Dynamics on multiple timescales in centrally coupled disordered spin systems

Many-body localization, though robust to local perturbations, is not believed to be a stable phase of matter upon introducing infinite range interactions. Such situations naturally arise in systems with central coupling to an auxiliary degree of freedom like a single mode cavity. Here, we consider a 1D disordered Ising chain globally coupled to d-level qudit, which has connections to systems under monochromatic external drive and display Floquet MBL. Through simulations using exact diagonalization and the multilayer multiconfigurational Hartree method, we find that the time evolution can be parsed according multiple timescales, which can be formalized within perturbation theory. Qualitatively, we find changes to the dynamics near a hypothesized breakdown of Floquet physics, leading to different behaviors for spin- and qudit-based observables as well as a logarithmically slow growth of entanglement entropy.