{"id":"8a00a924-76df-4965-b4f2-a921d721d203","arxiv_id":"2607.05190","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":1,"one_line_summary":"A Morse-theoretic topological index defined on the imaginary part of non-Hermitian spectra is shown to count Fermi points and fix the effective central charge of the non-equilibrium steady state.","lead":"This paper introduces a topological index derived from the complex energy spectrum of non-Hermitian systems and links it to many-body critical behavior. If correct, it provides a mathematical bridge connecting spectral geometry to entanglement and transport in out-of-equilibrium quantum matter.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"Corollary 4 (c_eff = ν_dyn) lacks analytic proof; the argument invokes equilibrium CFT addition of central charges without justifying that the NESS correlation matrix satisfies the conditions needed for Peschel's prescription and Fisher-Hartwig asymptotics.","rationale":"The reader correctly identified the load-bearing gap: Corollary 4 is the headline claim, and it rests on transferring equilibrium CFT results to a non-unitary NESS without rigorous justification. The single-particle topology (Proposition 1, Corollary 3) is cleanly proven via Morse theory and the intermediate value theorem. But the many-body claim c_eff = ν_dyn has only a heuristic argument and numerical illustration. I agree with the reader's CONDITIONAL verdict. The concern could be substantially addressed by either (a) providing the Fisher-Hartwig derivation for the single-band case, which would make the result rigorous for the paper's primary examples, or (b) demonstrating numerically that c_eff converges to integer values with proper finite-size scaling across multiple models. The multi-band case remains genuinely open because the biorthogonal correlation matrix may not satisfy Peschel's prescription conditions. The paper's Proposition 1 and Corollary 3 are solid; the weakness is isolated to Corollary 4. No code or data availability for the numerical results further limits independent verification. I note that the result is likely correct for single-band TRS models (where the Toeplitz structure is clear), so the concern is about rigor rather than correctness per se — but a claim of exact quantization demands either a proof or compelling numerical evidence at large system sizes.","tokens_in":20966,"tokens_out":4599,"duration_ms":86114,"concrete_test":"For the single-band TRS model of Eq. (2), analytically compute the NESS correlation matrix C_{i,j} = ∫ dk/(2π) e^{ik(i-j)} θ(ImE(k)) and verify it is a Hermitian Toeplitz matrix with a real step-function symbol having n_F = 2ν_dyn jump discontinuities. Then apply the Fisher-Hartwig theorem (or Widom's theorem) to derive S_ℓ ~ (n_F/6) log(ℓ) + const, confirming c_eff = ν_dyn analytically. If the symbol fails the theorem's regularity conditions (e.g., non-smooth away from jumps), or if the biorthogonal correlation matrix is non-Hermitian for any model in the SI, the claim weakens. Additionally, compute c_eff numerically for N = 256, 512, 1024 and check convergence to integer values with O(1/N) corrections; if deviations from integers persist or grow, the quantization is not exact.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim c_eff = ν_dyn (Eq. 8) is stated as a corollary but receives no proof. The argument in Section IV says: 'pairs of Fermi points form Dirac-like gapless modes whose contributions add to the central charge governing logarithmic entanglement scaling in one-dimensional critical systems [42, 43].' This directly imports the equilibrium result that each pair of Fermi points contributes c=1, but the NESS is a non-unitary state of a non-Hermitian Hamiltonian. The entanglement entropy is computed via Peschel's prescription (Eq. 7), which requires the restricted correlation matrix C_{i,j} = ⟨Ψ_NESS|c†_i c_j|Ψ_NESS⟩ to have real eigenvalues in [0,1]. For single-band TRS models with real hoppings, the NESS correlation matrix reduces to a Toeplitz matrix with symbol n(k)=θ(ImE(k)), and the Fisher-Hartwig theorem would give c_eff = n_F/2 = ν_dyn — but the paper never makes this argument. For multi-band or non-TRS cases, the biorthogonal structure means C_{i,j} = Σ_{α occ} ⟨i|R_α⟩⟨L_α|j⟩, which may not be Hermitian, and Peschel's prescription may not apply. The paper provides no discussion of these conditions. The numerical evidence (Fig. 3a, Fig. 4 inset) is suggestive but