{"id":"35802a61-2c19-469e-a1f2-de81193ecbef","arxiv_id":"2505.01786","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Ballistic propagation bounds for all moments and thermodynamically stable Lieb-Robinson bounds are proven for long-ranged Bose-Hubbard Hamiltonians under suitable initial-state conditions.","lead":"Long-range interacting bosons on a lattice are proven to obey a finite speed limit: particles and quantum information cannot spread faster than a fixed velocity, provided the initial gas is neither too empty nor too crowded. The proof supplies the first such bounds that remain valid as the system size grows to infinity.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.2 states α>3d+1 but (2.23) gives β=0 for 3d+1<α<3d+2, so the claimed ξ^{-β} light-cone error does not decay; the proof only supports a meaningful LRB for α>3d+2, as the introduction itself says.","rationale":"I read the paper as proving two conditional ballistic-propagation statements: moment bounds for two-sided density initial states, and a thermodynamically stable light-cone approximation for states with a particle-free shell. The shell assumption is restrictive but it is an explicit hypothesis, and the paper correctly flags in Remark 2(ii) the extra term that appears without it; I do not regard the initial-state geometry as an internal error. The most concrete load-bearing problem is the α-threshold inconsistency in Theorem 2.2: the displayed exponent β=⌊α−3d−1⌋ is zero on an open parameter range allowed by (2.18), so the asserted ξ^{−β} decay does not exist there. The reader's rationale also identifies this threshold issue, although the reader's 'weakest assumption' centers on initial-state geometry rather than on this threshold. My reading therefore agrees with the CONDITIONAL verdict but routes the condition through Theorem 2.2's statement: the proof appears capable of delivering a meaningful LRB for α>3d+2, matching the introduction, but the formal theorem as written claims a non-vacuous light cone for α>3d+1 only. I did not find a separate fatal flaw in the multiscale ASTLO induction; the paper is long and I did not machine-check every inequality, but the threshold mismatch is sufficient to require correction before acceptance.","tokens_in":51620,"tokens_out":21509,"duration_ms":230778,"concrete_test":"Re-derive the final error estimate in Proposition 7.3 for the borderline case d=1, α=4.5, which satisfies α>3d+1 but has β=0. Track the exponent in (7.23) and (8.20): with n≤α−d−1=2.5, take n=2, and check whether the final line (7.40) can produce any factor ξ^{−β} with β>0. If the computation yields only C‖A‖‖B‖|t|, the theorem must be restated with α>3d+2 (or with β replaced by max(1,⌊α−3d−1⌋)) before the claimed thermodynamically stable Lieb-Robinson bound is non-vacuous in the stated range.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The critical defect is the threshold stated in Theorem 2.2. The theorem assumes α>3d+1 in (2.18) and defines the error exponent as β=⌊α−3d−1⌋ in (2.23). For every α in the open interval 3d+1<α<3d+2, this gives β=0, so the right-hand side of (2.22) and (2.24) is C‖A‖‖B‖|t|ξ^0=C‖A‖‖B‖|t|. That error is independent of ξ: enlarging the separation between X and Y does not shrink the bound, and the 'light-cone approximation' becomes vacuous. A Lieb-Robinson bound with power-law error requires β≥1. The proof's own estimates confirm that β=0 is not an accidental cosmetic issue: the base case and Lemma 8.1 produce errors of order ξ^{−n+dp} with p=2 and n≤α−d−1 (see (7.19), (7.23), and (8.20)). Replacing ξ^{−n+2d} by the claimed ξ^{−β} forces n≥2d+β, hence β≤α−3d−1. To have β≥1 one must have α>3d+2. This matches the informal statement in the introduction, which says 'Assume α>3d+2' for the Lieb-Robinson result. Thus Theorem 2.2 is internally inconsistent with both its advertised result and the proof mechanism: as written, it claims a nontrivial light cone in a parameter range where the displayed error has no separation decay. The shell assumption (2.20) is strong but explicitly stated and used honestly; Remark 2(ii) correctly identifies the uncontrolled term when the shell is dropped. The sharper issue is the threshold: the theorem should either be restated with α>3d+2, or the exponent β needs to be redefined with a floor that is at least 1, and the proof adjusted accordingly. This does not destroy the central method, but it is a load-bearing correctness issue in the main theorem statement.