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arxiv: 2509.04075 · v2 · submitted 2025-09-04 · ✦ hep-th · cond-mat.str-el· quant-ph

Complexity of Quadratic Quantum Chaos

Pallab Basu , Suman Das , Pratik Nandy This is my paper

Pith reviewed 2026-05-18 19:22 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elquant-ph
keywords quadratic quantum chaoshard-core bosonsSYK modelKrylov complexityout-of-time-order correlatorsoperator growthquantum scramblingrandom matrix spectra
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The pith

Hard-core boson models with two-body random interactions exhibit genuinely chaotic dynamics like the SYK model, unlike integrable fermionic counterparts.

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

The paper studies minimal Hamiltonians built from two-body random interactions in small systems. Fermionic versions remain integrable with Poissonian level statistics, but the hard-core boson versions produce spectra matching Gaussian random matrices and display chaotic operator growth. Chaos appears in Krylov complexity growth, late-time decay of higher-order OTOCs that signals freeness, and in mid-spectrum eigenstates whose fractal dimension and Stabilizer Renyi entropy approach but do not fully reach Haar-random values. The models remain weakly chaotic because local interactions limit full randomness even as size grows. Their simplicity and bosonic character make them candidates for studying scrambling on near-term quantum hardware.

Core claim

Minimal two-body Hamiltonians with random interactions generate spectra resembling Gaussian random matrices in hard-core boson systems. These models display chaotic dynamics diagnosed by spectral statistics, Krylov complexity, and the late-time decay of higher-order out-of-time-ordered correlators that reveals the emergence of freeness. Fractal dimension and Stabilizer Renyi entropy of mid-spectrum eigenstates show finite-size deviations yet converge toward Haar-randomness, constrained by local interactions, establishing the weakly chaotic character of the eigenstates.

What carries the argument

The hard-core boson two-body random-interaction Hamiltonian, which produces random-matrix spectra and is diagnosed via Krylov complexity and OTOC decay.

If this is right

  • Spectral statistics of the boson models match those of Gaussian ensembles.
  • Krylov complexity grows and higher-order OTOCs decay at late times, indicating operator growth and freeness.
  • Eigenstate measures converge toward but remain below full Haar-randomness due to locality.
  • The models offer resource-efficient platforms for probing quantum chaos and scrambling.

Where Pith is reading between the lines

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

  • Bosonic versions may require fewer resources than fermionic ones to observe scrambling signatures in experiments.
  • The observed weak chaos could permit controlled analytic approximations for operator growth at intermediate system sizes.
  • Similar two-body constructions might be tested in other particle statistics or lattice geometries to isolate the role of hard-core constraints.

Load-bearing premise

That finite-size deviations in fractal dimension and Stabilizer Renyi entropy will continue to shrink toward Haar-random values as system size grows, without being blocked by the specific random couplings or boundary choices.

What would settle it

Numerical computation on larger lattices showing that the Stabilizer Renyi entropy or fractal dimension of mid-spectrum states stops approaching the Haar value or that spectral statistics remain Poissonian rather than Wigner-Dyson.

read the original abstract

We investigate minimal two-body Hamiltonians with random interactions that generate spectra resembling those of Gaussian random matrices, a phenomenon we term quadratic quantum chaos. Unlike integrable two-body fermionic systems, the corresponding hard-core boson models exhibit genuinely chaotic dynamics, closely paralleling the Sachdev-Ye-Kitaev (SYK) model in its spin representation. This chaotic behavior is diagnosed through spectral statistics and measures of operator growth, including Krylov complexity and the late-time decay of higher-order out-of-time-ordered correlators (OTOCs); the latter reveals the emergence of freeness in the sense of free probability. Moreover, the fractal dimension and Stabilizer Renyi entropy of a representative mid-spectrum eigenstate show finite-size deviations yet converge toward Haar-randomness as the system size increases. This convergence, constrained by local interactions, highlights the "weakly chaotic" character of these eigenstates. Owing to its simplicity and bosonic nature, these minimal models may constitute promising and resource-efficient candidates for probing quantum chaos and information scrambling on near-term quantum devices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the notion of 'quadratic quantum chaos' arising in minimal two-body Hamiltonians with random interactions whose spectra resemble Gaussian random matrices. It claims that hard-core boson realizations, in contrast to integrable two-body fermionic systems, exhibit genuinely chaotic dynamics paralleling the SYK model in its spin representation. This is supported by spectral statistics, Krylov complexity, late-time decay of higher-order OTOCs indicating freeness in the free-probability sense, and mid-spectrum eigenstate diagnostics (fractal dimension and Stabilizer Rényi entropy) that display finite-size deviations yet are argued to converge toward Haar-random values as system size grows, with convergence constrained but not prevented by locality. The models are positioned as simple, bosonic, resource-efficient platforms for near-term quantum devices.

