Monitored fermions with conserved U(1) charge

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 timescales. We focus on a multiflavor 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) nonlinear sigma model (NLsM) coupled to fluctuating hydrodynamics. Contrary to the case of noninteracting 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 NLsM and also discuss higher-dimensional generalizations.