Does a critical phase exist for monitored free fermions in one dimension
In plain words
For particles that do not interact, some simulations show a whole range of measurement rates with slowly growing entanglement, while a field theory predicts that this range disappears in very large systems. The two pictures disagree.
Precise statement
For 1D free fermions with particle-number conservation under random projective (or continuous weak) measurements of local occupation numbers at rate $\gamma$, decide whether the half-chain entanglement $S(L) \sim c(\gamma) \log L$ persists as $L$ goes to $\infty$ for $\gamma$ below some $\gamma_{c}$, or whether $S$ saturates (area law) for all $\gamma > 0$ beyond a length $\ell(\gamma) \sim \exp(C/\gamma)$. The nonlinear sigma model analysis of Poboiko, Popperl, Gornyi and Mirlin (Phys. Rev. X 13, 041046, 2023) predicts the latter.
What would settle it
Numerics at system sizes beyond the predicted crossover length, or a rigorous asymptotic analysis of the monitored dynamics.