{"id":"ea3a9f15-9a5a-4f72-b044-d2b2c39e7bc1","arxiv_id":"2412.01076","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For critical-to-critical quenches in the long-range Kitaev chain, the stationary state shows the same effective central charge as the ground state, so universality can be read from the late-time state.","lead":"This paper studies quantum quenches in a long-range Kitaev chain and asks whether the final steady state remembers the universality class of the ground state. It finds that for a critical-to-critical quench, the effective central charge extracted from mutual information and log-negativity matches the ground-state value for α=0 and α=2.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central-charge comparison assumes stationary-state mutual information and log-negativity upper bound scale with ground-state CFT prefactors; neither relation is derived or documented with fits, so the agreement with Table I may be coincidental.","rationale":"The reader identified the same load-bearing assumption: the stationary-state scaling of mutual information and log-negativity is assumed to follow the ground-state CFT prefactors. My stress-test sharpens this in two ways. First, the stationary state is a GGE, not a CFT ground state; the log coefficient of I in a free-fermion GGE is fixed by the structure of n_k, and there is no a priori reason it equals c/3. Second, the log-negativity result is computed from an upper bound, and the slope of an upper bound need not be the slope of the true negativity. The paper provides no fits, error bars, or system-size scans, so the agreement with Table I could be coincidental. This is not an attack on the physics or on the authors; the exact solvability and the soft-mode argument give real support to the presence of a peak in the stationary-state entanglement measures. But the quantitative universality claim is exactly the step where the evidence is missing. A conditional verdict remains appropriate because the missing analysis is straightforward to supply: recompute the scalings and report the fits, or derive the GGE coefficient analytically. Therefore I would keep the reader's CONDITIONAL verdict unchanged.","tokens_in":16260,"tokens_out":11644,"duration_ms":114606,"concrete_test":"Recompute the scaling behind Figs. 3-4 for the α=2, critical-to-critical case (µi=µf=1, ∆i=-1→∆f=1): for N=200,400,800,1600 with adjacent blocks of length L=N/4, evaluate IA1:A2 and ξ^u from Eqs. (14)-(22) and fit I = a + b ln L and ξ^u = a' + b' ln L with jackknife errors. Separately, derive the exact asymptotic coefficient of I from the Toeplitz symbol n_k = (1/2)(1 - cos 2δθ_k) in Eq. (23) using Fisher-Hartwig/Widom asymptotics. If the extrapolated b differs from (1/2)/3 or b' differs from (1/2)/4, the claimed equality with the ground-state central charge is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that cI_eff and cN_eff extracted from the stationary state equal the ground-state central charges. The extraction depends on two asserted analogies: Eq. (10) applies the CFT ground-state form I ~ (c/3) ln L to the post-quench stationary state, and Eq. (13) applies the CFT form ξ ~ (c/4) ln L, although the quantity actually computed is only the upper bound ξ^u in Eqs. (18)-(20), whose asymptotic coefficient can differ from that of the true log-negativity. For a free-fermion generalized Gibbs ensemble, the log coefficient of the mutual information is controlled by the Fisher-Hartwig singularities of the mode-occupation symbol n_k = (1/2)(1 - cos 2δθ_k) in Eq. (23), not automatically by the ground-state central charge. The paper shows no scaling curves, no fit details, no system sizes, and no error bars: Figures 3 and 4 are phase maps of the fitted coefficients, and Appendix A only restates the claimed values rather than providing the promised finite-size analysis. Thus the exact numerical agreement with Table I is not independently supported, and the statement that universality can be inferred from the stationary state rests on this unvalidated identification.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies quench dynamics in the one-dimensional long-range Kitaev chain, focusing on the stationary state after sudden quenches of the chemical potential and pairing strength. Using free-fermion correlation-matrix techniques, the authors compute mutual information, tripartite mutual information, and an upper bound on logarithmic negativity in the long-time limit. They report that a critical-to-critical quench produces stationary-state signatures of criticality for all pairing exponents α considered, and they extract effective central charges cI_eff and cN_eff from the logarithmic scaling of mutual information and log-negativity. The central quantitative claim is that, for α=0 and α=2, these effective central charges agree with the ground-state central charges compiled in Table I, so that