Vortexable Chern Bands and Fractional Chern Insulators in Moire Graphene and Transition Metal Dichalcogenides

Fractional Chern insulators realize the remarkable physics of the fractional quantum Hall effect (FQHE) in crystalline systems with Chern bands. The lowest Landau level (LLL) is known to host the FQHE, but not all Chern bands are suitable for realizing fractional Chern insulators (FCI). Previous approaches to stabilizing FCIs focused on mimicking the LLL through momentum space criteria. Here instead we take a real-space perspective by introducing the notion of vortexability. Vortexable Chern bands admit a fixed operator that introduces vortices into any band wavefunction while keeping the state entirely within the same band. Vortexable bands admit trial wavefunctions for FCI states, akin to Laughlin states. In the absence of dispersion and for sufficiently short-ranged interactions, these FCI states are the ground state -- independent of the distribution of Berry curvature. Vortexable Chern bands emerge naturally in chiral twisted graphene, and fractional Chern insulators were subsequently observed experimentally. Recently, zero-field fractional Chern insulators, and potentially a zero-field composite Fermi liquid, were also observed in the nearly-vortexable twisted MoTe_2. New and exciting nearly-vortexable platforms are also appearing, including periodically strained graphene and helically twisted graphene.