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REVIEW 3 major objections 6 minor 85 references

Weak ergodicity breaking with isolated integrable sectors

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper argues that every non-trivial embedding of an integrable model into a chaotic spin chain found here is a perturbation of a Hilbert-space-fragmented model, with the perturbation preserving the integrable sector and ergodizing the…

desk verdict A genuinely useful toolbox for embedding integrable sectors into chaotic chains, with a valuable unifying fragmented-model story, but the MM and dipole examples need connectivity checks before 'weak ergodicity breaking' is fully established. read the letter →

arxiv 2412.13951 v1 pith:PA5Z2IUK submitted 2024-12-18 cond-mat.stat-mech quant-ph

Hosho Katsura , Chihiro Matsui , Chiara Paletta , Balázs Pozsgay This is my paper · ORCID
classification cond-mat.stat-mechquant-ph MSC 81R1282B2082B23 PACS 05.30.-d05.45.Mt75.10.Pq
keywords weakergodicitybreakingisolatedintegrablesectorHilbertspacefragmentationirreduciblestringseigenstatethermalizationhypothesisXXCmodelsspinchainsTrotterization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that weak ergodicity breaking by an isolated integrable sector is a generic construction: start from a model with Hilbert space fragmentation, select one fragment that realizes an integrable model, and add local perturbations that vanish on that fragment while kinetically connecting all the other fragments. In every non-trivial example considered, including a previously known spin ladder, the integrable sector survives exactly, grows exponentially with system size, and occupies a vanishing fraction of the full Hilbert space. The paper presents this as a unified mechanism distinct from quantum many-body scars and from trivial tensor-product embeddings, and it verifies numerically that the complement of the integrable sector shows ergodic signatures.

What carries the argument

The load-bearing object is the irreducible string: a sequence obtained by deleting vacuum or reference states from a computational basis state, which is conserved because the unperturbed kinetic terms never let two particles of different colors cross. Combined with color-charge separation, this splits the Hilbert space into exponentially many fragments, some of which realize an integrable model. The paper's perturbation terms—pair-flip, block-hopping, and number-breaking terms—are chosen so that they annihilate the distinguished pattern but restore hopping and mixing among all other fragments. Projectors onto the integrable sectors are written as matrix product operators with nonzero operator entanglement, showing that the sectors are genuinely entangled subspaces rather than product spaces.

What would settle it

Compute the full commutant of the perturbed Hamiltonian in the largest sector: if any local or matrix-product conserved operator survives in the complement, or if the adjacent-gap ratio of the largest sector departs from GOE at L>25, the ergodicity of the complement fails. A cheaper check is to count connected components of the adjacency graph at larger L and see whether $D_{\rm chaos}$ still equals $2^L$ minus the Lucas number.

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Extended reading notes

Core claim

The central claim is that all non-trivial examples in the paper of weak ergodicity breaking by isolated integrable sectors can be understood as perturbations of fragmented models. The perturbation is engineered to act as zero on a selected integrable sector, which is identified by a conserved 'irreducible string' pattern, while acting generically elsewhere to connect formerly disjoint fragments. The restriction of the Hamiltonian to the selected sector reproduces an integrable model—XX, XXZ, constrained XXZ, or an integrable RSOS chain—whereas the adjacent-gap statistics in the largest remaining sector match GOE, and the relative dimension of the integrable sector decays exponentially. For the folded-XXZ and RSOS examples the paper derives the exact dimension $D_{\rm chaos}(L)=2^L-L_L$ (with $L_L$ the Lucas number), giving $D_{\rm chaos}/2^L \sim 1-(0.809)^L$.

Load-bearing premise

The load-bearing premise is that the perturbations connect all fragments outside the integrable sector, so the complement really is ergodic and carries no hidden conservation laws; the paper verifies this numerically up to L=25 but does not prove it.

