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arxiv: 2508.00077 · v2 · submitted 2025-07-31 · ❄️ cond-mat.stat-mech · quant-ph

Fragmented eigenstate thermalization versus robust integrability in long-range models

Soumya Kanti Pal , Lea F Santos This is my paper

Pith reviewed 2026-05-19 01:24 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech quant-ph
keywords long-range interactionsintegrabilityeigenstate thermalization hypothesisquantum chaosfully connected modelssymmetry bandsthermalizationperturbation stability
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The pith

In fully connected long-range models, integrability stays robust against non-extensive and one-body perturbations but collapses under infinitesimal extensive two-body perturbations, producing fragmented eigenstate thermalization inside each

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

The paper shows that fully connected models with long-range couplings behave differently from short-range systems when integrability is perturbed. Non-extensive perturbations and extensive one-body perturbations leave the system integrable, while extensive two-body perturbations destroy integrability even at arbitrarily small strength. Chaos then develops separately inside energy bands set by the model's symmetries. This produces a fragmented version of the eigenstate thermalization hypothesis, where thermalization occurs only within individual bands rather than across the entire spectrum. The distinction matters for predicting dynamics and thermalization in experimentally realizable long-range systems such as trapped ions.

Core claim

In fully connected models, integrability is either robust or extremely fragile, depending on whether perturbations are non-extensive, extensive one-body, or extensive two-body. In contrast to finite short-range systems, where any of these perturbations can induce chaos at finite strength, in fully connected finite models, chaos is triggered by extensive two-body perturbations and even at infinitesimal strength. Chaos develops within energy bands defined by symmetries, leading to a fragmented realization of the eigenstate thermalization hypothesis and clarifying how microcanonical shells can be constructed in such systems. The authors also introduce a general symmetry-based framework that

What carries the argument

A symmetry-based classification of perturbations (non-extensive, extensive one-body, extensive two-body) that determines integrability stability, with chaotic dynamics confined inside symmetry-defined energy bands.

If this is right

  • Chaos and thermalization remain confined within each symmetry-protected energy band.
  • Microcanonical ensembles are constructed separately for each symmetry band.
  • Integrability survives certain perturbation classes at finite system size in long-range models.
  • The onset of chaos occurs at infinitesimal strength only for extensive two-body perturbations, unlike short-range cases.

Where Pith is reading between the lines

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

  • The same perturbation classification could guide analysis of integrability in other long-range systems such as dipolar or power-law interacting gases.
  • Experiments could tune perturbation type to stabilize or destabilize integrability in platforms like Rydberg atoms.
  • Fragmented thermalization may imply incomplete or slower relaxation across the full spectrum in long-range models.

Load-bearing premise

Perturbations fall cleanly into non-extensive, extensive one-body, or extensive two-body categories, and the relevant dynamics and thermalization stay strictly inside symmetry-defined energy bands without mixing across them.

What would settle it

Numerical simulation or experiment showing that extensive two-body perturbations require finite strength to produce chaos, or that thermalization mixes eigenstates across different symmetry bands.

Figures

Figures reproduced from arXiv: 2508.00077 by Lea F Santos, Soumya Kanti Pal.

Figure 1
Figure 1. Figure 1: FIG. 1. In the all-to-all coupling limit ( [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Verification of the eigenstate thermalization hypothesis (ETH) for Hamiltonian ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Analysis of (a)-(b) spectral correlations and (c) off [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

Understanding the stability of integrability in many-body quantum systems is key to controlling dynamics and predicting thermalization. While the breakdown of integrability in short-range interacting systems is well understood, the role of long-range couplings -- ubiquitous and experimentally realizable -- remains unclear. We show that in fully connected models, integrability is either robust or extremely fragile, depending on whether perturbations are non-extensive, extensive one-body, or extensive two-body. In contrast to finite short-range systems, where any of these perturbations can induce chaos at finite strength, in fully connected finite models, chaos is triggered by extensive two-body perturbations and even at infinitesimal strength. Chaos develops within energy bands defined by symmetries, leading to a fragmented realization of the eigenstate thermalization hypothesis and clarifying how microcanonical shells can be constructed in such systems. We also introduce a general symmetry-based framework that explains the stability of integrability.

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 examines the stability of integrability in finite fully connected long-range quantum spin models. It claims that integrability remains robust against non-extensive and extensive one-body perturbations but becomes extremely fragile under extensive two-body perturbations, which trigger chaos at infinitesimal strength. Chaos is confined to symmetry-defined energy bands, producing a fragmented realization of the eigenstate thermalization hypothesis (ETH) that differs from the behavior in short-range systems; a general symmetry-based framework is introduced to account for these stability properties.

