Field theory of monitored interacting fermion dynamics with charge conservation

Measurement-induced phase transitions (MIPTs) in monitored quantum dynamics are nonequilibrium phase transitions between quantum-chaotic (volume-law entangled) and entanglement-suppressed, area-law phases. Here we reveal how monitored dynamics are situated within the framework of general far-from-equilibrium, quantum condensed-matter physics. Measurement-induced heating effects scramble the distribution function in generic (interacting) monitored fermion systems, and this enables a simplified symmetry-based description of the dynamics. We demonstrate the equivalence of the Keldysh technique with the conventional Statistical-Mechanics Model for circuits, resulting from a doubled Hilbert-space (Choi-Jamio\l{}kowski) mapping. We illustrate this using the monitored dynamics of interacting fermions with a conserved charge, deriving a unified effective-field theory that captures all phases and phase transitions. The noninteracting counterpart in one-dimensional space only has an area-law phase, with no MIPT. This was explained via an effective nonlinear sigma model replica field theory possessing a very large symmetry. We show that other phases and phase transitions emerge when the replica symmetry is reduced by interactions. The reduced symmetry combines a replica permutation symmetry and charge-conservation within each replica. The former and its spontaneous breaking govern the MIPT, which can be recognized via a separatrix in the renormalization-group flow. The replica-resolved charge conservation dictates the ``charge-sharpening'' transition between two kinds of dynamics, where the global charge information is either hidden or reconstructible from the measurements. The field theory explains why the charge-sharpening transition should occur only in the volume-law phase. Our framework provides a template for other classes of MIPTs and situates these within the arena of nonequilibrium condensed-matter physics.