Interacting isolated quantum systems generically heat up when subjected to a periodic drive, tending at long times to a featureless state maximizing the local entropy. However, in the presence of sufficiently strong disorder this heating can circumvented. Indeed, the many-body localized (MBL) phase is known to be robust under weak, high-frequency drives. On the other hand, for the same driving amplitude, but at sufficiently low-frequency, this robustness is generically lost, and the system becomes ergodic.
We compare the critical driving amplitude for the transition between the Floquet-MBL phase and the ergodic phase when the system is subjected to either a sinusoidal or a square wave drive. At large frequencies we find that both drives exhibit a similar dependence of the critical driving amplitude on the driving frequency. However, for low frequencies, we find a qualitatively different behavior of the critical amplitude as a function of frequency between the two types of drives. While for a square wave drive we a find a critical amplitude which depends on 〖~ω〗^(1/2), for sinusoidal drives the critical driving amplitude shows a weak frequency dependence. We provide a qualitative argument for the difference in the frequency dependence of the critical amplitude between the type of drives.