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REVIEW 4 major objections 5 minor 3 references

Staggered Dzyaloshinskii-Moriya and canting angle in centrosymmetric altermagnetic and ferromagnetic phases: influence on the anomalous Hall effect and Weyl points

T0 review · 4 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

Pith's one-line read A symmetry-preserving Wannier model reproduces SrRuO3's near-zero anomalous Hall conductivity and shows that a 10° spin canting flips its sign.

desk verdict Useful method and symmetry table, but the SrRuO3 sign-change claim needs a relativistic benchmark and a working point inside the paper's own DFT angle range. read the letter →

arxiv 2602.10879 v1 pith:32I4IK6O submitted 2026-02-11 cond-mat.mtrl-sci cond-mat.other

classification cond-mat.mtrl-scicond-mat.other
keywords anomalousHalleffectspincantingDzyaloshinskii-MoriyainteractionaltermagnetismSrRuO3WannierfunctionWeylpointsperovskiteoxides
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

Using SrRuO3 as a test case, the authors show that the anomalous Hall conductivity of a magnet can be computed as a function of spin-canting angles by starting from a nonmagnetic Wannier Hamiltonian and adding on-site spin splitting, spin-orbit coupling, and tunable canting. In the ferromagnetic phase, the calculated intrinsic AHC at the Fermi level is close to zero, matching experiments, and a canting angle of about 10 degrees is sufficient to flip its sign. They map the symmetry-allowed components of the AHC tensor for ferromagnetic and altermagnetic orders in ABO3 perovskites with space group 62, finding e.g. a spontaneous in-plane Hall component for in-plane ferromagnets and a pure altermagnetic C-type phase with no allowed AHC. The paper also shows that canting angles shrink as correlations increase, and that the AHC is most parameter-sensitive in the middle of the band, where the Weyl points reside. The overall claim is that spin canting is a real, tunable lever on the anomalous Hall response in this material class.

What carries the argument

The central device is a rigid spin-splitting construction: a nonmagnetic t2g Wannier Hamiltonian for SrRuO3 is dressed with an on-site exchange field h(θ, φ) and spin-orbit coupling, so the magnetic configuration is controlled by two canting angles while the original space-group symmetries (including those that generate altermagnetism) are preserved. This makes the Berry-curvature integral, and hence the anomalous Hall conductivity, a tunable function of (θ, φ) that can be scanned without repeating relativistic DFT for each configuration — and the same symmetry analysis transfers the allowed AHC components to the whole space-group-62 family.

What would settle it

A full relativistic DFT calculation of the intrinsic anomalous Hall conductivity of ferromagnetic SrRuO3, without the rigid spin-splitting shortcut, should yield a value close to zero at the Fermi level and reverse sign under the calculated canting; if it does not, the model's central agreement with experiments is coincidental.

Watch

Extended reading notes

Core claim

The central claim is that a rigid spin-splitting construction — a nonmagnetic t2g Wannier Hamiltonian for SrRuO3 plus on-site spin splitting Δ, spin-orbit coupling λ = 100 meV, and two canting angles (θ, φ) — preserves enough of the electronic structure to reproduce the intrinsic anomalous Hall conductivity and its sign changes. For ferromagnetic SrRuO3 with magnetization along z, the paper finds σxy at the Fermi level is close to zero, consistent with experiments, and that θ = 10° (at φ = 0) or φ = 50° (at θ = 15°) flips the sign. The same model yields a table of the only nonzero AHC components for each collinear magnetic order: ferromagnetic, A-, C-, and G-type; the C-type altermagnet with

Load-bearing premise

The argument assumes that adding a uniform, isotropic on-site spin splitting to the nonmagnetic Wannier Hamiltonian — rather than computing the self-consistent relativistic magnetic state — faithfully preserves the band crossings near the Fermi level that determine the anomalous Hall conductivity and its sign flip.

