{"id":"f56c8ee2-0924-4726-9433-78d69635690a","arxiv_id":"2505.23951","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Coupled-resonator phononic chains emulate trans-polyacetylene (Su-Schrieffer-Heeger model) and trans-polyaminoborane (κ-deformed Dirac equation), including topological edge states, in finite-element simulations.","lead":"This paper shows how to build chains of aluminum resonators, connected by periodic blocks, that mimic the electronic behavior of two polymer molecules. The phononic versions reproduce the Su-Schrieffer-Heeger model and the κ-deformed Dirac equation, including topologically protected edge states.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No test shows the fitted SSH parameters transfer to the finite chain; the claimed topological protection rests on an untested assumption.","rationale":"The reader's weakest_assumption identifies the single-mode tight-binding description as the load-bearing premise. My concern sharpens this to a specific, testable gap: the paper never checks whether the TB parameters fitted to the infinite chain predict the finite-chain spectrum that is used to claim topological protection. This is more concrete than the general worry about residual hoppings, and it is the most direct way the central claim could fail. I considered the κ-deformed Dirac mapping in Appendix B as an alternative concern; the expansion in Eq. (B.5) contains a possible typo ('ka + π' vs 'ka - π') and the mapping to Eq. (B.2) is only a low-energy approximation, but the paper states this explicitly, so it is not an unacknowledged overclaim. I therefore kept the finite-chain transferability as the single most load-bearing issue. The paper deserves credit for reporting quantitative band agreement and for a clear design methodology, but the topological conclusion requires the proposed predictive test. The reader's conditional verdict already captures this uncertainty, so no change in verdict is needed.","tokens_in":10674,"tokens_out":8070,"duration_ms":79516,"concrete_test":"Extract the hopping parameters v and w from the infinite-chain FEM band fit for the topological configuration in Fig. 5(d) (c1=15 mm, c1'=16 mm). Construct the finite SSH chain with the same number of sites and boundary terminations as the FEM model in Fig. 8, and overlay the TB eigenfrequencies on the FEM spectrum as a function of v. Accept the SSH/topological claim only if the TB edge-state frequencies and their v-range agree with FEM within the same tolerance used for the infinite band (<0.2%). If they do not, the SSH model does not describe the finite system and the 'topologically protected' label is not justified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is quantitative emulation of the SSH model and, consequently, topologically protected edge states in a finite chain. The only quantitative evidence for the SSH description is the fit of the infinite-chain FEM bands to Eq. (A.2) in Figs. 5 and 7. The paper does not use the parameters extracted from that fit to predict the finite-chain spectrum shown in Fig. 8; instead, the topological phase is inferred from the persistence of edge states in the FEM spectrum alone. This leaves untested the assumption that the same tight-binding model (same v, w, and on-site frequencies) describes the finite geometry with its actual boundary terminations. If the fitted parameters do not transfer, the edge states could be termination-dependent defect modes rather than SSH solitons, and the claim of topological protection is unsupported. The paper's own admitted residual second-neighbor hoppings and non-orthogonal basis (Section 3) make this transferability test necessary. Without it, the error below 0.2% in the infinite band structure does not certify the finite-chain prediction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a computational design methodology for quasi-one-dimensional coupled-resonator phononic metamaterials and applies it to two molecular analogs: trans-polyacetylene and trans-polyaminoborane. Using finite-element simulations of aluminum resonator-coupler structures, the authors compare the computed band structures with a tight-binding Su-Schrieffer-Heeger model and with a κ-deformed Dirac equation, claiming quantitative agreement. A finite realization of the phononic polyacetylene chain is then studied, and midgap localized states are