Periodically and aperiodically Thue-Morse driven long-range systems: From dynamical localization to slow dynamics

We investigate the electric-field driven power-law random banded matrix (PLRBM) model where a variation in the power-law exponent $\ensuremath{\alpha}$ yields a delocalization-to-localization phase transition. We examine the periodically driven PLRBM model with the help of the Floquet operator. The level spacing ratio and the generalized participation ratio of the Floquet Hamiltonian reveal a drive-induced weak multifractal (fractal) phase accompanied by diffusive (subdiffusive) transport on the delocalized side of the undriven PLRBM model. On the localized side, the time-periodic model remains localized---the average level-spacing ratio corresponds to Poisson statistics and logarithmic transport is observed in the dynamics. Extending our analysis to the aperiodic Thue-Morse driven system, we find that the aperiodically driven clean long-range hopping model (clean counterpart of the PLRBM model) exhibits the phenomenon of exact dynamical localization on tuning the drive parameters at special points. The disordered time-aperiodic system shows diffusive transport followed by relaxation to the infinite-temperature state on the delocalized side, and a prethermal plateau with subdiffusion on the localized side. Additionally, we compare this with a quasiperiodically driven Aubry-Andr\'e-Harper model that also undergoes a localization-delocalization transition. Unlike the disordered long-range model, it features a prolonged prethermal plateau followed by subdiffusion to the infinite temperature state, even on the delocalized side.