{"id":"14da6697-44dc-48b7-84d4-644558af754e","arxiv_id":"2412.10526","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A winding number around bulk eigenvector degeneracy points predicts the number of edge states in asymmetric one-dimensional two-band models without symmetry protection.","lead":"The authors prove a winding number invariant that counts edge states in one-dimensional two-band chains with arbitrary complex couplings and open boundaries, even when no symmetry protects the states. The result provides a bulk-only way to predict and understand edge states in asymmetric Hermitian and non-Hermitian lattices, beyond the standard symmetry-based classification.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The general invariant is not yet verified against the quantity it claims to predict: the OBC edge-state count, especially in braided GBZ phases; Fig. 3 compares W to the same analytic criterion rather than to independently counted OBC edge states.","rationale":"The reader's weakest_assumption focuses on the smooth-torus/no-repeated-roots condition, and I agree that this is a genuine domain restriction on the proof of Eq. (10). However, the more load-bearing issue for the central claim is that the numerical verification does not independently test the quantity W is supposed to predict. The phase diagrams compare W against an analytic edge-state criterion derived from the same determinant expansion that defines edge states, so agreement between them is partly an internal consistency check. The paper does provide some direct OBC checks for individual models, and the algebraic proof is substantial, so I am not claiming the result is false. But the general non-Hermitian claim, especially for braided GBZ phases, currently rests almost entirely on the proof, with numerical support deliberately avoiding the difficult regimes. A direct OBC-spectrum comparison across braided phases would settle whether the concern lands and would strengthen the paper substantially.","tokens_in":37307,"tokens_out":9083,"duration_ms":95103,"concrete_test":"For a grid in Fig. 3(b) that explicitly includes parameter points inside the braided GBZ regions mentioned in SM Sec. IX.B, directly diagonalize the OBC Hamiltonian for N=32, count edge states as discrete eigenvalues outside the OBC continuum using the Iedge indicator from SM Sec. IX.C, and compare that count with W from Eq. (10). A single mismatch would falsify the general invariant in the regime it was designed to cover; a full match would resolve the concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that W from Eq. (10) equals the number of edge states in the OBC spectrum. In Fig. 3, however, both panels compare W to Nedge computed from the M1=M2/M3=M4 condition with an epsilon tolerance, which is the same leading-term analysis used to define edge states, rather than to an independent count of discrete OBC eigenvalues. Direct OBC comparisons are limited to single examples (Fig. 1 and SM Secs. IV, IX). Moreover, SM Sec. IX.B states that the non-Hermitian phase diagram is restricted to an unlinked GBZ phase and that braiding transitions occur only over parameter ranges smaller than the grid size, so those transitions are effectively approximated away. These are precisely the regimes where the proof's inside/outside GBZ construction and the loop-sorting procedure are nontrivial. Thus the numerical support for the claimed validity 'regardless of the underlying GBZ eigenvalue topology' is not yet delivered, and the analytical proof carries the entire weight there. If that proof has a hidden assumption in braided or degenerate regimes, the mismatch would not be visible in the current phase diagrams.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces a winding-number invariant for one-dimensional nearest-neighbor two-band tight-binding models with arbitrary complex couplings. The invariant is built from the image of the (generalized) Brillouin zone on a Riemann surface of the eigenvector ratio M, using the two bulk eigenvector degeneracy points Mdeg and, for the general non-Hermitian case, the M-plane branch points Mbranch. The authors prove analytically for generic models, via a Weierstrass elliptic parametrization of the band-structure torus, that the parity of the winding counts whether Mdeg corresponds to an edge state satisfying M1=M2 or M3=M4, and they verify the invariant numerically against that same analytic condition in phase diagrams for Hermitian and non-Hermitian random models. The paper claims a bulk-boundary correspondence that does not rely on symmetry protection and reduces to known invariants when symmetries are imposed.","tokens_in":37513,"tokens_out":5628,"duration_ms":51603,"significance":"If the proof is fully correct for the claimed domain, this is a significant contribution: it provides a symmetry-independent bulk invariant for edge states in a large class of one-dimensional two-band models, covering nonzero-energy edge states and non-Hermitian systems, and it explicitly connects the bulk-boundary correspondence to the Riemann-surface structure of the Bloch