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REVIEW 4 major objections 6 minor 2 cited by

The odd-parity altermagnetism: A spin group study

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Collinear antiferromagnets can produce odd-parity spin splitting without spin-orbit coupling when three symmetry conditions are met.

desk verdict A useful spin-group framework for odd-parity altermagnetism, but the abstract's [C2||M] criterion is not sufficient as stated and needs amending. read the letter →

arxiv 2507.09906 v5 pith:MPMAKFRN submitted 2025-07-14 cond-mat.str-el

Minghuan Zeng , Zheng Qin , Ling Qin , Shiping Feng , Lin Wu , Dong-Hui Xu , Rui Wang This is my paper · ORCID
classification cond-mat.str-el
keywords altermagnetismodd-parityspinsplittinggroupcollinearantiferromagnetHaldane-Hubbardmodelnonmagnetictime-reversalsymmetrysublatticecurrentsp-wavemagnet
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 tries to establish exactly when a collinear antiferromagnet can show odd-parity spin splitting—band splitting that changes sign under inversion in momentum space—without any spin-orbit coupling. Using spin-group analysis, the authors derive three sufficient conditions: time-reversal symmetry in real space must be broken, the magnetic order must be long-range, collinear, and compensated, and opposite-spin sublattices must be connected by a 180-degree spin rotation combined with either inversion or a mirror reflection. These conditions produce what they call odd-parity altermagnetism, with high-harmonic ($l\ge3$) or $p$-wave ($l=1$) spin splitting. They then show that the Haldane-Hubbard model with opposite sublattice currents satisfies the conditions and exhibits the predicted splitting in its collinear altermagnetic ground state. If correct, this gives a route to Rashba-like spin-momentum locking in materials without heavy elements or spin-orbit coupling.

What carries the argument

The carrying object is the spin group, a symmetry group whose elements $[R_i||R_j v]$ act independently on spin space ($R_i$) and real space ($R_j$ followed by translation $v$). For collinear magnets the spin-only subgroup is $\{C_\infty, \bar{C}_2\}$. The paper classifies nontrivial spin groups by how the coset decomposition of the crystallographic Laue group $G = H + AH$ is paired with the spin rotation $C_2$; in even-parity altermagnets $A$ cannot be inversion, but in odd-parity altermagnets $A$ is inversion $\bar{E}$ or mirror $M$. The Haldane-Hubbard model supplies the microscopic realization: opposite sublattice currents from the complex next-nearest-neighbor Haldane hopping break nonmagnetic TRS, and the symmetry $[C_2||\bar{E}]$ connects opposite-spin sublattices. The cluster slave-spin computation then yields the staggered magnetization and the four-regime phase diagram.

What would settle it

Measure spin-resolved photoemission on a collinear compensated antiferromagnet with sublattice currents and $[C_2||\bar{E}]$ symmetry: if the bands satisfy $E_{k\sigma}=E_{-k\sigma}$ rather than $E_{k\sigma}=E_{-k,-\sigma}$, the predicted odd-parity splitting is absent.

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Extended reading notes

Core claim

The central claim is that odd-parity altermagnetism is a distinct class of collinear compensated magnetism, not a variant of conventional antiferromagnetism. The paper derives that a collinear antiferromagnet exhibits odd-parity spin splitting when (i) nonmagnetic time-reversal symmetry—time reversal acting only on real space—is broken; (ii) long-range collinear compensated magnetic order is present; and (iii) the symmetry operation $[C_2||\bar{E}]$ (a 180-degree spin rotation around the axis perpendicular to the spins combined with spatial inversion) or $[C_2||M]$ (the same spin rotation combined with a mirror reflection) connects opposite-spin sublattices. The first criterion removes the combined symmetry $[\bar{C}_2||T]$ that forces even-parity bands, while the third supplies the odd-parity connection. The inversion-type operation yields high-order harmonics ($l\ge3$) and the mirror-type yields $p$-wave ($l=1$) spin splitting. The Haldane-Hubbard model with opposite sublattice currents from Haldane hopping is identified as a concrete realization: the currents break nonmagnetic TRS, and the reversed currents on opposite-spin sublattices enforce $[C_2||\bar{E}]$, producing the predicted odd-parity splitting in the collinear altermagnetic ground state.

