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REVIEW 3 major objections 5 minor 56 references

Effect of symmetry breaking on altermagnetism in CrSb and Formation of fragmented nodal curves

T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read In altermagnetic CrSb, reducing six-fold rotational symmetry to two-fold replaces the three diagonal nodal planes with band-specific fragmented nodal curves, and this enables anomalous Hall conductivity for both in-plane and out-of-plane Né

desk verdict Useful AHC prediction under strain, but the symmetry-enforced FNC claim collapses: Eq. (10) relates opposite momenta, not same-k degeneracies. read the letter →

arxiv 2602.21135 v1 pith:Q5FPS6NF submitted 2026-02-24 cond-mat.mtrl-sci quant-ph

Arindom Das , Arijit Mandal , Nayana Devaraj , B.R.K. Nanda This is my paper · ORCID
classification cond-mat.mtrl-sciquant-ph
keywords altermagnetismCrSbfragmentednodalcurvesmomentum-spacespinpolarizationanomalousHalleffectspacegroupsymmetryloweringstrainengineering
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 sets out to show what happens to altermagnetism in CrSb when the crystal symmetry is lowered from six-fold to two-fold rotation. It claims that the three diagonal nodal planes of the pristine material disappear and are replaced by fragmented nodal curves (FNCs): band-specific lines in the Brillouin zone along which pairs of opposite-spin bands are exactly degenerate, with non-degenerate splitting elsewhere. The authors argue this follows from the combined action of a two-fold spin-space rotation and the non-relativistic time-reversal/inversion operation, and they support the claim with first-principles calculations on intentionally modified model structures, 5% uniaxially strained CrSb, and the compound RbMnPO4. A key consequence they stress is that the symmetry lowering allows a finite anomalous Hall conductivity for both in-plane and out-of-plane Néel vector orientations, which pristine CrSb forbids for the out-of-plane orientation.

What carries the argument

The central object is the fragmented nodal curve (FNC): a band-specific curve in the Brillouin zone along which two opposite-spin sublattice bands are degenerate, replacing the rigid nodal planes of the high-symmetry altermagnet. The argument is carried by the spin-space-group operations [C2||C2z] and [C2||Mz] together with the non-relativistic dual operation [E||TR,L] (spin-preserving reversal of all momenta, equivalent to inversion in the non-relativistic limit). The paper uses these operations to derive an eigenvalue relation that, in its reading, forces the spin-opposite pair bands to cross at points in the kx-ky plane for each kz; as k varies, these points trace out the FNC. The work al

What would settle it

Check whether [E||TR,L] belongs to the spin space group of the relaxed MS-V or 5%-strained CrSb crystal by querying the actual magnetic space group (the paper does not list it for strained CrSb, and for MS-V it lists P21/m without inversion in the chemical space group). If it is absent, then compute the band eigenvalues along one of the plotted FNCs: if the opposite-spin bands are not exactly degenerate, the FNC is not a symmetry-required crossing. A more direct test is to add small spin-orbit coupling, which breaks [E||TR,L]; if the degeneracy lifts, the curve is a non-relativistic symmetry a

Watch

Extended reading notes

Core claim

The central discovery is the formation of fragmented nodal curves (FNCs) in an altermagnet when the spin-space symmetry is restricted to a two-fold rotation. In pristine CrSb, the high-symmetry operations [C2||C6z] and [C2||Mz] generate four nodal planes (three diagonal and one basal). Once the symmetry is lowered to C2z, the paper shows that the three diagonal planes are no longer symmetry-forced; instead, for each pair of spin-opposite sublattice bands there appear curves in reciprocal space along which the two bands are degenerate. The paper argues from spin-space-group operations that these curves must exist, and it verifies their presence in density-functional band structures of model s

Load-bearing premise

The argument's load-bearing step is the assumption that the combined operation [E||TR,L] (spin-preserving time reversal plus inversion) is a symmetry of the C2-symmetric, non-centrosymmetric structures used for MS-V and strained CrSb; if that operation is not actually present, the eigenvalue relation derived in Section V does not force opposite-spin bands to be degenerate at the same momentum, and the fragmented nodal curves would not be symmetry-guaranteed.