Hamiltonian. The analytic proof is substantial, including a torus parametrization and an argument-principle counting argument, and the paper gives a falsifiable prediction (W equals the number of OBC edge states) that can be checked by independent numerical diagonalization. However, the current numerical verification is not independent of the analytic edge-state condition and does not exercise the braided GBZ regimes where the proof's geometry is most delicate, so the claimed universality is not yet fully supported.","major_comments":[{"comment":"The phase diagrams in Fig. 3 compare W from Eq. (9) or Eq. (10) to Nedge obtained from the analytic condition |M1/M2-1|<epsilon or |M3/M4-1|<epsilon, which is the same leading-term criterion (Eq. (11)) used to define edge states in the proof. This is an internal consistency check, not an independent verification against the open-boundary spectrum; direct OBC comparisons appear only for individual examples in Fig. 1 and SM Sec. IV. Please add a systematic comparison of W to an independently counted number of discrete OBC eigenvalues, for example using the signed indicator Iedge defined in SM Sec. IX.C or an IPR-based counting of finite-N spectra.","section":"Fig. 3, SM Sec. IX.A-B"},{"comment":"The proof of Eq. (10) explicitly assumes that z^2 Delta(z) has no repeated roots (SM Sec. V A, text after Eq. (S24)), so that the band structure is a smooth genus-1 curve. The abstract and main text nevertheless claim validity for 'any complex couplings' and 'regardless of the underlying GBZ eigenvalue topology.' Degenerate cases such as the SSH model are excluded by this assumption and are only reached as limits; the proof as written does not establish the invariant for them. Please either prove the degenerate cases by a limiting argument or restrict the universality claims to generic models.","section":"SM Sec. V.A"},{"comment":"The non-Hermitian phase diagram is computed under the approximation that the GBZ is unlinked, and the braiding transitions are stated to occur over parameter ranges smaller than the grid size. The proof's inside/outside GBZ construction and the loop-sorting procedure are precisely the nontrivial parts in braided or linked phases, so the numerical support for the 'regardless of the underlying GBZ eigenvalue topology' claim does not cover those regimes. Please add explicit tests in a parameter region with nontrivial GBZ braiding, or state the domain of validity of the numerical verification.","section":"SM Sec. IX.B, Fig. 3(b)"}],"minor_comments":[{"comment":"The main text defines OBC as setting the wavefunction to zero at n=0 and n=N+1, but the determinant in Eq. (14) and the coefficient expressions in Eqs. (16)-(19) use powers z^N (i.e., a boundary at n=N); please reconcile this notation or clarify the convention.","section":"End Matter, Analytical OBC"},{"comment":"The footnote for the Supplemental Material contains a placeholder text '[todo] will be inserted by publisher' that should be replaced with an actual URL before publication.","section":"Reference [19]"},{"comment":"The phrase 'the two links of the unlink are the subGBZ loops' should read 'the two loops of the unlink are the subGBZ loops' for clarity.","section":"SM Sec. IX.B"}],"recommendation":"major_revision","confidential_remarks":"The paper is a well-written Letter with a substantial supplementary analysis. The main concerns are fixable: the numerical verification in Fig. 3 is circular relative to the analytic edge-state condition, and the proof and numerics leave out braided GBZ regimes and degenerate band structures. I would not reject, but the abstract and main-text claims should be qualified to match the actual domain of the proof and verification."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper's new result is the Mdeg-based winding number in Eqs. (9) and (10), which assigns a bulk topological count to edge states in 1D two-band chains with arbitrary complex couplings and no symmetry. That is a genuine generalization of the SSH and non-Hermitian sublattice-symmetric invariants, and the Riemann-surface construction with the two Mdeg points is novel. The analytical work is substantial: they parametrize the band structure as an elliptic curve and reduce the edge-state condition to a winding integral. I have not verified every step, but the proof sketch is coherent and the SM is detailed.\n\nThe soft spots are real but not fatal. The numerical verification in Fig. 3 compares W to the same analytic edge-state condition (|M1/M2 - 1| < epsilon) derived from the OBC determinant, not to an independent count of discrete OBC eigenvalues. That makes the agreement partly a consistency check. The paper does show direct OBC spectra for individual examples, but not a systematic sweep. And SM Sec. IX.B states that the non-Hermitian phase diagram is restricted to the unlinked GBZ phase; braided phases are effectively excluded by the parameter grid. So the claim that the invariant works 'regardless of GBZ eigenvalue topology' rests on the proof, not on the numerics.