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Phase transitions are driven by collective fluctuations of a system'sconstituents that emerge at a critical point. This mechanism has beenextensively explored for classical and quantum systems in equilibrium, whosecritical behavior is described by a general theory of phase transitions.Recently, however, fundamentally distinct phase transitions have beendiscovered for out-of-equilibrium quantum systems, which can exhibit criticalbehavior that defies this description and is not well understood. Aparadigmatic example is the many-body-localization (MBL) transition, whichmarks the breakdown of quantum thermalization. Characterizing quantum criticalbehavior in an MBL system requires the measurement of its entanglementproperties over space and time, which has proven experimentally challenging dueto stringent requirements on quantum state preparation and system isolation.Here, we observe quantum critical behavior at the MBL transition in adisordered Bose-Hubbard system and characterize its entanglement properties viaits quantum correlations. We observe strong correlations, whose emergence isaccompanied by the onset of anomalous diffusive transport throughout thesystem, and verify their critical nature by measuring their system-sizedependence. The correlations extend to high orders in the quantum criticalregime and appear to form via a sparse network of many-body resonances thatspans the entire system. Our results unify the system's microscopic structurewith its macroscopic quantum critical behavior, and they provide an essentialstep towards understanding criticality and universality in non-equilibriumsystems.

Quantum critical behavior at the many-body-localization transition