{"id":"eafebf9d-371b-4d2b-965c-0cad2be556df","arxiv_id":"2411.10398","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Nonlinear hopping in both the Hermitian and non-Hermitian halves of a topological interface lattice fully delocalizes the zero mode and allows its profile to be designed arbitrarily.","lead":"This paper shows that adding nonlinearity to both halves of a hybrid topological chain lets a protected zero-energy mode spread across the entire lattice without precise parameter tuning, and lets researchers sculpt the mode's shape. The result is a path toward compact, reconfigurable topological devices in photonics and circuits, where modes usually cling to surfaces or need careful engineering.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The spectral localizer analysis is underspecified: after the similarity transform S, the nonlinear hoppings λ_j depend on original amplitudes, so Hs must be evaluated with r^{-2(j−N)} rescaled intensities; the paper never states which convention Fig. 4 uses.","rationale":"I read the paper in good faith. The central physical claim, that nonlinearity combined with non-Hermiticity can delocalize topological zero modes across the whole lattice without requiring δ = δc, is supported by self-consistent numerics and by the analytic plateau arguments in Supplementary Note 4. Those parts are not the weak point. The load-bearing weakness is the spectral localizer argument used to certify topological protection. The localizer requires a Hermitian Hamiltonian Hs obtained by a similarity transformation, but the nonlinear hopping coefficients in Eq. (1) are functions of the original amplitudes. After conjugation by S, those coefficients acquire position-dependent factors r^{-2(j−N)} when written in terms of the transformed state |ψbar⟩. The paper does not specify whether Fig. 4 and Eq. (5) use this corrected form or the naive |ψbar|^2 form. This is not a mere cosmetic ambiguity: the two conventions define different nonlinear problems, so the claimed changes in Cζ and μζ may not witness the topology of the original non-Hermitian nonlinear eigenproblem. The reader identified this same issue, and I agree it is the single most load-bearing concern. I would not change the CONDITIONAL verdict, because the concern is concrete and testable rather than a demonstrated contradiction; a straightforward recomputation would settle it. The secondary overgeneralization of 'unattainable in Hermitian systems' is worth noting but is not the deciding issue. The absence of public code and the lack of formal verification increase residual uncertainty but do not by themselves overturn the numerical evidence.","tokens_in":26014,"tokens_out":7379,"duration_ms":73591,"concrete_test":"Recompute the spectral localizer data in Fig. 4 and Supplementary Figure S7 for a representative state, e.g., δ = 1, β = 0.05, I = 40^2, using the consistent convention. First solve H|ψ⟩ = ω|ψ⟩ in the original basis; set ψbar = Sψ; then define Hs with λ_j = λ~_j + β(|a_{j+1}|^2 + |b_j|^2) expressed through ψbar via the inverse transformation, i.e., with the explicit r^{-2(j−N)} intensity weights. Compare the resulting eigenvalues σ(L~ζ), invariant Cζ, and local gap μζ against the naive convention that uses |ψbar|^2 without rescaling. If Cζ or the zero-crossing pattern of σ(L~ζ) changes, the topological-protection claim in Fig. 4 is not established for the stated model. If the two conventions coincide numerically, the paper should state explicitly that the r^{-2(j−N)} factors are included, resolving the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest load-bearing step is the spectral localizer construction in Methods and Fig. 4. Eq. (1) defines λ_j = λ~_j + β(|a_{j+1}|^2 + |b_j|^2) in the original basis. The similarity transformation S in Eq. (12) has diagonal entries r^{k} on the non-Hermitian chain, with r = sqrt(|J−δ|/|J+δ|), so after transformation one has |a_{j+1}|^2 = r^{-2(j−N)} |ψbar_{a,j+1}|^2 and |b_j|^2 = r^{-2(j−N)} |ψbar_{b,j}|^2. Therefore the Hermitian Hamiltonian Hs in Eq. (13), when expressed through the transformed state |ψbar⟩ = S|ψ⟩, contains the spatially varying nonlinearity λ_j = λ~_j + β r^{-2(j−N)}(|ψbar_{a,j+1}|^2 + |ψbar_{b,j}|^2), not λ_j = λ~_j + β(|ψbar_{a,j+1}|^2 + |ψbar_{b,j}|^2). The paper never states which convention is used to construct Hs 'accounting for its current occupations |ψbar⟩'. If Fig. 4 uses the naive |ψbar|^2 convention, the resulting Cζ and μζ describe a different nonlinear eigenproblem and cannot certify topological protection of the original TZM. The same ambiguity affects the disorder criterion in Eq. (5): physical disorder added to H is conjugated by S, so it does not correspond to an arbitrary perturbation of Hs. The delocalization numerics in Figs. 2 and 3 are separate and are not called into question by this issue, but the topological-protection claim is load-bearing for the paper's stated rigorous verification.