{"id":"2ff2d9ab-27fe-4353-9ad3-f0ada862f784","arxiv_id":"2412.16041","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"In boundary-driven spin chains, steady-state integrability is independent of Liouvillian integrability and is best diagnosed by the operator-size distribution of the steady-state Hamiltonian, not by local simplicity.","lead":"The paper separates two types of integrability in open quantum systems, showing that a chaotic Liouvillian can still produce an integrable steady state. It uses level statistics and the operator-size distribution of the steady-state Hamiltonian to tell the two cases apart, and finds that even integrable steady states contain long-range many-body operators.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The B1-B3 classification as nonintegrable steady states depends entirely on the conjectured gamma_deph ~ 1/N dephasing scale; if the true scale is O(1), the central P(S) discriminator becomes an artifact of parameter choice.","rationale":"The reader's weakest assumption identifies exactly the load-bearing point. The paper has genuine strengths: exact steady states for models A and B, multiple independent diagnostics (CSR, r_tilde, NESS ETH) that are internally consistent, and a clear demonstration that integrable steady states are not operator-local. My concern is not with the internal consistency of these diagnostics but with the external validity of the B1-B3 nonintegrability classification. The 1/N dephasing conjecture is not a minor technical detail; it selects the parameter regime in which the central discriminator is tested. At gamma_deph = 1 the same models give Poisson statistics, so without the conjecture the paper's claim reduces to 'after tuning dephasing to the conjectured scale, nonintegrable steady states show GUE and integrable ones show Poisson,' which is not a robust discriminator unless the scale is independently established. A direct check of the NESS as a function of N and gamma_deph would settle the issue. Conditional acceptance remains the appropriate verdict, with this check as a natural condition.","tokens_in":17624,"tokens_out":6335,"duration_ms":61978,"concrete_test":"Compute the NESS of model B1 (and B2) by time evolution to convergence for N = 10, 12, 14, 16 at fixed gamma_deph = 0.1 and at fixed gamma_deph = 1.0, and measure the normalized distance to the infinite-temperature state, delta_N = ||rho_N - 1/2^N||_HS, together with the steady-state spin current. If the 1/N scaling is correct, delta_N and the current should decay with N at fixed gamma_deph = 1.0, while at fixed gamma_deph = 0.1 they should remain finite or grow. Alternatively, compute the Liouvillian gap as a function of N and gamma_deph and test whether the gap obeys gap = gamma_deph * f(N gamma_deph); if the gap saturates at fixed gamma_deph = 1.0, the O(1) scale is the relevant one and the B1-B3 classification as nonintegrable steady states is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that P(S) distinguishes chaotic from integrable steady states, and specifically that B1-B3 have nonintegrable steady states while their Liouvillians are chaotic, hinges on the dephasing scale. Section III B conjectures that the nontrivial boundary-driven NESS is reached only for gamma_deph = gamma_tilde/N, and the numerical results use gamma_tilde = 1 (gamma_deph = 0.1 at N ~ 10). The prefactor is not derived; it is chosen to match the gamma_deph = 0.1 runs. The heuristic supporting the conjecture is an operator-growth lower bound on the Liouvillian gap (g ~ gamma N), which does not directly locate the crossover of the NESS from boundary-dominated to dephasing-dominated. At gamma_deph = 1, the same models B1-B3 show Poisson Hss statistics and no NESS ETH (Fig. 3c,e), which the paper attributes to a trivial infinite-temperature steady state. If the dephasing scale for a nontrivial NESS is actually O(1), then B1-B3 are integrable-steady-state (or at least non-chaotic) at all studied sizes, and the separation in Fig. 3(b,d) is an artifact of tuning gamma_deph. The operator-size discriminator is only demonstrated in this tuned regime and only visually; no quantitative separation criterion is given. Thus the load-bearing assumption is the 1/N dephasing scaling.