Monitored fermions with conserved $\mathrm{U}(1)$ charge

TL;DR

This paper develops a field theory description for measurement-induced entanglement phases in U(1) symmetric free fermion systems, revealing an area law for entanglement entropy across monitoring rates.

Contribution

It introduces an effective field theory combining a non-linear sigma model and hydrodynamics for U(1) symmetric fermions, extending previous approaches to systems with conserved charge.

Findings

Entanglement entropy follows an area law at all monitoring rates.

Large correlation length leads to nontrivial entanglement scaling.

Numerical evidence supports the theoretical predictions.

Abstract

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.