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the quantum dynamics of Bose-Hubbard Hamiltonians on finite lattices with power-law hopping and interactions, |J_xy|, |V_xy| ≤ C|x−y|^{−α}. The two main results are: (1) for α > d+1 and initial states satisfying a two-sided density bound (2.10), all p-th moments of local particle numbers satisfy ballistic growth bounds uniformly in the particle number N (Theorem 2.1); and (2) for initial states with a particle-free shell, the Heisenberg evolution of a local observable can be approximated by the evolution restricted to a light-cone region, with an N-independent power-law error, yielding a thermodynamically stable Lieb-Robinson bound (Theorems 2.2–2.3). The proofs develop a multiscale ASTLO method with logarithmic renormalization and are presented in considerable detail. The main central issue is that Theorem 2.2 is stated with α > 3d+1 while its error exponent β defined in (2.23) vanishes for 3d+1 < α < 3d+2, so the advertised light-cone approximation has no separation decay in that parameter range.","tokens_in":52044,"tokens_out":12377,"duration_ms":120838,"significance":"The results are potentially significant: Theorem 2.1 removes the p-dependent decay threshold of earlier work and reaches the conjecturally sharp condition α > d+1 for all moments, while Theorem 2.2 aims to give the first thermodynamically stable Lieb-Robinson bound for long-range bosons. The paper is careful about its initial-state hypotheses and explicitly acknowledges the cost of dropping the particle-free shell in Remark 2(ii), which is a strength. The multiscale induction for Theorem 2.1 is developed in detail and appears internally coherent. However, the Lieb-Robinson theorem contains a load-bearing threshold inconsistency: for the range 3d+1 < α < 3d+2 the defined error exponent in (2.23) is zero, so the bound does not decay in the separation ξ; the introduction states α > 3d+2, and the proof mechanism also points to α ≥ 3d+2 for a positive exponent. This must be corrected before the central LRB claim can be accepted as stated.","major_comments":[{"comment":"Theorem 2.2 is stated for α > 3d+1 in (2.18), but the error exponent is defined as β = ⌊α−3d−1⌋ in (2.23). For every α in (3d+1, 3d+2), β = 0, so the right-hand sides of (2.22) and (2.24) are C‖A‖‖B‖|t|, independent of ξ. Such a bound has no light-cone content: increasing the separation between X and Y does not suppress the error, and the claimed approximation by the localized evolution becomes vacuous. This is not a cosmetic issue. The proof's own estimates show why: the base case and Lemmas 7.5–7.7 produce errors of order ξ^{−n+dp} (see (7.23) and (8.20)), while the available n is constrained by n ≤ α−d−1 through (7.19). Requiring ξ^{−n+2d} to decay as ξ^{−β} forces β ≤ α−3d−1; a positive exponent β ≥ 1 therefore requires α ≥ 3d+2. This agrees with the informal statement in the introduction, which says “Assume α > 3d+2” for the Lieb-Robinson result. Theorem 2.2 and its corollary Theorem 2.3 should be restated with the corrected threshold (α > 3d+2, or the exact condition needed for β ≥ 1), and the proof adjusted accordingly.","section":"Section 2.3, Theorem 2.2 and Eq. (2.23)"},{"comment":"The generalization in Proposition 7.8 inherits the same threshold problem. For Q = 1 the assumptions give α > max{3d/2+1, 2d+1} = 2d+1 and the error exponent is ⌊α−2d−1⌋, which is zero for α in (2d+1, 2d+2). For Q = 2 the stated threshold is α > 3d+1 and the exponent is ⌊α−3d−1⌋, exactly the zero-exponent range identified for Theorem 2.2. Moreover, if the Q = 1 case of Proposition 7.8 is intended to cover the same Hamiltonian as Theorem 2.2, the thresholds in (2.18) and (7.41) are inconsistent with each other. The proof of Proposition 7.8 is only sketched as a “straightforward adaption,” so it currently cannot resolve which threshold is the correct one. This needs to be clarified and corrected.","section":"Section 7.6, Proposition 7.8 and Eq. (7.45)"}],"minor_comments":[{"comment":"The density condition is written “∞ > λ2 > λ1 > 0”; please use the more conventional 0 < λ1 < λ2 < ∞, and clarify that B_r(x) denotes the intersection with Λ so that the lower bound is meaningful near the boundary of a finite lattice.","section":"Section 2.2, Eq. (2.10)"},{"comment":"Theorem 7.1, quoted from the authors' previous work [32], is a load-bearing input for the annular propagation estimates used in the proof of Theorem 2.2. Since the present paper does not prove this theorem, please state its proof status and verify that the version imported here has hypotheses matching the corrected threshold of Theorem 2.2.","section":"Section 7.1, Theorem 7.1"},{"comment":"There are several small presentation inconsistencies: the abstract says the second result is a Lieb-Robinson bound, while the body first states a light-cone approximation and derives the commutator bound as a corollary; the notation “α > 3d+1” in Theorem 2.2 and “α > 3d+2” in the introduction should be reconciled; and there are typographical slips such as “thermodynamicall” in Remark 2(i). None of these affect the mathematics, but they should be fixed in a revision.