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies boundary-driven spin chains with bulk dephasing and asks whether the notion of integrability of the nonequilibrium steady state is independent of integrability of the Liouvillian. Using exact benchmarks (model A, whose steady state is known exactly, and model B, which is Bethe-ansatz integrable) and perturbations thereof (A', B1, B2, B3), the authors compute complex spacing ratios of the Liouvillian, level-spacing ratios and NESS-ETH scaling of the steady-state Hamiltonian H_ss = -ln ρ_ss, and the Pauli-string size distribution P(S) of H_ss. They report that steady-state integrability is signaled by Poisson statistics of H_ss and failure of NESS ETH, even when the Liouvillian is chaotic, and that although integrable steady states are not simple in the Pauli basis, the shape of P(S) can distinguish chaotic from integrable steady states. The dephasing strength is argued to have a natural scale γ_deph ~ 1/N, and the choice γ_deph = 1/N is used for the nonintegrable B-type cases.","tokens_in":17965,"tokens_out":13548,"duration_ms":135677,"significance":"If the central claims hold, the paper makes a useful conceptual contribution: it separates 'Liouvillian integrability' from 'steady-state integrability' and provides concrete diagnostics for the latter. The use of known exact steady states as benchmarks is a strength, as is the combination of spectral statistics, ETH scaling, and operator-size analysis. The numerical evidence is extensive and clearly presented, and the finding that integrable steady states contain long Pauli strings of all lengths is a genuinely nontrivial observation. The main weakness is that the classification of the B-type models as having chaotic steady states depends on a conjectured dephasing scale that is not derived and is only tested at one value.","major_comments":[{"comment":"The classification of models B1-B3 as having nonintegrable steady states rests on the conjecture in Sec. IIIB that the nontrivial boundary-driven NESS requires γ_deph = γ~_deph/N, with the prefactor implicitly set to unity to match the earlier γ_deph = 0.1 results. The heuristic gap argument (g ~ γ N) does not directly locate the NESS crossover, and no systematic scan over γ_deph at fixed N is presented. At γ_deph = 1, the same models show Poisson H_ss statistics and no NESS ETH (Fig. 3c,e), which the paper attributes to a trivial infinite-temperature steady state; that attribution is part of the same conjectural scaling. If the relevant scale were instead O(1), the B1-B3 steady states would be classified as integrable (or at least non-ETH) at all sizes studied, and the separation in Figs. 3(b,d) and 4 would be a parameter artifact. The authors should either derive the 1/N scale more directly or provide a finite-size crossover study (e.g., varying N γ_deph through values 0.1, 1, 10 for the level statistics and NESS ETH) to demonstrate that γ_deph = 1/N is indeed the correct operational scale.","section":"Sec. IIIB, Fig. 3"},{"comment":"The NESS-ETH scaling claims are made without any estimate of statistical uncertainty. The standard deviation σ of the observable matrix elements is plotted at single values of N, with no error bars and no statement of how many eigenstates fall in the spectral window used for the fit. Consequently, the fitted exponents α in Fig. 2(b) have unknown confidence intervals, and the statements that A' is 'consistent with' the D^{-1/2} prediction and that the deviation is a 'small systematic' finite-size effect are not quantitatively supported. Please report the standard error of σ (e.g., from eigenstate-to-eigenstate fluctuations) or at least the sample sizes, and show the linear fits with confidence intervals.","section":"Sec. IIIA, Figs. 1(c), 2(b), 3(d-f)"},{"comment":"The abstract claims that the operator-size distribution can be used to distinguish chaotic and integrable steady states, but the evidence in Fig. 4 is visual only. No quantitative separation criterion is defined: there is no metric such as the decay rate of P(S) at large S, the weight ratio P(S)/P(1), or a threshold based on the average operator size. Given that the B-model comparison is entangled with the