Disorder averaging in random lattice models with periodic boundary conditions: Application to models with uncorrelated and correlated disorder

Periodic boundary conditions are not always used in the study of disordered systems, but it can be advantageous to apply them to mimick thermodynamically large systems. In this case, polarization and its cumulants can not be obtained directly, but through the tools of the modern theory of polarization. This theory casts the polarization in crystalline systems as a geometric phase, rather than an operator expectation value. We develop disorder averaging techniques within the context of this theory which can calculate the variance of the polarization, its higher order moments, and the excess kurtosis (or Binder cumulant). We also derive an indicator of delocalization based on the degeneracy as a function of boundary conditions. We apply the computational techniques to two model systems. To test localization, we use a one-dimensional disordered model which is fully Anderson localized. Our calculations verify this. We also apply our techniques to the one dimensional de Moura-Lyra model, developed to study power law correlated (controlled by a parameter, $α$) disorder. While this model is a pathological one, our method is validated. We also point out the significance of pairwise degeneracies found in the parameter range, $α>2$ and near the band center (or near half filling), where the model was conjectured to exhibit a mobility edge.