Zero Flux Localization: How flat bands magically appear and disappear

Flat bands correspond to the spatial localization of a quantum particle moving in a field with discrete or continuous translational invariance. The canonical example is the flat Landau levels in a homogeneous magnetic field. Several significant problems---including flat bands in moir\'e structures---are related to the problem of a particle moving in an inhomogeneous magnetic field with zero total flux. We see that while perfectly flat bands in such cases are impossible, the introduction of a ``non-Abelian component''---a spin field with zero total curvature---can lead to perfect localization. It is shown that for doubly periodic systems, flat bands are only possible for certain magic values of the field corresponding to a quantized flux through an individual tile. These exact solutions clarify the simple structure underlying flat bands in moiré materials and provide a springboard for constructing a novel class of fractional quantum Hall states.