Quantum Phases of Vortices in Rotating Bose-Einstein Condensates

We investigate the ground states of weakly interacting bosons in a rotating trap as a function of the number of bosons, $N$, and the average number of vortices, $N_V$. We identify the filling fraction $\nu \equiv {N/N}_V$ as the parameter controlling the nature of these states. We present results indicating that, as a function of $\nu$, there is a zero temperature phase transition between a triangular vortex lattice phase, and strongly correlated vortex liquid phases. The vortex liquid phases appear to be the Read-Rezayi parafermion states.