Pith. sign in

REVIEW 3 major objections 4 minor 55 references

Tuning the coupling range in a random quantum hopping model reveals an intermediate localized regime, attributed to structural disorder.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 21:05 UTC pith:FTN5KKHT

load-bearing objection The single-particle intermediate-range localization looks real and worth engaging; the finite-density claim rests on a free-fermion surrogate that is not the model in Eq. (3). the 3 major comments →

arxiv 2607.16394 v1 pith:FTN5KKHT submitted 2026-07-17 quant-ph

Entanglement Entropy in Quantum Networks with Tunable Geometry

Andrea Azzali , Sridevi Kuriyattil , Andrew J. Daley , Maril\`u Chiofalo This is my paper
classification quant-ph PACS 03.65.Ud72.15.Rn
keywords entanglement entropyrandom graphAnderson localizationpower-law rangequantum dynamicshard-core bosonsfree fermionsstructural disorder
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 studies quantum hopping on random graphs where the probability of linking two nodes decays as a power of their distance, controlled by an exponent α. Varying α interpolates between a nearest-neighbor chain (α→∞) and an all-to-all network (α→0). The authors claim that, unexpectedly, the asymptotic entanglement entropy is not monotonic: the system is delocalized at both limits but enters a localized regime for intermediate α, visible in both single-particle and half-filling dynamics. They attribute this localization to structural disorder—randomness in which long-range links are present—rather than to on-site noise or removed links. If correct, this is a minimal model of Anderson localization generated purely by the geometry of the interaction graph, with implications for amorphous materials and tunable-range quantum simulators.

Core claim

Central claim: the random graph with link probability c|j−ℓ|^{−α} and c=1 shows an intermediate localized regime between the chain (α→∞) and all-to-all (α→0) limits. The asymptotic entanglement entropy S∞ is non-monotonic in α, dropping to a minimum at intermediate α, as corroborated by the inverse participation ratio. This is attributed to structural disorder: random long-range links act as scattering impurities, and a large-α impurity model fits S∞(α). The effect also appears at half-filling, with S∞/(N log 2) reaching about 0.28 for delocalized regimes, consistent with a random Gaussian state—but the finite-density calculation uses a free-fermion Hamiltonian obtained by deleting the strin

What carries the argument

The key object is the random graph ensemble defined by independent link probability c|j−ℓ|^{−α}, with c=1 for the main results. The quantum dynamics is the hopping Hamiltonian H=J∑ A_{jℓ} σ_j^+ σ_ℓ^- (hard-core bosons), and the central diagnostic is the asymptotic entanglement entropy S∞ of a spatial bipartition, complemented by the inverse participation ratio. The explanatory mechanism is a large-α approximation in which the graph is viewed as a one-dimensional chain (nearest-neighbor links) with rare length-2 links acting as scattering impurities; treating the wavefunction as exponentially localized with decay length λ set by the impurity density 2^{-α} yields a fit to S∞(α). This impurity

Load-bearing premise

The finite-density claim rests on replacing the hard-core bosons by spinless fermions and deleting the string terms of a Jordan–Wigner transformation, an operation the paper admits changes the Hamiltonian except in the single-particle sector.

What would settle it

Run exact diagonalization of the hard-core boson Hamiltonian at half-filling for N up to about 20 and see whether the intermediate-α dip in S∞/(N log 2) survives; if the dip vanishes or shifts, the finite-density claim is not supported. Alternatively, measure the single-particle inverse participation ratio for N=200,400,800 at α≈3: a true localized phase has IPR saturating to a constant, while a finite-size crossover would show IPR decreasing with N.

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

If this is right

  • The intermediate-α localized regime persists at half-filling (in the free-fermion surrogate), so the effect is not an artifact of the single-particle sector.
  • The localized wavefunction's decay length depends only on α, not on system size N, so larger systems exhibit smaller S∞ and a clearer localized signal.
  • The localization is caused by off-diagonal (structural) disorder in the coupling pattern, not by on-site noise or by the classical percolation transition.
  • The model offers a concrete mechanism for Anderson localization in amorphous materials and for experiments with tunable-range Rydberg or dipolar simulators.

