I discuss the quantum dynamics of strongly disordered quantum systems with critically long range interactions, decaying as $1/r^{2d}$ in $d$ spatial dimensions. I argue that, contrary to expectations, localization in such systems is stable at low orders in perturbation theory, giving rise to an unusual “critically many-body localized (MBL) regime.” I discuss the phenomenology of this critical MBL regime, which includes distinctive signatures in entanglement, charge statistics, noise, and transport. Experimentally, such a critically localized regime can be realized in three-dimensional systems with van der Waals interactions, such as Rydberg atoms, and in one-dimensional systems with $1/r^2$ interactions, such as trapped ions. I estimate timescales on which high-order perturbative and nonperturbative (avalanche) phenomena may destabilize this critically MBL regime and conclude that the avalanche sets the limiting timescale, in the limit of strong disorder or weak interactions.

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• Localization
• van der Waals interaction
• Trapped ions