Editorial extensions

If this is right

  • The near-zero AHC of ferromagnetic SrRuO3 at the Fermi level is an intrinsic property of this Wannier model, and the sign change follows from spin canting alone, meaning interface or strain effects are not the only possible route to the experimental sign reversal.
  • The symmetry table applies to all ABO3 perovskites with space group 62, so any ferromagnet with spins in the xy plane will show both σyz and σzx, the latter being a spontaneous in-plane anomalous Hall effect.
  • The C-type altermagnet with Néel vector along z is the only purely altermagnetic phase in this family: time reversal is broken but no AHC component is symmetry-allowed.
  • Raising electron correlations suppresses canting, which implies the canting-induced sign-change mechanism weakens in samples with stronger correlation.
  • Weyl points in the ferromagnetic phase are concentrated in the central band region and respond an order of magnitude more strongly to the polar canting angle than to the azimuthal one, tying AHC sensitivity to polar tilt.

Reading between the lines

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

  • Because the rigid spin-splitting shortcut omits momentum-dependent spin-momentum locking (d-wave magnetism in the subdominant spin components), the near-zero AHC and the 10° sign flip should be tested against a full relativistic DFT AHC calculation; until such a benchmark, the 'agreement with experiments' is conditional on the model's band crossings being faithful.
  • If the symmetry transfer is as clean as claimed, the same Wannier-plus-splitting recipe could be used to screen vanadates, chromites, and other space-group-62 perovskites for sign-change AHC behavior without expensive relativistic DFT, by simply scanning (θ, φ).
  • The strong θ-sensitivity of Weyl-node energies suggests that angle-resolved measurements of the AHC in single-domain samples could serve as a probe of the staggered DMI direction, since the polar tilt selectively shifts the nodes.
  • The 'collinear states dominate AHC' result implies that in materials where the collinear AHC is sizable, the Hall sign is robust, but near zeros of the collinear AHC the response becomes exquisitely sensitive to small canting — exactly the regime where SrRuO3 sits, making it a plausible tunable Hall switch.
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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

4 major / 5 minor

Summary. The paper introduces a methodology for computing the anomalous Hall conductivity (AHC) of collinear and canted magnetic phases starting from a nonmagnetic Wannier Hamiltonian. Magnetism is injected via an on-site spin splitting Δ, spin-orbit coupling λ L·S, and tunable spin-canting angles (θ, φ), with the crystal symmetry inherited from the nonmagnetic DFT electronic structure. The method is applied to SrRuO3 in its ferromagnetic and A-, C-, and G-type altermagnetic orders. The paper reports the allowed AHC tensor components for each magnetic order (Table I), a near-zero σxy at the Fermi level for the ferromagnetic phase with magnetization along z, a canting-induced sign change of σxy at a selected spin splitting Δ = 0.84 eV, and the evolution of Weyl points under canting. DFT+U is used to estimate the physical range of canting angles and to show that stronger correlations suppress canting.

Significance. If the central claims hold, the paper offers a practical, symmetry-preserving route to compute AHC for arbitrary spin canting, a useful addition to the altermagnetism and anomalous Hall toolbox. The manuscript has several strengths: the pipeline (VASP → Wannier90 → in-house SOC code → WannierTools) is standard and clearly described; the code is made publicly available; k-grid convergence is checked (101^3 vs 201^3); and the symmetry table for space-group-62 perovskites is a potentially useful classification. However, the quantitative claims about SrRuO3 rest on an unbenchmarked rigid spin-splitting model, and the sign-change demonstration is performed at hand-selected parameters that partly fall outside the DFT-derived canting range. These issues make the central 'agreement with experiments' claim and the generality of the conclusions provisional.

