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REVIEW 3 major objections 4 minor 105 references

On the non-hermitian Kitaev chain

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that the open non-Hermitian Kitaev chain's infinite-size spectrum is exactly the solutions of three Vieta equations with an ordering constraint, and that the zero-mode region is fixed by a single inverse-cosine inequality.

desk verdict A solid analytic treatment of the non-Hermitian Kitaev chain spectrum, but the zero-mode criterion (Eq. 29) rests on a skipped derivation with a contradictory sign remark, and the abstract overstates the skin-effect results for complex parameters. read the letter →

arxiv 2411.14776 v2 pith:6P6ZDS77 submitted 2024-11-22 quant-ph cond-mat.mes-hall

Eddy Ardonne , Viktor Kurasov This is my paper · ORCID
classification quant-phcond-mat.mes-hall MSC 81Q1282B23
keywords non-HermitianKitaevchainskineffectzeromodethermodynamiclimitVietaequationsBistritzalgorithmcomplexeigenvaluesMajoranamodes
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies the non-Hermitian Kitaev chain, a one-dimensional fermion model with complex hopping and pairing amplitudes, and aims to give a complete, numerically stable description of its open-chain spectrum in the infinite-size limit. It claims that every eigenvalue curve can be obtained from three Vieta equations for two complex parameters $\kappa$ and $s$ and a real angle $\alpha$, subject to the ordering constraint $\frac{1}{|s|} \leq |\kappa|^2 \leq |s|$, so the spectrum can be computed without the ill-conditioned diagonalisation of large non-Hermitian matrices. It also claims to fully determine the parameter region where a zero mode exists, via one inequality involving the imaginary parts of two inverse-cosine expressions built from the model parameters. Using these results, it characterises when the skin effect is absent and which eigenvectors are delocalised. The payoff is that three delicate questions—eigenvalue curves, zero modes, and skin-effect localisation—are reduced to algebraic conditions that can be checked directly.

What carries the argument

The central object is the fourth-order bulk equation for the plane-wave factor $x$, obtained by substituting the ansatz $\psi = (x, a x, x^2, a x^2, \ldots)$ into the Bogoliubov-de Gennes equations. Because the quartic is palindromic, its four roots satisfy $x_1 x_2 x_3 x_4 = 1$. In the thermodynamic limit the boundary determinant is dominated by the roots of largest and second-largest modulus, and a non-trivial solution forces $|x_2| = |x_3|$; writing the roots as in Eq. (31) and applying the three remaining Vieta relations yields the eigenvalue equations (37). The zero-mode analysis uses the fact that at $\lambda = 0$ the four roots factor into two reciprocal pairs, and the boundary conditions select the pair with both members inside the unit circle; the arccos inequality (29) encodes exactly when that happens. The skin-effect analysis uses the Bistritz algorithm, a root-counting algorithm for polynomials relative to the unit circle, to locate parameter choices for which two roots lie on the unit circle—the signature that the corresponding eigenstates are delocalised.

What would settle it

Take parameter sets near the boundary of the inequality (29), for instance with small nonzero $d_1, d_2$ and $m, t_1, t_2$ chosen so the two sides of the inequality are nearly equal, and compute the smallest eigenvalue modulus for $L = 100, 200, 400, 800$ using high-precision arithmetic; check that it decays to zero exactly on the predicted side of the boundary. The criterion is falsified if the boundary shifts when the square-root branch convention in Eq. (27) is changed, or if a zero-energy eigenvalue appears on the side predicted to have none.

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Extended reading notes

Core claim

In the limit $L \to \infty$, the eigenvalues $\lambda$ of the open chain with arbitrary complex $m, t_1, t_2, d_1, d_2$ are exactly the solutions of the three Vieta equations (37), where the four roots of the bulk quartic are written as $x_1 = s/\kappa$, $x_2 = \kappa e^{i\alpha}$, $x_3 = \kappa e^{-i\alpha}$, $x_4 = 1/(s\kappa)$, with $\kappa$ and $s$ complex and $0 \leq \alpha < 2\pi$, and only solutions obeying $|x_1| \geq |x_2| = |x_3| \geq |x_4|$ (equivalently $\frac{1}{|s|} \leq |\kappa|^2 \leq |s|$) are physical eigenvalues. The zero mode $\lambda = 0$ exists if and only if $\left|\operatorname{Im}\arccos\left(\frac{-m}{2\sqrt{t_1t_2 - d_1d_2}}\right)\right| < \left|\operatorname{Im}\arccos\left(\frac{t_1+t_2}{2\sqrt{t_1t_2 - d_1d_2}}\right)\right|$. In the hermitian limit this reduces to the familiar Kitaev conditions, while for $d_1d_2 = 0$ a zero mode can appear only at infinite system size. For real parameters the paper fully classifies the absence of the skin effect: it is absent when $m = 0$, when $t_1 = \pm t_2$, or for the purely imaginary eigenvalue branch when $d_1d_2 < 0$ with the radicand negative; for complex parameters it gives sufficient conditions that it argues are also necessary. A further result is that the branch points visible in the spectrum arise from switching among three ordering branches of the Vieta solutions, with the non-physical branches explaining the geometry of the physical curves.

