{"id":"cc03fef6-ce6d-488a-a18b-988106d86149","arxiv_id":"2411.14144","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"For a non-Hermitian quasicrystal with power-law hopping, p-wave pairing creates oscillating near-zero edge modes that localize as pairing grows and progressively destroys the fractal-dimension plateaus.","lead":"This paper uses computer simulations to study a one-dimensional chain where particles can hop between distant sites, with a non-Hermitian (gain/loss-like) potential and superconducting p-wave pairing. It reports how pairing turns near-zero-energy edge states into localized Majorana-like modes and erases the fractal plateaus seen in the spectrum.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Plateau-reduction claim rests on a single N=1597, θ=0 realization with box sizes ≤20; without finite-size scaling or phase averaging, the five-to-two-to-none sequence may be a commensurability/resolution artifact.","rationale":"The reader's weakest assumption—finite-size and single-phase control for the plateau-disappearance claim—is indeed the most load-bearing issue. The paper's central new quantitative result is the reduction of fractal-dimension plateaus from five to two to none as Δ increases (Sec. IV, Fig. 3). This result is presented through visual inspection of D2 color maps at one Fibonacci system size, one potential phase, and a box-size range capped at d=20. The paper elsewhere demonstrates awareness of finite-size checks (Appendix C checks N=2584, 4181, 6765 for the Δ=0 multifractal Dq), but no such check is provided for the pairing-induced plateau changes. This is not an internal inconsistency, but it is a gap between the numerical evidence and the thermodynamic claim. The BdG doubling of the spectrum for Δ>0 raises an additional definitional ambiguity: the plateau fractions i/N from the Δ=0 single-particle problem must be mapped carefully onto the 2N BdG eigenstates, and the manuscript does not specify how this mapping is done. If the mapping is inconsistent, the plotted plateau positions could be shifted or mislabeled. A finite-size and phase-averaged recomputation with a quantitative plateau-detection threshold would settle whether the five-to-two-to-none sequence is physical or an artifact. Because the concern is addressable and does not by itself invalidate the other results (qMZM/MDM observations and phase diagrams), the appropriate verdict remains conditional, matching the reader's assessment.","tokens_in":13618,"tokens_out":7298,"duration_ms":74513,"concrete_test":"Recompute the D2 maps of Fig. 3 for N=2584, 4181, and 6765 at θ=0, and also at several other phases (e.g., θ=π/2 and a generic irrational phase), using box sizes d up to 100 while keeping d≪N. For each Δ=0.1, 0.13, 0.16, 0.17, and 0.25, detect plateaus by a threshold criterion (e.g., contiguous eigenstate ranges where D2 varies by less than 0.02) and record the plateau count and the Δ values at which plateaus vanish. If the plateau count or the annihilation thresholds change with N or θ, or if the Δ=0.25 'no plateaus' case shows plateaus at larger N or smaller d, the central claim would need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing premise is that the pairing-induced plateau reduction reported in Sec. IV is a thermodynamic property of the model. The evidence is exclusively color maps of D2 at N=1597, θ=0, h=0.1, with box sizes d=2..20. In AAH-type systems with β chosen so that βN is close to an integer (Fibonacci N), finite-N commensurability can both create and hide plateaus; a single phase θ=0 is not obviously representative because the system is not self-averaging. Moreover, for Δ>0 the BdG basis doubles the matrix to 2N×2N, but the text does not state whether the eigenstate index i in the plateau fractions i/N is measured against N or 2N; if half the spectrum is discarded, the claimed fractions β, β², ... require explicit justification. The specific statement that no plateaus are discernible at Δ=0.25 is especially sensitive: with d up to 20, any plateau whose width is smaller than the smallest box size cannot be detected, so the apparent disappearance may be a resolution artifact rather than a physical effect. Appendix C checks finite-size invariance only for the Δ=0 multifractal Dq, not for the Δ>0 plateau counts, so the extrapolation to larger systems is unsupported for the central plateau claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":13892,"tokens_out":4409,"duration_ms":41687,"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":[{"comment":"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.","section":"Sec. IV, Figs. 3(c)-(h)"},{"comment":"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.","section":"Sec. IV, Eqs. (3)-(5)"},{"comment":"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.","section":"Sec. III and Appendix B"}],"minor_comments":[{"comment":"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.","section":"Sec. III A"},{"comment":"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.","section":"Appendix A, Eq. (A2)"},{"comment":"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).","section":"Sec. IV"},{"comment":"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.","section":"Fig. 3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of the journal and the numerical methods are appropriate for a first exploration. The main risk is that the headline plateau-reduction result and the MZM/MDM classifications are not yet supported by finite-size or phase-averaged evidence; these are addressable with additional exact diagonalization runs and quantitative diagnostics, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper adds p-wave pairing to the non-Hermitian GAAH model with power-law hopping. That combination is new, and the qualitative story—oscillating quasi-Majorana modes turning into edge-localized Majorana modes for short-range hopping, oscillating massive Dirac modes for long-range hopping, and pairing eroding the fractal-dimension plateaus—is a natural but useful extension of earlier work by the same group and others.