Apparent slow dynamics in the ergodic phase of a driven many-body localized system without extensive conserved quantities

We numerically study the dynamics on the ergodic side of the many-body localization transition in a Floquet model with no global conservation laws. We describe and employ a numerical technique based on the fast Walsh-Hadamard transform that allows us to perform an exact time evolution for large systems and long times. As in models with conserved quantities we observe a slowing down of the dynamics as the transition into the many-body localized phase is approached. More specifically, our data is consistent with a subballistic spread of entanglement and a stretched-exponential decay of the return probability, with the appropriately defined exponents, for a fixed system size, seeming to smoothly go to zero at the transition. However, with access to larger system sizes, we observe a clear flow of the exponents towards faster dynamics and can not rule out that the slow dynamics is a finite-size effect. Furthermore, we observe examples of non-monotonic dependence of the exponents with time, with dynamics initially slowing down but accelerating again at even larger times, reminiscent of what is observed in large scale simulations of random regular graphs and consistent with the slow dynamics being a crossover phenomena with a localized critical point.