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Superconducting $p$-wave pairing effects on one-dimensional non-Hermitian quasicrystals with power law hopping

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

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Pith's one-line read Pairing turns quasicrystal edge modes into Majorana zeros

desk verdict A plausible numerical study of a new model combination; the central gap physics is credible, but the plateau-reduction claim needs finite-size and phase-averaged support before I'd trust it. read the letter →

arxiv 2411.14144 v1 pith:ZSURX2O7 submitted 2024-11-21 cond-mat.supr-con cond-mat.dis-nncond-mat.otherquant-ph

Shaina Gandhi , Jayendra N. Bandyopadhyay This is my paper · ORCID
classification cond-mat.supr-concond-mat.dis-nncond-mat.otherquant-ph
keywords non-HermitianAubry-André-Harpermodelpower-lawhoppingp-wavesuperconductingpairingMajoranazeromodesmassiveDiracfractaldimensionplateausquasicrystallocalizationreal-to-complextransition
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 asks what happens when superconducting $p$-wave pairing is added to a one-dimensional non-Hermitian Aubry-André-Harper model whose hopping decays as a power law. It claims that pairing acts as a tuning knob for both edge physics and spectral structure: for short-range hopping, weak pairing produces oscillating quasi-Majorana zero modes that become separated edge-localized Majorana zero modes as pairing increases, while for long-range hopping the system develops massive Dirac modes that localize at the edges. It also claims that pairing destroys the fractal-dimension plateaus that characterize the delocalized-to-multifractal transition in the long-range model, reducing them from five to two to none as pairing strength grows. A sympathetic reader would care because the result points to a single parameter, pairing strength, controlling both topological zero modes and the multifractal spectrum of a quasicrystal.

What carries the argument

The machinery is the Bogoliubov-de Gennes Hamiltonian of a chain with power-law hopping $t/s^\xi$, a complex onsite potential $f(j)=\cos(2\pi\beta j)+ih\sin(2\pi\beta j)$, and real nearest-neighbor $p$-wave pairing $\Delta$. Particle-hole symmetry forces the spectrum to come in $(E,-E^*)$ pairs and is what makes zero-energy edge modes possible, while $\xi$ controls whether hopping is effectively short range or long range. To distinguish genuine crossings from avoided ones the authors compute the ground-state fermion parity from the Pfaffian of the Hamiltonian in a Majorana basis, and to quantify localization they use the box-counting fractal dimension $D_2$ over eigenstates with box sizes $d=2$ to $20$.

What would settle it

Compute the same spectra and fractal dimensions at larger Fibonacci sizes such as $N=2584$, $4181$, and $6765$ and for several phases $\theta$; if the five-to-two-to-none plateau sequence no longer holds, or the short-range edge modes do not separate into two non-overlapping zero modes as $\Delta$ grows, the central claim would be shown to be a finite-size artifact. A second check is the ground-state fermion parity at the alleged crossings: if the sign stops switching at larger $N$, the oscillating quasi-Majorana modes are not genuine.

Watch

Extended reading notes

Core claim

The central discovery is a pairing-controlled crossover in the non-Hermitian power-law AAH chain. With short-range hopping set by $\xi=5.0$, even an infinitesimal pairing $\Delta=10^{-10}$ opens a particle-hole symmetric spectrum with oscillating quasi-Majorana zero modes; these oscillations are accompanied by genuine zero-energy crossings where the ground-state fermion parity switches, and increasing $\Delta$ to $0.5$ fully separates the two edge modes into non-overlapping Majorana zero modes. For long-range hopping set by $\xi=0.2$, the central gap contains no exact zero modes; instead the near-gap states are massive Dirac modes with oscillatory character that progressively localize at both edges as $\Delta$ grows. In the long-range case with weak non-Hermiticity $h=0.1$, the fractal dimension $D_2$ of energy eigenstates shows five plateaus at fractions $\beta^l$ when $\Delta=0$, only two plateaus for up to $\Delta=0.13$, destruction of the first plateau by $\Delta=0.16$--$0.17$, and complete disappearance by $\Delta=0.25$. The paper further maps real-to-complex transitions and shows that for $h\lesssim 1.0$ the real-energy window and delocalized states coincide for short-range hopping but not for long-range hopping.

