Pith. sign in

REVIEW 2 major objections 4 minor 103 references

The paper claims that for unitary systems with extremely strong eigenvalue repulsion, the spectral form factor keeps crystal-like periodic peaks damped by a Debye-Waller factor until a new plateau time t* ≈ t_H√(β/4), much later than the He

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

2026-08-03 16:57 UTC pith:LD7TJQDV

load-bearing objection Defines a genuinely new large-β SFF regime with a Debye-Waller damped crystal and a new plateau time scale; the claims mostly hold up and the paper deserves refereeing, with requests to clarify C≈3.6 and test the deep tail. the 2 major comments →

arxiv 2512.11054 v3 pith:LD7TJQDV submitted 2025-12-11 quant-ph cond-mat.dis-nncond-mat.stat-mechnlin.CDnlin.CG

Crystalline Spectral Form Factors

Dmitrii A. Trunin , David A. Huse This is my paper
classification quant-ph cond-mat.dis-nncond-mat.stat-mechnlin.CDnlin.CG
keywords spectral form factorcircular beta-ensembleCoulomb gasDebye-Waller factorlevel repulsionHeisenberg timepermutation circuitsrandom matrix theory
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 argues that the spectral form factor (SFF) of quantum systems with extremely strong eigenvalue repulsion does not settle to its plateau at the Heisenberg time t_H. Instead, at integer multiples of t_H the SFF keeps periodic 'Bragg' peaks, damped by a Debye-Waller factor e^{-2W} that the authors derive within a quadratic approximation for the circular β-ensemble Coulomb gas. The predicted time scale t* ≈ t_H√(β/4) marks when the peaks vanish and the plateau is reached, which is much later than t_H when β is large. The same crystalline oscillation appears in perturbed permutation circuits and in a random Lax-matrix ensemble, suggesting a broader universality class of intermediate level statistics. If correct, strongly level-repelling quantum systems have a new long-time dynamical regime.

Core claim

The paper's central claim is that the SFF of the circular β-ensemble at inverse temperature β ≫ 1 behaves near multiples of the Heisenberg time as K(t_H τ) ≈ d + e^{-2W} d^2, with Debye-Waller exponent e^{-2W} ≈ d^{-4τ²/β} for 4τ²/β ≪ 1, log(d)/d at the crossover 4τ²/β = 1, and (β/τ²d)(Cπ)^{-4τ²/β} for 4τ²/β ≫ 1. The plateau is therefore reached only at t* ≈ t_H√(β/4). The paper further estimates that the SFF's singularities at integer multiples of the Heisenberg time have order γ = 4τ²/β − 1, recovering the known β = 1, 2, 4 results at τ = 1 (and τ = 2 for β = 4), and it reproduces the same damped crystalline oscillations in a perturbed permutation circuit and in a random Lax-matrix ensembl

What carries the argument

The central object is the Debye-Waller factor e^{-2W}, borrowed from Bragg diffraction, which suppresses the periodic SFF peaks as thermal fluctuations move eigenvalues away from exactly evenly spaced positions. It is computed from a quadratic expansion of the log-gas (Coulomb gas) Hamiltonian around the equidistant saddle point, treating eigenvalue displacements as independent momentum modes; the same damping arises in the other two models from the perturbation parameter g (permutation circuits) and the rod length g (Lax ensemble).

Load-bearing premise

The derivation assumes the fluctuations of eigenvalues around their evenly spaced positions stay small enough that only the quadratic part of the repulsion energy matters up to the predicted plateau time; if cubic and higher-order terms become significant earlier, the Debye-Waller exponent, the singularity orders, and the new time scale t* would all change.

What would settle it

Measure the SFF of the circular β-ensemble numerically for a fixed large β (e.g., β ≈ 500, d ≈ 512) and check the heights of the peaks at τ = 2 and 3: the ratio K(t_H τ)/d^2 should follow e^{-2W} ≈ d^{-4τ²/β}. If the peaks decay faster than this, or if the SFF reaches its plateau before t_H√(β/4), the quadratic approximation is not valid and the central prediction fails.

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

If this is right

  • The SFF of strongly level-repelling unitary systems does not plateau at the Heisenberg time; it keeps periodic crystal-like peaks up to t* ≈ t_H√(β/4).
  • The singularities of the SFF at integer multiples of the Heisenberg time have order γ = 4τ²/β − 1; for β = 1, 2, 4 this reproduces known features at τ = 1 (and τ = 2 for β = 4), showing those standard cases are remnants of Bragg peaks.
  • The perturbed permutation circuit U = e^{−igH}S and the random Lax-matrix ensemble provide explicit models that interpolate between β = 2 (CUE) and β = ∞ (picket fence), with SFF peak damping controlled by g.
  • The delay of the plateau to t* sets a new, longer time window over which eigenbasis dephasing is incomplete, affecting correlation functions and entanglement dynamics in these systems.
  • The Debye-Waller formula is testable on quantum processors, since the models can be implemented with shallow circuits.

