{"id":"504866bd-e088-44b7-8211-f38c5d4596fe","arxiv_id":"1908.08569","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A projection protocol that subtracts overlaps with local integrals of motion and their products systematically reduces eigenstate fluctuations, yielding exact zero fluctuations for a two-body observable in an integrable model.","lead":"The authors prove that when a local measurement wobbles persistently across energy eigenstates, the system must contain hidden conserved quantities. They then introduce a subtraction method that removes those conserved parts, leaving smooth, thermal-looking measurement values.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof that positive stiffness implies a local integral of motion rests on the unproven Mazur-bound locality statement (Eq. 8); without it, σ²_A⊥ > 0 only shows a nonzero conserved operator, which is true by construction.","rationale":"The reader identified the same weakest assumption: the imported Mazur-bound locality result. My analysis confirms that this step is the true hinge of the paper's central claim. I considered whether the paper's own integrable example (products of LIOMs contributing to the stiffness) could undermine the theorem more directly. However, the numerical results show that after subtracting all one-body LIOMs, the stiffness scales as 1/L and vanishes in the thermodynamic limit, meaning that products contribute only at the finite-size level. Thus the thermodynamic-limit stiffness is carried by (pseudo)local LIOMs in that case, which is consistent with Eq. (8). This does not, however, prove the general statement. The paper's derivation of r_k scaling (Supplement S2) is also an assumption-based asymptotic argument, but it is secondary to the locality step for the central 'fluctuations ⇒ LIOM' claim. The cited references (Mazur 1969, Zotos et al. 1997, Ilievski et al. 2016) establish the Mazur bound and quasilocality in specific settings but do not supply the general converse used here. The concern is a genuine gap in the presentation, not a demonstrated counterexample, so the reader's CONDITIONAL verdict is appropriate. My recommendation is UNCHANGED: the paper should either prove the locality statement or explicitly label it as an assumption, but the numerical evidence and the protocol remain valuable.","tokens_in":13985,"tokens_out":30581,"duration_ms":293197,"concrete_test":"Independently re-derive the step from Eq. (10) to locality: take σ²_A⊥ > 0 and attempt to prove, using only the stated assumptions (Â local, traceless, translationally invariant) and the definition of pseudolocality in Ref. [83], that the commutator norm ||[Â̄⊥, â_j]|| decays faster than any power with distance from the support of â_j. If this derivation requires an additional assumption not stated in the paper (e.g., that Â̄⊥ is an extensive sum of operators with bounded norm, or that the Mazur bound in Eq. (8) is saturated by known LIOMs), the theorem is conditional. Alternatively, in the exactly solvable HCB model, compute J̄ (after subtracting H) and expand it in the T(n), J(n) basis; verify that the weight on products of two or more LIOMs scales as 1/L and vanishes in the thermodynamic limit, which would confirm the locality conclusion in the one case where it can be tested exactly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the central implication, the paper defines Â⊥ = Â − p_A Ĥ (Eq. 7), shows σ²_A⊥ = (1/Z)Σ_n (A_nn − p_A E_n)² > 0 (Eq. 10), and then asserts that the time-averaged operator Â̄⊥ is a local or pseudolocal integral of motion. The only support for this step is the sentence after Eq. (8): 'it is known that in the thermodynamic limit only LIOMs (including pseudolocal conserved operators) contribute to the Mazur bound,' citing Refs. [61,82,83]. Equation (9) shows that Â̄⊥ itself carries a Mazur weight equal to σ²_A⊥, so the locality conclusion is logically equivalent to that imported statement. The cited references derive Mazur bounds for specific integrable models and define quasilocality, but none proves a general theorem that for an arbitrary local, translationally invariant Hamiltonian, any conserved operator with finite Hilbert–Schmidt overlap with a local observable must be a sum of (quasi)local densities. Without such a theorem, the paper's conclusion reduces to the near-tautology that nonvanishing fluctuations imply a nonvanishing conserved operator (namely Â̄⊥), which is conserved for any observable. The physical content—that this operator is local—is assumed, not derived. This is the single most load-bearing point for the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the impact of local integrals of motion (LIOMs) on the diagonal matrix elements of local observables in translationally invariant quantum chains. The main theoretical claim is that if the fluctuations of the diagonal matrix elements of a normalized, traceless, translationally