major comments (4)
  1. [Sec. II, Eqs. (1)-(2)] The load-bearing approximation is a rigid, isotropic on-site spin splitting Δ added to a nonmagnetic Wannier Hamiltonian, with all spin texture captured by the scalar angles (θ, φ). This model is not benchmarked against a fully relativistic DFT AHC calculation or against prior quantitative intrinsic-AHC computations for SrRuO3 (e.g., Fang et al., Science 302, 92 (2003), which is not cited). The omission matters because the same group's ref. [49] reports d-wave subdominant spin-momentum locking in this material; such momentum-dependent exchange fields are not represented by a scalar Δ and can contribute to Berry curvature near the Fermi level. I recommend either providing a benchmark vs a full relativistic Wannier/DFT calculation or substantially tempering the 'agreement with experiments' claim.
  2. [Sec. IV B, Figs. 10-11] The sign-change demonstration is made at a specific working point, Δ = 0.84 eV, with θ = 10° and φ = 0° (Fig. 10) and θ = 15°, φ = 50° (Fig. 11). No physical justification is given for choosing Δ = 0.84 eV among the 0.5-2.0 eV range studied in Sec. III, and no quantitative value of 'close to zero' σxy at E_F is reported, nor the smearing parameter used in the WannierTools Kubo computation. More importantly, the angles used in Figs. 10-11 lie outside the DFT-derived realistic ranges stated in Sec. IV A (θ = 2°-7°, φ = 5°-33°). The claim that 'a canting angle of 10° was sufficient' is therefore not demonstrably realistic for the bulk phase studied in Sec. IV A. The authors should either restrict the sign-change demonstration to the DFT-derived range, provide a physical motivation for the larger angles, or present the result as a model prediction pending experimental constraints.
  3. [Sec. III, Table I] Table I is presented as a symmetry-allowed classification for 'the large family' of ABO3 perovskites with space group 62, but the entries are inferred from numerical AHC calculations for SrRuO3 alone. The paper identifies the relevant magnetic space group (62.448) in Sec. IV A but does not provide a group-theoretic derivation of the nonzero AHC components. Since the statement that the same components 'will be symmetry-allowed for systems of the same material class' is one of the paper's general claims, it should be substantiated by an explicit symmetry analysis (e.g., using magnetic point-group or multipole arguments), rather than by extrapolation from a single compound.
  4. [Sec. IV B and Conclusions] The claimed agreement with experiments is only qualitative. No measured AHC magnitude or temperature/field conditions are cited, and the experimental sign reversal in SrRuO3 has been observed mainly in heterostructures/interfaces (refs. 29-34), whereas the present calculation is for the bulk ferromagnetic phase. The phrase 'in agreement with experimental observations' is therefore ambiguous: it could refer to the sign change under interface engineering or to the near-zero bulk AHC. The manuscript should state explicitly which experiments are being compared and, ideally, provide a quantitative comparison of the computed σxy with measured values where they exist.
minor comments (5)
  1. [Eq. (3)] The equation 'ρxy ≠ R0Bz + Rs μ0 Mz = 0' is typeset ambiguously. It should be made clear that the right-hand side is zero while ρxy is nonzero (or vice versa).
  2. [Introduction] The Introduction says 'Section 6 summarizes the main conclusions', but the paper has only Section V. The section numbering should be corrected.
  3. [Sec. II, Eq. (2)] The symbol Δ used throughout the paper is not explicitly defined in Eq. (2), where the spin-splitting is written as the vector h(θ_S, φ_S). Please state the relation |h| = Δ and the sign convention.
  4. [References] Several references are listed as 'Submitted', 'In preparation', or 'In manuscript' (refs. 49, 69, 77). These should be updated or marked as unpublished in a way that is clearly dated, as they are used for important physical input (especially ref. 49).
  5. [Fig. 4] The caption for Fig. 4 says '(a,b) x-direction with two non-zero components: σyz and σzx respectively', but the panels are labeled (a), (b), (c). Please make the correspondence explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity — AHC is computed from an explicit model Hamiltonian, not fitted to experiment; self-citations are background only.

full rationale

The AHC derivation is self-contained: the model is built from a nonmagnetic DFT Wannier Hamiltonian (Eq. 1) plus on-site spin splitting and SOC (Eq. 2), and the AHC is evaluated with WannierTools; no computed AHC value is fed back into the model as an input or fitted parameter. The near-zero sigma_xy at E_F (Sec. III A, Fig. 4c) is a numerical output that persists across Δ ∈ [0.5,2.0] eV rather than a target used to select Δ, and the canting-induced sign change (Sec. IV B) is an existence result obtained by scanning the (θ,φ) parameter space at a fixed realistic Δ=0.84 eV; this is a sensitivity study, not a fit disguised as a prediction. The load-bearing symmetry statements (Table I) follow from the magnetic space group and standard tensor transformations, not from an author-specific theorem. Self-citations (refs 49, 51, 55, 60) supply background, prior code, or related demonstrations, but the central AHC and Weyl-point results do not reduce to them. Two flagged concerns are risks, not circularity: (i) the text says 'Limiting ourselves to realistic values of θ and φ' while Figs. 10-11 use θ=10° and φ=50°, outside the DFT ranges θ≈2°-7° and φ≈5°-33° from Fig. 8; (ii) the model is not benchmarked against full relativistic DFT AHC or prior quantitative AHC calculations, and the analytical DMI form is deferred to ref 77. These affect validity, not logical circularity.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

