{"id":"137f351f-f8dc-41ea-8a14-ae5586a66541","arxiv_id":"2508.11597","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"Non-Hermitian models built from a bipartite parent have p-th-root energy bands; with an SSH parent, their single-particle levels equal those of free parafermions.","lead":"This paper builds non-Hermitian lattice models from a simple two-band parent, so that the new energy bands are clean p-th roots of the parent's bands; for a Su-Schrieffer-Heeger parent, the single-particle levels exactly match those of free parafermions. The construction gives researchers a direct algebraic bridge between non-Hermitian topological lattices and parafermion physics, plus recipe-style models, including a square-root graphene variant, that photonic or electrical-","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The free-parafermion equivalence rests on an unproved isomorphism: Sec. II.D asserts H^(1,p-1) equals Fendley's M_n up to rescaling, and Eq. (12) is an open-boundary statement not implied by the periodic-boundary Eq. (5).","rationale":"The reader's weakest_assumption is exactly the load-bearing joint I find: the asserted isomorphism between H^(1,p-1) and Fendley's M_n matrices. The k-space derivation of Eq. (6) is clean and independently checkable, and the numerical spectra are plausible, but the title-level equivalence to free parafermions is an open-boundary statement that depends on an unproved matrix identification. I agree with the CONDITIONAL verdict: the physics may be correct, but the central bridging claim needs a proof or an explicit verification. I do not see a reason to move the verdict to REJECT or ACCEPT on the basis of the argument alone. The metadata mismatch is real and undermines provenance, but under the supplied review rule I treat the full text as the manuscript; it reinforces low confidence without changing the specific technical weakness. The proposed concrete test settles the concern: if the characteristic polynomials and the open-boundary power identity both hold, the missing link is supplied; if they fail, the claim must be qualified.","tokens_in":32628,"tokens_out":13505,"duration_ms":138195,"concrete_test":"Take p=3, L=2, with generic parameters (e.g., t=0.5, J=1, gamma=1). Construct H from Eq. (11) and the corresponding Fendley M_n matrices exactly as defined in Ref. [49] under the same open boundary conditions. In a symbolic CAS, compare the characteristic polynomials det(lambda I - H) and det(lambda I - M) after the claimed hopping rescaling, and also verify whether H^3 = gamma I_3 ⊗ (H_SSH)^2. If the polynomials are identical and the open-boundary identity holds, the bridge is sound; if either fails, the equivalence in Sec. II.D needs qualification or the parafermion claim fails. Repeat for p=4 to cover the even-p case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is not the Bloch-band formula (6), which is correctly derived from Eq. (5) and is self-contained. The headline claim is that the open-boundary position-space Hamiltonian H^(1,p-1) has the single-particle spectrum of free parafermions. In Sec. II.D this is made to rest on a single sentence: 'The matrices H^(1,p-1) are the same as the matrices in the construction [49] ... denoted M_n ... except for re-scaling.' No mapping is given: no site relabeling, no statement of which boundary conditions [49] uses, no derivation that the open-boundary H^(1,p-1) satisfies the same algebraic relations as M_n. Consequently Eq. (12) — the pL eigenvalues as p-th roots of gamma^{p-2}(epsilon_j^SSH)^2 — is asserted rather than derived. Eq. (5) cannot supply the missing step: it is a statement about the periodic Bloch matrix, which is a generalized permutation matrix, whereas the position-space matrix (11) is not (e.g., column 3 has two nonzero entries), so the open-boundary spectrum is not a direct corollary. Unless H^(1,p-1) is shown to be similar to (or to have the same characteristic polynomial as) the relevant Fendley matrix, or the equivalent open-boundary identity H^p = gamma^{p-2} I_p ⊗ (H_SSH)^2 is proved, the title claim is unsupported at its one load-bearing joint.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript, despite carrying an abstract on nonparametric learning of stochastic differential equations, is in fact a condensed-matter theory paper proposing a construction of non-Hermitian tight-binding models with p orbitals per unit cell and fully unidirectional hopping. The Bloch Hamiltonian H^{(m,n)}(k,1) is a generalized permutation matrix; the paper derives Eq. (5), (H^{(m,n)}(k,1))^p = γ^{p-2}|h(k)|^2 I_p, and hence the band formula Eq. (6). For the SSH parent, the paper further claims in Sec. II.D that the open-boundary position-space Hamiltonian H^{(1,p-1)} has the same single-particle energy levels as free parafermions in Baxter's clock model, summarized by Eq. (12), (ε_j^{(p)})^p = γ^{p-2}(ε_j^{SSH})^2. The rest of the paper treats partial unidirectional hopping for p=3 and p=4, exceptional points, topology, solitons, and a graphene application. The body of the paper is self-contained in its Bloch-band analysis, but the parafermion equivalence is asserted rather than proved.","tokens_in":32915,"tokens_out":30219,"duration_ms":248641,"significance":"If the central claim of Sec. II.D is correct, the paper offers a simple algebraic mechanism for constructing non-Hermitian models whose single-particle spectra coincide with those of free parafermions, with potential implications for synthetic non-Hermitian lattices and root topological models. The Bloch-band derivation (Eqs. (5)–(6)) is explicit and, for the specific matrices written, correct; the p=3, L=2 position-space spectrum can be checked to satisfy Eq. (12). However, the connection to Fendley's free parafermion construction is a single unproved sentence, and this is the load-bearing joint for the title claim. The paper contains useful numerical studies of edge states and exceptional points, and the graphene square-root discussion is attractive. Yet without a proof of the matrix isomorphism or of the open-boundary polynomial identity, the main novelty is not established. The manuscript also suffers from a complete mismatch between the stated title/abstract and the actual content.","major_comments":[{"comment":"The central claim that H^{(1,p-1)} has the same single-particle energy levels as free parafermions rests entirely on the sentence 'The matrices H^{(1,p-1)} are the same as the matrices in the construction [49] ... denoted M_n in [49], except for re-scaling of the hopping parameters.' No mapping is given: no site relabeling, no statement of boundary conditions, no demonstration that the open-boundary matrix (11) satisfies the same algebraic relations as Fendley's M_n. Equation (5), derived for periodic Bloch matrices, does not imply the open-boundary statement (12), and the position-space matrix is not a generalized permutation matrix. This missing proof is the bridge from the tight-binding model to the free-parafermion result and must be supplied before the title claim can be accepted.","section":"Sec. II.D, Eq. (12)"},{"comment":"The submitted manuscript is titled 'Nonparametric learning of stochastic differential equations from sparse and noisy data' and its abstract describes an EM-SMC-RKHS estimation method. The full text, however, is a physics paper on a non-Hermitian Su-Schrieffer-Heeger model with free-parafermion energy levels. There is no SDE, no RKHS, and no EM algorithm anywhere in the body. As presented, the manuscript is internally inconsistent at the level of its central topic; the journal cannot evaluate or publish it in this form. This needs to be resolved editorially, either by correcting the submission or by reframing the physics paper with an appropriate title and abstract.","section":"Title and Abstract"},{"comment":"Even if the matrix identity with Fendley's construction were established, the text does not derive the open-boundary spectrum (12) from it. The parent SSH chain with L unit cells has L positive eigenvalues ε_j^{SSH}; the claim that the pL eigenvalues of H^{(1,p-1)} satisfy (ε_j^{(p)})^p = γ^{p-2}(ε_j^{SSH})^2 is a nontrivial polynomial identity. The paper provides no proof for general L, and the discussion in Sec. V only states the generalization for inhomogeneous parameters without justification. A direct proof of the characteristic-polynomial factorization, or a similarity transformation to a block-diagonal form, is required.","section":"Sec. II.D, Eq. (12)"}],"minor_comments":[{"comment":"The text labels some terms as 'intercell' and 'intracell' in a way that appears reversed relative to the standard SSH convention; the definitions in Eq. (9) are clear, but the wording is confusing.","section":"General"},{"comment":"The figures showing complex spectra (Figs. 2, 4, 5, 6) are information-dense and the captions would benefit from explicit statements of which curves are analytic bands and which are numerical eigenvalues, especially for the edge-state circles.","section":"Sec. III and figures"},{"comment":"The Supplemental Material statement in Ref. [103] says 'degree of universality' where 'unidirectionality' is intended. Also, the generalization discussion in Sec. V would be more useful with at least a sketch of why Eq. (12) remains valid for inhomogeneous t and J.","section":"Sec. V and Supplemental Material"},{"comment":"Ref. [49] is cited as the source of the free-parafermion construction, but the specific equations or section of that paper corresponding to the claimed matrix identity are not indicated. The reader cannot check the assertion without a pointer.