Universality of Boundary Charge Fluctuations

We establish the quantum fluctuations ΔQB2 of the charge QB accumulated at the boundary of an insulator as an integral tool to characterize phase transitions where a direct gap closes (and reopens), typically occurring for insulators with topological properties. The power of this characterization lies in its capability to treat different kinds of insulators on equal footing, being applicable to transitions between topological and nontopological band, Anderson, and Mott insulators alike. In the vicinity of the phase transition, we find a universal scaling ΔQB2(Eg) as a function of the gap size Eg and determine its generic form in various dimensions. For prototypical phase transitions with a massive Dirac-like bulk spectrum, we demonstrate a scaling with the inverse gap in one dimension and a logarithmic one in two dimensions.