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

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.

💡 Research Summary

This paper investigates the emergence of fractional quantum anomalous Hall (FQAH) states in a class of gapless flat bands that lack both a well‑defined Chern number and the ideal quantum‑geometric properties traditionally thought necessary for stabilizing fractional Chern insulators. The authors focus on “singular flat bands” (SFBs) in which a quadratic band touching at the Brillouin‑zone center produces a discontinuous Bloch wavefunction, divergent quantum‑metric tensor G(k), and a non‑integer Berry flux Φ↻ that depends on a tunable singularity parameter (δ for a honeycomb model, α for a kagome model). The singularity strength is quantified by the maximal Hilbert‑Schmidt distance d_max, which varies from 0 (regular flat band) to 1 (maximally singular). As d_max increases, the Berry phase Φ↻ = 1 − d_max² (in units of 2π) deviates from an integer, and both the Berry curvature Ω(k) and tr G(k) become highly inhomogeneous, concentrating near the touching point.

Using exact diagonalization (ED) and density‑matrix renormalization group (DMRG) simulations, the authors study spinless fermions at filling ν = 1/3 with a nearest‑neighbour repulsion H_int = U∑⟨i,j⟩ n_i n_j. They find robust FQAH phases in wide windows of the singularity parameters (0 ≤ δ ≲ 0.43 for the honeycomb lattice and 0.35 ≲ α ≲ 2.46 for the kagome lattice) that persist for any interaction strength 0 < U ≤ ∞. The Hall conductance is quantized to σ_H = e²/(3h), as demonstrated by charge‑pumping simulations and by the characteristic Laughlin counting {1,1,2,3,5,…} in the momentum‑resolved entanglement spectrum. Notably, the FQAH state survives both the weak‑coupling regime (U ≪ band gap Δ) and the strong‑coupling regime (U ≫ Δ), indicating that the usual requirement of projecting onto a single isolated band is not essential in these singular systems.

The many‑body gap Δ_mb is largest when d_max is small, correlating loosely with the overall uniformity of the quantum geometry (standard deviations σ_Ω, σ_tr G, and the trace‑condition‑violation integral TCV). However, Δ_mb does not scale directly with any single geometric metric. Instead, the authors uncover a self‑adjusting mechanism: the occupation distribution n(k) becomes anti‑correlated with tr G(k) away from the band‑touching point (regions of large tr G(k) are depleted), while surprisingly large occupation persists at the singular point despite the divergence of both Ω(k) and tr G(k). This leads to a decrease of the occupation‑weighted Berry flux ⟨Ω⟩_occ and of ⟨tr G⟩_occ as the singularity strengthens, which tracks the eventual collapse of the FQAH phase.

When the singularity parameter exceeds a critical value (δ ≈ 0.43 or α ≈ 2.46), the system undergoes a first‑order transition to a charge‑density‑wave (CDW) state with a √3 × √3 pattern. The transition is signaled by a discontinuity in the derivative of the ground‑state energy, a peak in the entanglement entropy, and a sharp rise in the occupation at the Brillouin‑zone corners. In the CDW regime, ⟨Ω⟩_occ is strongly suppressed while ⟨T⟩_occ (the occupation‑weighted trace‑condition violation) grows, reflecting electron localization that quenches the fractional topological order.

Overall, the work establishes three major insights: (1) fractional topological order can arise without a well‑defined Chern number or uniform quantum geometry; (2) the many‑body state can adapt to a highly fluctuating geometric landscape by developing an inhomogeneous carrier distribution that effectively “screens” the singularities; (3) the FQAH phase is remarkably robust against interaction strength, persisting from weak to strong coupling. These findings broaden the design space for fractional quantum Hall physics beyond the conventional ideal flat‑band paradigm, suggesting that engineered singularities and divergent quantum metrics in moiré materials, photonic lattices, or cold‑atom setups could be exploited to realize new strongly correlated topological phases.