New kinds of intermediate flux states should be accessible in high-$T_c$ superconductors, in fields slightly above $H_{c1\mathrm{}}$. Flux-line wandering leads to an entangled vortex state whose statistical mechanics is isomorphic to an interacting 2D Bose superfluid with cooperative ring exchanges. In sufficiently thin samples, this "braided flux" phase transforms into a liquid of disentangled rods.

• Received 28 March 1988

DOI:https://doi.org/10.1103/PhysRevLett.60.1973