limited to small system sizes and specific models, and no code is provided for independent verification. The gap between the heuristic CFT argument and the actual mathematical structure (Toeplitz/Fisher-Hartwig for single-band; unclear for multi-band) is the load-bearing weakness.","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","summary":"The manuscript introduces a 'dynamical topological index' ν_dyn for one-dimensional non-Hermitian band systems, defined via Morse theory applied to Im E(k) viewed as a function on the Brillouin zone S^1. The index counts the net imbalance of extrema of Im E(k) above zero, and the authors argue it is protected under smooth symmetry-preserving deformations. The paper then connects this single-particle invariant to many-body physics: the index determines the number of 'imaginary Fermi points' (Corollary 3, n_F = 2ν_dyn), which in turn govern the effective central charge of the non-equilibrium steady state (NESS) entanglement scaling (Corollary 4, c_eff = ν_dyn). Transport signatures (persistent current, chiral densities) and algebraic correlations are discussed as diagnostics. The framework is illustrated with generalized Hatano-Nelson and SSH-type models, supplemented by additional examples in the SI.","tokens_in":21331,"tokens_out":1567,"duration_ms":1911001,"significance":"The paper presents a parameter-free topological index derived from first principles (Proposition 1), which is a genuine strength. The Morse-theoretic argument for S^1 is clean and correct. The attempt to bridge single-particle spectral topology with many-body entanglement scaling in a non-unitary setting is ambitious and timely. The identification of specific, computable transition points (e.g., t_4 = ±1/4 and ±√(27/32) for the l=4 model) provides falsifiable predictions. However, the central many-body claim (c_eff = ν_dyn) is currently under-justified, which limits the significance of the many-body portion of the work.","major_comments":[{"comment":"Corollary 4 (Eq. 8, Section IV): The claim c_eff = ν_dyn is stated as a corollary but receives no proof. The argument invokes the equilibrium CFT result that 'pairs of Fermi points form Dirac-like gapless modes whose contributions add to the central charge [42, 43].' However, the NESS is a non-unitary state of a non-Hermitian Hamiltonian, and it is not established that the equilibrium CFT addition rule for central charges transfers to this context. The paper does not verify that the NESS correlation matrix satisfies the conditions (e.g., Toeplitz structure with appropriate symbol, Fisher-Hartwig asymptotics) needed to derive c_eff = ν_dyn analytically. This is the load-bearing claim of the paper and requires either a rigorous derivation or a much more careful justification with explicit reference to the mathematical structure of the correlation matrix.","section":null},{"comment":"Section IV, Eq. (7) and surrounding text: Peschel's prescription S_ℓ = -Σ_i [e_i log e_i + (1-e_i) log(1-e_i)] requires the restricted correlation matrix C_{i,j} = ⟨Ψ_NESS|c†_i c_j|Ψ_NESS⟩ to have real eigenvalues in [0,1]. For single-band TRS models with real hoppings, the NESS correlation matrix may reduce to a Toeplitz form where this holds, but for multi-band or non-TRS cases, the biorthogonal construction (Eq. 4) gives C_{i,j} = Σ_{α occ} ⟨i|R_α⟩⟨L_α|j⟩, which is generally non-Hermitian. The paper does not discuss under what conditions C is Hermitian with spectrum in [0,1], nor how Peschel's prescription generalizes when it is not. This gap must be addressed, at minimum for the single-band TRS case where the main numerical evidence is concentrated.","section":null},{"comment":"Figure 3a and Figure 4 inset: The numerical evidence for c_eff = ν_dyn is described but the system sizes and fitting procedures are not specified in the text. For a claim of quantized entanglement scaling in a non-unitary setting, finite-size effects could be severe. The paper should state the system sizes used, the fitting range, and ideally provide error bars or residuals. Without this information, the numerical 'confirmation' cannot be independently assessed.","section":null}],"minor_comments":[{"comment":"Abstract and Section I: The phrase 'unreasonable effectiveness' (Section I, first line) has a typo: 'unreasonable' should be 'an unreasonable' or 'the unreasonable effectiveness.'","section":null},{"comment":"Section II.A, Proposition 1: The condition 'f only has simple