Inspired by the discovery of a variety of correlated insulators in the moiré universe, controlled by interactions projected to a set of isolated bands with a narrow bandwidth, we examine here a partial sum rule associated with the inverse frequency-weighted optical conductivity restricted to low energies. Unlike standard sum rules that extend out to infinite frequencies, which include contributions from all interband transitions, we focus here on transitions associated only with the projected degrees of freedom. We analyze the partial sum rule in a nonperturbative but solvable limit for a variety of correlation-induced insulators. This includes (i) magic-angle twisted bilayer graphene at integer filling with projected Coulomb interactions, starting from the chiral flat-band limit and including realistic perturbations, (ii) fractional fillings of Chern-bands which support generalized Laughlin-like states, starting from a Landau level and including a periodic potential and magnetic-field, respectively, drawing connections to twisted ${\mathrm{MoTe}}_2$, and (iii) integer filling in toy models of nontopological flat bands with a tunable quantum geometry in the presence of repulsive interactions. The partial sum rule in all these examples is implicitly constrained by the form of the band quantum geometry via the low-lying excitation spectrum but is not related to it explicitly. For interacting Slater-determinant insulators, the partial sum rule is related to a “many-body projected quantum geometry”, obtained from the interaction-renormalized electronic bands. We also point out an intriguing connection between the partial sum rule and the quantum Fisher information associated with the projected many-body position operator. Finding alternative experimental routes that enable a direct measurement of the quantum geometry in correlated insulators remains an exciting and uncharted territory.

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• Berry curvature
• Optical conductivity
• Topological phases of matter
• Strongly correlated systems
• Twisted heterostructures