Quantum geometry and stability of fractional Chern insulators I: Perturbations from the ideal limit

Fractional Chern insulators have recently attracted significant attention, following their experimental observation in tMoTe₂ and also in multilayer graphene. In particular, low-energy continuum model descriptions of transition metal dichalcogenide (TMD) homobilayers display nearly flat Chern bands with almost ideal quantum geometry at a magic angle. This suggests a connection with Landau level physics, which is believed to be key for the emergence of FCI states in these systems. Here we study a model of Aharonov-Casher bands subject to a periodic potential, which is an approximate description of TMD homobilayers near the magic angle. We perform band-projected exact diagonalization calculations to obtain the many-body phase diagram of the model at fractional band fillings. Starting from the ideal band limit, we investigate how the introduction of Berry curvature fluctuations and violations of the trace condition affect the stability of the fractional Chern insulator ground state and also the possible competing phases.