Depinning and flow of a vortex line in a uniaxial random medium

Abstract

We study numerically and analytically the dynamics of a single directed elastic string driven through a three-dimensional disordered medium. In the quasistatic limit the string is super-rough in the direction of the driving force, with roughness exponent ζ∥=1.25±0.01, dynamic exponent z∥=1.43±0.01, correlation-length exponent ν=1.33±0.02, depinning exponent β=0.24±0.01, and avalanche-size exponent τ∥=1.09±0.03. In the transverse direction we find ζ⊥=0.5±0.01, z⊥=2.27±0.05, and τ⊥=1.17±0.06. Our results show that transverse fluctuations do not alter the critical exponents in the driving direction, as predicted by the planar approximation (PA) proposed by Ertas and Kardar (EK) [Phys. Rev. B 53, 3520 (1996)PRBMDO0163-182910.1103/PhysRevB.53.3520]. We check the PA for the measured force-force correlator, comparing to the functional renormalization-group and numerical simulations. Both random-bond (RB) and random-field (RF) disorder yield a single universality class, indistinguishable from the one of an elastic string in a two-dimensional random medium. While relations z⊥=z∥+1/ν and ν=1/(2-ζ∥) of EK are satisfied, the transversal movement is that of a Brownian, with a clock set locally by the forward movement. This implies ζ⊥=(2-d)/2, distinct from EK. Finally, at small driving velocities the distribution of local parallel displacements has a negative skewness, while in the transverse direction it is a Gaussian. For large scales, the system can be described by anisotropic effective temperatures defined from generalized fluctuation-dissipation relations. In the fast-flow regime the local displacement distributions become Gaussian in both directions and the effective temperatures vanish as Teff⊥∼1/v and Teff∥∼1/v3 for RB disorder and as Teff⊥≈Teff∥∼1/v for RF disorder.