Entanglement behavior and localization properties in monitored fermion systems

We study the stationary bipartite entanglement in various integrable and nonintegrable models of monitored fermions evolving along quantum trajectories. We find that, for the integrable cases, the entanglement versus the system size is well fitted, over more than one order of magnitude, by a function interpolating between a linear and a power-law behavior. Up to the sizes we are able to reach, a logarithmic growth of the entanglement can also be captured by the same fit with a very small power-law exponent. For the nonintegrable cases, such as the staggered 𝑡−𝑉 and the Sachdev-Ye-Kitaev (SYK) models, the numerical limitations prevent us from spanning different orders of magnitude in the system size. Here we fit the asymptotic entanglement versus the measurement strength with a generalized Lorentzian, finding a very good agreement, and then look at the scaling with the size of the fitting parameters. We find two different behaviors: for the SYK we observe a linear increase with the system size, while for the 𝑡−𝑉 model we see the emergence of traces of an entanglement crossover. In the latter models, we study the localization properties in the Hilbert space through the inverse participation ratio, finding an anomalous-delocalization behavior with no relation with the entanglement properties. Finally, we show that our function also fits very well the system-size dependence of the fermionic logarithmic negativity of a quadratic model in a two-leg ladder geometry, with stroboscopic projective measurements.