Quantum weight

We introduce the concept of quantum weight as a fundamental property of insulating states of matter that is encoded in the ground-state static structure and measures quantum fluctuation in electrons' center of mass. We find a sum rule that directly relates quantum weight -- a ground state property -- with the negative-first moment of the optical conductivity above the gap frequency. Building on this connection to optical absorption, we derive both an upper bound and a lower bound on quantum weight in terms of electron density, dielectric constant, and energy gap. Therefore, quantum weight constitutes a key material parameter that can be experimentally determined from X-ray scattering.