{"id":"af8d0e13-4ba5-45be-ad13-1f00d88c1187","arxiv_id":"2607.04279","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Nearly linear multiparticle sub-bands created by spatial modulation yield long-lived MWS revivals from spectral phase coherence among ETH-satisfying eigenstates, without nonthermal scars.","lead":"A modulated Bose-Hubbard lattice can produce long-lived collective revivals of multiparticle Wannier states even when every multiparticle Bloch eigenstate looks thermal. The effect comes from nearly equal energy spacings in engineered linear sub-bands, giving a route to weak ergodicity breaking that does not rely on scar-like nonthermal eigenstates.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond the reader's already-flagged finite-size ETH and residual-curvature premise.","rationale":"The reader's strongest_claim accurately restates the paper's contribution, and the weakest_assumption correctly identifies the only place where the argument is least secure: finite-size ETH proxies plus residual curvature in the thermodynamic limit. My second-pass check of the band-resolved fragmentation (Eq. 2, SM S1), the folding mechanism that isolates linear segments (Fig. 2(b), SM S2), the absence of scar-like outliers (Fig. 2(d), SM S3), and the multi-particle and Fock-superposition robustness (SM S4–S5) finds no deeper flaw. The phenomenon is explicitly weak and expected to merge with thermalization as the revival period diverges; that is not a contradiction but the definition of the claimed regime. Therefore the CONDITIONAL verdict with medium correctness_risk already reflects the right balance; no adjustment is warranted.","tokens_in":22569,"tokens_out":577,"duration_ms":6888,"concrete_test":"Recompute the fidelity revival contrast for the same nearly-linear dimer-monomer sector at M=120 (or the largest accessible size) with the identical moderate modulation used in Fig. 3; if the first few revival peaks remain above ~0.5 while the curved-band control continues to dephase, the finite-size claim is reinforced; if contrast collapses faster than 1/M scaling of the level spacing, residual curvature is already fatal at accessible sizes.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that long-lived MWS revivals arise from spectral phase coherence among ETH-satisfying multiparticle Bloch states that form a nearly linear sub-band (Eqs. 3–4, Figs. 2–4). The reader already isolates the weakest premise: that finite-size ETH diagnostics (smooth local observables and non-anomalous entanglement in Fig. 2(d) and SM S3) remain valid, and residual band-edge curvature does not destroy phase locking once level spacing vanishes. After re-examining the band-folding construction (SM S2), the projected Wannier-sector dynamics (SM S1), the OTOC and long-time beat oscillations (SM S4), and the multi-particle extensions (SM S5), I find no additional internal inconsistency or hidden assumption that is more load-bearing. The mechanism is self-consistently weak (revival period diverges with system size) and the authors state this explicitly. The hand-tuned modulation and finite-size proxies are real limitations, but they are already captured by the reader's weakest_assumption and do not introduce a new correctness risk.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript proposes a mechanism for weak ergodicity breaking that does not rely on nonthermal (scar-like) eigenstates. In a spatially modulated Bose-Hubbard model, cotranslation symmetry yields multiparticle Bloch bands and multiparticle Wannier states (MWSs). The Hamiltonian is block-diagonal in the multiparticle band index (band-resolved Wannier-sector fragmentation). Moderate spatial modulation folds multiparticle bands into sub-bands; nearly linear sub-bands inherit linear segments of the original dispersion and therefore possess approximately equal energy spacings. Although the multiparticle Bloch eigenstates in those sub-bands pass finite-size ETH diagnostics (level statistics, smooth local observables, non-anomalous entanglement), the corresponding MWSs exhibit long-lived collective revivals of fidelity, density, and entanglement, with frequencies matching the equal spacings. The authors contrast this spectral phase-coherence mechanism with quantum many-body scars and conventional Hilbert-space fragmentation, and they show that the revivals persist for larger systems, weak disorder, Fock-state superpositions, and multi-particle hybrid manifolds, while remaining weak (revival period diverges with system size).","tokens_in":22865,"tokens_out":1164,"duration_ms":16315,"significance":"If the finite-size ETH diagnostics continue to hold and residual band-edge curvature does not destroy phase locking, the work supplies a conceptually distinct route to weak ergodicity breaking: long-lived memory retention from spectral phase coherence among ETH-satisfying eigenstates rather than from a vanishing fraction of nonthermal eigenstates. The construction is concrete (superlattice Bose-Hubbard model, experimentally accessible platforms), the band-folding route to nearly linear multiparticle sub-bands is nontrivial (long-range hopping alone fails), and the Supplemental Material provides extensive supporting diagnostics (projected Wannier-sector