{"id":"28af8812-23d6-4738-b120-a05412742188","arxiv_id":"2501.08594","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A disordered non-Hermitian electrical circuit with a single impurity is reported to show scale-free eigenstate localization whose direction is set by the impurity, opposite to the bulk hopping direction.","lead":"This paper reports an electrical circuit experiment that realizes a disordered one-dimensional non-Hermitian model and claims to observe a size-dependent \"scale-free\" localization of voltage modes controlled by a single impurity, with direction opposite to the bulk hopping. The result matters because it would be the first experimental confirmation of a predicted anomalous skin effect in a nonreciprocal circuit platform.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section IV directly contradicts its own evidence: it states the measured profiles are 'not collapsed, indicating the absence of scaled localization,' yet immediately claims the linear ξ(N) fit proves scale-free localization.","rationale":"I read the paper as claiming an experimental realization of impurity-induced scale-free localization: measured voltage profiles should show eigenstate accumulation whose localization length grows linearly with system size, opposite to the bulk hopping direction. For that claim, the decisive observable is the collapse of normalized profiles and a quantitatively validated ξ ∝ N scaling. The paper's Section IV contains a direct self-contradiction on the first point: it says the experimental profiles are 'not collapsed, indicating the absence of scaled localization,' and then says a linear fit reveals scale-free localization. This is the most load-bearing weakness because, if taken literally, the experiment's own data contradict the conclusion; if taken as a typo, the paper still lacks the collapse analysis and fit statistics needed to distinguish scale-free behavior from finite-size or disorder effects. The reader's verdict of CONDITIONAL is appropriate: the concern is serious but could be resolved by releasing raw data and performing a collapse test. I therefore leave the verdict unchanged. I partially agree with the reader's weakest_assumption: the single-eigenvector assumption (Eq. B2) is also unvalidated, but the explicit collapse contradiction is more fundamental and should be addressed first.","tokens_in":12156,"tokens_out":5145,"duration_ms":52953,"concrete_test":"Request the raw peak-voltage profiles and measurement uncertainties for all four system sizes (N=7,11,15,22) and both impurity parameter sets; for each profile, fit Φ_n = A exp[-(n-1)/ξ]+c over the same fitted range and report ξ ± δξ. Then test the scale-free criterion directly: compute ξ/N for each N and check whether it is constant within propagated errors, and overlay all profiles on the normalized axis. If the profiles remain non-collapsed as Section IV currently states, or if ξ/N varies beyond errors, the linear ξ(N) fit is insufficient and the central claim is not supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing condition for the experimental claim is that the normalized voltage profiles in Fig. 4(a,b) collapse when plotted against (n-1)/(N-1), because scale-free localization means ξ ∝ N. Section IV states the opposite: '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...' It then concludes 'After performing a linear fit of the localization length at different sizes, we observe scale-free localization behavior.' These statements cannot both be true under the paper's own criterion used in Fig. 2(c,d), where collapse is taken as evidence of size-dependent localization length. If the profiles do not collapse, an extracted ξ that grows with N does not establish scale-free localization: the growth could be driven by disorder-realization differences, finite-size boundary offsets, or multi-mode contamination, and the exponential fit is not validated—no residuals, error bars, slope, intercept, or R² are reported, and only four small system sizes (N=7,11,15,22) are used. Thus the central claim currently rests on an unresolved internal contradiction and an unvalidated fitting procedure. A literal reading of Section IV is that the experiment did not observe scale-free localization; a charitable reading is that 'not collapsed' is a typo, but then the supporting data and collapse analysis must be supplied.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":12411,"tokens_out":3403,"duration_ms":32330,"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":[{"comment":"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.","section":"Section IV, Fig. 4(a,b)"},{"comment":"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).","section":"Section IV and Appendix B.2"},{"comment":"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.","section":"Section IV, Fig. 4(c,d) and experimental methods"}],"minor_comments":[{"comment":"The caption contains the typo 'Photographne' and should read 'Photograph'.","section":"Figure 1(d) caption"},{"comment":"The phrase 'the the hopping strength at the single-impurity site' contains a duplicated article and should be corrected.","section":"Section III, second paragraph"},{"comment":"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.","section":"Section IV, first paragraph"},{"comment":"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.","section":"Section IV, Fig. 3(b) reference"}],"recommendation":"major_revision","confidential_remarks":"The internal contradiction in Section IV is the central issue: the paper states the opposite of what its own criterion requires. This is fixable if the authors can supply corrected text and a proper collapse analysis, or alternatively if they tone down the claim to a demonstration of size-dependent localization without collapse. The absence of any error analysis or single-mode validation is also a serious gap for an experimental claim. The novelty relative to Ref. 54 (PT-symmetric defect) is present but incremental; the paper would benefit from a sharper statement of what is new experimentally."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Paolo—quick take. This paper reports the first circuit experiment on impurity-induced scale-free localization in a nonreciprocal disordered chain, following Molignini et al.'s prediction. That is a real and worthwhile thing to attempt, and the platform (INIC-based nonreciprocal hopping, single impurity at the boundary, on-site disorder) is appropriate. The simulation section is clean: profiles collapse under normalized coordinate and ξ(N) looks linear. The experiment correctly reproduces the left/right localization controlled by the impurity, opposite to the bulk hopping direction. So the basic phenomenology is likely right.