Editorial extensions

If this is right

  • A 5% uniaxial strain along the basal axis of CrSb reduces the symmetry to C2 and produces FNCs in the calculated constant-energy surfaces, offering a concrete experimental route to create them.
  • In the symmetry-lowered systems, anomalous Hall conductivity becomes finite for both out-of-plane and in-plane Néel vector orientations, not just in-plane as in the pristine case.
  • Because each band pair has its own FNC, the momentum-space spin polarization becomes selectable by energy and band index rather than fixed by symmetry planes.
  • The appearance of FNCs in RbMnPO4, which has only a two-fold screw symmetry, indicates the phenomenon is not restricted to CrSb but should occur in any altermagnet with C2 spin-space symmetry.
  • Within the non-relativistic picture, proper six-fold rotation and six-fold roto-inversion produce the same nodal planes, so improper rotations can sustain altermagnetism just as well.

Reading between the lines

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

  • If the FNC degeneracies are real, one could engineer the anomalous Hall response of CrSb by choosing strain magnitude and direction; the paper shows AHC around the Fermi level is already higher under 5% strain but does not map how it grows or changes sign with further strain.
  • The band-specific nature of FNCs implies that shifting the Fermi level by doping or gating will switch between different subsets of curves, making the AHC sharply tunable with carrier density — a prediction the paper does not directly test.
  • The same symmetry-lowering logic used here for CrSb could be applied to other hexagonal altermagnets; verifying FNCs in a second family would test whether the mechanism is generic.
  • A useful next step would be spin-resolved ARPES or a direct experimental probe of the constant-energy surfaces under strain; the predicted FNC pattern would show as spin-degenerate lines surrounded by opposite-spin splitting, which is measurable in principle.
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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 studies altermagnetic spin splitting in CrSb and in a set of hypothetical model structures built by vacancy/interstitial engineering, with the aim of understanding how lowering the crystal symmetry from sixfold to twofold affects the momentum-space spin polarization. The authors report that when the symmetry is reduced to a C2z rotation (together with Mz), the three diagonal nodal planes of pristine CrSb disappear and are replaced by what they call fragmented nodal curves (FNCs) — band-specific curves in the Brillouin zone along which opposite-spin sublattice bands become degenerate. They support this observation with DFT constant-energy surfaces for a model structure (MS-V), for uniaxially strained CrSb, and for RbMnPO4, and they show that the same symmetry lowering allows finite anomalous Hall conductivity for both in-plane and out-of-plane Néel-vector orientations.

Significance. If the symmetry analysis were correct, FNCs would be a genuinely new type of nodal feature in altermagnets, distinct from nodal planes and nodal axes, and the proposal that such curves can be engineered by strain or doping would be of practical interest. The paper also contains an independent, non-fitted symmetry analysis (spin-space-group based), a separate validation compound (RbMnPO4), and standard AHC calculations; these are positive features. However, the central derivation of the FNCs is not sound as written, and the paper's main conceptual novelty therefore rests on a claim that is not established. The AHC results may be correct regardless of the FNC interpretation, but the title and abstract hinge on the FNC mechanism.