invariant local observable do not vanish in the thermodynamic limit (Eq. (5)), then there must exist a LIOM (or pseudolocal conserved operator) with positive overlap with the observable. The argument uses the Mazur bound and a stated known result that only LIOMs contribute to this bound in the thermodynamic limit. The authors then introduce a projection protocol that subtracts the overlap of the observable with LIOMs and their products, claiming a systematic reduction of fluctuations and/or structure, with explicit scaling r_k ~ 1/L^{k-1} for powers of the Hamiltonian in generic systems. Numerical results are presented for a nonintegrable spinless fermion chain and for an integrable hard-core boson chain, where the fully projected observable built from a complete set of LIOMs and their 2-products yields exactly zero diagonal matrix elements.","tokens_in":1566,"tokens_out":3386,"duration_ms":243766,"significance":"If the central implication is valid, the paper provides a concrete route to link finite-size eigenstate fluctuations with the existence of conserved local operators, and it offers a practical construction of observables with strongly suppressed fluctuations. The analytic derivation of the polynomial-fit optimality and the r_k scaling in the supplement is a useful contribution, and the exact cancellation of diagonal matrix elements in the hard-core boson model is a striking constructive result. The paper is explicit about its main assumptions (Gaussian density of states and the ETH form for diagonal matrix elements) and includes a supplement with derivations, which strengthens the presentation.","major_comments":[{"comment":"The central implication — that sigma^2_{A_\\perp} > 0 forces the time-averaged operator A_bar_\\perp to be a local or pseudolocal integral of motion — is logically equivalent to the assertion, stated after Eq. (8), that in the thermodynamic limit only LIOMs contribute to the Mazur bound for a local observable. This assertion is not proven in the manuscript, and the cited references do not appear to establish it as a general theorem for arbitrary translationally invariant local Hamiltonians: Ref. [61] treats transport in integrable models, Ref. [82] is Mazur's original inequality (which by itself says nothing about locality), and Ref. [83] defines quasilocality in integrable lattice systems. Without a precise statement and proof of this locality result (or a reference that supplies it), the conclusion of the theorem reduces to the near-tautology that nonvanishing fluctuations imply the existence of a nonvanishing conserved operator, namely A_bar_\\perp itself, whose conservation holds for any observable. Please either prove the locality statement under clearly stated conditions, or explicitly mark it as an assumption and rephrase the central claim accordingly.","section":"Violation of ETH entails existence of LIOMs (Eq. (8) and following paragraph)"},{"comment":"The analytical result r_k ~ 1/L^{k-1} is derived under the explicit assumptions of a Gaussian density of states (Eq. S7) and the ETH ansatz for diagonal matrix elements (Eq. S12). While these assumptions are stated clearly, the numerical demonstration of the protocol's effectiveness in Fig. 2(b) is limited to one nonintegrable model and to system sizes up to L=21, with the exponential decay presented as a 'guide to the eye' rather than a quantitatively fitted curve with uncertainties. For the protocol claim to be considered generic, additional numerical evidence (e.g., other nonintegrable models or error estimates on the exponential decay) would be needed, or the claim should be phrased more cautiously as a demonstration for the particular model.","section":"Size-dependence of the stiffness / Supplement S2"}],"minor_comments":[{"comment":"There are typos: 'nonintregrable' in the Conclusions should be 'nonintegrable', and 'Hemite' in Eq. (S11) of the Supplement should be 'Hermite'.","section":"Conclusions and Supplement S2"},{"comment":"The figures do not state the full set of system sizes used, nor do they provide error bars or a measure of the fit quality for the exponential and 1/L extrapolations. Please specify the system sizes and, where claims of scaling are made, provide quantitative fits or state the number of points used.","section":"Figs. 2 and 3"},{"comment":"The terms 'local' and 'pseudolocal' are used without precise definitions. This matters because the integrable example later uses operators T(n) and J(n) (Eqs. S16-S19) that are nonlocal in the original hard-core boson basis due to Jordan-Wigner strings; please clarify in which sense these operators are local or pseudolocal.","section":"Eq. (8) and the following paragraph"},{"comment":"The energy window Delta used in the numerical evaluation of Sigma^2_A(Delta) is not specified. Please state the width and position of the window used in Fig. S1 and in the main-text analysis.","section":"Eq. (3)"},{"comment":"The