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A local quantized marker for topological magnons from circular dichroism

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Pith's one-line read A frequency-integrated circular-dichroism measurement reads out a local Chern marker in magnon bands.

desk verdict The marker derivation is clean and the ideal-state numerics are convincing, but the driven-dissipative preparation only reaches eC≈0.8, so the 'quantized' claim is not yet backed by the realistic protocol. read the letter →

arxiv 2504.17374 v3 pith:JILNZICF submitted 2025-04-24 cond-mat.mes-hall cond-mat.mtrl-scicond-mat.othercond-mat.quant-gasquant-ph

Baptiste Bermond , Anaïs Defossez , Gregor Jotzu , Nathan Goldman This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.mtrl-scicond-mat.othercond-mat.quant-gasquant-ph
keywords topologicalmagnonslocalChernmarkercirculardichroismdriven-dissipativepreparationDzyaloshinskii-Moriyainteractionhoneycomblatticebosonicbandssingle-siteresolution
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 that the topology of a magnon band can be read out locally through a circular-dichroic measurement. It defines a site-resolved marker $\hat{e}_C(\mathbf r_i)$ and shows that, as a function of position, this marker is quantized to $1$ deep in the bulk and becomes large and negative near the edges, exactly the behavior of a local Chern marker. The construction matters because magnons are charge-neutral bosonic excitations, so the usual Fermi-surface or charge-transport probes do not directly apply; the paper supplies both a local preparation scheme and a measurement protocol that work in the presence of losses. Demonstrated on a two-dimensional ferromagnetic Heisenberg spin system with Dzyaloshinskii–Moriya interactions, the protocol maps the marker with single-site resolution.

What carries the argument

The load-bearing object is the local marker $\hat{e}_C(\mathbf r_i)=-\frac{4\pi}{V_{\mathrm{cell}}}\operatorname{Im}\left(\sum_{g\in\mathrm{LB}}\langle\downarrow_i|g\rangle\langle g|\hat{x}\hat{Q}\hat{y}|g\rangle\langle g|\downarrow_i\rangle\right)$, where $\hat{Q}$ projects onto the upper magnon band and $V_{\mathrm{cell}}$ is the unit-cell area. It is carried by the identity $\hat{e}_C(\mathbf r_i)=\lim_{t\to\infty}\frac{\hbar^2}{V_{\mathrm{cell}}\epsilon^2}\Delta\Gamma_i(t)$, which connects a measured frequency-integrated chiral-excitation-rate difference to a geometric invariant. In the bulk the matrix elements of $\hat{x}\hat{Q}\hat{y}$ within the lowest band become diagonal in quasimomentum, so the marker reduces to the standard local Chern marker, while the driven-dissipative mean-field equations provide the approximately uniform band population that the measurement assumes.

What would settle it

Compute the frequency-integrated chiral excitation rates using the actual steady state of the mean-field equations, rather than the idealized lowest-band projection, on a 30×30 lattice and at the lattice center; if the resulting marker does not converge toward $1$ as the system grows, or if a multi-frequency pump cannot push it closer to $1$, the central link between preparation, dichroism, and quantization is falsified.

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Extended reading notes

Core claim

The central claim is that the quantity $\hat{e}_C(\mathbf r_i)=-\frac{4\pi}{V_{\mathrm{cell}}}\operatorname{Im}\left(\sum_{g\in\mathrm{LB}}\langle\downarrow_i|g\rangle\langle g|\hat{x}\hat{Q}\hat{y}|g\rangle\langle g|\downarrow_i\rangle\right)$ behaves as a local Chern marker, quantized to $1$ in the bulk and large and negative near the edges. The paper derives this marker from the long-time integrated circular-dichroic signal, showing that the frequency integral of the chiral excitation-rate difference converges to the geometric expression $\lim_{t\to\infty}\frac{\hbar^2}{V_{\mathrm{cell}}\epsilon^2}\Delta\Gamma_i(t)$. In the bulk, quasimomentum is a good quantum number and the relevant operator becomes diagonal within the lowest band, so off-diagonal lowest-band terms vanish and the marker coincides with the standard local Chern marker; finite-size calculations show exponential convergence to $1$ at the sample center. The preparation step is a driven-dissipative protocol in which a local pump and homogeneous losses produce a steady state that approximates a uniform population of the lowest magnon band, so losses are incorporated rather than avoided.

