Magnetic Dichroism in Rutile NiF$_2$: Separating Altermagnetic and Ferromagnetic Effects

A. Hariki; J. Kune\v{s}; K. Sakurai; T. Okauchi

REVIEW 4 major objections 5 minor 57 references

Magnetic Dichroism in Rutile NiF$_2$: Separating Altermagnetic and Ferromagnetic Effects

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read In NiF2, XMCD spectra decompose into two field-independent shapes, so the Néel vector orientation can be read directly from a field-rotation experiment.

desk verdict A solid computation that extends the altermagnet XMCD toolkit to the weakly ferromagnetic case, but the FM component in the central two-component formula is an admitted ad hoc term whose line shape is not well defined. read the letter →

arxiv 2501.08056 v2 pith:OTF3NZ4A submitted 2025-01-14 cond-mat.mtrl-sci cond-mat.str-el

A. Hariki , K. Sakurai , T. Okauchi , J. Kuneš This is my paper · ORCID

classification cond-mat.mtrl-scicond-mat.str-el PACS 78.70.Dm75.25.-z

keywords x-raymagneticcirculardichroismaltermagnetismrutilestructureNiF2NéelvectorweakferromagnetismHallspectraldecomposition

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

The paper's aim is to show that in the rutile altermagnet NiF2 the x-ray magnetic circular dichroism (XMCD) signal is carried by exactly two field-independent spectral shapes: one locked to the Néel vector (the altermagnetic term) and one proportional to the tiny net moment (the ferromagnetic term). This matters because altermagnets are collinear magnets with no net moment but with spin-split bands, and spin-orbit coupling contaminates most probes with weak ferromagnetism; x-ray dichroism is proposed as a clean separator. If the two-component formula is right, the orientation of the Néel vector can be read from the angular dependence of a field-rotation XMCD experiment, and the distinctive relation between the Néel vector and the x-ray Hall vector in rutile can be verified. The paper supports the claim with atomic-multiplet simulations that reproduce the measured absorption spectrum and match the two-component formula to better than 1% for fields from 0.3 to 40 T and all in-plane field angles.

What carries the argument

The load-bearing object is Eq. (1): the decomposition of the x-ray Hall vector into ΔALT(ω) M(110) L-hat(B) + ΔFM(ω) m(B). The altermagnetic term is fixed by the rutile crystal symmetry through the mirror operation M(110), which ties the direction of the Hall vector to the Néel vector; the ferromagnetic term follows the net moment. The two spectral functions are extracted from the field-dependent simulated spectra (using a Ni2+ atomic model with crystal field, spin-orbit coupling, Weiss mean-field exchange, and the external field acting on both spin and orbital moments), and then the formula is used to predict spectra for other field angles and amplitudes.

What would settle it

Measure XMCD on a single-domain NiF2 crystal at 10 T while rotating the in-plane field, and fit every spectrum with the two fixed components ΔALT and ΔFM determined at one reference angle; if residuals exceed 1% or the fitted ΔFM shape changes with the angle between m and the crystal axes, the two-component decomposition is wrong.

Watch

Extended reading notes

Core claim

At the Ni L2,3 edge, the x-ray Hall vector h(ω) — the vector whose projection onto the photon propagation direction gives XMCD — can be written as ΔALT(ω) M(110) L-hat + ΔFM(ω) m, where L is the Néel vector, m the net magnetization, M(110) the mirror across the (110) plane, and ΔALT and ΔFM are spectral functions that do not depend on the field. The first term is exact for a collinear rutile antiferromagnet without valence spin-orbit coupling; the second is an approximation that relies on the smallness of m. The paper shows by full multiplet calculations that this formula reproduces the XMCD spectra with relative accuracy better than 1% for magnetic fields up to 40 T and all studied in-plane orientations, and that as the field rotates, the altermagnetic and ferromagnetic terms acquire different angular weights. Because of those different weights, measuring the spectra at a few field angles separates the two contributions experimentally, and the direction of the altermagnetic part of h gives the Néel vector orientation, as predicted.

Load-bearing premise

The ferromagnetic term is assumed to be a single, field-independent spectrum multiplied by the net moment vector, so that the line shape of each moment's contribution does not change as moments cant or rotate; the authors call this an ad hoc approximation based on the smallness of the moment.

