Integer and Fractional Chern Insulators At Zero Magnetic Field

Recent experimental progress in topological states of matter has brought about the discovery of electron fractionalization into e/3 particles in the absence of any applied magnetic fields. 

I will review the literature starting from the early writing down of flat chern band models and further of fractional Chern insulators in these models. We will review the recent progress in finding fractional states in twisted MoTe2 and rhombohedral pentalayer graphene. We will show how the -4/7, -3/5, -2/3 states appear naturally in twisted MoTe2 but how the -1/3 and -4/3 states are absent. We will review the single particle models, show that they fall in two types of parameters of different quantum geometry and Berry curvature distribution, and show that one set of parameters can match the experiment, but only if band mixing is taken into account. 

On pentalayer graphene, we will show how that in the experimental regime of parameters the flat band is degenerate with dispersive bands at many points in the Brillouin zone. We will show how a Hartree Fock calculation at filling 1 can give rise to two types of chern=1 states, an anomalous hall crystal (for a given interaction normal ordering) and a moire Chern insulator (for another type of commonly used normal ordering). We will make predictions on how to distinguish the two phases, including the collective modes. In the fractional regime we will show that fractionally filling the band obtained at integer filling heavily biases the system (by hand) towards FCI and that an unbiased calculation brings about both questions and surprises.