Load-bearing premise

The load-bearing premise is that the driven-dissipative steady state spreads itself evenly across the lowest magnon band, so the dichroic signal comes out quantized; the paper's own simulations find only about 0.78 instead of 1 on a 20×20 lattice, so this uniformity is only approximate.

Editorial extensions

If this is right

  • Deep in the bulk of a topological magnon system, a single-site dichroic measurement yields the Chern number of the band, so band topology can be identified without resolving edge modes.
  • Because the recipe only needs a localized single-excitation state and a chiral perturbation, it transfers to any bosonic spin system with a topological band structure.
  • The driven-dissipative preparation converts homogeneous magnon losses into a resource for shaping the steady state rather than a source of decoherence.
  • Finite-size scaling at the sample center shows exponential convergence to the quantized value, so small lattices already display a plateau of $\hat{e}_C=1$ in the bulk.
  • Near the sample edges the marker takes large negative values, providing a local real-space signature of the topological boundary.

Reading between the lines

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

  • A natural next step, which the paper only sketches as a perspective, is to replace the single-frequency pump with a multi-frequency pulse; the size of the steady-state deviation ($\hat{e}_C\approx 0.78$ on a $20\times20$ lattice) suggests this may be required to reach genuinely quantized readings.
  • The same frequency-integrated dichroic construction could likely be adapted to other invariants, such as spin-Hall or $\mathbb{Z}_2$ markers, by choosing generalized position operators that couple to the relevant internal degrees of freedom.
  • Testing the marker beyond the single-magnon approximation, with magnon-magnon interactions or without magnon-number conservation, would show how much of the quantization survives in regimes closer to real materials.
  • The frequency integral and the small-drive constraint create an experimental trade-off between measurement time, frequency resolution, and signal size; the paper's error analysis bounds this trade-off but does not provide an end-to-end signal-to-noise estimate for a specific material.
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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 manuscript proposes a local topological marker for magnonic Chern bands, defined through the long-time integrated circular-dichroic response of a locally prepared single-magnon state. The authors derive an expression for the marker eC(r_i) from first-order time-dependent perturbation theory [Eqs. (6)-(9)], show that for an ideal lowest-band-localized state it coincides with the Bianco-Resta local Chern marker in the bulk, and demonstrate exponential finite-size convergence of the ideal-state marker in the supplemental material. They then introduce a driven-dissipative preparation scheme based on a local pump and homogeneous losses, optimize the pump/loss parameters by minimizing a band-uniformity factor U [Eq. (17)], and report that the marker measured on the resulting steady state deviates from quantization by about 15-22% (eC≈0.78 for 20x20 and 0.85 for 30x30). The paper suggests but does not implement a multi-frequency pumping protocol as a remedy.

Significance. The idea of extracting a quantized local Chern marker from circular dichroism in a bosonic, charge-neutral system is valuable and would be a genuine methodological advance if the full protocol is realized. The formal derivation from Eq. (6) to Eq. (9) is clean and standard, and the paper does a service by carefully itemizing the numerical limitations (finite frequency step, finite time, small epsilon) in the supplemental material. The exponential convergence of the ideal-state marker and the explicit comparison with the Bianco-Resta marker are strong points. The load-bearing weakness is the connection between the ideal-state construction and the experimentally proposed preparation scheme: the realistic protocol, as presented, does not produce a quantized marker.