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies a one-dimensional topological interface model formed by a Hermitian nonlinear SSH chain and a non-Hermitian nonlinear SSH chain, and reports that topological zero modes can be made to occupy the entire lattice for sufficiently large nonlinear intensity, even when the nonreciprocal hopping strength δ deviates from the linear critical value δc. The authors support this with self-consistent solutions of the nonlinear eigenproblem, derive simple plateau-height formulas from the eigenequations, demonstrate arbitrary wavefunction-profile shaping by engineering the hopping distributions, and use a spectral localizer to argue topological protection and robustness to disorder. The paper also studies dynamical preparation of extended zero modes by external pumping, stability against noise, and a two-dimensional extension built from stacked chains. The delocalization results are well illustrated numerically, but the spectral localizer analysis is under-specified in a way that affects the topological-protection claim.","tokens_in":26406,"tokens_out":6326,"duration_ms":63924,"significance":"If the central effect is as claimed, the paper offers a plausible route to extended, reconfigurable topological modes that avoids fine-tuning the linear critical condition, with potential relevance to nonlinear topological photonics and circuit implementations. The main strengths are that the delocalization is obtained from direct self-consistent solutions of the nonlinear Schrödinger equation rather than from fitted parameters, that the plateau amplitudes in Supplementary Note 4 are derived from the equations, and that numerical data are deposited on Zenodo. The two-dimensional extension adds breadth. However, the topological-protection argument rests on a spectral localizer construction whose convention for evaluating nonlinear hoppings after a similarity transformation is not specified, so the 'rigorous verification' claim is not currently established.","major_comments":[{"comment":"The construction of the Hermitian Hamiltonian H_S via H_S = S H S^{-1} is under-specified for the nonlinear terms. In Eq. (1), λ_j and t_j depend on the original amplitudes |a_{j+1}|^2 and |b_j|^2, but after defining |ψbar⟩ = S |ψ⟩, these amplitudes acquire position-dependent factors on the non-Hermitian chain (for the given diagonal form of S, |a_{j+1}|^2 and |b_j|^2 are multiplied by r^{-2(j-N)}). The manuscript never states whether H_S in Eq. (13) is evaluated with these rescaled amplitudes or with the naive |ψbar|^2 expressions. Without this specification, the spectra σ(L̃_ζ), the invariant C_ζ, and the local gap μ_ζ in Fig. 4 and Eqs. (17)-(19) are not tied to the original nonlinear eigenproblem. I request an explicit formula for the λ_j actually used in Eq. (13) and a demonstration that, under that convention, the localizer gap closes where the original zero mode is supported.","section":"Methods, Eqs. (12)-(13); Fig. 4"},{"comment":"Eq. (5) states a robustness condition for arbitrary perturbations ΔH_S(W) of the similarity-transformed Hamiltonian. However, physical disorder added to the original Hamiltonian, as in Eqs. (S4)-(S5), is not an arbitrary perturbation of H_S: it enters as S (δH) S^{-1}, which is a restricted set. As written, the inequality therefore certifies stability only against a different class of perturbations. Please either restrict ΔH_S to the image of physical disorder under the similarity transformation or recompute the disorder robustness using the actual S δH S^{-1} terms, and report whether the local-gap bound is satisfied for the disorder ranges tested in Fig. S3 and Fig. S6.","section":"Eq. (5) and Supplementary Note 3"},{"comment":"Because the localizer is built from H_S evaluated with the occupations of the already-computed zero mode, the observation that σ(L̃_ζ) crosses zero exactly where |ψbar_x| is large is in part a restatement of the input rather than an independent topological verification. The paper should state what the C_ζ change adds beyond the direct self-consistent solution. A concrete test would be to verify that the localizer gap remains open when the state is artificially removed from the nonlinear coefficients, or to check that the C_ζ changes occur only in regions supporting the zero mode and not for a corresponding trivial configuration. Without such a check, the phrase 'rigorously verified' overstates the evidence in the present text.","section":"Topological origin of zero modes; Fig. 4"}],"minor_comments":[{"comment":"The diagonal entries of R are listed with a length that does not obviously match the stated dimension L-2N for the parameters used (e.g., L=121 and N=31), and the pairing of amplitudes under the transformation is not defined; please clarify the indexing and the correspondence between the entries of R and the lattice sites.","section":"Methods, Eq. (12)"},{"comment":"The equations write the nonlinear hopping terms with a_j^2 rather than |a_j|^2; since the amplitudes may be complex in general, a brief statement that the relevant zero-mode amplitudes can be chosen real and nonnegative would remove