Entanglement in Dilute Flux Line Liquids

We construct, within the framework of classical statistical mechanics, a mean field theory of dilute flux line liquids which goes beyond linear hydrodynamics. Within our approach, we find that interactions between vortices produce a massive term in the Hamiltonian of the internal modes of the flux lines which confines their transverse fluctuations. This suggests that the flux line liquid, at least in the low density limit considered in this paper, might very well be in a weakly entangled state, where the average width $<u^2>^{{1/2}}$ of flux lines can be much larger than the average distance between the lines but does not diverge with the sample thickness L. Consequences on the physics of flux line liquids are briefly discussed.