{"id":"fdeceb5f-79a5-4e8f-80ae-98d1710f82a5","arxiv_id":"2504.15620","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"The winding number of a non-chiral non-Hermitian two-band system is measured experimentally from right-eigenstate spin textures alone, without simulating the conjugate Hamiltonian.","lead":"This paper reports an experiment that measures a topological winding number in a non-Hermitian quantum system without chiral symmetry, using a two-qubit nuclear magnetic resonance simulator. The key simplification is extracting the invariant from right-eigenstate spin patterns alone, which removes the need to simulate the conjugate Hamiltonian.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (9) fixes Re[φ_yx] only modulo nπ/2, and the paper never specifies how the integer branch n(k) is determined from measured spin textures; without a data-driven branch rule the claimed measurement of wt is underdetermined.","rationale":"The reader's conditional verdict is appropriate, but the specific weak assumption identified here is different from the reader's emphasis on dilation errors and fit sensitivity. The reader focused on experimental imperfections; this review identifies a more fundamental, protocol-level ambiguity in Eq. (9): the nπ/2 term is k-dependent in general, and the paper does not explain how to determine it experimentally. This does not invalidate the algebraic content of Appendix A, which is correct as a tangent identity, but it makes the headline claim of a complete measurement protocol incomplete. The missing branch-fixing step also connects to the reader's observation that no winding number with uncertainty is reported: without a stated unwrapping rule, the plotted Re[ϕ_yx] curves cannot be independently converted into wt from the data alone. A synthetic-data pipeline test can settle whether the ambiguity is benign for the specific model or requires an additional protocol step. Since the paper already receives a CONDITIONAL verdict for missing data and overclaiming, this concern reinforces that condition rather than changing the verdict.","tokens_in":14386,"tokens_out":24035,"duration_ms":225264,"concrete_test":"Generate noiseless synthetic time traces for both experimental parameter sets (J0 = 1 and J0 = 0.3, J1 = 1, δ = 0.3), fit them as described to extract ϕ++_yx and ϕ−−_yx at a dense k grid using standard atan2, then reconstruct Re[ϕ_yx] and wt using only a continuity/unwrapping rule that never references the known h(k). If the reconstructed wt differs from the true values 1 and 2, or if different legitimate unwrapping choices change the result, the protocol needs an explicit, data-driven rule for fixing n(k) before the claimed measurement can be considered well defined.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The derivation in Appendix A ends at tan(ϕ++_yx + ϕ−−_yx) = tan[2 Re(ϕ_yx)] (Eq. A5), so Eq. (9) — Re[ϕ_yx] = (ϕ++_yx + ϕ−−_yx)/2 + nπ/2 — holds only up to an integer multiple of π/2. The integer n(k) is not shown to be constant; it can change with k when the principal branches of arctan in Eq. (10) cross branch cuts. For the wt = 2 phase, Re[ϕ_yx] changes by 2π across the Brillouin zone, while wrapped values of ϕ++_yx and ϕ−−_yx are confined to bounded intervals, so n(k) must jump if Eq. (9) is used pointwise. The paper does not state an unwrapping rule or a method to fix n(k) from data. Section IV and Figs. 3(e,f) present Re[ϕ_yx] curves, but Appendix C shows only the wrapped ϕ++_yx and ϕ−−_yx, and the main text never reports the experimental wt value or the branch convention used. If n(k) was chosen to match the known theoretical Re[ϕ_yx], the measurement is circular; if n(k) was obtained by some unwrapping, that step is omitted. Thus the central claim — that wt can be measured from right-eigenstate spin textures — is not fully supported without an explicit branch-fixing procedure.