Fractional Quantum Hall Effect of Lattice Bosons Near Commensurate Flux

We study interacting bosons on a lattice in a magnetic field. When the number of flux quanta per plaquette is close to a rational fraction, the low-energy physics is mapped to a multispecies continuum model: bosons in the lowest Landau level where each boson is given an internal degree of freedom, or pseudospin. We find that the interaction potential between the bosons involves terms that do not conserve pseudospin, corresponding to umklapp processes, which in some cases can also be seen as BCS-type pairing terms. We argue that in experimentally realistic regimes for bosonic atoms in optical lattices with synthetic magnetic fields, these terms are crucial for determining the nature of allowed ground states. In particular, we show numerically that certain paired wave functions related to the Moore-Read Pfaffian state are stabilized by these terms, whereas certain other wave functions can be destabilized when umklapp processes become strong.

Figure 1

(a) The FQH gap $E$ at effective filling fraction $\tilde{\nu }=1$ as a function of $ϵ$, where the flux density is $n_{\varphi }=1/2+ϵ$. Data are shown for $N=8$, 10 bosons. Increasing $ϵ$ increases the strength of the pairing terms of the Hamiltonian and stabilizes this state. (b) The FQH gap for $\tilde{\nu }=2/3$ ($N=10$), $4/3$ ($N=12$), and 2 ($N=14$) as a function of $ϵ$. Increasing $ϵ$ decreases the FQH gap and destabilizes the corresponding states. (c) Overlap between the exact ground state of the system at effective filling fraction $\tilde{\nu }=1$ and the trial wave function Eq. (6) vs $ϵ$. The overlap for $ϵ>0.1$ exceeds 95%.