{"id":"f977cdd3-6b36-451e-a871-6e05e574b3fe","arxiv_id":"2411.12424","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"In nonlinear non-Hermitian lattices, open-boundary skin-mode energies are no longer a subset of the semi-infinite spectrum, and coupling impurities create new localized modes and dark solitons.","lead":"This paper studies a one-dimensional lattice with asymmetric hopping and nonlinearity, and shows that nonlinearity changes the energy range of edge-localized skin modes and can create new localized states. It offers a fixed-point view that may help design nonlinear photonic devices that trap or route light.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"OBC-vs-SIBC spectral claim hinges on accepting finite-N roots with nonzero infinite-lattice tails as localized skin modes; a thermodynamic-limit localization check is missing.","rationale":"The paper's fixed-point framework is internally coherent: for E inside the linear PBC ellipse the zero fixed point is linearly stable, producing SIBC-compatible skin modes, while for E>Ec finite-size OBC roots can occur because the right boundary imposes only a single-point condition. The shooting construction genuinely produces solutions of the stated finite boundary-value problem, so the non-subset statement is not a straightforward algebraic error. The load-bearing weakness is semantic and asymptotic: the central claim is labeled as a statement about skin modes, but the computation only proves existence of solutions to the N-point problem. Whether these solutions are localized left-edge modes in the thermodynamic limit is not established. The reader's conditional verdict already identified this as the most fragile premise, and I agree. A concrete test of localization versus system size would settle the issue: if the modes remain exponentially localized with N-independent width, the claim stands; if their support grows with N or they carry O(1) weight at the right edge, the headline should be downgraded to a finite-size spectral effect. Either way, the paper's useful fixed-point analysis and the impurity-induced soliton results are not invalidated. I therefore see no reason to move away from the reader's conditional acceptance, provided the localization check is supplied or the claims are suitably tightened.","tokens_in":9889,"tokens_out":15590,"duration_ms":165731,"concrete_test":"For the representative E=1.3>Ec modes of Fig. 2, repeat the shooting calculation for N=200 and N=400 at the same E and locate the psi_1 roots closest to those found at N=100. For each root, compute the fraction of total power in the first 20 sites and the ratio |psi_N|/|psi_1|. A genuine left-localized skin mode should have power fraction approaching 1 and |psi_N|/|psi_1| exponentially small as N grows. If instead the right-edge amplitude stays O(1) or the power fraction decays with N, the E>Ec objects are finite-size boundary solutions rather than skin modes, and the OBC/SIBC non-inclusion claim loses its physical content. As a secondary check, allow complex psi_1 and complex E for N=10 and search for roots of psi_{N+1}=0; any complex-energy root would falsify the unproved 'real OBC spectrum' remark.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that the OBC spectrum is not a subset of the SIBC spectrum rests on treating a shooting root psi_{N+1}(E, psi_1)=0 at E>Ec as a genuine nonlinear OBC skin mode, even when continuing the recurrence past site N+1 would drive the field to a nonzero fixed point rather than to zero (Sec. II, Fig. 2). These roots are legitimate solutions of the finite N-point boundary-value problem, since OBC imposes equations only for 1<=n<=N, but the paper's own definition of OBC spectrum requires 'stable skin modes localized at the left edge.' Satisfaction of one boundary equation at n=N+1 does not establish that the mode is a localized skin mode in the thermodynamic limit. In the linear system the OBC subset property is tied to the fact that both characteristic exponents have modulus sqrt(gamma)<1, so continuation automatically decays; here the E>Ec modes break that link by relying on a single right-boundary zero. The text gives no evidence that these modes remain exponentially localized with N-independent localization length, rather than being finite-size boundary artifacts or extended bulk modes of the finite problem. Without this check, the headline statement overstates what the shooting method demonstrates. The additional assertion that the nonlinear OBC spectrum is real-valued rests only on an unproved numerical observation about Nc for complex E, not on an argument or exhaustive scan.