\n\nThe soft spot is in Section IV, and it is not minor. The text says the left-side skin modes are 'not collapsed, indicating the absence of scaled localization' and says the same for right-side modes. Two sentences later it claims the linear fit of ξ(N) shows scale-free localization. These two statements cannot both be true under the paper's own criterion, where collapse is the evidence for size-dependent localization length. If the profiles do not collapse, the fitted growth of ξ could come from disorder-realization differences, finite-size offsets, or multi-mode contamination. The reader's stress-test note is accurate; I checked the actual wording and it is indeed a direct contradiction.\n\nThere are also supporting-data gaps. Only N = 7, 11, 15, 22; no error bars; no fit slope, intercept, or residuals; the disorder realizations are not specified, so I cannot tell whether each N uses a different realization, which would change the interpretation of the ξ(N) trend. Appendix B.2 gives the standard single-eigenmode voltage-response argument, but the paper never validates that the measured peak is dominated by one eigenvector for these sizes and parameters. That assumption might hold, but it is load-bearing and untested.\n\nThis is fixable. The authors need to either supply the collapsed data (i.e., the 'not collapsed' sentence is a typo and they have the plots) or explain why ξ(N) growth counts as scale-free without collapse. They should also report fit parameters, error bars, and the disorder configuration. If they can do that, this becomes a solid experimental confirmation of Ref. 55. As it stands, the central claim is not supported by the evidence presented.\n\nRecommendation: send to peer review, but with a clear request for the missing data and a reconciliation of the contradiction. It deserves referee time; the experimental platform and the target effect are significant enough that a serious editor should not desk-reject. I would not cite it in its current form.","headline":"A promising experimental test of impurity-induced scale-free localization undercut by an internal contradiction in the scaling evidence.","tokens_in":12945,"tokens_out":1638,"would_cite":false,"duration_ms":16365,"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":"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.","keywords":["non-Hermitian skin effect","scale-free localization","electrical circuit","Hatano-Nelson model","impurity","disorder","nonreciprocal hopping","topolectrical circuit"],"falsifier":"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.","tokens_in":11939,"feed_emoji":"⚡","tokens_out":9704,"duration_ms":85066,"temperature":0.7,"pith_summary":"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.","feed_headline":"A single impurity forces eigenstates to one side of a circuit","feed_subtitle":"Voltage maps show accumulation opposite the hopping direction, with localization length growing with system size.","key_machinery":"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 $N$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"This reference supplies the theoretical model and phase diagram for anomalous skin effects induced by a single non-Hermitian impurity, which the experiment is designed to verify.","marker":"[55]"},{"why":"This reference introduces impurity-induced scale-free localization and the defining idea that the localization length depends on system size.","marker":"[31]"},{"why":"This reference reports the earlier experimental observation of scale-free localized states from a non-Hermitian defect in a Hermitian circuit, which the present work extends to a nonreciprocal bulk.","marker":"[54]"},{"why":"This reference provides the INIC-based implementation of nonreciprocal hopping used to build the circuit's asymmetric couplings.","marker":"[63]"},{"why":"This reference establishes the topolectrical-circuit framework that maps circuit Laplacians to tight-binding Hamiltonians, so the measured voltages can be read as eigenstates.","marker":"[64]"},{"why":"This reference gives the measurement protocol for voltage response and Laplacian reconstruction, which underlies the eigenstate profiles and admittance spectra.","marker":"[65]"},{"why":"This reference supplies the method for measuring complex admittance in non-Hermitian topolectrical circuits, used here to obtain the complex eigenvalue spectra.","marker":"[70]"}],"fun_headline_variants":["Single impurity bends skin effect against hopping direction","Impurity-induced scale-free localization in non-Hermitian circuit","Anomalous skin effect: impurity overrides hopping direction","One impurity flips eigenstate localization in electrical circuit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Single impurity bends skin effect against hopping direction","Impurity-induced scale-free localization in non-Hermitian circuit","Anomalous skin effect: impurity overrides hopping direction","One impurity flips eigenstate localization in electrical circuit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000156,"raw_usage":{"total_tokens":1178,"prompt_tokens":865,"completion_tokens":313,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":481,"completion_tokens_details":{"reasoning_tokens":249}},"tokens_in":481,"tokens_out":313,"duration_ms":3738,"temperature":1.0,"reasoning_tokens":249,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:22:27.247909+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Molignini , author O","cited_arxiv_id":null,"evidence_quote":"This reference supplies the theoretical model and phase diagram for anomalous skin effects induced by a single non-Hermitian impurity, which the experiment is designed to verify."},{"cited_title":"Li , author C","cited_arxiv_id":null,"evidence_quote":"This reference introduces impurity-induced scale-free localization and the defining idea that the localization length depends on system size."},{"cited_title":"Xie , author G","cited_arxiv_id":null,"evidence_quote":"This reference reports the earlier experimental observation of scale-free localized states from a non-Hermitian defect in a Hermitian circuit, which the present work extends to a nonreciprocal bulk."},{"cited_title":"Helbig , author T","cited_arxiv_id":null,"evidence_quote":"This reference gives the measurement protocol for voltage response and Laplacian reconstruction, which underlies the eigenstate profiles and admittance spectra."}],"review_version":1}