Measurement-Protected Order in Monitored Quantum Circuits with Continuous Symmetry*

Monitored quantum circuits, composed of local unitary operators and projective measurements, have emerged as a rich setting for studying non-equilibrium quantum dynamics. In these systems, measurements can protect various monitored steady state phases with area-law entanglement, including phases which can host measurement-protected Ising spin-glass order. To begin exploring whether this measurement-protected order is a generic phenomenon or is reliant on the discrete Ising symmetry, we introduce a circuit model with continuous symmetry where unitary feedback is used to generate local order. In one dimension, we find that long-range order arises in the steady state of our model but that this order is fragile to symmetry-respecting perturbations. Additionally, we find that the continuous symmetry leads to diffusive dynamics that are otherwise not generally present.