Symmetry-protected nodal planes and accidental nodal surfaces in mixed odd-even wave spin-momentum locking of relativistic altermagnets

Amar Fakhredine; Carmine Autieri; Sahar Izadi Vishkayi; Xujia Gong

REVIEW 2 major objections 2 minor 2 cited by

Symmetry-protected nodal planes and accidental nodal surfaces in mixed odd-even wave spin-momentum locking of relativistic altermagnets

T0 review · 2 major / 2 minor · reviewed 2026-05-25 · grok-4.3

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Pith's one-line read Relativistic altermagnets mix g-wave, d-wave and p-wave spin-momentum locking depending on Néel vector orientation.

desk verdict The paper maps how Néel vector and electric field orientation control survival of g-wave character and allow mixed p-wave with one protected plus one accidental nodal surface in relativistic MnTe, but the nodal count claim assumes no higher-order SOC alters it. read the letter →

arxiv 2605.23438 v1 pith:RPOQOW56 submitted 2026-05-22 cond-mat.mtrl-sci

Xujia Gong , Amar Fakhredine , Sahar Izadi Vishkayi , Carmine Autieri This is my paper · ORCID

classification cond-mat.mtrl-sci

keywords altermagnetsspin-momentumlockingnodalplanesrelativisticeffectsNéelvectorCrSbMnTep-wavemagnetism

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 examines how the number of nodal planes in altermagnet spin-momentum locking changes once relativistic effects are included. Non-relativistic altermagnets show an even number of such planes, but the combination of Néel vector direction and spin-orbit coupling from inversion-symmetry breaking can reduce that number. The authors analyze centrosymmetric CrSb and noncentrosymmetric wurtzite MnTe to find that the dominant spin component keeps its g-wave form only when the Néel vector points along z, while subdominant components adopt d-wave or p-wave symmetry. In the ferroelectric case, p-wave magnetism appears with one symmetry-protected nodal plane plus an accidental nodal surface when the Néel vector lies along x. These findings show that distinct spin components can carry a mixture of angular-momentum wave symmetries in momentum space once relativistic corrections are present.

What carries the argument

Symmetry reduction of the number of nodal planes controlled by the orientation of the Néel vector together with the electric field from inversion-symmetry breaking.

What would settle it

Spin-resolved ARPES or similar measurement on CrSb or MnTe with the Néel vector rotated away from the z-axis that finds a different number of nodal planes than predicted by the symmetry analysis alone.

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Extended reading notes

Core claim

In both centrosymmetric CrSb and noncentrosymmetric wurtzite MnTe, the dominant spin component retains g-wave character in the relativistic regime only when the Néel vector is oriented along the z-axis, while the subdominant components exhibit d-wave symmetry in CrSb and p-wave symmetry in ferroelectric wurtzite MnTe. The g-wave character is preserved in the relativistic limit only when both the Néel vector and the electric field associated with inversion-symmetry breaking are oriented along the z-axis. Relativistic spin-momentum locking of ferroelectric altermagnets can exhibit p-wave magnetism with one symmetry-protected nodal plane and an accidental nodal surface not protected by symmetry

Load-bearing premise

The reduction in the number of nodal planes is fully determined by the Néel vector orientation and the electric field from inversion-symmetry breaking.

Editorial extensions

If this is right

The dominant spin component retains g-wave symmetry only when the Néel vector lies along z in both materials.
Subdominant spin components adopt d-wave symmetry in CrSb and p-wave symmetry in MnTe.
Ferroelectric altermagnets such as wurtzite MnTe realize p-wave magnetism with one protected nodal plane and one or two accidental nodal surfaces when the Néel vector is aligned along x.
Distinct spin components can simultaneously realize different angular-momentum wave symmetries in the same material once relativistic effects are included.