the universality class of the ground state can be inferred from the stationary state. The paper also explains the nonanalytic features at μ_f=1 through the soft-mode occupation probability n_k in Eq. (23).","tokens_in":16473,"tokens_out":3681,"duration_ms":36962,"significance":"If the claimed agreement is correct, the result is significant: it would show that for an integrable long-range fermionic model, the stationary state after a critical-to-critical quench retains enough information about the underlying conformal critical point to reproduce the ground-state central charge, even though fermionic correlators decay algebraically at noncritical points for α≤1. The manuscript has real strengths: the model is exactly solvable, the stationary-state correlation functions in Eq. (17) are analytic and clearly presented, the soft-mode occupation mechanism in Eqs. (23)-(25) gives a concrete and testable explanation of the peak/dip structure, and the comparison against the XY-model precedent in Ref. [40] is a natural methodological link. However, the central quantitative claim is currently supported only by fitted-coefficient phase maps and a table of final values; no scaling curves, system sizes, fit details, or uncertainties are shown, and the log-negativity analysis is based on an upper bound whose asymptotic prefactor is not established.","major_comments":[{"comment":"The central claim that cI_eff equals the ground-state central charge for α=0 and α=2 is not supported by the displayed evidence. Figures 3 and 4 present only color maps of the fitted coefficient over the (μ,Δ) plane, with no scaling curves of I_A1:A2 or ξ^u versus ln L, no statement of total system sizes N, no subsystem length range, no fit function, and no error bars. Without such information, the reader cannot check whether the quoted values are robust or whether the agreement with Table I is genuine.","section":"§IV B, Fig. 3"},{"comment":"The log-negativity claim is extracted from the upper bound ξ^u defined in Eq. (18), not from the true log-negativity of Eq. (11). The comparison with the ground-state central charge assumes that ξ^u scales with the same asymptotic coefficient c/4 as the true negativity in Eq. (13). No argument or numerical check is given that the bound is tight for the stationary state of the long-range Kitaev model; if the upper bound has a different coefficient, the reported cN_eff values would not measure the central charge. Appendix A also does not provide the promised finite-size analysis but merely restates the final values.","section":"§III, Eqs. (18)-(20), Appendix A"},{"comment":"The effective central charge cI_eff is defined through the assumed CFT scaling form I_A1:A2 ∼ (cI_eff/3) ln L for the stationary state, and no derivation or numerical verification of this form is provided for the post-quench generalized Gibbs ensemble of this model. Since the prefactor is fixed by assumption, the subsequent agreement with the ground-state central charges in Table I could be a consequence of the assumed scaling form rather than an independent test of universality. The authors should either derive this prefactor from the Fisher-Hartwig structure of the stationary-state mode occupation or demonstrate it with explicit finite-size scaling data.","section":"§III, Eq. (10)"}],"minor_comments":[{"comment":"The quantity plotted in the middle row is labeled ξ_A1:A2, but the method computes only the upper bound ξ^u via Eqs. (18)-(20); the caption should distinguish the bound from the true log-negativity.","section":"Fig. 1 caption"},{"comment":"The operators O_+ and O_- are used in Eq. (18) but are not explicitly defined in the text; a brief definition or a pointer to the decomposition used in Refs. [84-89] would improve clarity.","section":"Eq. (18)"},{"comment":"The main text says the details of the log-negativity finite-size scaling are in Appendix A, but the appendix reports only the extracted cN_eff values; the promised scaling analysis should either be included or the reference to it should be removed.","section":"Appendix A"},{"comment":"In the sentence discussing α=1, the text contains the garbled expression 'ΔΔ 0'; this should be corrected to a clear statement about the sign or magnitude of Δ_f Δ_i.","section":"§IV B"},{"comment":"The notation λ^f_α(k_n) in Eq. (16) and the surrounding text is not fully consistent with the earlier notation λ_α(k_n); unify the superscripts and subscripts used for the post-quench dispersion.","section":"Eq. (16)"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a timely and interesting question, and the analytic setup is sound. My main concern is that the central quantitative comparison with ground-state central charges is not independently verifiable from the manuscript as written. If the authors can provide scaling curves, fit details, system sizes, uncertainties, and an explicit check of the tightness of the log-negativity upper bound, the claim would become much more convincing. I would not reject the paper, but the missing evidence is load-bearing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has a real qualitative result and a shaky quantitative one. The qualitative result: in the long-range Kitaev chain, critical-to-critical quenches leave a stationary state whose mutual information and log-negativity peak at the critical point, even for α ≤ 1 where ground-state correlations are already algebraic. That extends the hidden quantum criticality idea from the short-range XY model to a long-range integrable model, and the soft-mode occupation argument gives a plausible mechanism. I believe that part holds up.