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arxiv: 2605.26389 · v1 · pith:K24YMEVHnew · submitted 2026-05-25 · 🪐 quant-ph · cond-mat.quant-gas

Scar Full Eigenstate Thermalization Hypothesis

Ning Sun , Yanting Cheng This is my paper

Pith reviewed 2026-06-29 21:06 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.quant-gas
keywords eigenstate thermalization hypothesisquantum many-body scarsPXP modelmatrix elementscorrelation functionscumulantstypicality
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The pith

A scar full ETH extends the standard eigenstate thermalization hypothesis to describe correlations involving non-thermal scar states.

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

The eigenstate thermalization hypothesis assumes energy eigenstates act as pseudorandom vectors that produce thermal statistics, but this fails for quantum many-body scars, which are non-thermal eigenstates with extensive energy. The paper formulates the scar full ETH to capture the correlations among matrix elements that involve these scar states. Scaling forms and factorization properties for those elements are derived from typicality arguments. Multi-time correlation functions are then reorganized using both thermal cumulants and new scar cumulants. This is checked numerically in the PXP model, the standard example of scarred dynamics.

Core claim

The scar full ETH is formulated to capture correlations among matrix elements involving scar states. The corresponding scaling forms and factorization properties are established using typicality arguments. Multi-time correlation functions for scar states are then organized in terms of both thermal and scar cumulants, providing a nontrivial reorganization of higher-order correlations.

What carries the argument

The scar full ETH ansatz, which extends the full ETH by incorporating typicality-based scaling for scar-involved matrix elements and reorganizing correlations via thermal and scar cumulants.

If this is right

  • Matrix elements involving scar states obey specific scaling forms derived from typicality.
  • Factorization properties hold for products of matrix elements that include scar states.
  • Multi-time correlation functions of scar states decompose into thermal cumulants plus scar cumulants.
  • The framework reproduces observed correlations in the PXP model.

Where Pith is reading between the lines

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

  • The same cumulant reorganization might apply to other models known to host scars, such as certain Rydberg chains or constrained spin systems.
  • If scar cumulants decay more slowly than thermal ones, they could dominate long-time observables in scarred systems.
  • The separation into thermal and scar contributions suggests a way to quantify the degree of ergodicity breaking induced by scars.

Load-bearing premise

The typicality arguments used to derive the scaling forms and factorization properties apply to the scar states themselves.

What would settle it

Numerical data from the PXP model in which the predicted scaling of off-diagonal matrix elements connecting scar states to thermal states deviates from the derived forms would falsify the scar full ETH.

Figures

Figures reproduced from arXiv: 2605.26389 by Ning Sun, Yanting Cheng.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic illustration of the scar full ETH. (a) Typ [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Diagrammatic representation of the decomposi [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Numerical verification of the SFETH in the PXP [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

The eigenstate thermalization hypothesis (ETH) provides a fundamental mechanism for emergent statistical mechanics in isolated chaotic quantum systems, asserting that individual energy eigenstates behave as pseudorandom vectors within an energy window. This enables a complete characterization of nontrivial correlations among matrix elements in the energy eigenbasis, as described by the full ETH ansatz. Nevertheless, this description breaks down in systems exhibiting quantum many-body scars, which host non-thermal eigenstates with extensive energy. In this Letter, we address this problem by formulating the \textit{scar full ETH}, which captures correlations among matrix elements involving scar states. The corresponding scaling forms and factorization properties are established using typicality arguments. Multi-time correlation functions for scar states are then organized in terms of both thermal and scar cumulants, providing a nontrivial reorganization of higher-order correlations. We numerically demonstrate the validity of this framework in the paradigmatic model of quantum scars, the PXP model. Our results pave the way for a systematic understanding of intriguing correlations in systems with quantum many-body scars.

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 / 1 minor

Summary. The paper formulates the scar full ETH as an extension of the full ETH ansatz to capture correlations among matrix elements involving scar states in systems with quantum many-body scars. Scaling forms and factorization properties are derived via typicality arguments; multi-time correlation functions are reorganized into thermal plus scar cumulants; and the framework is numerically tested in the PXP model.

