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Random long-range hopping forces volume-law entanglement in monitored 1D free fermions for any measurement strength when the hop decays slowly enough.

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2026-07-30 13:44 UTC pith:F3KPCEBX

load-bearing objection Solid α–γ phase diagram for monitored PRBM fermions; the “area law for any γ when α>3/2” leg is asserted harder than the weak-γ data support. the 4 major comments →

arxiv 2607.23871 v1 pith:F3KPCEBX submitted 2026-07-26 quant-ph

Entanglement transitions and multifractality in monitored free-fermions with random long-range hopping

Bo Fan , Antonio M. Garc\'ia-Garc\'ia This is my paper
classification quant-ph
keywords measurement-induced phase transitionentanglement entropylong-range hoppingfree fermionsmultifractalitypower-law random banded matrixquantum state diffusionarea-law to volume-law
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper maps how continuous monitoring of particle number competes with random power-law hopping in a one-dimensional chain of free complex fermions. When the hopping decays slowly (exponent α ≲ 1), classical superdiffusive jumps keep the steady-state entanglement entropy growing with system size for every monitoring strength, approaching a full volume law once α ≲ 1/2. When hops are shorter-ranged (α > 3/2) the system collapses to an area law for any nonzero monitoring. Between these regimes an α-dependent measurement-induced transition appears, with logarithmic entanglement and multifractal density correlations exactly at criticality. The result isolates the role of classical long-range transport as a source of entanglement that cannot be erased by local measurements, distinct from genuine quantum non-locality.

Core claim

For monitored free complex fermions with random power-law hopping of exponent α, the steady-state entanglement entropy is sub-volume (tending to volume law for α ≲ 1/2) for any monitoring strength when α ≲ 1, is strictly area-law for any monitoring strength when α > 3/2, and undergoes a measurement-induced transition between those phases when 1 < α ≲ 3/2; at the critical line the entropy is logarithmic and the density-density correlator is multifractal.

What carries the argument

The density-density correlator C(r) = |D_{i+r,i}|^2 extracted from the Gaussian correlation matrix, whose power-law decay C(r) ∝ r^{-(α+1/2)} is derived from the trajectory-averaged Itô equation for D and directly controls the second particle-number cumulant and hence the entanglement entropy via the cumulant expansion.

Load-bearing premise

The analytic decay of correlations assumes that off-diagonal unitary terms cancel and that the time derivative of the correlation matrix can be dropped, approximations the authors themselves say fail both for very long-range hops and for ordinary diffusion.

What would settle it

Measure the half-chain entanglement entropy versus system size for fixed weak monitoring (γ ≲ 0.3) at α = 1.6 and α = 1.4; if the former remains size-independent while the latter grows as a power of L, the claimed phase boundary at α = 3/2 is confirmed.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Superdiffusive classical hopping alone is sufficient to protect volume-law entanglement against arbitrary local monitoring in one dimension.
  • The critical line 1 < α_c(γ) ≲ 3/2 is a continuous family of multifractal fixed points whose singularity spectrum peak α_0 rises with monitoring strength.
  • For α > 3/2 the monitored long-range model collapses to the same area-law physics already known for short-range free fermions.
  • Any future analytic theory (e.g., nonlinear sigma model) must reproduce the explicit relation a_s ≃ 3/2 − α for the entanglement exponent in the sub-volume phase.

Where Pith is reading between the lines

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

  • The same α = 3/2 boundary that separates power-law localization from ordinary Anderson insulation in the unmonitored PRBM spectrum reappears as the boundary beyond which monitoring can enforce an area law, suggesting a deeper link between single-particle localization length and measurement-induced entanglement.
  • Because the derivation never uses fermionic statistics beyond Gaussianity, an analogous volume-law protection should appear for monitored bosonic or spin models with the same random long-range couplings.
  • Finite-size drifts at weak γ may still hide a very weak MIPT for α slightly above 3/2; larger-scale GPU simulations or an NLSM treatment would settle the issue.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 7 minor

Summary. The manuscript studies the steady-state entanglement of continuously monitored (QSD protocol) one-dimensional free complex fermions with random power-law hopping t_ij ~ |i-j|^{-α} (the PRBM ensemble). Using mutual-information crossings with finite-size collapse (L up to 3200), bipartite EE scaling, the density-density correlator C(r), and a multifractal singularity-spectrum analysis, the authors construct a phase diagram in the (α, γ) plane: sub-volume (approaching volume-law) EE scaling for any monitoring when α ≲ 1, an α-dependent MIPT line for 1 < α ≲ 3/2 with logarithmic EE and multifractal C(r) at criticality, and a claimed area-law phase for any γ when α > 3/2. An analytic treatment of the averaged Itô equation for the correlation matrix, under a steady-state and off-diagonal-cancellation approximation, yields C(r) ~ r^{-(α+1/2)} and hence S ~ L^{3/2-α}, in good agreement with the fitted exponents in the stated window of validity.

