  We study measurement-induced phases of free fermion systems with U(1)
symmetry. Following a recent approach developed for Majorana chains, we derive
a field theory description for the purity and bipartite entanglement at large
space and time scales. We focus on a multi-flavor one-dimensional chain with
random complex hoppings and continuous monitoring of the local fermion density.
By means of the replica trick, and using the number of flavors as a large
parameter controlling our approximations, we derive an effective field theory
made up of a SU(N) non-linear sigma model (NL$\sigma$M) coupled to fluctuating
hydrodynamics. Contrary to the case of non-interacting Majorana fermions,
displaying no U(1) symmetry, we find that the bipartite entanglement entropy
satisfies an area law for all monitoring rates, but with a nontrivial scaling
of entanglement when the correlation length is large. We provide numerical
evidence supporting our claims. We briefly show how imposing a reality
condition on the hoppings can change the NL$\sigma$M and also discuss higher
dimensional generalizations.
