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REVIEW 3 major objections 4 minor 70 references

Observation of Impurity-Induced Scale-Free Localization in a Disordered Non-Hermitian Electrical Circuit

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read In a disordered non-Hermitian electrical circuit, a single impurity controls the side where eigenstates accumulate, producing scale-free localization whose length grows with system size.

desk verdict A promising experimental test of impurity-induced scale-free localization undercut by an internal contradiction in the scaling evidence. read the letter →

arxiv 2501.08594 v1 pith:7PSSWU3N submitted 2025-01-15 cond-mat.dis-nn cond-mat.mes-hall

Hao Wang , Jin Liu , Tao Liu , Wen-Bo Ju This is my paper · ORCID
classification cond-mat.dis-nncond-mat.mes-hall
keywords non-Hermitianskineffectscale-freelocalizationelectricalcircuitHatano-Nelsonmodelimpuritydisordernonreciprocalhoppingtopolectrical
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 reports an experimental observation of impurity-induced scale-free localization in a disordered non-Hermitian electrical circuit. The authors build a one-dimensional chain with nonreciprocal hopping and a single non-Hermitian impurity, and measure voltage profiles that accumulate at one side of the chain. They find that the localization direction is controlled by the impurity rather than by the bulk hopping direction, and that the extracted localization length grows linearly with the number of sites. This matters because it demonstrates in hardware a form of skin localization whose scale depends on the sample size, confirming a theoretical scenario in an electronic-circuit platform.

What carries the argument

The central object is the single non-Hermitian impurity: an extra asymmetric hopping $(v+\delta)$ in one direction and $(v-\delta)$ in the other between the first and last nodes of the chain. In the circuit this impurity is a capacitor and an INIC (negative impedance converter with current inversion) connecting the two end nodes, while bulk nonreciprocal hopping $t\pm\gamma$ is realized with INICs between neighbouring nodes and disorder comes from random grounded capacitors. The readout is the spatial profile of the voltage response at the resonance frequency, which by Eq. (B2) approximates the right eigenvector of the Laplacian $J(\omega)$. Scale-free localization is identified from the normalized spatial distribution $\Phi_n$ and the linear growth of the fitted localization length $\xi$ with system size $N$.

What would settle it

Measure a longer chain, say $N=40$ or more, with the same impurity parameters and check whether the normalized peak profile stays collapsed and the fitted localization length $\xi$ keeps growing linearly; alternatively, reconstruct the full admittance matrix and diagonalize it to confirm that a single eigenvector dominates the resonance response. If the width becomes size-independent or the response is a superposition, the scale-free claim fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that a single non-Hermitian impurity, realized as an asymmetric hopping between the first and last sites of a disordered Hatano-Nelson chain (a one-dimensional tight-binding chain with asymmetric hopping), produces an anomalous skin effect: all bulk eigenstates accumulate at a boundary chosen by the impurity, even when this is opposite to the direction set by the nonreciprocal bulk hopping. In the electrical circuit, the voltage response at resonance is taken to represent the right eigenvector of the circuit Laplacian $J(\omega)$, which shares eigenstates with the model Hamiltonian $H$ through $J=i\omega[H-\varepsilon(\omega)]$. For two impurity parameter settings the measured peak voltages are localized on opposite sides, and the localization length $\xi$ extracted from exponential fits scales linearly with the number of sites $N$ for $N=7,11,15,22$. The authors take this size-dependent localization length as the hallmark of scale-free localization, distinct from the conventional non-Hermitian skin effect.

Load-bearing premise

The conclusion depends on treating the measured voltage profile at one peak frequency as a single mode of the circuit and reading its width as a localization length; if several modes mix in, or the profile is not exponential, the scale-free conclusion collapses.

Editorial extensions

If this is right

  • Switching the impurity's asymmetric boundary hopping should move the accumulated eigenstates from one side to the other without reversing the bulk hopping direction, giving a direct control knob for skin-mode localization.
  • Longer chains should show normalized voltage profiles that stay collapsed while the extracted localization length continues to grow, distinguishing the effect from the size-independent non-Hermitian skin effect.
  • The circuit platform makes the eigenstate profile directly measurable as node voltages, so other impurity shapes and strengths can be explored by rewiring only the end nodes.
  • The observation corroborates the theoretical phase diagram that places scale-free localization in the weak-disorder regime, beyond the reach of conventional Anderson localization.

