(4+N)-dimensional elastic manifolds in random media: A renormalization-group analysis

Motivated by the problem of weak collective pinning of vortex lattices in high-temperature superconductors, we study the model system of a four-dimensional elastic manifold with N transverse degrees of freedom (4 + N model) in a quenched disorder environment. We assume the disorder to be weak and short-range correlated, and neglect thermal effects. Using a real-space functional renormalization-group (FRG) approach, we derive a RG equation for the pinning-energy correlator up to a two-loop correction. The solution of this equation allows us to calculate the size R-c of collectively pinned elastic domains as well as the critical force F-c, i.e., the smallest external force needed to drive these domains. We find R-c proportional to delta(alpha 2) exp(alpha(1)/delta(p)) and F-c proportional to delta(p)(-2 alpha 2) exp(-2 alpha(1)/delta(p)), where delta(p) much less than 1 parametrizes the disorder strength alpha(1) = (2/pi)(N/2)8 pi(2)/(N + 8), and alpha(2) = 2(5N + 22)/(N + 8)(2). In contrast to lowest-order perturbation calculations which we briefly review, we thus arrive at determining both alpha(1) (one loop) and alpha(2) (two loop).