Quantum Hall Fractions in Rotating Bose-Einstein Condensates

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 low-lying excited states of a small number of bosons as a function of the filling fraction $\nu$, the ratio of the number of bosons to the number of vortices. We show the occurrence of the Jain principal sequence of incompressible liquids for $\nu =2/3,3/4,4/3,5/4$ in addition to the Laughlin state $\nu =1/2$ as well as the Pfaffian state for $\nu =1$. We give gap estimates by finite-size scaling of both charged and neutral excitations.

Figure 1

Energy spectrum for (a) $\nu =1/2$ ($N=8,2S=14$), (b) $\nu =2/3$ ($N=8,2S=9$), (c) $\nu =3/4$ ($N=12,2S=12$), (d) $\nu =4/5$ ($N=8,2S=5$). Energies are in units of $g$ and the horizontal axis is total angular momentum.

Figure 2

Two-particle correlation function $g(r)$ as a function of great circle distance in units of $\ell$. The $\nu =1/2$ curve is plotted for sizes $N=8,9$ and for $2/3$ $N=8,10$.

Figure 3

Gaps for the Bose-Jain sequence $\nu =1/2$ and $2/3$. The lines are linear fits.

Figure 4

(a) Spectrum at the Pfaffian matching condition $N=12,2S=10$, with a collective mode above an isolated singlet ground state (b) with one extra flux quantum, two quasiparticles give rise to a degenerate set of states $L=0,2,4,6$.

Figure 5

Gaps for the Pfaffian at $\nu =1$. The dashed line is a linear fit to the charged excitations of the three largest systems, the solid (respectively, dotted) line is a linear (respectively, quadratic) fit of the neutral excitations.

Figure 6

The two-particle correlation function for the exact ground state at the Pfaffian point $2S=N-2$. It has an extra concentration of bosons at the origin.