Low-energy optical absorption in correlated insulators: Projected sum rules and the role of quantum geometry

The authors study here a low\leavevmode\hbox{-}energy optical absorption sum rule for correlated insulators whose dynamics are governed by interactions projected to isolated (non)topological flat bands. Applying the framework to magic\leavevmode\hbox{-}angle twisted bilayer graphene and fractional Chern insulators in theoretically solvable limits, they provide a systematic understanding of what microscopic properties control the extent to which the correlated insulators are optically ``dark.''