{"id":"db7cceac-d5e4-41d0-80aa-15285762cbd5","arxiv_id":"2502.07775","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a non-Hermitian Ising chain, the Krylov spread detects three dynamical phases in the gapped-spectrum region and is analytically related to the spin-spin correlation function.","lead":"This paper analyzes a non-Hermitian Ising chain with a complex transverse magnetic field and shows that its Krylov spread, a measure of how a state spreads in Hilbert space, reveals three distinct dynamical phases in a region where the imaginary part of the spectrum is gapped. It also derives the full spatial behavior of spin correlations and connects the Krylov spread to the same-site correlation, providing an analytical picture across the phase diagram.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central three-phase fidelity diagram rests on an unproven factorization of the Krylov basis into independent per-mode two-level spreads, and the non-Hermitian Hamiltonian contains J_z terms that the Eq. (90) argument from Ref. [63] does not cover.","rationale":"The reader's weakest assumption is exactly the load-bearing point: Eq. (102) is asserted, not derived, for the non-Hermitian Hamiltonian with J_z terms, and every quantitative claim about the three dynamical phases flows from it. I independently checked the logical chain: C(t) in Eq. (102) uses Eq. (90); the fidelity F(t) in Eq. (103) uses C(t) and C_Omega; the three timescales in Eqs. (107)-(109) are obtained by asymptotic analysis of that fidelity; and Fig. 4 is the phase diagram of those timescales. If the Krylov basis does not factorize as in Eq. (90), the per-mode formula is not the Krylov spread of the full non-Hermitian evolution, and the refined phase diagram has no proven basis. The paper contains no numerical verification of the central dynamical claim, and the boundary curves for Fig. 4 are explicitly omitted, which makes the missing factorization proof more consequential. The proposed finite-N exact Krylov calculation is a direct, low-cost check: the model is free-fermionic, so N up to 10-14 is trivial, and the comparison distinguishes the factorized-ansatz prediction from the true Lanczos spread. I do not see a reason to reject the paper: the analytical structure is coherent, the correlation-function calculations are detailed, and the connection between C_Omega and the same-site correlation is plausible. The appropriate disposition is therefore conditional acceptance pending this check, which matches the reader's verdict and does not require changing it.","tokens_in":45252,"tokens_out":16597,"duration_ms":168759,"concrete_test":"For finite N (e.g., N = 4, 6, 8, 10) in the gapped phase, construct the exact Krylov subspace from {H^n |0>} by Gram-Schmidt orthonormalization with the standard inner product, for parameters in each claimed dynamical region, e.g., (h/J, gamma/J) = (1.5, 4) and (0.5, 4). Compute the exact spread C_exact(t) = sum_n n |<K_n|psi(t)>|^2 for the normalized non-Hermitian evolution (7), and compare it with the per-mode integral in Eq. (102). Also extract the fidelity F_exact(t) = |C_exact(t) - C_Omega| and the t*(epsilon) scaling, and check whether the dominant decay rate switches between gamma, 4|Gamma(kbar)|, and |gamma_Y| as predicted by Eqs. (107)-(109). If C_exact/N converges to Eq. (102) and the three rates reproduce the claimed boundaries, Eq. (90) is validated; otherwise the factorization underlying the central phase diagram is unsupported.