Load-bearing premise

The numerical results assume that exact diagonalization at $N=1597$ with quasiperiodic phase $\theta=0$ and box sizes $d=2$ through $20$ captures the thermodynamic-limit behavior, without finite-size scaling or phase averaging.

Editorial extensions

If this is right

  • Short-range chains with weak pairing host oscillating quasi-Majorana zero modes whose zero-energy crossings are genuine parity-switching events, so weak pairing already equips the non-Hermitian AAH chain with the spectral signature of Majorana physics.
  • Raising the pairing strength from $\Delta=10^{-10}$ to $0.5$ converts those oscillating near-zero modes into two spatially separated Majorana zero modes, one at each edge.
  • Long-range hopping replaces exact zero modes with massive Dirac modes, and increasing $\Delta$ drives them from oscillatory nonlocal states to edge-localized states, so long-range hopping changes the topological edge content qualitatively.
  • The five $\beta^l$ plateaus in the fractal dimension of the long-range AAH model are progressively erased by pairing: two survive to $\Delta=0.13$, one to about $\Delta=0.17$, and none at $\Delta=0.25$.
  • In the $h\lesssim 1$ regime the real-to-complex energy boundary and the delocalized-to-multifractal boundary overlap for short-range hopping but not for long-range hopping, meaning measurement of the energy spectrum alone does not predict localization in the long-range case.

Reading between the lines

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

  • The paper implicitly treats pairing as a controller of the delocalized-to-multifractal edge; a natural test is whether the plateau destruction collapses onto a scaling curve when $N$ is varied, which would show whether pairing acts like an effective non-Hermiticity or like a distinct symmetry-preserving perturbation.
  • Because only $N=1597$ and phase $\theta=0$ are used, the MZM/MDM classification and the plateau sequence are finite-size statements unless verified at larger Fibonacci sizes and other phases; that is an extension rather than a claim of the paper.
  • For long-range hopping the absence of exact zero modes suggests that power-law hopping effectively breaks the chiral condition protecting MZMs even though the pairing term is local; a topological invariant in the BdG band would test whether the massive Dirac modes are topologically protected.
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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 a one-dimensional non-Hermitian Aubry-André-Harper model with power-law hopping and p-wave pairing. For short-range hopping it reports that weak pairing produces oscillating quasi-Majorana zero modes that evolve into edge-localized Majorana zero modes as the pairing strength increases; for long-range hopping it reports massive Dirac modes with oscillatory behavior that localize at the edges with increasing pairing. The main quantitative claim is that superconducting pairing reduces the number of fractal-dimension plateaus from five to two to none as Δ increases, and the paper also presents phase diagrams for the real-to-complex transition and the delocalized-to-multifractal transition. The analysis is numerical, based on exact diagonalization at N=1597 with θ=0, box-counting fractal dimensions, and a Pfaffian parity check for short-range zero-energy crossings.

Significance. If the reported effects are robust, the paper would extend the understanding of how p-wave pairing modifies localization, multifractality, and edge modes in non-Hermitian quasicrystals. The work has clear strengths: it uses a concrete model with explicit symmetry analysis, provides direct numerical evidence for the central-gap modes, and includes a Pfaffian-based parity check that supports the genuineness of the short-range zero-energy crossings. The phase diagrams in Fig. 4 are useful as a first mapping of the h-Δ plane. The principal weakness is that the central plateau-reduction claim and the MZM/MDM classifications rest on a single system size and a single potential phase, with no finite-size scaling for the Δ>0 regime; this limits the current support for the thermodynamic-limit conclusions.