No new particles, forces, or conserved quantities are introduced; the staggered DMI and altermagnetic phases are drawn from prior literature, and even the 'spontaneous in-plane AHC' is a symmetry-allowed tensor component, not a new entity. The load-bearing assumptions are the rigid spin-splitting model (Eqs. 1-2), the (θ,φ) parametrization of magnetic space group 62.448, and the t2g-only Wannier description. The free parameters (Δ, λ, θ, φ) are tuned within DFT-anchored ranges; the AHC values themselves are computed, not fitted.

free parameters (4)
  • On-site spin splitting Δ = 0.84 eV used for the sign-change demonstration; scanned 0.5–2.0 eV
    Chosen by hand within a range said to match the LSDA spin splitting of ~1 eV (ref 42). The AHC at E_F and the sign-change mechanism depend strongly on this value (Figs. 4, 10-11).
  • Spin-orbit coupling λ_Ru = 100 meV
    Extracted by the authors from their own relativistic Wannierization; stated to agree with ref 61. The staggered DMI and canting physics enter through this coupling.
  • Canting angles θ, φ = θ=10°, φ=0° and θ=15°, φ=50° for sign-change demonstration; realistic DFT ranges θ≈2–7°, φ≈5–33°
    Scanned over the full parameter space; the paper notes 'multiple sign changes can be observed throughout the parameter space' (Sec. IV B), so the demonstration points are selected working points, not unique predictions.
  • Coulomb U (DFT+U) and Hund's coupling = U = 0–3 eV; U=3 eV, J_H=0.15U for the band-structure calculations
    Standard DFT+U model parameters; the DFT-canting curve vs U (Fig. 8) and the reported suppression of canting with correlations depend on them.
assumptions (6)
  • standard math Intrinsic anomalous Hall conductivity is given by the Kubo–Berry-curvature formula and is computed correctly by WannierTools on the model Hamiltonian.
    Used throughout Secs. III-IV; standard linear-response result for the intrinsic AHC.
  • domain assumption SrRuO3 crystallizes in Pbnm (space group 62) with the stated experimental lattice constants, and the low-energy physics is captured by the Ru t2g manifold alone.
    Appendix A and Fig. 1; the symmetry table for the ABO3 family inherits this structure assumption.
  • domain assumption A rigid, uniform on-site spin splitting added to the nonmagnetic Wannier Hamiltonian reproduces the magnetic electronic structure — including the band crossings that set the AHC — of the self-consistently magnetized system.
    Section II, Eqs. (1)-(2); the load-bearing model assumption. No comparison against a full relativistic DFT AHC is provided.
  • domain assumption The magnetic configurations generated by Eqs. (4)-(7) with parameters (θ, φ) exhaust the symmetry-allowed non-collinear states of magnetic space group 62.448 (BNS).
    Section IV A; the entire canting-dependence study rests on this parametrization.
  • domain assumption The staggered DMI is linear in λ (Moriya) and is implicitly captured by tuning the canting angles; SOC preserves the assumed symmetry constraints.
    Section IV, citing ref 73; the mapping 'tuning canting angles = manipulating staggered DMI parameters' is assumed, and the analytical staggered-DMI form is deferred to ref 77.
  • domain assumption The Weyl points found in the model band structure are genuine touching points whose evolution tracks the AHC changes.
    Section IV C; only representative node trajectories are shown (Figs. 13-14), and the connection to the AHC sign change is qualitative.

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Cite this review

Pith. "Pith review of Staggered Dzyaloshinskii-Moriya and canting angle in centrosymmetric altermagnetic and ferromagnetic phases: influence on the anomalous Hall effect and Weyl points." pith.science (2026). https://pith.science/paper/32I4IK6O

@misc{pith2026260210879,
  author       = {Pith},
  title        = {Pith review of: Staggered Dzyaloshinskii-Moriya and canting angle in centrosymmetric altermagnetic and ferromagnetic phases: influence on the anomalous Hall effect and Weyl points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/32I4IK6O}},
  note         = {Machine review of arXiv:2602.10879}
}
abstract