Load-bearing premise

The zero-mode criterion Eq. (29) assumes that the complex square-root factor introduced in Eq. (27) can only swap the labels of the four $x$ solutions, not change the parameter region where a zero mode exists; the text itself remarks that the sign might introduce an additional sign and that it 'does influence' the range, so if that caveat is literal the criterion may not be complete.

Editorial extensions

If this is right

  • The eigenvalue curves for any parameter set can be generated by numerically solving three polynomial equations over $\alpha \in [0, 2\pi)$, avoiding the instability of brute-force diagonalisation of large non-Hermitian matrices.
  • The zero-mode region of the non-Hermitian Kitaev chain is fixed by a single explicit inequality, so the topological phase boundary can be drawn without solving the full spectrum.
  • The zero mode itself is not captured by the Vieta equations; it sits at $\lambda = 0$ and is governed by the separate inequality (29), so a complete spectral picture needs both results.
  • The skin effect is absent exactly when two of the four bulk roots lie on the unit circle; for real parameters the paper lists all such cases, including the counterintuitive case of real eigenvalues with skin effect.
  • Phase-rotating the parameters rigidly rotates the entire spectrum, giving a one-parameter family of parameter sets with identical eigenvalue structure up to an overall phase.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same Vieta-branch method is likely transferable to other one-dimensional non-Hermitian models whose bulk polynomial is palindromic, since only the root-product identity and the ordering constraint are essential to the construction.
  • The zero-mode inequality may provide a direct non-Hermitian analogue of the topological phase boundary that could be compared with winding-number or biorthogonal-polarization invariants for the same model.
  • For complex parameters, the paper's skin-effect classification is only proven sufficient; if the stated belief that it is also necessary is wrong, there should exist parameter regions with extended $k$-intervals where the Bistritz algorithm is singular at level 4 but the phase conditions (52) fail.
  • The $d_1d_2 = 0$ case, where a zero mode appears only in the infinite-size limit, invites a closer check of whether the $L \to \infty$ limit commutes with the vanishing-pairing limit for boundary-driven phenomena.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the non-Hermitian Kitaev chain with arbitrary complex parameters m, t1, t2, d1, d2. The main results are: (i) a characterization of the eigenvalue curves in the infinite-size limit in terms of three Vieta equations (Eq. (37)) together with an ordering constraint 1/|s| ≤ |κ|^2 ≤ |s|; (ii) a criterion for the absence of skin effect, claimed to be complete for real parameters and conjectured sufficient for complex parameters; and (iii) a condition (Eq. (29)) that allegedly fully determines the parameter region in which a zero mode exists. The method combines a generalized Bloch ansatz, boundary determinant analysis, Vieta relations, and the Bistritz algorithm. The paper includes finite-size numerical checks (Figs. 1, 6, 7) supporting the Vieta-equation characterization.

Significance. If all three claims hold, the paper provides a valuable exact tool for non-Hermitian lattice models: a numerically stable characterization of the thermodynamic-limit spectrum, a zero-mode phase diagram, and a criterion for skin-effect absence. The Vieta-equation method is potentially generalizable to other one-dimensional non-Hermitian systems with longer-range couplings. A notable strength is the explicit comparison with finite-size numerics (Figs. 1, 6, 7), which demonstrates the practical utility of the eigenvalue characterization. However, the zero-mode section contains an unresolved internal contradiction about branch choices, and the skin-effect results for complex parameters are explicitly stated to be only conjectural. These points directly affect the paper's advertised completeness.