\n\nThe quantitative claim in the abstract—that effective central charges extracted from the stationary state match ground-state central charges—is not supported by what is shown. Figures 3 and 4 are phase plots of fitted coefficients, but there are no scaling curves, no system sizes, no fit ranges, and no error bars. The paper refers to Appendix A for the log-negativity analysis, but that appendix just restates the values. More importantly, the log-negativity calculation uses the upper bound from Eq. (18), and the paper assumes the bound scales with the same c/4 ln L coefficient as the true log-negativity without checking that. The same assumption underlies the mutual information scaling in Eq. (10). For a free-fermion generalized Gibbs ensemble, the log-coefficient of these quantities is controlled by the singularities of the mode-occupation function, not automatically by the ground-state central charge. So the agreement with Table I could be coincidental.\n\nI don't think the central idea is wrong. The qualitative signatures are robust, the model is exactly solvable, and the stationary-state correlators are standard, so the paper is not circular and it engages honestly with the previous work. The missing piece is documentation: a few scaling plots with fits and errors would probably settle it. The authors may well be right, but the evidence as presented is incomplete.\n\nThis paper deserves a serious referee. The question—whether the ground-state universality class can be read off a stationary state—is worth answering, and the long-range twist is new. I would send it to peer review with the expectation that the reports ask for the actual scaling analyses, not just the phase maps. I would not cite the central-charge claim yet, but I would cite the qualitative criticality result once it is in the literature.","headline":"Plausible qualitative extension of hidden quantum criticality to the long-range Kitaev chain, but the central charge match is not yet documented.","tokens_in":17010,"tokens_out":2424,"would_cite":false,"duration_ms":22294,"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":"The paper claims that after a critical-to-critical quench in the long-range Kitaev chain, the stationary state's mutual information and logarithmic negativity scale with the same effective central charge as the ground state, identifying…","keywords":["quantum criticality","quench dynamics","long-range Kitaev model","stationary state","central charge","mutual information","logarithmic negativity","universality class"],"falsifier":"Take the stationary state after a critical-to-critical quench for a moderate system size, construct the fermionic reduced density matrix, perform the partial transpose numerically, and compute the exact logarithmic negativity; if the exact $c^N_{\\rm eff}$ deviates from the upper-bound estimate or from the ground-state central charge, the quantitative match claimed here would not survive. A second check is to vary the pairing quench amplitude $\\Delta_f$ at fixed $\\Delta_i=-1$ and verify that the fitted $c^I_{\\rm eff}$ remains constant; any drift with $\\Delta_f$ or with subsystem size $L$ would show that the logarithmic scaling is not the CFT form.","tokens_in":2072,"feed_emoji":"⚛️","tokens_out":2497,"duration_ms":75579,"temperature":0.7,"pith_summary":"This paper investigates whether quantum criticality and its universality class survive in the long-time stationary state of an integrable long-range Kitaev chain after a sudden quench. The model is a one-dimensional fermionic chain with pairing interactions decaying with distance as $1/l^\\alpha$, and the authors study quenches of the chemical potential and pairing strength. Their central claim is that when both the pre- and post-quench Hamiltonians are at the critical point, the stationary state shows the same effective central charge as the critical ground state: $c=1$ for the all-to-all pairing limit $\\alpha=0$ and $c=1/2$ for $\\alpha=2$. This matters because it would mean the universality class of a quantum critical point can be read off from a nonequilibrium steady state, even in systems where fermionic correlators decay algebraically away