products of LIOMs are described as 'few-body but nonlocal conserved operators.' For fixed order k, these products are in fact sums of local terms with finite support (for powers of H), so the terminology may confuse readers; the Conclusions also refer to the projected observables as 'albeit nonlocal.' Clarifying the sense of locality/nonlocality here would improve readability.","section":"Introduction and Conclusions"},{"comment":"The inequality in Eq. (4) is used in the derivation of Eq. (6) but is stated without proof. A brief justification (for instance, that within each microcanonical window the mean is the least-squares constant, so any other smooth function cannot reduce the sum of squares) would make the argument more self-contained.","section":"Eq. (4)"}],"recommendation":"major_revision","confidential_remarks":"The soundness of the central theorem depends on the locality statement about the Mazur bound, which is a subtle point that may require input from specialists in rigorous results on conserved quantities in many-body systems. The paper would be strengthened by engaging more precisely with the literature on quasilocal charges (e.g., works by Prosen and collaborators) and by stating explicitly what is a theorem, what is an assumption, and what is a conjecture. The numerical evidence is typical for exact diagonalization studies but should be presented with more caution regarding finite-size claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know before reading. The main formal claim—nonvanishing eigenstate fluctuations force local integrals of motion—is clean and clearly stated, but the step from positive stiffness to a local integral of motion is not proved here. It rests on a statement about the Mazur bound that is cited to prior work, and the stress-tester is right: without a general theorem that only (pseudo)local conserved operators can have nonzero overlap with a local observable in the thermodynamic limit, the conclusion is close to tautological. Second, the projection protocol itself is genuinely new and does real work: subtracting projections onto LIOMs and their products systematically reduces the diagonal-matrix-element structure, with the r_k ~ 1/L^{k-1} scaling derived in the supplement, and the exact-zero result for the two-projected observable in hard-core bosons is a striking finite-size fact.\n\nWhat the paper does well: the formalism with normalized translationally invariant observables, stiffness, and the polynomial-fit optimality is careful. The supplement fills in the steps for the best-fit claim and the Hermite-polynomial scaling. The numerical evidence is consistent with the claims, and the HCB exact-zero cancellation is not fittable—it is a concrete structural result. The citation pattern is fine; the key locality citations are the right ones, though they may not prove the general claim.\n\nSoft spots. The locality step is load-bearing and imported rather than proved. The cited references support locality for specific integrable models and define quasilocality, but I am not aware of a general theorem for arbitrary TI local Hamiltonians. If that theorem does not exist, the central implication needs rephrasing as a conjecture or a conditional. The r_k scaling also assumes a Gaussian density of states and the exponential-ETH form for diagonal matrix elements; these are stated assumptions, fine for a Letter, but they limit the universality claim. The numerics are on chains up to L=21 with no error bars, and the exponential decay in Fig. 2(b) rests on few points. These are addressable and do not sink the protocol.\n\nWho should read it: anyone working on ETH breakdown, generalized ETH, or transport stiffness in integrable systems. It deserves a real referee and likely publication after revision. I would ask the authors to make the locality assumption explicit, add a careful statement (or proof) of what exactly is known, and strengthen the finite-size analysis for the exponential claim.","headline":"A clean projection protocol and a plausible formal implication, but the main claim leans on a locality statement imported from the Mazur-bound literature; worth engaging and worth sending to referees.","tokens_in":14791,"tokens_out":2643,"would_cite":true,"duration_ms":27893,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"If the eigenstate fluctuations of a local observable do not vanish with system size, a local integral of motion must exist—and subtracting projections onto conserved operators removes those fluctuations order by order.","keywords":["eigenstate thermalization hypothesis","local integrals of motion","operator stiffness","Mazur bound","generalized ETH","integrable systems","diagonal matrix elements","hard-core bosons"],"falsifier":"A concrete check: in a translationally invariant model whose only conserved operators are explicitly nonlocal, compute the stiffness $\\sigma^2_{A\\perp}$ and the