Quantum weight: A fundamental property of quantum many-body systems

We introduce the concept of quantum weight as a ground state property of quantum many-body systems that is encoded in the static structure factor and characterizes density fluctuation at long wavelengths. The quantum weight carries a wealth of information about dielectric responses and optical properties of the system, and is closely related to its quantum geometry. For systems with short-range interactions or low-dimensional Coulomb systems, we show that the many-body quantum metric (which measures the change of the ground state under twisted boundary conditions) can be determined directly from the quantum weight. Notably, the quantum weight is a property of a single ground state and independent of boundary conditions in the thermodynamic limit. Our finding thus enables direct experimental measurement and numerical calculation of many-body quantum metric. On the other hand, for three-dimensional Coulomb systems, we show that the quantum weight is distinct from the many-body quantum metric due to dielectric screening in three dimensions. We further use dielectric sum rules to derive upper and lower bounds on their quantum weight in terms of electron density, static dielectric constant, and plasmon energy. Our work highlights quantum weight as a fundamental material parameter, which can be experimentally determined by x-ray scattering or electron loss spectroscopy.