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 $\nu =p∕(p\pm 1)$ 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.

Figure 1

Energy spectrum for $N=5$ bosons. Energies are in units of $g$ and the horizontal axis is the total angular momentum. The Laughlin state appears at zero energy for $L=20$.

Figure 2

Candidates for incompressible states in the plane (number of particles, flux). Dots indicate a $L=0$ ground state. The lines show the flux number of the particle relationships of some FQHE states.

Figure 3

Energy spectrum for $N=10$ bosons at $2S=18$ corresponding to the filling $\nu =1∕2$. The Laughlin state is the $L=0$ ground state at zero energy. There is a clear collective mode branch that extends up to $L=N=10$. The upper part of the spectrum has been truncated for clarity but extends continuously. Energies are in units of $g$ and the horizontal axis is total angular momentum.

Figure 4

Energy spectrum for $N=12$ bosons at $2S=15$ corresponding to the filling $\nu =2∕3$. Above the $L=0$ ground state there is a collective mode branch that extends up to $L=7$.

Figure 5

Incompressible states related to the Pfaffian when the number of particles is even: (a) $N=8$, (b) $N=10$, (c) $N=12$.

Figure 6

States with one pair-breaking excitation when the number of particles is odd: (a) $N=7$, (b) $N=9$, (c) $N=11$.

Figure 7

The gaps of the candidate states for a hierarchical $\nu =2$ fraction.

Figure 8

Gaps for the possible Read-Rezayi states on the sphere: in (a) the fraction $\nu =3∕2$ is displayed and (b) shows the fraction $\nu =2$.