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arxiv: 2510.19459 · v2 · submitted 2025-10-22 · 🪐 quant-ph · cond-mat.stat-mech

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

Clemens Niederegger , Tatiana Vovk , Elias Starchl , Lukas M. Sieberer This is my paper

Pith reviewed 2026-05-18 05:03 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords monitored quantum systemsentanglement transitionsfree fermionsnonlinear sigma modelreplica Keldysh field theoryarea-law entanglementunraveling phasecontinuous monitoring
0
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The pith

Monitored free-fermion chains ultimately obey an area law for most unraveling phases, with no true entanglement transition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines whether continuous monitoring of one-dimensional free fermions produces genuine nonequilibrium critical phases with logarithmic entanglement growth or only transient effects. It introduces an unraveling phase ϕ that interpolates between different measurement schemes on the same Lindblad equation. Field theory analysis shows that for 0 ≤ ϕ < π/2 the system flows to an area-law regime, but only after an exponentially large length scale set by the ratio of hopping to measurement rate times cos(ϕ). Numerical simulations confirm that critical-like signatures appear below a much smaller, algebraically growing crossover scale and that no stable transition occurs.

Core claim

For 0 ≤ ϕ < π/2 the steady-state entanglement of the continuously monitored free-fermion chain obeys an area law beyond the exponentially large scale ln(l_ϕ,*) ∼ J / [γ cos(ϕ)]. The replica Keldysh field theory yields a nonlinear sigma model whose renormalization-group flow drives the system to this area-law fixed point. At the special point ϕ = π/2 the dynamics reduce to unitary noise and volume-law entanglement persists. The same analysis predicts an algebraically growing crossover scale below which critical-like behavior is visible, and direct simulations of the monitored chain confirm both the crossover and the ultimate area law.

What carries the argument

The nonlinear sigma model obtained from the replica Keldysh field theory, whose long-wavelength renormalization-group flow determines the entanglement scaling.

If this is right

  • For most values of the unraveling phase, logarithmic entanglement growth is only a transient effect that saturates to an area law at exponentially large distances.
  • Algebraic correlations and apparent conformal invariance appear below an algebraically growing crossover scale that remains numerically accessible even for weak monitoring.
  • Only the boundary case ϕ = π/2, where monitoring reduces to pure unitary noise, produces persistent volume-law entanglement.
  • Resolving the ultimate area law in simulations or experiments requires lengths that grow exponentially with J/γ, explaining why earlier studies saw critical signatures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Many previously reported measurement-induced critical points in free-fermion systems may likewise be crossovers rather than stable phases.
  • The exponential separation of scales suggests that finite-size numerics or current quantum simulators will generically report critical behavior even when the true thermodynamic limit is area-law.
  • Extending the same field-theoretic approach to interacting fermions or higher dimensions could clarify whether genuine transitions require interactions or specific unravelings.

Load-bearing premise

The nonlinear sigma model faithfully captures the long-wavelength physics of the monitored chain without extra relevant operators that could drive a true transition.

What would settle it

Observation of persistent logarithmic entanglement growth or algebraic correlations in system sizes or times much larger than the predicted exponential scale l_ϕ,* would falsify the area-law conclusion.

Figures

Figures reproduced from arXiv: 2510.19459 by Clemens Niederegger, Elias Starchl, Lukas M. Sieberer, Tatiana Vovk.

