Quantum Hall states in rapidly rotating two-component Bose gases

We investigate strongly correlated phases of two-component (or pseudospin-$1/2$) Bose gases under rapid rotation through exact diagonalization on a torus geometry. In the case of pseudospin-independent contact interactions, we find the formation of gapped spin-singlet states at the filling factors $\nu =k/3+k/3$ ($k/3$ filling for each component) with integer $k$. We present numerical evidences that the gapped state with $k=2$ is well described as a non-Abelian spin-singlet (NASS) state, in which excitations feature non-Abelian statistics. Furthermore, we find the phase transition from the product of composite fermion states to the NASS state by changing the ratio of the intercomponent-to-intracomponent interactions.

Figure 1

“Charge” gap $\Delta \left(N\right)$ [Eq. (2)] vs the filling factor per component, $\nu /2=N/\left(2N_V\right)$, in the $\mathrm{SU}\left(2\right)$-symmetric case $g_{\uparrow \downarrow }=g$. The aspect ratio is set to $L_x/L_y=3/\left(2\sqrt{3}\right)$ and $3/\left(3\sqrt{3}\right)$ for $N_V=6$ and 9, respectively. These ratios are chosen to be compatible with the rectangular vortex lattice expected for large $\nu$ [9].

Figure 2

Energy spectra vs the pseudomomentum $\mathit{K}$ at the filling factor $\nu =2/3+2/3$ for the $\mathrm{SU}\left(2\right)$-symmetric case $g_{\uparrow \downarrow }=g$. The ground-state energy is subtracted from the spectrum. For $N_V=6$ and 9, the same aspect ratios as in Fig. 1 are used; for $N_V=12$, we set $L_x/L_y=4/\left(3\sqrt{3}\right)$. Upward ($▵$) and downward ($▿$) triangles, representing ${\psi }_1$ and ${\psi }_2$, respectively, indicate the two lowest-energy states in the sector with $\mathit{K}=0$, ${\rho }_{\pi }=+1$, and $p_{\uparrow \downarrow }=+1$. Diamonds, representing ${\psi }_3$, indicate the lowest-energy state in the sector with $\mathit{K}=0$, ${\rho }_{\pi }=-1$, and $p_{\uparrow \downarrow }=-1$. Circles indicate other eigenstates in the equal-population case $N_{\uparrow }=N_{\downarrow }=N/2$. Crosses show eigenstates for the minimally imbalanced case $N_{\uparrow }=N/2+1$ and $N_{\downarrow }=N/2-1$. Only the two lowest energies are displayed in each sector.

Figure 3

(a) and (b) Energy spectra vs the aspect ratio $L_x/L_y$, at the filling factor $\nu =2/3+2/3$ in the $\mathrm{SU}\left(2\right)$-symmetric case $g_{\uparrow \downarrow }=g$. The same symbols as in Fig. 2 are used. Solid curves are guides for the eyes for the states ${\psi }_{1,2}$. (c) and (d) Squared overlaps of ${\psi }_{1,2}$ with the $\mathrm{SU}{\left(3\right)}_2$ states, plotted against the aspect ratio.

Figure 4

(a) Energy spectrum vs coupling ratio $g_{\uparrow \downarrow }/g$ for $N_V=6$ and $N=8$ with $L_x/L_y=1$. The states ${\psi }_{1,2,3}$ and symbols are defined in a similar manner as in Fig. 2. The energy of ${\psi }_1$ is subtracted from the whole spectrum. (b) Squared overlaps of the lowest-energy states ${\psi }_{1,2,3}$ with the states at $g_{\uparrow \downarrow }=0$ and the $\mathrm{SU}{\left(3\right)}_2$ states, plotted against $g_{\uparrow \downarrow }/g$.