Impact of long-range interactions on the disordered vortex lattice

The interaction between the vortex lines in a type-II superconductor is mediated by currents. In the absence of transverse screening this interaction is long ranged, stiffening up the vortex lattice as expressed by the dispersive elastic moduli. The effect of disorder is strongly reduced, resulting in a mean-squared displacement correlator $〈{u}^{2}(\mathbf{R},L)〉\ensuremath{\equiv}〈[\mathbf{u}(\mathbf{R},L)\ensuremath{-}\mathbf{u}(0,0){]}^{2}〉$ characterized by a mere logarithmic growth with distance. Finite screening cuts the interaction on the scale of the London penetration depth $\ensuremath{\lambda}$ and limits the above behavior to distances $Rl\ensuremath{\lambda}.$ Using a functional renormalization-group approach, we derive the flow equation for the disorder correlation function and calculate the disorder-averaged mean-squared relative displacement $〈{u}^{2}(\mathbf{R})〉\ensuremath{\propto}{\mathrm{ln}}^{2\ensuremath{\sigma}}{(R/a}_{0}).$ The logarithmic growth $(2\ensuremath{\sigma}=1)$ in the perturbative regime at small distances [A. I. Larkin and Yu. N. Ovchinnikov, J. Low Temp. Phys. 34, 409 (1979)] crosses over to a sub-logarithmic growth with $2\ensuremath{\sigma}=0.348$ at large distances.