Measurement-induced transitions for interacting fermions

Effect of measurements on interacting fermionic systems with particle-number conservation, whose dynamics is governed by a time-independent Hamiltonian, is studied. We develop Keldysh field-theoretical framework that provides a unified approach to observables characterizing entanglement and charge fluctuations. Within this framework, we derive a replicated Keldysh non-linear sigma model (NLSM), which incorporates boundary conditions specifically designed to produce generating functions for charge cumulants and entanglement entropies directly in the NLSM language. By using the renormalization-group approach for the NLSM, we determine the phase diagram and the scaling of physical observables. Crucially, the interaction-induced terms in the NLSM action reduce its symmetry, which affects the physics of the problem in a dramatic way. First, this leads to the "information-charge separation": charge cumulants get decoupled from entanglement entropies. Second, the interaction stabilizes the volume-law phase for the entanglement. Third, for spatial dimensionality d=1, the interaction stabilizes the phase with logarithmic growth of charge cumulants (in the thermodynamic limit). Thus, in the presence of interaction, there are measurement transitions in any d, at variance with free fermions, for which a d=1 system is always in the area-law phase. Analytical results are supported by numerical simulations using time-dependent variational principles for matrix product states, which, in particular, confirm the separation of information and charge as a hallmark of the delocalized phase.