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The abstract/title mismatch is so severe that I suspect a submission error; the editor may want to verify whether the intended manuscript is the physics paper or the SDE paper. If the physics paper is the intended submission, the free-parafermion equivalence needs a real proof, as the current single-sentence assertion is insufficient for a paper whose title makes that equivalence its headline. The Bloch-band part is sound and could be published on its own, but the parafermion claim is currently unsupported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The arXiv ID and abstract describe a statistical machine learning paper on SDE learning, but the full text is a condensed matter paper by Edward McCann on non-Hermitian SSH models. I reviewed the full text as the manuscript. The physics core is clean and largely correct: a generalized permutation-matrix Bloch Hamiltonian gives Eq. (5), H^p = gamma^{p-2}|h|^2 I_p, so the p bands are the p-th roots of a common real factor. I checked the p=3 case; the algebra holds. The paper is also honest about its own scope: it explicitly says the many-body spectrum differs from parafermions and that the models are fermionic.\n\nWhat is genuinely new: the general p-orbital construction with complex chiral symmetry ZHZ^{-1}=omega H, connecting the p-th-root program to free parafermion single-particle levels, plus a detailed treatment of partial unidirectionality for p=3 and p=4 and the graphene square-root example. The numerics track the analytic bands across u, and the exceptional point analysis looks consistent with the cited literature. The citation pattern is appropriate; the parafermion connection is to Fendley's published construction, not a self-citation.\n\nThe soft spot is exactly where the reader and the stress-test note put it. The title claim rests on Sec. II.D, where the position-space matrices H^(1,p-1) are declared to be \"the same as the matrices in the construction [49] ... except for re-scaling of the hopping parameters.\" No mapping is given: no site relabeling, no statement of boundary conditions, no proof that the open-boundary H satisfies the same algebraic relations as Fendley's M_n. Consequently Eq. (12), the pL eigenvalues as p-th roots of gamma^{p-2}(epsilon_j^{SSH})^2, is asserted rather than derived. The periodic Bloch identity (5) cannot supply that step by itself, because the open-boundary matrix is not a generalized permutation matrix. This is a genuine gap, and it is the load-bearing joint of the paper's central equivalence. It may well be repairable, but as written the equivalence is unsupported.\n\nSeparately, the metadata mismatch is severe: this upload is not the paper it claims to be. That alone would justify desk rejection of the submission as received, but the physics content deserves better.\n\nThis paper is for condensed matter theorists working on non-Hermitian topology, root Hamiltonians, and parafermions. A serious referee should engage with it, because the construction is checkable and the free-parafermion claim is worth pinning down. My recommendation: send the physics content to peer review once the submission metadata is corrected and the Sec. II.D isomorphism is either proven or softened.","headline":"The submission metadata is a different paper entirely; the actual text is a solid, checkable p-th-root non-Hermitian SSH construction whose one load-bearing claim—free-parafermion equivalence—is asserted rather than proven.","tokens_in":33509,"tokens_out":2167,"would_cite":true,"duration_ms":26703,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Fully unidirectional hopping makes the $p$ complex bands of a non-Hermitian chain exactly the $p$-th roots of the parent band; with an SSH parent, the single-particle levels equal those of free parafermions in Baxter's clock model.","keywords":["non-Hermitian SSH model","complex chiral symmetry","unidirectional hopping","free parafermions","Baxter clock model","square-root Hamiltonian","exceptional points","tight-binding lattice"],"falsifier":"Diagonalize the open-boundary chain $H^{(1,p-1)}$ for $p=3$ or $p=4$ with a small number of unit cells and compare every eigenvalue of the $pL\\times pL$ matrix — zero modes, algebraic multiplicities, and edge-state degeneracies included — against the single-particle spectrum of the free-parafermion Hamiltonian built from the $M_n$ matrices with matched couplings; any mismatch in the finite-size level pattern or its scaling with $L$ would falsify Eq. (12). A more direct check: compare the matrix in Eq. (11) with $M_n$ element by element to confirm the asserted identity, which the paper does not","tokens_in":32421,"feed_emoji":"⚛️","tokens_out":22801,"duration_ms":219104,"temperature":0.7,"pith_summary":"The paper aims to show that a simple modification of a Hermitian tight-binding model — replacing one hopping direction with fully unidirectional hopping and enlarging the unit cell to $p$ orbitals — produces a non-Hermitian