Particle entanglement spectra for quantum Hall states on lattices

We use particle entanglement spectra to characterize bosonic quantum Hall states on lattices, motivated by recent studies of bosonic atoms on optical lattices. Unlike for the related problem of fractional Chern insulators, very good trial wave functions are known for fractional quantum Hall states on lattices. We focus on the entanglement spectra for the Laughlin state at $\nu =1/2$ for the non-Abelian Moore-Read state at $\nu =1$. We undertake a comparative study of these trial states to the corresponding ground states of repulsive two-body or three-body contact interactions on the lattice. The magnitude of the entanglement gap is studied as a function of the interaction strength on the lattice, giving insight into the nature of Landau-level mixing. In addition, we compare the performance of the entanglement gap and overlaps with trial wave functions as possible indicators for the topological order in the system. We discuss how the entanglement spectra allow to detect competing phases such as a Bose-Einstein condensate.

Figure 1

Particle entanglement spectra for $N=6$ bosons on the torus of unity aspect ratio at filling factor $\nu =N/N_{\phi }=1/2$, for a particle partition with $N_A=3$. Left: Bosons interact through hardcore interaction. Right: Bosons interact through hardcore interaction and an additional longer-range interaction.

Figure 2

Particle entanglement spectra for $N=6$ bosons on a lattice at filling factor $\nu =N/N_{\phi }=1/2$, for a particle partition with $N_A=3$ and for different lattice geometries. The spectra are calculated for the two-fold degenerate ground-state manifold of the Hamiltonian (1) with $U/t=1$ and $V=0$.

Figure 3

Particle entanglement spectra for $N=6$ bosons on a lattice of $6\times 8$ sites at filling factor $\nu =N/N_{\phi }=1/2$ for particle partitions with (a) $N_A=1$ and (b) $N_A=2$ [$N_A=3$ shown in Fig. 2d]. For $N_A=1$, the upper blue dashed line marks the entanglement gap below which the counting matches the lowest Landau level state counting of 12 states, and the lower black dashed line marks the entanglement gap that isolated the two low-lying eigenstates. For $N_A=2$, the black dashed line marks the entanglement gap that isolated the two low-lying eigenstates.

Figure 4

Energy gap $\Delta$ and entanglement gap ${\Delta }_{\xi }$ in the $N_A=2$ and in the $N_A=3$ sectors as a function of $U$ for $\nu =1/2$, $N=5$, $L_x=6$, and $L_y=10$. The vertical purple line is the band gap. Note the different offsets on the scales for $\Delta$ and ${\Delta }_{\xi }$.

Figure 5

Overlap $\mathcal{O}$ and entanglement gap ${\Delta }_{\xi }$ in the $N_A=3$ sectors as a function of $U$ for $\nu =1/2$, $N=5$, $L_x=6$, and $L_y=10$. The vertical purple line is the band gap. Note the different offsets on the scales for $\mathcal{O}$ and ${\Delta }_{\xi }$.

Figure 6

Entanglement gap ${\Delta }_{\xi }$ as a function of flux density $n_{\phi }$ for the Laughlin state with $N=5$ particles. The vertical dotted line indicates the critical value $n_{\phi }^c\simeq 0.4$ up to which the ground state exhibits the Chern number of the Laughlin state (Ref. 10).

Figure 7

Particle entanglement spectra for $N=6$ bosons on a lattice at filling factor $\nu =N/N_{\phi }=1$, for a particle partition with $N_A=3$ and for different lattice geometries. The spectra are calculated for the three-fold degenerate ground-state manifold of the Hamiltonian (1) with $U=0$ and $V/t=1$. Left: The counting of the states below the gap is $(7,6,6,7,6,6)$. It matches the one of MR quasiholes states on the torus as $K_y^{\max }=N_{\phi }$.

Figure 8

Energy gap $\Delta =E_3-E_2$ and ground-state energy splitting $\delta =E_2-E_0$ as a function of $U$ for $\nu =1$, $N=6$, $L_x=6$, and $L_y=6$, as well as their dimensionless ratio.

Figure 9

Overlap $\mathcal{O}$ and entanglement gap ${\Delta }_{\xi }$ in the $N_A=3$ sectors as a function of $U$ for $\nu =1$, $N=6$, $L_x=6$, and $L_y=6$. Note the different offsets on the axes for the overlap $\mathcal{O}$ and entanglement gap ${\Delta }_{\xi }$.