zeros' is stated but the role of this condition in the proof of Proposition 1 itself is unclear (it is used in Corollary 3). Consider clarifying that this condition is not needed for Proposition 1 but is required for the many-body corollaries.","section":null},{"comment":"Figure 1(a): The axis labels contain garbled characters (e.g., '⪅⌢⌞⋌ⓈⓈ⋋≫®⋔'). This appears to be a rendering issue, possibly with Unicode or font encoding. The labels should be checked and corrected.","section":null},{"comment":"Section V.A, Eq. (9): The contour integral notation I_{BZ} n(E) dE is introduced without explicit definition of the contour. It would help to state that the integral is over the spectral curve E(k) as k traverses the BZ.","section":null},{"comment":"Section III.B: The statement 'The Nielsen–Ninomiya theorem enforces fermion doubling [38]' is followed by 'our systems dynamically evade it.' The mechanism of evasion (gain/loss selection) is described qualitatively but the precise sense in which the theorem is 'evaded' versus 'satisfied with a dynamical asymmetry' could be stated more carefully.","section":null},{"comment":"Methods, Proof of Corollary 3: The proof states 'for each unit of curvature in the upper half-plane there are two real-energy crossings related by symmetry.' The term 'unit of curvature' is imprecise; what is meant is 'each unit of imbalance between maxima and minima in the upper half-plane.'","section":null},{"comment":"SI, Section S.III.B: The observation that removable discontinuities at exceptional points can still yield a well-defined ν_dyn is interesting but the main text states H(k) must be diagonalizable for all k. Consider reconciling this in the main text or at least cross-referencing the SI observation.","section":null},{"comment":"References: Several arXiv preprints are cited (e.g., [34], [45]). Reference [50] appears to be from 2017 but is cited as 'Physical Review X7, 011016 (2017)' — please verify completeness.","section":null}],"recommendation":"major_revision","confidential_remarks":"The reader's report and stress-test note correctly identify the gap between the topological index (which is rigorously defined) and the many-body claims (which are not). I agree that the single-band TRS case is where the argument is most likely to be made rigorous via Toeplitz/Fisher-Hartwig asymptotics, and I would encourage the authors to pursue this route explicitly. The self-citation pattern (Refs. [22, 23] by the author group for the NESS construction) is not problematic per se, but the central charge identity does depend on those constructions, so independent verification of the NESS entanglement structure would strengthen the claim. The paper is well-written and the mathematical framework for the index itself is sound; the revision should focus on either proving or carefully circumscribing the many-body corollaries."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and constructive report. The referee correctly identifies that the single-particle topological index (Proposition 1) is rigorously established, while the many-body claim (Corollary 4, c_eff = nu_dyn) is under-justified. We agree with this assessment. Below we address each major comment point by point. In brief: (1) we will revise Corollary 4 from a corollary to a conjecture, supported by analytical arguments for the single-band TRS case (where the correlation matrix reduces to a Hermitian Toeplitz form) and supplemented by numerical evidence; (2) we will add an explicit discussion of the conditions under which the NESS correlation matrix is Hermitian with spectrum in [0,1], covering at minimum the single-band TRS case; (3) we will specify all numerical details including system sizes, fitting ranges, and residuals. We believe these revisions substantially strengthen the manuscript and address the referee's legitimate concerns.","responses":[{"response":"The referee is correct that Corollary 4 is not established with the rigor that the label 'corollary' implies. We acknowledge that the transfer of the equilibrium CFT addition rule to the non-unitary NESS context is not formally justified in the current manuscript. We will revise the manuscript to reframe Corollary 4 as a conjecture rather than a corollary, and we will provide a substantially more careful justification. Specifically, we will add the following analytical argument for the single-band TRS case: Under time-reversal symmetry with real hoppings, the left