerically estimate the effective ($m$-dependent) elasticity $c$, and the effective KPZ non-linearity $\lambda$. This allows us to define a dimensionless universal KPZ amplitude ${\cal A}:=\rho \lambda /c$, which takes the value ${\cal A}=1.10(2)$ in all systems considered in $d=1$. This proves that qKPZ is the effective field theory for all these models. Our work paves the way for a deeper understanding of depinning in the qKPZ class, and in particular, for the construction of a field theory that we describe in a companion paper.