Universality of Boundary Charge Fluctuations

We establish the quantum fluctuations \(\Delta Q_B^2\) of the charge \(Q_B\) 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 non-topological band, Anderson, and Mott insulators alike. In the vicinity of the phase transition we find a universal scaling \(\Delta Q_B^2(E_g)\) as function of the gap size \(E_g\) 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.