Depinning of an anisotropic interface in random media: The tilt effect

We study the tilt dependence of the pinning-depinning transition for an interface described by the anisotropic quenched Kardar-Parisi-Zhang equation in $2+1$ dimensions, where the two signs of the nonlinear terms are different from each other. When the substrate is tilted by m along the positive sign direction, the critical force $F_c\mathrm{}\left(m\right)\mathrm{}$ depends on m as $F_c\mathrm{}\left(m\right)\mathrm{}-F_c\mathrm{}\left(0\right)\mathrm{}\sim -|m{|}^{1.9\left(1\right)\mathrm{}}.$ The interface velocity $v$ near the critical force follows the scaling form $v\sim |f{|}^{\theta }{\Psi }_{\pm }{(m}^2/|f{|}^{\theta +\phi })$ with $\theta =0.9\mathrm{}\left(1\right)\mathrm{}$ and $\phi =0.2\mathrm{}\left(1\right),$ where $f\equiv F-F_c\mathrm{}\left(0\right)\mathrm{}$ and F is the driving force.