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