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REVIEW 3 major objections 5 minor 64 references

Observation of topology of non-Hermitian systems without chiral symmetry

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2504.15620 v1 pith:K5SNKZ4U submitted 2025-04-22 quant-ph

Shuo Wang , Zhengjie Kang , Hao Li , Jiaojiao Li , Yuanjie Zhang , Zhihuang Luo This is my paper · ORCID
classification quant-ph
keywords non-HermitiantopologywindingnumberspintexturechiralsymmetrybreakingNMRquantumsimulationdilationmethodtopologicalinvariantscomplexenergybands
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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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 / 5 minor

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.

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 (3)
  1. [Section II, Eq. (9); Appendix A, Eq. (A6)] 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).
  2. [Section IV, Figs. 3(e,f)] 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.
  3. [Section IV and Section V] 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.
minor comments (5)
  1. [Section II, after Eq. (9)] 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.
  2. [Appendix A] 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.
  3. [Fig. 1 caption] In the caption, 'actan(hy/hx)' should be 'arctan(hy/hx)'.
  4. [Section II and Appendix C] 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.
  5. [General] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (9) is derived algebraically from the right-eigenstate parameterization, and the winding number is reconstructed from measured spin textures rather than fitted to the target.

full rationale

The central identity Eq. (9) is proven in Appendix A from the explicit right-eigenstate parameterization Eqs. (A1)-(A5); it is a trigonometric identity and does not import the target winding number. The experiment fits time traces of spin textures to extract the right-eigenstate spin textures and then computes the angles via Eq. (10); the winding number is obtained by integrating the derivative of the reconstructed angle, not by fitting the winding number itself. The dilation construction in Appendix B is built from the known non-Hermitian Hamiltonian H(k) and not from the target invariant, so simulating H is standard quantum simulation and does not presuppose the result. The cited input for the topological invariant wt, Eq. (6), is external work [24], and the self-citations [64,65] concern state preparation and unrelated experiments rather than the load-bearing identity. The only notable gap is that Eq. (A5) fixes Re[phi_yx] only modulo n*pi/2, and the paper does not state the branch/unwrapping rule used to convert the measured angles into the continuous Re[phi_yx] curves of Figs. 3(e,f); this is a procedural underdetermination, not a demonstrated circular reduction. If the branch were chosen by matching the theoretical Re[phi_yx], the measurement would become circular, but the text does not say that such a fit was performed.

Assumptions & free parameters 6 free parameters · 3 assumptions · 0 invented entities

The central formula has no free parameters, but the experimental demonstration depends on hand-chosen Hamiltonian parameters and several unstated numerical choices (eta0, evolution time, Trotter steps). The paper also depends on prior definitions of non-chiral winding numbers and on the validity of the dilation method on an NMR platform.

free parameters (6)
  • J0 = 1.0 (wt=1); 0.3 (wt=2)
    Hamiltonian parameter chosen by hand to realize the two target topological phases; not fitted to the data.
  • J1 = 1.0
    Hamiltonian parameter chosen by hand; sets the momentum-dependent modulation amplitude.
  • delta = 0.3
    Non-Hermitian gain/loss parameter chosen by hand for the complex hy component.
  • hz = 0.5
    Chiral-symmetry-breaking field chosen by hand; nonzero value is required for the non-chiral regime.
  • eta0 = not stated
    Initial dilation parameter controlling M(0) and the ancilla rotation angle alpha; must be chosen so M(t)-I remains positive, but its value is not reported.
  • Evolution time T and Trotter steps M = not stated
    Total evolution time and number of time slices in the Trotter approximation are not specified; these affect simulation fidelity and reproducibility.
assumptions (3)
  • domain assumption The topological invariant wt is defined by wt = (1/pi) ∮ dk ∂k Re[phi_yx], as established for non-chiral non-Hermitian systems in prior work.
    The paper relies on Refs. [23,24] for the validity of this winding-number definition and does not re-derive it from first principles. Section II, Eq. (6).
  • domain assumption The dilation method of Refs. [45,62] exactly represents the non-Hermitian Hamiltonian dynamics via a Hermitian dilated Hamiltonian on an enlarged Hilbert space.
    The experimental implementation assumes Hs,a in Appendix B correctly encodes the non-Hermitian evolution for all k and t, including the positivity condition M(t)-I >= 0. Section III and Appendix B.
  • domain assumption The NMR pseudo-pure-state preparation, GRAPE pulses, and projection measurements faithfully realize the circuit in Fig. 2(b).
    The experiment assumes state preparation and readout errors are small enough that fitted spin textures and eigenvalues are reliable; Appendix D quantifies errors but does not eliminate the assumption. Section III and Appendix C.

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Pith. "Pith review of Observation of topology of non-Hermitian systems without chiral symmetry." pith.science (2026). https://pith.science/paper/K5SNKZ4U

@misc{pith2026250415620,
  author       = {Pith},
  title        = {Pith review of: Observation of topology of non-Hermitian systems without chiral symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K5SNKZ4U}},
  note         = {Machine review of arXiv:2504.15620}
}
read the original abstract

Topological invariants are crucial for characterizing topological systems. However, experimentally measuring them presents a significant challenge, especially in non-Hermitian systems where the biorthogonal eigenvectors are often necessary. We propose a general approach for measuring the topological invariants of one-dimensional non-Hermitian systems, which can be derived from the spin textures of right eigenstates. By utilizing a dilation method, we realize a non-Hermitian system without chiral symmetry on a two-qubit nuclear magnetic resonance system and measure the winding number associated with the eigenstates. In addition to examining the topology of the eigenstates, our experiment also reveals the topological structure of the energy band, which differs from that in chiral systems. Our work paves the way for further exploration of complex topological properties in non-Hermitian systems without chiral symmetry.

Figures

Figures reproduced from arXiv: 2504.15620 by the authors.

Figure 2
Figure 2. (a). Its NMR Hamiltonian in a rotating frame is HNMR = − X 2 i π(νi − ν RF i )σ i z + πJ 2 σ 1 zσ 2 z . (11) Here νi and ν RF i (i = 1, 2) represent the chemical shifts and the reference frequencies of RF fields, and the J is the coupling strength between 13C and 1H. Each nuclear spin can be individually controlled by radio-frequency (RF) pulses. The corresponding control Hamiltonian reads Hc = X 2 i=1 πBi(cos Φiσ i… view at source ↗
Figure 3
Figure 3. FIG. 3: (a-b) Time evolutions of spin textures [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4: Time-dependent control parameters [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: FIG. 5: (a) Experimental [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The resulted [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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