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arxiv: 2606.10418 · v1 · pith:XT6IE6YYnew · submitted 2026-06-09 · ❄️ cond-mat.mes-hall

Higher-winding phases in one-dimensional non-Hermitian topological superconductors

Yung-Yeh Chang , Xiang-Yu Li , Ken Shiozaki , Chen-Hsuan Hsu This is my paper

Pith reviewed 2026-06-27 12:22 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords non-Hermitian topological superconductorspoint-gap topologywinding numbersMajorana zero modesnon-Hermitian skin effectphase boundarieslonger-range hoppings
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0 comments X

The pith

A coefficient-based method produces analytical phase boundaries for higher-winding phases in one-dimensional non-Hermitian topological superconductors.

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

The paper introduces a coefficient-based approach that extracts winding numbers directly from the polynomial form of the Bloch Hamiltonian. This yields closed-form expressions for the boundaries between phases carrying different integer winding numbers under point-gap topology. The method is applied to lattice models that include longer-range hoppings and optional sublattice structure, thereby reaching phases with winding numbers larger than one. Weak perturbations are shown to remove the non-Hermitian skin effect while leaving the sublattice-protected invariant intact, so that zero-energy boundary modes survive. Open-boundary spectra and inverse-participation-ratio calculations confirm that the bulk winding number correctly counts the number of protected modes.

Core claim

The coefficient-based approach enables derivation of analytical expressions for phase boundaries in higher-winding phases of 1D non-Hermitian topological superconductors characterized by point-gap topology with Z invariants. Weak perturbations suppress the skin effect while preserving the sublattice-symmetry-protected invariant associated with Majorana zero modes. The predicted winding numbers are verified by open-boundary spectra, where one or multiple pairs of zero-energy boundary modes appear consistently with the bulk invariant.

What carries the argument

The coefficient-based approach that reads winding numbers from the coefficients of the characteristic polynomial of the Bloch Hamiltonian.

If this is right

  • Longer-range hoppings generate higher-order polynomials and therefore phases with winding numbers greater than one.
  • Analytical phase diagrams can be constructed for both sublattice and non-sublattice models without numerical diagonalization of the bulk Hamiltonian.
  • One or more pairs of zero-energy boundary modes appear at open ends exactly as dictated by the bulk winding number.
  • A weak perturbation can eliminate the skin effect while the symmetry-protected invariant and its associated boundary modes remain.
  • The inverse participation ratio of the boundary modes stays low under onsite disorder, confirming their topological protection.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same coefficient extraction may be applied to other point-gapped non-Hermitian chains whose characteristic polynomials are accessible.
  • Device designs could deliberately add weak terms to suppress skin localization while retaining Majorana-mode counting.
  • The method supplies a direct route to the Hermitian limit by continuously dialing the non-Hermitian parameters to zero.

Load-bearing premise

The winding number obtained from polynomial coefficients continues to serve as the correct bulk invariant that fixes the number of zero-energy boundary modes after longer-range hoppings and weak perturbations are introduced.

What would settle it

An open-boundary calculation for a model whose polynomial coefficients predict winding number two that instead shows zero or one pair of zero-energy modes would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.10418 by Chen-Hsuan Hsu, Ken Shiozaki, Xiang-Yu Li, Yung-Yeh Chang.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic illustration of the 1D lattice model described by [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Energy spectra under PBC (gray curves) and OBC (brown [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Spatial density profiles for the particle ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Evolution of OBC (colored) spectra for different disorder strengths [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Energy spectra under PBC (gray curves) and OBC (brown [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a–d) Phase diagrams characterized by the subsystem and composite winding numbers, [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. (a)–(h) Phase diagrams in the ( [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Similar plots to Fig. 5, but with different parameter sets; points A, B, D, and E correspond to the parameter sets indicated in Fig. 9(c). [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
read the original abstract

Non-Hermitian topological superconductors provide a setting in which point-gap topology, non-Hermitian skin effects, and Majorana zero modes are strongly intertwined. In this work, we adopt a coefficient-based approach for computing winding numbers and deriving analytical expressions for phase boundaries in one-dimensional non-Hermitian topological superconductors characterized by point-gap topology with $\mathbb{Z}$ invariants. We apply this approach to two non-Hermitian topological superconducting lattice models, with and without sublattice degrees of freedom, including longer-range hoppings, thereby accessing a much broader parameter space. These extensions generate higher-order polynomials and support phases with higher winding numbers, reflecting the underlying $\mathbb{Z}$ topology. We further clarify how a weak perturbation suppresses the non-Hermitian skin effect while preserving the sublattice-symmetry-protected invariant associated with Majorana zero modes. The predicted winding numbers are verified by open-boundary spectra, where one or multiple pairs of zero-energy boundary modes appear consistently with the bulk invariant. We also examine the stability of these modes against onsite disorder through the inverse participation ratio. Our results provide a systematic and efficient route to constructing topological phase diagrams for higher-winding non-Hermitian topological superconductors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a coefficient-based method to extract Z winding numbers from the characteristic polynomial of 1D non-Hermitian topological superconductors with point-gap topology. It applies the method to two lattice models (with and without sublattice symmetry) that include longer-range hoppings, thereby generating higher-degree polynomials and phases with higher winding numbers, and derives analytical expressions for the corresponding phase boundaries. A weak perturbation is introduced that suppresses the non-Hermitian skin effect while preserving the sublattice-symmetry-protected invariant tied to Majorana zero modes. The predicted winding numbers are stated to be confirmed by open-boundary spectra (showing the expected number of zero-energy boundary modes) and by inverse-participation-ratio analysis under onsite disorder.