Where Pith is reading between the lines

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

  • The same Debye-Waller mechanism should appear in other systems with large-β level statistics, such as the Gaussian β-ensemble or weakly disordered Anderson chains; a direct check would be to compute the SFF at τ = 1, 2 and fit the peak envelope to Eq. (11).
  • The paper's Gaussian truncation is expected to fail for non-rational β/2 at times τ ≫ β; a higher-order expansion, including the cubic vertex computed in the supplement, could predict how the SFF peak heights deviate from the pure Debye-Waller form at intermediate τ, giving a sharper falsifier.
  • Because the SFF is the Fourier transform of the two-level correlation function, the prediction of periodic peaks up to t* implies that the eigenvalue density autocorrelation retains sharp Bragg-like features out to distances ~√β times the mean spacing, which could be measured in level-spacing histograms.
  • The hard-rod gas underlying the Lax ensemble suggests an alternative experimental route: instead of tuning β, one tunes the rod length g; observing the late-time period d/g with amplitude ~d/((1−g)t) would distinguish this universality class from the circular β-ensemble.

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

2 major / 4 minor

Summary. The paper studies the spectral form factor (SFF) of unitary systems with very strong eigenvalue repulsion. It models the eigenvalues of the circular β-ensemble as a low-temperature Coulomb gas, expands the log-gas Hamiltonian around the equidistant 'crystal' configuration, and derives a Debye-Waller damping of the crystalline SFF: K(t_H τ) ≈ d + e^{-2W} d², with e^{-2W} given by the three asymptotic regimes in Eq. (11). It predicts that the plateau is reached only at t* ≈ t_H√(β/4), estimates the order of SFF singularities at integer multiples of the Heisenberg time via Eq. (12), and reproduces the crystalline oscillation in two further models: the perturbed permutation circuit U = e^{-igH}S and the random Lax-matrix ensemble. Numerical checks using d = 512 and β > 10 are reported for the circular β-ensemble, with stated agreement to a few percent in the regime e^{-2W} > 0.01.

Significance. If correct, the predicted long-lived crystalline regime with periodically peaked SFF up to t* ≫ t_H is a substantial addition to spectral statistics beyond the standard Wigner-Dyson classes, interpolating between β = ∞ permutation circuits and finite-β random matrix behavior. The derivation is internally consistent and not circular: the Debye-Waller factor follows from the Coulomb-gas Hamiltonian, and the Lax result is taken from the independent work of Bogomolny et al. The paper also provides reproducible numerical checks based on the Killip-Nenciu representation of the circular β-ensemble. The main weaknesses are the unexplained numerical constant C in Eq. (9)/(11) and the fact that the numerical verification does not extend into the deep-tail/plateau-onset regime; both need to be addressed before the central time-scale claim is fully supported.

major comments (2)
  1. [SFF of the Coulomb gas, Eq. (9) and Supplemental Eq. (8)] The constant C ≈ 3.6 is simply stated as a 'numerical coefficient'. Because C enters the exponent of the tail prediction (β/τ²d)(Cπ)^{-4τ²/β} in Eq. (11), an unexplained fitted C would reduce the claimed derivation to a two-parameter fit. The elementary evaluation of the momentum sum in Supplemental Eq. (8) gives C = 2e^γ ≈ 3.56; please show that derivation explicitly (or state precisely which regularization of the divergent sum is used).
  2. [Eq. (11), Fig. 2, and t* claim] The numerical support is limited to β > 10 and e^{-2W} > 0.01. However, the central plateau-time prediction t* ≈ t_H√(β/4) concerns the regime in which the Debye-Waller damping has already reduced the peaks to a small fraction of d²; for d = 512 the crossover is at the edge of the tested window, and for larger d it lies outside. Please either provide numerical data extending to e^{-2W} ≲ 0.001 (including larger d), or give an analytic bound showing that anharmonic terms beyond the one-loop estimate in Supplemental Sec. II do not shift e^{-2W} by a nonperturbative amount up to τ ≈ √β/2. This is load-bearing because t* is one of the headline results.
minor comments (4)
  1. [Eq. (4)] The first line appears to be a typographical error: S|n⟩|−⟩ = S|n⟩|+⟩ should presumably read S|n⟩|−⟩ = |n⟩|+⟩, i.e., the particle flips but does not move. Please correct.
  2. [Eq. (11)] The three asymptotic expressions do not match at the formal boundary 4τ²/β = 1: the first regime gives d^{-1}, the second gives log(d)/d. State the matching convention used, or define exactly which quantity is plotted in Fig. 2 so the reader can see how the crossover is obtained.
  3. [Fig. 3 inset and Eq. (14)] The model predicts α = d/g² for the pinning potential, but the inset fits α to the SFF instead of comparing with this prediction. Please report the fitted values alongside d/g² so the Debye-Waller factor in the perturbed permutation circuit is tested without a fitted parameter.
  4. [General] There is a typo 'Supplemental Meterial' in the Conclusion; also the sample counts in figure captions (e.g., '106 samples') should be typeset as 10^6.

