Anisotropic Scaling in Threshold Critical Dynamics of Driven Directed Lines
The dynamical critical behavior of a single directed line driven in a random medium near the depinning threshold is studied both analytically (by renormalization group) and numerically, in the context of a Flux Line in a Type-II superconductor with a bulk current $\vec J$. In the absence of transverse fluctuations, the system reduces to recently studied models of interface depinning. In most cases, the presence of transverse fluctuations are found not to influence the critical exponents that describe longitudinal correlations. For a manifold with $d=4-ε$ internal dimensions, longitudinal fluctuations in an isotropic medium are described by a roughness exponent $ζ_\parallel=ε/3$ to all orders in $ε$, and a dynamical exponent $z_\parallel=2-2ε/9+O(ε^2)$. Transverse fluctuations have a distinct and smaller roughness exponent $ζ_\perp=ζ_\parallel-d/2$ for an isotropic medium. Furthermore, their relaxation is much slower, characterized by a dynamical exponent $z_\perp=z_\parallel+1/ν$, where $ν=1/(2-ζ_\parallel)$ is the correlation length exponent. The predicted exponents agree well with numerical results for a flux line in three dimensions. As in the case of interface depinning models, anisotropy leads to additional universality classes. A nonzero Hall angle, which has no analogue in the interface models, also affects the critical behavior.