We present a simple methodology to compute the anomalous Hall conductivity (AHC) as a function of the canting angles in ferromagnets and altermagnets, starting from a nonmagnetic Hamiltonian obtained from first-principles calculations that preserves the full symmetry of the crystal structure. Magnetism is introduced by including on-site spin splitting, spin-orbit coupling, and spin-canting angles. As a representative material, we study SrRuO$_3$, which supports spin canting and exhibits a sign change of the AHC. In the ferromagnetic phase, the low-energy AHC is found to be close to zero at the Fermi level, in agreement with experimental observations. We show that the dependence of the AHC on the relevant physical parameters is most pronounced in the central region of the electronic bandwidth. We determine the symmetry-allowed components of the AHC for different magnetic orders in the large family of transition-metal perovskite ABO$_3$ compounds with space group $62$, including the spontaneous in-plane anomalous Hall effect. Within density functional theory, we evaluate the range of spin-canting angles in SrRuO$_3$ and demonstrate that it is suppressed as electronic correlations increase. By analyzing the AHC as a function of the canting angle, we find that the collinear magnetic configurations contribute most to the AHC, while spin canting plays a secondary role in determining its magnitude in non-collinear ferromagnets and altermagnets. However, canting can become relevant and induce a sign change of the AHC when the collinear magnetic state exhibits an AHC close to zero. Finally, we investigate the locations of Weyl points in the Brillouin zone and their evolution as a function of the canting angle.

Figures

Figures reproduced from arXiv: 2602.10879 by the authors.

Figure 1
Figure 1. FIG. 1. Comparison of the non-magnetic electronic band [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The altermagnetic phases of SrRuO [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. A depiction of the three regions observed in the band [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: FIG. 4. AHC as a function of the spin splitting for the ferromagnetic phase with magnetization along the (a,b) [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. AHC as a function of the spin splitting for the A-type altermagnet with N´eel vector along the (a) [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. AHC as a function of the spin splitting for the C-type altermagnet with N´eel vector along the (a,b) [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Evolution of the total spin [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 7
Figure 7. Figure 7: FIG. 7. AHC for the G-type magnetic order with (a) N´eel [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The variation of the AHC, [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. (a) The sign change of the AHC at the Fermi level [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]
Figure 12
Figure 12. Figure 12: FIG. 12. The nodes of the system in reciprocal space plotted [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]
Figure 3
Figure 3. Figure 3: We observe an extended distribution of several [PITH_FULL_IMAGE:figures/full_fig_p009_3.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Evolution of a Weyl point in the system as a func [PITH_FULL_IMAGE:figures/full_fig_p010_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Evolution of a Weyl point in the system as a function [PITH_FULL_IMAGE:figures/full_fig_p010_14.png]

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Works this paper leans on

3 extracted references · 2 linked inside Pith

  1. [12]

    It has been predicted that the anomalous Hall response can also serve as a sensitive probe of the N´ eel vector15,16

    The ori- entation of the Hall vector is strongly dependent on the N´ eel vector, which is defined as the difference between the magnetization vectors of the two inequivalent mag- netic sublattices 13,14. It has been predicted that the anomalous Hall response can also serve as a sensitive probe of the N´ eel vector15,16. Recently, the link between anomalou...

  2. [14]

    k z 0 10 20 30 40 0.00550.0050 0.0045 0.0040 0.0035 -0.050 -0.054 -0.046 0.039 0.038 0.037 0.036 0.035 50 Azimuthal Angleϕ(°) -0.042 k x k y y k FIG

    Nevertheless, the trajectories traced by the Weyl nodes as the spin canting is varied are node dependent. k z 0 10 20 30 40 0.00550.0050 0.0045 0.0040 0.0035 -0.050 -0.054 -0.046 0.039 0.038 0.037 0.036 0.035 50 Azimuthal Angleϕ(°) -0.042 k x k y y k FIG

  3. [62]

    Insulator-to-metal transition via magnetic reconstruction at oxide interfaces,

    For the ferromagnetic phase and for the C-type order with magnetization in thexyplane, bothσ zx andσ yz are allowed. For the ferromagnetic phase with the dominant component in thexyplane, this is the spontaneous in- plane anomalous Hall effect and we tested through DFT calculations that the subdominant component produces a weak ferrimagnetism by spin cant...

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Reviewed August 4, 2026 · model on record in the stance chip above.