Localization dynamics in a centrally coupled system

We investigate a disordered Ising chain in which all spins couple to a central qudit, through which spin flip interactions are mediated. For small but finite coupling, the trivial localization of the non-coupled limit survives as a form of many-body localization (MBL). By using an exact wavefunction propagation method, we are able to observe the dynamics of large system sizes (L~50-100) to intermediate timescales and establish the existence of slow or vanishing relaxation. This allows us to address the question of the proper scaling of the qudit size with L in the thermodynamic limit, the suppression of effective all-to-all coupling to protect MBL, and the dependence of localization on energy density.