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Eigenstates of interacting integrable systems behave as random superpositions of polynomially many Gaussian states.

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2026-07-03 20:28 UTC pith:277LBUFW

load-bearing objection The paper's main move is to argue that integrable eigenstates act like random superpositions of polynomially many Gaussians while nonintegrable ones need exponentially many, with the distinction shown through matches on one-body purity, non-Gaussianity, and entanglement entropy. the 2 major comments →

arxiv 2607.01326 v1 pith:277LBUFW submitted 2026-07-01 quant-ph cond-mat.stat-mech

One-Body Purity, Non-Gaussianity, and Entanglement in Interacting Integrable Models

Rafa{\l} \'Swi\k{e}tek , Maksymilian Kliczkowski , Lev Vidmar , Marcos Rigol This is my paper
classification quant-ph cond-mat.stat-mech
keywords integrable systemsentanglement entropyGaussian statesone-body puritynon-Gaussianitymany-body eigenstatesquantum integrabilityeigenstate thermalization
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes that eigenstates in interacting integrable models match the one-body purity, non-Gaussianity, and entanglement entropy of random superpositions involving only a polynomial number of Gaussian states. This ensemble accounts for both the volume-law entanglement and the persistent non-Gaussian character that single Gaussian states cannot explain. In contrast, the same three quantities in nonintegrable systems align with superpositions of an exponential number of Gaussian states. The comparison is made through analytical formulas and numerical checks on lattice Hamiltonians.

Core claim

When describing entanglement in typical midspectrum eigenstates of many-body lattice Hamiltonians, two paradigms have emerged that capture the behavior observed in integrable and nonintegrable systems, Haar-random fermionic Gaussian states and Haar-random pure states, respectively. Remarkably, the former capture the behavior of interacting integrable systems, whose eigenstates are non-Gaussian. We argue that the paradigm that captures both the entanglement properties and the lack of Gaussianity in integrable systems is that of random superpositions of polynomially many Gaussian states. In contrast, eigenstates of nonintegrable systems are consistent with being described by random superpositi

What carries the argument

random superpositions of polynomially many Gaussian states, which simultaneously reproduce the measured one-body purity, non-Gaussianity, and entanglement entropy of integrable eigenstates

Load-bearing premise

Matching values of one-body purity, non-Gaussianity, and entanglement entropy between eigenstates and the proposed random superpositions is enough to conclude that the eigenstates are built from those superpositions.

What would settle it

Direct computation of the one-body purity for midspectrum eigenstates of the XXZ chain or similar integrable model that deviates from the analytic prediction for random superpositions of polynomially many Gaussian states.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • One-body purity of integrable eigenstates equals the ensemble average over polynomially many Gaussian states.
  • A non-Gaussianity measure computed on the same eigenstates agrees with the polynomial superposition ensemble.
  • Entanglement entropy scaling in integrable systems is reproduced by the polynomial number of states in the superposition.
  • Nonintegrable eigenstates require an exponential number of Gaussian states to match the same three quantities.

Where Pith is reading between the lines

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

  • The polynomial-versus-exponential distinction may organize a broader classification of eigenstate complexity across different classes of Hamiltonians.
  • Higher-order correlation functions not studied in the paper could be predicted from the same polynomial superposition construction.
  • Time-dependent observables after a quench might be approximated by evolving the underlying Gaussian states before superposing them.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 0 minor

Summary. The paper claims that midspectrum eigenstates of interacting integrable lattice Hamiltonians are captured by random superpositions of polynomially many fermionic Gaussian states (explaining both their entanglement and non-Gaussianity), while nonintegrable eigenstates correspond to superpositions of exponentially many such states. This is supported by deriving analytical expressions and performing numerical comparisons for one-body purity, a non-Gaussianity measure, and entanglement entropy between the proposed ensembles and actual Hamiltonian eigenstates.

Significance. If the identification of the ensembles holds, the work offers a useful intermediate paradigm between Haar-random Gaussian states (which fail to capture non-Gaussianity in integrable systems) and full Haar-random states, potentially clarifying the structure of eigenstates in integrable models. The provision of both analytical formulas and numerical benchmarks for the three observables is a concrete strength that allows direct testing.

