Exact results for low-energy many-body optical sum-rule in moir'e graphene

Starting from the strong-coupling limit in twisted bilayer graphene, we present an exact analytical computation of the interaction-induced low-energy optical spectral weight at integer fillings. Remarkably, the emergent symmetries in this limit lead to the complete vanishing of the low-energy spectral-weight. Away from this solvable limit, we systematically analyze the effects of strain and finite interlayer tunnelings between the same sublattice sites, and comment on the role of electron-phonon interactions. Our results have direct implications for the physics of superconductivity and non-Fermi-liquid-like phases obtained upon doping these ``parent" correlated insulating states.