The Aubry-André model as a hobbyhorse for understanding the localization phenomenon

We present a thorough pedagogical analysis of the single particle localization phenomenon in a quasiperiodic lattice in one dimension. Beginning with a detailed derivation of the Aubry-André Hamiltonian we describe the localization transition through the analysis of stationary and dynamical observables. Emphasis is placed on both the properties of the model and technical aspects of the performed calculations. In particular, the stationary properties investigated are the inverse participation ratio, the normalized participation ratio and the energy spectrum as a function of the disorder strength. Two dynamical quantities allow us to discern 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. The present manuscript could be useful in bringing advanced undergraduate and graduate students closer to the comprehension of localization phenomena, a topic of current interest in fields of condensed matter, ultracold atoms and complex systems.