Origin of model fractional Chern insulators in all topological ideal flatbands: Explicit color-entangled wave function and exact density algebra
Jie Wang, Semyon Klevtsov, Zhao Liu
Abstract
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 this common lore by presenting a theory for all topological ideal flatbands with nonzero Chern number $\mathcal{C}$. The smooth single-particle Bloch wave function is proven to admit an exact color-entangled form as a superposition of $\mathcal{C}$ lowest Landau level type wave functions 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 the Berry curvature is, as long as the quantum geometry is ideal and the repulsion is short-ranged. The key reason is the existence of an emergent Hilbert space in which Berry curvature can be exactly flattened by adjusting the wave function's normalization. In such space, the flatband-projected density operator obeys a closed Girvin-MacDonald-Platzman type algebra, making exact mapping to $\mathcal{C}$-layered Landau levels possible. In the end, we discuss applications of the theory to moir\'e flatband systems, with a particular focus on the fractionalized phase and the spontaneous symmetry-breaking phase recently observed in graphene-based twisted materials.