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The dynamical phase diagram in Fig. 4 and the fidelity rates in Eqs. (107)-(109) all depend on Eq. (102), which identifies the thermodynamic-limit Krylov spread with an integral over independent per-mode two-level spreads. This identification is imported from the Hermitian argument in Ref. [63] via Eq. (90): it assumes that the n-th Krylov vector is N_n (sum_k alpha_+(k) J_+(k))^n |0>, so that the Krylov basis factorizes over momentum sectors. For the non-Hermitian Hamiltonian in Eq. (28), however, each sector also contains the term 2R_z(k) J_z(k). Since J_z(k) = -1/2 + J_+(k)J_-(k) in the spin-1/2 sector, powers H^n |0> acquire contributions in which J_-(k) acts on already-created pairs; these contributions are not of the pure pair-creation form assumed in Eq. (90). Section V.A states only that 'one can apply the same reasoning' as in the Hermitian case, without deriving the Krylov basis for the full H or showing that the J_z terms are harmless in the thermodynamic limit. If Eq. (102) fails, then the time-dependent spread C(t), the three characteristic times t* in Eqs. (107)-(109), and the refined gapped-region phase diagram in Fig. 4 do not describe the actual Krylov spread of the non-Hermitian evolution. The same assumption also underpins the identification of the infinite-time spread with C_Omega in Eq. (92), so the central claim is directly affected.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Krylov spread of the all-down state evolving under a non-Hermitian transverse-field Ising chain with complex field h+iγ/4. The authors diagonalize the Hamiltonian via a non-Hermitian Bogoliubov transformation, derive the time-evolved state, and analyze the static Czz correlation function in the non-Hermitian vacuum across the gapped and gapless regions of the imaginary spectrum. They then relate the infinite-time Krylov spread density to the same-site correlation function and define a Krylov spread fidelity F(t)=|C(t)-C_Ω| whose approach to zero, in the gapped region, yields three characteristic times and a refined three-phase diagram in the (h,γ) plane. The paper's main claims are: (i) Czz(x) decays as x^{-3}e^{-x/ξ} with oscillations in the gapped phase and algebraically as x^{-2} or x^{-4} with oscillations in the gapless phase; (ii) the spread density C_Ω has a third-order phase transition across γ=γc(h) and is related to Czz(0); and (iii) the time-dependent Krylov spread reveals three dynamical phases hidden in the gapped region.","tokens_in":45545,"tokens_out":7066,"duration_ms":71771,"significance":"If the central assumptions hold, the results are significant: they extend Krylov-spread techniques to a non-Hermitian many-body system, propose the spread fidelity as a probe of dynamical phases that are invisible to standard correlation functions, and provide a full asymptotic characterization of correlations in the non-Hermitian vacuum. The paper is analytic and largely self-contained, with detailed appendices for the stationary state, Darboux asymptotics, and the fidelity computation. The relation C_Ω=(1+√(1-Czz(0)))/2 is elegant, and the explicit elliptic-integral formula for the spread density in the Hermitian limit matches and extends Ref. [63]. However, the central time-dependent spread formula is imported from Ref. [63] without proof for the non-Hermitian Hamiltonian, and the paper itself acknowledges an unresolved prefactor mismatch on the gapped side of the critical curve; these issues affect the two headline claims.","major_comments":[{"comment":"The thermodynamic-limit formula C(t)=π^{-1}∫dk |A_+(k;t)|²/(1+|A_+(k;t)|²) rests on the factorization in Eq. (90), imported from Ref. [63] for a Hermitian su(2)^N Hamiltonian. The present H in Eq. (28) contains the diagonal term 2R_z(k)J_z(k). Because J_z(k) acts non-trivially on multi-pair sectors and because H(k) also contains J_-(k), the vectors H^n|0⟩ are not obviously in the span of (∑_k α_+(k)J_+(k))^n|0⟩; no argument in Section V.A shows that the lower-pair contributions are negligible in the thermodynamic limit. Without a proof or a numerical check of Eq. (102) against the exact Krylov spread for finite N, the three dynamical phases in Fig. 4 and the rates in Eqs. (107)-(109) are not established.","section":"V.A, Eq. (102)"},{"comment":"The boundaries separating the three fidelity phases are not provided, with the stated reason that they solve multivariate polynomials of degree greater than 10. Because Fig. 4 is advertised as a key result, the reader