model whose $p$ complex bands are completely determined by the parent band: each band is a $p$-th root of the parent's $|h(k)|^2$, up to constants. The load-bearing identity is that the $p$-th power of the Bloch Hamiltonian is a scalar matrix, $(H^{(m,n)}(k,1))^p=\\gamma^{p-2}|h(k)|^2 I_p$, so the spectrum needs no further calculation. When the parent is the SSH chain, the paper further claims that the position-space Hamiltonian coincides, up to rescaling, with the matrices of a free-parafermion construction, giving the same single-particle energy levels as Baxter's clock model — a concrete bridge from an ordinary fermionic chain to parafermion physics. The paper then charts the transition as hopping becomes partially unidirectional, showing how time-reversal and sublattice symmetries force the real Hermitian spectrum to evolve into a complex one, with exceptional points at edges, solitons, and graphene's Dirac point. A reader should care because the construction turns spectral design into a power identity and yields checkable predictions — including a square-root non-Hermitian graphene — in platforms where near-unidirectional hopping is achievable.","feed_headline":"Non-Hermitian SSH bands are exactly p-th roots of the parent band","feed_subtitle":"One hopping twist pins all p complex bands to the parent spectrum, and matches free-parafermion energy levels.","key_machinery":"The central object is the complex chiral symmetry $ZHZ^{-1}=\\omega H$ with $Z=\\mathrm{diag}(1,\\omega,\\dots,\\omega^{p-1})$, $\\omega=\\exp(2\\pi i/p)$; $H^{(m,n)}(k,1)$ is a generalized permutation matrix — one non-zero entry per row and column, a single $k$-dependent element $h(k)$ or $h^*(k)$, the rest constant $\\gamma$ hoppings. The workhorse identity is $(H^{(m,n)}(k,1))^p=\\gamma^{p-2}|h(k)|^{2}I_p$, fixing the $p$ complex bands as the distinct $p$-th roots of a real factor set entirely by the parent band. A second load-bearing element is the Sec. II.D assertion that the open-boundary matrices $H^{(1,p-1)}$ coincide, up to rescaling, with the free-parafermion matrices $M_n$, transferring Bax","core_discovery":"Central claim: from a Hermitian bipartite parent, $p$ orbitals per cell plus fully unidirectional hopping $\\gamma$ gives a Bloch matrix whose $p$-th power is a scalar, $(H^{(m,n)}(k,1))^p=\\gamma^{p-2}|h(k)|^2 I_p$. Hence the $p$ complex bands are fixed by the parent band alone: the distinct solutions of $(\\epsilon_j)^p=\\gamma^{p-2}|h(k)|^2$ times the $p$-th roots of unity. For an SSH parent this becomes $(\\epsilon_j^{(p)})^p=\\gamma^{p-2}(\\epsilon_j^{\\mathrm{SSH}})^2$, and the paper asserts that the position-space matrices $H^{(1,p-1)}$ are, up to rescaling, the matrices $M_n$ of the free-parafermion construction, so the single-particle levels match free parafermions in Baxter's clock model.","pith_inferences":["Beyond the paper: the $p$-th-root identity is effectively a spectral-design rule — choose a parent band $|h(k)|$ and its winding, and the unidirectional construction hands you an $n$-th-root non-Hermitian model with that band; parents with flat bands or higher winding numbers would be natural untested cases.","Beyond the paper: the open-boundary equivalence may extend past eigenvalues — level-spacing statistics, zero-mode degeneracies, and the response power $P(\\epsilon)$ of the chain should match the clock model's, giving circuit and photonic simulators a measurable fingerprint to verify the parafermion claim.","Beyond the paper: applying the same construction to other integrable parents (Kitaev-like chains, diamond ladders) would produce non-Hermitian models with other exactly known spectra, extending the unification to new symmetry classes.","Beyond the paper (editorial note): the abstract prefixed to this record describes a different manuscript, on nonparametric learning of stochastic differential equations, with no connection to the body text; this extraction summarizes the body text."],"forward_implications":["If the construction is right, any Hermitian bipartite parent yields a non-Hermitian $p$-band model whose entire complex spectrum is fixed by the parent's $|h(k)|$ plus the constants $\\gamma$, $t$, $J$ — the $p$ bands are $p$-th roots of the parent band.","For an SSH parent, the open-boundary chain $H^{(1,p-1)}$ has exactly the single-particle energy levels of free parafermion solutions of Baxter's clock model, so fermionic tight-binding chains and topolectrical-circuit simulations can reproduce that integrable model's single-particle spectrum and topology.","Fully unidirectional hopping produces defective eigenvalues (algebraic multiplicity $p$, geometric multiplicity two) at edges, solitons, and graphene's Dirac point; near the Dirac exceptional point the response power diverges as $|\\epsilon|^{-2n}$ and a loop around it acquires Berry phase $\\pi$.","Partial unidirectionality $(0