{"id":"c21810bd-f0b9-4be1-994b-31d4e7f027f6","arxiv_id":"2412.13951","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Isolated integrable subspaces can be embedded non-trivially into chaotic spin chains; all examples are perturbations of Hilbert-space-fragmented models.","lead":"This paper builds spin-chain models where most eigenstates are thermal, yet a rare integrable subspace survives inside an otherwise chaotic Hamiltonian. It shows that all such non-trivial embeddings can be understood as perturbations of fragmented models, and provides several new explicit examples.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Complement ergodicity is the weakest step: the Maassarani-Mathieu perturbation (Sec. IVD) is claimed to connect all non-integrable sectors without any numerical check, so hidden conservation laws could leave a non-vanishing non-ETH fraction.","rationale":"The reader's verdict ACCEPT is well supported for the folded XXZ and RSOS constructions, where exact dimension formulas and GOE statistics are given. My concern is narrower: the paper's central mechanism is claimed for all nontrivial examples, and the MM model is presented as the first new mechanism, yet its complement ergodicity is asserted without even finite-size evidence. Because the weak ergodicity breaking definition is a thermodynamic-limit statement about the fraction of non-ETH states, a hidden conservation law in the MM complement would not just be a technicality; it would mean that example is not an example of weak ergodicity breaking. The proposed adjacency-graph check is the same diagnostic the authors use elsewhere and is computationally trivial at these sizes. If it passes, the paper's central claim is strengthened and ACCEPT stands; if it fails, the MM section needs revision or removal. Hence I recommend CONDITIONAL rather than outright ACCEPT.","tokens_in":30331,"tokens_out":14967,"duration_ms":145462,"concrete_test":"Enumerate the Hamiltonian adjacency graph for the perturbed Maassarani-Mathieu chain (IV.16) with γ = 0.789 and open boundary conditions, for L = 8, 9, 10, 11. Fixing total particle number N_tot and the parity of N^{(1)}_tot (both conserved), check that within each (N_tot, parity) sector the subgraph of all states outside the alternating-color integrable subspace \\tilde{H} is a single connected component, and that its dimension is exactly the sector dimension minus dim(\\tilde{H} ∩ sector). Compute the adjacent-gap ratio in the largest such component; it should follow GOE. If additional disconnected components appear whose total dimension is not exponentially small relative to the sector, the claim that the perturbation removes fragmentation fails and the example is not weakly ergodic.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim requires that the complement of the exact integrable subspace is thermal. For the folded XXZ (Sec. VIID) and RSOS (Sec. IXB) examples the authors verify by exact adjacency-graph enumeration that the complement is a single connected sector of dimension 2^L - L_L, which is strong evidence. However, the Maassarani-Mathieu example (Sec. IV) has no such check: the statement that h_pair (IV.17) 'destroys fragmentation for all the other subspaces' is made without numerics. Because h_pair conserves N_tot and changes N1 by ±2, the quantities N1 mod 2 and N2 mod 2 are exactly conserved, so at best the complement splits into two parity sectors per N_tot; whether each such sector is internally connected is untested. The dipole model (Sec. VIII) is also weaker: the authors fit D_chaos/3^L to 1-exp(-αL) and explicitly leave 'many small subspaces' unclassified, so a hidden nonlocal conservation law could keep a non-vanishing fraction of states outside the thermal sector. These gaps do not contradict the exact embeddings, but they leave the label 'weak ergodicity breaking' resting on finite-size numerical evidence for some examples and on no evidence for the MM chain.