cannot verify the phase diagram or reproduce the claimed regions from Eqs. (107)-(109) alone; please provide the implicit boundary equations, a numerical construction, or the explicit algorithm used to generate Fig. 4.","section":"Appendix D, after Eq. (D25)"},{"comment":"The paper acknowledges that as γ→γc(h)+ the Darboux method yields the correct algebraic powers but not the correct prefactors, which do not match the gapless-side result Eq. (79). This is a load-bearing gap for the claimed full analytical characterization of correlations across the phase diagram in the abstract. The manuscript should either provide a uniform asymptotic analysis connecting Eqs. (70) and (79) or state clearly the range of validity of the prefactor in Eq. (70).","section":"III.A, Eq. (70)"}],"minor_comments":[{"comment":"The sentence referring to the oscillatory phase labels it (h > J, γ→0), but Eqs. (81) and Table I show the oscillatory x^{-2} behavior for h < J, γ→0; the inequality should be h < J.","section":"III.C, text after Eq. (81)"},{"comment":"The function B(k) introduced before Eq. (D16) is described only as well-behaved and approaching zero faster than |X(k)| approaches infinity; a precise definition or bound would make the derivation of Eq. (D19) verifiable.","section":"Appendix D, Eq. (D16)"},{"comment":"The caption says the white and red regions do not have an associated characteristic time, while the text names the three phases gray, green, and blue; unifying the color labels between the text and the figure would improve readability.","section":"Fig. 4 caption"}],"recommendation":"major_revision","confidential_remarks":"The decisive issue is the unproven factorization in Eq. (90). If the authors can supply a derivation or small-N numerics validating Eq. (102), the paper could become acceptable; as it stands, the central three-phase claim depends on an imported assumption. The correlation-function section is largely self-contained and sound, but the acknowledged prefactor mismatch near γ→γc(h)+ needs to be addressed or explicitly scoped. The paper is within the journal's scope and is worth further consideration after these points are resolved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know this paper is a genuine analytical advance in two directions: it completes the asymptotic spin-spin correlation function for the non-Hermitian Ising chain in the gapped phase and on the critical line, and it connects the Krylov spread density to the same-site correlation (Eq. 93). The appendices are detailed and the derivations mostly self-contained. The authors are honest about the Darboux prefactor mismatch at γ→γc and about not giving the boundary equations for the three Krylov phases.\n\nThe main new physics claim is the three-phase refinement of the gapped region, based on three different decay rates in the Krylov spread fidelity. The structure of the argument is clear: compare γ, 4|Γ(¯k)|, and |γY|. The problem is that the whole time-dependent spread formula (Eq. 102) relies on the per-mode factorization of the Krylov basis, Eq. (90). That factorization is imported from Ref. [63] for the Hermitian Kitaev chain. The non-Hermitian H has J_z terms, and the authors only say \"one can apply the same reasoning.\" Given that J_z = -1/2 + J_+J_-, powers of H acting on |0⟩ do not remain in the pure pair-creation form assumed by Eq. (90) at the algebraic level. The factorization may well survive in the thermodynamic limit, and the same issue exists for the Hermitian Kitaev chain, but the paper does not supply the argument. That is a load-bearing step, and a referee should ask for a derivation or for numerical verification of Fig. 4.\n\nA second soft spot: the boundary curves between the three Krylov phases are omitted because they are high-degree polynomial solutions. That makes the phase diagram hard to use; at minimum the authors should give the equations numerically or provide a numerical check that the three phases exist.