major comments (3)
  1. [Initial state preparation, Fig. 3] The central claim that the protocol can measure a quantized local Chern marker is not supported by the preparation scheme actually implemented. In Fig. 3(b) and the surrounding text, the steady-state marker is eC≈0.78 on a 20x20 lattice and eC≈0.85 on a 30x30 lattice, which are 15-22% below the quantized value of 1. The proposed multi-frequency pump is only described verbally and is not implemented or numerically tested. Since the abstract and introduction promise that the driven-dissipative preparation plus dichroic measurement maps a local Chern marker with single-site resolution, this discrepancy is load-bearing and must be addressed: either demonstrate a preparation that yields a marker arbitrarily close to 1 in the thermodynamic limit, or explicitly reframe the central claim as measuring an approximate, size-dependent quantity.
  2. [Eqs. (8)-(9) and the role of the ideal initial state] The derivation of the quantized marker relies on the initial state being exactly the lowest-band projected state |↓i⟩ of Eq. (4). The driven-dissipative steady state |imf⟩ from Eq. (16) is not such a state, and the paper's uniformity factor U [Eq. (17)] is minimized rather than driven to zero. The statement that 'inevitable deviations from the ideal state (U<1) introduce a small residual error in the Chern marker as the system approaches the thermodynamic limit' is not quantified. I ask the authors to provide a quantitative relation between U and the deviation of eC from 1, or to show a protocol in which U→0 in the thermodynamic limit; without this, the 'quantized marker' terminology applied to the realistic protocol is not justified.
  3. [Discussion of multi-frequency pumping] The suggestion that a multi-frequency pulse would 'naturally lead to a uniform population of the band' is a conjecture. Given that the entire experimental accessibility claim rests on this step, the manuscript should at least provide a proof-of-principle simulation of such a pulse, or a clear argument why it can eliminate the inhomogeneity that causes the 15-22% deviation. In the absence of either, the realistic protocol remains incomplete and the title's promise of a locally measurable quantized marker is not met.
minor comments (4)
  1. [Supplemental Material, 'Spin waves'] There is a typo: 'Hamiltonien' should be 'Hamiltonian' in the sentence after Eq. (S4).
  2. [Eq. (13)] The replacement x̂ = -Σ_j x_j ŝ^z_j is justified only for matrix elements between orthogonal states; the main text states this briefly, but it would help to add a sentence noting that the constant term SΣ x_j is dropped exactly for this reason, as done in the supplement.
  3. [Fig. 3(a) and Eq. (17)] The uniformity factor U is defined as a sum of absolute deviations, with ideal value 0; the figure caption says 'its minimal value is marked by a white cross,' but it would be clearer to state explicitly that U=0 for perfect uniformity and that the cross marks the minimum of U over the shown parameter range.
  4. [Fig. 3(b) caption] The caption states 'eC∼0.8' while the text reports eC≈0.78; for consistency, the same value with the same number of significant digits should be used.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the circular-dichroism marker is derived from perturbation theory and independently connected to the Bianco-Resta marker; the preparation protocol is optimized on band uniformity, not on the marker value.

full rationale

The derivation chain is self-contained. The central relation between the integrated circular-dichroic signal and the local marker is derived in the paper and in the Supplemental Material from first-order time-dependent perturbation theory (Eq. (6), Supplemental Eqs. (S6)-(S12)), followed by a frequency integral using the orthogonality limit (Eq. (S13)). The marker eC is then defined by Eq. (9) as the late-time limit of this derived signal; it is not an input parameter. The equality eC = C in the bulk is justified by an explicit argument: under periodic boundary conditions the lowest-band states are momentum eigenstates, so xQy is diagonal and the off-diagonal terms that distinguish Eq. (9) from the Bianco-Resta marker Eq. (10) vanish. The numerics in Figs. 1(d) and 2(d) are checks of this derivation, not fitted outputs. The driven-dissipative preparation is optimized by minimizing the uniformity factor U of Eq. (17), which measures how uniformly the steady state populates the target band; it does not directly target the marker value. The reported result eC ~ 0.78 (20x20) and ~0.85 (30x30) is presented honestly as a deviation from quantization, and the multi-frequency pump is described only as a possible refinement, not as a demonstrated result. Self-citations [21,23,24] concern earlier dichroism formalism, but the key formulas are re-derived in this paper, so those citations are not load-bearing. The skeptical concern about the preparation scheme is a correctness or feasibility issue, not a circularity issue, and does not affect the circularity score.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard perturbation theory and spin-wave theory, but the preparation protocol introduces optimized parameters (omega0, gamma, epsilon) and a mean-field approximation that are not independently justified. No new physical entities are postulated.

