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REVIEW 3 major objections 6 minor 51 references

Investigating topological in-gap states in non-Hermitian quasicrystal with unconventional $p$-wave pairing

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read In a non-Hermitian Aubry-André-Harper chain with p-wave pairing, asymmetric hopping replaces Majorana zero modes with central and bulk in-gap edge states that the authors argue are topologically protected and resistant to disorder.

desk verdict Genuine new model combination and a plausible triple transition at weak pairing, but the topological protection claim for the in-gap states rests on a PBC invariant in a skin-effect regime and a trivial disorder test. read the letter →

arxiv 2501.14481 v1 pith:OE35CNWN submitted 2025-01-24 cond-mat.dis-nn quant-ph

Shaina Gandhi , Jayendra N. Bandyopadhyay This is my paper · ORCID
classification cond-mat.dis-nnquant-ph
keywords non-HermitianquasicrystalAubry-André-Harpermodelp-wavepairingin-gapstatestopologicalphasetransitionmetal-insulatorreal-to-complexskineffect
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 a one-dimensional non-Hermitian Aubry-André-Harper model with p-wave superconducting pairing and two sources of non-Hermiticity: a complex on-site quasiperiodic potential and asymmetric hopping. It claims that at weak pairing strength the system undergoes a triple phase transition in which a topological transition, a metal-insulator transition, and a transition from real to complex energy eigenvalues coincide at the same boundary. Under open boundary conditions, asymmetric hopping eliminates the Majorana zero modes seen in the symmetric-hopping limit and instead produces two kinds of in-gap states, one centered in the spectral gap and one inside a bulk gap, both localized at the edges. The paper reports that these states survive disorder strengths up to near the size of the superconducting gap and interprets this robustness as topological protection, proposing the states as more accessible resources for topological quantum computation than Majorana modes.

What carries the argument

The load-bearing object is the periodic-boundary winding number of Eq. (6), w(h1,h2)=(1/2πi)∫ dθ ∂_θ ln det[H_BdG(θ/2L,h1,h2)-E_B] with E_B=0, which counts how many times the complex spectral trajectory encircles the base energy as the phase θ winds around. This is paired with two diagnostics: the averaged generalized fractal dimension D2 from Eq. (7) for the metal-insulator transition, and the largest imaginary part of the energy eigenvalues for the real-to-complex transition. The coincidence of the boundaries of all three quantities at weak pairing is the triple phase transition, and the in-gap states are characterized through open-boundary spectra and their wavefunction localizations.

What would settle it

Compute the open-boundary spectrum using the generalized Brillouin zone (replacing the real quasimomentum with a complex one) and check whether the bulk in-gap state survives and whether it shifts to the left side when h2 changes sign; if the state disappears in the non-Bloch spectrum or follows the hopping bias, the topological-protection claim is refuted.

Watch

Extended reading notes

Core claim

The central claim is that the phase diagram of this model in the plane of the two non-Hermitian parameters, for weak pairing Δ=0.01, has a single boundary where three quantities change simultaneously: the periodic-boundary winding number w computed at base energy E_B=0 jumps between trivial and non-trivial values, the averaged generalized fractal dimension D2 crosses from delocalized to critical or localized, and the largest imaginary part of the energy eigenvalues becomes nonzero. When the hopping asymmetry h2 is switched on, the open-boundary spectrum no longer hosts Majorana zero modes; instead the authors find two central in-gap states, each localized at one end of the chain, and one bulk in-gap state localized at the right end. They report that these states remain intact under disorder strengths up to ξ=0.35, compared with a superconducting gap of roughly 0.4, and they conclude on that basis that the in-gap states are topologically protected.

Load-bearing premise

The topological-protection argument assumes that the winding number computed under periodic boundary conditions at base energy zero correctly counts the open-boundary in-gap states, even though the model's asymmetric hopping is expected to produce non-Hermitian skin effects that require a generalized-Brillouin-zone treatment.