dynamics, OTOCs, multi-particle extensions, robustness). The explicit acknowledgment that the phenomenon is weak (revival period diverges in the thermodynamic limit) is a strength. The result would broaden the taxonomy of ergodicity breaking and suggest spectral engineering of multiparticle bands as a design principle.","major_comments":[{"comment":"The central claim that the multiparticle Bloch states in the nearly linear sub-band satisfy ETH rests on finite-size diagnostics (Fig. 2(c,d); SM S3 C,D). Level statistics show repulsion but deviate from GOE because of strong interactions; local densities and entanglement show no scar-like outliers within the available sizes (M ≤ 90 for bands, M ≤ 33 for N=4,5). The Summary already notes that the revival period diverges as level spacing vanishes. The manuscript should state more explicitly what would constitute a falsifying finite-size trend (e.g., growth of entanglement outliers or loss of equal-spacing fidelity peaks with M) and, if feasible, add one larger-size check of the equal-spacing structure or of the entanglement diagnostic for the selected sub-band, so that the ETH premise is not left solely to the present system sizes.","section":null},{"comment":"SM S2 C and the main-text discussion of the trade-off between gap size and linearity show that residual curvature at band edges is inevitable for any finite modulation that opens a gap. Figs. 3 and S10 already exhibit long-time beat oscillations from those edge deviations. The claim of 'long-lived' revivals would be strengthened by a quantitative measure of how the revival contrast or dephasing time scales with residual curvature (or with δU) and with system size, rather than relying only on visual persistence of oscillations. Without that, it remains unclear how much of the observed coherence is protected by the linear bulk versus limited by edge curvature once M increases further.","section":null}],"minor_comments":[{"comment":"The term 'spectral phase coherence' is introduced without a formal definition. A short equation or paragraph linking it to the phase factors in Eqs. (3)–(4) would help readers distinguish it from ordinary dephasing language.","section":null},{"comment":"Fig. 2(c): the level-statistics sample is restricted to dimer-monomer states with weak disorder. Clarifying the energy window and the number of states retained would make the comparison to Poisson/GOE more transparent.","section":null},{"comment":"Notation for the multiparticle Wannier states switches between |Wm(R)\rangle and 'maximally localized MWS'; a single consistent abbreviation in the main text would reduce ambiguity.","section":null},{"comment":"SM S1 B gives the thermodynamic scaling of Dfrag/Dtotal; a one-sentence pointer in the main text would help readers see immediately that the fragmentation is weak in the same sense as the revivals.","section":null},{"comment":"A few typos and typesetting issues (e.g., 'Bandre-solved', 'Mul tip ar ticle', missing spaces in SM headings) should be cleaned for the final version.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a solid conceptual contribution with careful numerics and an honest statement of the weak character of the breaking. The two major points are load-bearing for the ETH-without-scars claim but are addressable by clarification and modest additional diagnostics rather than a redesign of the work. Fit for a high-quality quant-ph / condensed-matter theory journal is good; I would not recommend rejection or major overhaul."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The real news here is a concrete construction: spatial modulation folds multiparticle bands so that a nearly linear sub-band appears, and the corresponding multiparticle Wannier states revive because their Bloch components have nearly equal energy spacings. The eigenstates themselves look ETH-compliant on the diagnostics they run. That is a different route from scars or constraint-induced fragmentation, and the authors are explicit that the effect is weak (revival period diverges as level spacing vanishes).\n\nWhat they do well is keep the argument tight. Cotranslation gives exact band-resolved Wannier-sector fragmentation (Eqs. 1–2, SM S1); the projected Hamiltonian reproduces the full dynamics, so numerics stay inside the sector. Band folding under moderate modulation isolates the linear segment (Fig. 2, SM S2); level statistics show repulsion, local observables and entanglement vary smoothly with no scar-like outliers (Fig. 2c,d and SM S3); fidelity, entanglement, and FFT peaks track the equal spacings (Figs. 3–4); OTOCs stay ballistic rather than scrambling (SM S4); and the same pattern appears for 4- and 5-particle hybrid manifolds and for Fock superpositions that only partially overlap the MWS (SM S5). Self-citations are to their earlier multiparticle Wannier machinery; the new claim is checked independently.\n\nSoft spots are real but already flagged by the authors and the reader. Modulation amplitude and pattern are hand-tuned (trade-off between gap size and residual curvature). ETH is only finite-size. Residual edge curvature produces long-time beats, and the thermodynamic limit is expected to kill the revivals. No code is shipped. None of that invents a hidden contradiction; the mechanism is self-consistently weak.