The Aubry-Andr\'e model as the hobbyhorse for understanding localization phenomenon

We present a thorough pedagogical analysis of the single particle localization phenomenon in a quasiperiodic lattice in one dimension. Description of disorder in the lattice is represented by the Aubry-Andr\'e model. Characterization of localization is performed through the analysis of both, stationary and dynamical properties. The stationary properties investigated are the inverse participation ratio (IPR), the normalized participation ratio (NPR) and the energy spectrum as a function of the disorder strength. As expected, the distinctive Hofstadter pattern is found. Two dynamical quantities allow discerning the localization phenomenon, being the spreading of an initially localized state and the evolution of population imbalance in even and odd sites across the lattice.