{"id":"119f7df4-1e76-4539-9486-211941fa9550","arxiv_id":"2608.08687","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In a 1D amorphous non-Hermitian chain, a singular-value-based real-space winding number stays quantized and predicts protected edge states even when the eigenvalue spectrum is unstable.","lead":"A team proposes using singular values, rather than energy eigenvalues, to reveal topologically protected boundary states in amorphous non-Hermitian quantum chains, where the eigenvalue spectrum is too unstable to read. The result is a real-space invariant that stays sharply quantized under disorder, giving experimentalists a practical probe for waveguides, photonic quasicrystals, and cold-atom arrays.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's central correspondence assumes a vanishing smallest singular value implies an eigenstate, but that is false for non-normal operators; its own pseudospectra show small singular values far from any eigenvalue.","rationale":"The central claim requires that singular-value zero modes of H-E_b I correspond to actual eigenstates of H at E_b in the thermodynamic limit. The only argument offered is the norm-convergence statement in Eqs. (8)-(9). That argument is insufficient: for non-normal operators, approximate eigenvectors need not converge to normalizable eigenstates, and the unilateral-shift example shows s_min tending to zero in the residual spectrum without any eigenstate. The paper's pseudospectrum plots (Fig. 3) explicitly demonstrate small singular values far from eigenvalues, so this is not a pedantic objection; it is the same regime the paper studies. The topological invariant V(E_b) counts zero modes of the SVD of H-E_b I, so if those zero-mode singular vectors are not eigenstates of H, the claim of topologically protected edge states does not follow. This concern does not by itself show the numerical conclusions are wrong: in the nontrivial-winding sector the underlying infinite-volume operator may be Fredholm with nonzero index, giving genuine kernel states. But that requires proof rather than assertion, and the paper does not supply it. The missing piece is exactly what a CONDITIONAL verdict should demand: an independent check that edge-localized singular modes approach true eigenstates as L grows, or a rigorous Fredholm/index statement. Because the reader already conditions acceptance on such a check, my assessment leaves the verdict at CONDITIONAL, while sharpening the reason the condition is necessary.","tokens_in":14026,"tokens_out":13244,"duration_ms":158949,"concrete_test":"Choose a single disorder realization in the topological phase (e.g., t=-1.5, gamma=0.2) and system sizes 2L=100, 200, 400. Compute the smallest singular value s_L and smallest singular vector v_L of H (or H-E_b I). Numerically diagonalize H and compute delta_L = min_lambda |lambda-E_b| and the maximum overlap O_L = max_lambda |<v_L|psi_lambda>| between v_L and normalized eigenstates of H. If delta_L does not decay to zero as L grows, or if O_L decays with L, the zero-mode singular state is not becoming an eigenstate and the correspondence fails. As a control, repeat at a point in the pseudospectral region outside the V(E_b) nonzero region of Fig. 6; if s_L decays there too but the singular vector is delocalized or has zero overlap with eigenstates, then only edge-localized modes tied to true eigenvalues should count as physical edge states.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the assertion around Eqs. (8)-(9) that if s_min(H-E_b I) tends to zero as L grows, then the right singular vector v_n becomes an eigenstate of H with eigenvalue E_b. This implication is not valid for non-normal operators. A vanishing smallest singular value only places E_b in the approximate point spectrum; in the thermodynamic limit it may lie in the residual or continuous spectrum, with no normalizable eigenvector. The standard forward-shift example has finite-truncation smallest singular values |z|^N tending to zero for every |z|<1, yet the infinite operator has no eigenstate in the open unit disk. The paper's own Fig. 3 uses exactly the equivalence s_min(zI-H)<epsilon to exhibit pseudospectra extending far beyond the discrete eigenvalues; those regions are not evidence of eigenstates. Thus the proof of the central correspondence is incomplete. What is needed is a Fredholm-index or spectral-projection argument showing that the zero-mode singular vectors selected by V(E_b) (edge-localized and stable in L) lie in the kernel of the infinite-volume H-E_b I, and that residual-spectrum pseudomodes are excluded. Without such an argument, Figs. 4 and 6 cannot be read as demonstrating protected eigenstates of H.