Fractional Chern Insulators in Twisted Bilayer MoTe$_2$: A Composite Fermion Perspective

The discovery of Fractional Chern Insulators (FCIs) in twisted bilayer MoTe$_2$ 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 MoTe$_2$, uncovering a family of FCIs with Abelian anyonic quasiparticles. Using a composite fermion approach, we identify a sequence of FCIs with fractional Hall conductivities $\sigma_{xy} = \frac{C}{2C + 1} \frac{e^2}{h}$ linked to partial filling $\nu_{\,\text{h}}$ 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 $\sigma_{xy} = (2/3) e^2/h$ and $\sigma_{xy} = (3/5) e^2/h$ at fractional fillings $\nu_{\,\text{h}} = 2/3$ and $\nu_{\,\text{h}} = 3/5$ 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.