Circularity Check

0 steps flagged

No significant circularity: the Debye-Waller factor is computed from the stated log-gas Hamiltonian, with external benchmarks and no load-bearing self-citation.

full rationale

The central derivation is self-contained and not circular. The Debye-Waller damping of the crystalline SFF is obtained by expanding the circular β-ensemble Coulomb-gas Hamiltonian (2) around the β=∞ picket-fence saddle point, retaining the quadratic terms (Eq. 8), and evaluating the Gaussian average in Eq. (9). The damping factor e^{-2W} is the result of that calculation, not an input to it; the paper's own self-consistency estimate (footnote [48] and Supplemental Sec. II) checks the Gaussian truncation, and the singularities (12) are benchmarked against the exact β=1,2,4 results. The numerical coefficient C≈3.6 in Eq. (9) is a constant inside a logarithm of an explicitly evaluated mode sum; even if its provenance is not fully displayed, the leading exponent and the plateau time t*≈t_H√(β/4) do not depend on it, so it is not a fitted surrogate for the claimed prediction. The toy model prediction (14) is derived by perturbation theory in Supplemental Sec. III, not extracted from the data; the α value in the Fig. 3 inset is explicitly a fit to the local-circuit SFF and is used only for a qualitative comparison, not as a prediction of the new time scale. The random Lax ensemble SFF (17) is taken from the independent work of Bogomolny, Giraud, and Schmit [75], and the connection between the random-walk model (3) and the circular β-ensemble is imported from Killip-Nenciu and Killip-Stoiciu [44,45], both external and reproducible. The only self-citation involving an author (Ref. [47]) concerns quantum-circuit realizations and is not load-bearing. The paper explicitly flags the limitation that its Gaussian approximation is only controlled for τ≪β and that for rational β/2=p/q the exact SFF has no singularities for τ>p (after Eq. 12 and footnote [49]); this is a validity caveat, not evidence of circularity. The central claim therefore has independent mathematical content with respect to its inputs.

Axiom & Free-Parameter Ledger

3 free parameters · 7 axioms · 0 invented entities

No new particles, forces, or conserved quantities are introduced. 'Eigenvalue crystal' and 'pinning potential' are effective-model descriptions of existing ensembles, not new entities. The main empirical inputs are the constants C and α listed above; the remaining assumptions are standard log-gas results, the harmonic expansion, and external exact formulas.

free parameters (3)
  • C (Debye-Waller cutoff constant) = ≈3.6
    Appears in Eq. (9) and Eq. (11) inside the Debye-Waller exponent |C d sin(πk/d)|^{-4t²/(βd²)}. The paper labels it 'numerical coefficient' without derivation; it may be 2e^γ, but this is not stated, so its value is an empirical input to the large-time tail.
  • α (pinning-potential strength in perturbed permutation circuit inset) = fitted (no value in text)
    Fig. 3 inset says α is 'determined by fitting the SFF with Eq. (14)', rather than predicted; the analytic expression α = d/g² is mentioned in the main text, but the shown comparison uses a fit.
  • α_local (Supplemental Fig. 3 numerical factor) = ≈8
    Supplemental Sec. IV reports 'The numerical factor α ≈ 8' determined from the local circuit SFF, an empirical matching factor.
axioms (7)
  • domain assumption The circular β-ensemble level statistics are exactly realized by the random-walk-like model U = SLS†M with block distributions (7).
    Invoked in the main text around Eqs. (3)-(7); relies on Refs. [44,45] for the proof. It is an external theorem, not rederived.
  • domain assumption At large β, fluctuations around the equidistant crystal are small and the log-gas Hamiltonian can be truncated at quadratic order (Eq. (8)).
    Central to the SFF derivation; the paper supplies a self-consistency check [48] and Supplemental Sec. II giving the τ ≪ β validity bound.
  • domain assumption For t/d = τ ≪ β the Gaussian average (9) is accurate and can be differentiated to locate singularities.
    Used to obtain Eq. (12); the paper itself limits the Gaussian approximation to τ ≪ β and to τ ≤ p for β/2 = p/q rational.
  • domain assumption For U = e^{-igH}S with g ≪ 1 and non-degenerate permutation spectrum, first-order perturbation theory gives eigenvalue correlations ⟨x_m x_n⟩ = (g²/d) δ_{mn}.
    Supplemental Sec. III; assumes GUE normalization and random phases on S.
  • standard math Large random permutations have a largest cycle of length O(d).
    Used to estimate Heisenberg time t_H = O(d) for the toy model; cited to Ford [73].
  • domain assumption The exact SFF of the Lax ensemble in the d → ∞ limit is Eq. (17) from Ref. [75].
    The paper imports this result rather than deriving it; it is independent external support.
  • standard math For large d, lattice sums can be replaced by logarithms/integrals, including the constant C ≈ 3.6.
    Used in Eq. (9) and Supplemental Eq. (8); the constant's origin is not displayed.

