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

Motivated by the problem of weak collective pinning of vortex lattices in hightemperature 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 Rc of collectively pinned elastic domains as well as the critical force Fc, i.e., the smallest external force needed to drive these domains. We find Rc ∝ � � 2 p exp(�1/�p) and Fc ∝ � 2�2 p exp(−2�1/�p), �