{"id":"1b95f49d-4526-4bc0-8a0d-a5a62ce04a01","arxiv_id":"2507.17227","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper derives a finite-size condition for exactly zero-energy topological edge modes in non-Hermitian SSH circuits and proposes impedance-peak detection of the modes.","lead":"This paper studies electrical circuit versions of a topological chain, and shows that adding gain and loss can make the edge states sit exactly at zero energy when the chain has a specific length. The authors derive formulas for that critical length and say it produces a large impedance peak in the circuit, which could be used for sensing and tunable devices.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The critical-size formula Eq. 7 is derived from an invalid large-M approximation and is numerically wrong at small M; for the paper's own parameters it predicts Mc≈3 where the exact transcendental equation gives ≈8.8.","rationale":"The reader's verdict of REJECT is well-founded. My independent check of the central formula shows the problem is not merely a missing justification: the approximation in Methods Eqs. 26-27 produces a quantitatively wrong Mc in the small-M regime that the paper emphasizes. The exact transcendental equation has no solution for M=3 in the γ=0.1 example, while Eq. 28 yields a spurious ϕ, leading to Mc≈3 instead of the true ≈8.8. The cos/cosh discrepancy between main-text Eq. 5 and Methods Eq. 29 is a related but separate typo; both point to the same conclusion that Eqs. 5-7 as printed are not reliable. I do not dispute the qualitative phenomenon, which is consistent with prior work, but the manuscript's stated exact results are unsupported. A corrected derivation and quantitative comparison with exact numerics would be required. Therefore I concur with the reader's rejection.","tokens_in":17709,"tokens_out":16582,"duration_ms":153683,"concrete_test":"For the Fig. 4 parameters (C1=0.9, Cλ1=0.1, C2=1.0, Cλ2=0) and γ=0, 0.05, 0.1, 0.15, numerically solve the exact transcendental equation sinh((M+1)ϕ)/sinh(Mϕ)=ξ for real M, then impose the zero-energy condition coshϕ=(A+B-γ^2)/(2√AB). Compare the resulting M to Eq. 7. If the exact M differs from Eq. 7 by more than one unit cell in any case (e.g., γ=0.1 gives ≈8.8 vs ≈3.0), the critical-size formula is refuted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central quantitative claim is the critical size Mc (Eq. 7) at which non-Hermitian edge states become exactly zero-energy. The derivation in Methods replaces (ξ+Δ)^(2M) by Δ^(2M) (Eqs. 26-27). This is invalid: for |Δ|≪ξ and large M, (ξ+Δ)^(2M) ≈ ξ^(2M), not Δ^(2M). The resulting ϕ in Eq. 28 is reliable only when ξ^(2M) ≫ 1, i.e., at large M. But the claimed effect — Mc shifting to shorter chains with increasing γ — is precisely the small-M regime where the approximation breaks down. Concretely, for the parameters of Fig. 4 (C1=0.9, Cλ1=0.1, C2=1.0, Cλ2=0, γ=0.1): exact solution of Eq. 24 with the zero-energy condition coshϕ=(A+B-γ^2)/(2√AB) gives a real critical M≈8.8, while Eq. 7 gives Mc≈3.0. For γ=0.2, σ≈0.984<1, so no real zero-energy solution exists at all, contradicting the trend shown in Fig. 5. The exact zero-mode recovery may still occur for some parameters, but the paper's stated exact formula and its numerical comparison in Fig. 5 are unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a finite non-Hermitian Su-Schrieffer-Heeger (SSH) chain realized as a topolectrical circuit, with asymmetric (non-reciprocal) couplings and staggered gain/loss. It claims exact analytical expressions for the finite-size edge-state admittance eigenvalues (Eq. 5) and for a critical system size Mc (Eq. 7) at which the topological zero modes (TZMs) recover exactly zero energy, with the effect observable as a large impedance peak. The Methods section derives these results from a boundary-condition determinant, a transcendental equation for the decay parameter, and an asymptotic inversion. The paper also proposes a grounding-capacitor scheme to tune edge-state energies at fixed resonance frequency, and discusses experimental feasibility and component tolerances.","tokens_in":18052,"tokens_out":8646,"duration_ms":80549,"significance":"The subject is timely: finite-size and non-Hermitian effects in topological circuits are of active interest, and an impedance-based signature of exactly