dephasing-scale issue, a quantitative discriminator applied to the A/A' pair and to the B-family would make the claimed distinction testable and would remove the ambiguity about what 'effectively use' means.","section":"Sec. IV, Fig. 4"}],"minor_comments":[{"comment":"The abstract says the steady state is expanded in Pauli strings, but the analysis is actually of H_ss = -ln ρ_ss. The authors mention that H_ss and ρ_ss share the same level statistics, but they do not address whether their Pauli-size distributions are qualitatively similar; please clarify the relationship or add a sentence justifying why P(S) of H_ss is the relevant object.","section":"Abstract and Sec. IV"},{"comment":"For model A the CSR values (⟨r⟩ = 0.701, -⟨cos θ⟩ = 0.100) are closer to Poisson than to GinUE. The text attributes this to a finite-size effect; a short N-dependence of the CSR values (or a reference to Ref. [14] for larger N) would make this more convincing.","section":"Table II and Fig. 1(a)"},{"comment":"The sentence 'For the system sizes available (N ≈ 10), this results in γ_deph ≈ 1/10' is imprecise because Fig. 3(f) uses γ_deph = 1/N, which ranges from 1/6 to 1/12 over the plotted N values; please state the actual range.","section":"Sec. IIIB"},{"comment":"The symbol D is used both for the dissipator superoperator (Eqs. 4-5) and for the Hilbert-space dimension (Eq. 15); although standard in context, a notational distinction would help readers.","section":"Eq. (15)"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the central idea is interesting. The main risk is the dephasing-scale conjecture: if the authors can provide a convincing crossover analysis or a derivation of the 1/N scale, the conclusions would be solid. The citation of Ref. [102], a closely related preprint by two of the coauthors, is appropriate but the overlap should be made explicit in the revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth your time. It does two useful things. First, it shows that steady-state integrability and Liouvillian integrability are genuinely independent: in model A, the Liouvillian is nonintegrable while the steady state is integrable, and the level statistics of Hss and the NESS ETH test follow the steady state, not the Liouvillian. Second, it tests the natural guess that integrable steady states are 'simple' in operator space, and finds they are not: the Pauli-string expansion of Hss has weight at all lengths, including long-range and many-body terms. That negative result is real and clearly demonstrated.\n\nThe operator-size distribution P(S) is the most interesting addition. In the A vs A' comparison, the integrable steady state has an approximately exponential decay of P(S) while the chaotic one is much flatter. This is a clean, dephasing-free distinction, and it supports the abstract's claim on its own.\n\nThe soft spots are real but not fatal. The B-family classification into chaotic steady states relies on the conjecture that the relevant dephasing scale is gamma_deph ~ 1/N. The prefactor is not derived; it is chosen so that gamma_deph ~ 0.1 at N ~ 10, matching the earlier data. If the true crossover to the trivial infinite-temperature state occurred at O(1), then B1-B3 would look Poissonian at every accessible N, and the bottom row of Fig. 3 would be an artifact. The stress-test note pushes this concern hard, but it overstates the damage: the A models involve no dephasing, so the P(S) discriminator and the non-locality of integrable steady states do not depend on the conjecture. What the B family adds is an example where the Liouvillian is chaotic but the steady state is only chaotic at the special scale. That is interesting, but it needs a clearer derivation or a sweep of gamma_deph across N to pin the crossover.\n\nA few smaller issues: the sigma scaling fits in Figs. 1(c) and 3(d) have no error bars and only a few points; model A's Liouvillian CSR sits between Poisson and GinUE, which the authors attribute to finite size without much evidence; and the P(S) discrimination is visual only, with no quantitative threshold separating the two classes.\n\nThe citation pattern looks fair. The paper leans on Refs. [43,50,51] for exact steady states and [14,29,68] for the diagnostics, which is appropriate. It also cites the operator-growth literature for the 1/N argument.