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a method to measure the winding number wt of one-dimensional non-Hermitian two-band systems without chiral symmetry using only right-eigenstate spin textures. The central identity is Eq. (9), derived in Appendix A: Re[φ_yx(k)] = (φ++_yx(k) + φ−−_yx(k))/2 + nπ/2, where φ±±_yx are obtained from spin textures of the right eigenstates of H(k), avoiding the need to implement H†. The authors implement a non-Hermitian Hamiltonian (hx = J0 + J1 cos k, hy = J1 sin k − iδ, hz = 0.5) on a two-qubit NMR platform using a dilation method, extract the two right-eigenstate spin textures and complex eigenvalues by fitting time traces, and present Re[φ_yx(k)] for two parameter sets corresponding to wt = 1 and wt = 2. They also present the complex energy bands and state that the phase boundaries of wt and νE differ because chiral symmetry is broken.","tokens_in":14669,"tokens_out":5084,"duration_ms":45557,"significance":"If the central claim holds, the paper provides a practical simplification: topological winding numbers of non-chiral non-Hermitian systems can be obtained without separately realizing H†, which is a real experimental advantage. The algebraic derivation in Appendix A is clean and does not rely on measured data, and the time-trace fitting extraction is a standard and generally sound procedure. The experiment is a concrete demonstration of a non-Hermitian non-chiral Hamiltonian on a quantum simulator, with a detailed pulse-sequence implementation and an error budget for the spin textures. However, the manuscript does not yet supply a complete data-driven procedure for fixing the branch integer n(k) in Eq. (9), nor does it report a numerical value of wt with an uncertainty. These gaps directly affect the strength of the claim that wt was measured, so the result is significant but currently under-supported.","major_comments":[{"comment":"The branch integer n(k) in Eq. (9) is never specified. The derivation in Appendix A establishes only tan(φ++_yx + φ−−_yx) = tan[2 Re(φ_yx)], so Eq. (9) determines Re[φ_yx] only modulo π/2. For the wt = 2 case, Re[φ_yx] must change by 2π across the Brillouin zone, while principal-branch values of φ++_yx and φ−−_yx are periodic and bounded; therefore n(k) cannot be a constant and must jump as k crosses branch cuts. The paper does not state an unwrapping rule or any data-driven method to fix n(k). If n(k) was chosen to match the known theoretical Re[φ_yx], the measurement is circular; if it was obtained by unwrapping the measured angles, that step is omitted. This is load-bearing because the claimed measurement of wt rests on Eq. (9).","section":"Section II, Eq. (9); Appendix A, Eq. (A6)"},{"comment":"The paper never reports the measured winding number wt numerically, nor its uncertainty. The text states that the topological invariants were successfully measured, but Figs. 3(e,f) show only Re[φ_yx(k)] curves, and no value for (1/π)∮∂_k Re[φ_yx] dk is given. Given the branch ambiguity in Eq. (9), a numerical wt with a propagated error is essential to support the claim. The error analysis in Appendix D reports root-mean-square deviations for individual spin textures, but does not propagate these errors to φ±±_yx, to Re[φ_yx], or to the winding number.","section":"Section IV, Figs. 3(e,f)"},{"comment":"The conclusion claims that the experiments demonstrate a discrepancy between the phase boundaries characterized by wt and νE. The data, however, consist of only two parameter sets, both with νE = 0 and with wt = 1 and wt = 2, respectively. No crossing of either phase boundary is tracked, and no parameter point with νE ≠ 0 is measured. The two points are consistent with the predicted boundary discrepancy, but they do not by themselves demonstrate it. The claim should either be tempered to a consistency check or supported with data across the relevant boundary.","section":"Section IV and Section V"}],"minor_comments":[{"comment":"The sentence 'Similarly, Re(φ_yx) can be rewritten in another way (see the proof in Appendix A)' is confusing because Eq. (9) is the formula being introduced; Appendix A actually derives Eq. (A6), which is the same relation up to the branch term. Please clarify the cross-reference.","section":"Section II, after Eq. (9)"},{"comment":"The sentence 'it is easy to derive the relationship of Eq. (10) in the main text' appears to refer to Eq. (9) of the main text, since Eq. (10) is only the definition of φµµ_yx. Please correct the reference.","section":"Appendix A"},{"comment":"In the caption, 'actan(hy/hx)' should be 'arctan(hy/hx)'.","section":"Fig. 1 caption"},{"comment":"The condition c± ≠ 0 is stated as necessary for extracting both right-eigenstate spin textures from the time traces, but no check is reported that this condition actually holds at every measured k, nor is the sensitivity to near-zero denominators in Eq. (10) quantified. A sentence reporting the fitted coefficients or the smallest observed values of ⟨σx⟩± would address this.","section":"Section II and Appendix C"},{"comment":"There are minor typographical errors, including 'constract' in Section II, 'biorthonornal' in the introduction, and 's' used inconsistently for indices in the pulse-sequence description in Section III.