Driven Interface Depinning in a Disordered Medium

TL;DR

This paper analyzes the depinning transition of a driven interface in a disordered medium using a functional renormalization group approach, deriving critical exponents and validating them with numerical simulations.

Contribution

It provides a detailed perturbative calculation of critical exponents for interface depinning and compares analytical results with simulations across different dimensions.

Findings

Critical exponents are derived to first order in epsilon.

Numerical results closely match analytical predictions for epsilon=1.

The roughness exponent exceeds the thermal equilibrium value in lower dimensions.

Abstract

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.