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript studies point-gap topology in a one-dimensional amorphous non-Hermitian chain with random site positions, exponential hopping, and nonreciprocal hopping strength γ. Starting from the observation that the eigenvalue spectrum of the disordered non-Hermitian Hamiltonian is highly unstable, with large ε-pseudospectra extending far beyond the eigenvalues (Fig. 3), the authors propose that stable zero-mode singular states of H−E_b I are in one-to-one correspondence with mid-gap eigenstates of H in the thermodynamic limit (Eqs. (8)–(9)). They construct an enlarged Hermitian operator H̃(E_b) with chiral symmetry and a real-space winding number V(E_b) = (1/2πi) Tr ln(P_A P_B†) built from the SVD of H−E_b I, and claim that |V(E_b)| predicts the number of topologically protected edge states of H at energy E_b. The numerics show sharply quantized V(0) for several system sizes (Fig. 4), a (t, γ) topological phase diagram (Fig. 5), and coincidence of the nontrivial V(E_b) region with the small-s_min(H−E_b I) region (Fig. 6). An experimental detection protocol based on the Loschmidt echo is sketched.","tokens_in":14251,"tokens_out":23472,"duration_ms":245823,"significance":"The proposed strategy is potentially valuable: singular values of H−E_b I are Lipschitz-stable under bounded perturbations, a Hermitian-stability property the paper correctly emphasizes, so they offer a route to probing point-gap topology in systems where the eigenvalue spectrum is unusably unstable. Extending the SVD-based approach of Refs. [85–87] from translationally invariant and Floquet systems to amorphous systems without any periodicity is a natural and worthwhile generalization, and the paper states its model and diagnostics clearly. The finitary numerical claims — near-integer V(0), edge-localized lowest singular vectors, and a clean phase boundary — are plausible and constitute a falsifiable prediction if supported by error statistics. However, the manuscript's central interpretive step, the thermodynamic-limit correspondence between zero singular values and true eigenstates, is asserted without proof and is false for general non-normal operators; the paper's own pseudospectra demonstrate the danger. The contribution is therefore conditional on repairing that step.","major_comments":[{"comment":"The thermodynamic-limit correspondence asserted in the paragraph containing Eqs. (8)–(9) is the load-bearing step of the paper, and the implication that lim_{L→∞} s_min(H−E_b I)=0 implies a normalizable eigenstate v_n of H with eigenvalue E_b is not valid for non-normal operators. The standard counterexample is the unilateral shift: for the finite truncations S_N, s_min(zI−S_N) decays to zero for every |z|<1, yet the infinite shift has no eigenvalue in the open unit disk, and the limiting singular vectors converge to eigenvectors of S*, not of S. This is not a remote pathology here, because the paper's own Figs. 3(e)–(h) show large regions where s_min(zI−H)<10^{-2} far from any eigenvalue; by the proposed implication those regions would be filled with eigenstates, contradicting Figs. 3(a)–(d). Consequently, the reading of Fig. 6(b) as evidence that the small-s_min region hosts protected eigenstates of H at energy E_b is unproven; those modes could equally be pseudomodes of the residual or continuous spectrum. I ask the authors to either (i) supply a convergence argument (for example via strong resolvent convergence plus a spectral-projection or Fredholm-index argument) showing that the edge-localized zero-mode singular vectors counted by V(E_b) lie in the point spectrum of the infinite-volume H−E_b I, or (ii) provide a direct numerical verification for this model: for the E_b values with V(E_b)≠0, compute the distance from E_b to the spectrum of H and the overlap of the singular vector with the corresponding eigenvector as functions of L. The scaling of s_min with L (algebraic versus exponential) should be reported, since exponential decay is the signature of pseudomodes rather than