Does a distinct quasi many-body localized phase exist? A numerical study of a translationally invariant system in the thermodynamic limit

We consider a quench in an infinite spin ladder describing a system with two species of bosons in the limit of strong interactions. If the heavy bosonic species has infinite mass the model becomes a spin chain with quenched binary disorder which shows true Anderson localization (AL) or many-body localization (MBL). For finite hopping amplitude $J^{\mathrm{\prime }}$ of the heavy particles, on the other hand, we find an exponential polarization decay with a relaxation rate which depends monotonically on $J^{\mathrm{\prime }}$. Furthermore, the entanglement entropy changes from a constant (AL) or logarithmic (MBL) scaling in time $t$ for $J^{\mathrm{\prime }}=0$ to a sub-ballistic power-law, $S_{\text{ent}}\sim t^{\alpha }$ with $\alpha <1$, for finite $J^{\mathrm{\prime }}$. We do not find a distinct regime in time where the dynamics for $J^{\mathrm{\prime }}\mathrm{\ne }0$ shows the characteristics of an MBL phase. Instead, we discover a time regime with distinct dephasing and entanglement times, different both from a localized and a fully ergodic phase.