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a one-dimensional tight-binding lattice with asymmetric couplings (JL=1, JR=γ<1) and several nonlinearities. Using a numerical shooting method and a fixed-point analysis of the recurrence, the author claims that (i) the semi-infinite (SIBC) spectral region is identical to the linear PBC loop irrespective of nonlinearity; (ii) the open-boundary (OBC) spectrum is not a subset of the SIBC spectrum, with OBC modes at real energies outside the linear interval ±(1+γ); (iii) nonlinear OBC skin modes exhibit a continuum of energies, degeneracies, and power-energy discontinuities; and (iv) a coupling impurity produces localized modes that are neither skin nor scale-free, as well as dark and anti-dark solitons. The central conceptual tool is the stability of zero and nonzero fixed points of the nonlinear recurrence.","tokens_in":10135,"tokens_out":12774,"duration_ms":118272,"significance":"If the non-subset result were rigorously established, it would be a notable qualitative difference between linear and nonlinear non-Hermitian systems and would provide a constructive fixed-point perspective for building nonlinear skin modes. The paper has clear strengths: the zero-fixed-point stability argument in Sec. II is analytic and shows that nonlinear parameters do not enter the linearized perturbation equation for a0=0; the Ablowitz-Ladik example yields a closed-form stability condition for the nonzero fixed point; and the impurity-induced dark/anti-dark soliton construction is simple and analytically grounded. However, the central spectral claims rest on a numerical shooting study whose thermodynamic-limit interpretation is not provided, and at least one key assertion (the real-valued OBC spectrum) is presented as a numerical suggestion rather than a demonstrated fact. The significance is therefore conditional on a more careful treatment of finite-N versus thermodynamic-limit modes.","major_comments":[{"comment":"The abstract's claim that the OBC spectrum is not a subset of the SIBC spectrum is not established by the presented evidence. In Sec. II the author defines the OBC spectrum as the set of energies of 'stable skin modes localized at the left edge.' The modes used to support the non-subset claim, shown in Fig. 2, satisfy ψ_{N+1}(E,ψ1)=0 for finite N while the infinite-lattice continuation of the same recurrence tends to a nonzero fixed point rather than decaying to zero. Such a continuation is not localized at the left edge in any thermodynamic sense, and the finite-N boundary condition alone does not make it a skin mode. The author notes that the roots are 'almost the same' for N=5, 10, and 100 but gives no localization-length analysis, no N→∞ limit, and no argument that the modes are not finite-size artifacts of the right boundary. Without a demonstration that these modes persist as exponentially localized states in the thermodynamic limit, the central 'non-subset' statement overstates what the shooting calculation shows.","section":"Section II, Fig. 2"},{"comment":"The assertion that the nonlinear OBC spectrum is real-valued is supported only by the sentence 'Our numerical calculations reveal that Nc does not take finite values when E is complex, suggesting that the nonlinear OBC spectrum is real valued.' This is a conjecture based on an undocumented numerical scan, not a proof or even a quantified numerical result. No complex-energy grid, tolerance, or convergence criterion is reported. Since the reality of the OBC spectrum is used to compare the nonlinear OBC interval with the linear interval ±2√γ, this claim is load-bearing and needs either an analytic argument or a fully documented numerical study.","section":"Section II"},{"comment":"The numerical shooting method is not described to the standard needed for the central claims to be assessable. The text does not state the root-finding algorithm, the tolerance for 'satisfying' ψ_{N+1}=0, the number of iterations, the sampling of the E and ψ1 continua, or the numerical precision used for Figs. 1(c), 1(d), and 2(a). The OBC spectrum, the continuum of energies, the degeneracies, and the power-energy discontinuities all follow from these roots, so the absence of any accuracy or convergence analysis is a load-bearing gap. A reader cannot reproduce or verify the spectral claims from the information given.","section":"Section II, Figs. 1(c)-(d)"},{"comment":"The claim that impurity-induced modes form 'a family of localized modes that are neither skin nor scale free localized modes' is not quantitatively substantiated. The profile shown in Fig. 3(a) is essentially a step: the field is zero up to N_c, rises sharply to a nonzero fixed-point value, and stays at that value until the right edge. This profile's support grows with N, and its 'localization length' in the direction away from the right edge is infinite (the field does not decay). The distinction from scale-free modes (whose localization length scales as N) is asserted without extracting any length scale from the data. A precise definition of localization for these modes and a scaling analysis in N are needed before the claimed new class of localized modes is established.","section":"Section III, Fig. 3(a)"}],"minor_comments":[{"comment":"The notation is inconsistent, e.g., 'ψN +1' appears alongside 'ψ_{N+1}', and several boundary conditions are written with spaces (ψ0 = ψN +1 = 0). The manuscript would benefit from a consistent formatting pass.","section":"Throughout"},{"comment":"The 'scaled ψ_{N+1}' is not defined; the reader cannot tell whether the plotted function is normalized, and by what factor.","section":"Fig. 1(c)"},{"comment":"The