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REVIEW 3 major objections 4 minor 104 references

Nonlocal Majorana polarization in non-Hermitian topological superconductors

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read In non-Hermitian superconductors, the biorthogonal nonlocal Majorana polarization is a reliable real-space indicator that takes the value -1 for true Majorana zero modes, near 0 for trivial zero-energy states, and ±∞ or NaN at exceptional p

desk verdict Useful extension of the Majorana polarization to non-Hermitian chains, but the central indicator is normalization-dependent as written. read the letter →

arxiv 2607.25424 v1 pith:HDLCTTK3 submitted 2026-07-28 cond-mat.supr-con cond-mat.mes-hall

Arjun S. Kumar , Jorge Cayao , Oladunjoye A. Awoga This is my paper
classification cond-mat.supr-concond-mat.mes-hall
keywords non-HermitiansuperconductorsMajoranazeromodespolarizationbiorthogonaleigenstatesexceptionalpointstrivialzero-energystatesKitaevchaintopologicalindicator
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

This paper extends the nonlocal Majorana polarization — a quantity that compares particle-hole expectation values at opposite halves of a wire — to non-Hermitian superconductors, where left and right eigenstates differ. The authors show that, computed with biorthogonal eigenvectors, this polarization takes the value -1 for genuine Majorana zero modes, approximately 0 for trivial zero-energy states, and ±∞ or NaN at exceptional points. They also introduce a 'sensitivity' R, the determinant of the four polarization components, which isolates the non-Hermitian contribution to the polarization. The diagnostic is demonstrated on minimal and long Hatano-Nelson-Kitaev chains, superconductor-normal junctions, and a non-Hermitian Rashba superconductor. A central finding is that non-Hermiticity (via nonreciprocal hopping) enhances the robustness of Majorana zero modes, a property captured by the polarization and its sensitivity.

What carries the argument

The central object is the biorthogonal nonlocal Majorana polarization: for each half of the chain one sums, over sites, the expectation value of the particle-hole operator C taken between a left (L) and right (R) eigenvector of the lowest-energy state (Eq. (1)), then multiplies the left-half and right-half sums (Eq. (3)). Because non-Hermitian Hamiltonians have distinct left and right eigenvectors, the off-diagonal components P^{LR}, P^{RL} carry the non-Hermitian information; the sensitivity R = det(P) then quantifies how much of the polarization is of non-Hermitian origin. The paper's main use is as a classifier: P^{LR}=-1 indicates a fully nonlocal, non-overlapping Majorana zero mode; P^{

What would settle it

Compute P^{LR} at the two-site HNK sweet spot t=sqrt(Γ²+Δ²) for a range of Γ using two different normalizations of the left eigenvectors (e.g., ⟨Ψ^L|Ψ^L⟩=1 versus ⟨Ψ^L|Ψ^R⟩=1); if P^{LR} does not stay exactly -1, the claimed universal value is a convention artifact rather than a property of the zero mode.

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Extended reading notes

Core claim

At the paper's core is the claim that the biorthogonal nonlocal Majorana polarization P^{ab} (a≠b), defined as the product of the particle-hole expectation values in the left and right halves of the system, functions as a topological indicator in non-Hermitian superconductors. In the sweet spot of the Hatano-Nelson-Kitaev model with zero chemical potential, fully nonlocal Majorana zero modes satisfy P^{ab}=-1 when Γ≠0 and a≠b, while the orthogonal components P^{aa} deviate from -1; the paper also provides analytic expressions for two- and three-site minimal chains showing that non-Hermiticity drives P^{LR} and R toward -1, restoring Majorana robustness degraded by a chemical potential. Away

Load-bearing premise

The load-bearing premise is that the off-diagonal polarization P^{LR}, computed with a particular normalization of the left eigenvectors, is a genuine state property whose value -1 for Majorana zero modes is independent of that normalization; if the value changes when the left eigenvectors are rescaled, the classification would be an artifact of the convention.