\n\nWho should read it: anyone working on quantum chaos or thermalization in open systems, and people who use NESS ETH as a probe. It deserves a serious referee. The main fixes are cosmetic-to-moderate, and a revised version with a dephasing sweep and error bars would be noticeably stronger.\n\nMy recommendation: send it to peer review, with the dephasing-scale question as the main referee issue.","headline":"A careful numerical study that cleanly separates Liouvillian from steady-state integrability and shows integrable steady states are not operator-local; the main weakness is a conjectured dephasing scale for the B models, but the A models carry the central claim.","tokens_in":18474,"tokens_out":4095,"would_cite":true,"duration_ms":34746,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["05.30.-d","05.45.Mt","03.65.Yz"],"model":"deepseek-v4-flash","headline":"Integrable steady states of open quantum systems are not built from few-body local operators, yet the operator-size distribution of their Pauli-string expansion still separates them from chaotic steady states.","keywords":["Lindblad master equation","steady-state integrability","operator size distribution","Pauli strings","level statistics","NESS ETH","boundary-driven spin chains","dephasing"],"falsifier":"Compute the Liouvillian gap or the decay rate of the slowest operator for the bulk-dephased chains as a function of $N$ and $\\gamma_{\\mathrm{deph}}$: if the crossover to the trivial infinite-temperature steady state occurs at $\\gamma_{\\mathrm{deph}} \\sim 1/N$, the chaotic classification of B1--B3 is supported; if it occurs at $\\gamma_{\\mathrm{deph}} \\sim O(1)$, the Poisson statistics seen at $\\gamma_{\\mathrm{deph}}=1$ would persist at all system sizes and the central claim would fail.","tokens_in":17449,"feed_emoji":"⚛️","tokens_out":11426,"duration_ms":87476,"temperature":0.7,"pith_summary":"The paper studies open quantum systems governed by a Lindblad master equation and distinguishes two notions of integrability: the full Liouvillian dynamics may be integrable, or only the long-time steady state may be integrable while the dynamics is chaotic. Using boundary-driven spin chains, it shows that steady-state integrability is consistently signalled by Poisson level statistics of the steady-state 'Hamiltonian' $H_{\\mathrm{ss}} = -\\ln \\rho_{\\mathrm{ss}}$ and by failure of an extension of the eigenstate thermalization hypothesis, regardless of whether the Liouvillian itself is chaotic. Counter to natural expectation, an integrable steady state is not a simple operator: expanded in Pauli strings, it contains contributions from strings of all lengths, including long-range and many-body terms. However, the relative weight of strings of different sizes, the operator-size distribution, decays roughly exponentially for integrable steady states and much more weakly for chaotic ones, so the distribution's shape works as a practical discriminator. The result gives concrete diagnostic tools for classifying the nonequilibrium steady states of open many-body systems.","feed_headline":"Operator-size distribution tells integrable and chaotic steady states apart","feed_subtitle":"Boundary-driven spin chains show integrable steady states are full of many-body terms, yet P(S) still flags them.","key_machinery":"The central object is the 'steady-state Hamiltonian' $H_{\\mathrm{ss}}$, defined through $\\rho_{\\mathrm{ss}} = e^{-H_{\\mathrm{ss}}}$; being Hermitian, its eigenvalues have conventional (real) level statistics comparable to Poisson or GUE predictions, and its eigenbasis is used to test the NESS extension of the eigenstate thermalization hypothesis. The structural analysis rests on expanding $H_{\\mathrm{ss}}$ in the basis of Pauli strings, products of single-site Pauli matrices, where each string has a length equal to the number of sites it acts on nontrivially. From that expansion the paper defines the operator-size distribution $P(S)$, the normalized total weight of strings of each length $S$, which is the quantity that separates integrable from chaotic steady states. Complementary machinery includes complex