from criticality.","feed_headline":"Quench between critical points preserves central charge","feed_subtitle":"Mutual information and log-negativity of the stationary state match ground-state values for alpha=0 and alpha=2.","key_machinery":"The exactly solvable free-fermion structure of the long-range Kitaev chain is the central object: after a Fourier transform and Bogoliubov rotation, the Hamiltonian becomes a sum of independent modes with dispersion $\\lambda_\\alpha(k)=\\sqrt{(\\mu+\\cos k)^2+(\\Delta g_\\alpha(k))^2}$, where $g_\\alpha(k)$ is expressed through polylogarithms. Time-dependent correlations after a quench are computed from the difference of pre- and post-quench Bogoliubov angles, and the long-time stationary state is obtained by dropping the oscillatory terms. Entanglement is extracted from the correlation matrix; mutual information cancels the volume-law part of the entropy and isolates the logarithmic scaling $I \\sim (c^I_{\\rm eff}/3)\\ln L$, while logarithmic negativity is estimated through the upper bound of Eq. (18) for fermionic Gaussian states, with scaling $\\xi \\sim (c^N_{\\rm eff}/4)\\ln L$. The soft-mode occupation probability $n_k$ at $k=\\pi$, given in Eq. (23), is the mechanism that produces the characteristic peak for critical-to-critical quenches and dip for noncritical-to-critical quenches.","core_discovery":"The central discovery claimed is that, in the long-range Kitaev chain, a critical-to-critical quench produces a stationary state whose entanglement scaling identifies the same universality class as the ground state. Extracting effective central charges from the logarithmic growth of mutual information, $I \\sim (c^I_{\\rm eff}/3)\\ln L$, and from the upper bound on logarithmic negativity, $\\xi \\sim (c^N_{\\rm eff}/4)\\ln L$, the authors find $c^I_{\\rm eff}=c^N_{\\rm eff}=1$ at $\\alpha=0$ and $1/2$ at $\\alpha=2$, matching the ground-state values. The same measures show a nonanalytic peak at the post-quench critical point $\\mu=1$ for all studied $\\alpha$, and this is explained through the occupation probability of the soft mode at $k=\\pi$, which controls the entanglement of the union of the two subsystems. For noncritical-to-critical and critical-to-noncritical quenches, the effective central charge takes the noncritical ground-state value, so the authors do not attribute those nonanalyticities to criticality. For $\\alpha=1$ the stationary state, like the ground state, has no universal central charge.","pith_inferences":["A testable extension would be to measure mutual information after a critical-to-critical quench in a trapped-ion simulator of the long-range Ising chain, whose Jordan-Wigner form is this model; observing the predicted logarithmic slope would provide an experimental route to the central charge from a nonequilibrium steady state.","Because the paper uses only an upper bound for logarithmic negativity, an exact partial-transpose calculation for small system sizes would show whether $c^N_{\\rm eff}$ genuinely equals $c^I_{\\rm eff}$ or whether the match is an artifact of the bound.","The paper's distinction between nonanalyticity and universality suggests a general methodological lesson: in long-range interacting systems, where correlation decay is algebraic even off criticality, scaling slopes rather than peaks or dips should be used to identify criticality in quench experiments."],"forward_implications":["For $\\alpha=0$ and $\\alpha=2$, the effective central charge extracted from the stationary state after a critical-to-critical quench equals the ground-state central charge, independent of the quench amplitude in the pairing strength $\\Delta$.","For noncritical-to-critical and critical-to-noncritical quenches, the effective central charge takes the noncritical ground-state value ($1/2$ for $\\alpha=0$, $0$ for $\\alpha=2$), so the nonanalytic behavior of entanglement measures at $\\mu=1$ is not by itself a signature of universality.","Logarithmic negativity confirms that the long-range correlations detected in the stationary state are genuinely quantum, since $c^N_{\\rm eff}$ matches $c^I_{\\rm eff}$ for both $\\alpha=0$ and $\\alpha=2$.","Tripartite mutual information also peaks at the critical point for a critical-to-critical quench, and its sign distinguishes delocalized information ($\\mu<1$) from information redundancy ($\\mu>1$) for $\\alpha=1$.","For $\\alpha=1$, the stationary state, like the ground state, cannot be assigned a universal central charge, but long-range correlations still develop for the critical-to-critical quench protocol."],"supporting_citations":[{"why":"Supplies the ground-state phase diagram and effective central-charge values for the long-range Kitaev model against which the stationary-state values are compared.","marker":"[44]"},{"why":"Establishes the critical-to-critical quench protocol and the soft-mode occupation-probability