major comments (3)
  1. [Sec. IV.B] The derivation of the zero-mode criterion Eq. (29) is internally inconsistent. The text states that putting 1/√(−4D2) under the square roots 'might introduce an additional sign' and that 'this does influence the range of parameters for which there is a zero mode', but immediately adds that 'the values of the various x±,± might be swapped'. If the four x values are only permuted, the existence condition (two of them having |x|<1) is invariant, so the sign choice would not change the zero-mode region; if instead the sign choice can change which pair of x values is small, then Eq. (29), derived from one particular assignment, is not established as the full condition. The paper never resolves this contradiction. Moreover, the step from the conditions |x−,−|<1 ∧ |x+,−|<1 or |x−,+|<1 ∧ |x+,+|<1 to the inequality |Im arccos(y1)| < |Im arccos(y2)| is stated without a derivation ('Analysing these conditions, one finds...'). The finite-size checks at the end of Sec. IV.B address only the d1d2→0 regularization and do not test Eq. (29) against a direct numerical zero-mode search in the complex parameter region where the branch ambiguity is live. Thus the abstract's claim to 'fully determine the region in parameter space for which the model has a zero mode' is not supported by the presented derivation.
  2. [Sec. V] The characterization of eigenvalues via the Vieta equations is stated to hold for 'arbitrary complex parameters', but the derivation explicitly excludes 'ten special values of λ' (plus the cases t2=-t1 and d1d2=0) with the remark 'We do not consider these ten special values, because we are interested the generic eigenvalues of the model'. The text does not show that these special values are not eigenvalues of the open chain in the L→∞ limit. If they are eigenvalues, they are missing from the claimed complete characterization; if they are not, a proof is needed. This is a load-bearing gap because the abstract and Sec. V claim a complete characterization of the eigenvalue curves.
  3. [Sec. VI.B] For complex parameters, the skin-effect characterization is only conjectural. The text states: 'although we believe we determined all cases for which there are at least two roots on the unit circle, we do not have a proof for this in the general case with complex parameters' and later 'we believe that the conditions provided, exhaust all these cases.' The abstract, however, claims 'we characterise under which conditions the skin effect is absent' without specifying that this is only proven for real parameters. The paper should either limit the claim to the proven real-parameter case or provide a proof of necessity for the complex case; otherwise the abstract overstates the result.
minor comments (4)
  1. [Table I] In Table I, the condition 't2 = -t2' should read 't2 = -t1'.
  2. [Sec. VI] The phrase 'Therefor' should be 'Therefore'.
  3. [Sec. V] In Eq. (31), the parametrization x1 = s/κ, x2 = κ eiα, x3 = κ e−iα, x4 = 1/(sκ) would be clearer if it stated explicitly that α is real with 0 ≤ α < 2π, while κ and s are complex.
  4. [Sec. IV.B] The definitions of y1 and y2 in Eq. (27) are missing punctuation in the typeset display, making them ambiguous. Please add a comma or semicolon between the two definitions.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the Sec. IV.B branch caveat is a deductive gap, not a circular step.

full rationale

The paper's central derivation is self-contained. The eigenvalue characterization in the thermodynamic limit starts from the model Hamiltonian (1), the bulk equations (7)-(8), the determinant condition (30), and an ordering argument that forces |x2|=|x3|; this leads to the three Vieta equations (37) with the physical branch constraint 1/|s| ≤ |kappa|^2 ≤ |s|. No parameter in (37) is fitted from the eigenvalues it predicts, and the finite-L spectra in Figs. 1, 6 and 7 are independent checks, not inputs. The zero-mode criterion (29)/(54) is derived from the explicit lambda=0 solutions (18)-(21) and the boundary inequalities on |x_{±,±}|; it is an algebraic consequence, not an assumed target result. The only self-citation, Ref. [88], appears in Sec. III for the finite-L t1=t2 case and is explicitly described as repeating the calculation rather than importing a result; it is not load-bearing for the infinite-size, zero-mode, or skin-effect claims. The admitted sign ambiguity following Eq. (27), where the text says the sign choice 'does influence the range of parameters' but also says the x_{±,±} values 'might be swapped', is an omitted branch analysis that could affect the validity of the proof of Eq. (29); however, this is a rigor gap, not a case where a prediction is equivalent to an input by construction. The skin-effect analysis uses the Bistritz algorithm as an external, machine-checkable algebraic method and compares with the periodic eigenvalues (11) derived separately. Overall, no fitted input is renamed as a prediction, no load-bearing self-citation chain forces the conclusion, and no target result is assumed at the start of a derivation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted; the central results rest on standard algebraic tools and on two unproved genericity/necessity assumptions: the exclusion of special parameter values in the Vieta branch selection, and the claim that the complex-parameter skin-effect conditions are necessary. A textual contradiction about a sign choice in the zero-mode derivation is also listed.

