  Monitored quantum circuits (MRCs) exhibit a measurement-induced phase
transition between area-law and volume-law entanglement scaling. MRCs with a
conserved charge additionally exhibit two distinct volume-law entangled phases
that cannot be characterized by equilibrium notions of symmetry-breaking or
topological order, but rather by the non-equilibrium dynamics and steady-state
distribution of charge fluctuations. These include a charge-fuzzy phase in
which charge information is rapidly scrambled leading to slowly decaying
spatial fluctuations of charge in the steady state, and a charge-sharp phase in
which measurements collapse quantum fluctuations of charge without destroying
the volume-law entanglement of neutral degrees of freedom. By taking a
continuous-time, weak-measurement limit, we construct a controlled replica
field theory description of these phases and their intervening
charge-sharpening transition in one spatial dimension. We find that the charge
fuzzy phase is a critical phase with continuously evolving critical exponents
that terminates in a modified Kosterlitz-Thouless transition to the short-range
correlated charge-sharp phase. We numerically corroborate these scaling
predictions also hold for discrete-time projective-measurement circuit models
using large-scale matrix-product state simulations, and discuss generalizations
to higher dimensions.