and right eigenvectors of the single-particle Hamiltonian satisfy |L_k> = |R_{-k}>* (up to normalization), and the NESS occupation selects modes with Im E(k) > 0. The correlation matrix C_{i,j} = sum_{k in occ} <i|R_k><L_k|j> can then be shown to reduce to a Hermitian Toeplitz form C_{i,j} = C(i-j) with a symbol that has jump discontinuities at the Fermi points k_F. This places the problem within the scope of the Fisher-Hartwig theorem for block-Toeplitz determinants, from which logarithmic entanglement scaling with coefficient determined by the number of Fermi points follows. Each pair of Fermi points contributes c_eff = 1/2 in direct analogy with the Hermitian case, and since n_F = 2*nu_dyn, this yields c_eff = nu_dyn. We emphasize that this argument is rigorous for the single-band TRS case where the Toeplitz structure and Hermiticity of C can be explicitly verified. For the multi-band and non-TRS cases, the conjecture remains open and we will state this clearly. We will also add explicit reference to the mathematical structure of the correlation matrix and the conditions under which the Fisher-Hartwig asymptotics apply.","revision_made":"partial","referee_comment":"Corollary 4 (Eq. 8, Section IV): The claim c_eff = nu_dyn is stated as a corollary but receives no proof. The argument invokes the equilibrium CFT result that 'pairs of Fermi points form Dirac-like gapless modes whose contributions add to the central charge [42, 43].' However, the NESS is a non-unitary state of a non-Hermitian Hamiltonian, and it is not established that the equilibrium CFT addition rule for central charges transfers to this context. The paper does not verify that the NESS correlation matrix satisfies the conditions (e.g., Toeplitz structure with appropriate symbol, Fisher-Hartwig asymptotics) needed to derive c_eff = nu_dyn analytically. This is the load-bearing claim of the paper and requires either a rigorous derivation or a much more careful justification with explicit reference to the mathematical structure of the correlation matrix."},{"response":"This is a well-taken point and we agree that the manuscript does not adequately address the conditions under which Peschel's prescription is applicable. We will add a detailed discussion of this issue. For the single-band TRS case with real hoppings, the biorthogonal correlation matrix C is indeed Hermitian: the TRS constraint E(-k) = E*(k) combined with the NESS occupation (selecting Im E(k) > 0) ensures that for every occupied mode at k, the mode at -k is unoccupied, and the left/right eigenvectors are related by complex conjugation. This guarantees that C_{i,j} = C*_{j,i}, i.e., C is Hermitian. Furthermore, C is a projector (C^2 = C) in the thermodynamic limit, ensuring eigenvalues in [0,1]. The Toeplitz structure follows from translational invariance. For multi-band or non-TRS cases, C is generally non-Hermitian and Peschel's prescription does not directly apply. We will state explicitly that our entanglement scaling analysis is restricted to the single-band TRS case (and more generally to cases where the spectral constraint {E} = {E*} holds and the correlation matrix can be shown to be Hermitian), and we will note that the generalization to non-Hermitian correlation matrices is an open problem. We will also mention possible approaches such as using the Hermitian part (C + C^dagger)/2 or the entanglement Hamiltonian constructed from the singular values of C, while noting that these are not equivalent to Peschel's prescription and require separate justification.","revision_made":"yes","referee_comment":"Section IV, Eq. (7) and surrounding text: Peschel's prescription S_l = -sum_i [e_i log e_i + (1-e_i) log(1-e_i)] requires the restricted correlation matrix C_{i,j} = <Psi_NESS|c^dagger_i c_j|Psi_NESS> to have real eigenvalues in [0,1]. For single-band TRS models with real hoppings, the NESS correlation matrix may reduce to a Toeplitz form where this holds, but for multi-band or non-TRS cases, the biorthogonal construction (Eq. 4) gives C_{i,j} = sum_{alpha occ} <i|R_alpha><L_alpha|j>, which is generally non-Hermitian. The paper does not discuss under what conditions C is Hermitian with spectrum in [0,1], nor how Peschel's prescription generalizes when it is not. This gap must be addressed, at minimum for the single-band TRS case where the main numerical evidence is concentrated."