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

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.