Editorial extensions

If this is right

  • At weak pairing, crossing the single boundary changes topological, localization, and spectral-reality properties at once, so tuning one parameter controls all three.
  • The disorder-robust in-gap states provide a candidate platform for topological quantum computation that avoids the strict experimental conditions needed for Majorana zero modes.
  • Because the in-gap states appear only under open boundary conditions, the paper's results confirm that they are edge states tied to the system's boundaries.
  • For stronger pairing, the three transitions no longer coincide and the analytic phase boundaries fail, indicating that asymmetric hopping and pairing strength compete in determining the phases.
  • The re-entrant real-energy region seen at Δ=1.5 and 2.0 shows that the real-to-complex transition is not monotonic in pairing strength.

Reading between the lines

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

  • Editorial inference: the bulk in-gap state's localization at only the right end is a signature of the non-Hermitian skin effect; whether it is a topologically protected edge mode rather than a skin mode has not been established because the paper does not compute a generalized-Brillouin-zone invariant.
  • Editorial inference: a direct test would be to reverse the sign of h2; a skin mode would migrate to the left end, whereas a topological edge mode would remain pinned by the topological invariant.
  • Editorial inference: the disorder robustness could also arise from boundary pinning by non-Hermitian pumping rather than from a bulk topological invariant; computing biorthogonal polarization would distinguish the two.
  • Editorial inference: the proposed quantum-computational platform is plausible only if the in-gap states are shown to be non-Abelian or otherwise manipulable, which this paper does not demonstrate.
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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 / 6 minor

Summary. The manuscript studies a one-dimensional non-Hermitian Aubry-André-Harper model with p-wave pairing, complex on-site quasiperiodic potential (h1), and asymmetric hopping (h2). It reports a triple phase transition for weak pairing Δ=0.01, at which a topological transition, a metal-insulator transition, and an unconventional real-to-complex spectral transition coincide, based on numerical PBC phase diagrams. Under OBC, the asymmetric hopping is claimed to replace the Majorana zero modes of the symmetric-hopping limit with two kinds of in-gap states (central and bulk), which are reported to persist under multiplicative disorder and are interpreted as topologically protected and promising for topological quantum computation.

Significance. If established, the triple phase transition would be an interesting interplay effect, and a non-Majorana route to topologically protected in-gap states would be a useful alternative to MZMs. The numerical work is transparent, the paper honestly reports that the analytic phase-boundary formulas fail for Δ=0.1 and 0.5, and no parameter fitting is used. However, the central topological-protection claim is not established by the present evidence: the only topological invariant is a PBC winding number, and the disorder test, as implemented, is degenerate. The significance therefore remains conditional on a non-Bloch bulk-boundary analysis and a meaningful disorder study.

