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Cumulants associated with geometric phases

The Berry phase can be obtained by taking the continuous limit of a cyclic product $-\mbox{Im} \ln \prod_{I=0}^{M-1} \langle Ψ_0({\boldsymbol ξ}_I)|Ψ_0({\boldsymbol ξ}_{I+1})\rangle$, resulting in the circuit integral $i \oint \mbox{d}{\boldsymbol ξ} \cdot \langle Ψ_0({\boldsymbol ξ})|\nabla_{\boldsymbol ξ}|Ψ_0({\boldsymbol ξ}\rangle$. Considering a parametrized curve ${\boldsymbol ξ}(χ)$ we show that the product $\prod_{I=0}^{M-1} \langle Ψ_0(χ_I)|Ψ_0( χ_{I+1})\rangle$ 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.