Editorial extensions

If this is right

A field-rotation XMCD experiment on single-domain NiF2 can separate the altermagnetic from the ferromagnetic response, because the two components have distinct angular weights as the field angle φB changes.
The measured direction of the altermagnetic Hall vector provides a direct readout of the Néel vector L via h_ALT = M(110) L-hat, allowing verification of the predicted rutile relation.
The two-component decomposition is quantitative to better than 1% up to 40 T, so the altermagnetic spectral function ΔALT(ω) can be determined without subtracting the troublesome ferromagnetic background at each field.
The approximation does not satisfy the standard XMCD sum rules accurately (15% for spin, up to 30% for orbital moments over the studied range), so the two components should not be used to extract quantitative spin or orbital moments.
The result places NiF2 in contrast with MnTe, where the XMCD amplitude depends strongly on the Néel vector orientation; in rutile the shape of the altermagnetic spectrum is essentially fixed by symmetry.

Reading between the lines

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

The same symmetry-based two-component decomposition should hold for other rutile altermagnets such as MnF2 and FeF2, where the altermagnetic term is again locked to the Néel vector by M(110); the practical difference will be in the size of the ferromagnetic contamination.
One could extend the analysis to out-of-plane magnetic fields or to other absorption edges; the symmetry argument fixes the form of the altermagnetic term, while the ferromagnetic term would need to be re-tested for isotropy.
The approximation's failure to satisfy the sum rules suggests a useful experimental diagnostic: any fitted component that violates the sum rules beyond the predicted 15-30% would indicate the onset of per-moment line-shape changes, i.e., breakdown of Eq. (1).
Since the ferromagnetic term is an assumption rather than a symmetry result, the cleanest tests are rotation experiments at high fields; if the two-component fits degrade as the moments cant, the separation scheme will need a field-dependent correction factor.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know up front. First, this is a genuine step forward: the authors show that XMCD in the rutile altermagnet NiF2, including its weak-ferromagnet background, can be described as two spectral components with different field-angle weights, and they propose a route to read off the Néel vector orientation. Second, the load-bearing second component, ΔFM(ω)m(B), is an admitted ad hoc ansatz, and the paper's own sum-rule analysis shows it fails to capture the actual spin and orbital moments. That gap is real, but the authors disclose it and it does not sink the paper.

What is new: the explicit two-component decomposition of the Hall vector into an altermagnetic term, symmetry-derived and checked against the no-SOC limit, and a ferromagnetic term that scales with the net moment. They validate Eq. 1 against full atomic-model calculations over fields up to 40 T and all in-plane angles, with better than 1% relative error. The robustness check against two SOC values and the honest reporting of sum-rule deviations (15% spin, 30% orbital) give the paper a fair-minded tone. The GitHub data release is a plus.

The soft spots are the FM term and the validation strategy. The sum-rule results in the supplement show that ml/ms changes from about 1:1 at zero field to 1:5 at 40 T. If the FM line shape were exactly one spectral function times the moment vector, that ratio should not matter; the fact that it does means ΔFM(ω) is not a well-defined physical function. The <1% agreement likely just reflects how small the FM term is relative to the altermagnetic signal – so the fit is insensitive to the FM line shape. That is a lighter criticism than it sounds, because the protocol still has a practical logic: you can measure the FM term separately in a geometry where the altermagnetic part vanishes, or at high field where the FM part dominates, and then subtract it. But the paper's language about 'uniquely distinguishing' the two contributions is stronger than what is shown. The basis spectra are also extracted from the same simulation family used for validation, so there is residual circularity, and Eq. 2's leading-order law uses fitted α and m0.

None of this is fatal. The altermagnetic term has a symmetry derivation, the protocol is testable, and the paper is honest about its limitations. It is a solid computational contribution for the altermagnet XMCD toolkit. It deserves a serious referee, and with modest revisions (better framing of the FM term's validity, perhaps an explicit error budget for the FM subtraction), it could be a useful reference. I would cite it and bring it to the reading group.

