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

## TL;DR

This paper provides a comprehensive pedagogical analysis of localization in a one-dimensional quasiperiodic lattice using the Aubry-André model, exploring stationary and dynamical properties to understand the phenomenon.

## Contribution

It offers a detailed analysis of localization phenomena in the Aubry-André model, combining stationary and dynamical perspectives to enhance understanding.

## Key findings

- Hofstadter butterfly pattern observed in energy spectrum
- Localization characterized by IPR and NPR metrics
- Dynamical spreading distinguishes localized from delocalized states

## Abstract

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.