Quantum Hall effect of two-component bosons at fractional and integral fillings

We investigate the feasibility of many candidate quantum Hall states for two-component bosons in the lowest Landau level. We identify interactions for which spin-singlet incompressible states occur at filling factors $\nu =2/3$, $4/5$, and $4/3$, and partially spin-polarized states at filling factors $3/4$ and $3/2$, where “spin” serves as a generic label for the two components. We study ground states, excitations, edge states, and entanglement spectrum for systems with up to 16 bosons and construct explicit trial wave functions to clarify the underlying physics. The composite fermion theory very accurately describes the ground states as well as excitations at $\nu =2/3$, $4/5$, and $3/4$, although it is less satisfactory for the $\nu =3/2$ state. For $\nu =4/3$ a “non-Abelian spin-singlet” state, which is the exact ground state of a three-body contact interaction, has been proposed to occur even for a two-body contact interaction; our trial wave functions are very accurate for the excitations of the three-body interaction, but they do not describe the excitations of the two-body interaction very well. Instead, we find that the $\nu =4/3$ state is more likely to be a spin-singlet state of reverse-flux-attached composite fermions at filling ${\nu }^{*}=4$. We also consider incompressible states at integral filling factors $\nu =1$ and 2. The incompressible state at $\nu =1$ is shown to be well described by the parton-based Jain spin-singlet wave function, and the incompressible state at $\nu =2$ as the spin-singlet state of reverse-flux-attached composite fermions at ${\nu }^{*}=2$, which provides an example of the bosonic integer topological phase.

Figure 1

Energy spectra (lines) of the $\nu =2/3$ state for the two-body contact Hamiltonian $H_2^{\mathrm{con}}$. The lines are colored according to their spin quantum numbers and are also shifted in the horizontal direction for clarity. The same conventions are used in all other figures. The crosses represent the energies of the wave functions ${\Psi }_{2/3}^{[1,1]}$ for the ground and excited states. The panels correspond to (a) $N_{\uparrow }=4$, $N_{\downarrow }=4$, and $2Q=10$; (b) $N_{\uparrow }=5$, $N_{\downarrow }=5$, and $2Q=13$; and (c) $N_{\uparrow }=5$, $N_{\downarrow }=5$, and $2Q=12$. The inset in panel (a) shows the color scheme for all panels. Panels (a) and (b) correspond to incompressible states where the uniform ground state has $L=0$ and $S=0$, and the excitations are neutral particle-hole pairs of composite fermions. Panel (c) corresponds to a system containing two quasiparticles; the low-energy band contains all possible states of these quasiparticles.

Figure 2

Energy spectra of the $\nu =4/5$ state for the Hamiltonian $H_2^{\mathrm{con}}$. The crosses represent the energies of the wave functions ${\Psi }_{4/5}^{[2,2]}$. (a) $N_{\uparrow }=4$, $N_{\downarrow }=4$, and $2Q=7$; (b) $N_{\uparrow }=6$, $N_{\downarrow }=6$, and $2Q=12$; (c) $N_{\uparrow }=5$, $N_{\downarrow }=5$, and $2Q=10$. The inset in panel (a) shows the color scheme for all panels. Panels (a) and (b) correspond to incompressible states where the uniform ground state has $L=0$ and $S=0$, and the excitations are neutral particle-hole pairs of composite fermions. Panel (c) corresponds to a system containing two quasiholes; the low-energy band contains all possible states of these quasiholes.

Figure 3

(a) Energy spectrum of the $\nu =3/4$ state for the Hamiltonian $H_2^{\mathrm{con}}$ with $N_{\uparrow }=3$, $N_{\downarrow }=8$, and $2Q=12$. The crosses represent the energies of the wave functions ${\Psi }_{3/4}^{[1,2]}$. (b) Energy spectrum of the $\nu =3/2$ state for the Hamiltonian $H_2^{\mathrm{con}}$ with $N_{\uparrow }=4$, $N_{\downarrow }=10$, and $2Q=10$. The crosses represent the energies of the states ${\Psi }_{3/2}^{[-1,-2]}$. The insets show the color schemes for the panels.

Figure 4

Energy spectra of the $\nu =4/3$ state for the Hamiltonian $H_2^{\mathrm{con}}$. The crosses represent the energies of the wave functions ${\Psi }_{4/3}^{[-2,-2]}$. (a) $N_{\uparrow }=6$, $N_{\downarrow }=6$, and $2Q=10$; (b) $N_{\uparrow }=7$, $N_{\downarrow }=7$, and $2Q=8$; (c) $N_{\uparrow }=7$, $N_{\downarrow }=7$, and $2Q=11$. The inset of panel (a) shows the color scheme for all panels.

