Monitored fermions with conserved <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> charge

We study measurement-induced phases of free fermion systems with <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>U</a:mi><a:mo>(</a:mo><a:mn>1</a:mn><a:mo>)</a:mo></a:mrow></a:math> 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 (<b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>NL</b:mi><b:mi>σ</b:mi><b:mi mathvariant="normal">M</b:mi></b:mrow></b:math>) coupled to fluctuating hydrodynamics. Contrary to the case of noninteracting Majorana fermions, displaying no <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>U</d:mi><d:mo>(</d:mo><d:mn>1</d:mn><d:mo>)</d:mo></d:mrow></d:math> 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi>NL</e:mi><e:mi>σ</e:mi><e:mi mathvariant="normal">M</e:mi></e:mrow></e:math> and also discuss higher-dimensional generalizations. Published by the American Physical Society 2024