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

TL;DR

This work demonstrates that fractional Chern insulator states can emerge from an isolated flat band with zero Chern number by leveraging the intrinsic quantum geometry, specifically the interplay between the quantum metric and Berry curvature. Using large-scale exact diagonalization and two-band iDMRG on an anisotropic fluxed dice lattice (reducible to a honeycomb-like model), the authors show a robust $1/3$ Laughlin-type FCI at filling $oldsymbol{ u_ ext{F}=2/3}$ with Hall conductance $oldsymbol{oldsymbol{ extstyle oldsymbol{oldsymbol{-1/3}}} e^2/h$ and a threefold ground-state degeneracy, despite a trivial single-particle topology. The FCI arises because holes preferentially occupy regions with nearly ideal quantum geometry away from the sharp peak at the $oldsymbol{oldsymbol{ m abla}}$-point, an effect that is mirrored by a particle-hole transformation that cancels Hartree-Fock dispersion and yields an effectively flat band for holes. As $oldsymbol{ u_ ext{F}}$ is tuned or the anisotropy $oldsymbol{ ilde{ u}}$ is varied, the system transitions to a charge-density-wave state, with iDMRG confirming fractional charge pumping and the characteristic edge-state spectrum of the $oldsymbol{1/3}$ Laughlin state. These results broaden the landscape of FCIs beyond nonzero Chern bands and suggest a general route to stabilize topological order via quantum geometry in isolated trivial flat bands.

Abstract

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.

Figures (20)

• Figure 1: Lattice models and band structures. (a) Schematic illustration of the dice lattice with anisotropic hopping, with hopping along $x$ direction set to $\eta t$, otherwise $t$. (b) The effective two-orbital model where the high-energy A orbitals are perturbatively eliminated. (c) Band structures at $\eta=0.4$ as an example, with both bands having $C=0$. (d) Zero modes that span the isolated flat band. (e) Schematic diagram of hoppings among nearest, next nearest, and next-next nearest neighbor sites of the effective two-orbital model in (b). $t^{\prime}=-t^2/E_A$. The isolated band limit will be considered, where interactions - being much smaller than the gap - leave the quantum geometry and trivial topology of the flat band unchanged.
• Figure 2: Hartree-Fock results at $\nu=1$ (filled lower dispersive band and empty flat band). (a) Hartree-Fock band gap $\Delta_{\rm HF}$ as a function of interaction strength $U$ and anisotropy parameter $\eta$. (b) Band width $w$. (c) Minimum Bloch function fidelity $F_{\rm min} = \min_{\rm BZ} F(\bm k)$, where $F(\bm k) \equiv | \langle \psi_{\rm HF} (\bm k) | \psi_{\rm flat} (\bm k) \rangle |$, with $\psi_{\rm HF}$ the Hartree-Fock wavefunction of the upper flat band and $\psi_{\rm flat}$ the Bloch function in the non-interacting limit as given in Eq. (\ref{['Eq_wave']}). (d)-(e) Dependence of the Hartree-Fock band gap and minimum fidelity on $\eta$ for $U=0.05$ (d), $U=0.01$ (e), and $U=0.001$ (f). The gray shaded area denotes the isolated band limit with $F_{\rm min} \geq 0.99$.
• Figure 3: Propoerties of isolated $C=0$ flat band at $\eta=0.7$. (a) Berry curvature $\Omega_{\mathbf{k}}$, (b) Trace of quantum metric tensor $\mathcal{G}$, (c) ${\rm Tr} \mathcal{G}+\Omega_{\mathbf{k}}$, (d) ${\rm Tr} \mathcal{G}-|\Omega_{\mathbf{k}}|$, and (e) Hartree Fock renormalized dispersion relation $\epsilon_\mathbf{k}$, under an interaction strength $U=0.001$. Such a small $U$, to be adopted in ED calculations, ensures the isolated band limit where $\mathcal{G}$ and $\Omega_{\mathbf{k}}$ remain the same as in non-interacting case. ${\rm Tr} \mathcal{G}$ and $\Omega_{\mathbf{k}}$ are in unit of $a^2$.
• Figure 4: ED results at $\nu_\mathrm{F}=2/3$ with $\eta=0.7$ and $U=0.001$. (a) Illustration of the $4\times6$$k$-mesh, with all momenta indexed. (b) ED spectrum calculated at $\theta_1=0$ and $\theta_2=0$ for flat band filling $\nu_{\rm F}=2/3$, exhibiting three quasi-degenerate ground states at K=0, 8 and 16, with a total Chern number of $-1$. (c) Spectral flow as a function of phase $\theta_1$, with $\theta_2=0$. (d) Spectral flow as a function of phase $\theta_2$, with $\theta_1=0$. In the spectral flow figures here and hereafter, we have subtracted a background energy from the spectrum at every given phase $\theta$, for a better visualization of the spectral gap as a function of $\theta$ (c.f. \ref{['APP_Background']}). (e) Distribution of many-body Berry curvature $F(\theta_1,\theta_2)$ (c.f. text). (f)-(j) Similar plots for a $6\times6$$k$-mesh.
• Figure 5: Momentum-space carrier distributions of the FCI. These results are obtained from ED simulations at $\nu_\mathrm{F}=2/3$ with $\eta=0.7$ and $U=0.001$. (a) Schematic of the $4\times6$$k$-mesh, with BZ color coded with the value of ${\rm Tr} \mathcal{G}-|\Omega_{\mathbf{k}}|$. (b) Momentum-space electron distribution, $n(\mathbf{k})$, obtained as an average over the three ground states. (c) The corresponding hole density $1-n(\mathbf{k})$, which exhibits nearly zero weight at the $\Gamma$ point of the BZ. The background of the BZ in panels (b) and (c) is the Berry curvature of the flat band. (d)-(f) Similar plots but for the $6\times6$$k$-mesh.