major comments (2)
  1. [Abstract] Abstract and the central comparison: agreement on one-body purity, non-Gaussianity, and entanglement entropy between Hamiltonian eigenstates and the proposed random-superposition ensembles is shown analytically and numerically, but the manuscript does not demonstrate that these three quantities uniquely identify the ensembles (as opposed to other structures that could reproduce the same values). No additional observables (e.g., two-body purity or Schmidt coefficient distributions) are checked to test the identification.
  2. The mapping from microscopic eigenstates to the polynomially/exponentially many Gaussian superposition ensembles is inferred from the match on the three diagnostics rather than derived from a microscopic construction; this leaves the central claim that the ensembles 'capture' or 'describe' the eigenstates under-determined without further evidence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive evaluation of its significance. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the central comparison: agreement on one-body purity, non-Gaussianity, and entanglement entropy between Hamiltonian eigenstates and the proposed random-superposition ensembles is shown analytically and numerically, but the manuscript does not demonstrate that these three quantities uniquely identify the ensembles (as opposed to other structures that could reproduce the same values). No additional observables (e.g., two-body purity or Schmidt coefficient distributions) are checked to test the identification.

    Authors: We agree that agreement on these three quantities does not establish uniqueness of the ensemble identification. These observables were chosen because they are the central diagnostics used in the literature to distinguish the entanglement and Gaussianity properties of integrable versus nonintegrable eigenstates. The manuscript derives closed-form expressions for the ensembles and shows quantitative agreement with Hamiltonian eigenstates; this supports the claim that the ensembles provide a useful intermediate paradigm. We do not assert that the ensembles are the only possible description, only that they capture the reported behavior. No additional observables are examined in the present work. revision: no

  2. Referee: The mapping from microscopic eigenstates to the polynomially/exponentially many Gaussian superposition ensembles is inferred from the match on the three diagnostics rather than derived from a microscopic construction; this leaves the central claim that the ensembles 'capture' or 'describe' the eigenstates under-determined without further evidence.

    Authors: The manuscript presents the ensembles as a phenomenological paradigm whose predictions for one-body purity, non-Gaussianity, and entanglement entropy match those of the eigenstates, both analytically and numerically. The central claim is therefore that the ensembles reproduce these key properties, thereby offering an understanding intermediate between Haar-random Gaussian states and full Haar-random states. While a direct microscopic construction that derives the superposition structure from the Hamiltonian is not provided, the consistent numerical agreement across multiple system sizes and the exact analytical formulas constitute the supporting evidence. A full microscopic derivation lies outside the scope of the work. revision: no

Circularity Check

0 steps flagged

No circularity: independent ensemble construction and observable matching

full rationale

The paper defines random superposition ensembles of Gaussian states independently of the Hamiltonian eigenstates, derives closed-form or numerical expressions for one-body purity, non-Gaussianity, and entanglement entropy on those ensembles, and then compares the results to direct computations on the eigenstates. No parameters are fitted from the eigenstate data and then relabeled as predictions; no self-citations are invoked to justify uniqueness or load-bearing premises; the three observables are computed from first principles on the ensembles rather than defined in terms of the target quantities. The central claim rests on empirical agreement rather than definitional reduction, so the derivation chain contains no self-definitional, fitted-input, or self-citation circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.1-grok · 5692 in / 1198 out tokens · 35817 ms · 2026-07-03T20:28:10.263208+00:00 · methodology

0 comments
read the original abstract

When describing entanglement in typical midspectrum eigenstates of many-body lattice Hamiltonians, two paradigms have emerged that capture the behavior observed in integrable and nonintegrable systems, Haar-random fermionic Gaussian states and Haar-random pure states, respectively. Remarkably, the former capture the behavior of interacting integrable systems, whose eigenstates are non-Gaussian. We argue that the paradigm that captures both the entanglement properties and the lack of Gaussianity in integrable systems is that of random superpositions of polynomially many Gaussian states. In contrast, eigenstates of nonintegrable systems are consistent with being described by random superpositions of exponentially many Gaussian states. We gain this understanding by comparing analytical and numerical results for the one-body purity, the non-Gaussianity, and the entanglement entropy of the random superpositions and the Hamiltonian eigenstates.

Figures

Figures reproduced from arXiv: 2607.01326 by Lev Vidmar, Maksymilian Kliczkowski, Marcos Rigol, Rafa{\l} \'Swi\k{e}tek.

Figure 1
Figure 1. Figure 1: FIG. 1. One-body purities vs the filling [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: we show how the bound is approached for n = 1 2 [ [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Average non-Gaussianity [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Average one-body purity for random superposi [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Average entanglement entropy for integrable [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

discussion (0)

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Typical Entanglement of Superpositions

    quant-ph 2026-07 conditional novelty 7.0

    An m-fold superposition of typical sub-maximally entangled states gains a universal ln(m) entanglement enhancement, while maximally entangled states relax to the Haar limit via N-independent scaling laws.

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