We study the Quantum Hall phases that appear in the dilute limit of rotating Bose-Einstein condensates. By exact diagonalization in a spherical geometry we obtain the ground-state and lowlying excited states of a small number of bosons as a function of the filling fraction ν, ratio of the number of bosons to the number of vortices. We show the occurrence of the Jain principal sequence of incompressible liquids for ν = 1/2, 2/3, 3/4, 4/3, 5/4 as well as the Pfaffian state for ν = 1. The collective excitations are well described by a composite-fermion scheme.PACS numbers: 03.75Kk, 05.30.Jp, 73.43.Cd, 73.43.Lp Bose-Einstein condensates in dilute atomic gases offer a unique opportunity to investigate the physics of vortex matter when they undergo rotation [1,2]. Indeed, recent experiments [3,4] have observed the appearance of large vortex arrays at sufficient high angular velocity ω. In addition to this phase akin to the Abrikosov lattice of type-II superconductors, there is the possibility that at larger ω the lattice melts [5] and is replaced by a quantum Hall liquid. Consider a trap with strong confinement in the z direction such that the system is effectively twodimensional (2D). Then if the rotation frequency is tuned to the characteristic frequency of the harmonic confining potential in the xy plane, the bosons feel only the Coriolis force and the system is equivalent to 2D charged bosons in a magnetic field, i.e. the conditions of the quantum Hall effect. In this regime, it has been pointed out [6,7] that the celebrated Laughlin wavefunction is the exact ground state for the filling fraction ν = 1/2 where ν is the ratio of the number of bosons to the number of vortices. Some of the excitations above this ground state are quasiparticles with fractional statistics which may eventually be probed by laser manipulations [8]. Investigations by exact diagonalization have given evidence [9] for even more exotic states of matter [10,11], some involving parafermionic wavefunctions introduced in the context of the fractional quantum Hall effect for fermions [12].In this Letter we investigate the quantum Hall states of bosons as a function of the filling ν by use of exact diagonalizations in the spherical geometry [13,14]. This allows to separate bulk from edge excitations. We show the appearance of the Bose analog of the Jain principal sequence of fractions, ν = n n+1 , p p−1 . The excited states show collective modes well described by a composite fermion picture in which there is binding of one vortex per boson. We obtain evidence for the Pfaffian state [10,11] at ν = 1 by displaying its peculiar half-vortex excitations. For higher fillings, ν ≥ 3/2, we observe some states with properties of the Read-Rezayi (RR) parafermionic states. However they show no clear tendency to convergence to the thermodynamic limit.In the rotating frame [15], the Hamiltonian describing N bosons of mass m is given by :where the xy trap frequency is ω 0 , the axial frequency is ω z and the angular velocity vector is ωẑ. In the ultracold at...

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Most of the present understanding of the Sϭ1 quantum spin chains displaying the Haldane gap is coming from the so-called valence-bond-solid ͑VBS͒ Hamiltonian which has an exactly known ground state. We show that this point is characterized by the onset of short-range incommensurate spin correlations in the oneparameter family of Hamiltonians H ϭcos͚ i S i •S iϩ1 ϩsin͚ i (S i •S iϩ1 ) 2 . This gives a physical meaning to this special point. We establish precise values for the gaps, correlations, the string order parameter, and identify the VBS point as a disorder point in the sense of classical statistical mechanics. It is a quantum remnant of the classical transition between a ground state with long-range Néel order and a ground state with incommensurate long-range order.

We find the finite width, i.e., the layer thickness, of experimental quasi-two-dimensional systems produces a physical environment sufficient to stabilize the Moore-Read Pfaffian state thought to describe the fractional quantum Hall effect at filling factor nu=5/2. This conclusion is based on exact calculations performed in the spherical and torus geometries, studying wave function overlap and ground state degeneracy.

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