\n\nThe proof also assumes the characteristic polynomial z^2 Δ(z) has no repeated roots, so the band structure is a smooth torus. SSH and other degenerate cases are excluded, covered only as limits. That is a stated limitation, but it cuts against the 'any complex couplings' framing. If the authors can extend the proof to degenerate limits or at least show the limit is well-behaved, that would close the gap.\n\nOverall: this is a serious, novel contribution. The main claim is plausible and the analytical machinery is impressive. I would send it to peer review. A careful referee should ask for direct OBC eigenvalue counting across a broader parameter range, including braided GBZ phases, and for a clearer treatment of the degenerate-root limit. I'd bring it to a reading group only if the group is actively working on non-Hermitian topology. I likely won't cite it in my own work this year, but that's a field-specific choice.","headline":"A serious generalization of edge-state invariants to asymmetric 1D two-band models, but the numerics verify the invariant against the same analytic condition it is built on rather than against independent OBC spectra, and the proof excludes degenerate-root limits.","tokens_in":38077,"tokens_out":3057,"would_cite":false,"duration_ms":28992,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A winding number computed from bulk data counts edge states in one-dimensional two-band chains with arbitrary complex couplings.","keywords":["bulk-boundary correspondence","edge states","non-Hermitian topology","winding number","Riemann surface","generalized Brillouin zone","two-band model","bulk eigenvector degeneracy"],"falsifier":"Compute $W$ from Eq. (10) for a large random sample of complex couplings of the form in Eq. (1) with non-repeated roots, diagonalize the open-boundary chain with thousands of unit cells, and count the discrete eigenvalues lying away from the GBZ continuum; a single parameter point with $W$ different from that count would falsify the claimed correspondence.","tokens_in":37081,"feed_emoji":"🌀","tokens_out":8003,"duration_ms":73093,"temperature":0.7,"pith_summary":"This paper proves a bulk-boundary correspondence for one-dimensional, nearest-neighbor two-band tight-binding chains with arbitrary complex couplings: a winding number built from bulk data predicts exactly how many edge states appear when the chain is cut. The invariant uses two special \"bulk eigenvector degeneracy\" points—pairs of distinct wavevectors at one energy that share the same eigenvector ratio—and winds the image of the (generalized) Brillouin zone around them on the M-Riemann sphere. No symmetry assumption is needed, so the result covers Hermitian and non-Hermitian models, including edge states at nonzero energy and transitions that do not coincide with gap closing. If correct, the criterion replaces symmetry-protected classification for this model class with a purely geometric count that reduces to the familiar SSH invariant when symmetry is present.","feed_headline":"Winding number predicts edge states in 1D chains without symmetry","feed_subtitle":"Bulk data alone—two degeneracy points on the M-Riemann sphere—count zero, one, or two edge states.","key_machinery":"The carrying object is the bulk eigenvector degeneracy point $M_{\\rm deg}$: a value of the eigenvector ratio $M = a/b$ shared by two distinct generalized Bloch wavevectors $z$ at the same energy. There are exactly two such points for the nearest-neighbor model, given by the quadratic in Eq. (8), and an edge state appears when the two preimages of $M_{\\rm deg}$ on the band-structure torus both lie inside ($M_1=M_2$) or both outside ($M_3=M_4$) the generalized Brillouin zone. The invariant winds the GBZ image $M(C_{\\rm GBZ})$ on the M-Riemann sphere around each $M_{\\rm deg}$, subtracting the winding around a branch point $M_{\\rm branch}$ to fix the parity; Eq. (9) is the Hermitian version integrated over the ordinary Brillouin zone. The Riemann-surface proof shows the band structure is a torus when $z^2\\Delta(z)$ has no repeated roots and uses the elliptic-function parametrization to turn the edge-state condition into an argument-principle count.","core_discovery":"The paper's claim is that for the model $H(z)=h_- z^{-1}+h_0+h_+ z$, the total winding number $W = W_1 + W_2$ from Eq. (10) (or Eq. (9) for Hermitian models) equals the number of open-boundary edge states, which can only be 0, 1, or 2. An edge state forms exactly when two of the four generalized Bloch solutions at an energy have equal eigenvector ratio $M(z,E)=a/b$; the two bulk eigenvector degeneracy points $M_{\\rm deg}$ are where two distinct $z$ values at the same energy share that ratio. The invariant winds the image of the generalized Brillouin zone, $M(C_{\\rm GBZ})$, on the M-Riemann sphere around each $M_{\\rm deg}$, subtracting a branch-point winding $M_{\\rm branch}$ to fix parity, and the mod-two structure tracks whether the two preimages of $M_{\\rm deg}$ lie on the same side of the GBZ. Under sublattice or chiral symmetry the invariant reduces to the standard SSH winding number, and it is unchanged by $z$-independent unitary or