(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 $R_c\propto {\delta }_p^{{\alpha }_2}\exp ({\alpha }_1/{\delta }_p)$ and $F_c\propto {\delta }_p^{-2{\alpha }_2}\exp (-2{\alpha }_1/{\delta }_p)$, where ${\delta }_p\ll 1$ parametrizes the disorder strength, ${\alpha }_1=(2/\pi {)}^{N/2\mathrm{}}8{\pi }^2/(N+8\mathrm{})\mathrm{}$, and ${\alpha }_2{=2\mathrm{}(5N+22)/(N+8\mathrm{})\mathrm{}}^2$. 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).