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within scope for a quantum-simulation and topological-physics journal. The main technical concern is the unstated branch-fixing procedure for Eq. (9); this is fixable in revision but is currently load-bearing. Please ask the authors to supply the explicit unwrapping algorithm, report the measured wt numerically with uncertainty, and either add data near the phase boundary or soften the claim about demonstrating the boundary discrepancy."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe paper has one genuinely useful new idea: for a 1D two-band non-Hermitian Hamiltonian, the real part of the complex angle φ_yx can be pulled from the spin textures of the two right eigenstates alone. No left eigenstates, no H† simulation. The proof in Appendix A is short and correct. The NMR experiment on a non-chiral model is the first of its kind, and the measured time traces agree with theory. That is the good news.\n\nThe problems sit in the extraction of the winding number. Equation (9) determines Re[φ_yx] only up to an integer multiple of π/2. To get wt, you need the continuous branch of Re[φ_yx] across the Brillouin zone. The paper never says how that branch is chosen. There is no unwrapping rule, no discussion of branch cuts, no criterion for fixing n(k). If n(k) was chosen to match the theoretical curve, the measurement is circular. If it came from unwrapping the data, that step is missing. This is not a small omission. The central claim is that wt was measured from right-eigenstate spin textures, and the manuscript does not show how the modulo ambiguity was resolved.\n\nSecond, the paper never reports a numerical winding number. It shows Re[φ_yx] curves and states it is evident that wt was obtained. An experimental result needs wt = 1 ± something. The error analysis gives RMS deviations for spin textures, but no uncertainty on the winding number itself. Third, the phrase \"demonstrated a discrepancy between the phase boundaries\" is stronger than the data support. Two measured points with νE=0 and wt=1,2 are consistent with the predicted mismatch, but they do not directly observe the boundary.\n\nThe theory survives. The identity is correct, and the approach is plausible if the branch issue can be solved. But as written, the experimental claim is underdetermined. A referee should ask for the unwrapping procedure, numerical winding numbers with error bars, and probably the raw fitted curves. Without those, the paper is not a complete demonstration.\n\nWorth engaging. Needs serious revision before I'd trust the experimental conclusion.","headline":"A correct algebraic shortcut for non-Hermitian winding numbers, but the NMR demonstration omits the branch-unwrapping rule needed to make the measured wt non-circular.","tokens_in":15228,"tokens_out":6869,"would_cite":false,"duration_ms":62313,"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 winding number of a non-Hermitian system without chiral symmetry can be measured from right eigenstates alone, using the identity that ties the azimuthal angle to spin-texture averages.","keywords":["non-Hermitian topology","winding number","spin texture","chiral symmetry breaking","NMR quantum simulation","dilation method","topological invariants","complex energy bands"],"falsifier":"At a non-degenerate $k$, compute the analytic right eigenstates of the model Hamiltonian in Eq. (13) and compare $\\mathrm{Re}[\\varphi_{yx}(k)]$ with $\\frac{1}{2}(\\varphi^{++}_{yx}(k)+\\varphi^{--}_{yx}(k))$; if the difference is not an integer multiple of $\\pi/2$ at each point, the central identity fails. Experimentally, repeating the spin-texture fit at a $k$ where one of the denominator components $\\langle\\phi^R_\\pm|\\sigma_x|\\phi^R_\\pm\\rangle$ nearly vanishes should produce the $\\pi$-jump structure predicted by Eq. (9); its absence would indicate that the dilation or fitting procedure corrupts the extracted angle.","tokens_in":14178,"feed_emoji":"⚛️","tokens_out":8243,"duration_ms":67523,"temperature":0.7,"pith_summary":"This paper establishes and demonstrates a general measurement protocol for the topological winding number of one-dimensional non-Hermitian systems that lack chiral symmetry. The usual route requires evolving both the Hamiltonian and its conjugate to access biorthogonal left and right eigenstates; the paper shows this is unnecessary. A new identity expresses the real part of the complex angle that winds around the Brillouin zone as the average