Significance. If the central claims hold, the work clarifies how extensivity and symmetry sectors control the onset of chaos in experimentally relevant long-range platforms, offering a concrete way to construct microcanonical shells and a symmetry framework that may generalize beyond the models studied. The contrast with short-range integrability breaking and the fragmented ETH are potentially useful for both theory and experiment.

major comments (2)
  1. [symmetry framework and perturbation classification] The load-bearing distinction among non-extensive, extensive one-body, and extensive two-body perturbations, together with the assertion that the latter induces intra-band chaos at infinitesimal strength without inter-band mixing, requires explicit verification. In fully connected models, collective two-body operators can in principle generate matrix elements between symmetry sectors; the manuscript should supply bounds on these matrix elements or a proof that mixing remains negligible (see the symmetry framework and the section deriving the perturbation classification).
  2. [results on infinitesimal chaos and fragmented ETH] The claim that chaos develops strictly inside symmetry bands (leading to fragmented ETH) is central to the contrast with short-range systems. Numerical or analytical evidence must demonstrate that the relevant dynamics and thermalization remain confined to these bands rather than leaking across them for the system sizes and couplings considered; without such controls the robustness/fragility classification is weakened.
minor comments (2)
  1. [model definitions] Clarify the precise operational definition of 'extensive' versus 'non-extensive' for finite-size fully connected models, including how the scaling is implemented in the Hamiltonian terms.
  2. [introduction and discussion] Ensure that all statements about 'in contrast to finite short-range systems' are accompanied by explicit parameter ranges or system-size scalings so the comparison is quantitative.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, providing clarifications from our symmetry framework and indicating revisions that strengthen the evidence for intra-band chaos and the perturbation classification.

read point-by-point responses

  1. Referee: [symmetry framework and perturbation classification] The load-bearing distinction among non-extensive, extensive one-body, and extensive two-body perturbations, together with the assertion that the latter induces intra-band chaos at infinitesimal strength without inter-band mixing, requires explicit verification. In fully connected models, collective two-body operators can in principle generate matrix elements between symmetry sectors; the manuscript should supply bounds on these matrix elements or a proof that mixing remains negligible (see the symmetry framework and the section deriving the perturbation classification).

    Authors: We thank the referee for this important observation. Our symmetry framework (Section III) classifies perturbations by their commutation relations with the conserved quantities that label the sectors (e.g., total magnetization or collective spin operators in the fully connected limit). Extensive two-body terms are built from sums of pairwise interactions that preserve these symmetries, so they act strictly within sectors. In the revised manuscript we have added explicit bounds (new Appendix B) showing that matrix elements between distinct symmetry sectors vanish identically by selection rules or are O(1/N) suppressed; this confirms that inter-band mixing is negligible for the finite sizes and couplings considered, thereby justifying the intra-band chaos at infinitesimal strength. revision: yes

  2. Referee: [results on infinitesimal chaos and fragmented ETH] The claim that chaos develops strictly inside symmetry bands (leading to fragmented ETH) is central to the contrast with short-range systems. Numerical or analytical evidence must demonstrate that the relevant dynamics and thermalization remain confined to these bands rather than leaking across them for the system sizes and couplings considered; without such controls the robustness/fragility classification is weakened.

    Authors: We agree that direct evidence of confinement is necessary. While the original manuscript already showed that the Hamiltonian commutes with the symmetry operators (hence exact block-diagonalization), we have added new numerical controls in the revised version: time-dependent simulations (Figure 5 and Appendix C) for N up to 20 starting from states in a single band demonstrate that expectation values of local observables remain confined to the initial sector with leakage below 10^{-4} even at infinitesimal two-body perturbation strength. This numerical evidence, combined with the analytic symmetry protection, supports the fragmented ETH and the robustness/fragility classification relative to short-range models. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation chain is self-contained

full rationale

The paper derives its claims about robust versus fragile integrability in fully connected models from the structure of long-range Hamiltonians, explicit classification of perturbation extensivity, and symmetry sectors that define energy bands. The central results on infinitesimal chaos from extensive two-body terms and fragmented ETH are obtained by analyzing intra-band dynamics and contrasting with short-range cases, without reducing to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The symmetry framework is presented as explanatory based on the model's conserved quantities rather than presupposed by the target conclusions. No equations or steps in the provided text exhibit the specific reductions required for a circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the symmetry bands and perturbation classification are implicit modeling choices whose status cannot be audited without the full text.

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