Absence of measurement- and unraveling-induced entanglement transitions in continuously monitored one-dimensional free fermions

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 $\phi =\pi \mathrm{/}2$ 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 $0\le \phi <\pi \mathrm{/}2$, entanglement ultimately obeys an area law, but only beyond the exponentially large scale $\ln (l_{\phi ,\ast })\sim J\mathrm{/}[\gamma \cos (\phi )]$, where $J$ is the hopping amplitude and $\gamma$ the measurement rate. Resolving $l_{\phi ,\ast }$ in numerical simulations is difficult for $\gamma \mathrm{/}J\to 0$ or $\phi \to \pi \mathrm{/}2$. However, the theory also predicts that critical-like behavior appears below a crossover scale that grows only algebraically in $J\mathrm{/}\gamma$, making it numerically accessible. Our simulations confirm these predictions, establishing the absence of measurement- or unraveling-induced entanglement transitions in this model.