Theory of Free Fermions under Random Projective Measurements

We develop an analytical approach to the study of one-dimensional free fermions subject to random projective measurements of local site occupation numbers, based on the Keldysh path-integral formalism and replica trick. In the limit of rare measurements, γ/J≪1 (where γ is measurement rate per site and J is hopping constant in the tight-binding model), we derive a nonlinear sigma model (NLSM) as an effective field theory of the problem. Its replica-symmetric sector is described by a U (2) /U (1) ×U (1) ≃S^2 sigma model with diffusive behavior, and the replica-asymmetric sector is a two-dimensional NLSM defined on SU (R) manifold with the replica limit R→1. On the Gaussian level, valid in the limit γ/J→0, this model predicts a logarithmic behavior for the second cumulant of number of particles in a subsystem and for the entanglement entropy. However, the one-loop renormalization group analysis allows us to demonstrate that this logarithmic growth saturates at a finite value ∼ (J/γ) ^2 even for rare measurements, which corresponds to the area-law phase. This implies the absence of a measurement-induced entanglement phase transition for free fermions. The crossover between logarithmic growth and saturation, however, happens at exponentially large scale, lnl₂₎ₑₑ∼J/γ. This makes this crossover very sharp as a function of the measurement frequency γ/J, which can be easily confused with a transition from the logarithmic to area law in finite-size numerical calculations. We have performed a careful numerical analysis, which supports our analytical predictions.