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Motivated by the problem of weak collective pinning of vortex lattices inhigh-temperature superconductors, we study the model system of afour-dimensional elastic manifold with N transverse degrees of freedom(4+N-model) in a quenched disorder environment. We assume the disorder to beweak and short-range correlated, and neglect thermal effects. Using areal-space functional renormalization group (FRG) approach, we derive a RGequation for the pinning-energy correlator up to two-loop correction. Thesolution of this equation allows us to calculate the size R_c of collectivelypinned elastic domains as well as the critical force F_c, i.e., the smallestexternal force needed to drive these domains. We find R_c prop. todelta_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 tolowest-order perturbation calculations which we briefly review, we thus arriveat determining both alpha_1 (one-loop) and alpha_2 (two-loop).Comment: 18 pages, 2 figures, to appear in Phys. Rev.

physical review. b, condensed matter

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn /><mml:mo>(</mml:mo><mml:mn>4</mml:mn><mml:mo>+</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo><mml:mn /></mml:math>-dimensional elastic manifolds in random media: A renormalization-group analysis

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