Weak ergodicity breaking transition in a 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>μ</a:mi></a:math>. Upon varying <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>μ</b:mi></b:math>, we find a transition between thermal and weakly nonergodic (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 localization in Hilbert space.