Pinned vortex lattices: a variational study

Abstract

We study the effects of disorder on a flux lattice at equilibrium, in the absence of free dislocations, using a gaussian variational ansatz. This variational method allows us to obtain the physical properties of such a system in any dimension. For the case of point like disorder, we find universal logarithmic growth of displacements for 2 < d < 4: 〈u(χ) − u(0)〉 2 ∼ A d log|χ| and persistence of algebraic quasi-long range translational order. Close to four dimensions, a functional renormalization group calculation in ϵ = 4 − d agrees within 10% on the value of A d with the variational method. We compute the function describing the crossover between the short distance (“random manifold”) regime and the logarithmic regime. In d = 2, the variational method gives a logarithmic growth of displacement with a temperature independent prefactor at variance from previous RG calculations. Columnar disorder is studied by the same method. In d=2+1, describing vortex lattices with the field along the columns, one finds a logarithmic growth of the in-plane displacements. In d = 1 + 1 dimensions, this problem is related by bosonization to Anderson localization of interacting electrons by static disorder. The variational method predicts a localized-delocalized transition, for attractive enough interactions, in agreement with previous studies, as well as a conductivity σ( ω) ∼ ω 2 in the localized phase.