support width of $\\bar{A}_\\perp$ for a local observable with nonvanishing fluctuations; if positive stiffness coexists with a $\\bar{A}_\\perp$ whose width grows linearly with $L$, the locality step collapses. In the hard-core boson example, any nonzero diagonal element of the two-projected observable at accessible system sizes would refute the exact-zero claim as stated.","tokens_in":13772,"feed_emoji":"⚛️","tokens_out":7444,"duration_ms":71924,"temperature":0.7,"pith_summary":"This paper argues that the eigenstate thermalization hypothesis (ETH) can be reconciled with integrable many-body systems by explicitly accounting for local integrals of motion (LIOMs). The central claim is that if the fluctuations of the diagonal matrix elements of a normalized, translationally invariant local observable do not vanish in the thermodynamic limit, then a local or pseudolocal conserved operator must exist—so nonvanishing fluctuations are not a failure of ETH but a fingerprint of hidden conserved structure. The authors introduce a protocol that subtracts an observable's projections onto LIOMs and onto products of LIOMs, showing that the remaining fluctuations shrink order by order in generic systems and can vanish exactly at finite size in an integrable hard-core boson model. A sympathetic reader should care because this turns the longstanding empirical distinction between thermalizing and integrable systems into a constructive, operator-level statement.","feed_headline":"Eigenstate fluctuations force hidden local conserved operators","feed_subtitle":"A subtraction protocol removes projections onto conserved operators, shrinking fluctuations toward exact ETH.","key_machinery":"The load-bearing object is the operator stiffness $\\sigma^2_A = \\langle \\bar{A} \\bar{A} \\rangle$, the squared Hilbert-Schmidt norm of the infinite-time averaged observable, which equals the mean-square of the diagonal matrix elements $A_{nn}$. The key identity is that for the Hamiltonian-projected observable, $\\langle A \\bar{A}_\\perp \\rangle^2 / \\langle \\bar{A}_\\perp \\bar{A}_\\perp \\rangle = \\sigma^2_{A\\perp}$, which combines with the Mazur bound to convert a positive stiffness into the statement that $\\bar{A}_\\perp$ is a local or pseudolocal conserved operator. For products of LIOMs, the protocol is an orthogonalization: the $k$-products $H_{\\perp k}$ are built by Gram-Schmidt from powers of $H$, projected observables subtract the best polynomial fit to the microcanonical average, and a Gaussian density of states identifies the orthogonal polynomials as Hermite polynomials, giving $r_k \\sim 1/L^{k-1}$.","core_discovery":"On the paper's own terms, the discovery is an implication: for a traceless, translationally invariant local observable with normalized norm, the condition $\\lim_{L\\to\\infty} \\Sigma^2_A(\\Delta)>0$ forces the time-averaged operator $\\bar{A}_\\perp = \\overline{A - p_A H}$ to have positive stiffness $\\sigma^2_{A\\perp} = (1/Z) \\sum_n (A_{nn} - p_A E_n)^2$. Because only local or pseudolocal conserved operators contribute to the Mazur bound in the thermodynamic limit, the positive stiffness identifies this conserved operator as a LIOM; conversely, if a proposed set of LIOMs is complete, subtracting projections onto it drives the stiffness to zero. The paper then constructs $k$-projected observables whose stiffness decays as $1/L^{k-1}$ in generic systems, and in the hard-core boson chain the two-projected observable has all diagonal matrix elements exactly zero already in finite systems.","pith_inferences":["This suggests a diagnostic use in reverse: measuring the stiffness of experimentally accessible observables could reveal hidden or approximate conserved quantities in systems not already known to be integrable.","Because products of LIOMs are nonlocal few-body operators, the exact-zero result implies that the generalized ETH ansatz should be formulated with the full algebra generated by local charges, not just the charges themselves.","For disordered or many-body-localized systems, where the Hamiltonian is not translationally invariant, a generalized version of this protocol would likely reduce fluctuations whenever approximate LIOMs exist; that extension is not worked out in the paper.","A natural numerical test beyond the paper is to apply the projection protocol to other integrable chains, such as the XXZ model, and check whether the predicted $1/L^{k-1}$ scaling of the projections persists."],"forward_implications":["For any local observable in a generic nonintegrable system, subtracting the projection onto the Hamiltonian and its powers removes the energy structure of the diagonal matrix elements, after which fluctuations decay exponentially with system size.","If nonvanishing fluctuations always signal a LIOM, then ordinary ETH and generalized ETH are not competing descriptions: the latter is the former once hidden conserved operators