interpreted as topologically protected SSH edge states. The manuscript includes appendices defining the phononic tight-binding Hamiltonian and the κ-deformed Dirac mapping.","tokens_in":10863,"tokens_out":9497,"duration_ms":87783,"significance":"If the quantitative claims hold, this is a useful and timely demonstration of a classical mechanical platform for emulating one-dimensional molecular tight-binding models, including tunable first- and second-neighbor hoppings. The paper is refreshingly explicit about its own limitations, acknowledging residual second-neighbor couplings, a non-orthogonal basis, and a regime of very high second-neighbor coupling where the simple models fail. The FEM band structures and the analytical curves do appear to follow the same qualitative trends, and the topological-phase indicator based on the persistence of edge states is a reasonable qualitative diagnostic. However, the central quantitative claim is weakened by the fact that the tight-binding parameters appear to be chosen to match the FEM data, and by the absence of any test that these parameters transfer to the finite chain used for the topological-state claim.","major_comments":[{"comment":"Equation (A.2) as written is not the eigenvalue expression of the Hamiltonian in Eq. (A.1). The square root should contain (fA - fB)^2, not fc^2 = (fA + fB)^2. For the parameters of Fig. 7(a) (fA = fB = 56760 Hz, v = w = 52 Hz, u = 0), Eq. (A.2) gives f+ ≈ 113520 Hz and f- ≈ 0 Hz, whereas the correct eigenvalues of Eq. (A.1) are centered at 56760 Hz and match the plotted green curves. Please correct Eq. (A.2) and state explicitly which expression was used to generate the green curves in Figs. 5 and 7.","section":"Appendix A, Eq. (A.2)"},{"comment":"The manuscript does not state how the tight-binding parameters v, w, u, fA, and fB are determined. The values quoted in the Fig. 7 caption vary with the geometric parameter c2 in a way that is consistent with fitting to the FEM bands. If these parameters are free fitting parameters, then the 'agreement' between the FEM bands and Eqs. (A.2)/(B.5) is partly by construction and does not by itself establish the phononic system as a predictive analog of the SSH or κ-deformed Dirac models. Please show how the hopping amplitudes follow from the coupler geometry and evanescent decay, or provide an out-of-sample test, such as predicting the finite-chain spectrum using only the infinite-chain fitted parameters.","section":"Sections 3 and 4, Fig. 7 captions"},{"comment":"The claim of topologically protected edge states is not supported by a quantitative link between the infinite-chain SSH parameters and the finite-chain FEM results. Fig. 8 shows localized edge states in the finite chain, but the paper does not compare the finite-chain FEM spectrum or wave amplitudes to the SSH tight-binding prediction using the same v and w extracted from the infinite chain. Without this transferability test, the observed midgap states could be termination-dependent defect modes rather than SSH solitons, and the phrase 'topologically protected' is not fully justified. Please add this check or explicitly soften the topological-protection conclusion to a statement of consistency with the SSH picture.","section":"Section 5, Fig. 8"}],"minor_comments":[{"comment":"The caption says 'u = w = 52 Hz' in panels (b) and (c), but the Hamiltonian notation in Eq. (A.1) uses v and w for the first-neighbor hoppings and u for the second-neighbor hopping; 'u = w' is therefore ambiguous or contradictory, especially since the same sentence gives '-u = 5 Hz'. Please use distinct symbols consistently, e.g., 'v = w = 52 Hz' and 'u = -5 Hz'.","section":"Section 4, Fig. 7 caption"},{"comment":"The text says the blue dotted curves correspond to the κ-deformed Dirac equation from Eq. (B.3), while Appendix B states that the curves are obtained from the expanded Hamiltonian (B.5). Please align the figure caption with the equation actually plotted.","section":"Section 4 and Fig. 7 caption"},{"comment":"The sentence 'Regardless of the value of c2, the CRPnTPB band structure exhibits a gap' conflicts with the immediately preceding statement that panel (a), with c2 = 0, shows a linear (gapless) dispersion. Please rephrase to indicate that a gap opens once c2 differs from zero.","section":"Section 4, first paragraph"},{"comment":"The FEM calculations