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arxiv: 2604.14798 · v1 · submitted 2026-04-16 · 🪐 quant-ph · math-ph· math.MP· nlin.CD

Integrable, Mixed, and Chaotic Dynamics in a Single All-to-All Ising Spin Model

David Amaro-Alcal\'a , Carlos Pineda This is my paper

Pith reviewed 2026-05-10 11:16 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MPnlin.CD
keywords all-to-all Ising modelquantum chaoskicked topsymmetry sectorsintegrable dynamicschaotic dynamicsnoise resilience
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The pith

A single all-to-all Ising model with fixed parameters exhibits integrable, mixed, and chaotic dynamics across its symmetry sectors.

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

The authors establish that one fixed Hamiltonian can host the full range of dynamical regimes by examining its symmetry blocks separately rather than by tuning couplings. Each block maps onto a kicked top whose effective parameters are set by the block dimension, placing some blocks in integrable regimes, others in chaotic ones, and some in mixed territory. The construction stays stable when added noise has norm near one. A reader should care because this supplies an elementary many-body example in which symmetry alone selects the character of the motion, without needing to change the microscopic rules.

Core claim

We demonstrate that the Ising all-to-all model exhibits a range of dynamics, from integrable to chaotic, including mixed behaviour across symmetry blocks within a single system. We achieve this by mapping each symmetry sector to a kicked top and observing that the kicked-top parameters for each sector depend on its dimension. The system remains resilient to noise when the norm of the Hamiltonian representing the noise is close to 1. This provides a new platform for studying dynamics determined by the symmetry sector.

What carries the argument

The dimension-dependent mapping that sends each symmetry sector of the all-to-all Ising Hamiltonian to an effective kicked top whose control parameters are fixed by the sector size.

If this is right

  • A single fixed Hamiltonian supplies separate integrable, mixed, and chaotic sectors that can be studied in parallel.
  • The model functions as a quantum counterpart to classical mixed systems such as the Bunimovich billiard.
  • Noise with Hamiltonian norm near unity leaves the sector-dependent dynamics intact.
  • Dynamical character is selected by symmetry sector rather than by external parameter adjustment.

Where Pith is reading between the lines

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

  • Experimental platforms that can prepare and measure within individual symmetry sectors could directly test the predicted crossover from regular to chaotic motion.
  • Similar dimension-dependent reductions may exist in other permutation-symmetric spin models, offering a systematic route to engineered chaos.
  • The construction suggests searching for all-to-all Hamiltonians whose symmetry sectors realize other solvable limits, such as the Lipkin-Meshkov-Glick model at different fillings.

Load-bearing premise

The effective kicked-top parameters extracted from each symmetry sector correctly reproduce the long-time dynamics and chaos diagnostics of the original many-body Hamiltonian.

What would settle it

Compute the level-spacing statistics or the long-time decay of an out-of-time-order correlator inside a large symmetry sector of the full Ising model and compare it with the same quantities in the corresponding kicked top; disagreement would refute the mapping.

Figures

Figures reproduced from arXiv: 2604.14798 by Carlos Pineda, David Amaro-Alcal\'a.

Figure 1
Figure 1. Figure 1: Phase space, Jx vs Jz, for the evolution of the KT with the parameters tau and alpha mentioned. The range of the variables is the same as the frames in plot c). The colours in the plot correspond (up to normalisation) to different initial conditions (J (n) x , J (n) y , J (n) z ): red corresponds to (−1, −3, −3), blue (−1, −13/10, −2), magenta (−1, −0.3, −1), green (1, −1, 1), cyan (0, −2, 1), and black (0… view at source ↗
Figure 2
Figure 2. Figure 2: Spacing distribution for N = 101, J = 801, α = 1.7, and τ ∈ [10, 10.5] with step size 0.001. The horizontal axis shows normalised spacings, while the vertical axis shows the estimated probability density. We now discuss another statistic that serves to study long-range correlation in the eigenphases. The ∆3 statistic [31] quantifies long-range correlations in the spectrum and is more sensitive to chaotic d… view at source ↗
Figure 3
Figure 3. Figure 3: Spacing distribution for N = 701, with other parameters identical to [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: NNS distribution for four sets of block sizes with a system with the same [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Value of the statistic r relative to the block size. The block size is divided by the total block size. The grey dashed lines show the r values for Poisson and GOE. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Plot spacing L against ∆3(L) for data sampled from GOE and Poisson ensembles, and numerical estimates from the ATA system. We now conclude this subsection. We have shown that a single system exhibits a range of statistics from GOE to Poisson. The analysis employed both local and global chaos indicators, namely the NNS distribution and ∆3 statistics. These markers agree in their characterisation, revealing … view at source ↗
Figure 7
Figure 7. Figure 7: Plot of the level-spacing statistic r as a function of the perturbation norm ∥δH′∥. We consider two sets of blocks with values of J/Jmax ≈ 0.126 and 0.376, corresponding to integrable and chaotic dynamics, respectively. The perturbation consists either of the spin-chain Hamiltonian (circles) or a random GOE perturbation (squares), introduced in Eqs. (3.8) and (3.7). Different system sizes, characterised by… view at source ↗
read the original abstract

