Localisation Transition in the Driven Aubry-Andr\'e Model

A recent experiment by P. Bordia et al. (Periodically Driving a Many Body Localized Quantum System, Nat Phys, Jan 2017) has demonstrated that periodically modulating the potential of a localised many-body quantum system described by the Aubry-Andr\'e Hamiltonian with on-site interactions can lead to a many-body localisation-delocalisation transition, provided the modulation amplitude is big enough. Here, we consider the noninteracting counterpart of that model in order to explore its phase diagram as a function of the strength of the disordered potential, the driving frequency and its amplitude. We will first of all mimic the experimental procedure of the experiment and use the even-odd sites imbalance as a parameter in order to discern between different phases. Then we compute the Floquet eigenstates and relate the localisation-delocalisation transition to their Inverse Participation Ratio. Both these approaches show that the delocalisation transition occurs for frequencies that are low compared to the bandwidth of the time independent model. Moreover, in agreement with the results of the experiment there is an amplitude threshold below which no delocalisation transition occurs. We estimate both the critical values for the frequency and the amplitude.