ambiguity.","section":"Supplementary Note 4, Eqs. (S6)-(S8)"},{"comment":"Reference 72 has a malformed author list ('M. Padlewski H. Lissek P. Delplace R. Fleury X. Guo, L. Jezequel') and should be corrected to the standard citation for the arXiv preprint by Guo et al.","section":"References"},{"comment":"In the sentence describing the random disturbance, the text reads 'range10 [−3, 3]'; the superscript '10' appears to be a typographical artifact and should be removed.","section":"Dynamical evolution under external pumping"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of the journal and the central delocalization result is numerically well supported. The main risk is the spectral localizer analysis: the similarity-transformation convention for nonlinear hoppings is ambiguous and the robustness criterion in Eq. (5) does not obviously match the physical disorder tested in the Supplementary Material. These issues are fixable in revision, but they affect the advertised 'rigorous verification' of topological protection and should be resolved before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new thing here is putting Kerr nonlinearity inside the non-Hermitian chain, not just the Hermitian one. That removes the fine-tuning condition delta = lambda - J from Ref. 11 and goes beyond the partial delocalization of Ref. 69. The main delocalization result, including the flat, square, triangle, and cosine profiles, is backed by self-consistent numerical solutions and by the closed-form plateau heights in Supplementary Note 4 (|aR| = sqrt((J+delta-lambda)/beta), |aL| = sqrt((tau-t)/alpha)). That is real evidence; the formulas are derived from the eigenequations, not fitted. The authors also honestly report the cases where the NHSE wins, which lends credibility.\n\nThe soft spot is exactly the one the stress-test note flags. In the spectral localizer section, H_S = S H S^{-1} is introduced, but the nonlinear hoppings lambda_j depend on the original amplitudes |a_{j+1}|^2 and |b_j|^2. After the similarity transformation with position-dependent r^k, those become r^{-2(j-N)} times the transformed amplitudes. The paper never states which convention goes into H_S when the localizer is evaluated. If Fig. 4 uses the naive |psi_bar|^2, then C_zeta and mu_zeta describe a different nonlinear problem, and the disorder criterion in Eq. (5)—where physical disorder is conjugated by S—does not certify protection of the original mode. This is a genuine ambiguity in a load-bearing step, not a nitpick.\n\nA second, milder issue: the phrase \"unattainable in Hermitian systems\" is too broad. Supplementary Note 7(B) shows that a Hermitian chain can host a static extended TZM (Fig. S10a) but cannot fully excite it under pumping. So the honest claim is about dynamical excitation, not about the existence of delocalized modes. The text mostly says the right thing, but the abstract and conclusion overreach.\n\nThe circularity concern raised in the reader's report is minor: the localizer is applied to modes already found by direct solution, so the C_zeta jumps are partly built in. That is typical for nonlinear localizer studies, and the local gap computation still adds information. Not a fatal flaw.\n\nWho is this for: people working on nonlinear topological photonics or topolectrical circuits who want a design knob for wavefunction shaping. The 2D extension is a nice bonus. Code is not public, but the data is on Zenodo; given the analytic plateau formulas, that is acceptable.\n\nRecommendation: send it to peer review, but the referee should ask for an explicit statement of how H_S is evaluated after the similarity transform—ideally a worked example for one profile, or a reference to the exact convention. That is a fixable clarity issue, not a reason to reject.","headline":"The full-delocalization mechanism is real and well supported by self-consistent numerics and simple plateau formulas, but the spectral localizer section underspecifies how the nonlinearity transforms under the similarity map, which keeps the topological-protection claim provisional.","tokens_in":26961,"tokens_out":1120,"would_cite":true,"duration_ms":12680,"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":"A non-Hermitian lattice with nonlinear couplings can spread a topological zero mode across every site and shape it into arbitrary profiles without fine-tuning.","keywords":["non-Hermitian skin effect","topological zero modes","nonlinear SSH model","spectral localizer","nonreciprocal hopping","Kerr nonlinearity","wavefunction engineering","topological photonics"],"falsifier":"Evaluate the self-consistent zero-mode solution with the nonlinear coefficients $\\lambda_j$ computed from the similarity-transformed wavefunction instead of the original wavefunction. If the mode no longer fills the lattice at $\\delta = 1.5$, or if the spectral localizer's spectrum no longer crosses zero with a corresponding change in $C_{\\zeta}$, then the complete-delocalization or topological-protection claim would be refuted for that