Wannier functions, if the Wannier functions themselves are localized within the unit cell (non-overlapping among different unit cells). Indeed, in the construction of tight-binding based lattice models, one starts with a continuum description, and assumes a localized basis of non-overlapping Wannier functions (see for example, Ref. [@Essler05]). In practice, however, constructing such a localized basis is not trivial [@Marzari97].

The distribution of the polarization gauges the extent to which the system is localized in the full configuration space, a criterion [@Kohn64] which distinguishes an insulator from a conductor. The second GIC was shown [@Kudinov91; @Souza00] to be proportional to the integrated frequency dependent conductivity (sum rule). A gauge dependent definition of the spread (similar to the second GIC) was used to define the maximally localized Wannier function [@Marzari97]. Also, the second cumulant was proposed [@Resta99] to distinguish conductors from insulators. In Ref. [@Hetenyi14] the simplest system with a Berry phase, an isolated spin-$\frac{1}{2}$ particle in a magnetic field, was considered, and it was shown that (based on calculating the first four cumulants) the moments of this underlying distribution are all equal.

The Zak phase was measured in Ref. [@Atala13] in an optical lattice setup which corresponds to the experimental realization 