pith-pipeline@v1.3.0-alltime-deepseek · 16016 in / 16583 out tokens · 149764 ms · 2026-08-03T16:57:33.340887+00:00 · methodology

0 comments
read the original abstract

We investigate crystalline-like behavior of the spectral form factor in unitary quantum systems with extremely strong eigenvalue repulsion. Using a low-temperature Coulomb gas as a model of repulsive eigenvalues, we derive the Debye-Waller factor suppressing periodic oscillations of the spectral form factor and estimate the order of its singularities at multiples of the Heisenberg time. We also reproduce this crystalline-like behavior using perturbed permutation circuits and random matrix ensembles associated with Lax matrices. Our results lay a foundation for future studies of quantum systems that exhibit intermediate level statistics between standard random matrix ensembles and permutation circuits.

Figures

Figures reproduced from arXiv: 2512.11054 by David A. Huse, Dmitrii A. Trunin.

Figure 1
Figure 1. Figure 1: The SFF of the circular β-ensemble, which describes a Coulomb crystal of d repulsive eigenvalues at inverse tem￾perature β. We illustrate this for d = 512, β = 500, averaged over 106 samples. Dashed line shows the damping Debye￾Waller factor. Inset: deviation of the SFF from the plateau (zoom in the same graph). circular β-ensemble (Coulomb crystal of eigenvalues), perturbed permutation circuit, and random… view at source ↗
Figure 3
Figure 3. Figure 3: The SFF of toy model (13) interpolating between [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: SFF of the random ensemble of Lax matrices (15). [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 1
Figure 1. Figure 1: Quantum circuits implementing the random-walk-like model of the circular [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The SFF of the interpolating model U = e −igHS for g = 1/200 (a), g = 1/20 (b), and g = 1/8 (c). We set d = 512 and average over 1000 samples. Dashed lines denote analytical fits implied by Eq. (15) or circular unitary ensemble universality. Permutation S contains a single random cycle of length d. two-point correlators, which, in turn, decouple due to identity (13). This allows us to estimate the SFF: K(t… view at source ↗
Figure 3
Figure 3. Figure 3: The SFF of the local interpolating model for [PITH_FULL_IMAGE:figures/full_fig_p012_3.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

103 extracted references · 59 linked inside Pith

  1. [1]

    C. E. Porter, Statistical Theories of Spectra: Fluctu- ations, Perspectives in Physics (Academic Press, New York, NY, 1965)

    1965

  2. [2]

    M. L. Mehta, Random Matrices , Pure and Applied Mathematics, Vol. 142 (Elsevier, New York, 2004)

    2004

  3. [3]

    P. J. Forrester, Log-Gases and Random Matrices , Lon- don Mathematical Society Monographs, Vol. 34 (Prince- ton University Press, Princeton, NJ, 2010)

    2010

  4. [4]

    Haake, Quantum Signatures of Chaos (Springer- Verlag Berlin Heidelberg, 2010)

    F. Haake, Quantum Signatures of Chaos (Springer- Verlag Berlin Heidelberg, 2010)

    2010

  5. [5]

    Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, England, 1999)

    H.-J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, England, 1999)

    1999

  6. [6]

    T. Guhr, A. Müller-Groeling, and H. A. Weidenmüller, Random matrix theories in quantum physics: Com- mon concepts, Phys. Rept. 299, 189 (1998) , arXiv:cond- mat/9707301

    arXiv 1998

  7. [7]

    D’Alessio, Y

    L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65, 239 (2016) , arXiv:1509.06411

    Pith/arXiv arXiv 2016

  8. [8]

    A complete classification of random matrix ensembles reveals additional spectral features [ 98, 99]

  9. [9]

    Bohigas, M

    O. Bohigas, M. J. Giannoni, and C. Schmit, Character- ization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52, 1 (1984)

