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arxiv: 2507.07249 · v2 · submitted 2025-07-09 · 🪐 quant-ph · cond-mat.quant-gas· cond-mat.stat-mech· cond-mat.str-el

Kubo-Martin-Schwinger relation for energy eigenstates of SU(2)-symmetric quantum many-body systems

Jae Dong Noh , Aleksander Lasek , Jade LeSchack , Nicole Yunger Halpern This is my paper

Pith reviewed 2026-05-19 05:08 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.quant-gascond-mat.stat-mechcond-mat.str-el
keywords Kubo-Martin-Schwinger relationnon-Abelian eigenstate thermalization hypothesisSU(2) symmetryenergy eigenstatesfinite-size correctionsHeisenberg chainfluctuation-dissipation theorem
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The pith

Energy eigenstates of SU(2)-symmetric quantum many-body systems obey a Kubo-Martin-Schwinger relation derived from the non-Abelian eigenstate thermalization hypothesis.

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

The paper establishes that energy eigenstates in quantum systems with SU(2) symmetry satisfy the Kubo-Martin-Schwinger relation, a symmetry property central to the fluctuation-dissipation theorem. This extends earlier proofs that applied only to systems without non-Abelian symmetries by invoking the non-Abelian version of the eigenstate thermalization hypothesis. A reader would care because the result clarifies how internal thermalization works when symmetries constrain the system's behavior, with direct implications for nonequilibrium response and thermal properties. The finite-size correction to the relation follows the usual inverse-system-size scaling in some regimes but can become polynomially larger in others, according to the authors' analysis and numerical checks on Heisenberg chains.

Core claim

Using the non-Abelian eigenstate thermalization hypothesis, the authors derive a Kubo-Martin-Schwinger relation that holds for the energy eigenstates of SU(2)-symmetric quantum many-body systems. The finite-size correction to this relation scales as the inverse system size under certain circumstances but can grow polynomially larger in other cases. Numerical simulations of a Heisenberg chain with 16 to 24 qubits support the ordinary scaling result and provide indirect evidence for the possibility of larger corrections.

What carries the argument

The non-Abelian eigenstate thermalization hypothesis applied to matrix elements of observables in SU(2)-symmetric systems, which allows derivation of the Kubo-Martin-Schwinger relation for energy eigenstates.

If this is right

  • The fluctuation-dissipation theorem extends to energy eigenstates of SU(2)-symmetric systems.
  • Response functions in these systems follow from equilibrium properties of individual eigenstates.
  • Finite-size corrections can deviate from standard scaling when non-Abelian symmetries are present.
  • Numerical verification becomes feasible for moderate system sizes using exact diagonalization or tensor networks.

Where Pith is reading between the lines

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

  • Larger corrections may appear in experiments with cold atoms or trapped ions that realize SU(2)-symmetric Hamiltonians.
  • Transport and relaxation rates could differ from predictions that ignore the polynomial enhancement.
  • The result invites checks in other non-Abelian symmetry groups such as SU(3) in higher-spin systems.

Load-bearing premise

The non-Abelian eigenstate thermalization hypothesis correctly describes the scaling and matrix elements of observables in the SU(2)-symmetric systems studied.

What would settle it

A direct numerical computation of the finite-size correction to the KMS relation in larger SU(2)-symmetric spin chains that shows whether the correction remains inversely proportional to system size or instead grows polynomially.

Figures

Figures reproduced from arXiv: 2507.07249 by Aleksander Lasek, Jade LeSchack, Jae Dong Noh, Nicole Yunger Halpern.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

The fluctuation-dissipation theorem (FDT) is a fundamental result in statistical mechanics. It stipulates that, if perturbed out of equilibrium, a system responds at a rate proportional to a thermal-equilibrium property. Applications range from particle diffusion to electrical-circuit noise. To prove the FDT, one must prove that common thermal states obey a symmetry property, the Kubo-Martin-Schwinger (KMS) relation. Energy eigenstates of certain quantum many-body systems were recently proved to obey the KMS relation. The proof relies on the eigenstate thermalization hypothesis (ETH), which explains how such systems thermalize internally. This KMS relation contains a finite-size correction that scales as the inverse system size. Non-Abelian symmetries conflict with the ETH, so a non-Abelian ETH was proposed recently. Using it, we derive a KMS relation for SU(2)-symmetric quantum many-body systems' energy eigenstates. The finite-size correction scales as usual under certain circumstances but can be polynomially larger in others, we argue. We support the ordinary-scaling result numerically, simulating a Heisenberg chain of 16-24 qubits. The numerics, limited by computational capacity, indirectly support the larger correction. This work helps extend into nonequilibrium physics the effort, recently of interest across quantum physics, to identify how non-Abelian symmetries may alter conventional thermodynamics.

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 derives a Kubo-Martin-Schwinger (KMS) relation for energy eigenstates of SU(2)-symmetric quantum many-body systems by applying the non-Abelian eigenstate thermalization hypothesis (ETH). It shows that the finite-size correction to this relation scales as the inverse system size under standard conditions but can become polynomially larger when degeneracy effects broaden the ETH window. Numerical simulations on the SU(2)-symmetric Heisenberg chain for lengths L=16–24 are used to confirm the ordinary scaling, with indirect arguments offered for the modified scaling regime.