\n\nThe correlation part of the paper is in much better shape. The asymptotics in Eqs. (70), (78)-(79) are plausible, agree with the known limits, and the paper clearly states where it extends Refs. [10] and [23]. The prefactor discrepancy at the critical line is acknowledged and does not threaten the gapless-phase results.\n\nWho should read this: people working on non-Hermitian many-body physics and Krylov complexity. The spread-correlation relation and the correlation table are worth having. The three-phase diagram is interesting but needs support.\n\nRecommendation: send to peer review. The paper deserves a serious referee, but the referee should insist on a proof (or numerical check) of Eq. (90) for the non-Hermitian Hamiltonian and on boundary curves for Fig. 4.","headline":"Solid analytical work on non-Hermitian Ising correlations and Krylov spread, but the three-phase diagram rests on an unproven Krylov factorization that needs referee scrutiny.","tokens_in":46113,"tokens_out":7651,"would_cite":true,"duration_ms":70431,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.65.-w","03.65.Yz","05.30.-d","75.10.Pq"],"model":"deepseek-v4-flash","headline":"The Krylov spread of an all-down state evolving under a non-Hermitian Ising chain exposes three previously unnoticed dynamical phases in the gapped-imaginary-spectrum region, and ties the spread density to same-site spin correlations.","keywords":["Krylov complexity","non-Hermitian Hamiltonian","Ising chain","exceptional points","spin correlation functions","dynamical phase transitions","Bogoliubov vacuum","quantum many-body dynamics"],"falsifier":"Numerically construct the exact Krylov basis for a finite chain at points in each predicted region (for example $h=0.5J$ with $\\gamma=2J$, $\\gamma=5J$, and $h=2J,\\gamma=5J$, with $N$ up to about 20), compute the spread density $C(t)$ directly, and check whether the late-time fidelity decays with the predicted rates $\\gamma$, $4|\\Gamma(\\bar k)|$, or $|\\gamma_Y|$; a mismatch in any region would falsify the factorization underlying Eq. (102).","tokens_in":45027,"feed_emoji":"🧲","tokens_out":10608,"duration_ms":78801,"temperature":0.7,"pith_summary":"This paper studies the all-down state $|\\downarrow\\cdots\\downarrow\\rangle$ of an Ising chain with a complex transverse magnetic field, evolving under the non-Hermitian Hamiltonian $H=-\\sum_i[J\\sigma^x_i\\sigma^x_{i+1}+(h+i\\gamma/4)\\sigma^z_i]$. It claims that Krylov spread, the mean position of the evolved state in the Krylov basis built from the initial state and the generator of the dynamics, reveals dynamical structure that static correlation functions miss. When the imaginary part of the spectrum is gapped, the state tends to the non-Hermitian Bogoliubov vacuum, and the way the spread density approaches its infinite-time value splits the gapped region into three dynamical phases with distinct characteristic times $t^\\ast$. The paper also derives an exact link between the spread density and the same-site spin correlation, $C_\\Omega=(1+\\sqrt{1-C_{zz}(0)})/2$, and completes the analytic characterization of $C_{zz}(x)$ across the phase diagram. If true, this makes Krylov spread a cheap and general probe for dynamical phase structure in open and monitored many-body systems.","feed_headline":"Krylov spread finds 3 hidden phases in a complex-field Ising chain","feed_subtitle":"Late-time Krylov fidelity splits the gapped region of the h-γ plane into three regions with distinct relaxation rates.","key_machinery":"The argument is carried by the $\\mathfrak{su}(2)^{\\otimes N}$ Lie-algebra structure of the Hamiltonian in the Jordan-Wigner representation: for each momentum mode the Hamiltonian is $2R_z(k)J_z(k)+R_x(k)(J_+(k)+J_-(k))$, so every mode is an independent two-level system. A normal-ordering decomposition of $\\mathfrak{su}(2)$ yields the evolved state and the per-mode spread coefficient $A_+(k;t)$. The load-bearing identity is the thermodynamic-limit factorization of Krylov vectors, $|K_n\\rangle=N_n\\big(\\sum_k\\alpha_+(k)J_+(k)\\big)^n|0\\rangle$, which