Ensembles of vortex lines have been found in rotating superfluid
He 4, high
temperature superconductors and more recently in Bose-Einstein
condensates
of trapped ions, where they appear as quantized line defects. It
is well
known in the new vortex matter community, that at lower
temperatures, the
vortex line ensemble form a crystalline solid and has an elastic
modulus
associated with its 2D cross-sectional lattice structure. At high
temperatures $T > > 1$, this vortex lattice melts and Quantum
Monte-Carlo simulations
of Andersen and Lim show that it then behave like a vortex liquid
of hard
rods (non-interacting except for the non-overlap
condition).
This new phase transition is characterized by the change from a
internal energy dominated
low temperature phase to a entropy dominated high temperature phase.
The excluded volume calculation for hard rods produced analytic
expressions which compare well
with the vortex line melting data.