Topological Zero Modes in Non-Hermitian Topolectrical Systems: Size and Impedance Control

Mansoor B.A. Jalil; Md. Saddam Hossain Razo; S M Rafi-Ul-Islam; Zhuo Bin Siu

REVIEW 4 major objections 5 minor 73 references

Topological Zero Modes in Non-Hermitian Topolectrical Systems: Size and Impedance Control

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2507.17227 v1 pith:2Q4M3IFJ submitted 2025-07-23 cond-mat.mes-hall physics.atm-clus

S M Rafi-Ul-Islam , Zhuo Bin Siu , Md. Saddam Hossain Razo , Mansoor B.A. Jalil This is my paper · ORCID

classification cond-mat.mes-hallphysics.atm-clus

keywords non-HermitianSSHchaintopologicalzeromodestopolectricalcircuitsfinite-sizeeffectsasymmetriccouplinggainandlossimpedancespectraskineffect

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

More 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.

All 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.

My 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.

Referee Report

4 major / 5 minor

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.

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 (4)

[Section II (Eqs. 5-6)] 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.
[Methods Section III.A (Eqs. 26-28)] 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.
[Methods (text near Eq. 25)] 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 II (Fig. 5)] 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.

minor comments (5)

[Section II (Eq. 4)] 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 II (Eq. 3)] 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.
[Figure 2 caption] 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.
[Appendix D] There is a typographical error in Appendix D: 'C_lambda1 != 0,,' contains a double comma.
[Abstract and Introduction] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the edge-state eigenvalues and critical size are derived from the lattice Hamiltonian and checked against independent numerics; heavy self-citation is contextual, not load-bearing.

full rationale

The paper's central objects (E_edge and Mc) are not fitted inputs renamed as predictions. The finite-size eigenvalue formula Eq. 29 is obtained by solving the open-boundary eigenvalue problem of the Hamiltonian in Eq. 8: recurrence relations in Eqs. 9-12, the ansatz in Eq. 15, the boundary-condition determinant in Eq. 21, and the transcendental equation in Eq. 24 are all derived within the paper. Setting the derived band gap to zero (Eqs. 30-31) yields the critical-size formula Eq. 33, and Fig. 5 compares this formula with numerical diagonalization. The paper's approximation in Eqs. 26-27 may be mathematically questionable, but that is an accuracy/correctness concern, not a circular input-output identity. The self-citations (e.g., Refs. [38], [54], [73]) describe topolectrical-circuit implementations, prior size-dependent TZMs, and the non-Hermitian skin effect; they are contextual and do not supply the central Mc formula or forbid alternatives. No uniqueness theorem from the authors is imported, and the cosh/phi structure is derived explicitly rather than adopted by citation. Therefore no significant circularity is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central derivation rests on the circuit-to-lattice mapping, the assumed open boundary conditions, the asserted reduction of the boundary-condition determinant, and the approximate solution of the transcendental equation. No free parameters are fitted to data; all circuit parameters are inputs. No new physical entities are introduced.

assumptions (4)

domain assumption The circuit admittance matrix at resonance maps exactly to the tight-binding Hamiltonian H(kx) of Eq. 2, including the i-gamma gain/loss term.
Sec. II and Methods: the mapping requires the resonance condition Lg = 1/(omega^2(C1+C_lambda1+C_lambda2)) and neglects parasitic effects, NIC bandwidth, and any residual diagonal terms from grounding capacitors.
domain assumption The finite-chain eigenstates satisfy the open boundary conditions psi(0)_B = 0 and psi(M+1)_A = 0, and can be expanded as a sum of two components beta^j with a common modulus r.
Methods Eqs. 13-15 and surrounding text; the expansion assumes no other boundary terms such as the extra grounding capacitor -C2 altering the terminal equations.
ad hoc to paper The determinant of the 2x2 boundary-condition matrix reduces to sin(M theta)(xi + cos theta) + cos(M theta) sin theta = 0.
Methods Eq. 21; this algebraic reduction is asserted without derivation, and later steps in the same section contain sign errors, so the reduction is not transparently reliable.
ad hoc to paper The transcendental equation 2M phi = ln((e^{-phi} - xi)/(e^{phi} - xi)) can be solved by writing e^{phi} = xi + Delta phi with Delta phi small, and approximating (xi + Delta phi)^{2M} approximately (Delta phi)^{2M}.
Methods Eqs. 26-27; this approximation step is algebraically unjustified (for large M the left side is of order xi^{2M}) and the sign of the log argument for xi > 1 is not handled. The resulting phi (Eq. 28) feeds the edge-state energy and the critical-size formula, so the central result depends on this step.

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Pith. "Pith review of Topological Zero Modes in Non-Hermitian Topolectrical Systems: Size and Impedance Control." pith.science (2026). https://pith.science/paper/2Q4M3IFJ

@misc{pith2026250717227,
  author       = {Pith},
  title        = {Pith review of: Topological Zero Modes in Non-Hermitian Topolectrical Systems: Size and Impedance Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2Q4M3IFJ}},
  note         = {Machine review of arXiv:2507.17227}
}

read the original abstract

We investigate the size-dependent behavior of topological zero modes (TZMs) in finite non-Hermitian Su-Schrieffer-Heeger (SSH) chains implemented on a topolectrical circuit platform. By deriving exact analytical solutions for the eigenenergies and band gaps of TZMs, we reveal their sensitivity to system size and non-Hermitian parameters, such as asymmetric coupling and onsite gain or loss. Our results show that non-Hermiticity enables the recovery of exactly zero-energy TZMs at a critical system size, unlike Hermitian systems where finite-size effects cause energy splitting. These zero-energy modes produce pronounced impedance peaks in the circuit's admittance spectrum, providing a measurable signature of topological states. Additionally, a tunable grounding capacitor enables precise control of TZM energies at a fixed resonance frequency, enhancing practical tunability. Our findings offer insights into finite-size effects in non-Hermitian topological systems and guide the design of robust, reconfigurable topolectrical circuits for sensing and energy transfer applications.

Figures

Figures reproduced from arXiv: 2507.17227 by the authors.

**Figure 2.** Size-dependent topological edge states in a Hermi [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Admittance band structure of the two edge states of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Edge state admittance responses of the non-Hermit [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: Plot of the theoretical critical system size [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: (a-c) Full Admittance spectra (of Fig. 2) as a funct [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: (a-c) Real part of full admittance spectra (of Fig. [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: Absolute values of edge state energies as function [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: (a) Complex admittance eigenvalue spectrum for a n [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

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Pith tools

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