Where Pith is reading between the lines

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

  • A direct test of the hard-core boson model at half-filling (e.g., small-system exact diagonalization) is needed, since the paper's finite-density results come from an inequivalent free-fermion Hamiltonian.
  • If the structural-disorder mechanism is the cause, the same intermediate localization should appear in observables like mean-squared displacement or conductivity, which the paper does not compute.
  • The impurity model predicts a localization length scaling as 1/λ ∝ 2^{-α}; fitting the exponential tail of the single-particle wavefunction directly would provide a sharper test than the entropy.
  • On graphs with different degree distributions, a similar non-monotonic entropy could serve as a signature of non-ergodic, multifractal phases—an extension the paper mentions as future work.

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

3 major / 4 minor

Summary. The paper studies quantum hopping on a random graph model with link probability c |j-l|^{-alpha}, interpolating between all-to-all (alpha=0) and one-dimensional chain (alpha->infinity) geometries. For c=1, exact-diagonalization simulations of the single-particle sector and Gaussian-state simulations of the half-filled sector are used to compute asymptotic entanglement entropy, IPR, and wavefunction spreading. The central claim is that, as alpha is varied, the system exhibits an unexpected intermediate localized regime caused by structural disorder, and that this regime is robust in both single-particle and finite-density cases. A large-alpha approximate model treats length-2 links as impurities and fits a single parameter to reproduce the decreasing entropy.

Significance. If the result holds, the paper identifies a genuinely interesting mechanism: Anderson-like localization produced by randomness in the connectivity pattern of a graph, without on-site disorder. The single-particle numerical evidence is internally cross-checked: it uses sample-averaged data with error bars, two bipartitions, IPR corroboration, and an N-dependence consistent with a localized wavefunction. This part is a useful contribution. However, the finite-density leg of the claim is computed in a free-fermion model obtained by deleting Jordan-Wigner strings from the hard-core boson Hamiltonian, a model the authors themselves state is inequivalent to Eq. (3). As a result, the abstract's 'robust in both single-particle and finite-density cases' is not established by the reported numerics. The explanatory large-alpha model is also fit to the same data it explains, and no scaling collapse is provided to justify the thermodynamic-limit language. These gaps are load-bearing and require revision before the central claim can be accepted.

major comments (3)
  1. [Methods, 'Half-filling case'; Fig. 3; Figs. S5/S6] The finite-density results are obtained by replacing hard-core bosons in Eq. (3) with spinless fermions and 'deleting the Jordan-Wigner strings.' The manuscript explicitly states that this Hamiltonian 'is not equivalent to the previous one, except for the single-particle sector.' On generic non-path graphs, the omitted string operators do not cancel, so the simulated model is a different noninteracting theory. All half-filling claims, including S_inf/(N log 2) ~ 0.28 and the comparison to random Gaussian states, are therefore not evidence about the hard-core boson model of Eq. (3). The abstract and conclusions nevertheless present the localized regime as 'robust in both single-particle and finite-density cases.' This is an internal-consistency gap. The authors should either simulate the actual hard-core boson model at finite density (e.g., small-N ED, tensor networks, or other controlled
  2. [Supplemental Material, 'Large-alpha limit at c=1'; Fig. S7; Fig. 3] The large-alpha 'prediction' for S_inf(alpha) is a one-parameter least-squares fit of A in 1/lambda = 2^{-alpha} A against the same numerical data it is intended to explain. Moreover, the fitted A drifts with system size (0.36, 0.292, 0.233, 0.206 for increasing N), so the model is not a parameter-free predictive derivation. More importantly, the thermodynamic-limit statement is based only on the qualitative trend that S_inf decreases with N; no finite-size scaling collapse is presented, so this could be a finite-size crossover rather than a true localized phase. For the central 'new localization regime' claim, the authors should provide a scaling analysis or otherwise specify the regime of validity of the extrapolation.
  3. [Results, Case c=1, first paragraph] The text states that the alpha=0 limit is the 'all-to-all model, fully localized as N->infinity.' For a single particle on a complete graph with an initially localized state, the long-time state is delocalized and the asymptotic entanglement entropy is maximal, not zero. This statement is also inconsistent with the same paragraph's assertion that 'reaches maximally entangled values for alpha <= 2' and with Fig. 4a, where the probability spreading is 'almost uniform on all sites' at small alpha. This appears to be a typographical error, but as written it misstates a benchmark and should be corrected.
minor comments (4)
  1. [Fig. 3 and text below it] The half-filling curve is said to saturate at S_inf/(N log 2) ~ 0.28, which is consistent with S/N -> log 2 - 1/2 after normalization. This conversion is not explained in the main text; please define the normalization explicitly in the figure caption or text.
  2. [Supplemental Material, 'Results with c<1', Fig. S4] The dotted reference line in Fig. S4 is described as 'the fraction of sites in the cluster' computed at the largest N. Please state in the caption which N is used and whether the comparison across N is affected by this choice.
  3. [Methods, Eq. (3)] The notation [b_j, b_l^dagger] = delta_{jl}(I - 2 n_j) is the anti-commutator-like relation for hard-core bosons on the same site and a commutator on different sites. It would help to state explicitly that this is the standard hard-core boson algebra, as the mixed commutator/anti-commutator form can confuse readers.
  4. [Supplemental Material, 'Kac normalization'] The Kac normalization discussion is clear but would benefit from a statement of whether the main-text results are Kac-normalized. The sentence 'the results remain unaffected' at alpha>1 should be made more precise by specifying that time is rescaled and S_inf is unchanged.