of two angles obtained from right-eigenstate spin textures, so the topological invariant can be read off from dynamics under $H$ alone. The authors implement the scheme on a two-qubit NMR quantum simulator using a dilation method, extract winding numbers $w_t=1$ and $2$, and also measure the complex-energy winding, observing that its phase boundary differs from the eigenstate winding boundary in this non-chiral system.","feed_headline":"Winding numbers measured without chiral symmetry","feed_subtitle":"A dilation experiment on two nuclear spins extracts the winding number without needing H†.","key_machinery":"The load-bearing object is the azimuthal angle $\\varphi_{yx}(k)=\\arctan(h_y/h_x)$ of the complex vector field $h(k)$ in $H(k)=h(k)\\cdot\\sigma$; its winding around the Brillouin zone is the summed invariant $w_t$. The new identity Eq. (9) transfers the computation of $\\mathrm{Re}[\\varphi_{yx}]$ to the right-eigenstate spin-texture angles $\\varphi^{\\mu\\mu}_{yx}(k)=\\arctan(\\langle\\phi^R_\\mu|\\sigma_y|\\phi^R_\\mu\\rangle / \\langle\\phi^R_\\mu|\\sigma_x|\\phi^R_\\mu\\rangle)$, so the biorthogonal pair is replaced by quantities reachable through a single non-unitary evolution. The experimental machinery is the dilation method, which embeds the non-Hermitian evolution in a Hermitian Hamiltonian on a larger space via an ancilla, together with GRAPE-optimized NMR pulses and a fitting procedure that extracts both band contributions and complex eigenvalues from the observed spin-texture oscillations.","core_discovery":"The central claim is Eq. (9): $\\mathrm{Re}[\\varphi_{yx}(k)] = \\frac{1}{2}(\\varphi^{++}_{yx}(k) + \\varphi^{--}_{yx}(k)) + n\\pi/2$, where $\\varphi^{++}_{yx}$ and $\\varphi^{--}_{yx}$ are the azimuthal angles of the two right-eigenstate spin textures. Because the summed winding number $w_t = \\frac{1}{\\pi}\\oint \\partial_k \\mathrm{Re}[\\varphi_{yx}]\\,dk$ is topological, the invariant can be obtained from these right-eigenstate textures without ever implementing $H^\\dagger$. The paper verifies the identity analytically in Appendix A, then realizes it experimentally: a two-qubit NMR system, with the non-Hermitian Hamiltonian dilated into a Hermitian evolution of system plus ancilla, yields time traces from which both right-eigenstate spin textures and the complex eigenvalues are extracted by fitting. For the model $h_x = J_0 + J_1\\cos k$, $h_y = J_1\\sin k - i\\delta$, $h_z = 0.5$, the measured $\\mathrm{Re}[\\varphi_{yx}]$ winds as $w_t=1$ and $w_t=2$ in two parameter regimes, and the energy-band winding $\\nu_E=0$ in both, showing the two invariants change at different boundaries in the absence of chiral symmetry.","pith_inferences":["The identity Eq. (9) is purely analytic, so the protocol should transfer to other quantum simulators such as photonic, trapped-ion, or superconducting platforms where the dilated Hermitian evolution can be compiled, offering a cross-platform test of the experimental assumption.","A natural stress test is to apply the same procedure to a chiral model ($h_z=0$) and to a model with an exceptional point; where a spin-texture denominator vanishes, the reconstruction of $\\mathrm{Re}[\\varphi_{yx}]$ should show the branch structure predicted by the identity, exposing the practical limits of the fitting approach.","The observed separation of eigenstate and energy-band phase boundaries in non-chiral systems could serve as a general diagnostic for chiral-symmetry breaking in future non-Hermitian experiments, independent of the specific model.","Extending the right-eigenstate-only idea to multi-band or two-dimensional non-Hermitian models would require analogous angle decompositions for Chern numbers; the paper identifies this as the natural next step but leaves it open."],"forward_implications":["Experiments on non-Hermitian topology no longer need a separate implementation of $H^\\dagger$; any platform that can simulate $H$ can in principle measure $w_t$.","The same dataset yields both the eigenstate winding number and the complex band energies, enabling simultaneous study of the two distinct topological structures in one experiment.","For non-chiral systems, the measured mismatch between the $w_t$ and $\\nu_E$ phase boundaries gives a direct experimental signature of chiral-symmetry breaking.","The right-eigenstate spin-texture extraction works whenever both bands have non-zero initial amplitudes ($c_\\pm \\neq 0$), making the protocol applicable to generic one-dimensional two-band non-Hermitian Hamiltonians."],"supporting_citations":[{"why":"Introduced