Editorial extensions

If this is right

  • In the sweet spot of the HNK model, fully nonlocal MZMs are characterized by P^{ab}=-1 (a≠b) for any finite Γ, making the biorthogonal component a direct real-space signature of non-Hermitian Majorana physics.
  • The sensitivity R detects the proportion of non-Hermitian contribution to the Majorana polarization, with R=-1 in the pure non-Hermitian limit and R=0 in the Hermitian limit; this gives a quantitative measure of how much non-Hermiticity stabilizes a zero mode.
  • Non-Hermiticity restores Majorana zero modes degraded by a finite chemical potential in few-site Kitaev chains (two- and three-site), as captured by P^{LR} and R approaching -1 for large Γ.
  • In superconductor-normal junctions, P^{LR} distinguishes MZMs from spatially confined trivial zero-energy states that appear in the normal region, while exceptional points produce a distinct divergent or NaN signature.
  • In the non-Hermitian Rashba superconductor, the topological critical field is renormalized to sqrt(μ²+4(αΓ/t)²+Δ²), and nonreciprocity lowers the Majorana splitting and drives P and R toward -1.

Reading between the lines

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

  • If P^{LR}=-1 is indeed universal for non-overlapping Majorana zero modes under any form of non-Hermiticity (not just nonreciprocal hopping), the indicator could also diagnose Majorana modes in open-driven or dissipative systems where biorthogonality is the natural language—a direction the paper does not pursue.
  • The analytic treatment covers only minimal few-site chains; the numerics suggest the -1 signature persists for longer chains, so testing the diagnostic on disordered or multi-channel wires would be a natural next step.
  • Because the polarization is built from the lowest-energy eigenvector only, it may be possible to convert it into a post-selection-free measurement if the particle-hole expectation can be read out via local density-of-states or tunneling probes; the paper does not address the measurement protocol but the definition points to local observables.
  • The denominator in the polarization uses the diagonal overlap rather than the biorthogonal overlap; checking whether the -1 value is invariant under left-eigenvector normalization would test how robust the claimed universal value is.
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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 manuscript generalizes the nonlocal Majorana polarization (MP) to non-Hermitian topological superconductors. It defines biorthogonal MP components P^{ab} (a,b=L,R) in Eqs. (1)-(3), a nonlocal MP sensitivity R in Eqs. (4)-(5), and applies these to Hatano-Nelson-Kitaev chains (few-site, 16-site, and SN junctions) and to non-Hermitian Rashba chains. The main claim is that the off-diagonal component P^{LR} equals -1 for fully nonlocal Majorana zero modes, is approximately 0 for trivial zero-energy states, and becomes ±∞ or NaN at exceptional points; it also claims that non-Hermiticity enhances MZM robustness. Analytic few-site expressions are provided for HNK sweet spots.

Significance. If the proposed indicator were well defined, it would be a useful addition to the non-Hermitian Majorana toolbox, and the numerical exploration across several models is a positive feature. The paper also reproduces the Hermitian limit and provides analytic few-site expressions, which are helpful. However, the central observable P^{LR} is not invariant under the normalization freedom of left and right eigenvectors, so the quantitative signature P^{LR}=-1 is not yet established as a state property. The determinant R is better behaved under the reciprocal rescaling, but the paper's primary claims rely on P^{LR}.

