Fractional Chern insulator states in twisted bilayer graphene: An analytical approach

Recent experiments on bilayer graphene twisted near the magic angle have observed spontaneous integer quantum Hall states in the presence of an aligned hexagonal boron nitride substrate. These states arise from the complete filling of Chern bands. A natural question is whether fractional filling of the same bands would lead to fractional Chern insulators (FCIs), i.e., fractional quantum Hall states realized in the absence of a magnetic field. Here, we argue that the magic angle graphene bands have favorable quantum geometry for realizing FCIs. We show that in the tractable “chiral” limit, the flatbands wave functions are an analytic function of the crystal momentum. This remarkable property fixes the quantum metric up to an overall momentum-dependent scale factor, the local Berry curvature, whose variation is itself small. Thus the three conditions associated with FCI stability—(i) narrow bands, (ii) “ideal” quantum metric, and (iii) relatively uniform Berry curvature—are all satisfied. Our work emphasizes continuum real-space approaches to FCIs, in contrast to earlier works, which mostly focused on tight-binding models. This enables us to construct a Laughlin wave function on a real-space torus that is a zero-energy ground state of the Coulomb interaction in the limit of very short screening length. Finally, we discuss the evolution of the band geometry on tuning away from the chiral limit and show numerically that some of the desirable properties continue to hold at a quantitative level for realistic parameter values.