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Few layers of graphene at small twist-angles have emerged as a fascinatingplatform for studying the problem of strong interactions in regimes with anearly quenched single-particle kinetic energy and non-trivial band topology.Starting from the strong-coupling limit of twisted bilayer graphene with avanishing single-electron bandwidth and interlayer-tunneling between the samesublattice sites, we present an ${\it exact}$ analytical theory of the Coulombinteraction-induced low-energy optical spectral weight at all ${\it integer}$fillings. In this limit, while the interaction-induced single-particledispersion is finite, the optical spectral weight vanishes identically atinteger fillings. We study corrections to the optical spectral weight bysystematically including the effects of experimentally relevant strain-inducedrenormalization of the single-electron bandwidth and interlayer tunnelingsbetween the same sublattice sites. Given the relationship between the opticalspectral weight and the diamagnetic response that controls superconducting$T_c$, our results highlight the relative importance of specific parentinsulating phases in enhancing the tendency towards superconductivity whendoped away from integer fillings.

A theory for the low-energy optical sum-rule in moir\'e graphene