defined the associated moments can be defined based on the cumulants. Following this standard procedure we also define a set of moments. For the first four moments the expressions are $$\begin{aligned}
\label{eqn:muC}
\mu_C^{(1)} &=& C_1 \\ \nonumber 
\mu_C^{(2)} &=& C_2 + C_1^2 \\ \nonumber
\mu_C^{(3)} &=& C_3 + 3 C_2 C_1 + C_1^3 \\ \nonumber 
\mu_C^{(4)} &=& C_4 + 4 C_3 C_1 + 3 C_2^2 + 6 C_2 C_1^2 + C_1^4.\end{aligned}$$ As discussed below, when the Wannier functions of a particular model are localized within the unit cell, these moments correspond to the moments of the polarization, alternatively, to the distribution of the Wannier functions themselves.

We remark that in general, the Berry phase is a physically well-defined observable, which is thought not to correspond to an operator acting on the Hilbert space. The Zak phase, however, is known to correspond to the total position, and is the basic quantity in expressing the polarization in the modern theory [@King-Smith93; @Resta94; @Resta98].

Connection to the distribution of Wannier centers
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Cumulants of the type described in the previous section appear in the theory of polarization [@Souza00]. In this section we connect the cumulants to the distribution of Wannier centers. We consider 