\n\nWhat it does well: the spectrum and edge-mode plots are clean; the Pfaffian parity check in Appendix B distinguishes genuine crossings from avoided ones; and Appendix C shows the Δ=0 multifractal Dq is stable across four Fibonacci sizes. That last check is real evidence.\n\nThe soft spot is the plateau claim in Sec. IV. Everything rests on one realization: N=1597, θ=0, h=0.1, with box sizes d=2..20. The stress-test note is right: with d max 20, a plateau narrower than the smallest box cannot be detected, so the five-to-two-to-none sequence may be a resolution artifact. There is no finite-size scaling for Δ>0 and no phase averaging, and a single phase in a non-self-averaging quasiperiodic system is not obviously representative. The text also doesn't state clearly whether the eigenstate index i in the plateau fractions is measured against N or 2N once the BdG basis doubles the Hilbert space. That matters and should be explicit. These are addressable but load-bearing concerns, not deal-breakers.\n\nThe phase diagrams in Sec. V are more qualitative and less concerning, though they too use a single size and no data release. The paper ships no code or data, which makes the finite-size question harder to check.\n\nWho is this for: people working on non-Hermitian quasicrystals and topological superconductivity. It doesn't resolve a long-standing question, but it charts a new model class. With code and a proper finite-size scaling section, it could be a solid contribution. I'd send it to peer review rather than desk reject, but the referee should ask for major revision on the plateau claim.","headline":"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.","tokens_in":14429,"tokens_out":3195,"would_cite":false,"duration_ms":27038,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Pairing turns quasicrystal edge modes into Majorana zeros","keywords":["non-Hermitian Aubry-André-Harper model","power-law hopping","p-wave superconducting pairing","Majorana zero modes","massive Dirac modes","fractal dimension plateaus","quasicrystal localization","real-to-complex transition"],"falsifier":"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.","tokens_in":13421,"feed_emoji":"🧩","tokens_out":9188,"duration_ms":79078,"temperature":0.7,"pith_summary":"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.","feed_headline":"Pairing turns quasicrystal edge modes into Majorana zeros","feed_subtitle":"Pairing sharpens short-range Majorana modes, births massive Dirac modes at long range, and erases fractal plateaus.","key_machinery":"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 $20$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the non-Hermitian AAH model with p-wave pairing whose symmetry, localization, and topological transitions this paper extends to power-law hopping.","marker":"[53]"},{"why":"Establishes mobility edges in the non-Hermitian AAH model with power-law hopping, the baseline for the model studied here.","marker":"[42]"},{"why":"Documents the D2 plateau structure in the non-Hermitian power-law hopping model without pairing, which pairing is then shown to erase.","marker":"[43]"},{"why":"Determines the fractions of delocalized eigenstates and the beta^l plateau positions for the long-range AAH model that the pairing modulation acts on.","marker":"[17]"},{"why":"Introduces the one-dimensional quasicrystal with power-law hopping and the delocalized-to-multifractal transition underlying the plateau physics.","marker":"[16]"},{"why":"Explains oscillating Majorana wavefunctions in finite-length wires, used to interpret the quasi-Majorana zero-mode oscillations.","marker":"[60]"},{"why":"Introduces massive Dirac modes in long-range Kitaev chains, the reference for the edge modes seen in the long-range hopping case.","marker":"[63]"},{"why":"Discusses topological massive Dirac edge modes in long-range superconducting Hamiltonians, supporting the MDM identification.","marker":"[65]"},{"why":"Provides the unconventional real-to-complex transition scenario in non-Hermitian quasiperiodic lattices that the phase diagrams build on.","marker":"[55]"}],"fun_headline_variants":["Pairing drives non-Hermitian AAH chain to Majorana zeros","Short-range pairing yields Majorana zeros; long-range births Dirac modes","Pairing erases fractal plateaus in non-Hermitian quasicrystals","Crossover from quasi-Majorana to edge-localized Majorana modes","How pairing reshapes one-dimensional non-Hermitian quasicrystals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Pairing drives non-Hermitian AAH chain to Majorana zeros","Short-range pairing yields Majorana zeros; long-range births Dirac modes","Pairing erases fractal plateaus in non-Hermitian quasicrystals","Crossover from quasi-Majorana to edge-localized Majorana modes","How pairing reshapes one-dimensional non-Hermitian quasicrystals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000192,"raw_usage":{"total_tokens":1382,"prompt_tokens":1019,"completion_tokens":363,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":635,"completion_tokens_details":{"reasoning_tokens":268}},"tokens_in":635,"tokens_out":363,"duration_ms":4032,"temperature":1.0,"reasoning_tokens":268,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:29:10.079526+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Roy and A","cited_arxiv_id":null,"evidence_quote":"Determines the fractions of delocalized eigenstates and the beta^l plateau positions for the long-range AAH model that the pairing modulation acts on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the one-dimensional quasicrystal with power-law hopping and the delocalized-to-multifractal transition underlying the plateau physics."},{"cited_title":"Fleckenstein, F","cited_arxiv_id":null,"evidence_quote":"Explains oscillating Majorana wavefunctions in finite-length wires, used to interpret the quasi-Majorana zero-mode oscillations."},{"cited_title":"Vodola, L","cited_arxiv_id":null,"evidence_quote":"Introduces massive Dirac modes in long-range Kitaev chains, the reference for the edge modes seen in the long-range hopping case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the unconventional real-to-complex transition scenario in non-Hermitian quasiperiodic lattices that the phase diagrams build on."}],"review_version":1}