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":30631,"tokens_out":18943,"duration_ms":162284,"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":[{"comment":"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.","section":"Sec. IVD, Eq. (IV.17)"},{"comment":"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.","section":"Sec. VIII, Figs. 4-6"},{"comment":"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.","section":"Sec. XI (Conclusions)"}],"minor_comments":[{"comment":"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).","section":"Sec. IVC-D"},{"comment":"Typo: 'Krlylov' should be 'Krylov' (the misspelling occurs twice within the paragraph).","section":"Sec. IIC"},{"comment":"Formatting: 'the so-calledt-0 model' should read 'the so-called t-0 model'.","section":"Sec. IVB"},{"comment":"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').","section":"Fig. 3 caption"},{"comment":"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.","section":"Sec. VIII, Fig. 5"},{"comment":"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.","section":"Sec. XB"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the manuscript fits the scope of a general physics journal (PRB, SciPost Physics, or JSTAT) and is likely to be influential given the current interest in quantum scars and Hilbert space fragmentation. The citation practice is sound; several key references ([19], [23], [28], [52], [54]) include members of the author team, but the integrability results they rely on are established, published results used appropriately, so I see no circularity problem. The main substantive gap is the unverified complement-ergodicity claim for the Maassarani-Mathieu example, which a moderate numerical effort could close; softening the relevant sentences in Secs. IVD and XI would be an acceptable alternative if the authors prefer not to add numerics. The dipole-model claims in Sec. VIII should be brought in line with the evidence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline: this is a genuinely useful construction paper for weak ergodicity breaking with isolated integrable sectors, and the unifying observation that all the nontrivial embeddings are perturbations of fragmented models is the most valuable part. The exact mappings of the integrable sectors to XX, XXZ, constrained XXZ, and RSOS chains are clean, and the MPO projectors for the entangled sectors are a nice touch.\n\nThe strongest evidence in the paper is in Sec. VIID (folded XXZ) and Sec. IXB (RSOS), where adjacency-graph enumeration up to L=22 shows the complement of the integrable sector is a single connected component (up to trivial isolated states), so the 'weak' part of the claim rests on solid finite-size footing there. The dimension scalings (D_chaos = 2^L - L_L) also check out.\n\nThe soft spots are the Maassarani-Mathieu example (Sec. IVD) and the dipole model (Sec. VIII). For the MM chain, the statement that h_pair 'destroys fragmentation for all the other subspaces' is asserted without any check. Since h_pair conserves N_tot and changes N1, N2 by ±2, you automatically get at least two parity sectors per N_tot; whether each is internally connected is untested. That is a real gap in the support for calling the complement ergodic. For the dipole model, the paper openly leaves 'many small subspaces' unclassified and supports complement ergodicity with a fit of D_chaos/3^L to 1-exp(-αL). That fit does not rule out a hidden local conservation law that keeps a non-vanishing fraction of states outside the thermal sector.\n\nThese caveats do not undercut the central construction—the integrable sectors really are preserved and really do solve to Bethe-ansatz-integrable models—but they temper the label 'weak ergodicity breaking' for those two cases. The paper would be stronger with a connectivity statement, or at least an explicit caveat, for the MM and dipole examples.\n\nWho is this for? Anyone constructing models with embedded integrable sectors or studying the boundary between fragmentation and thermalization. It is a serious contribution that deserves referee time; I would accept with revisions that tighten the MM and dipole sections.","headline":"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.","tokens_in":31121,"tokens_out":4556,"would_cite":true,"duration_ms":38011,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81R12","82B20","82B23"],"pacs":["05.30.