zero-energy edge modes would be experimentally valuable. If the central analytical result were correct, the paper would provide a useful closed-form description of size-dependent TZM recovery. However, the derivation contains a mathematically invalid approximation in the central step, the main-text formulas are inconsistent with the Methods derivation, and the numerical validation in Fig. 5 does not actually compare the claimed critical sizes. The paper ships no code or machine-checkable proofs, and the central quantitative claim is currently unsupported.","major_comments":[{"comment":"The main-text edge-state energy, Eq. (5), is written with cos(phi), where phi is defined as the real positive number ln(xi + (xi - xi^3)/(1 + xi^{2M+2})). The Methods derivation, Eq. (29), gives the same quantity with cosh(phi), which follows from Eq. (23) after substituting theta = pi + i phi (since cos(theta) = -cosh(phi)). Because phi is real, cos(phi) oscillates and cannot describe the exponentially decaying edge-state energy. This is not a harmless typo, since Eq. (6) sets the zero-gap condition using cos(phi), whereas the Methods condition, Eq. (31), uses cosh(phi) = sigma.","section":"Section II (Eqs. 5-6)"},{"comment":"The step labeled exact in the solution of the transcendental equation is algebraically invalid. Equation (26) is correct, but the approximation (xi + Delta_phi)^{2M} approx (Delta_phi)^{2M} in Eq. (27) is not valid for xi > 1 and small |Delta_phi|; the leading behavior is (xi + Delta_phi)^{2M} approx xi^{2M}, not Delta_phi^{2M}. Consequently Eq. (27) for Delta_phi and Eq. (28) for phi do not follow, and the critical-size formula Eq. (7) (and Eq. (33)) is not an exact result. A concrete check for the Fig. 4 parameters (C1 = 0.9, C_lambda1 = 0.1, C2 = 1.0, C_lambda2 = 0, gamma = 0.1) shows that the exact condition sinh((M+1)phi)/sinh(M phi) = xi with cosh(phi) = sigma has a solution near M ~ 8.8, while Eq. (7) gives Mc ~ 3.0.","section":"Methods Section III.A (Eqs. 26-28)"},{"comment":"The claim that a solution to Eq. (25) exists only when the denominator e^phi - xi is close to zero is an asymptotic large-M condition, not a general property. For finite M the denominator need not be small, and the paper's advertised effect is precisely the shift of the critical size to smaller M with increasing gamma. This makes the 'exact' solution internally inconsistent with the parameter regime in which the claimed phenomenon occurs.","section":"Methods (text near Eq. 25)"},{"comment":"The numerical validation does not support the stated conclusion. The gray dots in Fig. 5 are numerical edge-state energies as functions of M and gamma, not numerically extracted critical sizes Mc, so the claimed 'close agreement between the analytical predictions and numerical results' is not actually demonstrated by the figure. Moreover, for the stated parameters with gamma = 0.2, sigma = 0.984 < 1, so cosh^{-1}(sigma) is not real and Eq. (7) does not yield a real critical size, which contradicts the monotonic trend asserted in the text and in the figure caption.","section":"Section II (Fig. 5)"}],"minor_comments":[{"comment":"In Eq. (4), the summation index is inconsistent: the numerator reads |psi_{l,p} - psi_{k,q}|^2, but the second wavefunction should be psi_{l,q} to correspond to the l-th eigenmode.","section":"Section II (Eq. 4)"},{"comment":"The relation defining phi in Eq. (3) is typeset incorrectly ('1+xi2M +2' should likely be '1+xi^{2M+2}'), and the condition xi > 1 that is needed for the subsequent formulae is not stated.","section":"Section II (Eq. 3)"},{"comment":"The caption refers to 'a = 1 and b = M' for the impedance measurement, while the main text and Fig. 3 use p = 1 and q = 30; the notation should be made consistent.","section":"Figure 2 caption"},{"comment":"There is a typographical error in Appendix D: 'C_lambda1 != 0,,' contains a double comma.","section":"Appendix D"},{"comment":"The paper repeatedly describes the formulas as 'exact analytical solutions,' but Eq. (28) is explicitly derived using an approximation; the terminology should be corrected throughout once the derivation is fixed.","section":"Abstract and Introduction"}],"recommendation":"reject","confidential_remarks":"The paper has a high fraction of self-citations (roughly 15 of 73 references are by the same group), which is not by itself