are not described with mesh-convergence details, boundary-condition checks, or numerical error bars. Reporting the mesh parameters and a convergence test would strengthen the quantitative claims.","section":"Sections 2-5"},{"comment":"The affiliation line contains a corrupted fragment ('country3 H. M. Simpson and P. J. Wolfe Youngs modulus...') that appears to be an editing artifact; it should be removed.","section":"Author affiliations"}],"recommendation":"major_revision","confidential_remarks":"The design methodology and the FEM band-structure comparisons are of genuine interest, and the paper's self-identified limitations are a positive sign. However, the incorrect Eq. (A.2), the unexplained parameter fitting, and the missing finite-chain transferability test all bear directly on the central claim of quantitative SSH emulation. These issues are fixable in revision, so I recommend major revision rather than rejection. The journal may also wish to ask whether the parameters in Fig. 7 were fitted or derived, as this determines how the results should be framed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nThe genuinely new piece is the phononic analog of trans-polyaminoborane: a coupled-resonator chain with controlled first- and second-neighbor hoppings, mapped to the κ-deformed Dirac equation. That is a real extension beyond the usual acoustic SSH emulations, and the paper is honest that photonic realizations struggle with the higher couplings. The FEM work looks competent; the infinite-chain band matching to the tight-binding model is genuinely good, with errors below 0.2% when the model applies.\n\nThe main soft spot is the finite-chain topological claim. The tight-binding parameters (v, w, u, fA, fB) are evidently fitted to the infinite-band FEM data; they are not derived from geometry. The paper then jumps to a finite chain and reads the presence of edge states in the FEM spectrum as topological protection. It never uses the fitted parameters to predict the finite-chain spectrum, and it never computes a topological invariant. So the edge states could in principle be termination-dependent defect modes rather than SSH solitons. The paper admits residual second-neighbor hoppings and a non-orthogonal basis in Section 3, which makes that transferability test necessary. This is a real gap, but I would call it moderate, not fatal. The infinite-system emulation claim is supported by the data; the topological part is plausible but under-evidenced.\n\nA few smaller issues: no mesh details or error bars, no code or data, and the κ-deformed Dirac part is a low-energy approximation of the same fitted Hamiltonian, so the nice blue-dot agreement in Fig. 7 is not an independent prediction. The citation pattern looks fine — the relevant acoustic SSH and photonic κ-Dirac literature is there, and the self-citations to the group's prior methodology are appropriate.\n\nWho is it for: people designing phononic or mechanical analogs of 1D molecular models. It is a useful design/tool paper, not a field reorientation.\n\nRecommendation: send it to peer review. A serious referee should ask for a finite-chain test using the same TB parameters, or a winding-number/Zak-phase computation, and for details on mesh/convergence. If the transferability check fails, the topological section should be downgraded to 'edge states observed'; if it passes, the paper is solid.","headline":"A useful phononic design paper: the trans-polyaminoborane emulation with next-nearest-neighbor hoppings is new, but the finite-chain topological claim needs a transferability test before it is fully supported.","tokens_in":11423,"tokens_out":2506,"would_cite":false,"duration_ms":25627,"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 phononic metamaterial built from coupled aluminum resonators quantitatively reproduces the band structures of trans-polyacetylene and trans-polyaminoborane, including topologically protected edge states in the polyacetylene analog.","keywords":["phononic metamaterials","Su-Schrieffer-Heeger model","polyacetylene","polyaminoborane","κ-deformed Dirac equation","tight-binding model","topological edge states","coupled-resonator waveguides"],"falsifier":"Fabricate a finite CRPnTPA chain with $c_1<c_1'$ and measure its mechanical spectrum; the claim fails if no mid-gap mode localized at the ends appears. A cleaner check