Geometric cumulants associated with adiabatic cycles crossing degeneracy points: Application to finite size scaling of metal-insulator transitions in crystalline electronic systems
In this work, we focus on two questions. One, we complement the machinery to calculate geometric phases along adiabatic cycles as follows. The geometric phase is a line integral along an adiabatic cycle and if the cycle encircles a degeneracy point, the phase becomes nontrivial. If the cycle crosses the degeneracy point, the phase diverges. We construct quantities which are well defined when the path crosses the degeneracy point. We do this by constructing a generalized Bargmann invariant and noting that it can be interpreted as a cumulant generating function, with the geometric phase being the first cumulant. We show that particular ratios of cumulants remain finite for cycles crossing a set of isolated degeneracy points. The cumulant ratios take the form of the Binder cumulants known from the theory of finite size scaling in statistical mechanics (we name them geometric Binder cumulants). Two, we show that the developed machinery can be applied to perform finite size scaling in the context of the modern theory of polarization. The geometric Binder cumulants are size independent at gapclosure points or regions with closed gap (Luttinger liquid). We demonstrate this by model calculations for a onedimensional topological model, several two-dimensional models, and a one-dimensional correlated model. In the case of two dimensions, we analyze to different situations, one in which the Fermi surface is one dimensional (a line) and two cases in which it is zero dimensional (Dirac points). For the geometric Binder cumulants, the gap-closure points can be found by one-dimensional scaling even in two dimensions. As a technical point, we stress that only certain finite difference approximations for the cumulants are applicable since not all approximation schemes are capable of extracting the size scaling information in the case of a closed-gap system.