major comments (3)
  1. [Sec. IV, Figs. 3(c)-(h)] The central plateau-reduction claim (five plateaus to two to none) rests on a single realization with N=1597, θ=0, h=0.1, and box sizes d=2..20. Appendix C verifies finite-size invariance only for Δ=0, not for the Δ>0 plateau counts. Because AAH-type systems with Fibonacci N and irrational β are known to exhibit finite-N commensurability effects, and because d≤20 cannot resolve plateaus narrower than about 1/20 in the fraction i/N, the apparent disappearance of P1 at Δ=0.16-0.17 and of all plateaus at Δ=0.25 could be a resolution or commensurability artifact. Please provide finite-size scaling for these Δ values, e.g., at N=2584, 4181, and 6765, and either phase-averaged results or an argument that θ=0 is representative.
  2. [Sec. IV, Eqs. (3)-(5)] For Δ≠0 the BdG Hamiltonian is 2N×2N, but the text states that the Hamiltonian is an N×N matrix and identifies plateau positions as i/N. It is not specified whether D2 is computed for all 2N eigenstates, for the N positive-energy eigenstates, or for some other subset. If all 2N eigenstates are used, the counting of states entering the fractions i/N requires explicit justification; if half the spectrum is used, the selection rule must be stated. This ambiguity directly affects the claimed β, β², ... plateau positions and should be clarified before the quantitative plateau statement can be assessed.
  3. [Sec. III and Appendix B] The identification of a qMZM-to-MZM crossover for short-range hopping and of MDM for long-range hopping is based on visual inspection of Re(E) versus h plots and density profiles at N=1597. The Pfaffian parity check in Appendix B validates genuine zero-energy crossings only for Δ=10^-10 and Δ=0.005 in the short-range case; it does not validate the Δ=0.5 MZM claim or the long-range MDM interpretation. Please provide a quantitative criterion, for example exponential decay of the edge-mode splitting with N, a Majorana polarization, or a topological invariant, to distinguish MZM and MDM from finite-size near-zero modes.
minor comments (4)
  1. [Sec. III A] In the paragraph after Fig. 1, the text says 'In Figs. 2 (k)-(n), we present the edge modes', but the edge-mode panels for the short-range case are in Fig. 1(k)-(n); please correct the cross-reference.
  2. [Appendix A, Eq. (A2)] The relation 'where N−j=j' used in the parity transformation is not a valid index substitution; the equality should be completed by relabeling the summation index and using fermionic anticommutation. Please rewrite this step for clarity.
  3. [Sec. IV] The sentence 'Consequently, the Hamiltonian of the system is a N × N matrix' is inaccurate for Δ≠0, where the BdG Hamiltonian is 2N×2N; this should be corrected for consistency with Eq. (3).
  4. [Fig. 3] The color maps in Fig. 3 would benefit from an explicit color bar or a statement of the D2 scale; the text says D2 is 'shown in color' but the range and mapping are not defined in the caption.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the MZM/MDM identifications, plateau-reduction sequence, and phase diagrams are direct numerical outputs with no fitted parameters; the only self-citation is contextual and non-load-bearing.

full rationale

All central claims are obtained by exact diagonalization of the model defined in Eqs. (1)-(4): the qMZM/MZM/MDM classification in Sec. III follows from computed spectra and eigenstate profiles (Figs. 1-2), corroborated by an independent Pfaffian/fermion-parity calculation in Appendix B; the plateau analysis in Sec. IV is a box-counting evaluation of D2 over computed eigenstates, with no parameter fitted to the reported five-to-two-to-none sequence. The plateau fractions i/N=β^l are prior known values from Ref. [17], used as benchmarks for the computed maps rather than being imposed by the model. The phase diagrams in Sec. V are direct scans in (h, Δ) of |Im E| and mean D2. The one self-citation (Ref. [68]) is invoked for background on unconventional real-complex transitions in similar systems; the present numerical phase diagrams do not reduce to that citation. Caveats such as fixed N=1597, θ=0, d=2..20, and the Appendix C finite-size check being restricted to Δ=0 are robustness/resolution concerns, not evidence that a result is equivalent to its inputs by construction. No quoted equation or parameter-fitting step exhibits self-definitional or forced circularity.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central results are numerical diagonalization outputs rather than derived equations. The main free choices are the fixed potential phase, the single system size, and the fixed weak non-Hermiticity in the plateau study. The paper leans on standard background assumptions about particle-hole symmetry, box-counting fractal dimensions, and the massive-Dirac-mode analogy.

