Absence of measurement- and unraveling-induced entanglement transitions in continuously monitored one-dimensional free fermions

TL;DR

This study investigates whether continuous monitoring induces true entanglement phase transitions in one-dimensional free fermions, finding no such transitions occur despite critical-like phenomena.

Contribution

The paper demonstrates that measurement schemes do not produce genuine entanglement phase transitions in monitored free fermionic chains, supported by theoretical analysis and numerical simulations.

Findings

Entanglement obeys an area law for most measurement schemes.

Critical-like behavior appears below a crossover scale growing algebraically with system parameters.

Numerical simulations confirm the absence of measurement-induced entanglement transitions.

Abstract

Continuous monitoring of one-dimensional free fermionic systems can generate phenomena reminiscent of quantum criticality, such as logarithmic entanglement growth, algebraic correlations, and emergent conformal invariance, but in a nonequilibrium setting. However, whether these signatures reflect a genuine phase of nonequilibrium quantum matter or persist only over finite length scales is an active area of research. We address this question in a free fermionic chain subject to continuous monitoring of lattice-site occupations. An unraveling phase $\varphi$ interpolates between measurement schemes, corresponding to different stochastic unravelings of the same Lindblad master equation: For $\varphi = 0$, measurements disentangle lattice sites, while for $\varphi = \pi/2$ they act as unitary random noise, yielding volume-law steady-state entanglement. Using replica Keldysh field theory, we obtain a nonlinear sigma model describing the long-wavelength physics. This analysis shows that for $0 \leq \varphi < \pi/2$, entanglement ultimately obeys an area law, but only beyond the exponentially large scale $\ln(l_{\varphi,*}) \sim J/[\gamma \cos(\varphi)]$, where $J$ is the hopping amplitude and $\gamma$ the measurement rate. Resolving $l_{\varphi, *}$ in numerical simulations is difficult for $\gamma/J \to 0$ or $\varphi \to \pi/2$. However, the theory also predicts that critical-like behavior appears below a crossover scale that grows only algebraically in $J/\gamma$, making it numerically accessible. Our simulations confirm these predictions, establishing the absence of measurement- or unraveling-induced entanglement transitions in this model.