  The Berry phase can be obtained by taking the continuous limit of a cyclic
product $-\mbox{Im} \ln \prod_{I=0}^{M-1} \langle \Psi_0({\boldsymbol
  \xi}_I)|\Psi_0({\boldsymbol \xi}_{I+1})\rangle$, resulting in the circuit
integral $i \oint \mbox{d}{\boldsymbol \xi} \cdot \langle \Psi_0({\boldsymbol
  \xi})|\nabla_{\boldsymbol \xi}|\Psi_0({\boldsymbol \xi}\rangle$. Considering
a parametrized curve ${\boldsymbol \xi}(\chi)$ we show that the product
$\prod_{I=0}^{M-1} \langle \Psi_0(\chi_I)|\Psi_0( \chi_{I+1})\rangle$ can be
equated to a cumulant expansion. The first contributing term of this expansion
is the Berry phase itself, the other terms are the associated spread, skew,
kurtosis, etc. The cumulants are shown to be gauge invariant. It is also shown
that these quantities can be expressed in terms of an operator.