    1984

  10. [10]

    M. V. Berry and M. Tabor, Level clustering in the reg- ular spectrum, Proc. R. Soc. Lond. A 356, 375 (1977)

    1977

  11. [11]

    H. A. Weidenmüller and G. E. Mitchell, Random matri- ces and chaos in nuclear physics. Part 1. nuclear struc- ture, Rev. Mod. Phys. 81, 539 (2009) , arXiv:0807.1070

    Pith/arXiv arXiv 2009

  12. [12]

    Srednicki, Chaos and quantum thermalization, Phys

    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994) , arXiv:cond-mat/9403051

    Pith/arXiv arXiv 1994

  13. [13]

    Rigol, V

    M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum sys- tems, Nature 452, 854 (2008) , arXiv:0708.1324

    Pith/arXiv arXiv 2008

  14. [14]

    Sachdev, Quantum Phase Transitions , Lecture Notes in Physics Monographs (Vol

    S. Sachdev, Quantum Phase Transitions , Lecture Notes in Physics Monographs (Vol. 18) (Cambridge University Press, Cambridge, England, 2011)

    2011

  15. [15]

    Nandkishore and D

    R. Nandkishore and D. A. Huse, Many body localiza- tion and thermalization in quantum statistical mechan- ics, Ann. Rev. Condensed Matter Phys. 6, 15 (2015) , arXiv:1404.0686

    Pith/arXiv arXiv 2015

  16. [16]

    D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Colloquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys. 91, 021001 (2019) , arXiv:1804.11065

    Pith/arXiv arXiv 2019

  17. [17]

    Collins and I

    B. Collins and I. Nechita, Random matrix techniques in quantum information theory, J. Math. Phys. 57, 015215 (2016), arXiv:1509.04689

    Pith/arXiv arXiv 2016

  18. [18]

    Carmichael, An Open Systems Approach to Quantum Optics, Lecture Notes in Physics Monographs (Vol

    H. Carmichael, An Open Systems Approach to Quantum Optics, Lecture Notes in Physics Monographs (Vol. 18) (Springer-Verlag Berlin Heidelberg, 1993)

    1993

  19. [19]

    C. W. J. Beenakker, Random-matrix theory of quantum transport, Rev. Mod. Phys. 69, 731 (1997) , arXiv:cond- mat/9612179

    arXiv 1997

  20. [20]

    Brézin, C

    E. Brézin, C. Itzykson, G. Parisi, and J. B. Zuber, Pla- nar diagrams, Commun. Math. Phys. 59, 35 (1978)

    1978

  21. [21]

    J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker, D. Stanford, A. Streicher, and M. Tezuka, Black holes and random matrices, J. High Energy Phys. 2017 (05), 118, [Erratum: J. High Energy Phys. 2022 (09), 002], arXiv:1611.04650

    Pith/arXiv arXiv 2017

  22. [22]

    P. Saad, S. H. Shenker, and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115

    Pith/arXiv arXiv 1903

  23. [23]

    Cotler and K

    J. Cotler and K. Jensen, AdS 3 gravity and random CFT, J. High Energy Phys. 2021 (04), 033, arXiv:2006.08648

    Pith/arXiv arXiv 2021

  24. [24]

    Belin, J

    A. Belin, J. de Boer, D. L. Jafferis, P. Nayak, and J. Son- ner, Approximate CFTs and random tensor models, J. High Energy Phys. 2024 (09), 163, arXiv:2308.03829

    Pith/arXiv arXiv 2024

  25. [25]

    D. L. Jafferis, L. Rozenberg, and G. Wong, 3d gravity as a random ensemble, J. High Energy Phys. 2025 (02), 208, arXiv:2407.02649

    Pith/arXiv arXiv 2025

  26. [26]

    Farrelly, A review of quantum cellular automata, Quantum 4, 368 (2020) , arXiv:1904.13318

    T. Farrelly, A review of quantum cellular automata, Quantum 4, 368 (2020) , arXiv:1904.13318

    Pith/arXiv arXiv 2020

  27. [27]

    Arrighi, An overview of quantum cellular automata, Natural Comput

    P. Arrighi, An overview of quantum cellular automata, Natural Comput. 18, 885 (2019) , arXiv:1904.12956

    Pith/arXiv arXiv 2019

  28. [28]

    Gopalakrishnan and B

    S. Gopalakrishnan and B. Zakirov, Facilitated quan- tum cellular automata as simple models with nonther- mal eigenstates and dynamics, Quantum Sci. Technol. 3, 044004 (2018) , arXiv:1802.07729

    Pith/arXiv arXiv 2018

  29. [29]