sentence 'a0 = 0 and a0 ≠ 0 ensure SIBC (or OBC when Nc is finite) and PBC, respectively [39]' is cryptic; the role of footnote [39] is unclear and it should either be integrated into the main text or removed.","section":"Sec. II"},{"comment":"The statement that 'nonlinear interactions are shown to have no impact on the spectral region' should be qualified as 'for the fixed-point skin modes constructed here,' since only the linear stability of the zero fixed point is analyzed.","section":"Abstract"},{"comment":"The parameters in the inset of Fig. 3(b) appear inconsistent with the caption: the main panel uses p = p' = 150, while the inset is described at p = 50 with different impurity strengths; the caption should be checked for consistency.","section":"Sec. III, Fig. 3(b)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's central numerical claims are not accompanied by a data/code availability statement. Given that the main spectral conclusions rest on shooting calculations, the editor may wish to request the code or full numerical details as part of the revision. The paper also leans on several self-citations; this is not inappropriate, but a careful check of the novelty relative to refs. [23] and [34] would be prudent. The core fixed-point construction is promising, but the OBC-vs-SIBC claim needs a clearly stated thermodynamic-limit definition before it can support the abstract's headline statement."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Know this paper for one thing: the fixed-point stability argument that explains why the semi-infinite spectrum is insensitive to nonlinearity. That part is clean and genuinely useful. Everything else in the abstract, particularly the claim that the nonlinear OBC spectrum is not a subset of the SIBC spectrum, is more fragile than it looks.\n\nWhat is new: the zero-fixed-point linear stability analysis is a nice organizing principle; it shows the SIBC spectrum is the linear PBC loop regardless of nonlinearity. The construction of nonlinear OBC modes with multiple powers at fixed energy and the power-energy branching are plausible and not in the cited literature. The impurity-induced modes that are neither skin nor scale-free are also interesting, though shown only for a few parameter sets.\n\nSoft spots: the central spectral claim rests on shooting roots psi_{N+1}(E,psi1)=0 for E>Ec. These are legitimate solutions of the finite-N boundary value problem, but the paper labels them “stable skin modes localized at the left edge.” For E>Ec the zero fixed point is unstable, so continuing the recurrence past N+1 drives the field to a nonzero fixed point. The mode is only zero at the artificial right boundary. In the linear system, OBC modes are automatically compatible with the semi-infinite tail because both characteristic exponents decay; here that link is broken. The paper provides no thermodynamic-limit analysis: no localization length vs N, no inverse participation ratio, no argument that these modes remain exponentially localized as N grows. The figures suggest the profile extends to the right edge, which would make them extended bulk modes of the finite system rather than skin modes. If “spectrum” is defined to require a decaying tail, the extra modes disappear. The real-valuedness of the OBC spectrum is also an unproved numerical observation. There is no reproducibility information (code, tolerances, convergence criteria).\n\nWho it’s for: researchers in nonlinear non-Hermitian lattice dynamics. The fixed-point picture deserves attention; the spectral claims need to be reworked with a proper thermodynamic-limit definition.\n\nMy recommendation: send to peer review. The idea is original enough and the fixed-point analysis solid enough to deserve referee time, but the referee should push for a rigorous definition of OBC spectrum and a numerical check of localization.","headline":"The fixed-point stability argument is a genuine contribution, but the headline claim that the nonlinear OBC spectrum is not a subset of the SIBC spectrum rests on finite-size shooting roots and needs a thermodynamic-limit check.","tokens_in":10689,"tokens_out":5074,"would_cite":false,"duration_ms":47061,"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":"Nonlinearity lets open-boundary skin modes in a one-dimensional lattice exist at energies that the semi-infinite spectrum forbids, so the open-boundary spectrum is not a subset of the semi-infinite spectrum.","keywords":["nonlinear skin effect","non-Hermitian lattice","open boundary conditions","fixed points","shooting method","scale-free localization","dark soliton","tight-binding model"],"falsifier":"Continue the shooting solution that satisfies $\\psi_{N+1}=0$ at some $E>1+\\gamma$ to $n=10N$ with no boundary imposed: if the field does not stay zero but grows to the nonzero fixed point $a_0$, the claimed open-boundary mode is an artifact of truncating at $N$. A second check is to solve the stationary equation on lattices of increasing $N$ and see whether energies outside $[-E_c,E_c]$ persist or recede; if they vanish as $N\\to\\infty$, the claimed non-containment of the