{"id":"a4d6fc95-1b03-48c6-aa1c-c6c909d61a42","arxiv_id":"2601.01524","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For a 2D non-Hermitian SSH model, the number of stable zero singular values of H (or of U(T)−I and U(T)+I) equals the number of topological corner states in the thermodynamic limit, restoring bulk-boundary correspondence.","lead":"The paper finds that corner states in a two-dimensional non-Hermitian lattice are easily destroyed by tiny disorder, so it switches to counting the Hamiltonian's singular values, which stay robust and correctly count the protected corner states. It extends this singular-value method to periodically driven (Floquet) lattices, including cases with all symmetries broken.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Tiny singular values need not imply zero-energy eigenstates for nonnormal H; the claimed singular-value/energy correspondence is unproven and contradicted by Fig. 2(b) vs Fig. 3(a).","rationale":"The reader's weakest assumption identifies the same load-bearing point: a zero singular value for a highly non-normal matrix does not imply a zero eigenstate. My stress-test reinforces this with an explicit counterexample and points out that the paper's own data (Fig. 2(b) vs Fig. 3(a)) show Min|E| and Min[s] disagreeing in finite systems under chiral-preserving disorder. The paper's central claim is conditional on proving that the singular-value zero modes are accompanied by true energy-space corner states in the thermodynamic limit — or, alternatively, on openly redefining 'corner state' in terms of singular vectors and abandoning the energy-spectrum formulation. Since the reader already assigned CONDITIONAL with this exact caveat, I do not move the verdict. The proposed numerical scaling-and-overlap test would directly settle whether the concern lands.","tokens_in":9676,"tokens_out":7757,"duration_ms":90574,"concrete_test":"For the same parameters as Fig. 3(a) at a topologically nontrivial v_x, apply the same chiral-preserving disorder d=0.05 and symmetry-breaking disorder d'=0.05, and for Lx=Ly=20,40,80,160 compute: (i) the smallest singular value s_min of H2D; (ii) the minimum |E| over all eigenvalues of H2D; (iii) the overlap |<ψ_min|v_min>| between the eigenvector with smallest |E| and the smallest singular vector; (iv) the number of singular values below a stated threshold. If s_min remains tiny while min|E| stays O(1), or the overlap tends to zero, the central correspondence fails. If min|E|→0 and the overlap→1, the concern is answered. Repeat for U(T)∓I for the Floquet claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The inference at Eqs. (7)-(8) — that s_n→0 implies 'the system supports zero-mode states with wave function v_n' — is not valid for non-normal operators. A 2×2 matrix with singular values {R, 1/R} can have eigenvalues ±1 (e.g., [[0,R],[1/R,0]]), so min(s)→0 does not imply that any eigenvalue of H tends to 0. The paper itself reports κ=2.93×10^9 (Eq. 6) and shows in Fig. 2(b) that chiral-preserving disorder makes Min|E| deviate significantly from zero, while Fig. 3(a) shows Min[s]=0 for the same kind of weak disorder. That is exactly the non-normal regime where the correspondence is nontrivial. No proof is supplied that in the thermodynamic limit exponentially small singular values are accompanied by true eigenstates of H2D localized at corners; the text simply asserts it. The count 'number of zero-mode singular values = 2V' therefore establishes, at best, a stable property of H†H, not a correspondence to protected energy corner states. Additionally, under fully symmetry-breaking disorder a generic finite matrix has no exact zero singular value, so the plotted 'Min[s]=0' must rely on an unspecified numerical threshold; the count of 'zero-mode singular values' is not well-defined without one.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies a two-dimensional non-Hermitian Su-Schrieffer-Heeger model and its Floquet-driven variant. It first shows that the conventional non-Bloch winding-number prediction of zero-energy corner states breaks down under weak chiral-preserving disorder in large systems, attributing the instability to the large condition number of the nonnormal Hamiltonian. It then proposes to restore a bulk-boundary correspondence using the singular-value spectrum of H2D (and U(T)±I for Floquet systems), defines a real-space winding number V (resp. V±), and claims that the number of zero-mode singular values equals 2V (resp. 2V±) and directly counts topologically protected corner states of the energy spectrum in the thermodynamic limit.","tokens_in":9925,"tokens_out":6863,"duration_ms":76460,"significance":"If the central correspondence were valid, the paper would provide a stable way to identify higher-order