Critical localization with Van der Waals interactions

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 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 non-perturbative (avalanche) phenomena may destabilize this critically MBL regime, and conclude that the avalanche sets the limiting timescale, in the limit of strong disorder / weak interactions.