{"id":"34f6528b-0346-495f-922c-e02302fc9421","arxiv_id":"2502.04922","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Non-Hermitian electric-circuit lattices with gain and loss host localized disclination states and a measured fractional charge near the defect core.","lead":"Scientists built special electric circuits with added energy (gain) and energy loss, creating a crystal defect that traps waves. The circuits show a half-integer localized charge near the defect, a first for non-Hermitian systems.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Fractional-charge claim rests on an omitted filling rule and an ill-defined biorthogonal charge; until the prescription is supplied and tested against alternative fillings, 1/2 mod 1 is not established.","rationale":"The reader's weakest assumption names the missing occupied-state prescription, and I agree that this is the load-bearing point. My pass sharpens it: even if Supplementary Note 2 supplied a filling rule, Eq. (3) is textually incomplete because biorthogonal products of left and right eigenvectors of the complex-symmetric Laplacian are not necessarily real; a mod-1 reduction of a complex number needs a stated convention. The paper quotes 0.57 (calc) and 0.47 (exp) without error bars or a stability analysis under the chosen cut, so the agreement with 1/2 is not quantified. I therefore do not think the charge-fractionalization part of the headline can be accepted as a measurement until the rule is specified and tested against plausible alternatives. I do not see a comparable flaw in the mode spectra: the five mid-gap disclination states in the topological C5 lattice, their absence in the trivial lattice, and the C4 degenerate states are consistent internal evidence, and the experimental and simulated spectra agree. The missing supplement is an addressable omission, so the reader's CONDITIONAL verdict is the right level; I would not reject the paper on the basis of the mode observations. My recommendation is UNCHANGED relative to the reader's verdict, with the added request that Eq. (3)'s convention and robustness be made explicit and checked before the fractional-charge claim is treated as established.","tokens_in":8592,"tokens_out":8378,"duration_ms":88214,"concrete_test":"Ask the authors for the raw measured admittance matrix (or the full eigenvector set) and the Supplementary Note 2 filling rule; then recompute Q_u mod 1 under (a) the stated rule, (b) all states with Re(j) below the bulk gap, (c) all states with Im(j) below the gap, (d) the stated rule with one state added/removed, and (e) the same rule but with Q_u defined as Re(Σ V^R V^L) vs |Σ V^R V^L|. Also propagate component tolerances (Ct, C0, L0, R0: ±1%–5%) to the charge. If any alternative filling or definition moves Q_u mod 1 from ~1/2 to ~0 by more than 0.1, the fractionalization conclusion is not robust.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that gain and loss produce a measured fractional disclination charge of 1/2 mod 1 stands or falls on Eq. (3). Two things needed to evaluate Eq. (3) are not given. First, the set of 'occupied states' is deferred to Supplementary Note 2 (main text, p. 4), which is absent from the preprint. For a non-Hermitian circuit Laplacian with complex spectrum there is no canonical filling: ordering by Re(j), Im(j), or by a point-gap projection can give different sets, and the reported 0.57/0.47 could simply reflect a set chosen after inspecting the measured eigenstates. Second, Eq. (3) as printed multiplies right and left biorthogonal eigenvectors, V^R V^L, with no complex conjugation, absolute value, or real-part instruction. Since the lattice is complex-symmetric rather than Hermitian, these products are generically complex, so 'mod 1' of their unit-cell sum is undefined; one must specify whether Q_u is Re(Σ V^R V^L), |Σ V^R V^L|, Σ|V^R V^L|, or something else. Neither convention nor an uncertainty on the 0.47/0.57 totals is stated. The mode spectra and localization are independent evidence for NH disclination states, but the headline charge fractionalization is not independently verifiable from the manuscript as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript reports an experimental implementation of non-Hermitian electric-circuit lattices with C5 and C4 symmetry, in which on-site gain and loss are claimed to produce topological disclination states without any Hermitian topological structure. The central experimental evidence consists of complex admittance spectra showing five mid-gap disclination modes in a C5 topological configuration versus a pure gap in a trivial control, direct voltage-profile visualization of the localized modes, and a