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 varphi interpolates between measurement schemes, corresponding to different stochastic unravelings of the same Lindblad master equation: For varphi = 0, measurements disentangle lattice sites, while for varphi = π/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 leq varphi < π/2, entanglement ultimately obeys an area law, but only beyond the exponentially large scale ln\(l_varphi,*\) sim J/\[γcos\(varphi\)\], where J is the hopping amplitude and γ the measurement rate. Resolving l_{varphi, *} in numerical simulations is difficult for γ/J → 0 or varphi → π/2. However, the theory also predicts that critical-like behavior appears below a crossover scale that grows only algebraically in J/γ, making it numerically accessible. Our simulations confirm these predictions, establishing the absence of measurement- or unraveling-induced entanglement transitions in this model.