free parameters (3)
  • Quasiperiodic phase θ = 0
    Set to zero for all calculations; finite-size quasiperiodic spectra can depend on the phase, and no phase averaging is performed.
  • System size N = 1597
    Fixed at the Fibonacci number 1597 for all main results; no finite-size scaling is shown for the pairing-induced plateau destruction or the MZM/MDM classification.
  • Weak non-Hermiticity h = 0.1
    Plateau and fractal-dimension results in Sec. IV are shown only for h=0.1; the claimed plateau counts and their Δ-dependence may vary with h.
assumptions (3)
  • domain assumption Particle-hole (PC) symmetry of the BdG Hamiltonian pairs energies as (E, -E*) and protects zero-energy edge modes.
    Invoked throughout Sec. III to interpret near-zero modes as Majorana-derived. The proof in Appendix A, Eq. (A2), contains a re-indexing step ('where N-j=j') that is not generally valid for finite N; the global symmetry may still hold.
  • domain assumption The box-counting fractal dimension D2 computed with d from 2 to 20 for N=1597 is a faithful proxy for the thermodynamic-limit multifractal dimension.
    The plateau identification in Sec. IV depends on D2 values at a single system size; Appendix C checks N scaling only for Δ=0, not for the pairing-induced changes.
  • domain assumption Nonzero-energy edge states localized at both ends of the long-range hopping chain correspond to massive Dirac modes of Refs. [63-66].
    Used in Sec. IIIB to label the observed edge states as MDM by analogy; no topological invariant is computed for this case.

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Cite this review

Pith. "Pith review of Superconducting $p$-wave pairing effects on one-dimensional non-Hermitian quasicrystals with power law hopping." pith.science (2026). https://pith.science/paper/ZSURX2O7

@misc{pith2026241114144,
  author       = {Pith},
  title        = {Pith review of: Superconducting $p$-wave pairing effects on one-dimensional non-Hermitian quasicrystals with power law hopping},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZSURX2O7}},
  note         = {Machine review of arXiv:2411.14144}
}
abstract

We study the effects of superconducting $p$-wave pairing on the non-Hermitian Aubry-Andr\'e-Harper model with power-law hopping. For the case of short-range hopping, weak pairing leads to oscillating quasi-Majorana zero modes, turning to edge-localized Majorana zero modes as pairing strength increases. For the case of long-range hopping, we observe the emergence of massive Dirac modes having oscillatory behavior, similar to Majorana modes with weak pairing. The massive Dirac modes localize at the edges as the pairing strength grows. The superconducting pairing spoils the plateaus observed in the fractal dimension of all the energy eigenstates of the Aubry-Andr\'e-Harper model with power-law hopping. The number of plateaus decreases with the increasing pairing strength for the weak non-Hermiticity in the system. The phase diagram of the system reveals that real and complex energy spectrums correlate differently with the localization properties of the eigenstates depending on the strength of pairing and hopping range.

Figures

Figures reproduced from arXiv: 2411.14144 by the authors.

Figure 1
Figure 1. FIG. 1: (a)-(e) Real parts of the eigenvalues are presented as a function of increasing pairing strength [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: (a)-(f) The real part of the energy spectrum is shown as a function of the non-Hermitian parameter [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Fractal dimension [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The phase diagrams of the unconventional real [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The energy spectrum and ground state fermion [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The average fractal dimension [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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    In a non-Hermitian Aubry-André-Harper chain with p-wave pairing, weak pairing makes topological, localization, and real-to-complex transitions coincide, while asymmetric hopping replaces Majorana zero modes with disor...

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