Driven Interface Depinning in a Disordered Medium

The dynamics of a driven interface in a medium with random pinning forces is analyzed. The interface undergoes a depinning transition where the order parameter is the interface velocity $v$, which increases as $v \sim (F-F_c)^\theta$ for driving forces $F$ close to its threshold value $F_c$. We consider a Langevin-type equation which is expected to be valid close to the depinning transition of an interface in a statistically isotropic medium. By a functional renormalization group scheme the critical exponents characterizing the depinning transition are obtained to the first order in $\epsilon=4-D>0$, where $D$ is the interface dimension. The main results were published earlier [T. Nattermann et al., J. Phys. II France {\bf 2} (1992) 1483]. Here, we present details of the perturbative calculation and of the derivation of the functional flow equation for the random-force correlator. The fixed point function of the correlator has a cusp singularity which is related to a finite value of the threshold $F_c$, similar to the mean field theory. We also present extensive numerical simulations and compare them with our analytical results for the critical exponents. For $\epsilon =1$ the numerical and analytical results deviate from each other by only a few percent. The deviations in lower dimensions $\epsilon = 2,3$ are larger and suggest that the roughness exponent is somewhat larger than the value $\zeta = \epsilon / 3$ of an interface in thermal equilibrium.