Origin of model fractional Chern insulators in all topological ideal flatbands: Explicit color-entangled wave function and exact density algebra

It is commonly believed that nonuniform Berry curvature destroys the Girvin-MacDonald-Platzman algebra and as a consequence destabilizes fractional Chern insulators. In this work, we disprove common lore by presenting theory for all topological ideal flatbands with nonzero number $\mathcal{C}$. The smooth single-particle Bloch wave function proven to admit an exact color-entangled form superposition of $\mathcal{C}$ lowest Landau level type functions distinguished boundary conditions. Including repulsive interactions, Abelian non-Abelian model insulators Halperin are stabilized zero-energy ground states no matter how is, long quantum geometry repulsion short-ranged. key reason existence emergent Hilbert space in which can be exactly flattened adjusting function's normalization. such space, flatband-projected density operator obeys closed algebra, making mapping $\mathcal{C}$-layered levels possible. end, discuss applications moir\'e flatband systems, particular focus on fractionalized phase spontaneous symmetry-breaking recently observed graphene-based twisted materials.