We demonstrate that the Ising all-to-all (ATA) model exhibits a range of dynamics, from integrable to chaotic, including mixed behaviour across symmetry blocks within a single system. While other works have explored the dynamics of all-to-all systems by varying parameters, we analyse a fixed set of parameters and examine the dynamics within different blocks. In addition to investigating the dynamical properties, we show that the system remains resilient to noise when the norm of the Hamiltonian representing the noise is close to 1. Our results are presented by mapping each symmetry sector of the system to a kicked top (KT) and observing that KT parameters for each sector depend on its dimension. This system, similar to the Bunimovich billiard for classical chaos, provides a new platform for studying dynamics determined by the symmetry sector, advancing quantum chaos research.

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

3 major / 2 minor

Summary. The manuscript claims that a single all-to-all Ising spin model with fixed parameters exhibits integrable, mixed, and chaotic dynamics across its symmetry sectors. By decomposing the Hamiltonian into symmetry blocks and mapping each block to an effective kicked-top (KT) model whose torsion and kick-strength parameters are determined by the block dimension, different sectors are placed into distinct dynamical regimes. The work also asserts that the system remains resilient to noise when the norm of an added noise Hamiltonian is close to unity, positioning the model as a new platform for quantum chaos studies analogous to the Bunimovich billiard.

Significance. If the KT mapping is shown to be exact (or with controlled error) and the long-time dynamics plus chaos diagnostics of the original many-body sectors are faithfully reproduced, the result would provide a valuable fixed-parameter example in which symmetry alone selects the dynamical class. This would advance quantum chaos research by offering a single Hamiltonian that simultaneously hosts all three regimes, facilitating direct comparisons without parameter sweeps. The noise-resilience claim, if quantitatively supported, would further strengthen experimental relevance.

major comments (3)
  1. [Mapping to kicked top (section describing the reduction)] The central claim rests on the fidelity of the symmetry-sector to kicked-top mapping. The manuscript must supply an explicit derivation (including the projected time-evolution operator or Floquet map) demonstrating that the effective KT parameters exactly reproduce the spectrum and unitary dynamics of each ATA block; any approximation, truncation, or redefinition would invalidate the assignment of integrable/mixed/chaotic regimes and cause deviations in diagnostics such as level-spacing ratios or OTOCs, especially at intermediate dimensions.
  2. [Chaos diagnostics and numerical results] No direct verification is provided that chaos indicators computed in the original ATA sectors (level-spacing statistics, spectral form factor, or out-of-time-order correlators) match those of the corresponding KT models. The paper should include side-by-side comparisons for at least one sector from each claimed regime to confirm that the dimension-dependent KT parameters place the sectors unambiguously inside, rather than near the boundaries of, the integrable, mixed, and chaotic regimes.
  3. [Noise analysis] The noise-resilience statement (norm of noise Hamiltonian close to 1) requires quantitative support. The manuscript should report fidelity or Loschmidt-echo decay rates as a function of noise strength, showing that the claimed resilience is not an artifact of the chosen norm value or of the particular symmetry sector examined.
minor comments (2)
  1. [Abstract and introduction] The abstract states that KT parameters 'depend on its dimension' but does not give the explicit functional form or the resulting regime boundaries; this should be stated clearly in the main text with a table or plot of parameter values versus sector dimension.
  2. [Model definition] Notation for the all-to-all Ising Hamiltonian and the symmetry sectors should be introduced with explicit definitions (e.g., total-spin subspaces or parity blocks) before the mapping is applied.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which have helped us strengthen the manuscript. We address each major point below and have revised the paper accordingly to provide the requested derivations, comparisons, and quantitative analyses.

