Low-Energy Optical Sum Rule in Moiré Graphene

Few layers of graphene at small twist angles have emerged as a fascinating platform for studying the problem of strong interactions in regimes with a nearly quenched single-particle kinetic energy and nontrivial band topology. Starting from the strong-coupling limit of twisted bilayer graphene with a vanishing single-electron bandwidth and interlayer tunneling between the same sublattice sites, we present an exact analytical theory of the Coulomb interaction-induced low-energy optical spectral weight at all integer fillings. In this limit, while the interaction-induced single-particle dispersion is finite, the optical spectral weight vanishes identically at integer fillings. We study corrections to the optical spectral weight by systematically including the effects of experimentally relevant strain-induced renormalization of the single-electron bandwidth and interlayer tunnelings between the same sublattice sites. Given the relationship between the optical spectral weight and the diamagnetic response that controls superconducting 𝑇𝑐, our results highlight the relative importance of specific parent insulating phases in enhancing the tendency towards superconductivity when doped away from integer fillings.