Weak ergodicity breaking transition in randomly constrained model

Experiments in Rydberg atoms have recently found unusually slow decay from a small number of special initial states. We investigate the robustness of such long-lived states (LLS) by studying an ensemble of locally constrained random systems with tunable range $\mu$. Upon varying $\mu$, we find a transition between a thermal and a weakly non-ergodic (supporting a finite number of LLS) phases. Furthermore, we demonstrate that the LLS observed in the experiments disappear upon the addition of small perturbations so that the transition reported here is distinct from known ones. We then show that the LLS dynamics explores only part of the accessible Hilbert space, thus corresponding to localisation in Hilbert space.