Circularity Check

1 steps flagged

The large-α explanation of the localized regime is a one-parameter fit to the data it is said to predict; the central numerical result and IPR remain independent, while the finite-density leg uses an inequivalent free-fermion surrogate.

specific steps
  1. fitted input called prediction [Results, 'Case c=1' paragraph; Supplemental Material, 'Large-α limit at c=1', Fig. S7]
    "From it, we obtain a localized wavefunction (see Supplemental Material for the details) and derive a prediction for S∞(α) that has been fit to the actual numerical data, with good agreement for large values of α (see Fig. S7). ... We perform a least-squares fit with A as only parameter (see Fig. S7). ... the best-fit values of A are (with increasing N) 0.36, 0.292, 0.233, 0.206."

    The 'prediction' for S∞(α) in the large-α Anderson-impurity model has one free parameter, the single-impurity absorption amplitude A in 1/λ = 2^{−α}A. A is determined by least-squares fitting to the very S∞(α) curves (Fig. 3 / Fig. S7) that the model is then said to predict. Thus the agreement in Fig. S7 is not an independent check: it demonstrates only that the chosen exponential ansatz can be matched to the tail once A is fitted. The core non-monotonic/intermediate-localization claim does not rest on this fit (the authors note it fails for α<2), but the 'prediction' label overstates what the fit establishes.

full rationale

The central single-particle result—non-monotonic S∞(α) with an intermediate localized regime—is obtained by exact diagonalization of the original Hamiltonian (3), with no fitted parameters, cross-checked by the IPR and by the known α=0 and α→∞ benchmarks. That part of the derivation is self-contained and does not reduce to its inputs. The only identifiable circular step is the large-α 'prediction' from the disordered-chain model: the decay-length amplitude A is fitted to the same numerical S∞ data the model is used to explain, so the agreement is a fitted post-hoc explanation rather than a parameter-free prediction. This step is supporting, not load-bearing for the main claim. Separately, the half-filling results are computed for spinless fermions obtained by 'deleting the Jordan–Wigner strings', a Hamiltonian the paper explicitly states 'is not equivalent to the previous one, except for the single-particle sector'; this is a model-validity gap for the abstract's 'robust in both single-particle and finite-density cases' claim, but it is not circularity. No load-bearing self-citation chain or imported uniqueness theorem was found.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

The paper contributes the numerical S_∞(α) curves and a graph-distance analysis; its analytic content is (i) the large-α impurity ansatz carrying one fitted parameter A, and (ii) the free-fermion substitution at half-filling, acknowledged by the authors as inequivalent to Eq. (3). The graph model and its percolation phase diagram are imported from [36]; Anderson localization is imported as background. No new entities are invented — the 'impurities' are existing length-2 graph edges reinterpreted as scatterers.