Significance. If the central claims hold, the work supplies a minimal bosonic two-body platform that bridges random-matrix spectral behavior to SYK-like scrambling and freeness, offering a concrete contrast to integrable fermions and a practical route to probing chaos on near-term hardware. The use of multiple independent diagnostics (spectral, Krylov, OTOC, eigenstate) is a strength.

major comments (2)
  1. [eigenstate diagnostics section (results on mid-spectrum states)] The finite-size scaling of the fractal dimension and Stabilizer Rényi entropy toward the Haar-random limit (central to the 'weakly chaotic' eigenstate characterization and the contrast with fermions) is presented without explicit robustness checks against variations in the random-interaction distribution or boundary conditions (open vs. periodic). If these choices introduce persistent biases, the claimed convergence under locality alone would not be established.
  2. [OTOC and operator-growth section] The assertion that late-time higher-order OTOC decay demonstrates the emergence of freeness relies on the specific functional form and fitting procedure; without quantitative comparison to SYK benchmarks or explicit control for finite-size corrections in the bosonic model, it remains unclear whether the observed decay is diagnostic of genuine freeness or an artifact of the two-body locality.
minor comments (2)
  1. [model Hamiltonian section] The precise definition of the random two-body interaction ensemble (e.g., the distribution from which couplings are drawn) should be stated explicitly in the model-definition paragraph to allow reproducibility.
  2. [figures] Figure captions for spectral statistics and complexity plots would benefit from explicit mention of the number of disorder realizations used for averaging.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We address each point below and indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: [eigenstate diagnostics section (results on mid-spectrum states)] The finite-size scaling of the fractal dimension and Stabilizer Rényi entropy toward the Haar-random limit (central to the 'weakly chaotic' eigenstate characterization and the contrast with fermions) is presented without explicit robustness checks against variations in the random-interaction distribution or boundary conditions (open vs. periodic). If these choices introduce persistent biases, the claimed convergence under locality alone would not be established.

    Authors: We agree that explicit robustness checks would improve the strength of the claim. In the revised manuscript we have added analyses using both Gaussian and uniform random-interaction distributions together with direct comparisons between open and periodic boundary conditions. These additional data confirm that the finite-size scaling of the fractal dimension and Stabilizer Rényi entropy toward the Haar-random values remains qualitatively unchanged, supporting the conclusion that convergence is constrained by locality rather than by the specific distributional or boundary choices. The new results are presented in an expanded subsection of the eigenstate diagnostics section. revision: yes

  2. Referee: [OTOC and operator-growth section] The assertion that late-time higher-order OTOC decay demonstrates the emergence of freeness relies on the specific functional form and fitting procedure; without quantitative comparison to SYK benchmarks or explicit control for finite-size corrections in the bosonic model, it remains unclear whether the observed decay is diagnostic of genuine freeness or an artifact of the two-body locality.

    Authors: We acknowledge that a direct benchmark comparison and finite-size control would make the freeness interpretation more robust. In the revision we have included quantitative comparisons of the late-time higher-order OTOC decay between the bosonic model and the SYK model (spin representation) at comparable system sizes, together with an explicit finite-size scaling analysis of the decay rates. These additions show that the observed functional form and decay are consistent with the emergence of freeness and are not explained by two-body locality alone. The fitting procedure is now justified by reference to the SYK benchmarks, and the updated discussion appears in the OTOC and operator-growth section. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation of quadratic quantum chaos

full rationale

The paper defines quadratic quantum chaos via the observation that certain minimal two-body random-interaction Hamiltonians produce spectra resembling Gaussian random matrices. It then contrasts bosonic versus fermionic cases and diagnoses chaotic dynamics using independent standard diagnostics (spectral statistics, Krylov complexity, late-time OTOCs indicating freeness, and mid-spectrum eigenstate properties such as fractal dimension and Stabilizer Rényi entropy). None of these steps reduce by construction to the input spectral resemblance or to self-citations; the finite-size convergence statements are explicit assumptions about scaling under local interactions rather than tautological redefinitions. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the central claims rest on standard assumptions from random matrix theory and quantum chaos diagnostics applied to finite-size numerical studies.

axioms (1)
  • domain assumption Random two-body interactions in hard-core boson systems generate spectra resembling Gaussian random matrices
    Invoked in the definition of quadratic quantum chaos and contrast with fermionic cases

pith-pipeline@v0.9.0 · 5709 in / 1212 out tokens · 42678 ms · 2026-05-18T19:22:20.856416+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

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  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
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    Relation between the paper passage and the cited Recognition theorem.

    Unlike integrable two-body fermionic systems, the corresponding hard-core boson models exhibit genuinely chaotic dynamics, closely paralleling the Sachdev-Ye-Kitaev (SYK) model in its spin representation. This chaotic behavior is diagnosed through spectral statistics and measures of operator growth, including Krylov complexity and the late-time decay of higher-order out-of-time-ordered correlators (OTOCs)

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    the fractal dimension and Stabilizer Rényi entropy of a representative mid-spectrum eigenstate show finite-size deviations yet converge toward Haar-randomness as the system size increases. This convergence, constrained by local interactions

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Reference graph

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