Entanglement behavior in fermionic chains subject to a finite-density local monitoring

We study the behavior of the entanglement entropy in two fermion models on a one dimensional lattice. It is known that in these systems the competition between unitary dynamics, which generate entanglement, and measurements, which destroy it, gives rise to a phase transition. We present evidence of this phase transition in the case where not every site of the chain is measured, in fact we consider a new parameter, the measurement density, which keeps track of the fraction of measured sites over the total number of sites. We show how by varying this parameter, together with the measurement rate, one can witness the different phases of the entanglement and the transitions between them. We conclude our work by characterizing the phase diagram of the models in the parameters space.