major comments (3)
  1. [Section IV, Eq. (6)] The OBC in-gap states in Figs. 2-4 are labeled topologically protected solely through the PBC winding number w(h1,h2) computed in Eq. (6) around EB=0. Since the hopping in Eq. (1) is asymmetric (e^{±h2}), the model generically has a non-Hermitian skin effect, and Refs. [6-8] on non-Bloch bulk-boundary correspondence are cited but never applied. The bulk in-gap state in Fig. 2(e) is localized only at the right end, which is the hallmark of a skin mode, and a non-zero PBC winding number does not by itself imply OBC topological edge modes when the generalized Brillouin zone is not the unit circle. The absence of these states under PBC, reported in Appendix B, is consistent with skin modes as well as with topological edge modes. A non-Bloch winding number or biorthogonal polarization, together with a justification of EB=0 for finite-energy states, is needed before the phrase 'topologically protected' can be used.
  2. [Section V] The disorder implementation in Sec. V is not a disorder test. The text replaces V, t, and Δ by rV, rt, and rΔ using the same random factor r, so for every realization H_dis = rH. This only rescales the entire spectrum and leaves all eigenstates unchanged, so the persistence of the in-gap states in Fig. 5 is tautological and cannot distinguish topologically protected states from any other state. The conclusion that the in-gap states are 'robust against disorder' is therefore unsupported; independent or site-dependent disorder on the potential, hopping, and pairing terms is required.
  3. [Section III] The coincidence claim for the triple phase transition at Δ=0.01 is supported numerically, but the analytical phase boundaries are asserted by substituting t = t e^{-h2} into formulas from Refs. [36,40] rather than derived for the present Hamiltonian. The paper honestly reports that these formulas fail for Δ=0.1 and 0.5, but the origin of the failure is not discussed, and the validity at Δ=0.01 rests entirely on the numerical comparison. This is not fatal for the numerical claim, but it should be stated more cautiously, and a derivation or at least a discussion of the breakdown would strengthen the paper.
minor comments (6)
  1. [Section III] The text says the winding number takes values w=0, 0.5, and 1, while the caption of Fig. 1(a) says the non-trivial regions have w=-0.5 and -1; the sign convention should be reconciled.
  2. [Eq. (6)] The notation H_BdG(θ/(2L), h1, h2) in Eq. (6) is not defined; please state explicitly how the phase θ enters the BdG Hamiltonian and whether this is a Peierls-type twist.
  3. [Appendix B] The text referring to 'Figure 2(d) presents the imaginary part of the energy eigenvalues' should refer to Fig. 9(d), which is the panel actually showing the imaginary part.
  4. [Section III] In the paragraph on Fig. 1(b), the sentence 'the localized region corresponds to w=-1' appears to be a typo; Fig. 1(b) shows the fractal dimension, while the winding number is shown in Fig. 1(a).
  5. [Section V] The statement that the disorder strengths are chosen so as to remain within the superconducting gap 'which is approximately 0.4' needs clarification, since ξ=0.35 is close to that value; please specify whether the relevant scale is the gap, the in-gap state separation, or something else.
  6. [Section IV] The captions of Figs. 2-4 do not state what the color scale represents in the spectrum panels; the reader is left to infer that it is the fractal dimension D2.

Circularity Check

1 steps flagged · score 6.0 of 10

The disorder-robustness claim for the in-gap states is tautological: the 'disorder' multiplies every Hamiltonian term by the same global factor r, so H_dis = r H and the eigenstates cannot change.

  1. self definitional [Section V, 'Effect of disorder on in-gap states' (definition of r and Fig. 5)]
    "Specifically, we introduce the random disorder factor r, where r = 1 + δr, and δr is a uniformly distributed random variable within the range [−ξ, ξ]... Thus, the onsite potential V, hopping strength t and the pairing strength Δ in the original Hamiltonian is replaced by Vdisordered = rV , tdisordered = rt and Δdisordered = rΔ... we observe that the central and bulk in-gap states remain largely unaffected... we conclude that the central and bulk in-gap states observed in the eigenspectra are robust against disorder and are topologically protected."

    Because the same global factor r multiplies every term of the Hamiltonian — on-site potential, both hopping amplitudes, and pairing — the disordered Hamiltonian is exactly r times the clean Hamiltonian: H_disordered = r H_original. Every eigenvector of the clean model is therefore an eigenvector of the disordered model, with eigenvalues merely rescaled by r. The persistence of the central and bulk in-gap states under this 'disorder' is guaranteed by the definition of the disorder model; it is an algebraic identity, not a numerical test of topological protection. The robustness claim, which is one of the paper's central conclusions, reduces by construction to the way the disorder perturbation was defined.

full rationale

The paper's main derivation — the triple phase transition — is computed directly from the Hamiltonian: the winding number (Eq. 6), the generalized fractal dimension (Eq. 7), and the largest imaginary part are all evaluated from the BdG kernel, and the analytic phase boundaries are imported from external Refs. [36,40]. There is no fitted parameter renamed as a prediction. The self-citations (Refs. [19,32,51]) describe prior related models and are not used as uniqueness theorems or to forbid alternative explanations; the re-entrant real-region statement in Appendix A is also reproduced numerically in Fig. 8, so Ref. [51] is not load-bearing. The one genuine circular step is in Sec. V: the 'disorder' multiplies all three Hamiltonian parameters (V, t, Δ) by the same global random factor r, making H_disordered = r H_original exactly. The in-gap states are then trivially unchanged, so the observation that they are 'robust against disorder' cannot support topological protection; it is a self-definitional reduction of the robustness claim. The separate concern that a PBC winding number around EB=0 is used to label OBC in-gap states in a model with asymmetric hopping (where the generalized Brillouin zone is not the unit circle) is a correctness risk, not a circularity, and is not scored here.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The model introduces no fitted constants; t and V are set to 1 as normalizations, and h1, h2, and Δ are scanned. The main assumptions are the applicability of the PBC winding number to OBC edge states, the transfer of phase-boundary formulas from earlier p-wave AAH papers, the sufficiency of L=377 for the D2 phase classification, and the disorder protocol as a meaningful robustness test. No new physical entities are postulated.

