Dislocation loops and bond-orientational order in the Abrikosov flux-line lattice

An Abrikosov flux-line lattice with an equilibrium concentration of unbound dislocation loops is considered as a way to describe the entangled flux liquid that arises in high-$T_c$ superconductors. The long-wavelength properties of this dislocation loop gas are discussed using continuum elastic theory. We show explicitly that edge dislocations drive the long-wavelength shear modulus to zero, i.e., melt the lattice, analogous to what happens in two dimensions. Dislocations do not, however, destroy the sixfold long-range orientational order of the crystal in the xy plane. The flux-line lattice with dislocations is therefore a hexatic liquid crystal of lines rather than an isotropic liquid. The expected signature of an entangled hexatic liquid from neutron diffraction is discussed. Long-range orientational order relaxes in the z direction with a correlation length due to flux-line entanglement mediated by screw dislocations.