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Monitored quantum circuits (MRCs) exhibit a measurement-induced phasetransition between area-law and volume-law entanglement scaling. MRCs with aconserved charge additionally exhibit two distinct volume-law entangled phasesthat cannot be characterized by equilibrium notions of symmetry-breaking ortopological order, but rather by the non-equilibrium dynamics and steady-statedistribution of charge fluctuations. These include a charge-fuzzy phase inwhich charge information is rapidly scrambled leading to slowly decayingspatial fluctuations of charge in the steady state, and a charge-sharp phase inwhich measurements collapse quantum fluctuations of charge without destroyingthe volume-law entanglement of neutral degrees of freedom. By taking acontinuous-time, weak-measurement limit, we construct a controlled replicafield theory description of these phases and their interveningcharge-sharpening transition in one spatial dimension. We find that the chargefuzzy phase is a critical phase with continuously evolving critical exponentsthat terminates in a modified Kosterlitz-Thouless transition to the short-rangecorrelated charge-sharp phase. We numerically corroborate these scalingpredictions also hold for discrete-time projective-measurement circuit modelsusing large-scale matrix-product state simulations, and discuss generalizationsto higher dimensions.

Field theory of charge sharpening in symmetric monitored quantum circuits