fractional disclination charge computed from biorthogonal eigenvectors, with reported total values of 0.57 (calculation) and 0.47 (experiment) for the topological phase and about 0.1 for the trivial phase. The paper also reports degenerate zero-energy disclination states in a C4-symmetric configuration. The mode spectra and localization data are presented in detail, but the fractional-charge claim depends on a filling rule and a definition of the biorthogonal charge that are not fully specified in the submitted text and that are partly deferred to supplementary notes not included in the preprint.","tokens_in":8873,"tokens_out":5677,"duration_ms":46370,"significance":"If the charge-fractionalization measurement is correct, this would be the first experimental observation of non-Hermitian disclination states induced solely by gain and loss, and it would extend the topological bulk-defect correspondence to non-Hermitian systems, with potential implications for active topological photonic devices. The manuscript's strengths are the clean topological/trivial control experiment, the good agreement between calculation and experiment for the complex spectra, the direct visualization of the disclination mode profiles, and the inclusion of a second C4-symmetric geometry. The paper also benefits from a concrete circuit platform with explicit component values. However, the headline fractional-charge result is not independently assessable from the manuscript as written: the occupied-state prescription is deferred to an absent supplementary note, Eq. (3) does not specify how the biorthogonal product is made real before taking the mod-1 operation, and the reported 0.57/0.47 values carry no uncertainty. Because the tested theoretical prediction (Ref.","major_comments":[{"comment":"The set of occupied states used in Eq. (3) is not specified in the main text; the reader is referred to Supplementary Note 2, which is not included in the arXiv preprint. For a non-Hermitian circuit Laplacian with a complex spectrum, ordering by Re(j), by Im(j), or by a point-gap projection can yield different occupied sets and therefore different values of Q_u. The authors should state the filling rule explicitly in the main text or in an included supplement, and should report Q_u for alternative reasonable fillings to demonstrate that the 1/2 mod 1 result is not an artifact of a post hoc choice.","section":"§Results, Eq. (3); main text p. 5"},{"comment":"Equation (3) multiplies the right and left biorthogonal eigenvectors as V^R V^L with no complex conjugation, absolute value, or real-part instruction. Because the circuit Laplacian is complex-symmetric rather than Hermitian, these products are generically complex, and 'mod 1' of the unit-cell sum is not defined. The authors must specify whether Q_u is Re Σ V^R V^L, |Σ V^R V^L|, Σ |V^R V^L|, or another convention, and must justify that convention against the theoretical prediction in Ref. 40.","section":"§Results, Eq. (3)"},{"comment":"The text states that the trapped disclination charge in each bulk unit cell clusters around 1/2 mod 1 and then reports 'total disclination charges of 0.57 and 0.47 for calculation and experiment', without defining how the total is aggregated over unit cells. If several unit cells each carry a charge near 1/2 mod 1, the meaning of a single total is ambiguous. The authors should define the aggregation region, report the per-cell values and their deviations from 1/2, and provide an uncertainty estimate that propagates from the measured admittance matrices to Q_u.","section":"§Results, p. 5"},{"comment":"No raw admittance matrices or eigenstates are deposited, and no data availability statement is provided. Since the fractional charge is computed numerically from the measured circuit Laplacian, readers cannot independently verify the 0.47 and 0.57 values or check that the chosen filling rule is robust. The measured J(ω0) matrices used in Eq. (3) should be made publicly available together with the analysis scripts.","section":"Methods and Code availability"}],"minor_comments":[{"comment":"References 20 and 41 are the same paper (Deng et al., Phys. Rev. Lett. 128, 174301 (2022)) and should be consolidated.","section":"References"},{"comment":"The 'appendant term' in Eq. (2) that is said to vanish at the resonance condition is not written out explicitly; the derivation should be shown in the main text or in Supplementary Note 1 rather than asserted.","section":"§Results, Eq. (2)"},{"comment":"The impedance transformation Z = Z₀(I + S)(I - S)⁻¹ is stated without specifying the port-normalization convention; the sign convention depends on the VNA's definition of S, so this should be