Elastic theory of pinned flux lattices

The effect of weak impurity disorder on flux lattices at equilibrium is studied in the absence of free dislocations using both the Gaussian variational method and, to O(ε=4-d), the functional renormalization group. We find universal logarithmic growth of displacements for 2<d<4:〈u(x)-u(0)${\mathrm{〉}}^2$¯∼${\mathit{A}}_{\mathit{d}}$ln‖x‖ and persistence of algebraic quasi-long-range translational order. When the two methods can be compared they agree within 10% on the value of ${\mathit{A}}_{\mathit{d}}$. We compute the function describing the crossover between the ‘‘random manifold’’ regime and the logarithmic regime. A similar crossover could be observable in present decoration experiments.