major comments (3)
  1. [Section V, Eq. (10) and following paragraph] Equation (10) states ε(kx,ky,kz,σ)=ε(-kx,-ky,kz,-σ)=ε(kx,ky,-kz,-σ). This is a relation between opposite-spin eigenvalues at different momenta, not a same-k degeneracy. For a generic (kx,ky) with kz≠0, the operation C2z maps k to (-kx,-ky,kz), so it does not leave the momentum invariant. The text then claims that 'the former implies that two spin-opposite bands have to intersect each other at a given point (nodal point) in the kx-ky plane.' This implication does not follow from Eq. (10) alone. To force a crossing at the same k, one would need an additional symmetry that leaves k invariant or a separate topological/band-representation argument. As written, the proof does not establish that the plotted FNCs are symmetry-enforced; they may be accidental crossings. This is load-bearing for the paper's central claim.
  2. [Section V and Figures 5-7] The FNCs are identified visually from constant-energy surfaces (e.g., Fig. 5 lower panel, Fig. 6, Fig. 7(f,g)). The text states that as kz varies continuously, nodal points form a nodal curve, but no direct calculation of the spin-resolved band gap or a definition of the degeneracy locus is provided. A constant-energy contour intersection at selected energies is not by itself a proof of a continuous 1D nodal curve in the full 3D Brillouin zone. To make the claim quantitative, the authors should compute the direct gap between the opposite-spin pairs, identify the zero-gap locus in 3D, and confirm its dimension and band-specificity. Without this, the 'fragmented nodal curves' remain a visual feature of constant-energy plots.
  3. [Section V and abstract] If, after the above is fixed, no symmetry-enforcing mechanism can be found, the paper should explicitly reframe the FNCs as accidental crossings that are compatible with, but not required by, the C2z and [E||TR,L] symmetries. The current abstract and Section V assert a 'discovery' of symmetry-driven FNC formation. A reframing would substantially lower the novelty claim but would make the paper internally consistent. The AHC results and the comparative study with RbMnPO4 can stand independently of the symmetry-enforcement issue.
minor comments (5)
  1. [Section V] The text says 'Eqs. (6) - (8)' when referring to Eqs. (7)-(9); please correct the equation numbering in the sentence above Eq. (10).
  2. [Section V, Eq. (7)] The notation [C2||M2z] is inconsistent with the earlier notation Mz used throughout the paper (e.g., in Section II and Fig. 1). Define whether M2z means Mz or a different mirror operation.
  3. [Abstract and Section V] The phrase 'when in an altermagnetic material when the symmetry is restricted' is grammatically awkward and should be revised.
  4. [Section V] The term 'random FNCs' is imprecise; the curves appear to be band-specific but not random. Clarify what is meant by 'random'.
  5. [Section V.A] For strained CrSb, the spin-space group and magnetic space group are not explicitly listed, although the text states that C2z and Mz are retained. Providing the exact SSG/MSG for the strained structure would help the reader connect the AHC analysis in Table II to the strained system.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: FNCs are derived from independent DFT calculations and an external spin-space-group formalism, not from fitted inputs or a self-citation chain.

full rationale

The central claim—formation of fragmented nodal curves when symmetry is reduced to C2z—is supported by DFT electronic-structure calculations and by a symmetry analysis framed in the independently established spin-space-group formalism (Refs. [35]–[38]) plus the non-relativistic [E||TR,L] operation. No parameter is fitted to the FNCs, and the constant-energy surfaces that display the FNCs are not used as inputs to the symmetry relations in Section V. The claim is also checked against an independently known compound, RbMnPO4, and against strained CrSb, both computed from first principles. The paper's self-citations, [16] and [17], are used for background on chemical bonding and quasi-altermagnetism, but they are not load-bearing for the FNC derivation; even if removed, the DFT results and the cited external spin-space-group references would remain. The symmetry argument in Section V does contain a logical gap—Eq. (10) relates opposite-spin eigenvalues at opposite momenta, and a same-k degeneracy is not forced at a generic point of the kx-ky plane—but this is a correctness or derivation concern, not circularity: the conclusion does not reduce by construction to the equations' inputs, nor is any fitted parameter being relabeled as a prediction. Accordingly, no circular step meets the evidentiary standard required here.

Assumptions & free parameters 2 free parameters · 5 assumptions · 2 invented entities

The central claim rests on the non-relativistic spin-space-group treatment of hypothetical or strained structures. The only fitted/hand-chosen numerical inputs are Ueff and the strain value; the more serious burden is the unjustified use of [E||TR,L] in non-centrosymmetric settings and the assumption that the chosen magnetic order is stable.

