Continuous monitoring of one-dimensional free fermionic systems can generate phenomena reminiscent of quantum criticality, such as logarithmic entanglement growth, algebraic correlations, and emergent conformal invariance, but in a nonequilibrium setting. However, whether these signatures reflect a genuine phase of nonequilibrium quantum matter or persist only over finite length scales is an active area of research. We address this question in a free fermionic chain subject to continuous monitoring of lattice-site occupations. An unraveling phase $\phi$ interpolates between measurement schemes, corresponding to different stochastic unravelings of the same Lindblad master equation; for $\phi =0$, measurements disentangle lattice sites, while for $\phi =\pi /2$ they act as unitary random noise, yielding volume-law steady-state entanglement. Using replica Keldysh field theory, we obtain a nonlinear sigma model describing the long-wavelength physics. This analysis shows that for $0\le \phi <\pi /2$, entanglement ultimately obeys an area law, but only beyond the exponentially large scale $ln\left(l_{\phi ,*}\right)\sim J/[\gamma cos\left(\phi \right)]$, where $J$ is the hopping amplitude and $\gamma$ the measurement rate. Resolving $l_{\phi ,*}$ in numerical simulations is difficult for $\gamma /J\to 0$ or $\phi \to \pi /2$. However, the theory also predicts that critical-like behavior appears below a crossover scale that grows only algebraically in $J/\gamma$, making it numerically accessible. Our simulations confirm these predictions, establishing the absence of measurement- or unraveling-induced entanglement transitions in this model.

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• Critical phenomena
• Entanglement entropy
• Fermions
• Nonlinear sigma model
• Open quantum systems
• Quantum measurements
• Quantum stochastic processes
• Lindblad equation