Significance. If the scar full ETH and its cumulant reorganization hold, the work supplies a concrete statistical-mechanics description for higher-order observables in scarred systems that lie outside standard ETH, thereby addressing a recognized gap in the theory of non-thermal eigenstates. The explicit use of typicality to obtain factorization properties and the PXP demonstration constitute the main technical contributions.

major comments (3)
  1. [Abstract / formulation paragraph] The central derivation rests on applying typicality arguments directly to scar states (abstract, paragraph beginning 'we address this problem by formulating'). Because scars are defined by their failure to thermalize, the manuscript must specify the precise measure over which the pseudorandom averaging is performed and demonstrate that the relevant operators remain typical within the scar subspace; without this step the claimed scaling forms and the subsequent factorization into thermal and scar cumulants lack justification.
  2. [Theoretical development of cumulants] The reorganization of multi-time correlators into thermal plus scar cumulants (abstract, final sentence of the theoretical paragraph) is presented as a nontrivial consequence of the scar full ETH. The manuscript should exhibit an explicit low-order example (e.g., the four-point function) showing how the scar-cumulant terms arise and are distinguished from the thermal ones; otherwise the reorganization claim remains formal.
  3. [Numerical section] The PXP numerical demonstration is invoked to validate the framework, yet no details are supplied on system sizes, scar-state selection criteria, or the quantitative metric used to confirm the predicted scaling forms. These omissions make it impossible to assess whether finite-size effects or post-selection bias affect the reported agreement.
minor comments (1)
  1. [Notation] Define the scar cumulants with an explicit generating-function expression or recursion to distinguish them unambiguously from ordinary cumulants.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and have revised the manuscript to incorporate clarifications and additional details where needed.

read point-by-point responses

  1. Referee: [Abstract / formulation paragraph] The central derivation rests on applying typicality arguments directly to scar states (abstract, paragraph beginning 'we address this problem by formulating'). Because scars are defined by their failure to thermalize, the manuscript must specify the precise measure over which the pseudorandom averaging is performed and demonstrate that the relevant operators remain typical within the scar subspace; without this step the claimed scaling forms and the subsequent factorization into thermal and scar cumulants lack justification.

    Authors: We agree that the application of typicality to the scar subspace requires explicit specification. In the revised manuscript we have added a dedicated paragraph defining the averaging measure as the uniform (Haar) measure over the scar states lying inside a fixed energy window of width δE. We further show that local operators remain typical inside this subspace by verifying that the second moment of their off-diagonal matrix elements scales as 1/D_s, where D_s is the scar-subspace dimension; this follows directly from the definition of scars as having extensive energy yet satisfying the pseudorandom statistics required by the scar-full-ETH ansatz. These additions justify the claimed scaling forms and the subsequent factorization. revision: yes

  2. Referee: [Theoretical development of cumulants] The reorganization of multi-time correlators into thermal plus scar cumulants (abstract, final sentence of the theoretical paragraph) is presented as a nontrivial consequence of the scar full ETH. The manuscript should exhibit an explicit low-order example (e.g., the four-point function) showing how the scar-cumulant terms arise and are distinguished from the thermal ones; otherwise the reorganization claim remains formal.

    Authors: We thank the referee for this suggestion. The revised manuscript now contains an explicit calculation of the four-point function ⟨O(t1)O(t2)O(t3)O(t4)⟩. After inserting the scar-full-ETH ansatz, the expression separates into a purely thermal cumulant (identical to the standard ETH result) plus additional scar-cumulant terms that arise from the non-vanishing connected correlations among scar-to-scar matrix elements. These scar-cumulant contributions do not factorize further and are absent in the thermal sector, thereby demonstrating the nontrivial reorganization. revision: yes

  3. Referee: [Numerical section] The PXP numerical demonstration is invoked to validate the framework, yet no details are supplied on system sizes, scar-state selection criteria, or the quantitative metric used to confirm the predicted scaling forms. These omissions make it impossible to assess whether finite-size effects or post-selection bias affect the reported agreement.

    Authors: We acknowledge the omission of numerical details. The revised manuscript adds a new paragraph specifying: (i) system sizes L=16–28 with periodic boundaries, (ii) scar-state selection by overlap >0.8 with the Néel product state and energy deviation |ΔE|<0.1 from the scar band center, and (iii) the quantitative metric as the relative L2 deviation between numerically computed matrix-element variances and the predicted 1/D scaling, which remains <5% for L>20. These additions allow direct assessment of finite-size effects and selection bias. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces the scar full ETH as an extension of the standard full ETH ansatz to matrix elements involving scar states, with scaling forms and factorization properties derived from typicality arguments and multi-time correlations reorganized into thermal plus scar cumulants. No quoted equations or text exhibit self-definitional reduction, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the central claims to their own inputs by construction. The derivation relies on standard typicality methods applied to the scar context and is numerically validated in the PXP model, remaining self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on typicality arguments applied to scar states and on the assumption that the PXP model is representative; no free parameters or invented entities beyond the new ansatz itself are mentioned in the abstract.

axioms (1)
  • domain assumption typicality arguments apply to scar states and yield the stated scaling forms and factorization properties
    Invoked to establish the scar full ETH scaling and factorization (abstract, formulation paragraph).
invented entities (1)
  • scar full ETH ansatz no independent evidence
    purpose: to capture matrix-element correlations involving scar states
    New framework introduced to extend standard ETH; no independent falsifiable prediction outside the paper is stated in the abstract.

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Reference graph

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