explanation that this paper extends to long-range pairing.","marker":"[40]"},{"why":"Shows that mutual information in a nonequilibrium steady state grows logarithmically with subsystem size, motivating the scaling form used here.","marker":"[39]"},{"why":"Provides the quench-evolved correlation functions used to construct the stationary-state correlation matrix.","marker":"[83]"},{"why":"Gives the partial-transpose difficulty for fermionic Gaussian states and the Gaussian-operator method underlying the log-negativity upper bound in Eq. (18).","marker":"[84]"},{"why":"Shows that mutual information acts as a probe of criticality in fermionic systems, supporting its use for the stationary state.","marker":"[71]"},{"why":"Supplies the conformal-field-theory expression $\\xi \\sim (c/4)\\ln L$ for logarithmic negativity used to define $c^N_{\\rm eff}$.","marker":"[81]"}],"fun_headline_variants":["Critical quench leaves universal signature in stationary state","Stationary state preserves central charge after critical quench","Long-range Kitaev quench: criticality survives in steady state","Quench to critical point keeps universality in Kitaev chain"],"cache_read_input_tokens":19200,"weakest_assumption_plain":"The argument assumes that the stationary state's mutual information and the upper bound on logarithmic negativity follow the same logarithmic scaling laws with the same prefactors as the critical ground state, so that the fitted slope can be identified with the ground-state central charge; for log-negativity, only a bound is computed, so the match also depends on that bound being tight.","fun_headline_variants_meta":{"raw":{"variants":["Critical quench leaves universal signature in stationary state","Stationary state preserves central charge after critical quench","Long-range Kitaev quench: criticality survives in steady state","Quench to critical point keeps universality in Kitaev chain"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000153,"raw_usage":{"total_tokens":1214,"prompt_tokens":962,"completion_tokens":252,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":578,"completion_tokens_details":{"reasoning_tokens":185}},"tokens_in":578,"tokens_out":252,"duration_ms":2743,"temperature":1.0,"reasoning_tokens":185,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:42:39.077226+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the stationary state after a critical-to-critical quench for a moderate system size, construct the fermionic reduced density matrix, perform the partial transpose numerically, and compute the exact logarithmic negativity; if the exact $c^N_{\\rm eff}$ deviates from the upper-bound estimate or from the ground-state central charge, the quantitative match claimed here would not survive. A second check is to vary the pairing quench amplitude $\\Delta_f$ at fixed $\\Delta_i=-1$ and verify that the fitted $c^I_{\\rm eff}$ remains constant; any drift with $\\Delta_f$ or with subsystem size $L$ would show that the logarithmic scaling is not the CFT form.","supporting_citations":[{"cited_title":"Kitaev chains with long- range pairing,","cited_arxiv_id":null,"evidence_quote":"Supplies the ground-state phase diagram and effective central-charge values for the long-range Kitaev model against which the stationary-state values are compared."},{"cited_title":"Hidden quantum criticality and entanglement in quench dynamics,","cited_arxiv_id":null,"evidence_quote":"Establishes the critical-to-critical quench protocol and the soft-mode occupation-probability explanation that this paper extends to long-range pairing."},{"cited_title":"Area-law violation for the mutual information in a nonequilibrium steady state,","cited_arxiv_id":null,"evidence_quote":"Shows that mutual information in a nonequilibrium steady state grows logarithmically with subsystem size, motivating the scaling form used here."},{"cited_title":"Quench dynamics across quantum critical points,","cited_arxiv_id":null,"evidence_quote":"Provides the quench-evolved correlation functions used to construct the stationary-state correlation matrix."},{"cited_title":"On the partial trans- pose of fermionic gaussian states,","cited_arxiv_id":null,"evidence_quote":"Gives the partial-transpose difficulty for fermionic Gaussian states and the Gaussian-operator method underlying the log-negativity upper bound in Eq. (18)."},{"cited_title":"Mutual information for fermionic systems,","cited_arxiv_id":null,"evidence_quote":"Shows that mutual information acts as a probe of criticality in fermionic systems, supporting its use for the stationary state."},{"cited_title":"Quantum quench in the transverse field ising chain: I. time evolution of order parameter correlators,","cited_arxiv_id":null,"evidence_quote":"Supplies the conformal-field-theory expression $\\xi \\sim (c/4)\\ln L$ for logarithmic negativity used to define $c^N_{\\rm eff}$."}],"review_version":1}