topological boundary states in non-Hermitian systems, going beyond fragile non-Bloch invariants. The observation that energy zero modes are unstable while singular values are stable is a useful and interesting numerical fact, and the Floquet extension is natural. However, the claimed equivalence between zero singular values and zero-energy eigenstates is not proved and, as stated, is not valid for nonnormal operators. Since this equivalence is the load-bearing step of the paper, the significance of the results for energy-spectrum topology is not established.","major_comments":[{"comment":"The inference from s_n→0 to 'the system supports zero-mode states with wave function v_n' is invalid for nonnormal H. For example, M=[[0,R],[1/R,0]] has singular values R and 1/R, so the smallest singular value tends to 0 as R→∞, while the eigenvalues remain ±1. The reported condition number κ=2.93×10^9 in Eq. (6) places the model precisely in the regime where eigenvalues and singular values differ strongly. The contrast between Fig. 2(b), where Min|E| leaves zero under chiral-preserving disorder, and Fig. 3(a), where Min[s] is plotted as zero for the same type of weak disorder, is direct evidence that the singular-value spectrum is not equivalent to the eigenvalue spectrum. A proof that the singular vectors converge to genuine zero-energy eigenstates of H2D in the thermodynamic limit for this model is required before the central claim can be accepted.","section":"§3, Eqs. (7)-(8)"},{"comment":"The equality 'number of zero-mode singular values equals 2V' is presented as a new bulk-boundary correspondence, but V is computed from the same open-boundary SVD (U and V in Eq. (11)) whose zero singular values are being counted. The construction via the doubled Hermitian operator H̃ in Eq. (9) is the standard chiral-index argument from Refs. [49,51], so the equality is at least partly a restatement of the index theory of H̃ rather than an independent correspondence between singular states and energy eigenstates. The manuscript should specify whether V is a bulk quantity obtained from periodic-boundary data and should provide the index-theoretic derivation step by step.","section":"§3, Eq. (10) and following"},{"comment":"For a finite matrix with fully random, symmetry-breaking disorder, no exact zero singular value is expected generically. The plotted 'Min[s]=0' therefore relies on an unspecified numerical threshold. Without a precise threshold, the count of zero-mode singular values is not well defined, and the claimed equality with 2V cannot be tested. The same issue affects Fig. 4 for the singular values of U(T)±I.","section":"§3, Fig. 3(a)"},{"comment":"The paper demonstrates exponential decay of Min[s] with system size, but the claimed correspondence is to energy corner states in the thermodynamic limit. It does not provide the analogous scaling of Min|E| for the same disordered systems; the finite-size data in Fig. 2(b) suggest that Min|E| remains finite. Thus the thermodynamic-limit statement about energy eigenstates is not supported by the numerical evidence presented.","section":"§3, Fig. 3(b) and thermodynamic limit"}],"minor_comments":[{"comment":"Typo: 'mordynamic' should be 'thermodynamic'.","section":"§3, text after Eq. (8)"},{"comment":"Typo: 'ia also equal' should be 'is also equal'.","section":"§3, Fig. 3(a) caption"},{"comment":"The notation C̃_k for the generalized Brillouin zone is not defined precisely; the orientation of the contour and the branch of the logarithm should be specified.","section":"Eq. (3)"},{"comment":"The weighted inverse participation ratio formula appears to lack a normalization factor and a clear summation convention; the present expression is not dimensionless as written.","section":"Eq. (5)"},{"comment":"The expression Tr ln(PA PB†) should be defined with a statement about the branch of the logarithm and why the result is quantized; as written the reader must infer the intended convention.","section":"Eq. (10)"}],"recommendation":"reject","confidential_remarks":"The central claim of the paper — that zero singular values of H2D correspond one-to-one to zero-energy corner states in the thermodynamic limit — is a load-bearing statement that is neither proved nor generally true for nonnormal operators. The manuscript would need to be substantially reframed around a different object (e.g., singular-value zero modes of H†H rather than energy eigenstates) to make the claim correct. As submitted, I recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this paper is worth reading for its Floquet construction, but the central theoretical claim is a gap. The genuinely new piece is using singular values of U(T)∓I to count 0- and π-modes in non-Hermitian Floquet systems, with winding numbers V± computed from the doubled operators. That is a clean idea and the numerics are internally consistent. The static part is largely a repackaging of the SVD bulk-boundary correspondence from Herviou et al. [49] and the real-space chiral invariant from Benalcazar and Cerjan [51]; the authors don't clearly state what is added beyond applying that framework to a 2D non-Hermitian SSH model and showing robustness to chiral-symmetry-breaking disorder. The demonstration that Min[s] stays small under disorder that makes Min|E| deviate from zero is the empirical core, and that observation is real.