open-boundary spectrum is false.","tokens_in":9633,"feed_emoji":"⚡","tokens_out":11867,"duration_ms":110262,"temperature":0.7,"pith_summary":"The paper studies a one-dimensional tight-binding lattice with asymmetric couplings and several types of on-site nonlinearity, asking how the non-Hermitian skin effect changes when nonlinearity is present. It argues that nonlinearity leaves the semi-infinite boundary spectrum unchanged—still the disk bounded by the linear periodic-boundary loop—but substantially reshapes the open-boundary spectrum, whose modes can appear at real energies outside that disk. The key mechanism is that on a finite lattice a field can satisfy the open-boundary condition $\\psi_{N+1}=0$ while its infinite continuation would approach a nonzero fixed point rather than decay to zero, so open-boundary skin modes are no longer constrained by the semi-infinite spectrum. If this is correct, nonlinear skin modes are power-dependent, form continuous energy bands, can be degenerate at a single energy, and show power-energy discontinuities, none of which occur in the linear case.","feed_headline":"Nonlinear skin modes appear outside the allowed linear spectrum","feed_subtitle":"On a finite non-Hermitian lattice, open-boundary skin modes occur at energies semi-infinite modes cannot reach, with degeneracies.","key_machinery":"The central object is a fixed point of the nonlinear recurrence in Eq. (1): a value $a_0 e^{i\\theta}$ such that continuing the field past some critical site $N_c$ leaves it unchanged, $\\psi_{N_c+1}=\\psi_{N_c+2}=\\dots=a_0 e^{i\\theta}$. There are zero and nonzero fixed points; the zero fixed point is stable exactly when $E$ lies inside the linear periodic-boundary loop $E=e^{ik}+\\gamma e^{-ik}$, independent of the nonlinear parameters, while nonzero fixed points have energy windows of stability set by the nonlinearity. The shooting method converts the boundary-value problem into iterating from a guessed amplitude $\\psi_1$ and scanning $(E,\\psi_1)$ for roots of $\\psi_{N+1}=0$, and the paper uses the stability of these fixed points to read off which energies admit skin modes. The load-bearing mechanism is that $\\psi_{N+1}=0$ no longer forces $\\psi_\\infty=0$ once nonlinearity is present: a trajectory can satisfy open boundary conditions and then settle onto a nonzero fixed point, which is exactly why open-boundary modes can live at energies outside the semi-infinite region.","core_discovery":"The central discovery, stated on the paper's own terms, is that the open-boundary-condition spectrum of a nonlinear non-Hermitian lattice is not a subset of its semi-infinite-boundary-condition spectrum. For the linear system, open-boundary energies are confined to the real interval $[-2\\sqrt{\\gamma},2\\sqrt{\\gamma}]$ inside the semi-infinite spectrum, but in the nonlinear case skin modes can occur at real energies $E>1+\\gamma$, outside the semi-infinite region. The reason is a failure of the implication $\\psi_{N+1}=0 \\Rightarrow \\psi_\\infty=0$: a shooting solution can pass through zero at the right boundary and then rise to a nonzero fixed point if continued further, so the finite-lattice mode satisfies open boundary conditions even though it would not be a semi-infinite mode. The paper also finds that the open-boundary spectrum is real, continuous in energy, and multiply degenerate, and that these nonlinear skin modes disappear when the couplings become Hermitian ($\\gamma=1$).","pith_inferences":["The power-energy discontinuity suggests that in an experiment sweeping pump power or energy, one would observe sudden jumps in the localization profile of open-boundary skin modes; such jumps could serve as a direct signature of the fixed-point mechanism.","If the open-boundary spectrum truly is not a subset of the semi-infinite spectrum, the standard non-Bloch band-theoretic characterization—which identifies open-boundary spectra with the generalized Brillouin zone—would need a nonlinear generalization, possibly phrased in terms of fixed-point basins rather than decay rates.","The paper constructs these outside-the-loop modes as stationary solutions but does not prove their dynamical stability; a time-dependent linearization around them would either confirm their physical stability or restrict them to a finite-time window.","The impurity-tunable dark solitons, whose width can be enlarged by placing neighboring impurities, could plausibly be tested in coupled optical waveguide arrays and might offer an optical switching or trapping mechanism."],"forward_implications":["For a finite lattice with asymmetric couplings, nonlinear open-boundary skin modes exist at real energies $E>1+\\gamma$, outside the semi-infinite spectrum, so the open-boundary spectrum is strictly larger than the semi-infinite spectrum in the nonlinear regime.","The nonlinear open-boundary spectrum is a continuum of real energies rather than isolated points, and two or more distinct skin modes with different powers can share the same energy, a degeneracy absent in the linear system.","The semi-infinite spectrum remains bounded by the linear periodic-boundary loop for both linear and