{"id":"3b0ccc71-d05b-445c-bcec-9f3c067a3c89","arxiv_id":"2510.09193","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a Floquet non-Hermitian SSH chain, edge-state counts remain well defined only through the singular values of U(T)±I in the thermodynamic limit, not through the raw quasienergy spectrum.","lead":"This paper studies a one-dimensional driven chain with gain and loss, where topological states at the ends of the chain vanish under tiny symmetry-preserving disorder. It proposes a way to recover a reliable topological count using singular values of the time-evolution operator instead of ordinary energy levels.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed stability of zero singular values is unproven and appears to conflict with the paper's own Fig. 2: a zero singular value of U(T)±I is equivalent to an exact 0/π quasienergy, which Fig. 2 shows is destroyed by chiral disorder.","rationale":"The reader's verdict (CONDITIONAL) already identifies the missing OBC/PBC proof for [U±I]†[U±I] and notes that Fig. 2 does not display singular spectra. My stress-test focuses on a more direct logical gap: the central claim requires zero singular values to be stable, but the paper's own Fig. 2 demonstrates that exact 0/π quasienergy edge states are not stable under chiral disorder. Since zero singular values of U±I are equivalent to exact eigenvalues ∓1 of U, the asserted robustness of the singular spectrum is either a different statement that needs its own argument, or it contradicts the paper's central observation. The paper provides neither a proof nor a numerical demonstration of the singular-spectrum robustness under disorder. This reinforces the CONDITIONAL verdict: the proposed bulk-boundary correspondence is plausible but lacking essential support. The concrete test would settle whether the stability claim holds as a finite-size scaling statement, which would repair the central argument. I do not recommend changing the reader's verdict, but the concern should be made explicit in any revision.","tokens_in":8743,"tokens_out":13961,"duration_ms":120492,"concrete_test":"Take the same parameters as Fig. 2 (w=1, γ=1.5, q=2, T1=T2=0.7, N=25) and disorder ΔH = d(α_ij a†_i b_j + β_ji b†_j a_i) with α,β∈[-0.5,0.5]. For d in {0.001,0.01,0.1} and L in {25,50,100}, compute the minimum singular values of U(T)-I and U(T)+I over 500 disorder realizations. If min s_n,± is bounded below by a positive constant independent of L for any d>0, the central 'stable zero-mode singular states' claim fails; if it decreases to 0 with L (e.g., as 1/L or exponentially), the claim is supported. Also verify whether Fig. 2 contains any singular spectrum; it does not, so the citation needs replacement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that stable zero singular values of U(T)±I count topologically protected edge states in the thermodynamic limit (Section 'Restoration of bulk-boundary correspondence in momentum space', Eqs. 6–9). This requires the zero singular values to be robust against symmetry-preserving perturbations. The paper asserts 'the singular spectrum is highly robust to perturbations, as demonstrated in Fig. 2' — but Fig. 2 shows quasienergy spectra and WIPR, not singular spectra, and in fact shows that 0-mode and π/T-mode edge states are suppressed by arbitrarily small chiral disorder. Since a zero singular value of U(T)±I is equivalent to U(T) having an exact eigenvalue ∓1 (quasienergy 0 or π/T), the robustness of zero singular values is logically the same as the robustness of exact 0/π quasienergy edges. The paper therefore appears to assert the opposite of what its own Fig. 2 demonstrates. No alternative mechanism (e.g., finite-size scaling closing the singular gap) is provided or tested. Without this, the proposed correspondence has no demonstrated object to count.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies a periodically driven non-Hermitian Su-Schrieffer-Heeger chain. It first shows that the OBC quasienergy spectrum, including its 0- and π/T-mode edge states, is highly sensitive to chiral-symmetric disorder, and argues that this finite-size instability causes a breakdown of the non-Bloch bulk-boundary correspondence based on the generalized Brillouin zone. To repair the correspondence, the authors propose to use the singular values of U(T)±I, asserting that stable zero singular values count the topologically protected 0- and π/T-mode edge states in the thermodynamic limit. They define momentum-space winding numbers V1,V2 and real-space invariants V'_1,V'_2, and use them to produce a phase diagram with up to three coexisting edge modes.","tokens_in":9053,"tokens_out":7569,"duration_ms":68511,"significance":"If correct, the proposed singular-value formulation would give a bulk-boundary correspondence for Floquet non-Hermitian systems that is robust against the quasienergy instabilities that plague the conventional non-Bloch approach. The algebraic identities in Eqs. (6)-(7) are straightforward and correct, and the model is simple enough to be a useful testbed. However, the