Characterizing fractional topological phases of lattice bosons near the first Mott lobe

The Bose-Hubbard model subjected to an effective magnetic field hosts a plethora of phases with different topological orders when tuning the chemical potential. Using the density matrix renormalization group method, we identify several gapped phases near the first Mott lobe at strong interactions. They are connected by a particle-hole symmetry to a variety of quantum Hall states stabilized at low fillings. We characterize phases of both particle and hole type and identify signatures compatible with Laughlin, Moore-Read, and bosonic integer quantum Hall states by calculating the quantized Hall conductance and by extracting the topological entanglement entropy. Furthermore, we analyze the entanglement spectrum of Laughlin states of bosonic particles and holes for a range of interaction strengths, as well as the entanglement spectrum of a Moore-Read state. These results further corroborate the existence of topological states at high fillings, close to the first Mott lobe, as hole analogs of the respective low-filling states.

Figure 1

Fractional quantum Hall phases of particles and holes. The average boson density $n_b$ of the ground state as a function of the chemical potential $\mu /U$ for strong interactions $U/t=50$ and flux $\varphi =\pi /3$, obtained on an infinite cylinder with circumference $L_y=6$ (inset, top left). We find gapped incompressible phases at various relevant filling factors. The plateaus labeled here correspond to the lowest states in the Jain sequence $\frac{p}{p+1}$ and the Read-Rezayi sequence $\frac{p}{2}$ with integers $p$. The pronounced symmetry around half filling suggests an approximate particle-hole symmetry emerging at strong interactions, with holes taking the place of particles in the wave function. We mark the hole filling factor with a star. Inset, bottom right: A sketch of the phase diagram indicating the $\mu$ cut of the main panel (red dashed line).

Figure 2

Selected incompressible phases. Phase diagram of the Bose-Hubbard model with a magnetic flux $\varphi =\pi /3$ through each plaquette on an infinite cylinder with circumference $L_y=6$, focused on the area between the vacuum and the first Mott lobe. Specific gapped states at hole filling factors ${\nu }^{(*)}=1/2,1,2$ are tracked as a function of the hopping strength $t/U$. We expect further gapped states (indicated by the hatched areas) corresponding, e.g., to the Jain and Read-Rezayi sequence states and their hole analogs. To obtain the phase diagram, we perform infinite-size DMRG calculations, which overestimate the size of the trivial Mott state as well as the nontrivial gapped states due to the finite size in the $y$ direction (see main text for details). The dotted line at $U/t=50$ and around $\mu /U=0$ indicates the cut considered in Fig. 1.

Figure 3

Charge pumping. (a) Charge pumping for the ${\nu }^{(*)}=1/2$ states for two periods. A single charge is transferred from one half to the other, as expected for the Laughlin state. The sign reversal between both states suggests an approximate particle-hole symmetry of the phases. The other states are analyzed over one pumping period: In (b), two charges are transferred per flux period for the ${\nu }^{(*)}=2$ states, again with different sign between the states. In (c), a single charge pumped is indicative of the Moore-Read state at ${\nu }^{(*)}=1$ at different interaction strengths. By contrast, the trivial Mott insulator in (d) does not allow any pumping, highlighting the difference between a trivial insulator and the more intricate topological states.

Figure 4

Finite-size flow of Rényi entropies. (a) and (b) Extrapolation of Rényi entropies for ${\nu }^{(*)}=1/2$ gives strong indications of Laughlin-type states at these filling factors. (c) and (d) States with ${\nu }^{(*)}=2$ are more difficult to interpret due to stronger deviations between the different entropy measures. However, the extrapolations of the low-order entropies are consistent with zero as expected for the bosonic integer quantum Hall state. The values for the Ising sector of the Moore-Read state in (e) suffer from the same problem; high-order entropies however are relatively close to the expected value. (f) Comparison for a Mott insulator, which yields zero topological entanglement entropy. Different flux densities $n_{\varphi }=1/6,1/7,1/8$ are analyzed, so the cylinder circumference $L_y$ is expressed in terms of the respective magnetic length ${\ell }_B$.

Figure 5

Entanglement spectra of the Laughlin states. The entanglement spectrum ${\varepsilon }_i$ as a function of the momentum in the $y$ direction $k_i$ for the ${\nu }^{(*)}=1/2$ Laughlin states at different interaction strengths $U/t$ on an $L_y=12$ cylinder. In the upper row, we show the spectrum for the ${\nu }^{*}=1/2$ hole state. While the hard-core limit clearly obtains the counting sequence $\{1,1,2,3,5,\cdots \}$ predicted by the conformal field theory of the Laughlin state (indicated by circles in the images), smaller values for $U/t$ lead to a background of low-lying Schmidt states that obscure this sequence. By contrast, the spectra for the particle state at $\nu =1/2$ are almost unaffected by weaker interactions due to the lower filling of bosons. Note that the entanglement spectra direction differs between particles and holes, which is consistent with the sign reversal of the Hall conductivity. For reasons of clarity, only the $Q_i^L=0$ charge sector is shown. Other charge sectors exhibit the same counting.

Figure 6

Entanglement spectra of the Moore-Read state. The entanglement spectrum ${\varepsilon }_i$ as a function of the momentum in the $y$ direction $k_i$ for the $\nu =1$ Moore-Read state at different flux densities $n_{\varphi }$ on an $L_y=10$ cylinder with interaction strength $U/t=2$, resolved by their charge value. In the upper row, we show the spectrum for the state at $n_{\varphi }=1/6$, as obtained by infinite DMRG. It clearly follows the counting $\{1,2,4,\cdots \}$ in every charge sector, which is characteristic of the Ising topological sector as given by the Moore-Read conformal field theory. In the lower row, we show the entanglement spectra obtained for $n_{\varphi }=1/7$. Here, the $\{1,2,4,\cdots \}$ counting is present only in the even charge sectors, while the odd sectors follow the sequence $\{1,1,3,\cdots \}$ in the low-energy space. This alternating behavior is characteristic of the other two sectors, the identity sector and the fermion sector.