assumptions (4)
  • domain assumption The PBC winding number in Eq. (6) is a valid topological invariant for the OBC in-gap states.
    Used to draw the topological phase diagram, while the in-gap states are studied under OBC. For non-Hermitian systems with asymmetric hopping, non-Bloch corrections are generally required; the paper does not apply them.
  • ad hoc to paper Phase boundaries from Refs [36,40] remain valid when the hopping is replaced by t e^{-h2} and a weak Δ term is added.
    The boundaries in Section III are imported by substitution rather than derived; the paper reports that they fail for Δ=0.1 and 0.5, so their validity at Δ=0.01 is an assumption supported only by numerical coincidence.
  • ad hoc to paper Correlated multiplicative disorder on V, t, and Δ is a meaningful test of topological protection.
    Section V applies the same random factor r to all three terms and checks one set of spectra per disorder strength. This is a special perturbation, and there is no comparison with trivial or skin-mode states.
  • domain assumption L=377 with β a Fibonacci approximant is large enough to distinguish delocalized, critical, and localized phases from D2.
    The D2 values in Fig. 1(b) are used to assert a metal-insulator transition, but no finite-size scaling or error bars are given for the fractal dimensions.

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Pith. "Pith review of Investigating topological in-gap states in non-Hermitian quasicrystal with unconventional $p$-wave pairing." pith.science (2026). https://pith.science/paper/OE35CNWN

@misc{pith2026250114481,
  author       = {Pith},
  title        = {Pith review of: Investigating topological in-gap states in non-Hermitian quasicrystal with unconventional $p$-wave pairing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OE35CNWN}},
  note         = {Machine review of arXiv:2501.14481}
}
abstract

The interplay of onsite quasiperiodic potential, superconductivity, and non-Hermiticity is explored in a non-Hermitian unconventional superconducting quasicrystal described by Aubry-Andr\'e-Harper (NHAAH) model with $p$-wave pairing. In previous studies, the non-Hermiticity was only considered at the onsite quasiperiodic potential of the NHAAH model, and Majorana zero modes (MZMs) were observed under open boundary conditions (OBC) in this model. In this work, we study an NHAAH model with $p$-wave pairing, where non-Hermiticity is considered onsite by introducing complex quasiperiodic potential and asymmetry at the hopping part. Our analysis uncovers triple-phase transitions, where topological, metal-insulator, and unconventional real-to-complex transitions coincide at weak $p$-wave pairing strength. Additionally, instead of the MZMs observed in the symmetric hopping case, we observe the emergence of in-gap states under OBC in this model. These in-gap states are robust against disorder, underscoring their topological protection. Therefore, unlike the MZMs, which are very challenging to experimentally realize, these in-gap states can be used in topological quantum computational protocols.

Figures

Figures reproduced from arXiv: 2501.14481 by the authors.

Figure 1
Figure 1. FIG. 1: The phase diagrams are presented for topological, MI, and unconventional real to complex transitions as a [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Here, the non-Hermitian parameter [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: We set the non-Hermitian parameter [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Once again, the same quantities are presented in this figure as in the last two figures. We also keep the value [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: (a)-(d) exhibits the real part of the energy spectrum for different values of disorder strength. The inset shows [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The phase diagrams are presented for topological, MI, and unconventional real to complex transitions as a [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The phase diagrams illustrate the topological, MI, and unconventional real to complex transitions as a function [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: The phase diagram highlights the unconventional real to complex transition in energy eigenvalues for [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: The energy spectrum is studied for the non-Hermitian parameter [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: The same results as in Fig [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Once again, the same results are presented as in the previous two figures for much stronger pairing strength [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

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