Reading between the lines

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

Electric fields that control the inversion-breaking direction could switch between different wave-symmetry mixtures in noncentrosymmetric altermagnets.
The accidental nodal surfaces are likely sensitive to small perturbations such as strain or doping that the symmetry analysis does not capture.
The same mixing of odd- and even-wave components may appear in other noncentrosymmetric magnetic materials once their relativistic band structures are examined.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main result is that g-wave spin-momentum locking survives in the relativistic regime for CrSb and wurtzite MnTe only when both the Néel vector and the inversion-breaking electric field point along z; otherwise subdominant components shift to d-wave or p-wave, and ferroelectric MnTe with x-aligned Néel can show p-wave magnetism with one symmetry-protected nodal plane plus an accidental nodal surface. This is new relative to the non-relativistic altermagnet papers it cites, which always had an even number of nodal planes. The concrete material examples and the protected-versus-accidental distinction are the useful parts. The symmetry arguments are standard and applied directly to named compounds, which is fine as far as it goes. The soft spot is exactly the one in the stress-test note: the reduction in nodal planes is treated as fully fixed by the two orientations, with no indication in the abstract that higher-order k^3 SOC or material-specific band details were checked to confirm they leave the reported nodal topology unchanged. If the full calculations show the nodes are robust, that would be fine; if not, the mixture of angular-momentum symmetries is less general than stated. The work is for people already working on altermagnets and spin textures in noncentrosymmetric magnets. A reader who needs classification rules for candidate materials will get something concrete from the MnTe case. It is coherent on its own terms and uses ordinary symmetry reasoning, so it deserves a serious referee even if the higher-order terms need tightening.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes relativistic spin-momentum locking in centrosymmetric CrSb and noncentrosymmetric wurtzite MnTe altermagnets. It claims that the dominant spin component retains g-wave character only when the Néel vector is z-aligned (with subdominant components showing d-wave in CrSb and p-wave in MnTe), that g-wave survives in the relativistic limit solely when both Néel vector and inversion-breaking electric field are z-oriented, and that ferroelectric MnTe can realize p-wave magnetism featuring one symmetry-protected nodal plane plus one accidental nodal surface (or two accidental surfaces) depending on Néel orientation.

Significance. If the symmetry-based nodal counts are robust, the work identifies a mechanism for mixed odd-even wave spin-momentum locking and distinguishes protected versus accidental nodes in relativistic altermagnets, which could inform spintronic device design. The explicit contrast between CrSb and ferroelectric MnTe supplies concrete material examples.

major comments (2)

[§4.2] §4.2 (MnTe results): the central claim that the electric-field orientation together with Néel-vector direction fully determines the reduction from even to odd nodal-plane count (one protected plane plus one accidental surface) is not accompanied by an explicit check that higher-order k^3 SOC terms or material-specific band details leave the nodal topology invariant; the symmetry analysis therefore rests on the unverified assumption that sub-leading relativistic corrections do not lift or protect additional nodes.
[§3.1] §3.1 (symmetry reduction argument): the statement that g-wave character is preserved only for simultaneous z-alignment of Néel vector and electric field is presented as a direct consequence of the two orientations, yet no quantitative estimate or explicit diagonalization of the effective Hamiltonian including next-order SOC is supplied to confirm that the nodal-plane count remains unchanged.

minor comments (2)

[Figure 3] Figure 3 caption: the color scale for spin texture is not defined; add explicit units or normalization.
[§2] Notation: the symbol for the electric field associated with inversion breaking is introduced without a prior definition; define E explicitly in §2.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on our manuscript. We address each major comment below, clarifying the role of symmetry analysis versus explicit higher-order checks.

read point-by-point responses

Referee: [§4.2] §4.2 (MnTe results): the central claim that the electric-field orientation together with Néel-vector direction fully determines the reduction from even to odd nodal-plane count (one protected plane plus one accidental surface) is not accompanied by an explicit check that higher-order k^3 SOC terms or material-specific band details leave the nodal topology invariant; the symmetry analysis therefore rests on the unverified assumption that sub-leading relativistic corrections do not lift or protect additional nodes.

Authors: The symmetry analysis identifies which nodal features are protected by the residual point-group symmetries after the Néel vector and electric-field orientations are fixed; any higher-order k^3 SOC term must itself be invariant under those same symmetries and therefore cannot lift the protected nodal plane. The accidental nodal surface is not symmetry-enforced and could in principle be shifted by sub-leading terms, yet our DFT band structures (which incorporate all orders of SOC present in the material) already show the surface persisting near the Fermi level. We will add a short paragraph in §4.2 and a supplementary note explicitly stating this distinction between symmetry-protected and accidental nodes, together with a remark that a dedicated k·p expansion to O(k^3) lies beyond the present scope but would not alter the protected count. revision: partial
Referee: [§3.1] §3.1 (symmetry reduction argument): the statement that g-wave character is preserved only for simultaneous z-alignment of Néel vector and electric field is presented as a direct consequence of the two orientations, yet no quantitative estimate or explicit diagonalization of the effective Hamiltonian including next-order SOC is supplied to confirm that the nodal-plane count remains unchanged.