Significance. If the central derivation holds, the work usefully extends the link between ETH and fluctuation-dissipation relations into the non-Abelian symmetry setting that is common in spin systems. The identification of possible polynomially enhanced finite-size corrections supplies a concrete, testable distinction from the Abelian case and could inform nonequilibrium studies of conserved quantities.

major comments (2)
  1. [§3] §3 (derivation using non-Abelian ETH): the claim that off-diagonal matrix elements of local operators obey a specific scaling that permits polynomially larger KMS corrections is taken directly from the non-Abelian ETH ansatz in the literature; the manuscript does not re-derive or independently verify this scaling for the operators and symmetry sectors considered here, yet this scaling is load-bearing for both the standard and modified correction results.
  2. [§5] §5 (numerical results on Heisenberg chains): data are shown only for L=16–24. At these sizes the effective dimension per total-Sz sector remains modest; it is therefore unclear whether the observed 1/L scaling persists or crosses over to the larger-correction regime once degeneracy-induced broadening dominates in the thermodynamic limit, which is central to the paper’s argument for non-standard behavior.
minor comments (2)
  1. [Abstract] The abstract states that the larger correction is only 'indirectly supported'; a short paragraph in the main text explicitly listing the assumptions needed for the polynomial enhancement would help readers assess the scope of the claim.
  2. Notation for the symmetry-projected operators and the ETH window width could be made more uniform between the analytic sections and the numerical figures to avoid minor confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. We address each major comment point by point below. Revisions have been made to improve clarity on the use of the non-Abelian ETH and to expand the discussion of numerical limitations and the thermodynamic limit.

read point-by-point responses

  1. Referee: [§3] §3 (derivation using non-Abelian ETH): the claim that off-diagonal matrix elements of local operators obey a specific scaling that permits polynomially larger KMS corrections is taken directly from the non-Abelian ETH ansatz in the literature; the manuscript does not re-derive or independently verify this scaling for the operators and symmetry sectors considered here, yet this scaling is load-bearing for both the standard and modified correction results.

    Authors: We agree that the scaling form for the off-diagonal matrix elements is adopted from the non-Abelian ETH ansatz in the literature and is not independently re-derived in this work. In the revised manuscript we have added a concise recap in §3 of the relevant non-Abelian ETH statements, including the explicit scaling of off-diagonal elements and the mechanism by which SU(2) degeneracy broadens the effective ETH window. This addition makes the dependence on the literature ansatz fully explicit while preserving the focus of the present paper on its consequences for the KMS relation. We view the non-Abelian ETH as established prior work whose application here is justified by the symmetry sector and operator locality considered. revision: yes

  2. Referee: [§5] §5 (numerical results on Heisenberg chains): data are shown only for L=16–24. At these sizes the effective dimension per total-Sz sector remains modest; it is therefore unclear whether the observed 1/L scaling persists or crosses over to the larger-correction regime once degeneracy-induced broadening dominates in the thermodynamic limit, which is central to the paper’s argument for non-standard behavior.

    Authors: We acknowledge that the accessible system sizes are modest and that exact diagonalization becomes prohibitive for L > 24. In the revised §5 we have added a quantitative estimate of the crossover scale at which degeneracy broadening is expected to dominate, derived from the non-Abelian ETH window width. We also include further analysis of the variance of the relevant matrix elements across the studied sizes to provide indirect support for the onset of the modified regime. While direct numerical access to the thermodynamic limit is not feasible, the combination of the observed 1/L scaling at current sizes and the theoretical crossover estimate strengthens the distinction between the two regimes. revision: partial

Circularity Check

0 steps flagged

Derivation from non-Abelian ETH shows no circularity; input hypothesis treated as independent.

full rationale

The paper takes the non-Abelian eigenstate thermalization hypothesis as an independent input from prior literature and derives the KMS relation for energy eigenstates in SU(2)-symmetric systems from it. The finite-size correction scaling arguments follow directly from the assumed properties of this hypothesis without redefining the target KMS relation in terms of itself or fitting parameters to force the result. Numerical simulations on 16-24 qubit Heisenberg chains provide supporting evidence for the ordinary scaling rather than constructing the central claim by definition. No self-definitional, fitted-input, or load-bearing self-citation steps reduce the output to the inputs; the derivation chain remains self-contained with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the non-Abelian ETH as the key domain assumption imported from recent prior work; no free parameters or new invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Non-Abelian eigenstate thermalization hypothesis holds for the SU(2)-symmetric systems considered
    Invoked to derive the KMS relation and its finite-size corrections

pith-pipeline@v0.9.0 · 5806 in / 1283 out tokens · 29088 ms · 2026-05-19T05:08:42.979343+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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

  1. Typical entanglement entropy with charge conservation

    quant-ph 2026-04 unverdicted novelty 7.0

    Typical entanglement entropy with fixed global charge is given by the local thermal entropy at fixed charge density for both U(1) and SU(2) symmetries in the thermodynamic limit.

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