First Landau level in moiré graphene system at zero magnetic field

Chern insulators, which are the lattice analogues of the quantum Hall states, provide a promising platform to explore fractional Chern insulators with partial band fillings, offering a pathway to manipulate non-Abelian excitations. Moiré materials, such as twisted bilayer graphene and twisted transition metal dichalcogenides, are ideal platform to realize fractional Chern insulators because they can be tuned to host narrow Chern bands with "vortexable" single particle wavefunctions, generalizing those of the lowest Landau level. Fractional Chern insulators were recently observed at zero magnetic field in twisted MoTe2. A natural next step is non-Abelian states, such as the Moore-Read Pfaffian state, which appear at half filling of the first Landau level. In this work, we provide a sharp definition of what first Landau level quantum geometry means when there is no external magnetic field, or magnetic translation symmetry. We find moiré graphene systems that realize this geometry, and compare with trial wavefunctions that have a manifest first Landau level character.