clarified.","section":"Methods"},{"comment":"The caption describes the experimental data as a 'green dashed line', while the text refers to 'green dots'; please reconcile the description and add error bars if uncertainty information is available.","section":"Fig. 3 caption"},{"comment":"The term 'charge' is used for a spectral quantity derived from circuit eigenvectors; the authors should clarify how this relates to the physical charge on the circuit nodes (capacitors), since readers may otherwise expect a direct charge measurement.","section":"General notation"},{"comment":"The phrase 'To obtain the less lossy spectra, making the admittance easier to measure over the noise' should be rephrased for clarity; presumably the gain compensates losses so that the relevant admittance features are not buried in noise.","section":"§Results, p. 4"}],"recommendation":"major_revision","confidential_remarks":"The main concern is verifiability rather than evident misconduct: the fractional-charge headline depends on a filling rule and a biorthogonal-charge convention that are not fully present in the submitted files. I recommend asking the authors to supply the missing supplementary notes (especially Note 2 and Note 5), to specify and justify the definition of Q_u, and to provide a robustness analysis with alternative fillings and proper uncertainty propagation. I would also note to the editor that the theoretical prediction being tested (Ref. 40) is from largely the same group, which does not invalidate the experiment but does raise the bar for presenting a pre-defined, unambiguous analysis protocol; if that protocol is supplied and the result survives alternative fillings, the paper could be acceptable after revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know about this paper. First, it is the first experimental realization of non-Hermitian topological disclination states, and the central evidence—the complex spectra and the voltage profiles of the localized modes—looks solid. Second, the headline fractional-charge claim is under-specified to the point of being unverifiable from the preprint. That is a real problem, but it is an addressable one.\n\nWhat it does well: the circuit platform is well engineered, the topological vs. trivial comparison is clean, and the agreement between calculation and experiment in Figs. 2 and 3 is convincing. The direct visualization of the disclination mode via monochromatic excitation is a nice touch. The C4-symmetric degenerate zero-energy states extend the acoustic disclination work of Deng et al. to the non-Hermitian regime, which is a reasonable new result.\n\nThe soft spot is the fractional charge. Equation (3) defines Q_u as the unit-cell sum of the product of right and left biorthogonal eigenvectors, summed over occupied states, taken modulo 1. As printed, that expression is not well defined for a complex non-Hermitian Laplacian: V^R V^L is generically complex, so 'mod 1' needs a real part, an absolute value, or some other stated convention. More importantly, the set of 'occupied states' is deferred to Supplementary Note 2, which is not included. For a non-Hermitian complex-spectrum system there is no canonical filling; ordering by real part, imaginary part, or a point-gap projection can give different sets, and the reported 0.57/0.47 values could simply reflect a choice made after inspecting the measured eigenstates. No uncertainty is quoted on these totals, which is troubling when the deviations from 1/2 are not tiny. None of this undermines the mode spectroscopy, but it does mean the paper currently overclaims when it says it observes charge fractionalization.\n\nI do not think this is deliberate obfuscation—the reference to Supplementary Note 2 suggests the authors have a prescription in mind—but a preprint must stand on its own. The same-group theory (ref. 40) is fine in itself, though it raises the bar for showing the charge measurement is not post-selected.\n\nWho is this for? Researchers in non-Hermitian topology, topolectrical circuits, and topological defect physics. It deserves serious peer review: the mode observations are publishable, and the fractional-charge part can be fixed with a clear definition, the missing supplement, and a robustness check against alternative filling rules. I would engage with the revised version.","headline":"A genuine first experimental report of non-Hermitian disclination states with convincing spectra and localization data, but the fractional-charge headline is not verifiable until the authors supply the occupied-state rule and a well-defined biorthogonal charge.","tokens_in":9453,"tokens_out":1824,"would_cite":true,"duration_ms":18526,"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":"The