  We analyze the quantum trajectory dynamics of free fermions subject to
continuous monitoring. For weak monitoring, we identify a novel dynamical
regime of subextensive entanglement growth, reminiscent of a critical phase
with an emergent conformal invariance. For strong monitoring, however, the
dynamics favors a transition into a quantum Zeno-like area-law regime. Close to
the critical point, we observe logarithmic finite size corrections, indicating
a Berezinskii-Kosterlitz-Thouless mechanism underlying the transition. This
uncovers an unconventional entanglement transition in an elementary, physically
realistic model for weak continuous measurements. In addition, we demonstrate
that the measurement aspect in the dynamics is crucial for whether or not a
phase transition takes place.
