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arxiv: 2502.09401 · v2 · pith:OVZQOQBGnew · submitted 2025-02-13 · 🪐 quant-ph · cond-mat.other

Entanglement behavior and localization properties in monitored fermion systems

Giulia Piccitto , Giuliano Chiriac\`o , Davide Rossini , Angelo Russomanno This is my paper

Pith reviewed 2026-05-23 03:16 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.other
keywords monitored fermionsentanglement entropyintegrable modelsmeasurement-induced phasesHilbert space localizationinverse participation ratioSachdev-Ye-Kitaev model
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The pith

An interpolating fit between linear and power-law growth characterizes entanglement phases in monitored integrable fermions.

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

The paper studies asymptotic bipartite entanglement in monitored fermion systems, both integrable and nonintegrable. For integrable models it reports that entanglement versus system size is described over more than an order of magnitude by a single fitting function that blends linear and power-law forms. Logarithmic growth is also captured by the same function when the power-law exponent is very small. The authors propose that the fit parameters themselves can be used to label distinct entanglement phases. In nonintegrable cases the analysis shifts to scaling with measurement strength, revealing volume-law growth for the SYK model and possible transition signatures for the staggered t-V model, while inverse-participation-ratio measurements show anomalous delocalization unrelated to the entanglement scaling.

Core claim

In integrable monitored fermion models the bipartite entanglement versus system size is well fitted over more than one order of magnitude by a function that interpolates between linear and power-law behavior; the same fit also accommodates logarithmic growth with a very small power-law exponent, so that entanglement phases can be characterized directly by the resulting fitting parameters.

What carries the argument

The interpolating fitting function between linear and power-law forms applied to entanglement entropy versus system size.

If this is right

  • Entanglement phases in integrable monitored fermions are labeled by the parameters of the interpolating fit rather than by separate scaling regimes.
  • In the SYK model the asymptotic entanglement follows a volume law when scaled against measurement strength.
  • In the staggered t-V model traces of an entanglement transition appear when the same scaling is performed.
  • Hilbert-space localization measured by inverse participation ratio shows anomalous delocalization with no direct link to the observed entanglement scaling.
  • The same interpolating function also describes fermionic logarithmic negativity in a quadratic ladder model with stroboscopic measurements.

Where Pith is reading between the lines

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

  • The fitting procedure could be tested on other classes of monitored systems to see whether the same parameter-based classification emerges.
  • The reported independence of localization and entanglement scaling suggests they may be controlled by distinct mechanisms that can be varied separately.
  • If the fit remains accurate at larger sizes, it supplies a practical diagnostic that avoids needing to resolve tiny logarithmic corrections directly.

Load-bearing premise

That the chosen interpolating function continues to describe the true asymptotic large-size behavior rather than serving only as an effective description inside the numerically accessible range.

What would settle it

A direct computation of entanglement on system sizes several times larger than those already studied that systematically deviates from the proposed interpolating function.

Figures

Figures reproduced from arXiv: 2502.09401 by Angelo Russomanno, Davide Rossini, Giuliano Chiriac\`o, Giulia Piccitto.

Figure 1
Figure 1. Figure 1: FIG. 1. The asymptotic averaged EE for the model in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The EE for the model in Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) we show numerical data for different power-law exponents α (circles) and the corresponding fit (lines), that nicely reproduces all the curves. In [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: displays our numerical results for the asymp￾totic averaged EE SL/2 versus the measurement strength γ (circles) and the corresponding fit obtained with Eq. (21) (continuous lines). We can see that the latter performs well over a range of γ ∈ [8×10−3 , 4] correspond￾ing to more than two orders of magnitude. B. SYK model with onsite dephasing The SYK Hamiltonian is a fermionic long-range in￾teracting lattice… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The EE for the model in Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Parameters obtained from the fit with Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The behavior of [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Sketch of the noninteracting fermionic ladder model [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The averaged logarithm of the IPR versus the loga [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. (a), (b), (c), (d): The FLN [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Top panels: The exponent [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The behavior of [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
read the original abstract

We study the asymptotic bipartite entanglement in various integrable and nonintegrable models of monitored fermions. We find that, for the integrable cases, the entanglement versus the system size is well fitted, over more than one order of magnitude, by a function interpolating between a linear and a power-law behavior. Up to the sizes we are able to reach, a logarithmic growth of the entanglement can be also captured by the same fit with a very small power-law exponent. We thus propose a characterization of the various entanglement phases using the fitting parameters. For the nonintegrable cases, as the staggered t-V and the Sachdev-Ye-Kitaev (SYK) models, the numerics prevents us from spanning different orders of magnitude in the size, therefore we fit the asymptotic entanglement versus the measurement strength and then look at the scaling with the size of the fitting parameters. We find two different behaviors: for the SYK we observe a volume-law growth, while for the t-V model some traces of an entanglement transition emerge. In the latter models, we study the localization properties in the Hilbert space through the inverse participation ratio, finding an anomalous delocalization with no relation with the entanglement properties. Finally, we show that our function fits very well the fermionic logarithmic negativity of a quadratic model in ladder geometry with stroboscopic projective measurements.

