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arxiv: 2605.08079 · v1 · submitted 2026-05-08 · ❄️ cond-mat.stat-mech · hep-th· math-ph· math.MP

Recognition: 2 theorem links

· Lean Theorem

Multiscale Structure of Eigenstate Thermalization

Pavel Orlov , Rustem Sharipov , Enej Ilievski
Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:59 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech hep-thmath-phmath.MP
keywords eigenstate thermalizationmatrix elementsintegrable field theorycharge fluctuationsalgebraic exponentsmultiscale structurequantum many-body systemsstatistical ensembles
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The pith

Matrix element distributions in energy eigenstates depend on the scale of charge fluctuations in the sampling ensemble, not only on macrostate densities.

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

The paper examines how off-diagonal matrix elements of local observables behave in energy eigenstates of an isolated quantum many-body system. It shows that these distributions are shaped not just by average values of conserved charges but also by how much fluctuation in those charges is permitted when selecting the eigenstates. Using an integrable field theory that allows exact sampling and thermodynamic extrapolation, the authors vary an ensemble parameter controlling fluctuation magnitude and observe that the power-law exponents in the matrix-element statistics shift in a non-analytic manner with this scale. This establishes a multiscale character to the eigenstate thermalization process. A reader would care because standard microcanonical treatments of thermalization may overlook this additional dependence on ensemble construction.

Core claim

In an integrable field theory, the statistical properties of matrix elements of local observables between eigenstates exhibit algebraic decay whose exponents depend non-analytically on the fluctuation scale of the ensemble used to sample the eigenstates, in addition to any dependence on the macrostate densities of local conserved charges. This reveals an underlying multiscale structure of matrix elements.

What carries the argument

The adjustable scale parameter that prescribes the magnitude of charge fluctuations in the statistical ensembles used to sample eigenstates, which governs the non-analytic change in the algebraic exponents of matrix-element distributions.

If this is right

  • Standard microcanonical windows are insufficient to characterize matrix-element distributions in macroscopic systems.
  • The suppression rate of matrix elements admits an explicit closed-form expression for a particular class of states.
  • Numerical sampling of matrix elements in the model permits reliable extrapolation to the thermodynamic limit.
  • Eigenstate thermalization acquires an additional dependence on ensemble properties beyond energy and charge densities.

Where Pith is reading between the lines

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

  • Numerical studies of thermalization in finite systems may need to report the fluctuation scale of their eigenstate sampling procedure.
  • The non-analytic dependence suggests that critical points or phase transitions in fluctuation parameters could produce sharp changes in thermalization rates.
  • Similar ensemble sensitivity might appear in other observables such as entanglement entropy or out-of-time-order correlators.
  • If the structure generalizes, it could refine predictions for how quickly local observables equilibrate after a quench.

Load-bearing premise

Results obtained for one specific integrable field theory and chosen class of states, including reliable extrapolation to the thermodynamic limit, apply to generic macroscopic quantum many-body systems.

What would settle it

Finding that the algebraic exponents governing matrix-element statistics remain independent of the charge-fluctuation scale when the same analysis is performed in a different integrable or non-integrable model at larger system sizes.

Figures

Figures reproduced from arXiv: 2605.08079 by Enej Ilievski, Pavel Orlov, Rustem Sharipov.

Figure 1
Figure 1. Figure 1: FIG. 1. Probing multiscale structure of eigenstate [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Introducing the distance [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Fluctuation scale dependence of ma [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Eigenstates as Young diagrams [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Introducing arms and legs [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Staircase diagrams [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Scaling behavior of suppression rate. [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Scaling behavior of distribution width. [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Probability distribution of matrix ele [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Probability distribution of ma [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Scaling analysis of suppression rate [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Scaling analysis of suppression rate [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Behavior of distribution width [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Scaling analysis of distribution mean [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
read the original abstract

The eigenstate thermalization hypothesis provides a framework for understanding thermalization in isolated quantum many-body systems by characterizing statistical properties of local observables in energy eigenstates. Here we demonstrate that distributions of matrix elements in macroscopic systems may depend not only on the macrostate parameters, such as the densities of local conserved charges, but generally also on the properties of ensembles used in sampling eigenstates. To this end, we depart from the conventional analysis of microcanonical windows and consider statistical ensembles with an adjustable scale parameter prescribing the magnitude of charge fluctuations. We specifically consider an integrable field theory that permits efficient numerical sampling of matrix elements and reliable extrapolation to the thermodynamic limit. Moreover, in this system, we identify a class of states that enables explicit closed-form computation of the suppression rate of matrix elements. Our findings reveal an underlying multiscale structure of matrix elements captured by a non-analytic fluctuation-scale dependence of algebraic exponents governing their statistical properties.