similarity transformations. The proof relies on the smooth band structure being a genus-one Riemann surface, parametrized by the Weierstrass elliptic function, on which the argument principle converts the edge-state condition into a winding count.","pith_inferences":["The same bulk-geometry picture suggests a testable route to longer-range two-band chains: the paper shows the edge-state condition becomes a vanishing of determinants $D_{\\rm left}$ and $D_{\\rm right}$, and if a finite algebraic set of degeneracy points can be defined there, an analogous winding invariant should count edge states; the paper leaves that extension open.","The distance between $M_{\\rm deg}$ and $M(C_{\\rm GBZ})$ could serve as a design parameter: one could engineer a lattice so that an edge state switches on at a chosen coupling value by tuning that distance to zero.","Because the edge-state energy is fixed by bulk quantities alone, an experiment on a photonic or electrical lattice of the form in Eq. (1) could predict edge-state frequency without diagonalizing the finite structure, complementing approaches that require a truncated-system Green's function.","One might expect an analogous invariant to exist for continuum models with polynomial band equations, but for models with transcendental band structures, such as photonic crystals, the algebraic degeneracy-point construction would need a different formulation."],"forward_implications":["For any nearest-neighbor two-band chain with complex couplings, the number of open-boundary edge states is fixed by bulk data and is always 0, 1, or 2.","Edge states may sit at nonzero energy, and their appearance or disappearance is marked by $M_{\\rm deg}$ crossing the GBZ image on the M-Riemann sphere, not by the bulk gap closing.","When sublattice or chiral symmetry is present, the new invariant reduces to the familiar SSH-type winding number, so the generalized criterion contains the symmetry-protected criterion as a special case.","The invariant is invariant under $z$-independent unitary and similarity transformations, matching the physical expectation that such basis changes do not alter the existence of edge states.","The two $M_{\\rm deg}$ points cannot be created or destroyed by continuous parameter changes, so edge states tied to them are stable against symmetry-preserving, nearest-neighbor, periodic perturbations; the distance between $M_{\\rm deg}$ and the GBZ image measures how much perturbation is required to create or remove an edge state."],"supporting_citations":[{"why":"Supplies the generalized Bloch theorem used to justify the bulk band structure and the four $z$-solutions for arbitrary boundary conditions.","marker":"[41]"},{"why":"Defines edge states and the generalized Brillouin zone for non-Hermitian chains, providing the GBZ condition $|z_2|=|z_3|$.","marker":"[8]"},{"why":"Establishes non-Bloch band theory and the GBZ as the continuum of bulk open-boundary states.","marker":"[9]"},{"why":"Introduces the M-Riemann sphere and the winding-number formulation for sublattice-symmetric non-Hermitian models that this work generalizes.","marker":"[56]"},{"why":"Provides the Riemann-surface band-structure framework and GBZ computation methods used throughout the paper.","marker":"[40]"},{"why":"Gives the symmetry-protected classification that the new invariant is defined against and reduces to in symmetric limits.","marker":"[2]"}],"fun_headline_variants":["Winding number counts 1D edge states without symmetry","Topological invariant for 1D edge states without symmetry","Complex-plane winding predicts 1D edge states","Counting edge states with a broader winding number"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes the characteristic polynomial $z^2\\Delta(z)$ has no repeated roots, so the band structure is a smooth genus-one torus that can be parametrized by the Weierstrass elliptic function; exactly degenerate models such as the SSH chain are excluded and only recovered as limits.","fun_headline_variants_meta":{"raw":{"variants":["Winding number counts 1D edge states without symmetry","Topological invariant for 1D edge states without symmetry","Complex-plane winding predicts 1D edge states","Counting edge states with a broader winding number"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000573,"raw_usage":{"total_tokens":2723,"prompt_tokens":980,"completion_tokens":1743,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":596,"completion_tokens_details":{"reasoning_tokens":1693}},"tokens_in":596,"tokens_out":1743,"duration_ms":11269,"temperature":1.0,"reasoning_tokens":1693,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:53:04.178387+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $W$ from Eq. (10) for a large random sample of complex couplings of the form in Eq. (1) with non-repeated roots, diagonalize the open-boundary chain with thousands of unit cells, and count the discrete eigenvalues lying away from the GBZ continuum; a single parameter point with $W$ different from that count would falsify the claimed correspondence.","supporting_citations":[],"review_version":1}