Significance. If the phase diagram holds, this is a useful and timely contribution: it is, to my knowledge, the first systematic entanglement phase diagram for monitored free fermions with random long-range hopping, and it cleanly connects the static PRBM classification (extended / power-law-localized / short-range-like at α = 1, 3/2) to the monitored dynamics. The work has several concrete strengths: multiple independent diagnostics (I_2 crossings and collapse, EE and I_2 L-scaling, C(r) exponents, P(ln I_2) scale invariance, f(α_q)) are mutually consistent; system sizes are large for this type of simulation; time-step convergence is checked explicitly (Fig. 11(a)); the critical exponent relation a_r = α + 1/2 is derived rather than fitted, and the data in Fig. 8(c) track it over three decades of γ; and the authors are unusually candid about the non-universality and instability of the extracted ν. The analytic derivation, while approximate, is transparent about its assumptions and produces a falsifiable, γ-independent exponent prediction that the numerics confirm within its stated window.

major comments (4)
  1. [§III.A, Fig. 12(a,b) and Fig. 1] The phase diagram's α > 3/2 leg — area law for *any* γ > 0 — is contradicted at face value by the paper's own weak-monitoring crossings: α_c = 1.59 ± 0.01 (γ = 0.1) and α_c = 1.52 ± 0.01 (γ = 0.3), with α_c monotonically increasing as γ decreases below 0.5 (α_c = 1.49 at γ = 0.5). The text attributes this to finite-size effects and notes that γ ≲ 0.3 is 'numerically challenging', but no analysis demonstrating the crossing actually drifts below 3/2 with increasing L is shown. Since the sizes used at γ = 0.1, 0.3 (L = 512–3200) are the same as those used to establish the transition at γ = 0.5, an unsupported drift assumption is doing load-bearing work for one of the three regions of Fig. 1. The authors should either (i) provide a crossing-drift analysis (α_c vs. minimum L in the collapse window, or vs. L directly), or an estimate of the crossover length near α → 3/2⁺ (where PRBM power-law-
  2. [§III.D, Eqs. (14)–(28), Fig. 9] The analytic derivation of C(r) ~ r^{-(α+1/2)} rests on three approximations whose support in the manuscript is thin: (a) dD_ij/dt ≈ 0 is justified only by Fig. 9 at a single system size L = 128 and a single parameter point (γ = 0.1, α = 1.5), which the caption itself describes as 'effectively the superdiffusive regime' at this size; (b) the cancellation of off-diagonal unitary terms (used both above Eq. (14) and in Eq. (17)) is asserted, not checked; (c) μ is treated as a positive constant, yet μ ∝ 1/γ, so the steady-state balance in Eq. (26) must fail as γ → 0 at fixed α — the regime where, per comment 1, the numerics are already most delicate. Given that the exponent prediction is compared against data across 1/2 < α ≲ α_c and is one of the paper's headline results, the authors should at minimum show Fig. 9-type diagnostics across the α window (including near α ≈ 1/2 where they state
  3. [§III.D, Eq. (30) and Fig. 8(a)] The EE prediction S ~ L^{3/2-α} is obtained by truncating the cumulant expansion Eq. (9) at second order. Appendix B (Fig. 11(b)) validates this truncation only at γ = 0.5, L = 2048, and the agreement is described as holding 'around the critical values of α'. Since the truncation is used to convert Eq. (28) into the sub-volume-law exponent that is then compared to fits over the whole superdiffusive region (and at γ = 0.1 and 2.0 in Fig. 8(a)), the validity of S ≈ (π²/3) C^(2) should be demonstrated in the sub-volume phase and at strong/weak monitoring, not only near criticality. This is presumably a straightforward extension of existing data but is currently a gap in the chain from Eq. (28) to Fig. 8(a).