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

1 major / 0 minor

Summary. The paper claims that distributions of matrix elements of local observables in energy eigenstates of macroscopic quantum many-body systems depend not only on macrostate parameters (e.g., densities of conserved charges) but also on the fluctuation scale of the sampling ensemble. This multiscale structure is manifested as a non-analytic dependence of algebraic exponents on the ensemble scale parameter. The claim is supported by efficient numerical sampling of matrix elements and closed-form suppression rates in a specific integrable field theory, with extrapolation to the thermodynamic limit.

Significance. If the multiscale dependence generalizes beyond the specific integrable model, the result would meaningfully extend the eigenstate thermalization hypothesis by identifying an additional controlling parameter (ensemble fluctuation scale) for matrix-element statistics. The provision of both numerical sampling and closed-form results in one model is a concrete strength that allows direct verification of the non-analytic exponent dependence.

major comments (1)
  1. [Abstract and final discussion] The central claim is phrased for generic macroscopic systems, yet all evidence is obtained in one integrable field theory with a tailored class of states. No derivation from general ETH assumptions, no comparison to non-integrable dynamics, and no argument for why the observed non-analytic scale dependence survives in chaotic systems are provided; this extrapolation is load-bearing for the stated scope and remains unproven.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the significance of our work and for the constructive comment on the scope of our claims. We address the major point below.

read point-by-point responses

  1. Referee: [Abstract and final discussion] The central claim is phrased for generic macroscopic systems, yet all evidence is obtained in one integrable field theory with a tailored class of states. No derivation from general ETH assumptions, no comparison to non-integrable dynamics, and no argument for why the observed non-analytic scale dependence survives in chaotic systems are provided; this extrapolation is load-bearing for the stated scope and remains unproven.

    Authors: We agree that the phrasing in the abstract and final discussion presents the multiscale structure as a general feature of macroscopic systems, while all concrete evidence (numerical sampling of matrix elements and closed-form suppression rates) is obtained in one integrable field theory with a specific class of states chosen to permit exact computations. The manuscript contains no derivation from general ETH assumptions, no data from non-integrable dynamics, and no explicit argument that the non-analytic scale dependence must persist in chaotic systems. To correct this, we will revise the abstract and the concluding discussion to state explicitly that the multiscale dependence on ensemble fluctuation scale is demonstrated in the present integrable model, that the underlying mechanism is tied to the adjustable ensemble scale rather than to integrability per se, and that whether the same non-analytic exponent dependence appears in non-integrable or chaotic systems remains an open question for future work. These changes will remove any load-bearing extrapolation while preserving the concrete results obtained in the model. revision: yes

Circularity Check

0 steps flagged

No circularity: results derived from explicit sampling and closed-form expressions in a specific model

full rationale

The paper's central finding—that matrix-element distributions exhibit non-analytic dependence on ensemble fluctuation scales—is obtained by introducing an adjustable scale parameter into the ensemble definition, then performing direct numerical sampling and closed-form calculations within one integrable field theory for a chosen class of states. No equations or claims reduce the observed algebraic exponents or multiscale structure to a fit of the target quantity itself, nor do they rely on self-citations for uniqueness or ansatz smuggling. The extrapolation to generic systems is stated as an open step rather than derived by construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the representativeness of the integrable field theory for general systems and the validity of thermodynamic-limit extrapolation; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Standard framework of the eigenstate thermalization hypothesis for local observables in energy eigenstates
    Invoked to define matrix elements and their statistical properties.
  • domain assumption Existence of an integrable field theory permitting efficient sampling and closed-form matrix element computations
    Used as the concrete system for numerical and analytical results.

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Reference graph

Works this paper leans on

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    Analytical results for the staircase diagrams The staircase diagrams, depicted in Fig. (6), enable us to derive an explicit analytic formula for the asymptotic behavior of the matrix elements as a function of distance. In the thermodynamic limit, the suppression rateκbe- comes a function of a single scaling variable, u= d2 L log L d ,(14) wheredhere is pr...

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