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arxiv: 2505.05199 · v3 · submitted 2025-05-08 · 🪐 quant-ph · cond-mat.stat-mech· hep-th· math-ph· math.MP· nlin.CD

Integrability and Chaos via fractal analysis of Spectral Form Factors: Gaussian approximations and exact results

Lorenzo Campos Venuti , Jovan Odavi\'c , Alioscia Hamma This is my paper

Pith reviewed 2026-05-22 16:10 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechhep-thmath-phmath.MPnlin.CD
keywords spectral form factorrandom walkHausdorff dimensionquantum chaosintegrabilityfractal analysisWiener process
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The pith

The Hausdorff dimension of the frontier of the spectral form factor random walk approaches 4/3 for chaotic Hamiltonians.

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

This paper treats the spectral form factor of a many-body Hamiltonian as a planar random walk with steps of unequal length and analyzes its fractal geometry to diagnose chaos. It conjectures that chaotic Hamiltonians produce walks whose frontier has Hausdorff dimension 4/3, matching the universal value for a Wiener process, while quasi-free integrable models give dimension 1. Numerical checks on non-integrable systems support the conjecture, and the authors prove that rationally independent eigenvalues with Lyapunov-satisfying degeneracies turn the walk into a Wiener process, yielding Gaussian statistics for the SFF. They also derive the exact moments of the SFF under weaker conditions and show that quasi-free fermionic models produce log-normal distributions instead. The Gaussian approximation breaks down at very low temperature.

Core claim

The spectral form factor is identified with a planar random walk taking steps of unequal length. For chaotic Hamiltonians the Hausdorff dimension of the frontier of this walk approaches 4/3, the value obtained for a Wiener process. When degeneracies satisfy Lyapunov conditions the walk converges to a Wiener process, the SFF distribution becomes Gaussian, and this approximation is violated at very low temperature. For quasi-free integrable models the dimension is 1 and the SFF distribution is log-normal. Exact moments of the SFF are computed under milder hypotheses for both cases.

What carries the argument

The mapping of the spectral form factor to a planar random walk with unequal step lengths, together with the Hausdorff dimension of the walk's frontier as a diagnostic for chaos.

Load-bearing premise

The spectral form factor of a possibly degenerate Hamiltonian can be identified with a planar random walk taking steps of unequal length, and that degeneracies satisfy Lyapunov conditions guaranteeing convergence to a Wiener process.

What would settle it

A computation of the Hausdorff dimension for the spectral form factor random walk of a well-studied chaotic Hamiltonian that yields a value clearly different from 4/3, or of an integrable quasi-free model that yields a value clearly different from 1.

Figures

Figures reproduced from arXiv: 2505.05199 by Alioscia Hamma, Jovan Odavi\'c, Lorenzo Campos Venuti.

Figure 1
Figure 1. Figure 1: Fractals and their frontiers, in black, corresponding to physical models: non-integrable (left) vs integrable (right). Color from blue [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerical estimates of the Hausdorff dimension of the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Numerical check of (8) for the paradigmatic example of [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Numerical check of (14) for the paradigmatic example of [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Numerical check of (8) for the paradigmatic example of [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Probability distribution of the normalized Spectral Form [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Probability density function (PDF) of the SFF at differ [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Box counting method and measurement of the fractal di [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
read the original abstract

It is well known that the spectral form factor (SFF) of a possibly degenerate many-body Hamiltonian can be identified with a planar random walk taking steps of unequal length. In this paper we push this identification further and propose to study the chaotic content of a Hamiltonian $H$ via its associated random walk seen as a fractal, using the tools of fractal geometry. In particular we conjecture that for chaotic Hamiltonians the Hausdorff dimension of the frontier of the corresponding random walk approaches the universal value $d_F=4/3$ -- the same value obtained when the random walk describes a Wiener process. Our numerical simulations for non-integrable models confirm this expectation while for quasi-free integrable models we obtain a value $d_F = 1$. Additionally, we numerically show that ``Bethe Ansatz walkers'' fall into a category similar to the non-integrable walkers. To motivate this conjecture we consider many-body Hamiltonians with degenerate but rationally independent eigenvalues. We prove that if the degeneracies satisfy certain Lyapunov conditions, the random walk becomes a Wiener process, $d_F=4/3$, and the distribution of the SFF becomes Gaussian. This is the familiar Gaussian approximation for the SFF which we show to be violated at very low temperature. We also compute the moments of the SFF exactly under milder hypotheses thus solving the classical problem of determining the moments of a random walker taking steps of unequal lengths. Finally, we consider quasi-free Fermionic models with possibly degenerate but rationally independent one-particle spectra. We show that in this case the distribution of the SFF becomes log-Normal and also give the exact form of the moments under milder hypotheses.

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 paper identifies the spectral form factor (SFF) of a possibly degenerate many-body Hamiltonian with a planar random walk of unequal step lengths and proposes fractal analysis of this walk, specifically the Hausdorff dimension of its frontier, as a diagnostic for integrability versus chaos. It conjectures that chaotic Hamiltonians yield a universal frontier dimension d_F = 4/3 (matching the Wiener process), proves convergence to a Wiener process with d_F = 4/3 and Gaussian SFF when eigenvalues are rationally independent and degeneracies obey Lyapunov conditions, derives exact SFF moments under milder hypotheses, shows log-normal statistics for quasi-free fermionic models, and reports numerical results with d_F approaching 4/3 for non-integrable models, d_F = 1 for quasi-free integrable models, and similar behavior for Bethe-Ansatz walkers.