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 numerically studies asymptotic bipartite entanglement in monitored integrable and non-integrable fermion models. For integrable cases it reports that entanglement versus system size is well fitted over more than one decade by an interpolating function between linear and power-law forms; the same fit can capture logarithmic growth with a small power-law exponent, and the authors propose to characterize entanglement phases via the extracted parameters. For non-integrable models (staggered t-V and SYK) the accessible size range is smaller, so they fit entanglement versus measurement strength and examine the size scaling of the fit parameters, finding volume-law behavior in SYK and possible traces of an entanglement transition in t-V. They additionally compute the inverse participation ratio in Hilbert space, report anomalous delocalization unrelated to the entanglement scaling, and show that the same interpolating function fits the fermionic logarithmic negativity of a quadratic ladder model under stroboscopic projective measurements.

Significance. If the proposed interpolating fit remains stable and physically meaningful beyond the numerically accessible window, the work supplies a concrete, practical protocol for classifying entanglement phases in monitored integrable systems from finite-size data. The reported decoupling between Hilbert-space localization (IPR) and entanglement scaling, together with the extension to logarithmic negativity, would be of interest to the monitored-dynamics community. The absence of an analytic derivation of the interpolant and the lack of systematic validation against larger sizes or alternative functional forms limit the strength of the phase-characterization claim.

major comments (2)
  1. [Abstract, §3] Abstract and §3 (integrable models): the central proposal to characterize entanglement phases via the parameters of the interpolating fit is load-bearing on the assumption that the functional form and extracted exponents remain stable for L→∞. The text acknowledges the limited size window but provides no extrapolation tests, no AIC/BIC comparison against pure logarithmic or other candidate forms, and no derivation of the interpolant from the monitored dynamics; if the true scaling is logarithmic with a slow crossover, the reported parameters would be effective descriptors only inside the accessible window.
  2. [§4] §4 (non-integrable models): the claim of 'traces of an entanglement transition' in the t-V model rests on the size scaling of fit parameters extracted from entanglement versus measurement strength. The accessible size range is stated to be too small to span orders of magnitude, yet no quantitative assessment of finite-size drift or sensitivity to the fitting window is supplied; this weakens the distinction drawn between SYK (volume-law) and t-V behaviors.
minor comments (2)
  1. [§2, figures] Figure captions and §2 should explicitly state the range of system sizes used for each model and the precise definition of the interpolating function (including any free parameters and their initialization).
  2. [§4] The IPR analysis in §4 would benefit from a brief statement of the Hilbert-space dimension and the precise definition of the participation ratio employed.

Simulated Author's Rebuttal

2 responses · 2 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major concerns point by point below, providing the strongest honest defense of our numerical observations while acknowledging limitations.

read point-by-point responses

  1. Referee: [Abstract, §3] Abstract and §3 (integrable models): the central proposal to characterize entanglement phases via the parameters of the interpolating fit is load-bearing on the assumption that the functional form and extracted exponents remain stable for L→∞. The text acknowledges the limited size window but provides no extrapolation tests, no AIC/BIC comparison against pure logarithmic or other candidate forms, and no derivation of the interpolant from the monitored dynamics; if the true scaling is logarithmic with a slow crossover, the reported parameters would be effective descriptors only inside the accessible window.

    Authors: We agree that an analytic derivation of the interpolant is absent, as the work is a numerical study identifying an empirical functional form that fits the data over more than one decade in L. We have added an AIC comparison in the revised manuscript demonstrating that the interpolating form outperforms a pure logarithmic fit across the accessible sizes. Additional finite-size stability checks for the extracted parameters have also been included. However, computational constraints prevent access to sizes sufficient for definitive L→∞ extrapolation or to rule out a very slow crossover; we therefore present the characterization as a practical protocol for the numerically reachable regime rather than a proven asymptotic classifier. revision: partial

  2. Referee: [§4] §4 (non-integrable models): the claim of 'traces of an entanglement transition' in the t-V model rests on the size scaling of fit parameters extracted from entanglement versus measurement strength. The accessible size range is stated to be too small to span orders of magnitude, yet no quantitative assessment of finite-size drift or sensitivity to the fitting window is supplied; this weakens the distinction drawn between SYK (volume-law) and t-V behaviors.

    Authors: We acknowledge the limited size range for non-integrable models. In the revision we have added explicit checks of fit-parameter sensitivity to the measurement-strength fitting window together with quantitative estimates of finite-size drift. These show that the volume-law signature in SYK remains robust while the t-V parameters exhibit a distinct trend consistent with an emerging transition, even within the accessible window. The language has been softened to describe these as preliminary indications rather than definitive evidence. revision: yes

standing simulated objections not resolved
  • Analytic derivation of the interpolating function from the underlying monitored dynamics
  • Numerical access to system sizes large enough to perform controlled extrapolations to L→∞ or to fully exclude slow crossovers

Circularity Check

0 steps flagged

Numerical fitting of entanglement scaling shows no circular derivation chain

full rationale

The paper is a numerical study that simulates monitored fermion models and empirically fits an interpolating function to entanglement-versus-size data in integrable cases, then extracts parameters for phase characterization. No derivation from first principles is presented that reduces by construction to the fitted quantities themselves; the fit is applied to observed data rather than used to predict the same data. No self-citations, uniqueness theorems, or smuggled ansatzes appear as load-bearing steps. The analysis remains self-contained as an empirical characterization within numerically accessible windows.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review prevents extraction of explicit free parameters or ad-hoc axioms; the work implicitly relies on standard assumptions of unitary evolution between measurements and projective measurement postulates in quantum mechanics.

pith-pipeline@v0.9.0 · 5774 in / 1317 out tokens · 42409 ms · 2026-05-23T03:16:37.917929+00:00 · methodology

discussion (0)

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