disqualifying but is worth noting. More importantly, the central quantitative result is not supported by the derivation as written: the main-text formulas are inconsistent with the Methods, the inversion of the transcendental equation uses an invalid algebraic approximation, and the numerical comparison in Fig. 5 does not validate the claimed exact Mc. The concrete discrepancy between Eq. (7) and the exact transcendental solution for the paper's own parameters indicates a load-bearing error. A resubmission could be considered if the authors re-derive the critical-size condition correctly, correct the cos/cosh inconsistency, and provide genuine numerical extraction of Mc."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my honest read. The paper's central claim—that non-Hermiticity can recover exact zero-energy edge modes at a critical system size—is plausible and consistent with Ref. 39, but the analytical machinery used to support it does not hold up. The concrete topolectrical circuit model, with component values, resonance tuning, and a tolerance analysis, is a genuine plus. That part looks like the authors know their experimental platform.\n\nThe math, however, is where it falls apart. Main-text Eq. 5 uses cos(φ), while the Methods derivation (Eqs. 23/29) correctly gives cosh(φ). Eq. 6 is also wrong: the condition should have cosh(φ), and the factor 1/2 in the denominator of σ is missing in Eq. 32. These are not cosmetic typos; they are load-bearing because Eq. 7 inherits them.\n\nMore seriously, the approximation step in Methods is invalid. The text says to replace (ξ+Δ)^(2M) with Δ^(2M), which is wrong; the final formula Eq. 27 only works if you use ξ^(2M). But even with that correction, the approximate φ(M) is accurate only for large M. For the parameters in Fig. 4 (C1=0.9, Cλ1=0.1, C2=1.0, γ=0.1), the exact transcendental equation gives Mc≈9, while Eq. 7 gives Mc≈3. For γ=0.2, the exact zero-energy condition has no real solution at all (σ<1), yet Fig. 5 plots a red dot. The numerical comparison is only qualitative; the gray cloud does not pin down where the crossing actually occurs.\n\nAll of this means the paper's central quantitative results, as printed, are unsupported. The qualitative phenomenon is likely real, and the circuit blueprint is worth keeping, but the derivation needs a thorough rework. The authors should correct the equations, provide a valid approximation scheme, and compare the analytic Mc against the actual numerical zero-energy crossing for several parameter sets.\n\nMy recommendation: do not publish this version. But do not desk-reject it either—send it to a referee with instructions to focus on the Methods derivation and the validity of Eq. 7. It is fixable, and the experimental framing has value.","headline":"The circuit design and qualitative picture are useful, but the central analytic formulas (Eqs. 5–7) contain internal contradictions and are numerically wrong, so the paper is not publishable as-is.","tokens_in":18530,"tokens_out":10363,"would_cite":false,"duration_ms":93153,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A finite non-Hermitian SSH chain has a critical length at which its edge modes sit at exactly zero admittance, and in a topolectrical circuit this appears as a large impedance peak.","keywords":["non-Hermitian SSH chain","topological zero modes","topolectrical circuits","finite-size effects","asymmetric coupling","gain and loss","impedance spectra","non-Hermitian skin effect"],"falsifier":"Numerically diagonalize the open-boundary Hamiltonian of Eq. (8) with the paper's parameter set, scan $M$ across the predicted $M_c$, and test whether the minimum edge-state energy reaches zero exactly and at the predicted integer. In a physical circuit, sweep the number of unit cells while holding all other parameters fixed and look for the impedance peak at $M_c$; if the peak appears at a different size or remains finite at the predicted size, the central claim is falsified.","tokens_in":17498,"feed_emoji":"⚡","tokens_out":8525,"duration_ms":76997,"temperature":0.7,"pith_summary":"This paper tries to establish that in a finite non-Hermitian Su-Schrieffer-Heeger chain, realized as a topolectrical circuit with non-reciprocal coupling and staggered gain and loss, the topological edge modes can be restored to exactly zero energy by choosing the right system