Significance. If the coefficient-based extraction is shown to be equivalent to the standard point-gap winding number (including under longer-range terms and the added perturbation), the work supplies an efficient route to analytical phase diagrams for higher-winding non-Hermitian superconductors. The explicit treatment of how a weak perturbation decouples the skin effect from the sublattice-protected Majorana modes is a useful clarification of the interplay between non-Hermitian topology and superconductivity.

major comments (2)
  1. [Section introducing the coefficient-based approach (likely §2 or §3)] The central claim that the winding number read directly from the coefficients of the characteristic polynomial equals the bulk point-gap invariant (and therefore correctly predicts the number of protected zero-energy boundary modes) is load-bearing for all higher-winding results. No derivation is supplied showing that this coefficient extraction coincides with the standard definition involving the complex-energy contour or the biorthogonal inner product, especially once the polynomial degree increases with longer-range hoppings. This equivalence must be demonstrated explicitly before the method can be used to assert higher-winding phases.
  2. [Section on weak perturbation and skin-effect suppression] In the discussion of the weak perturbation that suppresses the skin effect, the manuscript asserts that the sublattice-symmetry-protected invariant (and therefore the coefficient-derived winding number) remains unchanged. However, because the perturbation is added after the polynomial is constructed, an explicit check is required that the point gap and the topological invariant are unaffected; the open-boundary spectra alone do not suffice if the biorthogonal definition of the invariant is not recomputed.
minor comments (2)
  1. [Verification sections] The abstract and main text refer to “open-boundary spectra” and “inverse-participation-ratio checks” as verification, but no tables of winding numbers versus mode counts, no error bars, and no representative data for the higher-winding cases are mentioned; inclusion of such quantitative comparisons would strengthen the verification statements.
  2. [Method section] Notation for the characteristic polynomial and the coefficient extraction rule should be stated once with an explicit formula (e.g., Eq. (X)) rather than described only in prose, to allow readers to reproduce the winding-number calculation for arbitrary longer-range models.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments. We address the major comments point by point below, acknowledging where explicit demonstrations are needed, and will revise the manuscript to incorporate the required derivations and checks.

read point-by-point responses

  1. Referee: The central claim that the winding number read directly from the coefficients of the characteristic polynomial equals the bulk point-gap invariant (and therefore correctly predicts the number of protected zero-energy boundary modes) is load-bearing for all higher-winding results. No derivation is supplied showing that this coefficient extraction coincides with the standard definition involving the complex-energy contour or the biorthogonal inner product, especially once the polynomial degree increases with longer-range hoppings. This equivalence must be demonstrated explicitly before the method can be used to assert higher-winding phases.

    Authors: We acknowledge that an explicit derivation establishing equivalence between the coefficient-based winding number and the standard point-gap definition (via contour integral or biorthogonal inner product) is absent from the manuscript, particularly for higher-degree polynomials. While our numerical verifications with open-boundary spectra are consistent, a general proof is required. In the revised manuscript we will add a dedicated derivation in the section on the coefficient-based approach, showing that the coefficient extraction yields the same integer as the standard definition for arbitrary polynomial degree. revision: yes

  2. Referee: In the discussion of the weak perturbation that suppresses the skin effect, the manuscript asserts that the sublattice-symmetry-protected invariant (and therefore the coefficient-derived winding number) remains unchanged. However, because the perturbation is added after the polynomial is constructed, an explicit check is required that the point gap and the topological invariant are unaffected; the open-boundary spectra alone do not suffice if the biorthogonal definition of the invariant is not recomputed.

    Authors: We agree that an explicit recomputation of the biorthogonal winding number (and confirmation that the point gap remains open) is needed after introducing the perturbation. In the revised manuscript we will include this calculation, demonstrating that the invariant is unchanged, and present it together with the existing open-boundary spectra. revision: yes

Circularity Check

0 steps flagged

No circularity: coefficient winding numbers are computed from polynomials and independently verified by open-boundary spectra

full rationale

The paper computes winding numbers via a coefficient-based method applied to the characteristic polynomial of the non-Hermitian Hamiltonian, then verifies the resulting bulk invariants against explicit open-boundary spectra that count zero-energy modes. This verification step is external to the coefficient extraction itself. No self-citation chain, fitted-parameter renaming, or self-definitional loop is indicated in the abstract or skeptic summary; the method is presented as a computational shortcut whose output is cross-checked numerically rather than asserted by construction. The central bulk-boundary claim therefore retains independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no explicit list of fitted parameters, background axioms, or new postulated entities; full manuscript required for complete ledger.

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    eq:eigenstate_PHS_TRSd

    note In this footnote, we derive the first relation in Eq. eq:eigenstate_PHS_TRSd . From U_ C_- H_ NHTSC ^T U_ C_- ^ -1 =-H_ NHTSC , we obtain U_ C_- H_ NHTSC ^T=-H_ NHTSC U_ C_- , where H_ NHTSC H^ obc _ NHTSC +H_ pt . Acting on |E \! ^* , which satisfies H_ NHTSC ^T|E \! ^*=E|E \! ^* , gives U_ C_- H_ NHTSC ^T|E \! ^* = E U_ C_- |E \! ^* = -H_ NHTSC U_ ...

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    work page doi:10.5281/zenodo.20252705

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    note The theorem states that if two analytic functions \(f_1(z)\) and \(f_2(z)\) satisfy |f_2(z)|<|f_1(z)| on a closed contour, then \(f_1(z)\) and \(f_1(z)+f_2(z)\) have the same number of zeros inside the contour. Stop

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    @noop note The file containing the analytical expressions for the phase boundaries is available in the Zenodo record: https://doi.org/10.5281/zenodo.20292902. Stop

    work page doi:10.5281/zenodo.20292902