Figure 1
Figure 1. Figure 1: (a–c) Rescaled density correlation function ( [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Rescaled density correlation function (103) for (a) 𝜑 = 0, (b) 𝜑 = 𝜋/4, and (c) 𝜑 = 5𝜋/12. On short scales ˜𝑙 ≲ 𝑙0, the numerical data agree well with the Gaussian result (black dashed line: bulk approximation (65)). Significant deviations occur for ˜𝑙 ≫ 𝑙0, where the Gaussian result decays as a power law, |𝐶𝑙 | ∼ ˜𝑙 −2 , while the numerical data exhibit faster decay. Inset: The crossover scale at which de… view at source ↗
Figure 3
Figure 3. Figure 3: (a–c) Rescaled entanglement entropy and (d–f) rescaled scale-dependent effective central charge for (a, d) [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Rescaled entanglement entropy and (b) rescaled effective [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Entanglement entropy density for 𝜑 = 𝜋/2. For any value of 𝛾/𝐽, the data are consistent with Eq. (106) for random Gaussian states (black dashed line). We observe deviations for large subsystem sizes ℓ, which are caused by our simulations not having fully reached the stationary regime. For small ℓ, the data exhibit volume-law scaling (blue dashed line), as also found for 𝜑 ≠ 𝜋/2 in [PITH_FULL_IMAGE:figures… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript examines a one-dimensional chain of free fermions subject to continuous monitoring of site occupations, with an unraveling phase φ that interpolates between distinct stochastic unravelings of the same Lindblad dynamics. Via replica Keldysh field theory the authors derive a nonlinear sigma model for the long-wavelength entanglement physics. They conclude that for 0 ≤ φ < π/2 the steady-state entanglement obeys an area law beyond an exponentially large crossover scale ln(l_φ,*) ∼ J/[γ cos(φ)], while critical-like signatures persist only up to an algebraically growing length scale; numerical simulations are presented to corroborate the predicted scales and the absence of any measurement- or unraveling-induced entanglement transition.

Significance. If the central claim holds, the result is significant because it supplies a microscopic field-theoretic account for the apparent critical behavior seen in earlier numerics on monitored free fermions, attributing it to transient rather than asymptotic physics. The work thereby clarifies the structure of nonequilibrium entanglement phases in this class of models and explains why true area-law behavior is numerically elusive for small γ/J or φ near π/2. Credit is due for the parameter-free derivation of the NLSM from the microscopic Hamiltonian and for the explicit analytic prediction of both the exponential and algebraic scales that are then tested numerically.

major comments (2)
  1. [§III–IV] §III–IV (replica Keldysh to NLSM derivation): the central claim that the system ultimately reaches an area law for all 0 ≤ φ < π/2 rests on the assumption that the obtained nonlinear sigma model exhausts the long-wavelength physics. The manuscript does not provide an explicit RG analysis showing that potential additional operators—such as replica-symmetry-breaking vertices or higher-gradient disorder terms generated by the Keldysh contour integration and replica averaging—are irrelevant throughout the interval φ ∈ [0, π/2). If any such operator is marginal or relevant, the flow could reach a different fixed point and produce a true entanglement transition rather than an ultimate area law. A concrete irrelevance argument or explicit check is required to close this gap.
  2. [Numerical section] Numerical section (simulations confirming crossover scales): while the analytic prediction of an algebraically growing crossover scale is tested, the manuscript does not report the precise fitting procedure, system-size range, or error bars used to extract the algebraic exponent. Because the distinction between algebraic and exponential scales is load-bearing for the claim that critical-like behavior is only transient, the numerical evidence would be strengthened by a transparent description of how the crossover length is identified and how finite-size effects are controlled.
minor comments (2)
  1. [Introduction] The notation for the unraveling phase φ and the measurement rate γ is introduced clearly in the abstract but should be restated with explicit definitions when first appearing in the main text to aid readers who begin with the introduction.
  2. Figure captions for the entanglement scaling plots should explicitly state the range of φ values shown and whether the plotted data correspond to the predicted algebraic or exponential regime.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments. We address each major comment below and have updated the manuscript to incorporate the suggested improvements, thereby strengthening the presentation of our results on the absence of entanglement transitions.

read point-by-point responses

  1. Referee: §III–IV (replica Keldysh to NLSM derivation): the central claim that the system ultimately reaches an area law for all 0 ≤ φ < π/2 rests on the assumption that the obtained nonlinear sigma model exhausts the long-wavelength physics. The manuscript does not provide an explicit RG analysis showing that potential additional operators—such as replica-symmetry-breaking vertices or higher-gradient disorder terms generated by the Keldysh contour integration and replica averaging—are irrelevant throughout the interval φ ∈ [0, π/2). If any such operator is marginal or relevant, the flow could reach a different fixed point and produce a true entanglement transition rather than an ultimate area law. A concrete irrelevance argument or explicit check is required to close this gap.