Reading between the lines

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

  • Beyond the measured sizes, pushing $N$ past 22 would test the linear $\xi$ versus $N$ trend; a deviation would reveal where the scale-free regime ends.
  • The same circuit design could be adapted to two-dimensional lattices, where the impurity might control accumulation along a chosen edge, a direction the paper names only as future work.
  • Reconstructing the full Laplacian from multi-port measurements and diagonalizing it would test whether the resonance response is truly dominated by a single right eigenvector, which the paper assumes.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper reports an experimental realization of a disordered non-Hermitian Hatano-Nelson chain with a single non-Hermitian impurity, implemented as an electrical circuit with INIC-based nonreciprocal hopping and randomly chosen grounded capacitors as disorder. The authors claim to observe impurity-induced scale-free localization, meaning eigenstates accumulate at an impurity-controlled side opposite to the bulk hopping direction and the localization length grows linearly with system size. The manuscript presents simulations (Fig. 2) showing collapsed normalized voltage profiles and a linear ξ(N) relation, and experimental measurements (Fig. 4) that are claimed to reproduce this behavior. However, the experimental section explicitly states that the measured profiles are 'not collapsed, indicating the absence of scaled localization,' and then concludes that a linear fit of ξ(N) demonstrates scale-free localization; this is a direct internal contradiction that the paper does not resolve.

Significance. If the central claim were properly supported, this would be a valuable experimental confirmation of the theoretical prediction in Ref. 55, extending recent observations of scale-free localization from PT-symmetric defects (Ref. 54) to a disordered nonreciprocal chain with a single non-Hermitian impurity. The circuit platform and mapping between the Hamiltonian and the Laplacian are standard and clearly described, and the experimental measurement of voltage response and admittance is a reproducible approach. However, the current evidence is not self-consistent: the paper's own criterion for scale-free localization in the simulation (collapse of normalized profiles, Fig. 2(c,d)) is stated to be absent in the experiment, and the linear ξ(N) fit is presented without error bars, residuals, or any validation of the single-mode assumption. The manuscript therefore needs substantial revision before the claim can be accepted.

major comments (3)
  1. [Section IV, Fig. 4(a,b)] The text states: 'These left-side skin modes are not collapsed, indicating the absence of scaled localization for C1=9.4nF and Cv=47nF. There is also absence of scaled localization for right-side skin modes...' Immediately afterward it concludes: 'After performing a linear fit of the localization length at different sizes, we observe scale-free localization behavior.' This is a direct internal contradiction. Since the collapse of normalized profiles was used in the simulation (Sec. III, Fig. 2(c,d)) as the evidence for size-dependent localization length, the absence of collapse in the experimental data cannot be reconciled with the claim without additional explanation. The authors should either correct the text (if 'not collapsed' is a typo) and provide the collapse analysis, or revise the claim accordingly.
  2. [Section IV and Appendix B.2] The extraction of the localization length ξ from the measured voltage profiles relies on the assumption in Eq. (B2) that at the peak frequency the voltage response is dominated by a single right eigenvector of the Laplacian. No evidence is provided that this assumption holds for the four measured system sizes, and the exponential fit is not characterized: no residuals, error bars, R², or fitted slope and intercept for ξ(N) are reported. With only N = 7, 11, 15, 22, the apparent linear trend in ξ(N) could be an artifact of finite-size offsets, multi-mode contamination, or different disorder realizations. The authors should provide the full fit details and an analysis of single-mode dominance (e.g., by comparing the measured response with the full Green's function or by showing the frequency spacing of adjacent resonances).
  3. [Section IV, Fig. 4(c,d) and experimental methods] Each experimental data point in Fig. 4 corresponds to a single disorder realization (randomly chosen grounded capacitors), yet the claim of scale-free localization is an ensemble property: it requires that the localization length scales with N for typical disorder realizations. No disorder averaging or realization-to-realization fluctuation analysis is reported for the experimental data. The authors should either provide measurements over multiple samples with the same parameters or discuss how a single realization supports the statistical claim.
minor comments (4)
  1. [Figure 1(d) caption] The caption contains the typo 'Photographne' and should read 'Photograph'.
  2. [Section III, second paragraph] The phrase 'the the hopping strength at the single-impurity site' contains a duplicated article and should be corrected.
  3. [Section IV, first paragraph] The text says 'The experimental results indicate the existence of the anomalous skin-mode localization controlled by the single impurity in spite of the bulk hopping direction.' The phrase 'in spite of' is ambiguous; earlier in the paper the claim is that localization is 'opposite to' the bulk hopping direction. The authors should clarify the intended meaning.
  4. [Section IV, Fig. 3(b) reference] In the sentence 'for C1 = 22nF and Cv = 2.2nF, the voltage is peaked around the frequency of 164 kHz [see Fig. 3(b)]', the bracket notation is malformed ('[see Fig. 3(b)]' appears as 'Fig. 3(b)]' in the text) and should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the experiment tests an external theory (Ref. 55) through an explicit circuit mapping, and the only serious flaw is a non-circular internal contradiction in Sec. IV.