of an approaching eigenvalue.","section":"Eqs. (8)–(9) and Fig. 3"},{"comment":"The invariant V(E_b) in Eq. (11) is constructed from the SVD of the open-chain matrix H−E_b I over all sites, with the phase matrix P of Eq. (12) covering the full chain and the trace running over all singular vectors including the edge-localized ones; the lowest singular vectors of this same SVD are then identified as the predicted edge states. The text's claim that V(E_b) is computed solely from the bulk SVD data and predicts the edge-state count is therefore, as implemented, a self-consistency statement within a single decomposition rather than an independent bulk-boundary correspondence, in contrast to W(E_b) in Eq. (5), which at least uses the windowed trace Tr'. To support the predictive claim, the authors should compute V(E_b) from a bulk-windowed trace or from an auxiliary construction that does not use the edge data (for example, a periodically reordered arrangement of the same disorder realization), and show that this independent quantity still matches the number of edge-localized zero singular modes.","section":"Eqs. (10)–(12), Figs. 4 and 6"},{"comment":"The central empirical claims — that V(0) is sharply quantized and that the phase boundary in Fig. 5 is sharply resolved — are presented without statistical support: Figs. 2, 4, and 6 give disorder averages over 50–1000 realizations but no standard deviations, quantiles, or per-realization distributions, and Fig. 5 shows a single averaged color map with 100 realizations per grid point. Because V(E_b) is a phase-winding type quantity, its per-realization distribution is essential: the authors should state the quantization criterion (for example, the fraction of realizations with |V−1|<ε for a stated ε) and report the typical fluctuation scale. Without this, the reader cannot distinguish genuine quantization from an artifact of averaging.","section":"Figs. 4 and 5 (statistics)"}],"minor_comments":[{"comment":"The claimed advantage of V(E_b) over W(E_b) in terms of higher accuracy and less fluctuation is not demonstrated; no plot comparing the two quantities on the same disorder realizations is provided.","section":"Eq. (5) vs Eq. (11)"},{"comment":"The diagnostics mcom and WIPR weight by the site index j and |j−L/2|, although the physical positions x_j are randomly distributed over [0, 2L]; for large density fluctuations the index is not a faithful spatial coordinate, and the actual positions x_j should be used instead.","section":"Eqs. (3)–(4)"},{"comment":"The branch-cut convention used to evaluate Tr ln in Eq. (11) is not specified; a brief statement of how V(E_b) is evaluated as an integer in finite systems would help the reader reproduce the calculation.","section":"Eq. (11)"},{"comment":"The manuscript contains numerous rendering problems (for example, '2Lsites', 'H−E bI', 'values min' in the Fig. 6 caption, and missing spaces in the author list), which should be corrected.","section":"Entire manuscript"},{"comment":"The announced redefinition of the non-Hermitian skin effect is not stated as a precise criterion; the authors should give an operational definition (for example, scaling of boundary weight with system size) that distinguishes skin states from ordinary edge-localized states.","section":"Near Fig. 6 and Conclusion"},{"comment":"The counting of |2V(E_b)| zero-mode edge states of H̃ implicitly assumes dim ker(H−E_b I) = dim ker(H†−E_b^* I) = |V(E_b)|; in general H̃ has dim ker(A)+dim ker(A†) zero modes, which need not equal 2|V(E_b)|, so the statement should be qualified.","section":"Paragraph after Eq. (12)"}],"recommendation":"major_revision","confidential_remarks":"The central risk is that the thermodynamic-limit correspondence may fail for this model just as it fails for general non-normal operators: the forward-shift counterexample is standard, and the paper's own Fig. 3 already shows that small singular values do not imply nearby eigenvalues in this very system. If the authors can show directly that the modes counted by V(E_b) converge to genuine eigenstates of H (for example, by computing the distance from E_b to the spectrum and the singular-vector/eigenvector overlap as functions of L), the paper would be a solid contribution. I also recommend that the editor ask the authors to delineate crisply the novelty relative to Refs. [85–87], two of which are by the same group; the present version leans on those references for the core idea. The manuscript is within the journal's scope, and the topic is timely."