the dynamic winding number from long-time spin textures and the decomposition into right-right and left-left angles that this paper extends.","marker":"[35]"},{"why":"Proposed a quantum-circuit method for non-Hermitian winding numbers that this experiment improves upon by removing the need for $H^\\dagger$.","marker":"[36]"},{"why":"Established the summed winding number $w_t$ as the topological invariant for one-dimensional chiral non-Hermitian systems.","marker":"[24]"},{"why":"Gave the phase diagrams and topological invariants for one-dimensional two-band non-Hermitian systems without chiral symmetry, the theoretical basis of the model.","marker":"[23]"},{"why":"Supplied the dilation method for simulating non-Hermitian evolution with an ancilla.","marker":"[45]"},{"why":"Demonstrated the dilation approach in a real single-spin quantum system, serving as the experimental template.","marker":"[62]"},{"why":"Defined the energy-band winding number $\\nu_E$ that the experiment measures for the non-chiral bands.","marker":"[12]"},{"why":"Developed the topological band theory for non-Hermitian Hamiltonians underlying the complex-energy winding.","marker":"[13]"},{"why":"Provided the GRAPE technique used to compile the high-precision pulse sequences.","marker":"[63]"}],"fun_headline_variants":["NMR experiment reveals winding numbers without chiral symmetry","Spin textures map non-Hermitian topology without chiral symmetry","Two-qubit NMR reads winding numbers in non-Hermitian systems","No chiral symmetry needed: measuring topological invariants","Dilation method extracts winding numbers from spin textures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The protocol assumes that the dilated Hermitian evolution exactly reproduces the non-Hermitian dynamics at every quasimomentum and for the full evolution time, and that the measured spin-texture traces can be fitted to cleanly separate the two right-eigenstate contributions.","fun_headline_variants_meta":{"raw":{"variants":["NMR experiment reveals winding numbers without chiral symmetry","Spin textures map non-Hermitian topology without chiral symmetry","Two-qubit NMR reads winding numbers in non-Hermitian systems","No chiral symmetry needed: measuring topological invariants","Dilation method extracts winding numbers from spin textures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000191,"raw_usage":{"total_tokens":1354,"prompt_tokens":967,"completion_tokens":387,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":583,"completion_tokens_details":{"reasoning_tokens":310}},"tokens_in":583,"tokens_out":387,"duration_ms":3268,"temperature":1.0,"reasoning_tokens":310,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:22:28.960235+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"At a non-degenerate $k$, compute the analytic right eigenstates of the model Hamiltonian in Eq. (13) and compare $\\mathrm{Re}[\\varphi_{yx}(k)]$ with $\\frac{1}{2}(\\varphi^{++}_{yx}(k)+\\varphi^{--}_{yx}(k))$; if the difference is not an integer multiple of $\\pi/2$ at each point, the central identity fails. Experimentally, repeating the spin-texture fit at a $k$ where one of the denominator components $\\langle\\phi^R_\\pm|\\sigma_x|\\phi^R_\\pm\\rangle$ nearly vanishes should produce the $\\pi$-jump structure predicted by Eq. (9); its absence would indicate that the dilation or fitting procedure corrupts the extracted angle.","supporting_citations":[{"cited_title":"& Lee, C","cited_arxiv_id":null,"evidence_quote":"Introduced the dynamic winding number from long-time spin textures and the decomposition into right-right and left-left angles that this paper extends."},{"cited_title":"& Zhu, S.-L","cited_arxiv_id":null,"evidence_quote":"Proposed a quantum-circuit method for non-Hermitian winding numbers that this experiment improves upon by removing the need for $H^\\dagger$."},{"cited_title":"& Chen, S","cited_arxiv_id":null,"evidence_quote":"Established the summed winding number $w_t$ as the topological invariant for one-dimensional chiral non-Hermitian systems."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplied the dilation method for simulating non-Hermitian evolution with an ancilla."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Developed the topological band theory for non-Hermitian Hamiltonians underlying the complex-energy winding."},{"cited_title":"& Glaser, S","cited_arxiv_id":null,"evidence_quote":"Provided the GRAPE technique used to compile the high-precision pulse sequences."}],"review_version":1}