  4. [§III.C and Appendix D, Eqs. (D1)–(D5)] The multifractal analysis defines τ(q) = q(d+1) + Δ_q, so that the claim 'Δ_{q=1} ≈ 0 confirms C(r) ~ r^{-2}' is partially tautological: the (d+1) offset is chosen using the average C(r) exponent, and Fig. 15 then reports Δ_{q=1} = 0.02 ± 0.017, i.e., consistent with zero by construction of the reference. The genuinely non-trivial content is the curvature of τ(q) (equivalently α_0 > 2d in the parabolic fit). The presentation should be tightened to make clear which parts of the analysis are self-consistency checks and which constitute independent evidence of multifractality; error bars on f(α_q) away from q = 1 would also strengthen Fig. 7.
minor comments (7)
  1. [Fig. 9 caption] The statement 'the scale of ln D_ij is more than 20 times larger than ln dD_ij/dt' is misleading as phrased: the colorbars show ln D_ij ∈ (−12, −4) vs. ln dD_ij/dt ∈ (−12, −7), so the magnitudes of the logarithms differ by less than a factor of 2 in places. What matters is the ratio of the quantities themselves (e^Δln). Please reword and label the colorbars.
  2. [Fig. 8(b)] There is a stray trailing token in the legend ('a_p = 1.13 − 0.59α, 2.12' for γ = 2.0); presumably a fit-range annotation that should be cleaned up.
  3. [§III.D, after Eq. (27)] Typo: 'balance the terms in the right size of Eq. (26)' → 'right-hand side'.
  4. [Notation] α is overloaded for the hopping exponent and the singularity-spectrum variable α_q (Fig. 7, Appendix D); consider α → a or f(α_q) → f(ϑ) in the multifractal sections to avoid confusion, especially since both appear in the same figure discussion.
  5. [References] Ref. [55] is the same paper as Ref. [28] (Szyniszewski, Lunt, Pal, PRB 108, 165126) and should be merged. Refs. [47, 48, 61, 62] are 2026 preprints; the important input that short-range monitored disordered complex fermions are always area-law (used for the α > 3/2 leg) currently rests on the preprint [48] — a pointer to any peer-reviewed version would strengthen the citation.
  6. [§II] Please state how the saturation time t_f scales with L and α near criticality, and confirm that the largest systems (L = 2048–3200) were evolved well beyond the t_f estimated at L = 128; this matters for the reliability of the crossings used in Fig. 12.
  7. [Fig. 2 / Appendix A] The collapse ansatz y(x) = I_2(α, L) with no L-dependent rescaling of y assumes exact scale invariance of I_2 at criticality; a brief note on how sensitive α_c is to allowing a subleading correction (e.g., I_2 = I_2^* + c L^{-ω}) would be useful, given the acknowledged instability of ν.

Circularity Check

1 steps flagged

No significant circularity: phase boundaries and C(r)∼r^{-(α+1/2)} are independently obtained from numerics and an Itô derivation under stated approximations; only a minor non-load-bearing self-citation supports the α>3/2 analogy.

specific steps
  1. self citation load bearing [Sec. III.D, paragraph on α>3/2; also Introduction/Conclusion phase-diagram claims]
    "In the region α>3/2, the system crosses over to classical diffusion and the approximations leading to Eq. (28) no longer apply. The long-range tail is then irrelevant at long distances, and therefore the monitored dynamics is similar to that of monitored free fermions with short range hopping which results [48] in the absence of a MIPT."

    The strong claim that the system is area-law for any γ when α>3/2 is partly propped by citation [48] (Yin–Fan–García-García, overlapping authors) that short-range monitored 1d complex fermions have no MIPT. This is a minor supporting analogy, not a definitional circle or uniqueness import: the long-range numerics and the Itô derivation are independent, and [48] is a separate numerical study. Raises score only to 1.