formulation.","tokens_in":25802,"feed_emoji":"🔗","tokens_out":10479,"duration_ms":90084,"temperature":0.7,"pith_summary":"This paper proposes a one-dimensional interface model that joins a Hermitian nonlinear Su-Schrieffer-Heeger (SSH) chain to a non-Hermitian chain with nonreciprocal hopping, with Kerr-type nonlinear couplings whose strengths grow with local intensity. It aims to show that the topological zero mode, normally pinned to the interface or boundary, can be made to occupy every site of the combined lattice without tuning the system to the critical non-Hermitian condition $\\delta_c = \\lambda - J$. If correct, this removes a major obstacle to compact topological devices: the mechanism that delocalizes the mode also lets the user shape its wavefunction into flat, square, triangle, or cosine profiles by designing the hopping pattern. The paper further argues, via a real-space spectral localizer, that these extended modes remain topologically protected against disorder, and that external pumping can dynamically prepare the designed profiles, including long-range patterns that Hermitian systems cannot reach.","feed_headline":"Nonlinearity spreads topological modes across the entire lattice","feed_subtitle":"No fine-tuned hopping needed: the zero mode fills all sites and can be shaped into flat, square, triangle, or cosine profiles.","key_machinery":"The load-bearing object is the nonlinear non-Hermitian SSH interface model, whose intercell hoppings are intensity-dependent: $t_j = \\tilde{t}_j + \\alpha(|a_{j+1}|^2 + |b_j|^2)$ in the Hermitian chain and $\\lambda_j = \\tilde{\\lambda}_j + \\beta(|a_{j+1}|^2 + |b_j|^2)$ in the non-Hermitian chain, with nonreciprocal intracell hoppings $J \\pm \\delta$. The decisive mechanism is that the nonlinearity counteracts the exponential pinning produced by the skin effect: as the total intensity $I$ grows, the effective hopping in the non-Hermitian chain rises until the zero mode spreads across both chains without needing $\\delta = \\delta_c$. The topological-protection argument is carried by the real-space spectral localizer, a matrix that combines the position operator with the Hamiltonian after a similarity transformation $S$ makes the non-Hermitian Hamiltonian Hermitian; its smallest singular value $\\mu_{\\zeta}$ is the local gap, and its signature $C_{\\zeta} = \\tfrac{1}{2}\\mathrm{Sig}(\\tilde{L}_{\\zeta})$ is the local topological invariant. The paper uses the crossing of the localizer spectrum through zero and the associated change of $C_{\\zeta}$ as the real-space bulk-boundary correspondence for these nonlinear extended modes.","core_discovery":"The central claim is that nonlinearity and the non-Hermitian skin effect act together to release a topological zero mode from its boundary pinning: when nonlinear hopping is present in both chains, the zero mode of the SSH interface model spreads uniformly over the whole lattice even when the nonreciprocal hopping $\\delta = 1.5$ is far from the linear critical value $\\delta_c$, something neither the skin effect alone nor nonlinearity in a Hermitian chain can do. The paper shows that the spatial profile of the delocalized mode can be engineered at will by designing site-dependent hopping amplitudes $\\tilde{t}_j$ and $\\tilde{\\lambda}_j$, producing flat, square, isosceles-triangle, and cosine shapes, and that the plateau height in each chain is set by the nonlinear coefficients $\\alpha$ and $\\beta$. Using the spectral localizer on the similarity-transformed Hermitian Hamiltonian, the paper finds that the zero of the localizer spectrum moves with intensity, the local invariant $C_{\\zeta}$ changes sign at those points, and the local gap closes, which it reads as the real-space signature of a topological zero mode and the origin of the mode's protection against disorder. The same interplay is shown to work dynamically: pumping a single site drives the system into the designed steady-state profile, and in two dimensions, adding nonlinearity along both stacking directions delocalizes corner modes across the entire 2D lattice.","pith_inferences":["A testable check of the topological-protection argument: recompute the spectral localizer with the nonlinear coefficients $\\lambda_j$ evaluated using the similarity-transformed amplitudes $|\\bar{\\psi}|^2$ rather than the original amplitudes; if the zero crossing of $\\sigma(\\tilde{L}_{\\zeta})$ disappears, the localizer as written describes a different nonlinear problem and the protection statement ","Because the target profile is encoded in the site-dependent hopping pattern rather than in a global parameter, the same lattice could in principle be reconfigured between shapes by tuning $\\alpha$ and $\\beta$ externally, a natural route toward programmable topological photonic devices.","The 2D example suggests the mechanism may generalize to other higher-order topological lattices, but since only the BBH-type stacking is treated, that generalization remains conjecture rather than a claim of the paper."],"forward_implications":["A