nonlinear cases, meaning the nonlinearity affects the spectrum only through the boundaries.","A single coupling impurity can generate right-localized modes whose spatial extent grows with system size but whose profile is neither skin-like nor scale-free, and can also create discrete dark and anti-dark solitons under periodic boundary conditions.","At $\\gamma=1$ (Hermitian couplings), the zero fixed point loses stability and these nonlinear skin modes disappear, so the phenomena require broken reciprocity in the hopping amplitudes."],"supporting_citations":[{"why":"Defines impurity-induced scale-free localization, the class from which the new impurity modes are explicitly distinguished.","marker":"[9]"},{"why":"Shows that a single non-Hermitian defect can induce many size-dependent localized modes, providing the linear baseline for the impurity results.","marker":"[22]"},{"why":"Earlier construction of nonlinear skin modes that the fixed-point method here extends.","marker":"[23]"},{"why":"Supports treating semi-infinite skin modes as quasi-stationary open-boundary modes in sufficiently long lattices.","marker":"[34]"},{"why":"Provides the saturable nonlinearity form used in the generalized equation.","marker":"[35]"},{"why":"Provides the Ablowitz-Ladik nonlinearity, which admits the analytic nonzero fixed point used in the stability argument.","marker":"[36]"},{"why":"Provides the cubic-quintic-sextic nonlinearity form included in the model.","marker":"[37]"},{"why":"Supports the quasi-stationary interpretation of semi-infinite modes under open boundary conditions.","marker":"[38]"}],"fun_headline_variants":["Nonlinear skin modes break the spectral subset rule","Skin modes appear outside semi-infinite spectrum in nonlinear lattices","Open-boundary spectrum not subset of semi-infinite: nonlinear skin modes","Fixed-point view: nonlinear skin modes escape the linear band"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction treats a finite-lattice solution with $\\psi_0=\\psi_{N+1}=0$ as a genuine open-boundary skin mode even when continuing the recurrence beyond site $N+1$ would make the field rise to a nonzero fixed point rather than decay to zero; if 'spectrum' is required to mean modes whose tail is compatible with the semi-infinite boundary condition, those extra energies disappear.","fun_headline_variants_meta":{"raw":{"variants":["Nonlinear skin modes break the spectral subset rule","Skin modes appear outside semi-infinite spectrum in nonlinear lattices","Open-boundary spectrum not subset of semi-infinite: nonlinear skin modes","Fixed-point view: nonlinear skin modes escape the linear band"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000659,"raw_usage":{"total_tokens":2985,"prompt_tokens":890,"completion_tokens":2095,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":506,"completion_tokens_details":{"reasoning_tokens":2026}},"tokens_in":506,"tokens_out":2095,"duration_ms":14388,"temperature":1.0,"reasoning_tokens":2026,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:32:49.154461+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Continue the shooting solution that satisfies $\\psi_{N+1}=0$ at some $E>1+\\gamma$ to $n=10N$ with no boundary imposed: if the field does not stay zero but grows to the nonzero fixed point $a_0$, the claimed open-boundary mode is an artifact of truncating at $N$. A second check is to solve the stationary equation on lattices of increasing $N$ and see whether energies outside $[-E_c,E_c]$ persist or recede; if they vanish as $N\\to\\infty$, the claimed non-containment of the open-boundary spectrum is false.","supporting_citations":[{"cited_title":"Hybrid scale-free skin effect in non-Hermitian systems: A transfer matrix approach","cited_arxiv_id":null,"evidence_quote":"Shows that a single non-Hermitian defect can induce many size-dependent localized modes, providing the linear baseline for the impurity results."},{"cited_title":"Nonlinear non-Hermitian skin effect","cited_arxiv_id":null,"evidence_quote":"Earlier construction of nonlinear skin modes that the fixed-point method here extends."},{"cited_title":"Non-Hermitian excitations in nonlinear topological lat- tice","cited_arxiv_id":null,"evidence_quote":"Supports treating semi-infinite skin modes as quasi-stationary open-boundary modes in sufficiently long lattices."},{"cited_title":"Skin solitons","cited_arxiv_id":null,"evidence_quote":"Provides the saturable nonlinearity form used in the generalized equation."},{"cited_title":"A class of stable nonlin- ear non-Hermitian skin modes","cited_arxiv_id":null,"evidence_quote":"Provides the Ablowitz-Ladik nonlinearity, which admits the analytic nonzero fixed point used in the stability argument."},{"cited_title":"The discrete nonlinear Schrodinger equation: A survey of recent results","cited_arxiv_id":null,"evidence_quote":"Provides the cubic-quintic-sextic nonlinearity form included in the model."},{"cited_title":"Nonlinear differential- difference equations","cited_arxiv_id":null,"evidence_quote":"Supports the quasi-stationary interpretation of semi-infinite modes under open boundary conditions."}],"review_version":1}