\n\nThe soft spot is the step from s_n→0 to \"the system supports zero-mode states with wave function v_n.\" For a non-normal H, small ||H v|| does not imply a nearby eigenvector; the condition number here is ~10^9, so the pseudospectrum is large. The text calls the connection \"rigorous,\" but no proof is given. The count \"number of zero-mode singular values = 2V\" is also partly circular, since V is computed from the same open-boundary SVD data. More concretely, if Min[s] were exactly zero, then H would have an exact zero eigenvalue, contradicting Fig. 2(b); so the plotted \"zero\" must be a numerical threshold, and the paper never states it. That makes the correspondence somewhat definitional unless a thermodynamic-limit argument is supplied.\n\nWho is this for? Researchers working on non-Hermitian Floquet topology who might use the SVD spectrum as a practical diagnostic. It is not a finished theory paper. It deserves serious refereeing because the Floquet idea is new and the singular-value/eigenvalue question is subtle — a good referee could push the authors to either prove the correspondence or explicitly label it a conjecture and give analytic evidence. The paper is clearly written and honest about the breakdown of the energy-based bulk-boundary correspondence, so the right outcome is probably a major revision, not a desk reject.","headline":"Useful Floquet SVD proposal, but the central singular-value/energy correspondence is asserted rather than proved, and the authors' own data hint at a real discrepancy.","tokens_in":10537,"tokens_out":3284,"would_cite":true,"duration_ms":34822,"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":"This paper claims that even when disorder destroys the corner-state energies of a non-Hermitian higher-order topological insulator, the zero singular values of the Hamiltonian still count the topologically protected corner states in the the","keywords":["non-Hermitian topological insulator","higher-order topology","singular value spectrum","bulk-boundary correspondence","corner states","Floquet topological phases","point gap","skin effect"],"falsifier":"At the parameter point of Fig. 2(b) ($w_x = 1$, $v_x = -1.5$, $\\gamma_x = 1.5$, $w_y = 0$, $v_y = -9$, $\\gamma_y = 9$, system sizes growing up to $1000 \\times 1000$), compute the smallest singular vector $v$ of $H_{2D}$ with singular value $s$, and evaluate $\\|H_{2D} v\\|/s$ as a function of size. If this ratio grows without bound while $s$ decays, the singular vector is not approaching a zero-energy eigenstate, and the claimed correspondence between zero singular values and corner eigenstates would fail.","tokens_in":9461,"feed_emoji":"📐","tokens_out":4144,"duration_ms":207796,"temperature":0.7,"texified_at":"2026-08-05T20:46:30.457985+00:00","pith_summary":"The paper studies a two-dimensional non-Hermitian Su-Schrieffer-Heeger model whose corner-state energies are destroyed by arbitrarily weak disorder, even disorder that preserves chiral symmetry, because the Hamiltonian is far from normal. It argues that the stable topological content is not in the energy eigenvalues but in the singular-value spectrum: the number of zero-mode singular values equals $2V$, a real-space winding number, and this count remains intact under perturbations that destroy the spectral corner states. In the thermodynamic limit, each zero singular value corresponds to a topologically protected corner state. The same construction applies to Floquet systems, where singular values of $U(T)-I$ and $U(T)+I$ count 0- and π-modes.