Entanglement phases, localization and multifractality of monitored free fermions in two dimensions

We investigate the entanglement structure and wave function characteristics of continuously monitored free fermions with U$(1)$-symmetry in two spatial dimensions (2D). By deriving the exact fermion replica-quantum master equation, we line out two approaches: (i) a nonlinear sigma model analogous to disordered free fermions, resulting in an SU$(R)$-symmetric field theory of symmetry class AIII in (2+1) space-time dimensions, or (ii) for bipartite lattices, third quantization leading to a non-Hermitian SU$(2R)$-symmetric Hubbard model. Using exact numerical simulations, we explore the phenomenology of the entanglement transition in 2D monitored fermions, examining entanglement entropy and wave function inverse participation ratio. At weak monitoring, we observe characteristic $L\log L$ entanglement growth and multifractal dimension $D_q=2$, resembling a metallic Fermi liquid. Under strong monitoring, wave functions localize and the entanglement saturates towards an area law. Between these regimes, we identify a high-symmetry point exhibiting both entanglement growth indicative of emergent conformal invariance and maximal multifractal behavior. While this multifractal behavior aligns with the nonlinear sigma model of the Anderson transition, the emergent conformal invariance is an unexpected feature not typically associated with Anderson localization. These discoveries add a new dimension to the study of 2D monitored fermions and underscore the need to further explore the connection between non-unitary quantum dynamics in $D$ dimensions and quantum statistical mechanics in $D+1$ dimensions.