  Phase transitions are driven by collective fluctuations of a system's
constituents that emerge at a critical point. This mechanism has been
extensively explored for classical and quantum systems in equilibrium, whose
critical behavior is described by a general theory of phase transitions.
Recently, however, fundamentally distinct phase transitions have been
discovered for out-of-equilibrium quantum systems, which can exhibit critical
behavior that defies this description and is not well understood. A
paradigmatic example is the many-body-localization (MBL) transition, which
marks the breakdown of quantum thermalization. Characterizing quantum critical
behavior in an MBL system requires the measurement of its entanglement
properties over space and time, which has proven experimentally challenging due
to stringent requirements on quantum state preparation and system isolation.
Here, we observe quantum critical behavior at the MBL transition in a
disordered Bose-Hubbard system and characterize its entanglement properties via
its quantum correlations. We observe strong correlations, whose emergence is
accompanied by the onset of anomalous diffusive transport throughout the
system, and verify their critical nature by measuring their system-size
dependence. The correlations extend to high orders in the quantum critical
regime and appear to form via a sparse network of many-body resonances that
spans the entire system. Our results unify the system's microscopic structure
with its macroscopic quantum critical behavior, and they provide an essential
step towards understanding criticality and universality in non-equilibrium
systems.
