{"id":"110eafb2-c7bb-4645-aba2-8318e81cfafe","arxiv_id":"2501.14481","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"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 disorder-robust in-gap states.","lead":"By adding asymmetric hopping to a non-Hermitian Aubry-André-Harper superconductor with weak p-wave pairing, the paper reports that topological, metal-insulator, and real-to-complex transitions coincide, and that in-gap states appear instead of Majorana zero modes. A generalist might read it to see an attempt to find more experimentally accessible protected states for quantum computation, though that interpretation is not yet established.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"PBC winding in Eq. (6) is used to label OBC in-gap states in a model with asymmetric hopping, but non-Bloch/GBZ invariants are never computed; without them, the topological-protection and quantum-computing claims are unestablished.","rationale":"Good-faith reading: the paper makes a concrete prediction of triple phase transitions and supports it with winding-number, fractal-dimension, and spectral computations; the coincidence at Δ=0.01 is plausible, and the appendix honestly shows that the analytical boundaries fail at stronger pairing. The central interpretive claim, however, is that the OBC in-gap states are topologically protected. That claim is load-bearing for the abstract and for the quantum-computing proposal. The weakest point is exactly the bulk-boundary correspondence: Eq. (6) is a PBC winding around EB=0, while the model has h2≠0 asymmetric hopping, so the OBC eigenstates are controlled by the non-Bloch generalized Brillouin zone, and the bulk in-gap state's right-only localization is a skin-effect fingerprint. Refs [6,7] are cited but not implemented. The proposed GBZ/non-Bloch winding check would settle whether the PBC topological regions actually host OBC topological edge states. Separately, the Sec. V disorder test as written appears to multiply the entire Hamiltonian by one random scalar r; if so, it rescales eigenvalues without testing robustness. Together these issues justify keeping the verdict conditional: the numerics may be correct, but the topological-protection and quantum-computing conclusions should not be accepted until the non-Bloch invariant is computed and the disorder test is redone with site-resolved disorder and averaging.","tokens_in":11332,"tokens_out":8406,"duration_ms":85198,"concrete_test":"For h2=0.5, Δ=0.01, and h1=0.1, construct the generalized Brillouin zone from det[H_BdG(β)-E]=0 and compute the non-Bloch winding W = (1/2πi)∮_{GBZ} dβ ∂_β ln det[H_BdG(β)-0]. Compare W with the PBC w of Eq. (6) and with the OBC spectra of Fig. 2 at L=377 and L=610. If W=0 or differs from w while the in-gap states persist and shift with L, they are skin modes, not topologically protected edge states.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Eq. (6) defines a PBC Bloch winding number around base energy EB=0, and Fig. 1 uses it to delimit the 'topological' regions. Section IV then attributes the OBC in-gap states of Figs. 2-4 to these topological regions. This is the load-bearing step that is least secure. Because the hopping in Eq. (1) is asymmetric (e^{±h2}), the system generically exhibits the non-Hermitian skin effect; Refs [6-8] on non-Bloch bulk-boundary correspondence are cited but never applied. The bulk in-gap state is localized only at the right boundary (Fig. 2(e)), which is the signature of a skin mode, and the chosen base point EB=0 is not justified for the finite-energy in-gap states. A nonzero PBC winding does not, by itself, imply OBC topological edge modes when the generalized Brillouin zone is not the unit circle. Thus the statement that the in-gap states are 'topologically protected' is unsupported as written. Separately, the Sec. V disorder test uses a single random factor r multiplying V, t, and Δ; if r is global, then H_dis = rH, which only rescales the spectrum and gives no information about disorder robustness. The final quantum-computing proposal rests on this unresolved identification.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11609,"tokens_out":6548,"duration_ms":63373,"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":[{"comment":"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.","section":"Section IV, Eq. (6)"},{"comment":"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.","section":"Section V"},{"comment":"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.","section":"Section III"}],"minor_comments":[{"comment":"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.","section":"Section III"},{"comment":"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.","section":"Eq. (6)"},{"comment":"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.","section":"Appendix B"},{"comment":"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).","section":"Section III"},{"comment":"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.","section":"Section V"},{"comment":"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.","section":"Section IV"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a natural continuation of the authors' previous work and is within the journal's scope. My recommendation is driven by the two load-bearing gaps described in the major comments: the PBC-to-OBC topological inference and the degenerate disorder test. Both are fixable within the scope of the paper, so I do not recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe new model combination here is genuine, and the weak-pairing triple transition looks numerically solid, but the central claim — that the asymmetric-hopping in-gap states are topologically protected — is not established. I'd send it out for review, but the referee will need to push on two specific points.