topological zero mode can be spread over the entire lattice without satisfying the linear critical condition $\\delta_c = \\lambda - J$, so delocalization no longer requires precise parameter tuning.","The wavefunction profile of the extended mode is user-designable: flat, square, isosceles-triangle, and cosine shapes are achieved by engineering $\\tilde{t}_j$ and $\\tilde{\\lambda}_j$, with plateau heights set by $\\alpha$ and $\\beta$.","The extended modes keep topological protection: disorder in on-site energies or hoppings leaves the designed profiles essentially unchanged, consistent with the local invariant $C_{\\zeta}$ and local gap $\\mu_{\\zeta}$ from the spectral localizer.","Under external pumping with staggered losses, an initially localized excitation evolves into the designed steady-state profile; in the non-Hermitian model this works over much longer lattices than in the Hermitian case, enabling long-range pattern excitation.","Stacking the 1D chains into a 2D lattice delocalizes higher-order topological corner modes across the whole 2D system when nonlinearity is added along both directions."],"supporting_citations":[{"why":"Supplies the linear non-Hermitian morphing result and the critical condition $\\delta_c = \\lambda - J$ that this paper's nonlinear mechanism removes.","marker":"[11]"},{"why":"Shows partial delocalization of topological zero modes by nonlinearity in a Hermitian SSH chain and introduces configurable profiles and the similarity-function stability measure reused here.","marker":"[69]"},{"why":"Provides the spectral-localizer construction, reduced chiral form, local invariant $C_{\\zeta}$, and local gap $\\mu_{\\zeta}$ used to certify topological protection.","marker":"[76]"},{"why":"Extends the spectral localizer to nonlinear topological materials, supporting its use for the intensity-dependent Hamiltonian.","marker":"[75]"},{"why":"Applies the spectral localizer to photonic heterostructures, supporting its validity for the interface geometry.","marker":"[74]"},{"why":"The BBH lattice underlies the 2D stacking extension and the associated higher-order corner modes.","marker":"[80]"}],"fun_headline_variants":["Nonlinearity and skin effect release topological modes from edges","No fine-tuning: nonlinearity delocalizes topological states","Control topological mode shapes via nonlinear hopping","Skin effect meets nonlinearity to spread zero modes","Whole-lattice topological modes from nonlinearity without fine-tuning"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of topological protection assumes that the transformation used to convert the non-Hermitian lattice into a Hermitian one leaves the nonlinear hopping strengths exactly as they were, but the paper does not specify whether those strengths are evaluated from the original or the transformed wavefunction amplitudes.","fun_headline_variants_meta":{"raw":{"variants":["Nonlinearity and skin effect release topological modes from edges","No fine-tuning: nonlinearity delocalizes topological states","Control topological mode shapes via nonlinear hopping","Skin effect meets nonlinearity to spread zero modes","Whole-lattice topological modes from nonlinearity without fine-tuning"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000631,"raw_usage":{"total_tokens":2954,"prompt_tokens":1022,"completion_tokens":1932,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":638,"completion_tokens_details":{"reasoning_tokens":1857}},"tokens_in":638,"tokens_out":1932,"duration_ms":15314,"temperature":1.0,"reasoning_tokens":1857,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T19:41:00.582435+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the self-consistent zero-mode solution with the nonlinear coefficients $\\lambda_j$ computed from the similarity-transformed wavefunction instead of the original wavefunction. If the mode no longer fills the lattice at $\\delta = 1.5$, or if the spectral localizer's spectrum no longer crosses zero with a corresponding change in $C_{\\zeta}$, then the complete-delocalization or topological-protection claim would be refuted for that formulation.","supporting_citations":[{"cited_title":"Bai , author J.-Z","cited_arxiv_id":null,"evidence_quote":"Shows partial delocalization of topological zero modes by nonlinearity in a Hermitian SSH chain and introduces configurable profiles and the similarity-function stability measure reused here."},{"cited_title":"Cheng , author A","cited_arxiv_id":null,"evidence_quote":"Provides the spectral-localizer construction, reduced chiral form, local invariant $C_{\\zeta}$, and local gap $\\mu_{\\zeta}$ used to certify topological protection."},{"cited_title":"Wong , author T","cited_arxiv_id":null,"evidence_quote":"Extends the spectral localizer to nonlinear topological materials, supporting its use for the intensity-dependent Hamiltonian."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Applies the spectral localizer to photonic heterostructures, supporting its validity for the interface geometry."}],"review_version":1}