central physical claims — stability of the zero singular values and boundary insensitivity of the bulk singular spectrum — are asserted rather than proved, and the paper's own numerical results appear to contradict the stability claim. The manuscript currently lacks the evidence needed to support its main conclusion.","major_comments":[{"comment":"The statement \"the singular spectrum is highly robust to perturbations, as demonstrated in Fig. 2\" is not supported by Fig. 2. Figure 2(b)-(c) shows that the 0-mode and π/T-mode quasienergies are suppressed by arbitrarily small chiral-symmetric disorder, and Fig. 2 contains no singular-spectrum computation. Moreover, a zero singular value of U(T)±I is exactly equivalent to U(T) having an eigenvalue ±1, i.e., an exact 0 or π/T quasienergy. Thus robustness of zero singular values is logically the same as robustness of exact 0/π edge quasienergies. The paper must provide either a proof or a finite-size scaling analysis showing that the zero singular values survive in the thermodynamic limit despite the finite-size suppression shown in Fig. 2. Without this, the correspondence has no demonstrated object to count.","section":"Restoration of bulk-boundary correspondence in momentum space, after Eq. (7)"},{"comment":"The assertion that \"[U(T)±I]†[U(T)±I] has the same bulk band under both open and periodic boundary conditions\" is load-bearing but completely unproved. It is used to justify computing the PBC winding numbers V1 and V2 in Eqs. (8)-(9) and then comparing them with OBC singular spectra in Fig. 3. This is a nontrivial statement for a non-Hermitian system whose ordinary quasienergy spectrum is explicitly boundary-sensitive, as the paper itself demonstrates. The authors should either prove this boundary insensitivity for their model using an appropriate Toeplitz/Szegő-type theorem or provide a detailed numerical verification, including system-size scaling.","section":"Restoration of bulk-boundary correspondence in momentum space, fourth paragraph"},{"comment":"The real-space invariants V'_1 and V'_2 are claimed to be a more general scheme that retrieves the bulk-boundary correspondence, but no derivation or numerical validation is provided. The notation is also opaque: P^S_± = U^†_{S,±} P U_{S,±} and P_{ll'} = δ_{ll'}e^{-i2πl/L} do not clearly define the matrix being traced in Eqs. (11)-(12). The authors should state explicitly how these invariants count edge states, prove their quantization, and show that they reproduce the V1/V2 results in the clean limit.","section":"Restoration of bulk-boundary correspondence in real space, Eqs. (10)-(13)"},{"comment":"The claim that the proposed topology is intrinsic and exists \"even without symmetries\" is not established by the manuscript. All numerical examples use a chiral-symmetric model and chiral-symmetric disorder, and the derivation of the boundary-insensitive bulk singular spectrum is absent. While V1 and V2 are well-defined winding numbers that do not require chiral symmetry, the bulk-boundary correspondence for these invariants in the absence of symmetry is not demonstrated. At minimum, the authors should qualify this claim or provide a concrete symmetry-free example.","section":"Abstract and Conclusion"}],"minor_comments":[{"comment":"The caption says \"green and crimson line\" but the text refers to \"green and crimson line\" without clearly identifying which curve is which. Also \"when dis samll\" should be \"when d is small.\"","section":"Fig. 2 caption and text"},{"comment":"The definitions of P_A, P_B, and U_{S,±} should be written out more carefully; the current notation makes it difficult to reproduce the real-space invariant.","section":"Eqs. (11)-(13)"},{"comment":"The claim that the singular spectra \"can be well characterized\" by V1 and V2 is only qualitative. It would be helpful to state explicitly the number of zero singular values in each regime and the corresponding value of V1/V2.","section":"Fig. 3"},{"comment":"The sentence \"The topological modes can be detected by Loschmidt echo\" is unsupported by a reference or derivation. Please add a citation or briefly explain the detection protocol.","section":"Experimental realization"},{"comment":"There are several typographical errors (e.g., \"signifcantly\", inconsistent capitalization of \"we\" after commas). A careful proofread is recommended.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript addresses an important question and proposes an appealingly simple singular-value route to a Floquet non-Hermitian bulk-boundary correspondence. However, the central stability claim is not merely unproved; as written it is contradicted by the paper's own Fig. 2, because zero singular values of U±I are equivalent to exact 0/π quasienergies. A revision must either prove the thermodynamic-limit stability of the zero singular values (including a resolution of the apparent contradiction) or substantially weaken the central claim. The boundary-insensitivity of the bulk singular spectrum also needs a rigorous justification before the momentum-space winding numbers can be used to count OBC edge states."