Universal Shape Dependence of Charge Fluctuations in Three Dimensions

In two spatial dimensions, recent studies have shown that charge fluctuations within a subregion display interesting shape dependence when the region contains corners, reflecting universal properties of quantum critical points as well as observables of quantum geometry in insulators. In this work, we systematically extend this idea to three spatial dimensions. We identify the universal shape dependence at conformal quantum critical points associated with both smooth and singular geometries. We present analytical results for an arbitrary parallelepiped and benchmark them against Monte Carlo simulations of lattice models for the conventional O(3) transition. We also clarify the role of quantum geometry in three-dimensional insulators.