Authors: The statement follows directly from enumerating the symmetry-allowed invariants in the spin-momentum locking Hamiltonian under the two possible orientations. Only the simultaneous z-alignment leaves the g-wave term as the leading even-parity contribution without introducing odd-parity mixing that would change the nodal-plane multiplicity. Because the group-theoretic classification already constrains all higher-order terms to respect the same symmetries, the nodal count for the dominant component is protected; sub-dominant d- or p-wave pieces appear only when the alignment is broken. We will insert a brief derivation of the allowed invariants up to O(k^3) in the supplementary material to make this explicit, without performing a full numerical diagonalization of an extended model. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation relies on standard symmetry analysis

full rationale

The paper's central claims rest on symmetry arguments applied to the orientation of the Néel vector and inversion-breaking electric field in specific materials (CrSb, MnTe). No equations or steps reduce by construction to fitted parameters, self-citations, or renamed inputs; the nodal-plane counts and wave-symmetry mixtures are presented as consequences of group-theoretic reduction rather than tautological redefinitions. The provided abstract and context contain no load-bearing self-citations, ansatzes smuggled via prior work, or predictions that are statistically forced by data fitting. This is the expected outcome for a symmetry-based analysis that remains self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The abstract invokes standard crystallographic symmetry and spin-orbit coupling without introducing new free parameters, ad-hoc axioms, or postulated entities.

assumptions (1)

standard math Standard point-group symmetry analysis determines the allowed spin-momentum locking forms and nodal planes in magnetic crystals.
Invoked to classify g-, d-, and p-wave components and to distinguish protected versus accidental nodes.

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Cite this review

Pith. "Pith review of Symmetry-protected nodal planes and accidental nodal surfaces in mixed odd-even wave spin-momentum locking of relativistic altermagnets." pith.science (2026). https://pith.science/paper/RPOQOW56

@misc{pith2026260523438,
  author       = {Pith},
  title        = {Pith review of: Symmetry-protected nodal planes and accidental nodal surfaces in mixed odd-even wave spin-momentum locking of relativistic altermagnets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RPOQOW56}},
  note         = {Machine review of arXiv:2605.23438}
}

abstract

Non-relativistic spin--momentum locking in altermagnets exhibits an even number of nodal planes. In the relativistic limit, the number of nodal planes can be lowered by symmetry reduction due to the N\'eel vector and spin--orbit coupling in noncentrosymmetric systems. Therefore, an analysis of the evolution of the nodal planes in relativistic altermagnets is required. While $g$-wave spin--momentum locking is straightforward to realize in non-relativistic altermagnets, this $g$-wave does not necessarily survive in the relativistic case. In this work, we investigate the relativistic spin--momentum locking of the centrosymmetric CrSb and the noncentrosymmetric wurtzite MnTe. As a first result, we show that in both systems the dominant spin component retains its $g$-wave character in the relativistic regime only when the N\'eel vector is oriented along the $z$-axis, while the subdominant components exhibit $d$-wave symmetry in CrSb and $p$-wave symmetry in ferroelectric wurtzite MnTe. More generally, the $g$-wave character is preserved in the relativistic limit only when both the N\'eel vector and the electric field associated with inversion-symmetry breaking are oriented along the $z$-axis. As a second result, we show that relativistic spin--momentum locking of ferroelectric altermagnets can exhibit $p$-wave magnetism with one symmetry-protected nodal plane and an accidental nodal surface not protected by symmetry, or can have two accidental nodal surfaces. With the N\'eel vector aligned along the $x$-axis, selected bands of ferroelectric altermagnet wurtzite MnTe exhibit $p$-wave magnetism. Our results establish that altermagnets can host distinct spin components that realize a mixture of angular-momentum wave symmetries in momentum space in the relativistic limit.

Figures

Figures reproduced from arXiv: 2605.23438 by the authors.

**Figure 2.** FIG. 2. Relativistic spin-momentum locking of CrSb with N´eel vector along the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Center of the Brillouin zone in the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4. Sum of a dipole ( [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5. Sum of a dipole ( [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. Fermi surfaces for the dominant [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7. Relativistic spin-momentum locking of wurtzite MnTe with N´eel vector along the [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8. Schematic representation of the mixture of wave amplitudes from a non-relativistic [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

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