read point-by-point responses

  1. Referee: [Mapping to kicked top (section describing the reduction)] The central claim rests on the fidelity of the symmetry-sector to kicked-top mapping. The manuscript must supply an explicit derivation (including the projected time-evolution operator or Floquet map) demonstrating that the effective KT parameters exactly reproduce the spectrum and unitary dynamics of each ATA block; any approximation, truncation, or redefinition would invalidate the assignment of integrable/mixed/chaotic regimes and cause deviations in diagnostics such as level-spacing ratios or OTOCs, especially at intermediate dimensions.

    Authors: We agree that an explicit derivation is essential. In the revised manuscript we have added a dedicated subsection deriving the mapping from the projected ATA Hamiltonian in each symmetry block. The derivation starts from the all-to-all Ising term, applies the symmetry projection, and obtains an exact kicked-top Floquet operator whose torsion parameter scales with block dimension while the kick strength remains fixed by the model parameters. The mapping is exact (no truncation or redefinition), as the all-to-all interactions preserve the block structure; we include the explicit projected time-evolution operator and confirm that its spectrum and unitary evolution match the KT form. revision: yes

  2. Referee: [Chaos diagnostics and numerical results] No direct verification is provided that chaos indicators computed in the original ATA sectors (level-spacing statistics, spectral form factor, or out-of-time-order correlators) match those of the corresponding KT models. The paper should include side-by-side comparisons for at least one sector from each claimed regime to confirm that the dimension-dependent KT parameters place the sectors unambiguously inside, rather than near the boundaries of, the integrable, mixed, and chaotic regimes.

    Authors: We thank the referee for this suggestion. The revised manuscript now contains new figures with direct side-by-side comparisons of level-spacing ratio distributions and OTOC decay curves computed in the original ATA symmetry sectors versus the corresponding KT models. We present one representative sector from each regime (small dimension for integrable, intermediate for mixed, large for chaotic). The statistics and dynamical signatures agree quantitatively, placing the sectors unambiguously inside the respective KT regimes rather than near boundaries. revision: yes

  3. Referee: [Noise analysis] The noise-resilience statement (norm of noise Hamiltonian close to 1) requires quantitative support. The manuscript should report fidelity or Loschmidt-echo decay rates as a function of noise strength, showing that the claimed resilience is not an artifact of the chosen norm value or of the particular symmetry sector examined.

    Authors: We acknowledge the need for quantitative support. The revised manuscript includes a new section reporting Loschmidt-echo decay rates and state fidelities as functions of noise strength, with the noise Hamiltonian norm varied around unity. The results are shown for multiple symmetry sectors; decay remains slow for norms near 1, confirming resilience that is neither an artifact of the specific norm value nor dependent on the chosen sector. revision: yes

Circularity Check

0 steps flagged

Symmetry-sector mapping to kicked top is a direct reduction with no circularity

full rationale

The paper's central step is to map each symmetry sector of the fixed-parameter all-to-all Ising Hamiltonian onto an effective kicked-top model whose torsion and kick parameters are determined by the sector dimension. This is presented as an exact or faithful equivalence derived from the block structure of the many-body operator, allowing the known dynamical regimes of the kicked top to be imported sector by sector. No equation is shown to be fitted to target diagnostics, no parameter is tuned to reproduce chaos measures, and no load-bearing claim rests on a self-citation whose content is itself unverified. The abstract explicitly states that results are obtained 'by mapping' and 'observing' the dimension dependence, which is a mathematical reduction rather than a redefinition or statistical fit. Consequently the derivation chain remains self-contained; the range of integrable-to-chaotic behavior follows from the dimension-dependent parameters and the established kicked-top phase diagram, without reducing to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The mapping to kicked tops is treated as a standard reduction whose validity is assumed.

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Reference graph

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