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":7931,"prompt_tokens":787,"completion_tokens":7144,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":787,"completion_tokens_details":{"reasoning_tokens":6398}},"feed_headline":"Zero singular values count corner states that spectra lose","feed_subtitle":"An SVD-based winding number survives the disorder that breaks ordinary bulk-boundary correspondence.","key_machinery":"The central object is the singular-value decomposition of the non-Hermitian Hamiltonian, $H_{2D} = U S V^\\dagger$, together with the associated Hermitian chiral-symmetric matrix $\\tilde{H} = \\begin{bmatrix} 0 & H \\\\ H^\\dagger & 0 \\end{bmatrix}$, whose eigenvalues are $\\pm s_n$. Singular values $s_n$ are real, non-negative, and stable under perturbation, unlike the complex eigenvalues of a non-normal operator. A real-space winding number $V$, computed from the unitary factors $U$ and $V$ via a diagonal twist matrix $P$, counts the number $2V$ of zero singular values and therefore the number of protected corner states.","core_discovery":"The central claim is that for a non-Hermitian Hamiltonian $H_{2D}$ with a point gap at $E=0$, the number of zero singular values of $H_{2D}$ — equivalently the zero eigenvalues of the Hermitian doubled matrix $\\tilde{H} = \\begin{bmatrix} 0 & H \\\\ H^\\dagger & 0 \\end{bmatrix}$ — is exactly $2V$, where $V$ is a winding number computed from the SVD singular vectors via $P_A = U^\\dagger P U$, $P_B = V^\\dagger P V$. This number counts topologically protected corner states of the energy spectrum in the thermodynamic limit, and it survives both chiral-symmetric and fully symmetry-breaking disorder, in contrast to the fragile energy eigenvalues whose gaps are destroyed even by infinitesimal perturbations. The same logic applies to Floquet systems by replacing $H$ with $U(T)-I$","pith_inferences":["Because zero singular values of a non-normal operator do not generally imply zero eigenstates, the paper's mapping from singular vectors to corner states in the thermodynamic limit is an assumption that would need a separate proof for generic non-Hermitian Hamiltonians; the paper's numerical evidence suggests it, but no general theorem is established.","The method suggests a practical experimental protocol: measure the response matrix or scattering matrix of a finite lattice and inspect its singular values, rather than attempting to resolve fragile complex spectra, to detect higher-order topology.","If the singular-value winding number is the correct invariant, it may generalize to other non-Hermitian topological phases with point-gap topology, providing a classification scheme based on SVD rather than on generalized Brillouin zones."],"forward_implications":["If the correspondence holds, the number of protected corner states in a non-Hermitian higher-order topological insulator is given by the singular-value winding number 2V, not by the non-Bloch invariant, resolving the breakdown of bulk-boundary correspondence.","The invariant is robust to disorder that breaks chiral symmetry, so topological corner states can be certified without requiring symmetry protection.","In Floquet systems, 0- and π-modes are characterized by zero singular values of U(T)−I and U(T)+I, respectively, giving direct counting invariants V− and V+.","The exponential decay of the smallest singular value with system size provides a finite-size scaling tool for identifying topological phases in numerical and experimental data."],"fun_headline_variants":["Zero singular values count corner states despite fragile gaps","SVD winding number pins corner states in non-Hermitian lattices","Corner states counted by zero singular values, not energy gaps","Non-Hermitian corner states robust to disorder via SVD"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"A singular vector with a vanishing singular value in a finite open-boundary system is assumed to converge to an actual zero-energy corner eigenstate in the thermodynamic limit, even though for a non-normal Hamiltonian a zero singular value does not by itself guarantee a zero eigenstate.","fun_headline_variants_meta":{"raw":{"variants":["Zero singular values count corner states despite fragile gaps","SVD winding number pins corner states in non-Hermitian lattices","Corner states counted by zero singular values, not energy gaps","Non-Hermitian corner states robust to disorder via SVD"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001428,"raw_usage":{"total_tokens":5566,"prompt_tokens":681,"completion_tokens":4885,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":425,"completion_tokens_details":{"reasoning_tokens":4815}},"tokens_in":425,"tokens_out":4885,"duration_ms":30827,"temperature":1.0,"reasoning_tokens":4815,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T12:47:27.669981+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"At the parameter point of Fig. 2(b) ($w_x = 1$, $v_x = -1.5$, $\\gamma_x = 1.5$, $w_y = 0$, $v_y = -9$, $\\gamma_y = 9$, system sizes growing up to $1000 \\times 1000$), compute the smallest singular vector $v$ of $H_{2D}$ with singular value $s$, and evaluate $\\|H_{2D} v\\|/s$ as a function of size. If this ratio grows without bound while $s$ decays, the singular vector is not approaching a zero-energy eigenstate, and the claimed correspondence between zero singular values and corner eigenstates would fail.","supporting_citations":[],"review_version":1}