size. In an ordinary finite Hermitian chain the two edge modes hybridize and split away from zero; the authors claim that non-Hermiticity reverses this at a critical number of unit cells $M_c$, and they give a closed-form estimate for $M_c$. The electrical counterpart is that the admittance eigenvalue of the edge mode vanishes at $M_c$, which produces a very large impedance between the end nodes. The paper also shows that an added grounding capacitor makes the resonance frequency independent of the inter-cell capacitor, so the edge-state energy can be tuned at a fixed drive frequency.","feed_headline":"Edge modes hit exactly zero energy at one critical chain size","feed_subtitle":"Non-Hermitian gain and loss cancel finite-size splitting, producing a sharp impedance peak in a topolectrical circuit.","key_machinery":"The argument is carried by an Ansatz for edge eigenstates as superpositions of $\\beta^j$ modes, with $\\beta_1 = r e^{i\\theta}$ and $\\beta_2 = r e^{-i\\theta}$; for the topological edge states one takes $\\theta = \\pi + i\\phi$, where $\\phi$ is the decay parameter of the localized mode. Imposing the open-boundary conditions turns the secular problem into $\\sinh((M+1)\\phi)/\\sinh(M\\phi)=\\xi$, whose solution feeds into the edge eigenvalue through $\\cosh\\phi$. The critical size $M_c$ is the value of $M$ at which $\\phi = \\cosh^{-1}\\sigma$ makes the eigenvalue square root vanish, which is exactly where the two edge modes collapse to zero energy and the end-to-end impedance becomes very large.","core_discovery":"The central claim is that non-Hermiticity enables exact zero-energy topological zero modes in finite systems, not only in the thermodynamic limit. For the generalized non-Hermitian SSH model, the edge-state admittance eigenvalues are $E_{\\pm} = \\pm\\sqrt{-2\\sqrt{(C_1^2-C_{\\lambda1}^2)(C_2^2-C_{\\lambda2}^2)}\\cosh\\phi + (C_1^2-C_{\\lambda1}^2)+(C_2^2-C_{\\lambda2}^2)-\\gamma^2}$, with $\\phi$ a size-dependent decay parameter fixed by the equation $\\sinh((M+1)\\phi)/\\sinh(M\\phi)=\\xi$. Setting the square root to zero gives the condition that the edge-state gap vanishes, and solving for $M$ yields the critical size $M_c$ at which the two edge modes become degenerate at exactly zero admittance. In the topolectrical circuit this zero eigenvalue is read out as a large impedance peak between the chain ends, giving a direct measurement signature.","pith_inferences":["Editorial: if the effect is generic, similar size-tuning should appear in higher-order non-Hermitian topological circuits, where finite-size corner-mode splitting could be cancelled at a critical footprint.","Editorial: the impedance peak at $M_c$ could serve as a calibration observable, since measuring the peak position across lengths gives a direct estimate of the non-Hermitian parameters $\\gamma$ and $C_{\\lambda1}$.","Editorial: because $M_c$ comes from an approximate solution of the transcendental equation, checking the exact secular equation for large $M$, where the replacement $(\\xi+\\Delta\\phi)^{2M}\\approx(\\Delta\\phi)^{2M}$ becomes increasingly poor, would bound where the formula remains predictive."],"forward_implications":["If the critical-size claim is correct, a finite non-Hermitian SSH chain of length $M_c$ supports edge modes at exactly zero energy, so measurements along a chain of increasing length will show the edge-state gap closing precisely at $M_c$.","In a topolectrical circuit, the impedance between the two end nodes will show a pronounced peak at $M_c$, giving an electrical rather than spectroscopic signature of the zero mode.","Tuning the gain/loss parameter $\\gamma$ moves $M_c$ toward shorter chains, so the same circuit can be reconfigured by changing the resistor rather than by adding unit cells.","Adding the grounding capacitor $-C_2$ fixes the resonance frequency independently of $C_2$, so adjusting $C_2$ shifts the edge-state energies without retuning the drive."],"supporting_citations":[{"why":"Supplies the prior result that topological zero modes break up and recover in finite non-Hermitian optical lattices, which this paper extends to size-dependent recovery in topolectrical circuits.","marker":"[39]"},{"why":"Provides the non-Hermitian dimer-chain sensitivity analysis that motivates the finite-size deviation of edge states.","marker":"[70]"},{"why":"Gives the near-field