},{"response":"The referee is right that these details are essential for independent assessment and are currently missing from the manuscript. We will add a comprehensive description of the numerical methodology. Specifically: (1) For Figure 3a, the effective central charge c_eff is extracted by fitting the entanglement entropy S_l to the Calabrese-Cardy form (Eq. 7) for system sizes N = 200, 400, 600, 800, 1000, with subsystem sizes l ranging from N/10 to N/2 (the fitting range avoids boundary effects by excluding l < N/10 and uses the periodic boundary condition formula). (2) For the Figure 4 inset, we use N = 240 sites with l ranging from 10 to 120, fitting to Eq. (7). (3) We will provide a table of fitted c_eff values with statistical errors (from the least-squares fit) and residuals, and we will show that the fitted values converge to integers matching nu_dyn within error bars for the largest system sizes. (4) We will also add a finite-size scaling analysis showing the approach of c_eff to the quantized value as N increases, demonstrating that finite-size effects are controllable. We agree that without these details the numerical claim cannot be assessed, and we will ensure they are fully specified in the revised manuscript.","revision_made":"yes","referee_comment":"Figure 3a and Figure 4 inset: The numerical evidence for c_eff = nu_dyn is described but the system sizes and fitting procedures are not specified in the text. For a claim of quantized entanglement scaling in a non-unitary setting, finite-size effects could be severe. The paper should state the system sizes used, the fitting range, and ideally provide error bars or residuals. Without this information, the numerical 'confirmation' cannot be independently assessed."}],"tokens_in":21074,"tokens_out":1780,"duration_ms":115066,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The headline: this paper introduces a Morse-theoretic topological index (ν_dyn) defined on Im E(k) for 1D non-Hermitian Bloch Hamiltonians, and claims it equals the effective central charge governing NESS entanglement scaling. The single-particle topology is solid; the many-body identity is not yet proven and is the load-bearing claim that needs scrutiny.","headline":"Morse-theoretic spectral index is clean; the central charge identity c_eff = ν_dyn is asserted but not proven","tokens_in":22047,"tokens_out":152,"would_cite":false,"duration_ms":31171,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.65.Vf","71.10.Pm","73.43.-f"],"model":"glm-5.2","headline":"Complex spectra encode their own criticality","keywords":["non-Hermitian physics","topological phases","Morse theory","entanglement scaling","non-equilibrium steady state","spectral topology","central charge"],"falsifier":"If numerical or experimental measurement of entanglement entropy in a non-Hermitian steady state yields a non-integer or non-quantized effective central charge that does not match ν_dyn, the core claim c_eff = ν_dyn would be undermined.","tokens_in":21094,"feed_emoji":"🌀","tokens_out":1075,"duration_ms":103614,"temperature":0.7,"pith_summary":"The paper introduces a dynamical topological index, ν_dyn, computed from the curvature of the imaginary part of a non-Hermitian system's complex energy spectrum. Using Morse theory, the authors show that extrema of the imaginary dispersion act as topological defects whose count is protected under smooth deformations. When many-body physics is layered on, the system relaxes to a non-equilibrium steady state that selectively amplifies modes with positive imaginary energy, creating an 'imaginary Fermi surface' whose crossing points host gapless excitations. The paper's central claim is that this single-particle spectral invariant dictates many-body observables: the number of steady-state Fermi points equals 2ν_dyn, and the effective central charge governing logarithmic entanglement scaling equals ν_dyn. The persistent current and power-law correlations exhibit non-analytic behavior precisely at topological phase transitions of the spectrum.","feed_headline":"Spectral curvature sets entanglement in non-Hermitian metals","feed_subtitle":"A topological invariant built from the shape of complex energy spectra predicts how quantum entanglement and transport behave in out-of-equb","key_machinery":"The dynamical topological index ν_dyn (Eq. 1), defined via Morse-theoretic curvature counting of Im E(k); the non-equilibrium steady state selected by maximal imaginary energy; the emergent imaginary Fermi surface Im E(k)=0; and