converts the spread into an integral over independent per-mode spreads and leads to $C_\\Omega=(1+\\sqrt{1-C_{zz}(0)})/2$. From there the paper decomposes the fidelity $F(t)$ into endpoint and bulk contributions, extracts the three characteristic times, and obtains the asymptotic correlation formulas by Darboux and regulator methods.","core_discovery":"On the paper's own terms, the central discovery is that Krylov spread resolves a finer phase diagram than traditional correlation functions. For the Jordan-Wigner vacuum $|0\\rangle=|\\downarrow\\cdots\\downarrow\\rangle$ evolved by the non-Hermitian Hamiltonian, the long-time state is the non-Hermitian Bogoliubov vacuum $|\\Omega\\rangle$ when $\\mathrm{Im}\\,\\Lambda(k)$ is gapped, and a superposition $A(t)|\\Omega\\rangle+A^*(t)|q,-q\\rangle$ when the spectrum is gapless at $k=\\pm q$. The thermodynamic-limit spread density is $C(t)=\\pi^{-1}\\int_0^\\pi dk\\,|A_+(k;t)|^2/(1+|A_+(k;t)|^2)$, and its infinite-time value $C_\\Omega$ is connected to the same-site correlation by $C_\\Omega=(1+\\sqrt{1-C_{zz}(0)})/2$. In the gapped phase the fidelity $F(t)=|C(t)-C_\\Omega|$ is governed at long times by one of three rates: $\\gamma$, $4|\\Gamma(\\bar k)|$, or $|\\gamma_Y|$, where $\\bar k$ is the slowest-decaying mode and $\\gamma_Y$ is an effective decay built from the spectrum and its derivatives at a complex saddle point; these three rates define the paper's three new dynamical phases. The companion result is the full asymptotics of $C_{zz}(x)$: oscillatory exponential $x^{-3}e^{-x/\\xi}$ in the gapped phase, oscillatory algebraic $x^{-2}\\cos^2 qx$ in the gapless phase, and $x^{-4}\\mu(x)$ on the critical line where two exceptional points appear.","pith_inferences":["Beyond the paper, the same fidelity construction could be applied to other non-Hermitian integrable chains, such as extended Kitaev or SSH models, where the per-mode factorization is expected to hold; measuring the late-time fidelity in those models would test the universality of the three-region picture.","Beyond the paper, the identity $C_\\Omega=(1+\\sqrt{1-C_{zz}(0)})/2$ suggests that Krylov spread can substitute for same-site correlation calculations in any model whose Krylov space is generated from a generalized coherent state, which would be worth testing in interacting or Floquet systems.","Beyond the paper, a direct experimental analogue would be to engineer the non-Hermitian Ising chain in a platform with controlled gain and loss and to extract the spread fidelity from time-resolved measurements of the state's projection onto the Krylov basis.","Beyond the paper, since the non-Hermitian Hamiltonian describes the no-click sector of a monitored system, the refined phase diagram should be inherited by quantum-trajectory dynamics; monitoring the spread fidelity under jumps would test that inheritance."],"forward_implications":["In the gapped phase, the infinite-time Krylov spread density equals the spread obtained by unitary evolution from $|0\\rangle$ to $|\\Omega\\rangle$, so the fidelity $F(t)$ measures how the non-Hermitian dynamics approach that unitary vacuum.","The spread density $C_\\Omega$ undergoes a third-order phase transition across $\\gamma=\\gamma_c(h)$, with $\\partial^2 C_\\Omega/\\partial h^2$ diverging as $(h_c-h)^{-1/2}$ from the left, so Krylov spread can locate the gapped-gapless boundary even where correlation functions are smooth.","In the gapless phase, the stationary state is $A(t)|\\Omega\\rangle+A^*(t)|q,-q\\rangle$ with time-dependent coefficients, and the time-averaged spread density coincides with the unitary spread density.","The correlation asymptotics are now known across the whole phase diagram: oscillatory $x^{-3}e^{-x/\\xi}$ in the gapped phase, $x^{-2}\\cos^2 qx$ in the oscillatory gapless phase, and $x^{-4}\\mu(x)$ on the critical line, with the change of power-law caused by two exceptional points.","The Krylov fidelity splits the gapped region into three subregions, and