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this paper takes the SVD-based real-space winding number from Monkman-Sirker and the authors' prior Floquet work and applies it to a 1D amorphous non-Hermitian chain. The numerics show the winding number stays quantized and the smallest singular vectors are edge-localized, in a regime where the eigenvalue spectrum is a mess. That is a genuine extension, and the phase diagram in (t, gamma) is convincing. If the approach holds up, it gives experimental groups a way to look for point-gap edge states without relying on eigenvalues.\n\nWhat the paper does well: the pseudospectrum analysis in Fig. 3 makes the spectral instability concrete, the disorder averaging is careful, and the presentation is clear. The invariant V(E_b) is a sensible real-space generalization of the chiral winding number.\n\nThe soft spot is the load-bearing correspondence around Eqs. (8)-(9). The authors claim that if s_min(H - E_b I) -> 0 as L grows, then the right singular vector v_n becomes an eigenstate of H with eigenvalue E_b. For non-normal H that is false: a vanishing smallest singular value only means E_b lies in the approximate point spectrum. The standard forward-shift example has s_min = |z|^N, which goes to zero for every |z|<1, yet the infinite-volume operator has no eigenstates in the open disk. The paper's own Fig. 3 uses exactly this equivalence to plot pseudospectra: regions where s_min is small but no eigenvalue exists. So the claimed correspondence needs a spectral-projection or Fredholm-index argument that excludes residual-spectrum pseudomodes. Without it, Figs. 4 and 6 cannot be read as demonstrating protected eigenstates of H.\n\nThere is a related circularity: V(E_b) and the edge-state detection both come from the SVD of H - E_b I. The paper does not provide an independent bulk-vs-boundary test, e.g., computing the invariant on the bulk while checking edge modes separately. Error bars are also missing; the phase diagrams are averaged but no standard deviation is reported. And the 'redefine the non-Hermitian skin effect' language outruns the results: the singular vectors shown are not shown to be eigenstates.\n\nBottom line: this deserves a serious referee, not a desk reject. The method is plausible and the numerics are substantial. But the referee should ask for a rigorous statement of the thermodynamic-limit correspondence, an independent bulk-boundary check, error bars, and code/data. I would not cite it in its current form.\n\nBest,","headline":"A useful SVD-based probe for amorphous non-Hermitian chains, but the central correspondence between vanishing singular values and eigenstates is unproven and contradicted by the paper's own pseudospectra.","tokens_in":14833,"tokens_out":4633,"would_cite":false,"duration_ms":46688,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"In amorphous non-Hermitian chains, the real-space winding number computed from the singular-value decomposition stays quantized and predicts edge states even when the eigenvalue spectrum is unstable.","keywords":["non-Hermitian topology","point-gap","amorphous systems","singular value decomposition","skin effect","real-space winding number","spectral instability","pseudospectrum"],"falsifier":"Take a disorder realization of the amorphous chain and a reference energy $E_b$ where $s_{\\min}(H-E_b I)$ is numerically zero; compute the distance $\\min_n|\\lambda_n-E_b|$ from the eigenvalues to $E_b$ as the system size $L$ grows. If that distance stays of order one while $s_{\\min}$ keeps vanishing, the zero-mode singular vector is not converging to an eigenstate and the correspondence is falsified.","tokens_in":13785,"feed_emoji":"⚛️","tokens_out":7202,"duration_ms":64725,"temperature":0.7,"pith_summary":"This paper argues that in one-dimensional amorphous non-Hermitian systems the ordinary eigenvalue spectrum is too unstable to reveal point-gap topology, so topological edge states must be diagnosed through singular values instead. It establishes that stable zero-mode singular states of $H-E_b I$ correspond to mid-gap eigenstates of $H$ in the thermodynamic limit, and it constructs a real-space winding number $V(E_b)$ from the singular-value decomposition