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

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 zeta(parallel to) = 1.25 +/- 0.01, dynamic exponent z(parallel to) = 1.43 +/- 0.01, correlation-length exponent nu = 1.33 +/- 0.02, depinning exponent beta = 0.24 +/- 0.01, and avalanche-size exponent tau(parallel to) = 1.09 +/- 0.03. In the transverse direction we find zeta(perpendicular to)= 0.5 +/- 0.01, z(perpendicular to) = 2.27 +/- 0.05, and tau(perpendicular to) = 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)]. 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(perpendicular to) = z(parallel to)+ 1/nu and nu = 1/(2 - zeta(parallel to)) of EK are satisfied, the transversal movement is that of a Brownian, with a clock set locally by the forward movement. This implies zeta(perpendicular to)= (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 T-eff(perpendicular to) similar to 1/nu and T-eff(parallel to) similar to 1/v(3) for RB disorder and as T-eff(perpendicular to) approximate to T-eff(parallel to) similar to 1/v for RF disorder.

Automated summary

In this study, the dynamics of a single directed elastic string driven through a three-dimensional disordered medium is investigated numerically and analytically. The results show that in the driving direction, the string is super-rough with certain roughness, dynamic, correlation-length, depinning, and avalanche-size exponents. The transverse fluctuations do not affect the critical exponents in the driving direction. Random-bond and random-field disorder yield the same universality class as a two-dimensional random medium. The distribution of local displacements has different characteristics in the parallel and transverse directions.