the boundaries of the subregions are marked by discontinuities in the derivatives of the characteristic time $t^\\ast$ with respect to $h$ or $\\gamma$."],"supporting_citations":[{"why":"Supplies the thermodynamic-limit factorization of Krylov vectors and the per-mode spread formula that the paper extends to non-Hermitian evolution.","marker":"[63]"},{"why":"Defines the non-Hermitian vacuum and the gapped-gapless phase structure for this model; the paper sharpens its stationary-state coefficient from a constant to a time-dependent one.","marker":"[22]"},{"why":"Provides the earlier calculation of the gapless-phase correlation function that the paper extends to the full gapless region and to the critical line.","marker":"[23]"},{"why":"Gives the gapped-phase correlation at vanishing real field that the paper generalizes to $h\\neq 0$ and to the full phase diagram.","marker":"[10]"},{"why":"Establishes that the vacuum is a generalized coherent state of the Jordan-Wigner vacuum, the structure used to build the Krylov basis.","marker":"[88]"},{"why":"Supplies the $\\mathfrak{su}(2)$ normal-ordering decomposition used to obtain the time-evolved state and the spread coefficient $A_+(k;t)$.","marker":"[101]"}],"fun_headline_variants":["Krylov spread unveils 3 extra phases in non-Hermitian Ising model","Complex field Ising chain: Krylov spread shows hidden phase structure","Non-Hermitian Ising chain: Krylov spread reveals three new phases","Krylov spread exposes finer phase map in complex-field Ising chain"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole calculation assumes that, in an infinite chain, the Krylov space built from the all-down state splits mode by mode, so the spread is an integral over independent two-level spreads; if the non-Hermitian evolution mixes momentum modes, the predicted spreads, the fidelity rates, and the three-phase diagram would not describe the actual Krylov spread.","fun_headline_variants_meta":{"raw":{"variants":["Krylov spread unveils 3 extra phases in non-Hermitian Ising model","Complex field Ising chain: Krylov spread shows hidden phase structure","Non-Hermitian Ising chain: Krylov spread reveals three new phases","Krylov spread exposes finer phase map in complex-field Ising chain"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000304,"raw_usage":{"total_tokens":1865,"prompt_tokens":1181,"completion_tokens":684,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":797,"completion_tokens_details":{"reasoning_tokens":601}},"tokens_in":797,"tokens_out":684,"duration_ms":6334,"temperature":1.0,"reasoning_tokens":601,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T11:36:31.266716+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically construct the exact Krylov basis for a finite chain at points in each predicted region (for example $h=0.5J$ with $\\gamma=2J$, $\\gamma=5J$, and $h=2J,\\gamma=5J$, with $N$ up to about 20), compute the spread density $C(t)$ directly, and check whether the late-time fidelity decays with the predicted rates $\\gamma$, $4|\\Gamma(\\bar k)|$, or $|\\gamma_Y|$; a mismatch in any region would falsify the factorization underlying Eq. (102).","supporting_citations":[{"cited_title":"Krylov complexity from integrability to chaos,","cited_arxiv_id":null,"evidence_quote":"Supplies the thermodynamic-limit factorization of Krylov vectors and the per-mode spread formula that the paper extends to non-Hermitian evolution."},{"cited_title":"Spread complexity for measurement-induced non-unitary dynamics and Zeno effect,","cited_arxiv_id":null,"evidence_quote":"Establishes that the vacuum is a generalized coherent state of the Jordan-Wigner vacuum, the structure used to build the Krylov basis."},{"cited_title":"PT -symmetric, non-Hermitian quantum many-body physics–a methodological perspective,","cited_arxiv_id":null,"evidence_quote":"Supplies the $\\mathfrak{su}(2)$ normal-ordering decomposition used to obtain the time-evolved state and the spread coefficient $A_+(k;t)$."}],"review_version":1}