Intermediate superexponential localization with Aubry-Andr chains

We demonstrate the existence of an intermediate superexponential localization regime for eigenstates of the Aubry-Andre chain. In this regime, the eigenstates localize similar to the eigenstates of the Wannier-Stark ladder. The superexponential decay emerges on intermediate length scales for large values of the winding length-the quasiperiod of the Aubry-Andre potential. This intermediate length scale localization is present both in the metallic and insulating phases of the system. In the insulating phase, the superexponential localization is periodically interrupted by weaker decaying tails to form the conventional asymptotic exponential decay predicted for the Aubry-Andre model. In the metallic phase, the superexponential localization is only a transient phenomena and happens for states with energies away from the center of the spectrum and is followed by a superexponential growth into the next peak of the extended eigenstate. By adjusting the parameters, it is possible to extend the range of the superexponential localization to arbitrarily low values of the eigenfunction's amplitude. A similar intermediate superexponential localization regime is demonstrated in quasiperiodic discrete-time unitary maps.