full rationale

The paper's theoretical input is the disordered Hatano-Nelson chain with a single non-Hermitian impurity, Eq. (1), taken from Ref. 55 (Molignini, Arandes, and Bergholtz), with no author overlap; no load-bearing claim is justified by the present authors' own prior work. The circuit mapping is explicit and checkable: the circuit Laplacian in Eq. (4) and Appendix A is derived from Kirchhoff laws, and the identification J = i\omega[H - \epsilon(\omega)] follows from the component relations ±C2-C1 = t±γ, ±Cδ-Cv = v±δ, and Csn-CS/2 = Vn. This is not an ansatz that presupposes the observed localization. The voltage-response analysis in Appendix B.2, Eq. (B2), assumes single-eigenmode dominance at the resonance frequency; that assumption is uncontrolled and could affect the extracted profile, but it is an experimental-validity condition, not a circular definition. Likewise, extracting ξ by exponential fits and then fitting ξ(N) is ordinary data reduction, not a quantity defined in terms of the target result. The one serious flaw is an internal contradiction in Sec. IV: the text states the left- and right-side profiles "are not collapsed, indicating the absence of scaled localization" and then immediately claims "After performing a linear fit of the localization length at different sizes, we observe scale-free localization behavior." Under the paper's own collapse criterion used for Fig. 2(c,d), non-collapse is asserted to indicate absence, so the linear-fit statement does not restore the claim; this is a consistency/validation problem, but not a circularity, because no equation or fitted parameter is defined in terms of the target result. The derivation chain is therefore self-contained with respect to circularity, and the score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No new particles, forces, mediators, dimensions, or conserved quantities are introduced. The single non-Hermitian impurity is the boundary link of the model, not an invented entity. The load-bearing free parameters are the un-reported linear fit constants and the uncontrolled disorder realization; the key axioms are the ideal INIC mapping, single-eigenstate resonance dominance, and faithful disorder realization.

free parameters (2)
  • Localization length linear fit slope a in xi = a N + b = not reported
    The linear growth of xi with system size N is the main evidence for scale-free localization; the slope and intercept are fitted from four experimental points but no values or error bars are reported.
  • Per-node disorder capacitances Csn = random in [0, 10 nF], exact values not listed
    Onsite disorder is realized by choosing Csn randomly, but the exact realizations are not provided and manufacturing tolerances are unquantified, so the effective disorder strength V is not controlled or reported.
assumptions (4)
  • standard math Kirchhoff current laws and the circuit Laplacian relation I(omega) = J(omega) V(omega) apply to the network in Fig. 1(c).
    Used in Appendix A to derive the circuit Laplacian; this is standard linear circuit theory.
  • domain assumption The circuit Laplacian J and model Hamiltonian H share eigenstates because J = i omega [H - epsilon(omega)], with capacitance parameters mapped to t, gamma, v, delta.
    Requires the INIC elements to act as ideal linear capacitors of +/-C2 and neglects op-amp nonidealities, parasitic inductance, and frequency-dependent loss.
  • domain assumption At the measured resonance peak, the voltage response V is dominated by a single right eigenvector of J (Appendix B.2, Eq. B2).
    Assumes one eigenvalue j_n is near zero and that all other eigenstate contributions are negligible; this is not validated for the small, lossy circuits used.
  • domain assumption The random grounded capacitors Csn in [0, CS] reproduce the uniform disorder Vn in Eq. (1) with a controlled strength V.
    The paper does not measure the actual disorder distribution, and the realized disorder may not match the weak-disorder regime (V = 0.05) used in the phase diagram Fig. 1(b).

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Cite this review

Pith. "Pith review of Observation of Impurity-Induced Scale-Free Localization in a Disordered Non-Hermitian Electrical Circuit." pith.science (2026). https://pith.science/paper/7PSSWU3N

@misc{pith2026250108594,
  author       = {Pith},
  title        = {Pith review of: Observation of Impurity-Induced Scale-Free Localization in a Disordered Non-Hermitian Electrical Circuit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7PSSWU3N}},
  note         = {Machine review of arXiv:2501.08594}
}
read the original abstract

One of unique features of non-Hermitian systems is the extreme sensitive to their boundary conditions, e.g., the emergence of non-Hermitian skin effect (NHSE) under the open boundary conditions, where most of bulk states become localized at the boundaries. In the presence of impurities, the scale-free localization can appear, which is qualitatively distinct from the NHSE. Here, we experimentally design a disordered non-Hermitian electrical circuits in the presence of a single non-Hermitian impurity and the nonreciprocal hopping. We observe the anomalous scale-free accumulation of eigenstates, opposite to the bulk hopping direction. The experimental results open the door to further explore the anomalous skin effects in non-Hermitian electrical circuits.

Figures

Figures reproduced from arXiv: 2501.08594 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Schematic of HN model in the presence of onsite disorder and a single non-Hermitian impurity. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Simulated results for the scale-free localization in the electrical circuit. Frequency-resolved voltage distribution [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Experimentally measured voltages of the admittance [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Experimentally measured scale-free localization in electrical circuit. (a, b) Normalized spatial distribution Φ [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Experimental circuit board diagram containing [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

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