Elastic theory of flux lattices in the presence of weak disorder

The effect of weak impurity disorder on flux lattices at equilibrium is studied quantitatively in the absence of free dislocations using both the Gaussian variational method and the renormalization group. Our results for the mean-square relative displacements B̃(x)=〈u(x)-u(0)${\mathrm{〉}}^2$¯ clarify the nature of the crossovers with distance. We find three regimes: (i) a short distance regime (‘‘Larkin regime’’) whre elasticity holds, (ii) an intermediate reigme (‘‘random manifold’’) where vortices are pinned independently, and (iii) a large distance, quasiordered regime where the periodicity of the lattice becomes important. In the last regime we find universal logarithmic growth of displacements for 2<d<4: B̃(x)∼${\mathit{A}}_{\mathit{d}}$ln‖x‖ and persistence of algebraic quasi-long-range translational order. The functional renormalization group to O(ε=4-d) and the variational method, when they can be compared, agree within 10% on the value of ${\mathit{A}}_{\mathit{d}}$. In d=3 we compute the function describing the crossover between the three reigmes. We discuss the observable signature of this crossover in decoration experiments and in neutron-diffraction experiments on flux lattices. Qualitative arguments are given suggesting the existence for weak disorder in d=3 of a ‘‘Bragg glass’’ phase without free dislocations and with algebraically divergent Bragg peaks. In d=1+1 both the variational method and the Cardy-Ostlund renormalization group predict a glassy state below the same transition temperature T=${\mathit{T}}_{\mathit{c}}$, but with different B̃(x) behaviors. Applications to d=2+0 systems and experiments on magnetic bubbles are discussed.