Classification of Vortexable Bands

The fractional quantum Hall effect (FQHE) has traditionally be studied in the setting of Landau levels, which appear with large external magnetic fields. Recent experiments in twisted MoTe2 and graphene heterostructures have found bands that closely mimic the zeroth Landau level, giving rise to fractional quantum anomalous hall phases --- FQHE phases without a magnetic field. If one could instead mimic the first Landau level, then one could hope to realize the celebrated non-Abelian fractional quantum hall phases that are sought after for their topological properties and potential to store quantum information. To date, however, it is not clear what it means to mimic the first Landau level. This talk will present a classification of "vortexable" or "ideal" bands with higher Chern number via mathematics results from the holomorphic geometry. As an application, I will give a clear criterion for what it mean for a band to be close to the first Landau level without relying on magnetic translation symmetry.