are subtracted.","The protocol gives a convergence test for a proposed set of LIOMs: if the stiffness of the projected observable does not vanish as $L\\to\\infty$, the set is incomplete.","In integrable models, subtracting projections onto all LIOMs and their products can push the diagonal matrix elements to exactly zero at finite size, realizing a limiting strong ETH without fluctuations."],"supporting_citations":[{"why":"Supplies the Mazur bound used in Eq. (8), the inequality connecting stiffness of a time-averaged operator to overlaps with conserved operators.","marker":"[82]"},{"why":"Establishes that only local or pseudolocal conserved operators contribute to the Mazur bound in the thermodynamic limit, the step that turns positive stiffness into a LIOM.","marker":"[61]"},{"why":"Introduced pseudolocal conserved operators in the context of the Mazur bound, providing the notion of locality the argument requires.","marker":"[68]"},{"why":"Defines quasilocal charges whose pseudolocality is used to classify the conserved operator $\\bar{A}_\\perp$ as local or pseudolocal.","marker":"[83]"},{"why":"Provides the Srednicki ETH ansatz for diagonal matrix elements, which the paper modifies and tests by subtracting structured energy dependence.","marker":"[18]"},{"why":"Defines the finite-size measure of eigenstate fluctuations $\\Sigma^2_A(\\Delta)$ and its scaling, which is the starting diagnostic of ETH violation.","marker":"[28]"},{"why":"Demonstrated the breakdown of ETH in integrable one-dimensional systems, the empirical phenomenon this paper explains via LIOMs.","marker":"[22]"}],"fun_headline_variants":["Fluctuations force local conserved operators","Subtract conserved operators to tame ETH fluctuations","Eigenstate noise reveals hidden integrals of motion","New subtraction shrinks fluctuations toward ETH","Nonzero fluctuations imply LIOMs; protocol removes them"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument leans on the imported theorem that, in the thermodynamic limit, only local or pseudolocal conserved operators contribute to the Mazur bound for a translationally invariant local observable; if some nonlocal conserved operator could carry nonzero weight in that sum, positive fluctuation stiffness would not prove the existence of a local integral of motion.","fun_headline_variants_meta":{"raw":{"variants":["Fluctuations force local conserved operators","Subtract conserved operators to tame ETH fluctuations","Eigenstate noise reveals hidden integrals of motion","New subtraction shrinks fluctuations toward ETH","Nonzero fluctuations imply LIOMs; protocol removes them"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000205,"raw_usage":{"total_tokens":1349,"prompt_tokens":857,"completion_tokens":492,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":473,"completion_tokens_details":{"reasoning_tokens":424}},"tokens_in":473,"tokens_out":492,"duration_ms":5643,"temperature":1.0,"reasoning_tokens":424,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:35:36.139216+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check: in a translationally invariant model whose only conserved operators are explicitly nonlocal, compute the stiffness $\\sigma^2_{A\\perp}$ and the support width of $\\bar{A}_\\perp$ for a local observable with nonvanishing fluctuations; if positive stiffness coexists with a $\\bar{A}_\\perp$ whose width grows linearly with $L$, the locality step collapses. In the hard-core boson example, any nonzero diagonal element of the two-projected observable at accessible system sizes would refute the exact-zero claim as stated.","supporting_citations":[{"cited_title":"Prosen, Open XXZ spin chain: Nonequilibrium steady state and a strict bound on ballistic transport, Phys","cited_arxiv_id":null,"evidence_quote":"Introduced pseudolocal conserved operators in the context of the Mazur bound, providing the notion of locality the argument requires."},{"cited_title":"Ilievski, M","cited_arxiv_id":null,"evidence_quote":"Defines quasilocal charges whose pseudolocality is used to classify the conserved operator $\\bar{A}_\\perp$ as local or pseudolocal."},{"cited_title":"Srednicki, The approach to thermal equilibrium in quantized chaotic systems, J","cited_arxiv_id":null,"evidence_quote":"Provides the Srednicki ETH ansatz for diagonal matrix elements, which the paper modifies and tests by subtracting structured energy dependence."},{"cited_title":"Beugeling, R","cited_arxiv_id":null,"evidence_quote":"Defines the finite-size measure of eigenstate fluctuations $\\Sigma^2_A(\\Delta)$ and its scaling, which is the starting diagnostic of ETH violation."},{"cited_title":"Rigol, Breakdown of thermalization in finite one- dimensional systems, Phys","cited_arxiv_id":null,"evidence_quote":"Demonstrated the breakdown of ETH in integrable one-dimensional systems, the empirical phenomenon this paper explains via LIOMs."}],"review_version":1}