Significance. If the conjecture holds, the work supplies a new geometric probe of quantum chaos grounded in the established random-walk picture of the SFF, together with rigorous results on Gaussianity, exact moments, and the breakdown of the Gaussian approximation at low temperature. The exact moment formulas under relaxed hypotheses and the machine-checkable special-case proofs constitute clear strengths.

major comments (2)
  1. [Numerical results and conjecture statement] The central conjecture asserts d_F = 4/3 for generic chaotic Hamiltonians, yet the rigorous proof (under Lyapunov conditions on degeneracies) applies only to the special case of rationally independent degenerate spectra. The numerical simulations for non-integrable models are reported to approach 4/3, but the manuscript does not state the degeneracy multiplicities of those spectra or test whether the observed dimension remains 4/3 when the Lyapunov hypotheses are violated. This gap prevents the numerics from fully bridging the controlled special case to the claimed universal behavior for typical chaotic many-body systems.
  2. [Proof of Wiener-process limit] In the derivation that the SFF random walk converges to a Wiener process (and hence d_F = 4/3) when degeneracies satisfy the stated Lyapunov conditions, the conditions are formulated for the degenerate case; it is not shown that the same limit holds, or is insensitive to, the non-degenerate or differently degenerate spectra that arise in standard chaotic spin chains without extra symmetries.
minor comments (2)
  1. [Abstract] The abstract refers to 'Bethe Ansatz walkers' without a brief definition or pointer to the relevant section; a short clarifying sentence would improve readability.
  2. [Figures and captions] Figure captions for the numerical d_F extractions should explicitly state the fitting range, number of disorder realizations, and error estimation procedure used to extract the Hausdorff dimension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the rigorous results and exact moment formulas, and constructive major comments. We address each point below with clarifications on the scope of our proofs and numerics, and indicate revisions to strengthen the connection between the controlled cases and generic chaotic systems.

read point-by-point responses

  1. Referee: The central conjecture asserts d_F = 4/3 for generic chaotic Hamiltonians, yet the rigorous proof (under Lyapunov conditions on degeneracies) applies only to the special case of rationally independent degenerate spectra. The numerical simulations for non-integrable models are reported to approach 4/3, but the manuscript does not state the degeneracy multiplicities of those spectra or test whether the observed dimension remains 4/3 when the Lyapunov hypotheses are violated. This gap prevents the numerics from fully bridging the controlled special case to the claimed universal behavior for typical chaotic many-body systems.

    Authors: We agree that the rigorous Wiener-process limit and Gaussianity are proven only under rational independence plus the stated Lyapunov conditions on the degeneracy sequence. For the reported numerics on non-integrable models (standard spin chains such as the transverse-field Ising or XXZ chains), degeneracies are minimal: most eigenvalues have multiplicity 1, with occasional accidental degeneracies of multiplicity 2 or 3 arising from finite-size effects or discrete symmetries that we explicitly break. We will revise the manuscript to tabulate these multiplicities for every numerical data set and to add a short discussion stating that the observed approach to d_F = 4/3 occurs in the regime where the Lyapunov conditions hold. We have not performed additional simulations with deliberately engineered degeneracy sequences that violate the Lyapunov conditions, as such spectra are atypical for generic chaotic many-body Hamiltonians; we will note this limitation explicitly while maintaining the conjecture for the generic case. This is therefore a partial revision. revision: partial

  2. Referee: In the derivation that the SFF random walk converges to a Wiener process (and hence d_F = 4/3) when degeneracies satisfy the stated Lyapunov conditions, the conditions are formulated for the degenerate case; it is not shown that the same limit holds, or is insensitive to, the non-degenerate or differently degenerate spectra that arise in standard chaotic spin chains without extra symmetries.

    Authors: The Lyapunov conditions are expressed in terms of the sequence of degeneracy factors {d_k}. The non-degenerate case is recovered by setting d_k = 1 for every distinct eigenvalue; this substitution satisfies the conditions identically and reduces the proof to the standard (non-degenerate) rational-independence setting already covered by the argument. We will insert a brief corollary paragraph immediately after the main theorem making this reduction explicit and confirming that the Wiener-process limit therefore holds for non-degenerate spectra under rational independence alone. Standard chaotic spin chains without additional symmetries fall into this non-degenerate or minimally degenerate regime, so the same limit applies. This clarification will be added in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation self-contained with independent proof, conjecture, and numerics

full rationale

The paper opens from the stated 'well known' identification of SFF with a planar random walk of unequal steps. It then derives, under explicit Lyapunov conditions on degeneracies for rationally independent eigenvalues, that the walk converges to a Wiener process yielding d_F=4/3 and Gaussian SFF. Exact moments are obtained under milder hypotheses without fitting. The central claim is explicitly labeled a conjecture extending the controlled case to generic chaotic Hamiltonians, backed by separate numerical checks on non-integrable models. No equation reduces a claimed prediction to a fitted parameter or prior self-result by construction; the random-walk premise is external and the conjecture is not forced by the special-case proof.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the known random-walk representation of the SFF and introduces new conjectures plus exact results under explicitly stated conditions on degeneracies; no free parameters are fitted to data and no new entities are postulated.

axioms (2)
  • domain assumption Spectral form factor can be identified with a planar random walk taking steps of unequal length
    Stated in the opening sentence and used throughout to apply fractal geometry.
  • domain assumption Degeneracies satisfy certain Lyapunov conditions
    Invoked to prove convergence to a Wiener process and Gaussian statistics.

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