A fundamental property of any material is the electron filling, i.e., the number of electrons per unit cell. Within band theory, materials at odd filling must be metals, while those at even filling can be insulators.
In this talk I will show that such “filling constraints” can be made both more general - and far more constraining - than has long been thought. By generalizing the band criteria to strong interactions and complex crystal groups, we can infer when a material must have quantum entanglement over macroscopic length scales, with important implications for the interpretation of experiment. These constraints have interesting applications to the hunt for topological semi-metals and spin-orbit coupled spin-liquids.