Figure 5

Energy spectra of the $\nu =4/3$ state for the Hamiltonian $H_2^{\mathrm{con}}$. The crosses represent the energies of the wave functions ${\Psi }_{4/3}^{\mathrm{NASS}}$ obtained from the spinful bipartite CF theory. (a) $N_{\uparrow }=6$, $N_{\downarrow }=6$, and $2Q=7$; (b) $N_{\uparrow }=7$, $N_{\downarrow }=7$, and $2Q=8$; (c) $N_{\uparrow }=8$, $N_{\downarrow }=8$, and $2Q=10$. The inset of panel (a) shows the color scheme for all panels.

Figure 6

Energy spectra of the $\nu =4/3$ state for the three-body Hamiltonian $H_3$. The crosses represent the energies of the wave functions ${\Psi }_{4/3}^{\mathrm{NASS}}$ obtained from the spinful bipartite CF theory. (a) $N_{\uparrow }=6$, $N_{\downarrow }=6$, and $2Q=7$; (b) $N_{\uparrow }=7$, $N_{\downarrow }=7$, and $2Q=8$. The inset of panel (a) shows the color scheme for both panels.

Figure 7

RSES and edge excitations of the $\nu =4/3$ NASS state. (a) $N_{\uparrow }=8$, $N_{\downarrow }=8$, $N_{\uparrow }^A=4$, and $N_{\downarrow }^A=4$, using the exact NASS state; (b) $N_{\uparrow }=8$, $N_{\downarrow }=8$, $N_{\uparrow }^A=4$, and $N_{\downarrow }^A=4$, using the ground state of the two-body Hamiltonian $H_2$; (c) $N_{\uparrow }=4$ and $N_{\downarrow }=4$, energy spectrum on disk geometry of the Hamiltonian ${\tilde{H}}_2$ with confinement potential parameter ${\omega }_c=0.4$. The inset of panel (a) shows the color scheme for all panels. The arrow in panel (c) indicates the ground state and the arrows in panel (a) and panel (b) show the corresponding levels in the RSES.

Figure 8

Energy spectra of the $\nu =2$ ground states for the two-body Hamiltonian $H_2^{\mathrm{con}}$. The cross represents the energy of the wave functions ${\Psi }_2^{[-1,-1]}$. (a) $N_{\uparrow }=6$, $N_{\downarrow }=6$, and $2Q=6$; (b) $N_{\uparrow }=7$, $N_{\downarrow }=7$, and $2Q=7$; (c) $N_{\uparrow }=8$, $N_{\downarrow }=8$, and $2Q=8$. The inset of panel (a) shows the color scheme for all panels.

Figure 9

RSES and edge excitations of the $\nu =2$ state. (a) RSES for the exact ground state of the two-body Hamiltonian $H_2^{\mathrm{con}}$ for $N_{\uparrow }=8$, $N_{\downarrow }=8$, $N_{\uparrow }^A=4$, and $N_{\downarrow }^A=4$. (b) Energy spectrum on disk geometry of the Hamiltonian ${\tilde{H}}_2$ for $N_{\uparrow }=4$ and $N_{\downarrow }=4$; the confinement potential parameter is taken to be ${\omega }_c=0.4$. The inset of panel (a) shows the color scheme for both panels. The arrows in panel (b) indicate the ground state and backward-moving edge modes and the arrows in panel (a) show the corresponding levels in the RSES.

Figure 10

Energy spectra of the $\nu =1$ ground states for the two-body Hamiltonian $H_2$ with $c_0=1$, $c_2=0.3$ and all other $c_{\alpha }=0$ for $\alpha \ne 0,2$. The crosses represent the energies of the wave functions ${\Psi }_1^{\mathrm{JSS}}$. (a) $N_{\uparrow }=6$, $N_{\downarrow }=6$, and $2Q=9$; (b) $N_{\uparrow }=7$, $N_{\downarrow }=7$, and $2Q=11$. The inset of panel (a) shows the color scheme for both panels.

Figure 11

RSES and edge excitations of the $\nu =1$ state. (a) RSES for ${\Psi }_1^{\mathrm{JSS}}$ for $N_{\uparrow }=7$, $N_{\downarrow }=7$, $N_{\uparrow }^A=3$, and $N_{\downarrow }^A=3$. (b) RSES for the ground state of the two-body Hamiltonian $H_2$ for $N_{\uparrow }=7$, $N_{\downarrow }=7$, $N_{\uparrow }^A=3$, and $N_{\downarrow }^A=3$; the parameters of the Hamiltonian are $c_0=1$, $c_2=0.3$ and all other $c_{\alpha }=0$ for $\alpha \ne 0,2$. (c) Energy spectrum on disk for $N_{\uparrow }=3$ and $N_{\downarrow }=3$ for the Hamiltonian ${\tilde{H}}_2$ with ${\tilde{c}}_2=0.3$ and the confinement potential parameter ${\omega }_c=0.4$. The inset of panel (a) shows the color scheme for all panels. The arrows in (c) indicate the states obtained with four different choices for ${\Phi }_2$ in the wave function Eq. (10), which are $[3,3]$, $[4,2]$, $[5,1]$, and $[6,0]$ (from left to right). The arrows in (a) and (b) show the corresponding levels in the RSES, which nicely match the starting points of various edge branches.