Competing compressible and incompressible phases in rotating atomic Bose gases at filling factor ν=2

We study the groundstates of weakly interacting atomic Bose gases under conditions of rapid rotation. We present the results of large-scale exact diagonalization studies on a periodic geometry (a torus) which allows studies of compressible states with broken translational symmetry. Focusing on filling factor ν=2, we show a competition between the triangular vortex lattice, a quantum smectic state, and the incompressible k=4 Read-Rezayi state. We discuss the corrections arising from finite size effects, and the likely behavior for large system sizes. The Read-Rezayi state is stabilized by a moderate amount of additional dipolar interactions.