Fractional quantum Hall states of dipolar fermions in a strained optical lattice

We study strongly correlated ground states of dipolar fermions in a honeycomb optical lattice with spatial variations in hopping amplitudes. Similar to strained graphene, such nonuniform hopping amplitudes produce valley-dependent pseudomagnetic fields for fermions near the two Dirac points, resulting in the formation of Landau levels. The dipole moments aligned perpendicular to the honeycomb plane yield a long-range repulsive interaction. By exact diagonalization in the zeroth-Landau-level basis, we show that this repulsive interaction stabilizes a variety of valley-polarized fractional quantum Hall states such as Laughlin and composite-fermion states. The present system thus offers an intriguing platform for emulating fractional quantum Hall physics in a static optical lattice. We calculate the energy gaps above these incompressible states and discuss the temperature scales required for their experimental realization.

Figure 1

(top) Dirac spectra near the valleys ${\mathit{K}}_{\pm }$. (bottom) Relativistic Landau levels (14) in valley-dependent pseudomagnetic fields $\pm B$. Each level at each valley is $N_{\varphi }$-fold degenerate. We consider the case when the Fermi level lies near the ZLL $n=0$ and this level is partially populated. The number of fermions in the ZLL at the valley ${\mathit{K}}_{\pm }$ is denoted by $N_{\pm }$.

Figure 2

Pseudopotentials (34) for a dipole-dipole interaction $V\left(r\right)=C/r^3$ on a sphere with $2\tilde{S}=2S-1=12$. Here, the intravalley pseudopotentials (red circles) are calculated from Eq. (37), and the intervalley ones (blue stars; scaled by $1/10$) are obtained by inserting Eqs. (38) and (39) into Eq. (34). For the intravalley interactions, only the channels with odd (even) $I$ are allowed for even (odd) $2\tilde{S}$ because of the Fermi statistics. The intravalley pseudopotential with $I=2\tilde{S}=12$ is not shown because it diverges, but in any case it does not correspond to an allowed channel for fermions.

Figure 3

Dependence of the ground-state energy on the population imbalance $N_{+}-N_{-}$ between the two valleys for a dipole-dipole interaction on a sphere with $2\tilde{S}=2S-1=12$. We have used the pseudopotentials shown in Fig. 2. For each value of $N=N_{+}+N_{-}$, we determine the ground-state energy $E(N_{+},N_{-})$ for each distribution $(N_{+},N_{-})$ of particles at the two valleys, and plot $\delta E=E(N_{+},N_{-})-E(N,0)$ in units of $C/l_B^3$.

Figure 4

Candidates of incompressible ground states in the $(2\tilde{S},N)$ plane for a dipole-dipole interaction. The filled circles indicate ground states with total angular momentum $J=0$, where incompressible states can appear. The area of each filled circle is proportional to the neutral gap. Lines indicate the relation (40) for candidate FQH states with $0<\tilde{\nu }<1/2$ listed in Table 1. The empty circles indicates ground states with $J>0$, which are in general compressible. For each $2\tilde{S}$, the result is symmetric around $N=\tilde{S}+1/2$ because the particle-hole symmetry relates the particle numbers $N$ and $2\tilde{S}+1-N$.

Figure 5

Energy gaps ${\mathrm{\Delta }}_{\tilde{\nu }}^{\left(N\right)}$ above the ground state as a function of $1/N$ for $\tilde{\nu }=1/3$ (blue circles and a solid line) and $\tilde{\nu }=2/5$ (red triangles and a dashed line). Symbols and lines indicate the ED data and their fits given by Eq. (42), respectively.

Figure 6

Pair distribution function $G\left(r\right)$ for the ground state at $\tilde{\nu }=1/3$ for different number $N$ of particles.