spacing ratios for the Liouvillian spectrum, which detect repulsion among complex eigenvalues, and the conjecture that the natural dephasing scale is $\\gamma_{\\mathrm{deph}} \\sim 1/N$ rather than $O(1)$, motivated by a heuristic operator-growth and Liouvillian-gap argument.","core_discovery":"The paper shows that the steady state of an open many-body system can be integrable independently of whether the Liouvillian that generates the dynamics is integrable, and that the two kinds of integrability leave different spectral fingerprints. For the boundary-driven spin chains studied, full Liouvillian and steady-state integrability (models A and B) show Poisson level statistics both in the complex Liouvillian spectrum and in the real spectrum of the steady-state Hamiltonian $H_{\\mathrm{ss}} = -\\ln \\rho_{\\mathrm{ss}}$. When only the steady state is integrable (model A, whose Liouvillian is nonintegrable), the steady-state levels remain Poissonian while the Liouvillian levels repel as in the Ginibre unitary ensemble; when both are nonintegrable (models A$'$, B1--B3 at the dephasing scale $\\sim 1/N$), both spectra follow random-matrix statistics. Correspondingly, the NESS extension of the eigenstate thermalization hypothesis, tested through the finite-size scaling $\\sigma \\sim D^{-1/2}$ of expectation-value fluctuations, holds for nonintegrable steady states and fails for integrable ones. The paper further shows that the structure of the steady state defies the simple picture: the Pauli-string expansion of $H_{\\mathrm{ss}}$ contains strings of every length in both classes, and even the few-body coefficients are long-ranged. The quantity that nevertheless separates the two classes is the operator-size distribution $P(S)$, which decays approximately exponentially for integrable steady states and noticeably more slowly (flatter) for chaotic ones.","pith_inferences":["A natural next test is whether the exponential-versus-flat dichotomy in $P(S)$ generalizes to other integrable steady states, such as the boundary-driven Hubbard chain or models with more than one conserved charge; if it does, the shape of $P(S)$ could serve as a practical probe of steady-state integrability in numerical and experimental settings.","The conjectured $1/N$ dephasing scale has a direct thermodynamic-limit consequence the paper does not develop: if bulk dissipation couples to every site, boundary driving alone cannot sustain a nontrivial steady state at fixed coupling as $N \\to \\infty$, so nontrivial NESS in that limit requires scaling the bulk coupling down with system size.","Because $H_{\\mathrm{ss}}$ is not local even when integrable, a more promising characterization than operator locality may be the presence of quasilocal conserved quantities or a generalized-Gibbs-ensemble structure; the paper notes no such characterization is known, leaving that as an open avenue."],"forward_implications":["Poisson statistics of $H_{\\mathrm{ss}} = -\\ln \\rho_{\\mathrm{ss}}$ and violation of NESS ETH together identify an integrable steady state even when the Liouvillian is chaotic, so the two notions of integrability can be told apart spectroscopically.","Integrable steady states of the kind studied here receive weight from Pauli strings of all lengths, including long-range and many-body terms, so 'integrability' of a nonequilibrium steady state does not imply a local few-body structure.","Comparisons of steady-state chaos in bulk-dephased systems must be made at dephasing strengths $\\gamma_{\\mathrm{deph}} \\sim 1/N$; at $O(1)$ dephasing the steady state becomes the trivial infinite-temperature state and mimics integrable statistics.","The operator-size distribution $P(S)$ provides a structural discriminator between chaotic and integrable steady states that survives even though both contain strings of all sizes."],"supporting_citations":[{"why":"Supplies the complex spacing ratio method used to classify Liouvillian spectra as GinUE or Poisson.","marker":"[14]"},{"why":"Provides the NESS extension of the eigenstate thermalization hypothesis and the criterion that its scaling violation marks integrable steady states.","marker":"[29]"},{"why":"Establishes model B's Liouvillian and steady state as