Fractional Chern Insulators with spin and flavor

A number of recent experiments confirmed the observation of Fractional Chern insulator (FCI) phases in moir\’{e} superlattice of graphene and other material. We investigate the quantum geometry of such fractional Chern insulators (FCIs) phases in a class of moiré and super-moiré structures by combining analytical constructions in complex projective spaces with numerical techniques to construct maximally localized Wannier functions, and subsequent Berry curvature analysis. Within our proposed theoretical framework, we particularly consider spin and flavour degrees of freedom within the framework of complex projective manifolds $CP^{N-1}$ to analyse multi-component of FCI phases in such systems. Subsequently we extend this geometric formulation to explore projected many-body wavefunctions by generalizing Landau-level physics to lattice realizations.