Editorial extensions

If this is right

  • The XX ladder of [8] is re-interpreted as a perturbed XXC model, so its coexisting diffusive and ballistic sectors share the same mechanism.
  • The constructions yield local Hamiltonians whose integrable subspace has a projector with nonzero operator entanglement, separating them from trivial product-space embeddings.
  • In the folded-XXZ and RSOS examples the non-thermal sector has relative dimension $(0.809)^L$, so weak ergodicity breaking is exponential in system size.
  • The same embeddings survive in discrete time: brickwork quantum circuits with an integrable Trotterized sector and a chaotic complement can be built.
  • Perturbing the integrable sector itself, following the scar-construction recipe, turns selected eigenstates into quantum many-body scars while breaking integrability of that sector.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves open whether every weak-ergodicity-breaking integrable sector must arise from a fragmented parent; if the fragmented-parent mechanism is necessary, it would yield a classification of such models by their parent irreducible strings.
  • Because the perturbation that erases fragmentation is local and vanishes on the selected pattern, the same design could be applied to other constrained Hilbert spaces, such as other RSOS restrictions, to produce new chaotic models with isolated integrable sectors.
  • The dimension formula $D_{\rm chaos}=2^L-L_L$ suggests a sharp experimental probe: initial states inside the integrable sector should fail to thermalize for parametrically long times while typical states thermalize, which could be tested in Rydberg or cold-atom simulators.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This paper constructs families of local spin-chain Hamiltonians exhibiting weak ergodicity breaking in the sense of Definition 2: most eigenstates are expected to be thermal, while a distinguished integrable subspace, of exponentially large but vanishing relative dimension, hosts non-ETH states. The central structural claim is that every non-trivial example is a perturbation of a model with Hilbert space fragmentation, where the perturbation preserves a selected integrable sector (realizing the XX, XXZ, constrained XXZ, or off-critical RSOS chains, the latter under the Rydberg constraint) while undoing the fragmentation in the complementary space. Examples treated are trivial product-form embeddings (Sec. III); the Maassarani-Mathieu chain with a two-site color-flip perturbation (Sec. IV); the XXC models (Sec. V); Znidaric's XX spin ladder, rederived as a perturbed XXC model (Sec. VI); the folded XXZ model, where the integrable sector is the sector of even-length down-spin blocks (Sec. VII); the dipole-conserving spin-1 model (Sec. VIII); and RSOS-type chains extended to the full Hilbert space (Sec. IX). A Trotterized quantum-circuit analogue is given in Sec. X. The exact restrictions to the integrable sectors are derived analytically, the projectors are given as bond-dimension-2 MPOs in the non-trivial cases, and for the folded XXZ and RSOS examples the chaotic-sector dimension is exactly 2^L - L_L (Lucas numbers), verified by adjacency-graph enumeration.

Significance. The paper is a conceptually valuable and technically clean contribution. Its strengths include exact, parameter-free analytic mappings of integrable sectors to XX/XXZ/constrained-XXZ/RSOS models; explicit MPO projectors for the non-product integrable subspaces; exact counting of the chaotic-sector dimension in the folded XXZ and RSOS examples (D_chaos = 2^L - L_L, verified to L=22 by exact graph enumeration), which is a falsifiable, essentially machine-checkable statement; and GOE level-statistics and entanglement-entropy diagnostics that directly support the complement-ergodicity claim in those examples. The unified picture, that all non-trivial embeddings arise as perturbations of fragmented models, and the reinterpretation of the XX ladder as a perturbed XXC model, are genuinely useful and likely to be influential. However, the part of the central claim that distinguishes weak from strong ergodicity breaking, namely thermalization of the complement, is established numerically for two of the examples only; for the Maassarani-Mathieu model it is asserted without evidence, and for the dipole model it is supported by a short-range fit with unclassified residual subspaces.

