large variations in the maximum.'
author:
- 'M. Yahyavi and B. Hetényi'
title: 'Reconstruction of the polarization distribution of the Rice-Mele model'
---

\[sec:intro\]

Introduction
============

One way to derive the Berry phase [@Berry84; @Pancharatnam56; @Xiao10] is to form a product of scalar products between quantum states at different points of the space of external parameters (Bargmann invariant [@Bargmann64]) and to take the continuous limit along a cyclic curve. An extension [@Souza00; @Hetenyi14] of this derivation, keeping higher order terms, leads to gauge invariant cumulants (GIC) associated with the Berry phase. One is lead to ask two questions. The GICs give information of the distribution of what physical quantity? Can one reconstruct the probability distribution from the GICs?

The answer to the first question depends on the physical context in which the Berry phase is defined. In a crystalline solid the Berry phase (or Zak phase [@Zak89], in this context) corresponds to the macroscopic polarization. Zak showed [@Zak89] that the phase itself corresponds to the expectation value of the position over a Wannier function. For the higher order GICs Souza, Wilkens and Martin [@Souza00] showed that they only correspond to the cumulants of the distribution of the position associated with 