  We show that weak measurements can induce a quantum phase transition of
interacting many-body systems from an ergodic thermal phase with a large
entropy to a nonergodic localized phase with a small entropy, but only if the
measurement strength exceeds a critical value. We demonstrate this effect for a
one-dimensional quantum circuit evolving under random unitary transformations
and generic positive operator-valued measurements of variable strength. As
opposed to projective measurements describing a restricted class of open
systems, the measuring device is modeled as a continuous Gaussian probe,
capturing a large class of environments. By employing data collapse and
studying the enhanced fluctuations at the transition, we obtain a consistent
phase boundary in the space of the measurement strength and the measurement
probability, clearly demonstrating a critical value of the measurement strength
below which the system is always ergodic, irrespective of the measurement
probability. These findings provide guidance for quantum engineering of
many-body systems by controlling their environment.
