Quantum Hall fractions for spinless bosons

We study the quantum Hall phases that appear in the fast-rotation limit for Bose-Einstein condensates of spinless bosonic atoms. We use exact diagonalization in a spherical geometry to obtain low-lying states of a small number of bosons as a function of the angular momentum. This allows us to understand or guess the physics at a given filling fraction nu, the ratio of the number of bosons to the number of vortices. This is also the filling factor of the lowest Landau level. In addition to the well-known Bose-Laughlin state at nu=1/2, we give evidence for the Jain principal sequence of incompressible states at nu=p/(p+/-1) for a few values of p. There is a collective mode in these states whose phenomenology is in agreement with standard arguments coming, e.g., from the composite-fermion picture. At filling factor 1, the potential Fermi sea of composite fermions is replaced by a paired state, the Moore-Read state. This is most clearly seen from the half-flux nature of elementary excitations. We find that the hierarchy picture does not extend up to the point of transition towards a vortex lattice. While we cannot come to any definite conclusions, we investigate the clustered Read-Rezayi states and show evidence for incompressible states at the expected ratio of flux versus number of Bose particles.