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

Gauthier Mukerjee¹, Juan A. Bonachela², Miguel A. Muñoz³, Kay Jörg Wiese¹
¹ CNRS-Laboratoire de Physique de l'Ecole Normale Supérieure, PSL Research University, Sorbonne Université, Université Paris Cité, 24 rue Lhomond, 75005 Paris, France
²Department of Ecology, Evolution, and Natural Resources, Rutgers University, New Brunswick, NJ, United States
³Departamento de Electromagnetismo y Físíca de la Materia and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, Granada, Spain

Abstract

Depinning of elastic systems advancing on disordered media can usually be described by the quenched Edwards-Wilkinson equation (qEW) which encompasses a broad universality class. However, additional ingredients such as anharmonicity and forces that cannot be derived from a potential energy may generate a different scaling behavior at depinning. An example of the latter 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. Here, we show by using exact mappings supported by extensive numerical simulations that, at least for one- and two-dimensional interfaces, 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. On the other hand, we cannot rule out the existence of distinct universality classes for $d > 2$. We then develop scaling arguments for all critical exponents. Dimension $d=1$ is special since blocking configurations are directed-percolation (DP) clusters, allowing us to check scaling exponents against high-precision results from DP. To show that qKPZ is indeed the effective field theory for all these models, we develop an algorithm to measure the effective scale-dependent elasticity $c$, the effective KPZ non-linearity $\lambda$, and the effective disorder correlator $\Delta(w)$ across scales. This allows us to define the correlation length of the disorder $\rho:=\Delta(0)/|\Delta'(0)|$, and a dimensionless universal KPZ amplitude ${\cal A} :=\rho \lambda /c$, that takes the value ${\cal A}=1.10(2)$ in all systems considered in $d=1$. Using Graphical Processing Units, we are able to make further predictions in $d=1$ and $d=2$, and to some degree in $d=3$. Our work paves the way for a deeper understanding of depinning transitions in the qKPZ class, and in particular, for the construction of a field theory that we describe in a companion paper.

arXiv:2207.08341 [pdf]
Phys. Rev. E 107 (2023) 054136 [pdf]