Theory of free fermions under random projective measurements

Igor Poboiko; Paul Pöpperl; Igor V. Gornyi; Alexander D. Mirlin

doi:10.1103/PhysRevX.13.041046

paper

Theory of free fermions under random projective measurements

arXiv:2304.03138 · doi:10.1103/PhysRevX.13.041046

Abstract

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 non-linear sigma model (NLSM) as an effective field theory of the problem. Its replica-symmetric sector is described by a $U (2) / U (1) \times U (1) ≃ S_{2}$ sigma model with diffusive behavior, and the replica-asymmetric sector is a two-dimensional NLSM defined on $S U (R)$ manifold with the replica limit $R \to 1$ . On the Gaussian level, valid in the limit $γ / J \to 0$ , 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 $\sim (J / γ)^{2}$ 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, $ln l_{corr} \sim J / γ$ . 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.

24 pages, 5 figures

Theory of free fermions under random projective measurements

Abstract

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