major comments (3)
  1. [Sec. II, Eq. (1)] The off-diagonal MP in Eq. (1) is not invariant under the normalization freedom left by the only stated condition, biorthogonality ⟨Ψ^L_n|Ψ^R_m⟩=δ_nm. If |Ψ^R⟩→λ|Ψ^R⟩ and |Ψ^L⟩→λ^{-1}|Ψ^L⟩, the biorthogonality condition is preserved, but p_j^{LR}→λ²p_j^{LR} and therefore P^{LR}→|λ|^4P^{LR}. A real rescaling λ=2 changes the claimed value -1 to -16. The determinant R in Eq. (4) is invariant under this coupled rescaling, but the paper's primary diagnostic and the analytic values in Eqs. (7)-(9) are not. The text never states how left and right eigenvectors are normalized in the numerics or in the analytic derivations. Please specify a normalization convention or replace Eq. (1) by the standard biorthogonal expectation value ⟨L|C|R⟩/⟨L|R⟩, and recompute the affected results.
  2. [Sec. III A, Eqs. (7)-(9)] The two-site and three-site sweet-spot conditions in Sec. III A (t=√(Γ²+Δ²+μ²) and t=√(Γ²+Δ²+μ²/2), and their μ=0 limit) are asserted without derivation, and the resulting exact forms in Eqs. (7)-(9) are used as the analytic anchor for the numerical phase diagrams. A derivation of these conditions and the stated eigenvector normalizations should be included (or, at minimum, an explicit verification that the Hamiltonian has a zero eigenvalue at these parameters). The same applies to Eq. (11) for the Rashba critical field.
  3. [Sec. III A and Sec. V] The conclusion that 'non-Hermiticity enhances MZM robustness' is inferred from P^{LR} moving toward -1 and P^{aa} toward 0 as Γ increases. Since P^{LR} is normalization-dependent, this inference is not yet supported: a different normalization can produce the same trend by construction. Please base the robustness claim on a normalization-independent quantity (e.g., R, or a separately defined invariant) and state the criterion explicitly.
minor comments (4)
  1. [Abstract; Fig. 2 caption] Typographical errors: 'sensitiviy' in the abstract, 'biothorgonal' in the Fig. 2 caption, and 'Hemitian' elsewhere.
  2. [Eq. (3)] The identity P^{ab}=P_l^{ab}(P_r^{ab})^*=(P_l^{ab})^*P_r^{ab} holds only when the product is real. Please clarify whether P_l^{ab} and P_r^{ab} are real by construction, or define P^{ab} with a definite ordering.
  3. [Sec. III B and footnote [94]] The text says P^{LR} jumps around Γ/t=1, and footnote [94] says the value is shown as a jump to zero in Fig. 1(c,d) for clarity, but the relevant panels are in Fig. 3. This is confusing and should be corrected.
  4. [Sec. II, Eq. (5)] The phrase 'aa=ab contributions (Hermitian)' in the R-mapping is vague. Please spell out the conditions under which R=0 in terms of the actual MP matrix entries, especially since the off-diagonal entries can be normalization-dependent.

Circularity Check

1 steps flagged · score 4.0 of 10

The headline signature P^{LR}=-1 is not invariant under the only biorthogonality normalization stated; the central 'MZM detector' value is a normalization artifact, though R and the spectral evidence provide independent content.

  1. other [Eq. (1) and Eq. (7); Sections II and III.A]
    "we define the MP at each site j as the expectation value of the particle-hole operator as p^{ab}_j = Σ_σ ⟨Ψ^a_{jσ}|C|Ψ^b_{jσ}⟩ / Σ_σ ⟨Ψ^a_{jσ}|Ψ^a_{jσ}⟩ , (1) ... fully nonlocal MZMs localized at each edge are characterized by P^{ab}=−1 in the NH case when Γ≠0, provided a≠b."

    The only normalization stated is biorthogonality, ⟨Ψ^L_{n'}|Ψ^R_n⟩=δ_{n'n}. For a real rescaling λ, |Ψ^R⟩→λ|Ψ^R⟩ and |Ψ^L⟩→λ^{-1}|Ψ^L⟩ preserve this condition. Under that rescaling, the numerator of p^{LR}_j in Eq. (1) is unchanged while the denominator ⟨Ψ^L|Ψ^L⟩ scales as λ^{-2}, so p^{LR}_j→λ^2 p^{LR}_j. Since P^{LR} is the product of the left- and right-half sums, it scales as λ^4. Hence the exact value −1 in Eq. (7) can be turned into any positive multiple (e.g. −1/16 or −16) by an allowed choice of left-vector normalization. The paper never fixes this gauge, so the claimed signature of a robust nonlocal MZM is not a Hamiltonian-determined state property; it is an artifact of the arbitrary denominator in Eq. (1).

full rationale

The paper is mostly an application of an existing real-space indicator to specific non-Hermitian models, and most of the concrete calculations — sweet-spot conditions, two- and three-site analytic expressions, re-entrant phase diagrams, SN junctions, and the Rashba critical field — are not fitted to the desired classification. Ref. [9] is a self-citation by two of the present authors, but the analytic expressions and direct spectral comparisons give independent content, so the self-citation alone is not load-bearing circularity. The serious issue is different: Eq. (1) defines the biorthogonal MP with a denominator ⟨Ψ^L|Ψ^L⟩ while the only stated normalization is the biorthogonal overlap ⟨Ψ^L|Ψ^R⟩=δ. That leaves P^{LR}, and hence the central assertion P^{LR}=−1 for fully nonlocal MZMs, dependent on an unspecified rescaling of the left and right eigenvectors. This makes the headline quantitative prediction partially definitional rather than a robust derived invariant. The determinant R is better behaved (P^{LR}P^{RL} is gauge-invariant), and the energy spectra provide independent support, so the circularity is partial rather than total.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The central claims rest on the biorthogonal convention, unproved sweet-spot conditions, and an ad hoc determinant R; no data are fitted and no new physical entities are introduced. Model Hamiltonians are from previous literature.

