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Disorder in quantum systems can lead to the disruption of long-range order inthe ground state and to the localization of the elementary excitations - famousexamples thereof being the Bose glass of interacting bosons in a disordered orquasi-periodic environment, or the localized phase of spin chains mapping ontofermions. Here we present a two-dimensional quantum Ising model - relevant tothe physics of Rydberg-atom arrays - in which positional disorder of the spinsinduces a randomization of the spin-spin couplings and of an on-sitelongitudinal field. This form of disorder preserves long-range order in theground state, while it localizes the elementary excitations above it,faithfully described as spin waves: the spin-wave spectrum is partiallylocalized for weak disorder (seemingly exhibiting mobility edges betweenlocalized and extended, yet non-ergodic states), while it is fully localizedfor strong disorder. The regime of partially localized excitations exhibits avery rich non-equilibrium dynamics following a low-energy quench: correlationsand entanglement spread with a power-law behavior whose exponent is acontinuous function of disorder, interpolating between ballistic and arrestedtransport. Our findings expose a stark dichotomy between static and dynamicalproperties of disordered quantum spin systems, which is readily accessible toexperimental verification using quantum simulators of closed quantum many-bodysystems.

Anomalous diffusion and localization in a positionally disordered quantum spin array