that stays sharply quantized even under strong disorder. If the correspondence holds, singular-value spectra become a reliable experimental probe of non-Hermitian topology in systems, such as amorphous lattices, where translation symmetry is absent.","feed_headline":"Singular-value probe spots edge states hidden in unstable spectra","feed_subtitle":"In amorphous non-Hermitian chains, a real-space SVD winding number stays quantized and counts topological edge states.","key_machinery":"The engine of the argument is the singular-value decomposition $H-E_b I=U_A S U_B^\\dagger$ and the enlarged Hermitian Hamiltonian $\\tilde H(E_b)$ with chiral symmetry $\\Sigma\\tilde H(E_b)\\Sigma^{-1}=-\\tilde H(E_b)$. From $U_A$ and $U_B$ and a position matrix $P$ encoding real-space phases, the paper defines the winding number $V(E_b)=\\frac{1}{2\\pi i}\\operatorname{Tr}\\ln(P_A P_B^\\dagger)$, whose modulus counts the zero-mode edge states of $\\tilde H(E_b)$ and hence the edge states of $H$ at energy $E_b$. The singular values $s_n$ are square roots of eigenvalues of the positive Hermitian operator $(H-E_b I)^\\dagger(H-E_b I)$, which gives the singular spectrum the stability that the non-Hermitian eigenvalue spectrum lacks.","core_discovery":"The central claim is that for the amorphous non-Hermitian chain with random site positions and exponential hopping, the real-space winding number $V(E_b)=\\frac{1}{2\\pi i}\\operatorname{Tr}\\ln(P_A P_B^\\dagger)$ computed solely from the SVD of $H-E_b I$ is sharply quantized and counts the number of topologically protected edge states at energy $E_b$ in the thermodynamic limit. This remains true even though the eigenvalue spectrum is severely unstable: as the system size grows, the complex-plane eigenvalue distribution becomes structureless and the pseudospectrum balloons outward, while the singular values of $H-E_b I$ remain robust because they are eigenvalues of the Hermitian operator $(H-E_b I)^\\dagger(H-E_b I)$. The paper therefore proposes that the number of zero-mode singular states of $H-E_b I$ is linked to the number of stable $E_b$-mode states of $H$, and that the non-Hermitian skin effect should be identified through these singular vectors rather than through eigenstate localization.","pith_inferences":["If the correspondence is generic, singular-value invariants could replace spectral winding numbers for disordered non-Hermitian systems in higher dimensions and with multiple bands, since the same Hermitian-stability argument applies.","The proposed redefinition of the skin effect implies that skin modes should be diagnosed by zero-mode singular vectors rather than by eigenstate localization; this would change how experiments identify skin modes in systems with strong disorder.","A direct numerical test on small random matrices would be to fix $E_b$, add a weak perturbation to the hopping amplitudes, and compare how much the smallest singular values move versus how much the eigenvalues move; the paper's claim predicts the singular values remain nearly fixed.","The framework connects to pseudospectrum theory: it suggests that the low singular values inside the $\\varepsilon$-pseudospectrum carry topological integer data even where the spectrum itself is unstable."],"forward_implications":["Even when the eigenvalue spectrum shows no clear gap, the SVD-based winding number $V(E_b)$ identifies the topological phase and predicts edge states in finite amorphous samples.","The number of zero-mode singular values of $H-E_b I$ equals the number of stable mid-gap eigenstates in the thermodynamic limit, making singular values a practical diagnostic for non-Hermitian topology.","The real-space invariant extends point-gap bulk-boundary correspondence to disordered systems without translational symmetry, including amorphous lattices.","On experimental platforms such as waveguide arrays or cold atoms in optical tweezers, the topological edge states can be detected by preparing a zero-mode singular state and measuring a generalized Loschmidt echo that stays near unity."],"supporting_citations":[{"why":"establishes the correspondence between zero-mode singular states and stable zero-mode topological edge states in multiband non-Hermitian systems, which this work generalizes to amorphous systems.","marker":"[85]"},{"why":"demonstrates the pseudospectral origin of the non-Hermitian skin effect and