full rationale

The central results—finite-size I₂ crossings locating α_c(γ), EE and MI power-law/log/area scalings, C(r) decay exponents, and the parabolic multifractal spectrum—are extracted directly from the monitored PRBM trajectories and are not forced by construction from their inputs. The analytic chain in Sec. III.D starts from the averaged QSD equation for D_ij, imposes explicit approximations (neglect ∂_t D_ij off-diagonal, cancel off-diagonal unitary pieces, treat μ as a positive constant, power-law ansatz f_n∼n^{-z}), and balances powers to obtain z=α+1/2 and S∼L^{3/2-α}; these steps are derivations under stated assumptions that the paper itself flags as breaking down outside 1/2≲α≲α_c, and the resulting exponent is then compared to independent fits rather than fitted to force the claim. The short-range area-law analogy used for α>3/2 cites overlapping-author work [48], but that is ordinary supporting literature for a separate model, not a uniqueness theorem or a definitional reduction of the long-range phase diagram. Standard data-collapse fits for (α_c,ν) and parabolic f(α_q) fits are results, not circular predictions. Correctness concerns about weak-γ crossings sitting slightly above 3/2 are evidence/finite-size issues, not circularity. Score 1 only for the minor self-citation on the α>3/2 leg.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The central claim rests on the standard free-fermion Gaussian-state formalism under QSD, the known single-particle PRBM phase structure, and a set of controlled but non-rigorous approximations in the averaged correlation-matrix dynamics. No new physical entities are postulated; free parameters are the model knobs α and γ plus ordinary finite-size fitting parameters (ν, α_c).

free parameters (4)
  • α_c(γ) = e.g. 1.49±0.01 (γ=0.5), 1.36±0.01 (γ=2.0)
    Critical hopping exponent extracted from mutual-information crossings and data collapse for each fixed γ; central to locating the MIPT line.
  • ν (finite-size exponent) = ≈3–5.5 depending on γ
    Nuisance parameter in the scaling collapse of I_2; paper notes it is non-universal and window-dependent.
  • α_0 (singularity-spectrum peak) = 2.17±0.02 (γ=0.5), 2.49±0.02 (γ=2.0)
    Fitted peak of f(α_q) used to quantify weak multifractality at criticality.
  • μ (occupation inhomogeneity prefactor)
    Positive constant appearing in the analytic steady-state equation for f_n; not independently measured.
axioms (5)
  • domain assumption Continuous QSD monitoring of local occupation preserves Gaussianity of free-fermion states, so the correlation matrix D_ij fully determines the EE.
    Standard in the monitored free-fermion literature (Cao et al., Poboiko et al.); invoked from Sec. II onward.
  • domain assumption Single-particle PRBM eigenstate regimes (extended/multifractal/localized) are controlled solely by α, independent of disorder strength.
    Taken from Mirlin–Fyodorov–Evers literature; used to motivate the model and interpret α windows.
  • ad hoc to paper In the dynamical steady state, dD_ij/dt≈0 for i≠j and off-diagonal unitary contributions cancel, leaving D_ij≈(i H_ij/γ)(D_ii−D_jj).
    Key approximation of Sec. III.D (Eq. 14); justified numerically only inside 1/2<α≲α_c and stated to fail outside.
  • domain assumption Leading EE scaling is captured by the second particle-number cumulant, S≈(π²/3)C^{(2)}.
    Standard Klich–Levitov expansion; validated numerically in Appendix B for the present model.
  • ad hoc to paper For α>3/2 the long-range tail is irrelevant and the monitored dynamics reduces to short-range free fermions (always area-law).
    Theoretical extrapolation used to close the phase diagram where weak-γ numerics are inconclusive.

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read the original abstract

We study the entanglement dynamics of a one-dimensional chain of monitored non-interacting complex fermions with random power-law hopping characterized by a decay exponent $\alpha$. For $\alpha \lesssim 1$, in stark contrast with the case of hopping to nearest neighbors, the scaling of the entanglement entropy (EE) of the steady state with system size $L$ is faster than logarithmic for any monitoring or disorder strength and it tends towards a linear (volume-law) scaling for sufficiently small $\alpha \lesssim 1/2$. For $\alpha > 3/2$, the EE is in the area-law phase, namely, no scaling with $L$, for any monitoring strength. For $1 < \alpha \lesssim 3/2$, we identify an $\alpha$-dependent measurement-induced phase transition (MIPT) at a critical value of the monitoring strength separating the mentioned area-law and sub-volume-law phases. At this critical point, the EE scales logarithmically with system size, and the density-density correlation function, closely related to the EE, exhibits multifractal features. These results highlight the importance of superdiffusive classical hopping in the entanglement dynamic of quantum many-body systems and also help differentiate its role with respect to conventional sources of entanglement such as genuine quantum non-locality.

