Entanglement behavior and localization properties in monitored fermion systems

We study the asymptotic bipartite entanglement in various integrable and nonintegrable models of monitored fermions. 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 be also captured by the same fit with a very small power-law exponent. We thus propose a characterization of the various entanglement phases using the fitting parameters. For the nonintegrable cases, as the staggered t-V and the Sachdev-Ye-Kitaev (SYK) models, the numerics prevents us from spanning different orders of magnitude in the size, therefore we fit the asymptotic entanglement versus the measurement strength and then look at the scaling with the size of the fitting parameters. We find two different behaviors: for the SYK we observe a volume-law growth, while for the t-V model some traces of an entanglement transition emerge. In the latter models, we study the localization properties in the Hilbert space through the inverse participation ratio, finding an anomalous delocalization with no relation with the entanglement properties. Finally, we show that our function fits very well the fermionic logarithmic negativity of a quadratic model in ladder geometry with stroboscopic projective measurements.