integrable via Bethe ansatz, the base of the B-family.","marker":"[43]"},{"why":"Provides the exact integrable steady state of model A, the case of an integrable steady state with nonintegrable Liouvillian.","marker":"[50]"},{"why":"Motivates the use of Poisson level statistics of the steady-state density matrix as an integrability indicator.","marker":"[68]"},{"why":"Supplies the $\\sigma \\propto D^{-1/2}$ eigenstate thermalization scaling used to test NESS ETH.","marker":"[69]"},{"why":"Supplies the operator-size formalism and size superoperator used to define $P(S)$.","marker":"[73]"},{"why":"Motivates the $\\gamma_{\\mathrm{deph}} \\sim 1/N$ scaling via the Liouvillian gap $\\sim \\gamma N$ heuristic.","marker":"[76]"},{"why":"Provides the Wigner-like surmises for real spacing ratios used for $H_{\\mathrm{ss}}$.","marker":"[79]"}],"fun_headline_variants":["Steady-state integrability defies Liouvillian chaos","Operator-size distribution flags integrable vs chaotic steady states","Integrable steady states aren't simple, but P(S) tells them apart","Two flavors of integrability in open quantum systems","P(S) distinguishes chaos from integrability in steady states"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The classification of the dephased models B1--B3 as having chaotic steady states rests on the conjecture that the natural dephasing strength scales as $1/N$; if that scale were instead of order one, the same numerical data would classify those steady states as integrable.","fun_headline_variants_meta":{"raw":{"variants":["Steady-state integrability defies Liouvillian chaos","Operator-size distribution flags integrable vs chaotic steady states","Integrable steady states aren't simple, but P(S) tells them apart","Two flavors of integrability in open quantum systems","P(S) distinguishes chaos from integrability in steady states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000717,"raw_usage":{"total_tokens":3266,"prompt_tokens":1036,"completion_tokens":2230,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":652,"completion_tokens_details":{"reasoning_tokens":2148}},"tokens_in":652,"tokens_out":2230,"duration_ms":14105,"temperature":1.0,"reasoning_tokens":2148,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:51:02.393941+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Liouvillian gap or the decay rate of the slowest operator for the bulk-dephased chains as a function of $N$ and $\\gamma_{\\mathrm{deph}}$: if the crossover to the trivial infinite-temperature steady state occurs at $\\gamma_{\\mathrm{deph}} \\sim 1/N$, the chaotic classification of B1--B3 is supported; if it occurs at $\\gamma_{\\mathrm{deph}} \\sim O(1)$, the Poisson statistics seen at $\\gamma_{\\mathrm{deph}}=1$ would persist at all system sizes and the central claim would fail.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the NESS extension of the eigenstate thermalization hypothesis and the criterion that its scaling violation marks integrable steady states."},{"cited_title":"Žnidarič, A matrix product solution for a nonequi- librium steady state of an XX chain, J","cited_arxiv_id":null,"evidence_quote":"Establishes model B's Liouvillian and steady state as integrable via Bethe ansatz, the base of the B-family."},{"cited_title":"de Leeuw, C","cited_arxiv_id":null,"evidence_quote":"Provides the exact integrable steady state of model A, the case of an integrable steady state with nonintegrable Liouvillian."},{"cited_title":"Popkov, T","cited_arxiv_id":null,"evidence_quote":"Motivates the use of Poisson level statistics of the steady-state density matrix as an integrability indicator."},{"cited_title":"Hidden quasi-local charges and Gibbs ensemble in a Lindblad system","cited_arxiv_id":"2305.01922","evidence_quote":"Supplies the $\\sigma \\propto D^{-1/2}$ eigenstate thermalization scaling used to test NESS ETH."},{"cited_title":"Schuster, B","cited_arxiv_id":null,"evidence_quote":"Motivates the $\\gamma_{\\mathrm{deph}} \\sim 1/N$ scaling via the Liouvillian gap $\\sim \\gamma N$ heuristic."},{"cited_title":"Buča and T","cited_arxiv_id":null,"evidence_quote":"Provides the Wigner-like surmises for real spacing ratios used for $H_{\\mathrm{ss}}$."}],"review_version":1}