  In this work we focus on two questions. One, we complement the machinary 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 non-trivial. 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 machinery developed 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 gap closure points or regions with
closed gap (Luttinger liquid). We demonstrate this by model calculations for a
one-dimensional 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.