major comments (3)
  1. [Sec. IVD, Eq. (IV.17)] The statement that the perturbation h_pair 'destroys fragmentation for all the other subspaces' is not supported and, as written, is not literally correct. The operator S^-_j S^-_{j+1} changes two adjacent color-2 excitations into two color-1 excitations and S^+_j S^+_{j+1} does the reverse; hence N_tot is conserved while N^{(1)} is changed by ±2, so N^{(1)} mod 2 and N^{(2)} mod 2 are exact conserved quantum numbers in every sector. The complement therefore splits into at least two disconnected subspaces for each N_tot, and the phrase 'destroys fragmentation' must be qualified. More importantly, in contrast with Sections VIID and IXB, no numerical evidence is provided that each of these sectors is internally connected or ergodic: there is no adjacency-graph enumeration, no gap-ratio statistic, and no dimension scaling for this model. Since Sec. IVD is presented as 'a new mechanism that leads to weak ergodicity breaking', the authors should either add the missing finite-size checks (adjacency graphs in the sectors labeled by N_tot and N^{(1)} mod 2; a GOE test in the largest complement sector; scaling of the complement-sector dimensions) or explicitly demote the complement-ergodicity claim to a conjecture.
  2. [Sec. VIII, Figs. 4-6] For the dipole-conserving model, the evidence for weak ergodicity breaking is materially weaker than for the folded XXZ and RSOS examples. The authors acknowledge that 'many small subspaces' remain after the perturbation, and the central quantitative claim rests on the fit D_chaos/3^L = 1 - exp(-αL) with α ≈ 0.362, with data shown only up to L=12, plus a gap-ratio distribution at L=13 whose mean ⟨r⟩ = 0.526 is in only 'rough agreement' with the GOE value. Because a hidden local or nonlocal conservation law (the paper itself mentions a conserved local dipole moment between defects, following Ref. [39]) could keep a non-vanishing fraction of states outside the chaotic sector, a fit over L ≤ 12 is not conclusive. The authors should either extend the exact enumeration of sector dimensions to larger L, along the lines of Eq. (VII.20), or scale down the claim made for this model.
  3. [Sec. XI (Conclusions)] The sentence stating that the perturbations 'connect all (or almost all) other subspaces, making the model ergodic in the complement of the integrable subspace. These observations hold for all our examples, including the model of [8]' overstates what is established. Complement ergodicity is verified numerically for the folded XXZ and RSOS examples, asserted without evidence for the Maassarani-Mathieu example (Sec. IVD), and only partially established for the dipole example (Sec. VIII). The conclusion should be rewritten to separate the examples with exact dimension counting and level-statistics support from those for which complement ergodicity is a plausible conjecture rather than a demonstrated property.
minor comments (6)
  1. [Sec. IVC-D] The dimension of the alternating-string sector \tilde{H} is never given; to justify that this model has weak rather than strong ergodicity breaking per Definition 2, the paper should state the counting result, e.g., dim \tilde{H} ~ (1+√2)^L, which is exponentially large but o(3^L).
  2. [Sec. IIC] Typo: 'Krlylov' should be 'Krylov' (the misspelling occurs twice within the paragraph).
  3. [Sec. IVB] Formatting: 'the so-calledt-0 model' should read 'the so-called t-0 model'.
  4. [Fig. 3 caption] The notation '1s.u.' and the phrase 'corresponds to state with 0, 4, 7 particles' are unclear; please spell out the labels (e.g., 'one spin up' and 'states with 0, 4, and 7 particles').
  5. [Sec. VIII, Fig. 5] The fit to D_chaos/3^L = 1 - exp(-αL) should be described with its fit range and the uncertainty of α; the horizontal axis ends at L=12, which is short for a scaling claim. This is related to major comment 2.
  6. [Sec. XB] The assumption that L is a multiple of 3 (and of 4 for the perturbation part of the circuit) should be stated as a convention, and the treatment of general L should be mentioned.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the integrable sectors are explicit constructions, the complement ergodicity is tested numerically, and self-citations are to independent exact results.

full rationale

The paper's central constructions are not circular. Each integrable sector is defined by an explicit, checkable constraint (alternating colors in Sec. IV; even-length blocks of down spins in Sec. VII; Rydberg constraint in Sec. IX), and the added perturbation is engineered to annihilate that sector by inspection, e.g., h_pair in Eq. (IV.17) requires two adjacent same-color excitations, which are absent from the alternating-color subspace, and hkin(j), hX(j) in Eq. (IX.4) require adjacent down spins, which the Rydberg-constrained space forbids. The integrability of the restricted Hamiltonian is inherited from published Yang-Baxter and Bethe-ansatz results, and the paper cross-checks the resulting energies against the Bethe-ansatz formula (VII.15). The weak-ergodicity-breaking claim rests on the complement becoming ergodic; this is not derived from a fitted parameter but supported by adjacency-graph enumeration, Wigner-Dyson level statistics, and entanglement computations. The paper candidly flags its own residual gaps: in Sec. VIII it states that 'there still remain many small subspaces' and that 'the remaining fragmentation deserves further investigation', and the assertion in Sec. IVD that h_pair 'destroys fragmentation for all the other subspaces' is stronger than what is demonstrated, since total particle number and the parities of the two color numbers remain conserved. These are limitations of evidence about the complement, not reductions of the derivation to its inputs. The only fit in the paper, D_chaos/3^L ~ 1 - exp(-alpha L) in Fig. 5, is a numerical summary of a dimension ratio, not a Hamiltonian parameter renamed as a prediction. Self-citations, including Refs. [23], [28], [52], [54], and [76], are to exact published constructions and do not smuggle in the paper's target conclusion.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central constructions rely on known integrable models and standard statistical diagnostics; they do not introduce new particles or fitted parameters. The main entered axioms are citations to prior exact results, some from the same authors, and the numerical-to-thermodynamic extrapolation that the perturbed complement is ergodic.