    Gopalakrishnan, Operator growth and eigenstate en- tanglement in an interacting integrable Floquet system, Phys

    S. Gopalakrishnan, Operator growth and eigenstate en- tanglement in an interacting integrable Floquet system, Phys. Rev. B 98, 060302 (2018) , arXiv:1806.04156

    Pith/arXiv arXiv 2018

  30. [30]

    Gopalakrishnan, D

    S. Gopalakrishnan, D. A. Huse, V. Khemani, and R. Vasseur, Hydrodynamics of operator spreading and quasiparticle diffusion in interacting integrable systems, 6 Phys. Rev. B 98, 220303 (2018) , arXiv:1809.02126

    Pith/arXiv arXiv 2018

  31. [31]

    V. Alba, J. Dubail, and M. Medenjak, Operator entan- glement in interacting integrable quantum systems: The case of the rule 54 chain, Phys. Rev. Lett. 122, 250603 (2019), arXiv:1901.04521

    Pith/arXiv arXiv 2019

  32. [32]

    Iaconis, S

    J. Iaconis, S. Vijay, and R. Nandkishore, Anomalous subdiffusion from subsystem symmetries, Phys. Rev. B 100, 214301 (2019) , arXiv:1907.10629

    Pith/arXiv arXiv 2019

  33. [33]

    Iaconis, A

    J. Iaconis, A. Lucas, and R. Nandkishore, Multipole conservation laws and subdiffusion in any dimension, Phys. Rev. E 103, 022142 (2021) , arXiv:2009.06507

    Pith/arXiv arXiv 2021

  34. [34]

    Feldmeier, P

    J. Feldmeier, P. Sala, G. de Tomasi, F. Pollmann, and M. Knap, Anomalous diffusion in dipole- and higher- moment conserving systems, Phys. Rev. Lett. 125, 245303 (2020) , arXiv:2004.00635

    Pith/arXiv arXiv 2020

  35. [35]

    L. E. Hillberry, L. Piroli, E. Vernier, N. Y. Halpern, T. Prosen, and L. D. Carr, Integrability of Goldilocks quantum cellular automata, arXiv:2404.02994

    Pith/arXiv arXiv

  36. [36]

    Bertini, K

    B. Bertini, K. Klobas, P. Kos, and D. Malz, Quan- tum and classical dynamics with random permu- tation circuits, Phys. Rev. X 15, 011015 (2025) , arXiv:2407.11960

    Pith/arXiv arXiv 2025

  37. [37]

    Szász-Schagrin, M

    D. Szász-Schagrin, M. Mazzoni, B. Bertini, K. Klobas, and L. Piroli, Entanglement dynamics and Page curves in random permutation circuits, arXiv:2505.06158

    arXiv

  38. [38]

    Bertini, K

    B. Bertini, K. Klobas, P. Kos, and D. Malz, Random permutation circuits are quantum chaotic, arXiv:2508.10890

    arXiv

  39. [39]

    Possibly shifted by a constant phase

  40. [40]

    Note also an analogy between the eigenvalues of a ran- dom matrix ensemble and a system of spinless fermions with long-range repulsive potential [ 3], which also im- plies oscillations of the SFF [ 100, 101]

  41. [41]

    N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, New York, 1976)

    1976

  42. [42]

    We emphasize that this description does not rely on the specific properties of β = 1, 2, 4

    The equilibration of the Coulomb gas is described by the Dyson Brownian motion [ 1–3] and reproduces probabil- ity distribution (2). We emphasize that this description does not rely on the specific properties of β = 1, 2, 4

  43. [43]

    According to the Frobenius theorem, all such algebras are isomorphic to either real, complex, or quaternion numbers

    Physically sound basis transformations are described by an associative division algebra over real numbers. According to the Frobenius theorem, all such algebras are isomorphic to either real, complex, or quaternion numbers. These algebras correspond to the orthogonal (β = 1 ), unitary ( β = 2 ), and symplectic ( β = 4 ) basis transformations, respectively

  44. [44]

    Killip and I

    R. Killip and I. Nenciu, Matrix models for circular ensembles, Int. Math. Res. Not. 2004, 2665 (2004) , arXiv:math/0410034

    Pith/arXiv arXiv 2004

  45. [45]

    Killip and M

    R. Killip and M. Stoiciu, Eigenvalue statistics for CMV matrices: From Poisson to clock via random matrix en- sembles, Duke Math. J. 146, 361 (2009) , arXiv:math- ph/0608002

    arXiv 2009

  46. [46]

    M. J. Cantero, L. Moral, and L. Velazquez, Five- diagonal matrices and zeros of orthogonal polynomials on the unit circle, Linear Algebra Appl. 362, 29 (2003) , arXiv:math/0204300

