Fractional Chern insulator states in an isolated flat band of zero Chern number

A flat band with Chern number $C=0$, and well isolated from the rest of Hilbert space by a gap much larger than interaction strength, is a context that has not been regarded as relevant for fractional quantum Hall physics. In this work, we demonstrate the emergence of the fractional Chern insulator (FCI) states in such a trivial flat band, using large-scale exact diagonalization (ED) and infinite density matrix renormalization group (iDMRG) simulations. The $C=0$ isolated flat band is hosted by an anisotropic fluxed dice lattice. Both the quantum metric and Berry curvature of the $C=0$ flat band have a sharp peak at the $Γ$ point, whereas in the rest of the Brillouin zone (BZ) they mimic the quantum geometry of the lowest Landau level. We consider nearest-neighbor repulsion that is weak enough to ensure the isolated-band limit is always satisfied. From the projected ED simulations at $ν_\mathrm{F}=2/3$ electron filling of the flat band (i.e. $1/3$ hole filling), we find the unexpected FCI with 3-fold ground-state degeneracy and $σ_\mathrm{H}=-1/3 (e^2/h)$. The momentum space carrier distribution shows that the quantum metric peak tends to push the interacting holes away from $Γ$ point towards the BZ regions with the nearly ``ideal'' quantum geometry, underlying the formation of FCI in the $C=0$ flat band. Besides, when tuning the single-particle anisotropy such that the quantum geometry of the $C=0$ flat band becomes less sharp around $Γ$, we find the ground state becomes a charge density wave with tripled unit cell at $ν_\mathrm{F}=2/3$. Our two-band iDMRG simulations further corroborate the FCI in the isolated $C=0$ flat band, demonstrating in such parameter regime the fractionally quantized charge pumping upon flux insertion as well as the momentum-resolved entanglement spectrum characteristic of the $1/3$ Laughlin state.