free parameters (1)
  • Hamiltonian coupling constants (alpha, beta, gamma, kappa, kappa', a, b, Delta) = not fitted; arbitrary real numbers
    The constructions work for generic values of these couplings; the numerical checks use specific values (e.g., kappa=kappa'=0.789 or 1.0). No parameter is optimized to match a target result.
assumptions (5)
  • domain assumption The XXC models and the Maassarani-Mathieu chain are integrable in every fragment, with integrability established by Yang-Baxter techniques.
    Used in Sec. IV and V to assert the unperturbed models are integrable; based on Refs. [25,26].
  • domain assumption The folded XXZ model is integrable, and its even-block sector is described by a constrained XXZ model with a Bethe ansatz solution.
    Used in Sec. VII to identify the integrable sector and compute its energies via Bethe equations; based on Ref. [23].
  • domain assumption The off-critical RSOS integrable Hamiltonians acting on the Rydberg-constrained Hilbert space are exactly solvable.
    Used in Sec. IXA to claim the constrained subspace is integrable; based on Ref. [72].
  • domain assumption Wigner-Dyson level statistics in the largest adjacency-graph sector is a reliable indicator of ergodicity and ETH in the thermodynamic limit.
    Used in Secs. VIID, VIIIB, and IXB to conclude the complement is chaotic from finite-size gap-ratio distributions.
  • domain assumption The integrable sector plus trivially solvable small sectors exhaust the non-ETH subspaces.
    The dimension count D_chaos = 2^L - L_L treats everything outside the identified integrable and trivial sectors as chaotic; verified by adjacency graphs up to L=22 but not proven for all L.

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Pith. "Pith review of Weak ergodicity breaking with isolated integrable sectors." pith.science (2026). https://pith.science/paper/PA5Z2IUK

@misc{pith2026241213951,
  author       = {Pith},
  title        = {Pith review of: Weak ergodicity breaking with isolated integrable sectors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PA5Z2IUK}},
  note         = {Machine review of arXiv:2412.13951}
}
read the original abstract

We consider spin chain models with local Hamiltonians that display weak ergodicity breaking. In these models, the majority of the eigenstates are thermal, but there is a distinguished subspace of the Hilbert space in which ergodicity is broken. We achieve such a weak breaking by embedding selected integrable models into larger Hilbert spaces of otherwise chaotic models. The integrable subspaces do not have a tensor product structure with respect to any spatial bipartition, therefore our constructions differ from certain trivial embeddings. We consider multiple mechanisms for such an embedding, and we also review previous examples in the literature. Curiously, all our examples can be seen as perturbations of models with Hilbert space fragmentation, such that the perturbed models are not fragmented anymore.

Figures

Figures reproduced from arXiv: 2412.13951 by the authors.

Figure 1
Figure 1. Adjacency graph for the Hamiltonian (VII.16) with κ = κ ′ = 0.789 and L = 8. (a) chaotic sectors with 202 states; (b) integrable sector with 20 states with 2 pairs of nearby spin down particles; (c) integrable sector with 16 states and 3 pairs of nearby spin down particles; (d) integrable sector with 8 states, one pair of spin down; (e-f) trivially solvable sectors, 4 states, one spin up; (g-h) 1 state, all spins up… view at source ↗
Figure 2
Figure 2. Distribution of the adjacent-gap ratios of the Hamil [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Plot of the half-chain entanglement entropy as a [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Hamiltonian structure of (a) the original model [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Ratio of the dimension of the chaotic sector to that [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 7
Figure 7. Figure 7: Adjacency graph of the Hamiltonian H in Eq. (IX.4) for L = 8 with a = 1.5, b = 0.5, and κ = κ ′ = 0. The upper left sector corresponds to the integrable model in the constrained Hilbert space. For better visibility, we have removed the loops in the graph [PITH_FULL_IM…
Figure 8
Figure 8. Figure 8: Adjacency graph of the Hamiltonian H in Eq. (IX.4) for L = 8 with a = 1.5, b = 0.5, and κ = κ ′ = 1.0. (a) Chaotic sector with 209 states. (b) Integrable sector with 47 states, corresponding to the upper left sector in the adjacency graph of the undeformed model shown …
Figure 12
Figure 12. Figure 12: An example for a brickwork circuit with two-site [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 11
Figure 11. Figure 11: Plot of energy vs half-chain entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]

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