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper's main LRB result depends on Theorem 7.1 imported from the authors' earlier preprint [32]. Given the threshold inconsistency in Theorem 2.2, the editor may wish to verify that the exact version of Theorem 7.1 used here is proved in [32] with matching hypotheses and that the correction of the threshold does not invalidate the proof chain."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The real news here is Theorem 2.1: for Bose-Hubbard with power-law couplings, they control all p-th moments of local particle number under the sharp condition alpha > d+1, uniformly in the particle number and with the moment order entering only through the constants. That closes a named open problem and improves their own earlier work, where alpha had to grow with p. The logarithmic ASTLO idea is a legitimate new tool, and the proof is detailed and honest. I believe the moment half of the paper is a solid contribution.\n\nThe Lieb-Robinson half also aims at a real gap: previous long-range bosonic LRBs had errors growing with N, and here the error is N-independent. But the main theorem as printed is internally inconsistent. Theorem 2.2 states alpha > 3d+1 and defines beta = floor(alpha - 3d - 1). For every alpha in (3d+1, 3d+2), beta = 0, so the right-hand side is C ||A|| ||B|| |t|, with no decay in the separation xi. That is not a light cone; it is a vacuous bound. The proof itself shows why: the base case and Lemmas 7.5-7.7 give errors of order xi^{-n+2d} with n <= alpha-d-1, so a decaying error needs n >= 2d+1, which means alpha > 3d+2. The introduction says alpha > 3d+2 for the LRB. So this is not a harmless typo; the threshold in Theorem 2.2 must be corrected to alpha > 3d+2, or the exponent redefined accordingly. I do not see this as fatally damaging the method, but it is a load-bearing flaw in a central statement.\n\nThe particle-free shell assumption is strong, but it is explicitly stated and used honestly. Remark 2(ii) correctly notes the extra term if the shell is dropped. The heavy reliance on their previous Theorem 7.1 from [32] is not circular: that is a prior theorem with its own proof. No code or data, which is normal for this kind of analytic paper. The citation pattern looks appropriate.\n\nBottom line: Theorem 2.1 is worth serious attention. The LRB needs correction before it can be used. I would send this to peer review, with a clear request that the authors fix the threshold and re-check the exponent in Theorem 2.2 and Theorem 2.3.","headline":"The p-independent ballistic moment bounds are a genuine advance; the Lieb-Robinson theorem as stated has a threshold bug that needs fixing before the paper can be accepted.","tokens_in":713,"tokens_out":1534,"would_cite":true,"duration_ms":32996,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q40","81P45","82C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"When hopping and interaction amplitudes decay as |x−y|^(−α) with α > d+1, this paper proves that all moments of the local particle number propagate ballistically, and that correlations obey a Lieb-Robinson bound whose error is independent…","keywords":["Bose-Hubbard model","long-range interactions","Lieb-Robinson bounds","ballistic transport","many-body quantum dynamics","ASTLO method","thermodynamic limit","power-law decay"],"falsifier":"On a finite lattice with fixed density and a particle-free shell of width $\\xi$, compute the expectation of $[\\alpha_t(A),B]$ for observables separated by $2\\xi$ and increase $N$ at fixed $\\xi$; the theorem predicts an $N$-independent bound, so any growth of the commutator expectation with $N$ would falsify the thermodynamic-stability claim. For the particle-transport bound, the non-interacting case at $\\alpha=d+1$ is the natural test: if a single-particle wavepacket's local moments violate the claimed ballistic exponential, the sharp-threshold claim is