\n\nThis is for people who care about quantum thermalization, scars, and band engineering in cold atoms or circuits. It is solid enough for a serious referee. I would send it out.","headline":"Clean finite-size demonstration that nearly linear multiparticle sub-bands can produce MWS revivals from ETH-satisfying states via spectral phase coherence, without scars.","tokens_in":23453,"tokens_out":496,"would_cite":true,"duration_ms":6483,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Long-lived quantum revivals can arise from phase-locked thermal eigenstates alone, without scar-like nonthermal states.","keywords":["weak ergodicity breaking","eigenstate thermalization hypothesis","multiparticle Wannier states","band folding","spectral phase coherence","Bose-Hubbard model","quantum many-body scars"],"falsifier":"Enlarge the modulated lattice (or prepare a high-overlap Fock superposition of the target Wannier sector) and check whether the fidelity of a multiparticle Wannier state in the nearly linear band continues to show long-lived periodic peaks whose frequencies match the band's equal energy gaps, while generic Fock states in the same energy window thermalize; collapse of those peaks while the band remains approximately linear would falsify the claim.","tokens_in":23481,"feed_emoji":"⚛️","tokens_out":618,"duration_ms":7907,"temperature":0.7,"pith_summary":"Typical routes to ergodicity breaking in isolated interacting quantum systems rely on nonthermal eigenstates. This paper shows a different route: multiparticle Wannier states living in a nearly linear energy band can revive periodically even though every eigenstate that builds them satisfies the eigenstate thermalization hypothesis. Spatially periodic modulation of a Bose-Hubbard lattice folds multiparticle bands so that nearly linear sub-bands inherit equal energy spacings; those equal spacings lock the dynamical phases and produce collective revivals. The result matters because it separates long-lived memory retention from the requirement of scar-like or localized eigenstates, and it points to spectral engineering of multiparticle bands as a practical design principle.","feed_headline":"Revivals from thermal eigenstates, no scars required","feed_subtitle":"Equal energy spacings in a folded multiparticle band lock phases and keep memory alive.","key_machinery":"Band-resolved Wannier-sector fragmentation: multiparticle Wannier states constructed from a single multiparticle Bloch band form a dynamically isolated sector whose revival (or dephasing) is controlled solely by the energy-spacing structure of that band; spatial modulation folds an irregular band into nearly linear sub-bands that supply the equal spacings.","core_discovery":"Weak ergodicity breaking can occur without any ETH-violating eigenstates. In a spatially modulated Bose-Hubbard lattice, multiparticle Wannier states that are coherent superpositions of multiparticle Bloch states inside a nearly linear band exhibit long-lived collective revivals; the revivals come from emergent equal energy spacings that produce spectral phase coherence, not from nonthermal eigenstates.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Weak ergodicity breaking from phase-coherent thermal eigenstates","Collective MWS revivals via equal spacings in nearly linear bands","ETH eigenstates revive multiparticle Wannier states without scars","Band folding yields long-lived revivals from spectral phase coherence","Memory retained by coherent superpositions of thermal Bloch states"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"That the finite-size evidence that the multiparticle Bloch states inside the nearly linear band obey ETH remains valid, and residual band-edge curvature does not destroy phase locking when the system becomes large enough that level spacings vanish.","fun_headline_variants_meta":{"raw":{"variants":["Weak ergodicity breaking from phase-coherent thermal eigenstates","Collective MWS revivals via equal spacings in nearly linear bands","ETH eigenstates revive multiparticle Wannier states without scars","Band folding yields long-lived revivals from spectral phase coherence","Memory retained by coherent superpositions of thermal Bloch states"]},"model":"grok-4.5","effort":"low","cost_usd":0.005762,"raw_usage":{"total_tokens":1489,"prompt_tokens":745,"num_sources_used":0,"completion_tokens":87,"cost_in_usd_ticks":57620000,"prompt_tokens_details":{"text_tokens":745,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":657,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":745,"tokens_out":87,"duration_ms":9537,"temperature":1.0,"reasoning_tokens":657,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T20:24:39.778108+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Enlarge the modulated lattice (or prepare a high-overlap Fock superposition of the target Wannier sector) and check whether the fidelity of a multiparticle Wannier state in the nearly linear band continues to show long-lived periodic peaks whose frequencies match the band's equal energy gaps, while generic Fock states in the same energy window thermalize; collapse of those peaks while the band remains approximately linear would falsify the claim.","supporting_citations":[],"review_version":1}