coupling picture used to explain why short chains split edge modes away from zero energy.","marker":"[71]"},{"why":"Supplies the topolectrical-circuit realization and the negative impedance converter technique used to implement asymmetric couplings.","marker":"[41]"},{"why":"Establishes the non-Hermitian bulk-boundary correspondence framework used to characterize skin modes in such circuits.","marker":"[48]"},{"why":"Provides the model of non-reciprocal one-dimensional topolectrical circuits that underlies the asymmetric-coupling Hamiltonian.","marker":"[54]"},{"why":"Is the direct predecessor on system-size-dependent topological zero modes in coupled topolectrical chains.","marker":"[38]"},{"why":"Gives the impedance-response method used to predict large impedance peaks from near-zero admittance eigenvalues.","marker":"[52]"}],"fun_headline_variants":["Zero-energy edge modes appear at one critical chain length","Non-Hermitian trick pins edge modes to exact zero","Impedance peak signals exact zero-energy topological mode","Critical size unlocks exact zero modes in non-Hermitian chain","Tunable capacitor shifts edge-mode energy in topolectrical circuit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The formula for $M_c$ rests on an approximate solution of the transcendental equation $\\sinh((M+1)\\phi)/\\sinh(M\\phi)=\\xi$ in which terms like $(\\xi+\\Delta\\phi)^{2M}$ are replaced by $(\\Delta\\phi)^{2M}$; if that approximation fails, the predicted critical size at which zero energy is reached is not guaranteed.","fun_headline_variants_meta":{"raw":{"variants":["Zero-energy edge modes appear at one critical chain length","Non-Hermitian trick pins edge modes to exact zero","Impedance peak signals exact zero-energy topological mode","Critical size unlocks exact zero modes in non-Hermitian chain","Tunable capacitor shifts edge-mode energy in topolectrical circuit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000537,"raw_usage":{"total_tokens":2581,"prompt_tokens":948,"completion_tokens":1633,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":564,"completion_tokens_details":{"reasoning_tokens":1564}},"tokens_in":564,"tokens_out":1633,"duration_ms":13385,"temperature":1.0,"reasoning_tokens":1564,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:55:09.376948+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically diagonalize the open-boundary Hamiltonian of Eq. (8) with the paper's parameter set, scan $M$ across the predicted $M_c$, and test whether the minimum edge-state energy reaches zero exactly and at the predicted integer. In a physical circuit, sweep the number of unit cells while holding all other parameters fixed and look for the impedance peak at $M_c$; if the peak appears at a different size or remains finite at the predicted size, the central claim is falsified.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the prior result that topological zero modes break up and recover in finite non-Hermitian optical lattices, which this paper extends to size-dependent recovery in topolectrical circuits."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the non-Hermitian dimer-chain sensitivity analysis that motivates the finite-size deviation of edge states."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the near-field coupling picture used to explain why short chains split edge modes away from zero energy."},{"cited_title":"Imhof, C","cited_arxiv_id":null,"evidence_quote":"Supplies the topolectrical-circuit realization and the negative impedance converter technique used to implement asymmetric couplings."},{"cited_title":"Helbig, T","cited_arxiv_id":null,"evidence_quote":"Establishes the non-Hermitian bulk-boundary correspondence framework used to characterize skin modes in such circuits."},{"cited_title":"Rafi-Ul-Islam, Z","cited_arxiv_id":null,"evidence_quote":"Provides the model of non-reciprocal one-dimensional topolectrical circuits that underlies the asymmetric-coupling Hamiltonian."},{"cited_title":"Rafi-Ul-Islam, Z","cited_arxiv_id":null,"evidence_quote":"Is the direct predecessor on system-size-dependent topological zero modes in coupled topolectrical chains."},{"cited_title":"Sahin, Z","cited_arxiv_id":null,"evidence_quote":"Gives the impedance-response method used to predict large impedance peaks from near-zero admittance eigenvalues."}],"review_version":1}