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

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.

23 pages, 3 figures; v3: minor revision

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• Anderson Transitions
• Mapping out of equilibrium into equilibrium in one-dimensional transport models
• Measurement-induced phase transition for free fermions above one dimension
• Transitions in the learnability of global charges from local measurements
• Measurement-induced phase transition in a single-body tight-binding model
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• Absence of measurement- and unraveling-induced entanglement transitions in continuously monitored one-dimensional free fermions
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• Replica Keldysh field theory of quantum-jump processes: General formalism and application to imbalanced and inefficient fermion counting
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