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The discovery of Fractional Chern Insulators (FCIs) in twisted bilayer MoTe₂ has sparked significant interest in fractional topological matter without external magnetic fields. Unlike the flat dispersion of Landau levels, moir\'e electronic states are influenced by lattice effects within a nanometer-scale superlattice. This study examines the impact of these lattice effects on the topological phases in twisted bilayer MoTe₂, uncovering a family of FCIs with Abelian anyonic quasiparticles. Using a composite fermion approach, we identify a sequence of FCIs with fractional Hall conductivities ₗₘ = C2C + 1 e²h linked to partial filling \, ₇ of holes of the topmost moir\'e valence band. These states emerge from incompressible composite fermion bands of Chern number C within a complex Hofstadter spectrum. This approach explains FCIs with Hall conductivities ₗₘ = (2/3) e²/h and ₗₘ = (3/5) e²/h at fractional fillings \, ₇ = 2/3 and \, ₇ = 3/5 observed in experiments, and uncovers other fractal FCI states. The Hofstadter spectrum reveals new phenomena, distinct from Landau levels, including a higher-order Van Hove singularity (HOVHS) at half-filling, leading to novel quantum phase transitions. This work offers a comprehensive framework for understanding FCIs in transition metal dichalcogenide moir\'e systems and highlights mechanisms for topological quantum criticality.
Lu et al. (Wed,) studied this question.