free parameters (1)
  • A (single-impurity scattering amplitude in the large-α model) = 0.36, 0.292, 0.233, 0.206 (for increasing N)
    Least-squares fit parameter in the disordered-chain model 1/λ = 2^{−α}A, fit against the same S_∞(α) data it is used to explain (Supplemental Material, 'Large-α limit at c=1', Fig. S7). The fit is disclosed by the authors.
axioms (5)
  • domain assumption Graph model of Eq. (1): link probability c|j−ℓ|^{−α}, with the classical percolation phase diagram (α∈(1,2) transition)
    Imported from Gori et al. [36] and used to frame the quantum study; the paper's contribution is dynamics on top of this distribution.
  • domain assumption One-dimensional disordered quantum systems localize, with exponentially decaying wavefunction amplitude (Anderson localization)
    Used to build the large-α impurity ansatz |ψ(x)| ~ exp(−|x−N/2|/λ) in Supplemental Material 'Large-α limit at c=1', citing refs [40-42].
  • ad hoc to paper Deleting Jordan–Wigner strings yields a physically informative finite-density model
    Half-filling calculations replace Eq. (3) hard-core bosons with free spinless fermions; the paper states the models 'is not equivalent to the previous one, except for the single-particle sector,' yet the abstract's finite-density robustness claim rests on this replacement.
  • ad hoc to paper Links of length r>2 can be neglected in the large-α limit
    The impurity model keeps only r=1,2 links on the assumption that r≥3 links are negligible; reasonable for the tail, but the authors concede the model then fails for α<2 (Fig. S7).
  • domain assumption Saturation of dynamics by t = 10^4 ℏ/J uniformly across parameters
    Asymptotic values S_∞ and n_∞ are read off at a fixed time; the authors say this was checked (Fig. S1), but their own t* analysis shows the saturation timescale depends on α and N (linear in N for the clean chain).

pith-pipeline@v1.3.0-alltime-deepseek · 10517 in / 24681 out tokens · 226547 ms · 2026-08-01T21:05:34.268062+00:00 · methodology

0 comments
read the original abstract

Quantum many-body Hamiltonians with two-body interactions can be represented by graphs, with sites as nodes and two-site couplings as edges. We investigate how the geometry of these graphs affects transport and entanglement growth. To do so, we study dynamics for quantum hopping on a random graph model with an effective range parameter. Tuning it, we interpolate between nearest-neighbor and all-to-all graphs, identifying a new localization regime that is robust in both single-particle and finite-density cases. This provides a model of Anderson localization induced by structural disorder, with implications for amorphous materials and tunable-range analog quantum simulators.

Figures

Figures reproduced from arXiv: 2607.16394 by Andrea Azzali, Andrew J. Daley, Maril\`u Chiofalo, Sridevi Kuriyattil.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic view of the classical phase diagram [ [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic depiction of the simulation performed on [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Asymptotic entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a,b,c) Classical vs. quantum distributions for the [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

55 extracted references · 8 linked inside Pith

  1. [1]

    M. K. Joshi, A. Elben, B. Vermersch, T. Brydges, C. Maier, P. Zoller, R. Blatt, and C. F. Roos, Physical Review Letters124, 240505 (2020)

    2020

  2. [2]

    B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. A. Hazzard, A. M. Rey, D. S. Jin, and J. Ye, Nature501, 521 (2013)

    2013

  3. [3]

    Baier, M

    S. Baier, M. J. Mark, D. Petter, K. Aikawa, L. Chomaz, Z. Cai, M. Baranov, P. Zoller, and F. Ferlaino, Science 352, 201 (2016)

    2016

  4. [4]

    A. S. Buyskikh, M. Fagotti, J. Schachenmayer, F. Essler, and A. J. Daley, Physical Review A93, 053620 (2016), arXiv:1601.02106 [cond-mat]

    Pith/arXiv arXiv 2016

  5. [5]

    Defenu, T

    N. Defenu, T. Donner, T. Macr ` ı, G. Pagano, S. Ruffo, and A. Trombettoni, Reviews of Modern Physics95, 035002 (2023)

    2023

  6. [6]

    Defenu, A

    N. Defenu, A. Lerose, and S. Pappalardi, Physics Reports 1074, 1 (2024)

    2024

  7. [7]

    Bentsen, Y

    G. Bentsen, Y. Gu, and A. Lucas, Proceedings of the National Academy of Sciences116, 6689 (2019), arXiv:1805.08215 [cond-mat]

    Pith/arXiv arXiv 2019

  8. [8]

    Bentsen, T

    G. Bentsen, T. Hashizume, A. S. Buyskikh, E. J. Davis, A. J. Daley, S. S. Gubser, and M. Schleier-Smith, Physi- cal Review Letters123, 130601 (2019)

    2019

  9. [9]

    Hashizume, S

    T. Hashizume, S. Kuriyattil, A. J. Daley, and G. Bentsen, Symmetry14, 666 (2022)

    2022

  10. [10]

    Kuriyattil, T

    S. Kuriyattil, T. Hashizume, G. Bentsen, and A. J. Daley, PRX Quantum4, 030325 (2023)

    2023

  11. [11]

    Hashizume, G

    T. Hashizume, G. S. Bentsen, S. Weber, and A. J. Daley, Physical Review Letters126, 200603 (2021)

    2021

  12. [12]