Referee Report

4 major / 5 minor

Summary. This paper presents numerical exact-diagonalization simulations of Ni L2,3-edge XMCD in the rutile altermagnet NiF2 under in-plane magnetic fields up to 40 T. The central result is Eq. (1), which expresses the x-ray Hall vector as h(ω)=ΔALT(ω)M(110)L̂(B)+ΔFM(ω)m(B), with two field-independent spectral functions, and the authors demonstrate that this compact form reproduces their full numerical spectra with relative deviations better than 1% for all studied field magnitudes and in-plane field angles. The basis spectra are extracted from the φB=0 field series, and the formula is then checked at nonzero φB for k∥[100] and k∥[010]; Eq. (2) provides a simplified angular form for a rotating-sample experiment. The paper also compares calculated XAS with experiment and reports sum-rule deviations of the approximate formula, noting that the spin and orbital moments are not collinear and their ratio changes with field.

Significance. If Eq. (1) holds as a physical decomposition, the paper provides a practical route to measuring the orientation of the Néel vector in rutile altermagnets via XMCD, and it sharpens the distinction between altermagnetic and weak-ferromagnetic responses. The first term is symmetry-derived, the numerical tests are extensive, the XAS comparison anchors the model, and the code/data availability supports reproducibility. The authors are also commendably explicit about the ad hoc nature of the ferromagnetic term and about the sum-rule failures. However, the advertised 'unique separation' of altermagnetic and ferromagnetic contributions is not established: the ferromagnetic term is a single spectral function multiplying the total moment m, while the paper's own sum-rule analysis shows that the spin/orbital composition of m changes strongly with field. The result is best regarded at present as a highly accurate phenomenological parametrization of the simulated spectra, and the claims should be adjusted or the FM term subjected to a stronger test.

major comments (4)

[Sec. II, Eq. (1)] The second term of Eq. (1), ΔFM(ω)m(B), is introduced as an ad hoc approximation based on the smallness of m, and the paper's own sum-rule analysis (end of Sec. II and Supplementary Fig. S6) shows that the spin and orbital moments are not collinear and that their ratio ms/ml changes from roughly 1:1 at zero field to 5:1 at 40 T (Supplementary Fig. S2). Because spin and orbital contributions carry different spectral weights across the L2,3 edge, a single field-independent ΔFM(ω) cannot represent a ferromagnetic contribution whose spin/orbital composition changes with field. The Abstract's statement that the altermagnetic and ferromagnetic contributions 'can be uniquely distinguished' is therefore not supported. Please either test a generalized decomposition with separate spin and orbital spectral functions and show that the extracted ΔALT(ω) is unchanged, or rephrase the result as a phenomenological parametrization of the total spectra rather than a physical separation.
[Figs. 3, 5; Sec. II] The reported better-than-1% agreement is a consistency check within a single simulation family. ΔALT(ω) and ΔFM(ω) are extracted from the φB=0 field series of the same atomic model, and the nonzero-φB spectra are reconstructed using the same model's mean-field order parameters L(B) and m(B). This is a genuine out-of-sample test of the functional form, but it does not test the physical constancy of ΔFM against, for example, a model in which the ferromagnetic line shape depends on field direction or on the spin/orbital composition of m. A controlled test of that kind, or an explicit statement that the 1% accuracy is a model-internal property, is needed to support the proposed experimental protocol for separating the two effects.
[Sec. II, Fig. 3; Methods] The procedure for extracting ΔALT(ω) and ΔFM(ω) from the φB=0 data is not described. At φB=0 the two terms in Eq. (1) are collinear, so the separation relies on the assumed field-independence of both spectral functions and on the variation of |m(B)|; the manuscript should state how many field values are used, whether the fit is performed over the full spectrum, and what the residual is at each field for the basis decomposition. Without this information, the reader cannot judge whether the subsequent agreement at φB≠0 is a prediction of the two-component form or a consequence of the fitting freedom.
[End of Sec. II; Supplementary Note 4] The sum-rule deviations are reported inconsistently. The main text quotes 15% for the spin sum rule and 30% for the orbital sum rule, while Supplementary Fig. S6 quotes 9.11-11.7% for the effective spin and 14.6-36.1% for the orbital moment, depending on the projection and field angle. Please define precisely which quantity is being reported (total magnitude versus [100]/[010] component, spin versus effective spin) and reconcile the numbers, because these deviations are the main evidence of the limitations of Eq. (1).