assumptions (5)
  • domain assumption Non-Hermitian BdG Hamiltonians are diagonalizable and biorthogonal eigenstates exist away from exceptional points.
    Used in Sec. II to define P^{ab}; fails at EPs, where the paper switches to ±∞/NaN.
  • ad hoc to paper The biorthogonal normalization ⟨Ψ^L_n|Ψ^R_m⟩=δ_nm fixes the scale of left eigenvectors, making P^{LR} in Eq. (1) well-defined.
    The denominator uses ⟨Ψ^a|Ψ^a⟩, so P^{LR} depends on the left-vector normalization; the paper does not discuss invariance.
  • domain assumption The Hermitian interpretation P^{aa}=-1 ↔ robust nonlocal MZMs (Ref [9]) carries over to the NH biorthogonal components P^{LR}.
    Central inference of Secs. III-IV; no independent topological index proof.
  • ad hoc to paper Sweet-spot conditions (e.g., t=√(Γ²+Δ²+μ²) for two-site HNK, t=√(Γ²+Δ²+μ²/2) for three-site) produce exact zero-energy MZMs.
    Stated in Sec. III A without derivation; used to produce Eqs. (7)-(9).
  • ad hoc to paper The determinant R with regime mapping Eq. (5) captures the fraction of non-Hermitian contribution to MP.
    Defined in Eq. (4); the mapping to 'pure Hermitian' (0) and 'pure non-Hermitian' (-1) regimes is asserted, not derived from pseudospectrum theory.
invented entities (2)
  • Nonlocal biorthogonal Majorana polarization P^{ab} (a≠b)
    purpose: Topological diagnostic for NH superconductors
    Generalizes Hermitian MP from Ref [9]; no external falsifiable prediction beyond its role in the models studied.
  • Nonlocal MP sensitivity R
    purpose: Quantify the non-Hermitian contribution to MP and flag exceptional points
    Defined as determinant of MP matrix (Eq. 4); regime taxonomy Eq. (5) is asserted; no independent measurement predicted.

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Pith. "Pith review of Nonlocal Majorana polarization in non-Hermitian topological superconductors." pith.science (2026). https://pith.science/paper/HDLCTTK3

@misc{pith2026260725424,
  author       = {Pith},
  title        = {Pith review of: Nonlocal Majorana polarization in non-Hermitian topological superconductors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HDLCTTK3}},
  note         = {Machine review of arXiv:2607.25424}
}
read the original abstract

The nonlocal Majorana polarization, defined as the product of the expectation values of the particle-hole operator at opposite halves of the system, has been shown to be a reliable topological indicator that determines the presence and quality of Majorana zero modes in Hermitian topological superconducting setups. In this work, we extend the concept of nonlocal Majorana polarization to the non-Hermitian realm by taking into account the biorthogonal eigenstates and demonstrate its utility by exploring distinct non-Hermitian superconducting systems. In particular, we show that the Majorana polarization can distinguish between Majorana zero modes, trivial zero-energy states, and exceptional points in non-Hermitian superconductors. Also, we introduce the concept of nonlocal Majorana polarization sensitiviy for characterizing the contribution of non-Hermiticity to Majorana polarization. As a byproduct, we find that non-Hermiticity enhances Majorana zero modes robustness, a property captured by the nonlocal Majorana polarization.

Figures

Figures reproduced from arXiv: 2607.25424 by the authors.

Figure 1
Figure 1. Schematics of NH superconducting chains stud [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. (a-e) Nonlocal MP biothorgonal indicators ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Nonlocal MP indicators and spectral features of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Spectrum and nonlocal MP indicators in HNK junc [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Topology of zero-energy states in the NH Rashba [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Trivial and nontrivial zero-energy states in SN junc [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.