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper claims to restore bulk-boundary correspondence in Floquet non-Hermitian systems using singular values of U(T)±I, but the central robustness claim is contradicted by its own Fig. 2. The algebraic link between a zero singular value of U−I and a quasienergy-0 state is correct, which is exactly why the proposed fix cannot work as stated: if chiral disorder destroys the 0-mode quasienergy edge state, the corresponding singular value is no longer zero. The paper asserts the singular spectrum is robust \"as demonstrated in Fig. 2,\" but Fig. 2 shows quasienergy spectra, not singular spectra, and it shows edge states being suppressed by infinitesimal disorder. The stress-test note is right.\n\nWhat is actually new and useful: the demonstration that non-Bloch theory fails for this Floquet model due to finite-size instability of the quasienergy spectrum, and that symmetry-preserving disorder kills the 0 and π/T edge states despite topological invariants predicting them. That is a clean negative result worth recording. The authors also correctly identify that SVD-based methods are a promising alternative, following Herviou et al. But that's an imported idea, not new.\n\nThe soft spots are severe. The load-bearing claim that [U(T)±I]†[U(T)±I] has the same bulk band under OBC and PBC is asserted without proof, and it is not obvious for non-Hermitian Floquet systems with skin effect. The real-space winding formulas are introduced without derivation showing they count edge states. No code or data is provided, and the numerics are minimal. The phase diagram relies on the unproved equality.\n\nThe paper is not hopeless: the fragility observation is solid, and the SVD approach may be salvageable with a proper thermodynamic-limit analysis and a demonstration that the singular gap closes exactly when edge states exist. But as written, the central correspondence is unsupported and appears to conflict with the paper's own numerics. I would not cite it in its current form. It deserves a serious referee to sort out whether the singular-value method can be made rigorous, but it needs major revision before acceptance.\n\nRecommendation: send to peer review, since the negative result alone warrants scrutiny, but expect the authors to need to prove the OBC/PBC equality and reconcile the robustness claim with Fig. 2.","headline":"The proposed singular-value bulk-boundary correspondence for Floquet non-Hermitian systems is unsupported and appears to contradict the paper's own Fig. 2, which shows the very edge states the method is meant to count are destroyed by infinitesimal chiral disorder.","tokens_in":9512,"tokens_out":3408,"would_cite":false,"duration_ms":30041,"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":"Zero singular values restore the missing bulk–boundary count for Floquet non-Hermitian chains","keywords":["non-Hermitian Floquet topology","bulk-boundary correspondence","singular value decomposition","non-Bloch band theory","driven non-Hermitian SSH model","quasienergy spectrum instability","topological edge states","skin effect"],"falsifier":"Compute the low-lying singular spectrum of U(T)-I and U(T)+I for a long open chain and for the same chain under periodic boundary conditions at fixed parameters; if the bulk singular bands do not converge to a common limiting curve as L grows, the restored bulk–boundary correspondence fails. The claim would also be falsified by any parameter regime where zero singular values persist but no edge states appear in the thermodynamic quasienergy spectrum, or where chiral-symmetry-preserving disorder suppresses the zero singular values themselves.","tokens_in":8621,"feed_emoji":"⚛️","tokens_out":7556,"duration_ms":59381,"temperature":0.7,"pith_summary":"The paper studies a periodically driven non-Hermitian Su-Schrieffer-Heeger (SSH) chain—a one-dimensional lattice with two sites per unit cell and alternating, non-reciprocal hoppings. It argues that the usual story of topologically protected edge states can fail: infinitesimal perturbations that preserve sublattice symmetry can destroy the edge states, because the open-boundary quasienergy spectrum (the eigenvalues of the effective Hamiltonian defined over one driving period, modulo 2π/T) is unstable in finite systems. The authors locate this failure in the breakdown of non-Bloch bulk–boundary correspondence based on the generalized Brillouin zone. The repair is to track singular values of the one-period evolution operator U(T)±I rather than eigenvalues of U(T). The central claim is that the number of stable zero singular values of U(T)-I and U(T)+I equals the number of topologically protected edge states at quasienergies 0 and π/T in the thermodynamic limit. If true, this restores a bulk–boundary correspondence for Floquet non-Hermitian systems that does not require symmetry and survives disorder.","feed_headline":"Zero singular values restore the missing edge-state count","feed_subtitle":"When chiral disorder kills the