  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 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 prop. to
delta_p^alpha_2 exp(alpha_1/delta_p) and F_c prop. to delta_p^(-2 alpha_2)
exp(-2 alpha_1/delta_p), where delta_p <<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).

Superconductivity;Disordered Systems and Neural Networks