Cumulants associated with geometric phases

The Berry phase can be obtained by taking the continuous limit of a cyclic product $-\text{Im} \ln \prod_{I=0}^{M-1} \langle \Psi_0({\bm \xi}_I)|\Psi_0({\bm \xi}_{I+1})\rangle$ , resulting in the circuit integral $i \oint \text{d}{\bm \xi} \cdot \langle \Psi_0({\bm \xi})|\nabla_{\bm \xi}|\Psi_0({\bm \xi}\rangle$ . Considering a parametrized curve ${\bm \xi}(\chi)$ we show that a set of cumulants can be obtained from the product $\prod_{I=0}^{M-1} \langle \Psi_0(\chi_I)|\Psi_0(\chi_{I+1})\rangle$ . The first cumulant corresponds to the Berry phase itself, the others turn out to be the associated spread, skew, kurtosis, etc. The cumulants are shown to be gauge invariant. Then the spread formula from the modern theory of polarization is shown to correspond to the second cumulant of our expansion. It is also shown that the cumulants can be expressed in terms of the expectation value of an operator. An example of the spin-$\frac{1}{2}$ particle in a precessing magnetic field is analyzed