-d","05.45.Mt","75.10.Pq"],"model":"deepseek-v4-flash","headline":"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…","keywords":["weak ergodicity breaking","isolated integrable sector","Hilbert space fragmentation","irreducible strings","eigenstate thermalization hypothesis","XXC models","spin chains","integrable Trotterization"],"falsifier":"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.","tokens_in":1644,"feed_emoji":"🧩","tokens_out":1913,"duration_ms":66517,"temperature":0.7,"pith_summary":"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.","feed_headline":"Perturbed fragmented chains hide integrable islands in chaos","feed_subtitle":"Local terms preserve one integrable sector while the rest of the chain thermalizes, verified numerically to L=25.","key_machinery":"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.","core_discovery":"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 $D_{\\rm chaos}/2^L \\sim 1-(0.809)^L$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provided the first non-trivial weak-ergodicity-breaking integrable sector (the XX ladder); the paper re-derives it as a perturbed XXC model.","marker":"[8]"},{"why":"Supplies the Maassarani-Mathieu spin chain that serves as the base fragmented integrable model for the first construction.","marker":"[25]"},{"why":"Defines the XXC models whose perturbations give weak ergodicity breaking with integrable sectors.","marker":"[26]"},{"why":"Provides the folded XXZ model with Hilbert space fragmentation and its Bethe-ansatz sector solution.","marker":"[23]"},{"why":"Introduces the dipole-conserving model whose integrable XX sectors are the target of the perturbation construction.","marker":"[10]"},{"why":"Shows the spin-1 model embedding XXZ in fragmented sectors, used as a starting point for the new perturbations.","marker":"[19]"},{"why":"Introduces statistically localized integrals of motion, the conserved patterns that label fragments and identify integrable sectors.","marker":"[39]"},{"why":"Provides the integrable RSOS chain on the constrained Hilbert space used for the constrained-space construction.","marker":"[72]"},{"why":"Gives the integrable Trotterization technique used to lift the embeddings to quantum circuits.","marker":"[82]"}],"fun_headline_variants":["Integrable islands survive in chaotic seas via fragment perturbations","Perturbed fragmentation spawns isolated integrable sectors","Weak ergodicity breaking: integrable needles in chaotic haystacks","From fragmentation to integrability: hidden sectors in spin chains"],"cache_read_input_tokens":33280,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Integrable islands survive in chaotic seas via fragment perturbations","Perturbed fragmentation spawns isolated integrable sectors","Weak ergodicity breaking: integrable needles in chaotic haystacks","From fragmentation to integrability: hidden sectors in spin chains"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000615,"raw_usage":{"total_tokens":2809,"prompt_tokens":849,"completion_tokens":1960,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":465,"completion_tokens_details":{"reasoning_tokens":1904}},"tokens_in":465,"tokens_out":1960,"duration_ms":12792,"temperature":1.0,"reasoning_tokens":1904,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:37:10.105087+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Coexistence of diffusive and ballistic transport in a simple spin ladder","cited_arxiv_id":"1302.5509","evidence_quote":"Provided the first non-trivial weak-ergodicity-breaking integrable sector (the XX ladder); the paper re-derives it as a perturbed XXC model."},{"cited_title":"The su(N) XX model","cited_arxiv_id":"cond-mat/9709163","evidence_quote":"Supplies the Maassarani-Mathieu spin chain that serves as the base fragmented integrable model for the first construction."},{"cited_title":"The XXC Models","cited_arxiv_id":"solv-int/9712008","evidence_quote":"Defines the XXC models whose perturbations give weak ergodicity breaking with integrable sectors."},{"cited_title":"An integrable spin chain with Hilbert space fragmentation and solvable real time dynamics","cited_arxiv_id":"2105.02252","evidence_quote":"Provides the folded XXZ model with Hilbert space fragmentation and its Bethe-ansatz sector solution."},{"cited_title":"Exactly solvable subspaces of non-integrable spin chains with boundaries and quasiparticle interactions","cited_arxiv_id":"2309.13911","evidence_quote":"Shows the spin-1 model embedding XXZ in fragmented sectors, used as a starting point for the new perturbations."},{"cited_title":"Diffusing Reconstituting Dimers: A Simple Model of Broken Ergodicity and Ageing","cited_arxiv_id":"cond-mat/9702192","evidence_quote":"Introduces statistically localized integrals of motion, the conserved patterns that label fragments and identify integrable sectors."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the integrable RSOS chain on the constrained Hilbert space used for the constrained-space construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the integrable Trotterization technique used to lift the embeddings to quantum circuits."}],"review_version":1}