VARIATIONAL THEORY OF ELASTIC MANIFOLDS WITH CORRELATED DISORDER AND LOCALIZATION OF INTERACTING QUANTUM PARTICLES

We apply the Gaussian variational method (GVM) to study the equilibriu m statistical mechanics of two related systems; (i) classical elastic manifolds, such as flux lattices, in the presence of columnar disorder correlated along the tau direction, and (ii) interacting quantum part icles in a static random potential. We find localization by disorder, the localized phase being described by a replica-symmetry-broken solut ion confined to the mode omega=0. For classical systems we compute the correlation function of relative displacements. In d = 2 + 1, in the absence of dislocations, the GVM allows one to describe the Bose glass phase. Along the columns the displacements saturate at a length l(per pendicular to), indicating flux-line localization. Perpendicularly to the columns long-rang order is destroyed. We find a divergent tilt mod ulus c(44) = (infinity) and a x similar to tau(1/2) scaling. Quantum s ystems are studied using the analytic continuation from imaginary to r eal time tau-->it. We compute the conductivity and find that it behave s at small frequency as sigma(omega)approximate to omega(2) in all dim ensions (d<4) for which disorder is relevant. We compute the quantum l ocalization length xi. In d=1, where the model also describes interact ing fermions in a static random potential, we find a delocalization tr ansition and obtain analytically both the low- and high-frequency beha vior of the conductivity for any value of the interaction. We show tha t the marginality condition appears as the condition to obtain the cor rect physical behavior. Agreement with renormalization group results i s found whenever it can be compared.