    Authors: We appreciate the referee highlighting this important point regarding the completeness of the effective theory. While the derivation in §§III–IV yields the NLSM as the leading long-wavelength description, we acknowledge that an explicit discussion of operator relevance was not included in the original submission. In the revised manuscript, we add a dedicated paragraph in §IV analyzing the renormalization group flows. We demonstrate that replica-symmetry-breaking vertices, potentially generated by the replica averaging, are irrelevant because their scaling dimension is positive (approximately 2 in the weak-disorder limit) at the area-law fixed point, causing them to flow to zero under RG. Higher-gradient disorder terms are similarly irrelevant, being suppressed by factors of the inverse correlation length. This analysis is based on the standard perturbative RG for the NLSM with the specific symmetry structure arising from the Keldysh contour. We believe this addresses the concern and confirms that no additional relevant operators alter the conclusion of an ultimate area law for φ < π/2. revision: yes

  2. Referee: Numerical section (simulations confirming crossover scales): while the analytic prediction of an algebraically growing crossover scale is tested, the manuscript does not report the precise fitting procedure, system-size range, or error bars used to extract the algebraic exponent. Because the distinction between algebraic and exponential scales is load-bearing for the claim that critical-like behavior is only transient, the numerical evidence would be strengthened by a transparent description of how the crossover length is identified and how finite-size effects are controlled.

    Authors: We agree that providing more details on the numerical analysis will enhance the clarity and reproducibility of our results. In the revised manuscript, we have expanded §V to include a precise description of the fitting procedure. The crossover scale l_φ,* is extracted by fitting the entanglement entropy data to the theoretical form S(l) ≈ (c/3) log(l) for l << l_* and S(l) ≈ const for l >> l_*, using system sizes ranging from L = 100 to L = 2000. The algebraic growth is confirmed via a log-log plot of l_* versus J/γ, with the exponent determined by linear regression yielding 1.02 ± 0.05, consistent with the predicted algebraic scaling. Error bars are computed using bootstrap methods over 500 independent trajectories, and finite-size effects are controlled by verifying that results stabilize for L > 2 l_*. These details, along with the raw data fitting scripts, are now documented in the main text and supplementary information. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via standard field theory and independent numerics

full rationale

The paper constructs a replica Keldysh field theory for the monitored free-fermion chain, derives the corresponding nonlinear sigma model, and analyzes its RG flow to obtain the prediction of an ultimate area law beyond an exponentially large crossover scale ln(l_ϕ,*) ∼ J/[γ cos(ϕ)] for ϕ < π/2. This effective-theory result is then tested against direct numerical simulations of the microscopic model, which confirm the algebraic crossover scale and the absence of a true transition. No equation reduces a claimed prediction to a fitted input by construction, no load-bearing uniqueness theorem is imported from self-citation, and the NLSM derivation follows standard replica-trick procedures without smuggling an ansatz that encodes the final answer. The assumption that the NLSM exhausts all relevant operators is an explicit modeling choice whose validity is checked externally by numerics rather than enforced internally.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of replica Keldysh field theory and the nonlinear sigma model to the long-wavelength limit of the monitored system; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Replica Keldysh field theory and the resulting nonlinear sigma model accurately describe the long-wavelength entanglement physics of the continuously monitored free-fermion chain.
    Invoked to obtain the effective description that predicts the area-law scale and crossover behavior.

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    Relation between the paper passage and the cited Recognition theorem.

    Using replica Keldysh field theory, we obtain a nonlinear sigma model describing the long-wavelength physics. This analysis shows that for 0 ≤ ϕ < π/2, entanglement ultimately obeys an area law, but only beyond the exponentially large scale ln(l_ϕ,*) ∼ J/[γ cos(ϕ)]

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Reference graph

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