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

Abstract: 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 non-linear, 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. 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 non-linearity $\lambda$. 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.