We note that if the Wannier functions are localized in one unit cell, then the summation in the scalar product of Eq. (\[eqn:scalarprod\]) will be restricted to the term $\Delta p = 0$. In this case, the cumulants $C_M$ will correspond to those of the Wannier centers.

Relation to response functions {#sec:resp}
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The second GIC associated with the polarization gives a sum rule for the frequency-dependent conductivity. This was shown for a finite system by Kudinov [@Kudinov91], and the derivation was extended to periodic systems by Souza, Wilkens, and Martin [@Souza00], by replacing the ordinary matrix elements of the total position operator by their counterparts valid in the crystalline case. Their result is $$C_2 = \frac{\hbar}{\pi q_e^2 n_0} \int \frac{d \omega}{\omega} \bar{\sigma}(\omega),$$ where $q_e$ denotes the charge, $n_0$ the density, and $\bar{\sigma}(\omega)=(V/8\pi^3)\int d{\bf k} \sigma^{\bf k}(\omega)$.

For an insulating (gapped) system one can show that the second cumulant provides an upper bound for the dielectric susceptibility, $\chi$. This was shown by Baeriswyl [@Baeriswyl00] for an open system. This derivation is also easily extended to periodic systems by the appropriate replacement of the total position matrix elements, resulting in, $$\label{eqn:chi}
  \chi \leq \frac{2q_e}{V \Delta_g} C_2.$$ In this equation $\Delta_g$ denotes 