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We analyze the quantum trajectory dynamics of free fermions subject tocontinuous monitoring. For weak monitoring, we identify a novel dynamicalregime of subextensive entanglement growth, reminiscent of a critical phasewith an emergent conformal invariance. For strong monitoring, however, thedynamics favors a transition into a quantum Zeno-like area-law regime. Close tothe critical point, we observe logarithmic finite size corrections, indicatinga Berezinskii-Kosterlitz-Thouless mechanism underlying the transition. Thisuncovers an unconventional entanglement transition in an elementary, physicallyrealistic model for weak continuous measurements. In addition, we demonstratethat the measurement aspect in the dynamics is crucial for whether or not aphase transition takes place.

Entanglement transition in a monitored free fermion chain -- from extended criticality to area law