paper reports the first experimental observation of non-Hermitian topological disclination states and an associated fractional charge, produced in electric circuit lattices using only gain and loss.","keywords":["non-Hermitian topological phases","disclination states","charge fractionalization","electric circuit lattices","gain and loss","bi-orthogonal eigenvectors","bulk-defect correspondence"],"falsifier":"Compute the disclination charge from the measured circuit Laplacian using right eigenvectors only, omitting the bi-orthogonal product in the paper's Eq. (3); if the mod-1 charge moves away from approximately 1/2, the reported fractional value is a convention of bi-orthogonal calculus rather than a property of the lattice.","tokens_in":8390,"feed_emoji":"⚡","tokens_out":10513,"duration_ms":91641,"temperature":0.7,"pith_summary":"The paper claims the first experimental observation of non-Hermitian topological disclination states, together with the fractional charge they trap. The experiments use electric circuit lattices in which the only non-Hermitian ingredient is a pattern of gain and loss on lattice sites. In a C5-symmetric lattice with a disclination at the core, the topological phase shows five localized midgap modes and a measured disclination charge of about 1/2 modulo 1, while a trivial phase shows neither. The result matters because it extends the bulk-defect correspondence into non-Hermitian systems and indicates that gain and loss alone can engineer localized defect modes and fractional charge in two dimensions.","feed_headline":"Gain and loss alone trap half-integer charge at lattice defects","feed_subtitle":"Measured circuit Laplacians give a disclination charge of about 1/2 in the topological phase and 0 in the trivial phase.","key_machinery":"The central object is the non-Hermitian circuit Laplacian J evaluated at the LC resonance frequency, which maps the circuit equations to the tight-binding Hamiltonian with imaginary on-site masses ±iγ. Its complex eigenadmittance spectrum provides the quantity that calculation and experiment compare, and its bi-orthogonal right and left eigenvectors enter the charge formula that assigns a fractional charge to each bulk unit cell around the disclination core; the gain and loss themselves are produced by negative-impedance converters and ordinary resistors.","core_discovery":"In a circuit realization of a non-Hermitian tight-binding model with on-site gain and loss of magnitude γ relative to hopping t, the authors observe five midgap disclination states in a C5-symmetric lattice: one singlet and two doublets, all localized at the disclination core. From the measured circuit Laplacian, they compute the disclination charge of bulk unit cells near the core using a bi-orthogonal eigenvector sum over occupied states, obtaining a total charge of 0.57 in calculation and 0.47 in experiment, consistent with the predicted 1/2 modulo 1. In a comparison lattice with a different gain-loss pattern that is topologically trivial, the spectrum shows a pure bandgap and the same charge calculation gives 0.11 and 0.10. The authors also visualize the localized disclination mode by monochromatic field excitation and report degenerate zero-energy disclination states, without fractional charge, in a C4-symmetric non-Hermitian lattice.","pith_inferences":["An implication left implicit is that the validity of the 1/2 mod 1 charge depends on the filling rule for occupied states being fixed before inspecting the measured spectrum; re-deriving the charge with a stated rule would turn the two numbers into a quantitative test.","A testable extension is to compute the same charge using only right eigenvectors or only left eigenvectors; agreement would show the fractional value is independent of the bi-orthogonal convention, while disagreement would reveal the degree to which the result is a choice of inner product.","Varying the gain-loss parameter γ across an exceptional point could show whether the disclination charge jumps when the bulk gap closes, which would directly tie the fractional charge to non-Hermitian band topology rather than to the specific lattice geometry.","The observation of localised disclination modes under monochromatic excitation suggests a route to a disclination-based topological laser, a direction the authors mention but do not demonstrate."],"forward_implications":["Gain and loss alone can induce topological disclination modes in a two-dimensional lattice, without magnetic fields or engineered hopping phases.","The measured charge near 1/2 mod 1 in the topological phase, versus 0 mod 1 in the trivial phase, extends the