    Pith/arXiv arXiv 2003

  47. [47]

    Kolganov and D

    N. Kolganov and D. A. Trunin, Streamlined Krylov con- struction and classification of ergodic Floquet systems, Phys. Rev. E 111, L052202 (2025) , arXiv:2412.19797

    Pith/arXiv arXiv 2025

  48. [48]

    Hence, it is justified for β > 1 and any nu- merically feasible system size ( d < 108)

    Expansion near infinite-temperature eigenvalues is self- consistent if the fluctuation amplitude is smaller than average level spacing, i.e., ∆x2 k ≈ 2 log d/βd2 < (2π/d)2. Hence, it is justified for β > 1 and any nu- merically feasible system size ( d < 108)

  49. [49]

    (13.226) in [ 3], implies that the SFF has no singularities for τ > p , whereas Eq

    For a rational β/2 = p/q with p and q mutually prime, the asymptotic expansion of the density-density corre- lation function of the circular β-ensemble, Eq. (13.226) in [ 3], implies that the SFF has no singularities for τ > p , whereas Eq. (12) is valid for τ ≤ p

  50. [50]

    T. A. Brody, A statistical measure for the repulsion of energy levels, Lett. Nuovo Cimento 7, 482 (1973)

    1973

  51. [51]

    T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, and S. S. M. Wong, Random-matrix physics: spectrum and strength fluctuations, Rev. Mod. Phys 53, 385 (1981)

    1981

  52. [52]

    M. V. Berry and M. Robnik, Semiclassical level spac- ings when regular and chaotic orbits coexist, J. Phys. A: Math. Gen. 17, 2413 (1984)

    1984

  53. [53]

    Moshe, H

    M. Moshe, H. Neuberger, and B. Shapiro, A generalized ensemble of random matrices, Phys. Rev. Lett. 73, 1497 (1994), arXiv:cond-mat/9403085

    Pith/arXiv arXiv 1994

  54. [54]

    A. D. Mirlin, Y. V. Fyodorov, F.-M. Dittes, J. Quezada, and T. H. Seligman, Transition from localized to ex- tended eigenstates in the ensemble of power-law ran- dom banded matrices, Phys. Rev. E 54, 3221 (1996) , arXiv:cond-mat/9604163

    Pith/arXiv arXiv 1996

  55. [55]

    Bogomolny and O

    E. Bogomolny and O. Giraud, Eigenfunction entropy and spectral compressibility for critical random ma- trix ensembles, Phys. Rev. Lett. 106, 044101 (2011) , arXiv:1011.3686

    Pith/arXiv arXiv 2011

  56. [56]

    Evers and A

    F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80, 1355 (2008) , arXiv:0707.4378

    Pith/arXiv arXiv 2008

  57. [57]

    Pandey and M

    A. Pandey and M. L. Mehta, Gaussian ensembles of ran- dom Hermitian matrices intermediate between orthog- onal and unitary ones, Commun. Math. Phys. 87, 449 (1983)

    1983

  58. [58]

    Lenz and F

    G. Lenz and F. Haake, Transitions between universality classes of random matrices, Phys. Rev. Lett. 65, 2325 (1990)

    1990

  59. [59]

    Lenz and F

    G. Lenz and F. Haake, Reliability of small matrices for large spectra with nonuniversal fluctuations, Phys. Rev. Lett. 67, 1 (1991)

    1991

  60. [60]

    Dupuis and G

    N. Dupuis and G. Montambaux, Aharonov-Bohm flux and statistics of energy levels in metals, Phys. Rev. B 43, 14390 (1991)

    1991

  61. [61]

    Repulsion of energy levels

    N. Rosenzweig and C. E. Porter, “Repulsion of energy levels” in complex atomic spectra, Phys. Rev. 120, 1698 (1960)

    1960

  62. [62]

    V. E. Kravtsov, I. M. Khaymovich, E. Cuevas, and M. Amini, A random matrix model with localization and ergodic transitions, New J. Phys. 17, 122002 (2015), arXiv:1508.01714

    Pith/arXiv arXiv 2015

  63. [63]

    F. M. Izrailev, Simple models of quantum chaos: Spec- trum and eigenfunctions, Phys. Rept. 196, 299 (1990)

    1990

  64. [64]

    Prosen and M

    T. Prosen and M. Robnik, Energy level statistics in the transition region between integrability and chaos, J. Phys. A: Math. Gen. 26, 2371 (1993)

    1993

  65. [65]

    Prosen and M

    T. Prosen and M. Robnik, Semiclassical energy level statistics in the transition region between integrability and chaos: transition from Brody-like to Berry-Robnik behaviour, J. Phys. A: Math. Gen. 27, 8059 (1994)