minor comments (5)

[Methods, Sec. IV] The phrase 'N´ eel vecot L' contains a typo; it should read 'Néel vector L'.
[Fig. 2] The label 'F＋FーXMCD' is not explained in the caption or text; please clarify or remove it.
[Eq. (2)] The approximations φL≃φm+π/2, |m|≃m0, and φm≃αφB are asserted without stating the field range and angle range over which they hold; please add the relevant conditions or refer explicitly to the ranges in Fig. 4.
[Main text and Supplementary Material] The main text refers to 'Supplementary Figure 4' and 'Supplementary Figure 5', while the supplement labels the figures 'Fig. S4' and 'Fig. S5'; please unify the numbering.
[Sec. IV] The text says J = 1.47 meV is 'derived from the experiment' and cites Ref. [36], which is a theory paper; please clarify the original experimental source or adjust the wording.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. 1 is an openly parameterized two-component fit, cross-validated on independent field orientations; the admitted FM ansatz and author self-citations do not make the derivation circular.

full rationale

The central derivation is a numerical experiment: full L2,3 XMCD spectra are computed for each field from an atomic model with DFT-derived crystal field, SOC, Coulomb interactions, Weiss mean field, and external field. Eq. 1 is not derived from those spectra by construction; it is a two-component ansatz whose first term has a symmetry justification (supporting refs. [19,29] are by the same authors, but the paper independently shows the fitted Delta_ALT matches a no-valence-SOC calculation in Fig. 3(c)), and whose second term is explicitly labeled 'an ad hoc approximation based on the smallness of m.' The basis functions Delta_ALT and Delta_FM are extracted by fitting the phi_B = 0 field series, and the paper then uses these 'previously obtained spectral distributions' to reproduce spectra at phi_B = 10-40 deg (Fig. 5) and at varying phi_B with fitted alpha and m0 (Fig. 6), so the angular generalization is not statistically forced by the phi_B = 0 fit. The sum-rule deviations (15% spin, 30% orbital) and the field-dependent ml/ms ratio are admitted limitations on the physical interpretation of Delta_FM, not a reduction of the prediction to its inputs. The self-citations to prior altermagnet XMCD papers are contextual and are backed by in-paper numerical checks; no uniqueness theorem is imported from the authors' prior work. Therefore no circular step meeting the required evidence standard is present.

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

The central decomposition rests on modeling assumptions: atomic multiplet physics, Weiss mean-field exchange, dipole approximation, reduced spin-orbit coupling, and single-domain order. The free parameters are standard Coulomb and SOC constants plus a tuned SOC value and two fitted prefactors in the approximate angular formula. No new physical entities such as particles, forces, or dimensions are introduced.

free parameters (5)

Ni 3d valence spin-orbit coupling ξ = 50 meV (reduced from 77 to 83 meV atomic values)
Adjusted downward to reproduce the NiF6 cluster-model single-ion anisotropy splittings and the experimental magnetization; even then the zero-field net moment of 0.050 μB overestimates the experimental 0.03 μB. See Supplementary Note 2, Table I, and Fig. S3.
Heisenberg exchange J = 1.47 meV
Taken from experiment via Moriya's theory and used in the Weiss mean-field treatment of the magnetic ground state and field response. This controls the canting and field dependence of the Néel vector and net moment.
Hund coupling JH = 0.86 eV
A typical value for Ni2+ systems, not fitted specifically to NiF2 spectra; enters the multiplet structure of the 3d8 configuration.
Lifetime broadening = 0.30 eV HWHM
A single Lorentzian broadening used for all calculated spectra; the authors note it somewhat exaggerates the sharpness of the L2 features.
Angular prefactors α and m0 in Eq. 2 = α = 0.338, m0 = 0.166 μB at 10 T
Fitted to the leading-order field-angle dependence of the same simulations to produce the approximate cos(1±α)φB form; these are not independently predicted quantities.

