Fractional quantization by interaction of arbitrary strength in gapless flat bands with divergent quantum geometry

TL;DR

This work asks whether fractional quantum anomalous Hall states can emerge in gapless flat bands with divergent quantum geometry and non-integer Berry flux, beyond the ideal flat Chern-band paradigm. It answers affirmatively by constructing two gapless flat-band models (honeycomb and kagome) with tunable singularities and using ED and DMRG to demonstrate FQAH phases at filling $\nu=1/3$ that persist across interaction strengths $0<U\le\infty$. The many-body topological order adapts to the fluctuating quantum geometry, spontaneously producing inhomogeneous carrier distributions and a occupation-weighted Berry flux that tracks the phase stability, even when band topology is ill-defined. These results expand the landscape for FQAH and flat-band physics, showing that robust fractional topological order can survive in highly non-ideal, singular quantum-geometric environments.

Abstract

Fractional quantum anomalous Hall (FQAH) effect, a lattice analogue of fractional quantum Hall effect, offers a unique pathway toward fault-tolerant quantum computation and deep insights into the interplay of topology and strong correlations. The exploration has been successfully guided by the paradigm of ideal flat Chern bands, which mimic Landau levels in both band topology and local quantum geometry. Yet, given the near-infinite possibilities for Bloch bands in lattices, it remains a major open question whether FQAH states can emerge in scenarios fundamentally different from this paradigm. Here we turn to a class of gapless flat bands, featuring divergent quantum geometry at singular band touching, non-integer Berry flux threading the Brillouin zone (BZ), and ill-defined band topology. Our exact diagonalization and density matrix renormalization group calculations unambiguously demonstrate FQAH phase that is virtually independent of the interaction strength, persisting from the weak-interaction to the strong-interaction limit. We find the stability of the FQAH states does not uniquely correlate with the singularity strength or the BZ-averaged quantum geometric fluctuations. Instead, the many-body topological order can adapt to the singular and fluctuating quantum geometric landscape by spontaneously developing an inhomogeneous carrier distribution, while its quenching accompanies the drop in the occupation-weighted Berry flux. Our work reveals a profound interplay between quantum geometry and many-body correlation, and significantly expands the design space for exploring FQAH effect and flat-band correlation phenomena in general.

Figures (9)

• Figure 1: Single-particle results of the honeycomb model. (a) Schematics of the honeycomb model $\hat{H}_{\vcenter{\hbox{$\varhexagon$}}}(\boldsymbol{k})$, where $\xi=2\cos\theta_+$, $\eta=2\cos\theta_-$, $\phi=e^{i\theta_+}$ and $\varphi=e^{i\theta_-}$. (b) Band structure of $\hat{H}_{\vcenter{\hbox{$\varhexagon$}}}(\boldsymbol{k})$ for various $\delta$. (c) Evolution of $d_{\rm max}$ and Berry phase $\Phi_{\vcenter{\hbox{$\circlearrowright$}}}$ around the touching point with $\delta$. (d--f) Distribution of $\Omega(\boldsymbol{k})$, log$_{10}\text{tr}\,\mathcal{G}(\boldsymbol{k})$ and log$_{10}T(\boldsymbol{k})$ of the SFB in the $\boldsymbol{k}$ space for a few $\delta$. (g) Quantum geometry fluctuation as $\delta$ is varied. Orange circles in the middle panel denote standard deviation of particle occupation $n(\boldsymbol{k})$ of the many-body states with FQAH momenta. The shaded areas within $0\le\delta\lesssim0.43$ in panels (c) and (g) host FQAH effects.
• Figure 2: DMRG results of the honeycomb model. (a) Charge pumping under flux insertion in the FQAH phase with $\delta=0.2$ and various $U$ from weak- to strong-interaction limit. (b) Momentum-resolved entanglement spectrum $\epsilon$ of the charge sector $Q_L=0$ at $U= 10$ and $\delta=0.2$. (c) Variation of entanglement entropy $S$, first derivative of the ground state energy $\partial E/\partial \delta$, charge distribution at the BZ corner $\rho(\boldsymbol{K})$ with $\delta$. (d) Phase diagram in the $\delta$--$U$ parameter space and representative charge patterns in the FQAH and CDW phases.
• Figure 3: ED results of the honeycomb model. (a) The many-body energy spectrum with $n_{\rm up}=2$ at $U=1$ and $\delta =0.2$ in a rectangular system. (b) Orange curve denotes variation of the many-body gap $\Delta_{\rm mb}$ with $\delta$ evaluated at FQAH momenta. The blue curves with different symbols represent occupation-weighted $\braket{\text{tr}\,\mathcal{G}}_{\rm occ}$, $\braket{\Omega}_{\rm occ}$ and $\braket{T}_{\rm occ}$ averaged over the states with FQAH momenta. $\braket{\text{tr}\,\mathcal{G}}_{\rm occ}$ is shifted for clarity. (c--e) Particle occupation at $\delta=0.1$ (top row) and $0.3$ (bottom row) represented by dots, whose color and size denote the occupation averaged over the three states with FQAH momenta in a tilted system. The continuous background color display log$_{10}$tr$\,\mathcal{G}(\boldsymbol{k})$, $\Omega(\boldsymbol{k})$ and log$_{10}$tr$\,T(\boldsymbol{k})$, respectively.
• Figure 4: Single-particle results of the kagome model: (a) Schematic illustration of the kagome lattice and the hopping processes in $\hat{H}_{\vcenter{\hbox{$\davidsstar$}}}(\boldsymbol{k})$. Black/pink arrows denote NN hopping with complex amplitude $1\pm i\alpha$, blue arrows represent next-NN hopping with complex amplitude $i\alpha-\alpha^2$. (b) Band structures of $\hat{H}_{\vcenter{\hbox{$\davidsstar$}}}(\boldsymbol{k})$ for various $\alpha$. (c) Plots of $d_{\rm max}$ (blue) and Berry phase $\Phi_{\vcenter{\hbox{$\circlearrowright$}}}$ (yellow) around the touching point as functions of $\alpha$. (d) Quantum geometry fluctuations as function of $\alpha$. Orange circles in the middle panel denote standard deviation of particle occupation $n(\boldsymbol{k})$ of the many-body states with FQAH momenta. The shaded areas within $0.35\lesssim\alpha\lesssim2.46$ in panels (c) and (d) host FQAH effects.
• Figure 5: Many-body results of the kagome model: (a) Charge pumping simulation results obtained via DMRG at $\alpha=0.8$ and $U =1$. Inset: phase diagram obtained by DMRG. (b) Orange curve denotes variation of the many-body gap $\Delta_{\rm mb}$ with $\alpha$ evaluated at FQAH momenta in a rectangular system. The blue curves with different symbols represent occupation-weighted $\braket{\text{tr}\,\mathcal{G}}_{\rm occ}$, $\braket{\Omega}_{\rm occ}$ and $\braket{T}_{\rm occ}$ averaged over the states with FQAH momenta. $\braket{\Omega}_{\rm occ}$ is shifted for clarity. (c--e) Particle occupation at $\alpha=0.2$, 1 and 2 represented by dots, whose color and size denote the occupation averaged over the three states with FQAH momenta in a tilted system. The continuous background color display log$_{10}$tr$\,\mathcal{G}(\boldsymbol{k})$, $\Omega(\boldsymbol{k})$ and log$_{10}$tr$\,T(\boldsymbol{k})$, respectively.