    1994

  66. [66]

    Prosen, Time evolution of a quantum many-body system: transition from integrability to ergodicity in thermodynamic limit, Phys

    T. Prosen, Time evolution of a quantum many-body system: transition from integrability to ergodicity in thermodynamic limit, Phys. Rev. Lett. 80, 1808 (1998) , arXiv:cond-mat/9707180

    Pith/arXiv arXiv 1998

  67. [67]

    Prosen, On general relation between quantum ergod- icity and fidelity of quantum dynamics, Phys

    T. Prosen, On general relation between quantum ergod- icity and fidelity of quantum dynamics, Phys. Rev. E 65, 7 036208 (2002) , arXiv:quant-ph/0106149

    Pith/arXiv arXiv 2002

  68. [68]

    Bertini, P

    B. Bertini, P. Kos, and T. Prosen, Exact spectral form factor in a minimal model of many-body quantum chaos, Phys. Rev. Lett. 121, 264101 (2018) , arXiv:1805.00931

    Pith/arXiv arXiv 2018

  69. [69]

    A. Chan, A. De Luca, and J. T. Chalker, Spectral statistics in spatially extended chaotic quantum many- body systems, Phys. Rev. Lett. 121, 060601 (2018) , arXiv:1803.03841

    Pith/arXiv arXiv 2018

  70. [70]

    A. J. Friedman, A. Chan, A. De Luca, and J. T. Chalker, Spectral statistics and many-body quantum chaos with conserved charge, Phys. Rev. Lett. 123, 210603 (2019) , arXiv:1906.07736

    Pith/arXiv arXiv 2019

  71. [71]

    Sierant and J

    P. Sierant and J. Zakrzewski, Model of level statistics for disordered interacting quantum many-body systems, Phys. Rev. B 101, 104201 (2020) , arXiv:1907.10336

    Pith/arXiv arXiv 2020

  72. [72]

    Besides, lo- cal permutation circuits with exponentially large cycles can be constructed using primitive polynomials over the binary field, e.g., see [ 102, 103]

    We discuss a straightforward local generalization of model (13) in the Supplemental Material. Besides, lo- cal permutation circuits with exponentially large cycles can be constructed using primitive polynomials over the binary field, e.g., see [ 102, 103]

  73. [73]

    Ford, Cycle type of random permutations: A toolkit, Discrete Analysis 2022, 36 (2022) , arXiv:2104.12019

    K. Ford, Cycle type of random permutations: A toolkit, Discrete Analysis 2022, 36 (2022) , arXiv:2104.12019

    Pith/arXiv arXiv 2022

  74. [74]

    Bogomolny, O

    E. Bogomolny, O. Giraud, and C. Schmit, Random ma- trix ensembles associated with Lax matrices, Phys. Rev. Lett. 103, 054103 (2009) , arXiv:0904.4898

    Pith/arXiv arXiv 2009

  75. [75]

    Bogomolny, O

    E. Bogomolny, O. Giraud, and C. Schmit, Integrable random matrix ensembles, Nonlinearity 24, 3179 (2011), arXiv:1104.3777

    Pith/arXiv arXiv 2011

  76. [76]

    Bogomolny and C

    E. Bogomolny and C. Schmit, Spectral statistics of a quantum interval-exchange map, Phys. Rev. Lett. 93, 254102 (2004) , arXiv:nlin/0408062

    Pith/arXiv arXiv 2004

  77. [77]

    Giraud, J

    O. Giraud, J. Marklof, and S. O’Keefe, Intermediate statistics in quantum maps, J. Phys. A: Math. Gen. 37, L303 (2004) , arXiv:nlin/0403033

    Pith/arXiv arXiv 2004

  78. [78]

    Dumitriu and A

    I. Dumitriu and A. Edelman, Matrix models for beta ensembles, J. Math. Phys. 43, 5830 (2002) , arXiv:math- ph/0206043

    arXiv 2002

  79. [79]

    Jonathan Torres-Herrera, J

    E. Jonathan Torres-Herrera, J. A. Méndez-Bermúdez, and L. F. Santos, Level repulsion and dynamics in the finite one-dimensional Anderson model, Phys. Rev. E 100, 022142 (2019) , arXiv:1904.11989

    Pith/arXiv arXiv 2019

  80. [80]

    M. J. Cantero, F. A. Grünbaum, L. Moral, and L. Ve- lazquez, Matrix valued Szegő polynomials and quan- tum random walks, Commun. Pure Appl. Math. 63, 464 (2010), arXiv:0901.2244

    Pith/arXiv arXiv 2010

Showing first 80 references.