    Bluvstein, S

    D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, J. P. Bonilla Ataides, N. Maskara, I. Cong, X. Gao, P. Sales Rodriguez, T. Karolyshyn, G. Semegh- ini, M. J. Gullans, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Nature626, 58 (2024)

    2024

  13. [13]

    M. A. Valdez, D. Jaschke, D. L. Vargas, and L. D. Carr, Physical Review Letters119, 225301 (2017)

    2017

  14. [14]

    Nokkala, F

    J. Nokkala, F. Arzani, F. Galve, R. Zambrini, S. Manis- calco, J. Piilo, N. Treps, and V. Parigi, New Journal of Physics20, 053024 (2018)

    2018

  15. [15]

    Nokkala, J

    J. Nokkala, J. Piilo, and G. Bianconi, Journal of Physics A: Mathematical and Theoretical57, 233001 (2024), arXiv:2311.16265 [quant-ph]

    Pith/arXiv arXiv 2024

  16. [16]

    D. S. Bhakuni, R. Verdel, C. Muzzi, R. Andreoni, M. Aidelsburger, and M. Dalmonte, Physical Review B 110, 144204 (2024)

    2024

  17. [17]

    Ausilio, F

    S. Ausilio, F. Borgonovi, G. L. Celardo, J. Yago Malo, and M. L. Chiofalo, Physical Review B113, 064201 (2026)

    2026

  18. [18]

    De Luca, B

    A. De Luca, B. Altshuler, V. Kravtsov, and A. Scardic- chio, Physical Review Letters113, 046806 (2014)

    2014

  19. [19]

    Sierant, M

    P. Sierant, M. Lewenstein, and A. Scardicchio, SciPost Physics15, 045 (2023)

    2023

  20. [20]

    L. F. Cugliandolo, G. Schehr, M. Tarzia, and D. Ven- turelli, Physical Review B110, 174202 (2024)

    2024

  21. [21]

    Bhattacharjee, P

    S. Bhattacharjee, P. Sierant, M. Dudy´ nski, J. Wehr, J. Zakrzewski, and M. Lewenstein, Physical Review B 111, L180202 (2025)

    2025

  22. [22]

    Tikhonov and A

    K. Tikhonov and A. Mirlin, Annals of Physics435, 168525 (2021)

    2021

  23. [23]

    Lunkinet al., Hilbert space signatures of non-ergodic glassy dynamics (2026), arXiv:2601.01309

    A. Lunkinet al., Hilbert space signatures of non-ergodic glassy dynamics (2026), arXiv:2601.01309

    Pith/arXiv arXiv 2026

  24. [24]

    C.-T. Toh, H. Zhang, J. Lin, A. S. Mayorov, Y.-P. Wang, C. M. Orofeo, D. B. Ferry, H. Andersen, N. Kakenov, Z. Guo, I. H. Abidi, H. Sims, K. Suenaga, S. T. Pan- telides, and B. ¨Ozyilmaz, Nature577, 199 (2020)

    2020

  25. [25]

    H. Tian, Y. Ma, Z. Li, M. Cheng, S. Ning, E. Han, M. Xu, P.-F. Zhang, K. Zhao, R. Liet al., Nature615, 56 (2023)

    2023

  26. [26]

    Fujii and K

    K. Fujii and K. Nakajima, Physical Review Applied8, 024030 (2017)

    2017

  27. [27]

    Mujal, R

    P. Mujal, R. Mart ´ ınez-Pe˜ na, J. Nokkala, J. Garc ´ ıa-Beni, G. L. Giorgi, M. C. Soriano, and R. Zambrini, Advanced Quantum Technologies4, 2100027 (2021)

    2021

  28. [28]

    A. D. Mirlin and Y. V. Fyodorov, Nuclear Physics B366, 507 (1991)

    1991

  29. [29]

    Baroni, G

    M. Baroni, G. G. Lorenzana, T. Rizzo, and M. Tarzia, Physical Review B109, 174216 (2024)

    2024

  30. [30]

    Kirkpatrick and T

    S. Kirkpatrick and T. P. Eggarter, Physical Review B6, 3598 (1972)

    1972

  31. [31]

    Schubert and H

    G. Schubert and H. Fehske (Springer Berlin Heidelberg, Berlin, Heidelberg, 2009) pp. 1–28, in Quantum and Semi-classical Percolation and Breakdown in Disordered Solids

    2009

  32. [32]

