connection to the distribution associated with the Wannier functions. In section \[sec:resp\] we discuss the connection of the cumulants to response functions, after which the reconstruction procedure is presented. In section \[sec:SSH\] the Su-Schrieffer-Heeger and Rice-Mele models are introduced. Subsequently, the behavior of the moments for the fully dimerized limit is studied. Section \[sec:results\] contains our results and analysis before concluding our work.

Gauge invariant cumulants associated with the Zak phase {#sec:cml}
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Consider a one-dimensional system whose Hamiltonian which is periodic in $L$. We take Bloch functions parametrized by the crystal momentum, $\Psi_0(K)$ on a grid of $M$ points $K_I = 2\pi I /(M L) - \pi/L$, with $I=0,...,M-1$. The Zak phase can be derived from a product of the form $$\label{eqn:phi_dscrt}
\phi_{Zak} = \mbox{Im} \ln \prod_{I=0}^{M-1} \langle \Psi_0(K_I)|\Psi_0(K_{I+1})\rangle,$$ by taking the continuous limit ($M\rightarrow \infty$). The product in Eq. (\[eqn:phi\_dscrt\]) is known as the Bargmann invariant [@Bargmann64]. We will derive the Zak phase, as well as the associated gauge invariant cumulants (GIC). We start by equating the product in Eq. (\[eqn:phi\_dscrt\]) to a cumulant expansion, $$\label{eqn:cum_exp} 
\left[\prod_{I=0}^{M-1} \langle
  \Psi_0(K_I)|\Psi_0(K_{I+1})\rangle\right]^{\Delta K} = 
\exp\left(  \sum_{n=1}^\infty \frac{(i \Delta K)^n}{n!}\tilde{C}_n\right),$$ with $\Delta K = 2 \pi/M$. We now expand both sides and 