Cumulants associated with geometric phases

TL;DR

This paper introduces a cumulant expansion framework for geometric phases, including the Berry phase, revealing higher-order cumulants like spread and skewness, which are gauge invariant and expressible via operators.

Contribution

It presents a novel cumulant expansion approach to geometric phases, extending the Berry phase concept to include higher-order cumulants with gauge invariance and operator expressions.

Findings

Cumulants beyond the Berry phase are introduced, including spread, skewness, and kurtosis.

Higher-order cumulants are shown to be gauge invariant.

Geometric quantities can be expressed in terms of operators.

Abstract

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.