assumptions (5)

domain assumption The Ni2+ atomic model with a fixed 3d8 configuration and multiplet Coulomb interactions adequately describes Ni L2,3 XAS and XMCD without explicit charge transfer.
Used throughout the Methods; justified by comparison with the experimental XAS in Fig. 2, while ligand hybridization effects are absorbed into a reduced spin-orbit coupling.
domain assumption The magnetic ground state is obtained with Weiss mean-field theory for a nearest-neighbor Heisenberg exchange Hamiltonian.
Section IV: the exchange interaction is treated at static mean-field level, which is an approximation for a quantum magnet and affects the computed field dependence of L and m.
domain assumption The altermagnetic term in Eq. 1 is exact only for a collinear rutile antiferromagnet with no valence SOC and monopole-only core-valence interaction.
Stated in the Introduction after Eq. 1 and cited to Refs. 19 and 29; the full calculations include valence SOC, so the exactness of that term is only approximate in the actual model.
domain assumption The sample is a single magnetic domain, so the zero-field altermagnetic XMCD does not average to zero.
Introduction: the finite XMCD without external field is predicted 'assuming a single-domain sample'; a multidomain sample would cancel the altermagnetic contribution.
domain assumption X-ray absorption is computed in the electric dipole approximation, and dipole-dipole anisotropy is neglected because it is degenerate for all ab-plane Néel vector orientations.
Methods Section IV: Fermi's golden rule in the dipolar approximation is used, and the dipole-dipole energy is stated to be orientation-degenerate in the ab plane and therefore neglected.

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Pith. "Pith review of Magnetic Dichroism in Rutile NiF$_2$: Separating Altermagnetic and Ferromagnetic Effects." pith.science (2026). https://pith.science/paper/OTF3NZ4A

@misc{pith2026250108056,
  author       = {Pith},
  title        = {Pith review of: Magnetic Dichroism in Rutile NiF$_2$: Separating Altermagnetic and Ferromagnetic Effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OTF3NZ4A}},
  note         = {Machine review of arXiv:2501.08056}
}

abstract

We present numerical simulations of x-ray magnetic circular dichroism (XMCD) at the L$_{2,3}$ edge of Ni in the weakly ferromagnetic altermagnet NiF$_2$. Our results predict a significant XMCD signal for light propagating perpendicular to the magnetic moments, which are approximately aligned along the [100] easy-axis direction. The analysis shows that the altermagnetic and ferromagnetic contributions to the XMCD signal can be uniquely distinguished by their dependence on an applied magnetic field. By varying the angle of the field relative to the easy axis, the in-plane orientation of both the N\'eel vector and the net magnetization can be systematically controlled. We further demonstrate that the XMCD signal, even under fields as strong as 40 T and for any in-plane orientation, can be accurately described as a linear combination of two spectral components, with geometrical prefactors determined by the field magnitude and direction. This insight enables experimental validation of the distinctive relationship between the N\'eel vector orientation and the x-ray Hall vector in the rutile structure. Quantitative simulations supporting these findings are provided.

Figures

Figures reproduced from arXiv: 2501.08056 by the authors.

**Figure 2.** FIG. 2. The XAS calculated for the two circular polarizations [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 4.** These results align well with previous theoretical [PITH_FULL_IMAGE:figures/full_fig_p002_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 3.** Figure 3: FIG. 3. (a) Ni L2 approximations to (bllf) Case C [PITH_FULL_IMAGE:figures/full_fig_p003_3.png]

**Figure 5.** Figure 5: FIG. 5. (a) Ni [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

**Figure 6.** Figure 6: FIG. 6. (a) Ni [PITH_FULL_IMAGE:figures/full_fig_p004_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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[010][100] 0.00 0.02 0.04 0.06 0 10 20 30 40 Effective Spin Seff Magnetic Field B (T)0.00 0.10 0.20 0 10 20 30 40 Spin Seff Magnetic Field B (T)0.00 0.10 0.20 0 10 20 30 40 Effective Spin Seff Magnetic Field B (T) 0.00 0.05 0.10 0 10 20 30 40 Orbital L Magnetic Field B (T) ExactApprox. FIG. S6. The orbital angular momentum L (top) and effective spin Seff ...

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