free parameters (4)
  • pump frequency omega0 = hbar*omega0 = J
    Optimized to minimize the uniformity factor U in Eq. (17); not fitted to the marker value.
  • loss rate gamma = 2*hbar*gamma*S = 0.695 J
    Optimized alongside omega0 to maximize filling uniformity of the lowest band; affects the steady-state quality and hence the marker accuracy.
  • drive amplitude epsilon = 2e-4 J/a
    Chosen small to stay in the linear-response regime; too large values reduce the integrated rate (Fig. S1b,c).
  • frequency integration step delta_omega = varied in numerics
    Finite step causes deviations after t ~ 2pi/delta_omega (Fig. S1d); a balance of timescales is required.
assumptions (5)
  • domain assumption The ground state is ferromagnetic with global SO(3) symmetry for D <~ 0.7 J S, and a small magnetic field fixes the direction along z.
    Stated in the Model section; justifies the polarized Holstein-Primakoff expansion.
  • domain assumption Magnon losses are local and uniform, described by Lindblad jump operators L_k = s^+_k with a single rate gamma.
    Assumed throughout the preparation scheme; non-uniform losses would alter the steady state and marker.
  • domain assumption The system is restricted to the single-magnon sector; higher-order magnon-magnon interactions vanish in this sector.
    The Holstein-Primakoff expansion is truncated at quadratic order; the paper notes interactions are absent for single magnons, but multi-magnon effects are deferred to future work.
  • ad hoc to paper The driven-dissipative steady state is described by the mean-field, large-S equation of motion (Eq. 16) for b_i.
    This approximation is used to optimize the pump and loss parameters; the paper does not assess corrections beyond mean-field or large-S.
  • standard math The position operator in the spin language simplifies to x = -Sum_j x_j s^z_j when evaluated between orthogonal states.
    Derived from the Holstein-Primakoff mapping; exact within the single-magnon subspace when matrix elements are between orthogonal states.

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Pith. "Pith review of A local quantized marker for topological magnons from circular dichroism." pith.science (2026). https://pith.science/paper/JILNZICF

@misc{pith2026250417374,
  author       = {Pith},
  title        = {Pith review of: A local quantized marker for topological magnons from circular dichroism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JILNZICF}},
  note         = {Machine review of arXiv:2504.17374}
}
read the original abstract

The low-energy excitations of a spin system can display Bloch bands with non-trivial topological properties. While topological magnons can be identified through the detection of chiral propagating modes at the sample's edge, an intriguing approach would be to directly probe their topological nature via localized measurements deep within the bulk. In this work, we introduce a quantized topological marker suitable for topological spin systems, which can be experimentally accessed by combining a local driven-dissipative preparation scheme with a circular-dichroic measurement. Demonstrated on a 2D ferromagnetic Heisenberg spin system incorporating Dzyaloshinskii-Moriya interactions, this method effectively maps a local Chern marker with single-site resolution while inherently accounting for magnon losses. Our work offers a general strategy to access local topological markers in bosonic settings within a driven-dissipative framework.

Figures

Figures reproduced from arXiv: 2504.17374 by the authors.

Figure 1
Figure 1. Local Chern marker measurement in a topologi￾cal magnon system. (a) Illustration of the 2D spin model in Eq. (1), with nearest-neighbor ferromagnetic Heisenberg interactions J (black), and next-nearest-neighbor DM inter￾actions D (blue). (b) The corresponding energy spectrum displays two bands, with respective Chern numbers C = ±1; here J = 2D > 0. (c) Sketch of the preparation scheme: a localized spin excitation is… view at source ↗
Figure 2
Figure 2. Dichroism measurement for the S=1/2 bosonized Hamiltonian (3). (a) Single-magnon spectrum for a 20 × 20 lattice and D =J/2: the states in the lowest (highest) bulk band are represented in black (blue), while the edge states are shown in red. (b) Time evolution of the integrated differential rate [Eq. (8)] as a function of time, for a specific initial state located in the bulk; the driving amplitude is ϵ=2.10−4 J/a. … view at source ↗
Figure 3
Figure 3. Driven-dissipative preparation scheme and lo￾cal Chern marker measurement. (a) Uniformity factor U [Eq. (17)] for the steady-state solution of the mean-field equa￾tions [Eq. (S23)], as a function of the pump frequency ω0 and the loss rate 2ℏγS. Its minimal value is marked by a white cross, which is located at ℏω0 =J and 2ℏγS = 0.695J. (b) The local Chern marker Ce(⃗r), as obtained from a dichroic measure￾ment perfor… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Hall viscosity from metric-sensitive dichroic probes

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