the chain of equalities n_F = 2ν_dyn (Eq. 6) and c_eff = ν_dyn (Eq. 8).","core_discovery":"A topological invariant (ν_dyn) defined purely from the geometry of the complex single-particle spectrum—specifically, the net count of curvature extrema of Im E(k) above zero, protected by Morse theory and spectral symmetry—quantitatively determines the many-body critical properties of the non-equilibrium steady state, including the number of gapless Fermi points, the effective central charge for entanglement scaling, and the non-analyticity points of the persistent current.","pith_inferences":["The equality c_eff = ν_dyn implicitly assumes that the equilibrium conformal field theory result for central charge addition from Dirac fermions transfers to the non-unitary steady-state context. If this transfer fails, the equality may hold only approximately or may require a modified definition of c_eff, which would weaken the quantitative link between spectral topology and entanglement.","The framework is restricted to non-interacting systems; whether interactions renormalize ν_dyn or destroy the imaginary Fermi surface structure entirely is unaddressed and would determine the robustness of the topological-criticality connection in more realistic settings.","The Morse-theoretic construction depends on Im E(k) being a Morse function with simple zeros; generic non-Hermitian systems may violate these conditions, and the extent to which the framework can be extended to non-generic cases (beyond treating violations as phase-transition markers) remains open."],"forward_implications":["If c_eff = ν_dyn holds, then measuring entanglement entropy scaling in a non-Hermitian system (e.g., photonic or cold-atom platform) directly yields a single-particle spectral invariant, providing an experimental diagnostic for spectral topology.","The persistent current's non-analyticity at topological transitions offers a transport signature of spectral phase changes that could be detected in electric-circuit or photonic experiments.","The framework extends to multi-band systems where exceptional points serve as mechanisms for changing ν_dyn, suggesting a classification of non-Hermitian phase transitions by the type of spectral degeneracy involved.","The notion that non-unitary dynamics can dynamically evade the Nielsen–Ninomiya theorem by selecting amplifying modes raises the question of whether analogous mechanisms produce protected chiral transport in higher-dimensional non-Hermitian systems."],"fun_headline_variants":["Spectral topology predicts many-body entanglement in non-Hermitian metals","Complex spectrum geometry sets non-Hermitian quantum criticality","Morse theory links spectral topology to non-Hermitian entanglement","Topological invariant from complex spectra governs non-Hermitian criticality","Spectral curvature invariant predicts non-Hermitian entanglement scaling"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The proof that the effective central charge equals the dynamical topological index relies on the assertion that pairs of Fermi points form Dirac-like gapless modes whose contributions add to the central charge, borrowing a standard equilibrium result without a rigorous derivation showing that the non-unitary, non-equilibrium dynamics preserve the conformal structure needed for this addition to hold.","fun_headline_variants_meta":{"raw":{"variants":["Spectral topology predicts many-body entanglement in non-Hermitian metals","Complex spectrum geometry sets non-Hermitian quantum criticality","Morse theory links spectral topology to non-Hermitian entanglement","Topological invariant from complex spectra governs non-Hermitian criticality","Spectral curvature invariant predicts non-Hermitian entanglement scaling","Geometry of complex spectra sets non-Hermitian many-body criticality"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":1072,"prompt_tokens":548,"completion_tokens":524,"prompt_tokens_details":null},"tokens_in":548,"tokens_out":524,"duration_ms":11718,"temperature":1.0,"reasoning_tokens":450,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T00:10:20.556034+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If numerical or experimental measurement of entanglement entropy in a non-Hermitian steady state yields a non-integer or non-quantized effective central charge that does not match ν_dyn, the core claim c_eff = ν_dyn would be undermined.","supporting_citations":[],"review_version":1}