\n\nWhat's actually new: the paper adds p-wave pairing on top of the non-Hermitian AAH model with both complex onsite potential and asymmetric hopping. Ref. [21] had asymmetric hopping without pairing; Ref. [40] had pairing without asymmetric hopping. The reported triple phase transition at Δ=0.01 is supported by independent-looking checks: the winding number, the averaged fractal dimension, and the imaginary-spectrum data all change across the same boundary, and the paper openly says the analytic boundaries fail for Δ=0.1 and 0.5. That honesty is worth noticing.\n\nThe soft spots are proportionate to how much the conclusions lean on them. First, the PBC winding number of Eq. (6) is used to label the OBC in-gap states as topological. With asymmetric hopping e^{±h2}, the system generically has a non-Hermitian skin effect; the bulk in-gap state in Fig. 2(e) is localized only at the right end, which is the skin-mode signature. The authors cite Refs. [6-8] on non-Bloch bulk-boundary correspondence but never compute a generalized-Brillouin-zone invariant or biorthogonal polarization. A nonzero PBC winding around EB=0 does not, by itself, guarantee OBC topological edge modes when the GBZ is not the unit circle. So 'topologically protected' is currently unsupported, even though the triple transition may survive.\n\nThe second issue is the disorder test. Multiplying V, t, and Δ by a single global random factor r means the disordered Hamiltonian is just rH. That rescales the spectrum and leaves the eigenstates unchanged, so the observed robustness is trivial and says nothing about genuine disorder. A proper test would use independently distributed perturbations per site or per bond.\n\nThe quantum-computing suggestion in the abstract and conclusion rides entirely on the unresolved identification of these states as topologically protected rather than skin modes or trivial in-gap states.\n\nBottom line: the paper maps a new parameter regime and provides a clean numerical dataset for a non-Hermitian quasicrystal superconductor. It deserves a serious referee, but the referee should ask for a non-Bloch invariant and a real disorder average before the in-gap states are called topological.","headline":"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.","tokens_in":12142,"tokens_out":2742,"would_cite":false,"duration_ms":24034,"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":"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.","keywords":["non-Hermitian quasicrystal","Aubry-André-Harper model","p-wave pairing","in-gap states","topological phase transition","metal-insulator transition","real-to-complex transition","non-Hermitian skin effect"],"falsifier":"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.","tokens_in":11090,"feed_emoji":"⚛️","tokens_out":8335,"duration_ms":65469,"temperature":0.7,"pith_summary":"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.","feed_headline":"Asymmetric hopping turns Majorana modes into robust in-gap states","feed_subtitle":"Weak pairing makes topological, localization, and spectral-reality transitions coincide - a softer route to topological quantum computing.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the symmetric-hopping NHAAH+p-wave model that hosts MZMs and the analytic phase-boundary expressions the paper adapts for h1.","marker":"[40]"},{"why":"Gives the no-pairing asymmetric-hopping model whose double phase transition this paper extends to the paired case.","marker":"[21]"},{"why":"Establishes the triple phase transition in non-Hermitian quasicrystals, which this work generalizes to p-wave pairing.","marker":"[19]"},{"why":"Provides the mathematical formulation used to derive the analytical phase boundaries of the winding number.","marker":"[36]"},{"why":"Supplies the characterization of the unconventional real-to-complex transition used in the phase diagram.","marker":"[39]"},{"why":"Cited for non-Bloch bulk-boundary correspondence that would be needed to validate the open-boundary topological claim, though not employed in the paper's analysis.","marker":"[6]"}],"fun_headline_variants":["Triple phase transition yields robust in-gap states","Asymmetry unlocks triple transition and protected in-gap states","One boundary, three shifts: topological, localization, spectral","In-gap states offer robust alternative to Majorana modes","Weak pairing syncs three transitions, giving robust in-gap states"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Triple phase transition yields robust in-gap states","Asymmetry unlocks triple transition and protected in-gap states","One boundary, three shifts: topological, localization, spectral","In-gap states offer robust alternative to Majorana modes","Weak pairing syncs three transitions, giving robust in-gap states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000434,"raw_usage":{"total_tokens":2225,"prompt_tokens":975,"completion_tokens":1250,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":591,"completion_tokens_details":{"reasoning_tokens":1167}},"tokens_in":591,"tokens_out":1250,"duration_ms":8245,"temperature":1.0,"reasoning_tokens":1167,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T15:08:06.089318+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Cai, Localization and topological phase transitions in non-hermitian Aubry-André-Harper models withp-wave pairing, Phys","cited_arxiv_id":null,"evidence_quote":"Supplies the symmetric-hopping NHAAH+p-wave model that hosts MZMs and the analytic phase-boundary expressions the paper adapts for h1."},{"cited_title":"Gandhi and J","cited_arxiv_id":null,"evidence_quote":"Establishes the triple phase transition in non-Hermitian quasicrystals, which this work generalizes to p-wave pairing."},{"cited_title":"Cai, L.-J","cited_arxiv_id":null,"evidence_quote":"Provides the mathematical formulation used to derive the analytical phase boundaries of the winding number."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the characterization of the unconventional real-to-complex transition used in the phase diagram."}],"review_version":1}