is to compute the band structure with much longer couplers, suppressing residual second-neighbor hoppings, and verify that the SSH-predicted gap and the fitted $v/w$ remain unchanged.","tokens_in":10504,"feed_emoji":"🔊","tokens_out":5147,"duration_ms":45762,"temperature":0.7,"pith_summary":"The paper claims that a quasi-one-dimensional phononic metamaterial, built by coupling hexagonal aluminum resonators through finite phononic crystals, can act as a mechanical analog of two one-dimensional molecules. For trans-polyacetylene, the computed band structure matches the Su–Schrieffer–Heeger (SSH) model within 1% error, and a finite chain shows edge states whose number stays constant across the topological phase transition. For trans-polyaminoborane, including second-nearest-neighbor hoppings reproduces the band structure of the κ-deformed Dirac equation in its tight-binding approximation. If true, this gives a tunable, easily fabricated platform for studying molecular tight-binding physics, especially next-nearest-neighbor hoppings that are hard to realize in photonic analogs.","feed_headline":"Aluminum resonator chains mimic two molecular band structures","feed_subtitle":"A coupled-resonator waveguide matches the SSH model within 1% and mirrors the κ-deformed Dirac equation.","key_machinery":"The load-bearing object is the coupled-resonator phononic metamaterial unit cell: a hexagonal aluminum resonator whose normal-mode frequency (around 56 kHz) sits inside the complete bandgap of a finite phononic-crystal coupler, so that resonator modes overlap evanescently and behave like atomic orbitals. The argument is carried by the two-band tight-binding Hamiltonian $$H(ka)=\\begin{pmatrix} f_A+2u\\cos(ka) & v+w $e^{{-ika}}$\\\\ v+w $e^{{ika}}$ & f_B+2u\\cos(ka)\\end{pmatrix},$$ with site frequencies $f_A,f_B$, first-neighbor hoppings $v,w$, and second-neighbor hopping $u$. Setting $f_A=f_B$, $u=0$ gives the SSH chain; setting $v=w$, $f_A\\neq f_B$, $u>0$ gives the polyaminoborane chain, whose expansion around $ka=\\pi$ yields a Hamiltonian that maps to the κ-deformed Dirac Hamiltonian through $-u=a/2$, $w=1$, and $\\varepsilon=m$.","core_discovery":"On the paper's own terms, the central discovery is that a coupled-resonator phononic metamaterial with carefully chosen resonator and coupler dimensions quantitatively emulates the band structures of trans-polyacetylene and trans-polyaminoborane. The resonator normal mode falls in the coupler's complete bandgap, so modes localize and couple by evanescent overlap; changing the coupler widths $c_1$ and $c_1'$ tunes the hopping amplitudes $v$ and $w$, while changing $c_2$ tunes the second-neighbor hopping $u$. The finite-element band structures agree with the tight-binding Hamiltonian (A.1), and in the polyaminoborane case with the κ-deformed Dirac equation through the mapping described in Appendix B. A finite realization of the polyacetylene chain displays topologically protected mid-gap states in the topological phase and their disappearance in the trivial phase.","pith_inferences":["This suggests a route to mechanical simulators for other deformed relativistic wave equations, since the mapping from tight-binding parameters to the κ-deformed Dirac Hamiltonian is not specific to polyaminoborane.","One could test the robustness of the analogy by introducing controlled disorder in $c_1$ and $c_1'$ and measuring whether the mid-gap state survives; the paper does not report such a test.","Extending the same unit-cell method to two coupled chains could emulate the edge states of a topological insulator ladder, a step beyond the single-chain results reported here."],"forward_implications":["The same fabrication recipe should transfer to other one-dimensional tight-binding models by choosing resonator sizes and coupler widths that set site energies and hopping ranges.","The polyacetylene analog gives a mechanical testbed for topological invariants: the number of edge states in the gap stays fixed under adiabatic geometric changes.","Because second-neighbor coupling is set by a single width $c_2$, the polyaminoborane analog makes κ-deformed Dirac dispersion accessible in a classical mechanical system.","Finite chains with a domain wall, where $v>w$ on one side and $v<w$ on the other, should host localized states at the interface, a direct corollary of the SSH