assumptions (6)
  • standard math Vieta's formulas relate polynomial coefficients to roots (Sec. V).
    Used to derive the three equations (37) from the fourth-order bulk polynomial.
  • standard math Bistritz algorithm correctly counts roots inside, on, and outside the unit circle for polynomial P(x) (Sec. VI).
    Invoked to determine when two roots of the bulk equation lie on the unit circle, i.e., when the skin effect is absent.
  • domain assumption Open boundary conditions and the thermodynamic limit L→∞ define the model (Sec. II).
    The entire analysis targets open chains with free boundaries in the limit of infinite length; finite-size corrections are ignored except in numerical checks.
  • ad hoc to paper For generic parameters, the ten special λ values and degenerate cases (t2=-t1, d1d2=0, zero modes) can be excluded without losing eigenvalue branches (Sec. V).
    The authors state that these cases do not contribute generic eigenvalues, but no rigorous proof is given that all eigenvalue curves are captured by the |x2|=|x3| condition outside these exceptions.
  • ad hoc to paper For complex parameters, the sufficient conditions for absence of skin effect are also necessary (Sec. VI.B, Discussion).
    The paper says 'we believe that the conditions provided, exhaust all these cases', but necessity is not proven.
  • ad hoc to paper The sign choice in Eq. (27) does not change the zero-mode parameter region, despite the text saying 'does influence' (Sec. IV.B).
    The derivation of Eq. (29) relies on this assumption; the text is contradictory and needs clarification.

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Pith. "Pith review of On the non-hermitian Kitaev chain." pith.science (2026). https://pith.science/paper/6P6ZDS77

@misc{pith2026241114776,
  author       = {Pith},
  title        = {Pith review of: On the non-hermitian Kitaev chain},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6P6ZDS77}},
  note         = {Machine review of arXiv:2411.14776}
}
read the original abstract

We study the non-hermitian Kitaev chain model, for arbitrary complex parameters. In particular, we give a concise characterisation of the curves of eigenvalues in the complex plane in the infinite size limit, using a novel method which can be applied to other non-hermitian systems. Using this solution, we characterise under which conditions the skin effect is absent, and for which eigenstates this is the case. We also fully determine the region in parameter space for which the model has a zero mode.

Figures

Figures reproduced from arXiv: 2411.14776 by the authors.

Figure 1
Figure 1. FIG. 1: Plot for the non-hermitian Kitaev Chain with parameters [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The eigenvalues of the model with parameters [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The amplitude of eigenstate coefficients [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The eigenvalues for two sets of parameters. The colour coding is the same as in Fig. 2. The parameters for [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The eigenvalues for a set including complex parameters, namely [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Solutions for [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Finite size eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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    The blue squares (orange dots) correspond to the first (second) component of the eigenvector of a given site i. The left panel (a) corresponds to a real eigenvalue (λ ≈ 3.0182) showing skin effect (note the logarithmic scale), the right panel (b) corresponds to a purely imaginary eigenvalue (λ ≈ 4.3949i) without skin effect. T3(x) ≡ 0 reduces to |λ|2 sin(...

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    The gray lines correspond to the periodic case; the green dots correspond to the open finite chain with L = 100; the blue (red) lines correspond eigenvalues of the open infinite chain whose eigenstates do (do not) exhibit the skin effect. The eigenvalues along the imaginary axis extent to λ ≈ ±4.4495i. 13 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 0 20 40 60...

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    The black lines correspond to actual eigenvalues in the limit L → ∞. The different panels show the numerically obtained eigenvalues (using machine precision) for the sizes L = 100, L = 200, L = 400 and L = 800, showing the instability of the algorithm (green dots). By making use of the exact solution, we studied the presence of the skin effect. There exis...

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    Finally, T1(x) is given by N2N ∗ 2 T1(1) = 2(1 + x) h Re(λtdN ∗ 2 ) − Im(N2D∗ 2)2 Re(λtdD∗ 2) Re(λtdD∗ 2)2 + Im(mtsD∗ 2)2 i (A4) − 2i(1 − x) h Im(mtsN ∗ 2 ) + 4 Im(N2D∗

    − i Im(mtsD∗ 2) /(i Im(D2N ∗ 2 )) . Finally, T1(x) is given by N2N ∗ 2 T1(1) = 2(1 + x) h Re(λtdN ∗ 2 ) − Im(N2D∗ 2)2 Re(λtdD∗ 2) Re(λtdD∗ 2)2 + Im(mtsD∗ 2)2 i (A4) − 2i(1 − x) h Im(mtsN ∗ 2 ) + 4 Im(N2D∗

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    Re(λtdD∗ 2)2 Re(λtdD∗ 2)2 + Im(mtsD∗ 2)2 i , resulting in the following expression for T1(1), T1(1) = 4 N2N ∗ 2 Re(λtdN ∗ 2 ) − Im(D∗ 2N2)2 Re(λtdD∗ 2) Re(λtdD∗ 2)2 + Im(mtsD∗ 2)2 . (A5) Though we do not need it, we give an expression for the constant T0, in terms of T2(x) an T1(x), for completeness T0 = 2 Re(T2(0)/T1(0))T1(1) − T2(1) . (A6) 18

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.