fractional-charge bulk-defect correspondence to non-Hermitian circuits.","The disclination mode remains observable as a localized voltage profile under monochromatic excitation, and its resonance frequency shifts with the gain-loss parameter γ, indicating tunability.","The C4-symmetric lattice hosts degenerate zero-energy disclination states without fractional charge, establishing a distinct class of non-Hermitian defect states.","Because the circuits can be driven into a nonlinear regime with gain saturation, the same platform could develop into active topological devices such as defect lasers."],"supporting_citations":[{"why":"Supplies the theoretical prediction of non-Hermitian disclination states and charge fractionalization that this experiment is designed to observe.","marker":"[40]"},{"why":"Establishes the bulk-disclination correspondence and the local-density-of-states method used to compute fractional charges.","marker":"[13]"},{"why":"Demonstrates trapped fractional charges at bulk defects in Hermitian topological insulators, providing the direct Hermitian baseline for the charge measurement.","marker":"[14]"},{"why":"Provides the elastic-plate measurement of fractional charge at disclinations that the circuit LDOS analysis adapts.","marker":"[19]"},{"why":"Reports Hermitian degenerate zero-energy disclination states whose non-Hermitian analogue appears in the C4-symmetric lattice.","marker":"[41]"},{"why":"Introduces topolectrical circuits as a platform, grounding the mapping from tight-binding model to circuit Laplacian used throughout.","marker":"[49]"}],"fun_headline_variants":["Electric circuits show gain-loss induced disclination states with half-integer charge","Non-Hermitian circuits reveal topological disclination modes and fractional charge","Gain and loss alone generate disclination states with fractional charge in circuits","Experimental circuits: non-Hermitian disclination states yield half-integer charge","First circuits show non-Hermitian disclination states and fractional charge"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The 1/2 fractional charge is only meaningful if there is a single pre-agreed rule for which of the measured circuit modes count as occupied; that rule is deferred to a supplementary note that the preprint does not include.","fun_headline_variants_meta":{"raw":{"variants":["Electric circuits show gain-loss induced disclination states with half-integer charge","Non-Hermitian circuits reveal topological disclination modes and fractional charge","Gain and loss alone generate disclination states with fractional charge in circuits","Experimental circuits: non-Hermitian disclination states yield half-integer charge","First circuits show non-Hermitian disclination states and fractional charge"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000824,"raw_usage":{"total_tokens":3598,"prompt_tokens":934,"completion_tokens":2664,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":550,"completion_tokens_details":{"reasoning_tokens":2569}},"tokens_in":550,"tokens_out":2664,"duration_ms":17563,"temperature":1.0,"reasoning_tokens":2569,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T20:58:15.535441+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the disclination charge from the measured circuit Laplacian using right eigenvectors only, omitting the bi-orthogonal product in the paper's Eq. (3); if the mod-1 charge moves away from approximately 1/2, the reported fractional value is a convention of bi-orthogonal calculus rather than a property of the lattice.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the theoretical prediction of non-Hermitian disclination states and charge fractionalization that this experiment is designed to observe."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the bulk-disclination correspondence and the local-density-of-states method used to compute fractional charges."},{"cited_title":"W., Li, T., Jiang, W., Hughes, T","cited_arxiv_id":null,"evidence_quote":"Demonstrates trapped fractional charges at bulk defects in Hermitian topological insulators, providing the direct Hermitian baseline for the charge measurement."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the elastic-plate measurement of fractional charge at disclinations that the circuit LDOS analysis adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports Hermitian degenerate zero-energy disclination states whose non-Hermitian analogue appears in the C4-symmetric lattice."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces topolectrical circuits as a platform, grounding the mapping from tight-binding model to circuit Laplacian used throughout."}],"review_version":1}