Incommensurate-Stabilized Fractional Chern Insulator in Alternating Twisted Trilayer Graphene

Abstract: Fractional Chern insulators (FCIs) typically emerge in topological flat bands and are regarded as lattice analogs of fractional quantum Hall states. Conventionally, the flat-band wavefunctions that support FCIs are expected to mimic the lowest Landau level, a condition that can be quantified by the quantum-geometric indicators. In realistic systems, however, FCIs often compete with lattice symmetry-breaking orders, especially when the hosting flat bands not ideal. In this work, we propose stabilizing FCIs by exploiting the intrinsic incommensurability of alternating twisted trilayer graphene, which naturally suppresses competing charge-density-wave (CDW) phase while FCIs are less effected. Within an adiabatic approximation at the supermoiré scale, the effect of incommensuration on local physics can be quantified as phase shifts of interlayer coupling. Using exact diagonalization, we compute ground states in different local patches and uncover a strikingly counterintuitive result: the FCI gap increases as the quantum-geometric indicators worsen. Within certain parameter ranges, we further identify mixed phases where FCIs coexist with CDWs, but with CDWs confined only to patches of weak incommensurability. Finally, we provide experimental protocols and discuss how incommensuration enrich the system's topology and quantum geometry. Not only do our results establish incommensuration as a robust stabilizer of FCIs, but also provide a general paradigm for exploring strong-correlation physics in incommensurate systems.