Origin of Model Fractional Chern Insulators in All Topological Ideal Flatbands: Explicit Color-entangled Wavefunction and Exact Density Algebra

It is commonly believed that nonuniform Berry curvature ruins the Girvin-MacDonald-Platzman algebra and as a consequence destabilizes fractional Chern insulators. In this work we disprove such common sense by presenting a theory for all topological ideal flatbands with nonzero Chern number C. The smooth single-particle Bloch wavefunction is proved to admit an exact color-entangled form as a superposition of C lowest Landau level type wavefunctions distinguished by boundary conditions. Including repulsive interactions, Abelian and non-Abelian model fractional Chern insulators of Halperin type are stabilized as exact zero-energy ground states no matter how nonuniform Berry curvature is, as long as the quantum geometry is ideal and the repulsion is short-ranged. The key reason behind is the complete separation of the band-projected coordinate (guiding center) and the inter-band transition, in an unnormalized Hilbert space of ideal flatband where the Berry curvature is exactly flattened by scarifying normalization. In such Hilbert space, the flatband-projected density operator obeys a closed Girvin-MacDonald-Platzman type algebra, enabling an exact mapping to C-layered lowest Landau levels. In the end, we discuss applications of the theory, especially to the graphene and TMD based moire materials.