the instability of eigenvalue spectra, motivating the singular-value approach.","marker":"[84]"},{"why":"introduces the singular-value-based diagnostic scheme for topology in Floquet non-Hermitian systems, whose WIPR and SVD methods this paper adapts.","marker":"[86]"},{"why":"provides the singular-value-based bulk-boundary correspondence for a second-order topological insulator, used as a comparison benchmark.","marker":"[87]"},{"why":"supplies the pseudospectrum definition and the stability argument for Hermitian spectra used to justify singular-value robustness.","marker":"[88]"},{"why":"defines a bulk-edge correspondence for non-Hermitian Hamiltonians via singular-value decomposition, the conceptual basis for the SVD invariant.","marker":"[116]"},{"why":"gives the real-space representation of the winding number for a one-dimensional chiral-symmetric topological insulator, adapted here with SVD data.","marker":"[118]"},{"why":"supports the chiral-symmetry construction of the enlarged Hamiltonian used to define the winding number.","marker":"[119]"}],"fun_headline_variants":["Real-space SVD winding counts edge states despite spectral instability","Singular values probe topological edge states in amorphous non-Hermitian systems","SVD probe sees through spectral instability to count skin modes","Amorphous non-Hermitian topology from real-space singular values","Robust SVD winding number for edge states in amorphous chains"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument hinges on assuming that a vanishing smallest singular value of $H-E_b I$ forces a true eigenstate of $H$ at $E_b$ in the thermodynamic limit, even though non-Hermitian matrices can have tiny singular values far from any eigenvalue.","fun_headline_variants_meta":{"raw":{"variants":["Real-space SVD winding counts edge states despite spectral instability","Singular values probe topological edge states in amorphous non-Hermitian systems","SVD probe sees through spectral instability to count skin modes","Amorphous non-Hermitian topology from real-space singular values","Robust SVD winding number for edge states in amorphous chains"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000488,"raw_usage":{"total_tokens":2399,"prompt_tokens":933,"completion_tokens":1466,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":549,"completion_tokens_details":{"reasoning_tokens":1380}},"tokens_in":549,"tokens_out":1466,"duration_ms":12255,"temperature":1.0,"reasoning_tokens":1380,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:27:58.389168+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a disorder realization of the amorphous chain and a reference energy $E_b$ where $s_{\\min}(H-E_b I)$ is numerically zero; compute the distance $\\min_n|\\lambda_n-E_b|$ from the eigenvalues to $E_b$ as the system size $L$ grows. If that distance stays of order one while $s_{\\min}$ keeps vanishing, the zero-mode singular vector is not converging to an eigenstate and the correspondence is falsified.","supporting_citations":[{"cited_title":"Monkman and J","cited_arxiv_id":null,"evidence_quote":"establishes the correspondence between zero-mode singular states and stable zero-mode topological edge states in multiband non-Hermitian systems, which this work generalizes to amorphous systems."},{"cited_title":"Pseudospectral phenomena and the origin of the non-Hermitian skin effect","cited_arxiv_id":"2603.22643","evidence_quote":"demonstrates the pseudospectral origin of the non-Hermitian skin effect and the instability of eigenvalue spectra, motivating the singular-value approach."},{"cited_title":"Wu, X.-M","cited_arxiv_id":null,"evidence_quote":"introduces the singular-value-based diagnostic scheme for topology in Floquet non-Hermitian systems, whose WIPR and SVD methods this paper adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the singular-value-based bulk-boundary correspondence for a second-order topological insulator, used as a comparison benchmark."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the pseudospectrum definition and the stability argument for Hermitian spectra used to justify singular-value robustness."},{"cited_title":"Herviou, J","cited_arxiv_id":null,"evidence_quote":"defines a bulk-edge correspondence for non-Hermitian Hamiltonians via singular-value decomposition, the conceptual basis for the SVD invariant."}],"review_version":1}