mapping."],"supporting_citations":[{"why":"Supplies the SSH Hamiltonian and the topological phase transition that the phononic polyacetylene chain is designed to reproduce.","marker":"Su et al., 1979"},{"why":"Derives the κ-deformed Dirac equation and the tight-binding Hamiltonian whose expansion the phononic polyaminoborane bands are compared against.","marker":"Majari et al., 2021"},{"why":"Introduces coupled-resonator phononic metamaterials as mechanical analogs of tightly bound electrons, the foundation of the design methodology.","marker":"Ramírez-Ramírez et al., 2020"},{"why":"Proposes coupled-resonator optical waveguides, the conceptual template for using evanescent resonator coupling as a hopping channel.","marker":"Yariv et al., 1999"},{"why":"Provides the previous CRPnM emulation of cis-polyacetylene with first- and third-neighbor hoppings, the direct precedent this paper extends to trans isomers and second neighbors.","marker":"Betancur-Ocampo et al., 2024"},{"why":"Cited for the non-orthogonal tight-binding basis that accounts for the small deviations between FEM and SSH bands.","marker":"Santiago-García et al., 2025"},{"why":"Provides the tight-binding model for coupled-resonator phononic metamaterials used in the band-structure analysis.","marker":"López-Toledo et al., 2021"},{"why":"Defines the topological invariant whose constancy is used to identify the topological phase in the finite chain.","marker":"Asbóth et al., 2016"}],"fun_headline_variants":["Phononic chain mimics polyacetylene and polyaminoborane","Resonator chain emulates two molecular band structures","SSH and κ-Dirac bands from a resonator chain","Phononic metamaterial replicates SSH and κ-Dirac bands","Coupled resonators mirror two molecular band structures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that each resonator behaves as a single mode and that coupling only occurs through the designed first- and second-neighbor hoppings; if residual longer-range couplings or non-orthogonality were significant, the fitted hopping parameters would not transfer and the topological protection would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Phononic chain mimics polyacetylene and polyaminoborane","Resonator chain emulates two molecular band structures","SSH and κ-Dirac bands from a resonator chain","Phononic metamaterial replicates SSH and κ-Dirac bands","Coupled resonators mirror two molecular band structures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000871,"raw_usage":{"total_tokens":3713,"prompt_tokens":830,"completion_tokens":2883,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":446,"completion_tokens_details":{"reasoning_tokens":2799}},"tokens_in":446,"tokens_out":2883,"duration_ms":20057,"temperature":1.0,"reasoning_tokens":2799,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:38:31.599726+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fabricate a finite CRPnTPA chain with $c_1<c_1'$ and measure its mechanical spectrum; the claim fails if no mid-gap mode localized at the ends appears. A cleaner check is to compute the band structure with much longer couplers, suppressing residual second-neighbor hoppings, and verify that the SSH-predicted gap and the fitted $v/w$ remain unchanged.","supporting_citations":[{"cited_title":", author Sadurn\\' , E","cited_arxiv_id":null,"evidence_quote":"Derives the κ-deformed Dirac equation and the tight-binding Hamiltonian whose expansion the phononic polyaminoborane bands are compared against."},{"cited_title":", author Xu, Y","cited_arxiv_id":null,"evidence_quote":"Proposes coupled-resonator optical waveguides, the conceptual template for using evanescent resonator coupling as a hopping channel."},{"cited_title":", author Manjarrez-Monta\\ nez, B","cited_arxiv_id":null,"evidence_quote":"Provides the previous CRPnM emulation of cis-polyacetylene with first- and third-neighbor hoppings, the direct precedent this paper extends to trans isomers and second neighbors."},{"cited_title":", author M\\'endez-S\\'anchez, R.A","cited_arxiv_id":null,"evidence_quote":"Cited for the non-orthogonal tight-binding basis that accounts for the small deviations between FEM and SSH bands."},{"cited_title":", author Báez, G","cited_arxiv_id":null,"evidence_quote":"Provides the tight-binding model for coupled-resonator phononic metamaterials used in the band-structure analysis."}],"review_version":1}