Theory of free fermions under random projective measurements

TL;DR

This paper develops an analytical field-theoretic approach to free fermions under random measurements, revealing a saturation of entanglement growth and absence of measurement-induced phase transition, supported by numerical analysis.

Contribution

It introduces a novel non-linear sigma model framework for analyzing free fermions with measurements, showing entanglement saturation and clarifying the nature of measurement effects.

Findings

Entanglement entropy saturates at finite value for rare measurements.

No measurement-induced entanglement phase transition occurs for free fermions.

Crossover scale between growth and saturation is exponentially large in measurement rate.

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, $\gamma / J \ll 1$ (where $\gamma$ 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 $U(2) / U(1) \times U(1) \simeq S_2$ sigma model with diffusive behavior, and the replica-asymmetric sector is a two-dimensional NLSM defined on $SU(R)$ manifold with the replica limit $R \to 1$. On the Gaussian level, valid in the limit $\gamma / J \to 0$, 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 $\sim (J / \gamma)^2$ 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, $\ln l_\text{corr} \sim J / \gamma$. This makes this crossover very sharp as a function of the measurement frequency $\gamma / J$, 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.