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

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 cannot 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 Δ(w), and its correlation length ρ:=Δ(0)/|Δ^{'}(0)|. Finally, we present an algorithm to numerically estimate the effective (m-dependent) elasticity c, and the effective KPZ nonlinearity λ. This allows us to define a dimensionless universal KPZ amplitude A:=ρλ/c, which takes the value 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.

Depinning in the quenched Kardar-Parisi-Zhang class. II. Field theory.

There are two main universality classes for depinning of elastic interfaces in disordered media: quenched Edwards-Wilkinson (qEW) and quenched Kardar-Parisi-Zhang (qKPZ). The first class is relevant as long as the elastic force between two neighboring sites on the interface is purely harmonic and invariant under tilting. The second class applies when the elasticity is nonlinear or the surface grows preferentially in its normal direction. It encompasses fluid imbibition, the Tang-Leschorn cellular automaton of 1992 (TL92), depinning with anharmonic elasticity (aDep), and qKPZ. While the field theory is well developed for qEW, there is no consistent theory for qKPZ. The aim of this paper is to construct this field theory within the functional renormalization group (FRG) framework, based on large-scale numerical simulations in dimensions d=1, 2, and 3, presented in a companion paper [Mukerjee et al., Phys. Rev. E 107, 054136 (2023)10.1103/PhysRevE.107.054136]. In order to measure the effective force correlator and coupling constants, the driving force is derived from a confining potential with curvature m^{2}. We show, that contrary to common belief, this is allowed in the presence of a KPZ term. The ensuing field theory becomes massive and can no longer be Cole-Hopf transformed. In exchange, it possesses an IR attractive stable fixed point at a finite KPZ nonlinearity λ. Since there is neither elasticity nor a KPZ term in dimension d=0, qEW and qKPZ merge there. As a result, the two universality classes are distinguished by terms linear in d. This allows us to build a consistent field theory in dimension d=1, which loses some of its predictive powers in higher dimensions.