free parameters (2)
  • Ueff (Hubbard U) for RbMnPO4 = 3 eV
    Chosen via Dudarev DFT+U; affects the position of nodal features relative to the Fermi level in RbMnPO4.
  • Uniaxial strain magnitude on CrSb = 5% along a-axis
    Chosen to break C6z symmetry; the text states that increasing strain further fragments the curves, so the specific value is a hand-selected parameter.
assumptions (5)
  • domain assumption The non-relativistic limit (neglect of spin-orbit coupling) is valid for the symmetry analysis and for the non-AHC band structure.
    Sections IV and V use [E||TR,L] and spin-space-group relations valid only in the non-relativistic limit.
  • domain assumption The spin-space-group operations [C2||Mz] and [C2||C2z] are exact symmetries of MS-V and strained CrSb, with the listed MSGs/SSGs in Table I being correct.
    The FNC derivation assumes these symmetries survive relaxation and are not broken by further magnetic or structural distortions.
  • ad hoc to paper The operation [E||TR,L] acts as an inversion-like symmetry giving ε(k,σ)=ε(-k,σ) even in the non-centrosymmetric MS-V and strained CrSb structures.
    This is required for Eq. (9), but the paper states MS-V lacks an inversion center; the symmetry status of [E||TR,L] is not justified for these structures.
  • domain assumption The assumed A-type antiferromagnetic order with Néel vector along z (or at 30° to a) is the relevant magnetic configuration in each model structure.
    No total-energy comparison of different magnetic orders is presented; the AHC and FNC results depend on this ordering.
  • domain assumption DFT-GGA/PBE (plus U for RbMnPO4) accurately describes the electronic structure, magnetic moments, and band degeneracies of CrSb and its model structures.
    All central data come from DFT; exchange-correlation errors could affect band crossings and AHC magnitudes.
invented entities (2)
  • Fragmented nodal curves (FNCs)
    purpose: Label for the claimed band-specific degeneracy curves that form when C6z symmetry is lowered to C2z.
    No experimental prediction or external benchmark is provided to distinguish FNCs from accidental band crossings; they are identified from the paper's own constant-energy surfaces.
  • Hypothetical model structures MS-I to MS-V (Cr2Sb, Cr2Sb3, CrSb2)
    purpose: Designed vacancy and interstitial Sb configurations used to break the six-fold rotational symmetry of CrSb.
    These are not known to be synthesizable and no thermodynamic stability analysis is presented.

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Pith. "Pith review of Effect of symmetry breaking on altermagnetism in CrSb and Formation of fragmented nodal curves." pith.science (2026). https://pith.science/paper/Q5FPS6NF

@misc{pith2026260221135,
  author       = {Pith},
  title        = {Pith review of: Effect of symmetry breaking on altermagnetism in CrSb and Formation of fragmented nodal curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q5FPS6NF}},
  note         = {Machine review of arXiv:2602.21135}
}
abstract

Phenomena concerning altermagnets have opened up a window for unconventional analysis of the momentum space spin polarization (MSSP) of antiferromagnetic materials. Taking the example of one of the widely investigated altermagnets, CrSb, we explore the underlying mechanisms leading to the formation or breaking of altermagnetism. With the aid of DFT calculation and symmetry analysis, we study the behavior of MSSP in the altermagnetic bands of pristine CrSb, along with a few model structures designed from the pristine one by hypothetical vacancy engineering and interstitial doping. We show that the six-fold rotational symmetry of the pristine CrSb can be reduced to a two-fold rotational symmetry via vacancy and doping engineering. We discover the formation of fragmented nodal curves (FNCs) across the Brillouin zone when in an altermagnetic material when the symmetry is restricted to two-fold rotation. Unlike the typical nodal planes and axes, the location of the FNCs in the momentum space is found to be band-specific. The formation of FNCs is further validated by introducing uniaxial strain to CrSb and by examining the band structure of RbMnPO$_4$, as they both exhibit a two-fold rotational symmetry responsible for altermagnetism. We observe that, unlike the pristine case, these FNCs have the potential to manifest anomalous Hall conductivities (AHC), while the N\'eel vector orients along both in-plane and out-of-plane directions. This flexibility of the AHC will pave the way for the application of altermagnets in the futuristic quantum devices.

Figures

Figures reproduced from arXiv: 2602.21135 by the authors.

Figure 1
Figure 1. The effect of vacancy engineering and interstitial [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Schematic illustration of various altermagnetic [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Spin density contours of Cr atoms and their symme [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: Formation of fragmented nodal curves in MS-V. [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: Altermagnetism in strained CrSb and the formation [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 8
Figure 8. Figure 8: Anomalous Hall conductivity in pristine CrSb and [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

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