Vortexable bands: A unifying perspective on band geometry

Vortexable bands provide a unifying real-space perspective on quantum geometry. We define a band as vortexable if, roughly, one can choose complex coordinates on the plane so that multiplying the band by that coordinate operator does not cause interband transitions. A companion talk shows how vortexable bands are ideal for fractional Chern insulators and other many-body consequences. When this real-space concept is expressed in momentum space, it becomes a statement about the quantum geometry of the system. The subset of ``special" vortexable bands is shown to be equivalent to the momentum space ``trace condition" or ``ideal band condition." Alternatively, one can always choose a gauge so that the periodic Bloch wavefunctions of special vortexable bands are holomorphic in the Brillouin zone --- a powerful result linking to holomorphic geometry. We provide a number of examples of vortexable bands derived from chiral graphene and its variants with internal strain and moire potentials. ``General" vortexable bands, however, lie beyond the trace condition. We show the point of failure is the definition of periodic Bloch wavefunctions in terms of the laboratory coordinates, and introduce a modified momentum space measure that can detect any vortexable bands. Moreover, this measure quantifies deviations from vortexability, which can be applied to generic Chern bands to identify promising FCI platforms in moire systems.