predicted edge states, singular-value zeros still count the protected modes.","key_machinery":"The load-bearing object is the shifted one-period evolution operator U(T)±I and its singular-value spectrum. Rather than asking whether U(T) has eigenvalues on the unit circle—an unstable question for finite non-Hermitian systems—the paper asks whether U(T)-I or U(T)+I has zero singular values; such zeros persist under perturbation and mark edge states at quasienergy 0 or π/T in the thermodynamic limit. Two winding numbers, V1 and V2, count the winding of det(U(T)∓I) in momentum space, and real-space versions V'_1 and V'_2 are defined from the singular-value decomposition U(T)±I = U_A s U_B† through the winding number of a doubled chiral-symmetric operator. The singular-value spectra are the","core_discovery":"Using the one-dimensional non-Hermitian Floquet SSH model, the paper establishes that topologically protected edge states can be suppressed by arbitrarily small sublattice-symmetry-preserving disorder, and traces this to the instability of the finite-size quasienergy spectrum. It then shows that the singular spectrum of U(T)±I is stable under such perturbations, and that the number of zero singular values of U(T)-I (U(T)+I) counts the right and left edge states at quasienergy 0 (π/T) in the thermodynamic limit. The momentum-space invariants V1 and V2 are defined by the winding of det(U(T)∓I) across the ordinary Brillouin zone; real-space variants V'_1 and V'_2 are constructed from the singul","pith_inferences":["If the singular bulk band of [U(T)±I]†[U(T)±I] is truly insensitive to open versus periodic boundaries, the same construction could define non-Hermitian Floquet invariants on disordered lattices where translational-invariance-based winding numbers are undefined.","A natural extension would be to test whether the zero singular values remain pinned when the perturbation breaks sublattice symmetry or adds gain/loss beyond the non-reciprocal hop; the paper's symmetry-free claim suggests they should, but that goes beyond what is demonstrated.","The OBC/PBC invariance of the singular spectrum is mathematically a statement about large truncated versions of a single-particle operator; proving it explicitly would place the restoration on a theorem-level footing comparable to the index argument used for the momentum-space invariants."],"forward_implications":["In the thermodynamic limit, the number of stable zero singular values of U(T)±I gives the number of 0-mode and π/T-mode edge states, restoring a bulk–boundary correspondence where the generalized Brillouin zone method fails.","The invariants V1 and V2 require no chiral or sublattice symmetry, so driven non-Hermitian systems can carry symmetry-free topology.","0-gap and π/T-gap topology can coexist, producing phases with up to three edge states in the driven chain.","The singular-value counting method is claimed to generalize to multiband non-Hermitian Floquet models, offering a broad tool for classifying these phases."],"fun_headline_variants":["Singular zeros survive chiral disorder to count edge states","Chiral disorder kills states, but zero singularities recount them","Zero singular values restore edge-state counting","Fragile edge states rescued by singular-value zeros","Counting protected modes via zero singularities despite disorder"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The claim rests on the unproved assertion that [U(T)±I]†[U(T)±I] has the same bulk band under open and periodic boundary conditions; if that boundary-insensitivity of the singular bulk spectrum fails, the momentum-space winding numbers V1 and V2 would not count open-boundary edge states.","fun_headline_variants_meta":{"raw":{"variants":["Singular zeros survive chiral disorder to count edge states","Chiral disorder kills states, but zero singularities recount them","Zero singular values restore edge-state counting","Fragile edge states rescued by singular-value zeros","Counting protected modes via zero singularities despite disorder"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000155,"raw_usage":{"total_tokens":1029,"prompt_tokens":703,"completion_tokens":326,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":447,"completion_tokens_details":{"reasoning_tokens":252}},"tokens_in":447,"tokens_out":326,"duration_ms":53660,"temperature":1.0,"reasoning_tokens":252,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T10:39:08.981204+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the low-lying singular spectrum of U(T)-I and U(T)+I for a long open chain and for the same chain under periodic boundary conditions at fixed parameters; if the bulk singular bands do not converge to a common limiting curve as L grows, the restored bulk–boundary correspondence fails. The claim would also be falsified by any parameter regime where zero singular values persist but no edge states appear in the thermodynamic quasienergy spectrum, or where chiral-symmetry-preserving disorder suppresses the zero singular values themselves.","supporting_citations":[],"review_version":1}