Figures

Figures reproduced from arXiv: 2607.23871 by Antonio M. Garc\'ia-Garc\'ia, Bo Fan.

Figure 1
Figure 1. Figure 1: Phase diagram showing the scaling of the steady-state [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The mutual information I2 Eq. (6) as a function of α for γ = 0.5 (left) and γ = 2.0 (right). The sharp crossing at α = αc indicates a MIPT, where I2 becomes scale-invariant. The insets show the optimal finite-size data collapse around αc. The esti￾mated critical points are αc = 1.49±0.01 and αc = 1.36±0.01 for γ = 0.5 and 2.0, respectively. The critical exponent ν ex￾hibits larger fluctuations, due to the … view at source ↗
Figure 4
Figure 4. Figure 4: Steady-state entanglement entropy S(ℓA = L/2) Eq. (5) as a function of the system size L for γ = 0.5 (left) and γ = 2.0 (right), and different power-law exponents α. Fits to a power-law growth S ∼ L as(α) reveal a gradual increase of as with increasing α. For weak monitoring γ = 0.5, the system transits from as ≈ 1 (volume-law) for small α to a logarithmic growth α ≈ 3/2, and finally to an area-law phase f… view at source ↗
Figure 3
Figure 3. Figure 3: The probability distribution of the mutual information [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: , is in full agreement with that of the EE, providing further support for the phase diagram presented in [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Density-density correlation function C(r) Eq. (7) as a function of distance r for different values of α. The monitor￾ing strength is γ = 0.5 (left) and γ = 2.0 (right), with system size fixed at L = 2048. The dashed lines denote power-law fits C(r) ∼ 1/rar . The extracted power-law exponent ar is directly related to the growth of the EE with system size. Results for more values of α and γ are depicted in A… view at source ↗
Figure 8
Figure 8. Figure 8: (a). The scaling exponent as, characterizing the EE growth in the sub-volume-law phase, extracted from [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The spatial distribution of ln dDij dt (left) and ln Dij (right) for i ̸= j in the dynamical steady state. Note the scale of ln Dij is more than 20 times larger than ln dDij dt , even when the measurement strength is γ = 0.1 and α = 1.5. The system size is L = 128. Although (γ = 0.1, α = 1.5) corresponds to the critical parameter in the thermodynamic limit, considering the small system size, the results ar… view at source ↗
Figure 10
Figure 10. Figure 10: Heatmap of the cost function ln R(α, ν) Eq. (A1). The white circle and the errorbar correspond to the ν and αc and the corresponding error in [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: The mutual information I2 as a function of α for different system sizes L, and monitoring strengths ranging from weak (γ = 0.1) to strong (γ = 8.0). A sharp crossing at α = αc signals a MIPT, at a certain γ > 0, characterized by a scale-invariant I2. The inset shows the optimal data collapse around αc. While the obtained critical point αc is robust with only small uncertainties, the critical exponent ν ex… view at source ↗
Figure 13
Figure 13. Figure 13: Density-density correlation function C(r) Eq. (7) as a function of the distance r for different values of α and γ. The dashed lines show the corresponding power-law fits C(r) ∼ r −ar . described by a Gaussian whose mean grows and whose width narrows with increasing L, consistent with an av￾eraging MI that grows with system size in the thermody￾namic limit. In the area-law phase, the peak of P(ln I2) shift… view at source ↗
Figure 14
Figure 14. Figure 14: Probability distribution P(I2) (P(ln I2)) for different system sizes L in the (sub-)volume-law (area-law) phases. In the (sub-)volume-law phase, the distribution P(I2) are well approximated by a Gaussian, whose mean grows with system size L. In the area-law phase, the peak of P(ln I2) shifts towards ln I2 → −∞ with increasing system size, reflecting the vanishing of long-range correlations. Both behaviors… view at source ↗
Figure 15
Figure 15. Figure 15: Linear fits used to extract ∆q (left panels, Eq. (D2)) and αq (right panels, Eq. (D3)) at q = 1 for the critical points. (a) γ = 0.5, αc = 1.5. (b) γ = 2.0, αc = 1.35. In both cases, the extracted ∆q=1 ≈ 0, consistent with the critical scaling C(r) ∼ r −2 [PITH_FULL_IMAGE:figures/full_fig_p011_15.png] view at source ↗

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