refuted.","tokens_in":51390,"feed_emoji":"⚛️","tokens_out":13939,"duration_ms":123872,"temperature":0.7,"pith_summary":"This paper proves that long-range Bose-Hubbard models -- bosons on a lattice with hopping and interaction amplitudes decaying as $C|x-y|^{-\\alpha}$ -- do not transport particles or correlations arbitrarily fast, as long as the initial state has the right density profile. For every $\\alpha>d+1$, the sharp threshold known from single-particle dynamics, it bounds all moments of the local particle number by a ballistic exponential, uniformly in the particle number and in the thermodynamic limit. For initial states with a particle-free shell, it proves the first Lieb-Robinson bound for long-range bosons whose error term is independent of the total particle number, so the bound survives the thermodynamic limit. The results give rigorous meaning to the physical principle of locality for a class of strongly interacting, unbounded, long-ranged quantum many-body systems.","feed_headline":"Long-range bosons spread ballistically, with N-independent light cone","feed_subtitle":"Uniform particle moments for α>d+1 plus a Lieb-Robinson bound that survives the thermodynamic limit.","key_machinery":"The engine is the multiscale ASTLO (adiabatic space-time localization observable) method. ASTLOs are smoothed, time-dependent particle-counting operators $N_{f,t}^{(\\sigma)}=\\sum_x f_\\pm(|x|,t,\\sigma)n_x$ whose support moves with the light cone. The paper's new step is to differentiate the logarithm of $\\langle N_{f,t}^p\\rangle_t$ rather than the expectation itself; the resulting denominator makes the heavy tail of the power-law hopping controllable by a downward multiscale induction in length scale and an induction in the moment order $p$, which is what removes the $N$-dependence from the error terms.","core_discovery":"The central discovery is a pair of propagation bounds. First (Theorem 2.1): if the initial state satisfies the two-sided density bound $(\\lambda_1 r^d)^q \\le \\langle N_{B_r(x)}^q\\rangle_0 \\le (\\lambda_2 r^d)^q$ for $q\\le p$, then for $\\alpha>d+1$ and any velocity $v>12\\kappa$ the $p$-th moments satisfy, for $0\\le vt\\le R-r$, $\\langle N_{B_r}^p\\rangle_t \\le \\langle N_{B_R}^p\\rangle_0 \\exp\\{(C R^d + vt)/(R-r)\\}$ and the matching lower bound. Second (Theorems 2.2-2.3): if the initial state has a particle-free shell $N_{X_{2\\xi}\\setminus X}\\psi_0=0$ and a bounded density, then for $\\alpha>3d+1$ the Heisenberg evolution $\\alpha_t(A)$ is approximated by the localized evolution $\\alpha_t^{X_\\xi}(A)$ with error $O(\\|A\\|\\|B\\||t|\\xi^{-\\beta})$, $\\beta=\\lfloor\\alpha-3d-1\\rfloor$, with every constant independent of the total particle number $N$; this yields the thermodynamically stable Lieb-Robinson bound.","pith_inferences":["The logarithmic-ASTLO trick -- differentiating the log of the moment -- should transfer to other unbounded long-range many-body models, such as lattice oscillator chains or Fermi-Hubbard models with power-law hopping, because the denominator it creates is a general mechanism for taming heavy tails.","If the threshold $\\alpha>d+1$ is truly sharp for all moments as the paper argues, then the same threshold likely governs whether super-ballistic information transport is possible for general bounded-density initial states; the paper's LRB theorem covers only the particle-free-shell case, leaving the bounded-density super-ballistic regime as the natural next target.","The proof's uncontrolled term $\\langle N_{X_{2\\xi}\\setminus X}N_\\Lambda\\rangle_0$ when the shell is dropped suggests a quantitative trade-off: a partially filled shell with density decaying in $\\xi$ should yield a mixed bound interpolating between the $N$-independent and $N$-dependent regimes."],"forward_implications":["For any fixed $\\alpha>d+1$, the bound on all moments $p\\ge1$ is uniform in $N$, so arbitrarily many bosons can pile up on a single site without invalidating the ballistic speed; this directly controls local particle-number accumulation in the thermodynamic limit.","The Lieb-Robinson bound with $N$-independent error means that for states with a particle-free shell, the light cone is a genuine structural feature in the infinite-system limit, not an artifact of finite-size normalization.","The error exponent $\\beta=\\lfloor\\alpha-3d-1\\rfloor$ implies the light cone sharpens as $\\alpha$ grows; for very long-ranged interactions the bound is weaker, consistent with the known divergence of the single-particle velocity at $\\alpha=d+1$.","The maximal speed $\\kappa$ defined from the one-particle hopping matrix controls both particle and information transport for this class of Hamiltonians."],"supporting_citations":[{"why":"Supplies the multiscale ASTLO induction scheme that the paper refines; its p-dependent power-law threshold is the baseline being improved to the p-independent α>d+1.","marker":"[32]"},{"why":"Introduced the bosonic ASTLO method and the particle-free shell condition used in Theorem 2.2.","marker":"[13]"},{"why":"Established long-range Lieb-Robinson bounds for Bose-Hubbard models with errors depending on the total particle number; this paper removes that dependence.","marker":"[31]"},{"why":"Proved light-cone approximation for bosons with N-dependent remainder, providing the comparison baseline for the thermodynamically stable bound.","marker":"[42]"},{"why":"Demonstrated the single-particle threshold α>d+1 for finite group velocity, which establishes the sharpness of the particle-transport exponent.","marker":"[43]"},{"why":"Produced super-ballistic Lieb-Robinson bounds for finite-range interacting bosons at bounded density, used as the contrast for the fixed-speed regime.","marker":"[29]"}],"fun_headline_variants":["Ballistic spreads for long-range bosons: sharp α>d+1 bounds","Light cone survives thermodynamic limit in long-range Bose-Hubbard","Two propagation bounds for long-range Bose-Hubbard dynamics","Stable Lieb-Robinson bound for long-ranged bosons with power laws","α>d+1 sharp: all moments spread ballistically in long-range bosons"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The most fragile premise is the geometry of the initial state: the Lieb-Robinson result requires a completely empty shell around the region of interest, and the moment bounds require every ball at every scale to contain between $\\lambda_1 r^d$ and $\\lambda_2 r^d$ particles; if either condition fails, the proof's error terms are uncontrolled and may grow with the particle number.","fun_headline_variants_meta":{"raw":{"variants":["Ballistic spreads for long-range bosons: sharp α>d+1 bounds","Light cone survives thermodynamic limit in long-range Bose-Hubbard","Two propagation bounds for long-range Bose-Hubbard dynamics","Stable Lieb-Robinson bound for long-ranged bosons with power laws","α>d+1 sharp: all moments spread ballistically in long-range bosons"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001107,"raw_usage":{"total_tokens":4618,"prompt_tokens":949,"completion_tokens":3669,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":565,"completion_tokens_details":{"reasoning_tokens":3570}},"tokens_in":565,"tokens_out":3669,"duration_ms":28428,"temperature":1.0,"reasoning_tokens":3570,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:10:31.946323+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a finite lattice with fixed density and a particle-free shell of width $\\xi$, compute the expectation of $[\\alpha_t(A),B]$ for observables separated by $2\\xi$ and increase $N$ at fixed $\\xi$; the theorem predicts an $N$-independent bound, so any growth of the commutator expectation with $N$ would falsify the thermodynamic-stability claim. For the particle-transport bound, the non-interacting case at $\\alpha=d+1$ is the natural test: if a single-particle wavepacket's local moments violate the claimed ballistic exponential, the sharp-threshold claim is refuted.","supporting_citations":[{"cited_title":"3, 1011–1037","cited_arxiv_id":null,"evidence_quote":"Introduced the bosonic ASTLO method and the particle-free shell condition used in Theorem 2.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Established long-range Lieb-Robinson bounds for Bose-Hubbard models with errors depending on the total particle number; this paper removes that dependence."},{"cited_title":"On propagation of information in quantum many-body systems","cited_arxiv_id":"2212.14472","evidence_quote":"Proved light-cone approximation for bosons with N-dependent remainder, providing the comparison baseline for the thermodynamically stable bound."},{"cited_title":"C Tran, C.-F","cited_arxiv_id":null,"evidence_quote":"Demonstrated the single-particle threshold α>d+1 for finite group velocity, which establishes the sharpness of the particle-transport exponent."},{"cited_title":"Kuwahara, T","cited_arxiv_id":null,"evidence_quote":"Produced super-ballistic Lieb-Robinson bounds for finite-range interacting bosons at bounded density, used as the contrast for the fixed-speed regime."}],"review_version":1}