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

TL;DR

This work analyzes a 1D lattice of free fermions under continuous occupation measurements, introducing an unraveling phase $\varphi$ that interpolates between conventional quantum-state diffusion ($\varphi=0$) and unitary noise ($\varphi=\pi/2$). Using a replica Keldysh field theory and a nonlinear sigma model, the authors derive the long-wavelength physics and identify a crossover scale $l_{\varphi,*}$ beyond which entanglement obeys an area law, with $l_{\varphi,*}=l_0\exp[8\pi\beta g_{\varphi,0}]$ and $g_{\varphi,0}=\frac{l_0\rho_0(1-\rho_0)}{\cos(\varphi)}$, while for $0\le\varphi<\pi/2$ critical-like behavior appears only up to an algebraically growing scale in $J/\gamma$. Numerics validate the field-theory predictions, showing universal weak-localization corrections consistent with symmetry classes AIII ($\varphi\neq0$) and BDI ($\varphi=0$), and confirming the absence of a genuine entanglement transition, with $l_c\sim\gamma^{-2}$ and $l_m\sim\gamma^{-3/2}$ modulated by nonuniversal short-scale structure. The unitary unraveling at $\varphi=\pi/2$ yields volume-law entanglement and random Gaussian-state behavior. Overall, the KT-like transitions observed in some related models are best understood as finite-size crossovers rather than true nonequilibrium phases.” wrapped in $...$ where appropriate.

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 $\varphi = π/2$ 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 $0 \leq \varphi < π/2$, entanglement ultimately obeys an area law, but only beyond the exponentially large scale $\ln(l_{\varphi,*}) \sim J/[γ\cos(\varphi)]$, where $J$ is the hopping amplitude and $γ$ the measurement rate. Resolving $l_{\varphi, *}$ in numerical simulations is difficult for $γ/J \to 0$ or $\varphi \to π/2$. However, the theory also predicts that critical-like behavior appears below a crossover scale that grows only algebraically in $J/γ$, making it numerically accessible. Our simulations confirm these predictions, establishing the absence of measurement- or unraveling-induced entanglement transitions in this model.

Figures (5)

• Figure 1: (a--c) Rescaled density correlation function \ref{['eq:connected-density-correlation-function']} in momentum space and (d--f) weak-localization correction \ref{['eq:weak-localization-correction']} for (a, d) $\varphi = 0$, (b, e) $\varphi = \pi/4$, and (c, f) $\varphi = 5 \pi/12$. (a--c) The Gaussian correlation function (cyan dash-dotted line: full numerical solution \ref{['eq:C-l-Gaussian-full-numerical']}; black dashed line: bulk approximation \ref{['eq:C-q-Gaussian-bulk']}) decreases monotonically starting from $C_q/(g_{\varphi, 0} q) \to 1$ for $\tilde{q} l_0 \to 0$. In contrast, the numerical data attain a maximum at $q_c$ and decrease for $\tilde{q} l_0 \to 0$. Inset in (a--c): The position of the maximum scales as $q_c \sim \gamma^2$. (d--f) The weak-localization correction agrees well with the one-loop RG result \ref{['eq:weak-localization-correction']}. In particular, the analytically predicted difference in the slope by a factor of two between symmetry classes BDI for $\varphi = 0$ and AIII for $\varphi \neq 0$ is borne out by the data. Note that the analytical result \ref{['eq:weak-localization-correction']} is shifted for better visual comparison with the numerical data. Such a nonuniversal shift corresponds to a different value of the UV cutoff in the RG flow \ref{['eq:g-phi-RG']} and does not affect the universal slope of the weak-localization correction.
• Figure 2: Rescaled density correlation function \ref{['eq:connected-density-correlation-function']} for (a) $\varphi = 0$, (b) $\varphi = \pi/4$, and (c) $\varphi = 5 \pi/12$. On short scales $\tilde{l} \lesssim l_0$, the numerical data agree well with the Gaussian result (black dashed line: bulk approximation \ref{['eq:C-q-Gaussian-bulk']}). Significant deviations occur for $\tilde{l} \gg l_0$, where the Gaussian result decays as a power law, $\left\lvert C_l \right\rvert \sim \tilde{l}^{-2}$, while the numerical data exhibit faster decay. Inset: The crossover scale at which deviations from the Gaussian result become significant scales as $l_c \sim \gamma^{-2}$. For $\varphi = 5 \pi/12$, this scaling changes to $l_c \sim \gamma^{-2/3}$ at small values of $\gamma/J$.
• Figure 3: (a--c) Rescaled entanglement entropy and (d--f) rescaled scale-dependent effective central charge for (a, d) $\varphi = 0$, (b, e) $\varphi = \pi/4$, and (c, g) $\varphi = 5 \pi/12$. (a--c) For small values of $\gamma/J$, volume-law scaling of the entanglement entropy (blue dashed line) on short scales $\tilde{\ell} \lesssim l_0$ crosses over to apparently logarithmic growth on large scales. At larger values of $\gamma/J$, the data exhibit area-law scaling. (d--f) Unlike the Gaussian result (black dashed line), the numerical data for the effective central charge do not saturate to a finite value as $\tilde{\ell}/l_0 \to \infty$. This implies that the growth of the entanglement entropy is, in fact, not logarithmic on large scales. Inset: The effective central charge attains its maximum at $l_m \sim \gamma^{-3/2}$. For $\varphi = 5 \pi/12$, the scaling is modified to $l_m \sim \gamma^{-1}$ at small values of $\gamma/J$.
• Figure 4: (a) Rescaled entanglement entropy and (b) rescaled effective central charge for fixed $\gamma/J = 0.7$ and varying values of $\varphi$. The effect of increasing $\varphi$ in the range from 0 to $\pi/2$ is qualitatively similar to that of decreasing $\gamma/J$: In the behavior of the entanglement entropy on large scales, there is an apparent transition from area-law scaling to logarithmic growth. Concomitantly, the position of the maximum of the effective central charge shifts to the right.
• Figure 5: Entanglement entropy density for $\varphi = \pi/2$. For any value of $\gamma/J$, the data are consistent with Eq. \ref{['eq:S-rand']} for random Gaussian states (black dashed line). We observe deviations for large subsystem sizes $\ell$, which are caused by our simulations not having fully reached the stationary regime. For small $\ell$, the data exhibit volume-law scaling (blue dashed line), as also found for $\varphi \neq \pi/2$ in Fig. \ref{['fig:entanglement-entropy-central-charge']}(a--c).