    A. D. Mirlin, Y. V. Fyodorov, A. Mildenberger, and F. Evers, Physical Review Letters97, 046803 (2006)

    2006

  33. [33]

    Aizenman and C

    M. Aizenman and C. M. Newman, Communications in Mathematical Physics107, 611 (1986)

    1986

  34. [34]

    Leuzzi, G

    L. Leuzzi, G. Parisi, F. Ricci-Tersenghi, and J. J. Ruiz- Lorenzo, Physical Review Letters101, 107203 (2008). 6

    2008

  35. [35]

    Brezin, G

    E. Brezin, G. Parisi, and F. Ricci-Tersenghi, Journal of Statistical Physics157, 855 (2014), arXiv:1407.3358 [cond-mat]

    Pith/arXiv arXiv 2014

  36. [36]

    G. Gori, M. Michelangeli, N. Defenu, and A. Trombet- toni, Physical Review E96, 012108 (2017), publisher: American Physical Society

    2017

  37. [37]

    A. P. Mill´ an, G. Gori, F. Battiston, T. Enss, and N. Defenu, Physical Review Research3, 023015 (2021), arXiv:2006.10421 [cond-mat]

    Pith/arXiv arXiv 2021

  38. [38]

    Peschel, Journal of Physics A: Mathematical and Gen- eral36, L205 (2003)

    I. Peschel, Journal of Physics A: Mathematical and Gen- eral36, L205 (2003)

    2003

  39. [39]

    Surace and L

    J. Surace and L. Tagliacozzo, SciPost Physics Lecture Notes , 54 (2022)

    2022

  40. [40]

    P. W. Anderson, Physical Review109, 1492 (1958)

    1958

  41. [41]

    Landauer, Philosophical Magazine21, 863 (1970)

    R. Landauer, Philosophical Magazine21, 863 (1970)

    1970

  42. [42]

    P. W. Anderson, D. J. Thouless, E. Abrahams, and D. S. Fisher, Physical Review B22, 3519 (1980)

    1980

  43. [43]

    Bianchi, L

    E. Bianchi, L. Hackl, and M. Kieburg, Physical Review B103, L241118 (2021)

    2021

  44. [44]

    H. P. L¨ uschen, P. Bordia, S. S. Hodgman, M. Schreiber, S. Sarkar, A. J. Daley, M. H. Fischer, E. Altman, I. Bloch, and U. Schneider, Physical Review X7, 011034 (2017)

    2017

  45. [45]

    Stanzione, A

    V. Stanzione, A. Civolani, J. Y. Malo, and M. L. Chio- falo, Physical Review B112, 224209 (2025)

    2025

  46. [46]

    Hatano and D

    N. Hatano and D. R. Nelson, Physical Review Letters 77, 570 (1996)

    1996

  47. [47]

    Hatano and D

    N. Hatano and D. R. Nelson, Physical Review B56, 8651 (1997)

    1997

  48. [48]

    M. S. Rudner and L. S. Levitov, Physical Review Letters 102, 065703 (2009)

    2009

  49. [49]

    Yao and Z

    S. Yao and Z. Wang, Physical Review Letters121, 086803 (2018)

    2018

  50. [50]

    F. Song, S. Yao, and Z. Wang, Physical Review Letters 123, 246801 (2019)

    2019

  51. [51]

    Wegner, Zeitschrift f¨ ur Physik B Condensed Matter and Quanta36, 209 (1980)

    F. Wegner, Zeitschrift f¨ ur Physik B Condensed Matter and Quanta36, 209 (1980)

    1980

  52. [52]

    Evers and A

    F. Evers and A. D. Mirlin, Reviews of Modern Physics 80, 1355 (2008)

    2008

  53. [53]

    Calabrese and J

    P. Calabrese and J. Cardy, Journal of Statistical Me- chanics: Theory and Experiment2005, P04010 (2005), arXiv:cond-mat/0503393

    Pith/arXiv arXiv 2005

  54. [54]

    Alba and P

    V. Alba and P. Calabrese, Entanglement dynamics after quantum quenches in generic integrable systems (2018), arXiv:1712.07529

    Pith/arXiv arXiv 2018

  55. [55]

    M. Kac, G. E. Uhlenbeck, and P. C. Hemmer, Journal of Mathematical Physics4, 216 (1963). 1 Supplemental Material Convergence of entanglement entropy to its asymptotic value The asymptotic values are computed at timet= 10 4 ℏ/J, which was checked to be a long enough time to reach saturation for all choices of the parameters used in the numerical simulation...

    1963