Load-bearing premise

The load-bearing premise is that spin and real-space degrees of freedom decouple completely—spin-orbit coupling is absent—so the spin group factors into a direct product and the symmetry $[C_2||\bar{E}]$ can act independently on spins and lattice; if SOC is significant, the derived conditions are no longer sufficient.

Editorial extensions

If this is right

  • Odd-parity spin splitting can occur in collinear compensated magnets without spin-orbit coupling, so Rashba-like spin-momentum locking is not restricted to noncollinear or coplanar magnets.
  • The three criteria give a concrete materials search: look for collinear antiferromagnets with broken nonmagnetic TRS (via sublattice currents, light, or orbital order) and $[C_2||\bar{E}]$ or $[C_2||M]$ symmetry.
  • In the Haldane-Hubbard model, the odd-parity altermagnetic phase coexists with a Chern insulator ($C=2$) at intermediate interaction, so the phase diagram contains four regimes: AFMI, CI, odd-parity ALM CI, and odd-parity ALMI.
  • Quasiparticle scattering interference can identify the phase experimentally, because up- and down-spin constant-energy contours are centered on the two inequivalent Dirac points and do not intersect.

Reading between the lines

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

  • If the criteria are sufficient, they should transfer to other bipartite lattices with sublattice currents, so one can systematically scan two-dimensional magnets for odd-parity ALM by checking the three symmetries rather than by brute-force band calculation.
  • The analysis assumes negligible spin-orbit coupling; in real compounds with weak but finite SOC, the odd-parity splitting should acquire corrections and the exact symmetry labels may soften—a testable prediction for first-principles calculations.
  • The same spin-group logic could classify odd-parity spin splitting in three dimensions, where the rotation $C_{2z}$ is not automatically inversion, possibly yielding new harmonic orders beyond $l=1$ and $l=3$.
  • The phase diagram's ALM Chern insulator with $C=2$ suggests that odd-parity altermagnets could host chiral edge states, which the authors do not explicitly analyze.
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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 / 6 minor

Summary. The manuscript uses spin-group theory to propose sufficient conditions for odd-parity altermagnetism (ALM) in collinear compensated magnets: (i) broken nonmagnetic time-reversal symmetry, (ii) long-range collinear compensated magnetism, and (iii) a symmetry of the type [C2||Ebar] or [C2||M] connecting opposite-spin sublattices. The authors argue that these criteria produce the odd-parity spin-splitting relation E_{k,sigma}=E_{-k,-sigma}, and they illustrate the proposal with the Haldane-Hubbard model, where opposite sublattice currents break nonmagnetic TRS. Using the cluster slave-spin method, they present constant-energy contours, a phase diagram in the (lambda, U) plane, and the momentum dependence of spin-resolved gaps, identifying an odd-parity ALM Chern insulator and an odd-parity ALM insulator.

Significance. If the proposed criteria are correct and stated with the appropriate qualifications, the paper would provide a compact symmetry-based route to odd-parity spin splitting in collinear magnets without spin-orbit coupling, a topic of current interest. The formal derivation is a genuine strength: it is built from the cited spin-group formalism with no fitted parameters, and it shows clearly how breaking nonmagnetic TRS removes [Cbar2||T] while retaining the collinear spin-only group. The Haldane-Hubbard illustration is also useful as a concrete model, although it is an example chosen to satisfy the criteria rather than a test that could falsify them. The main weakness is that one of the central criteria, as stated in the abstract, is overbroad: [C2||M] alone does not enforce the odd-parity relation E_{k,sigma}=E_{-k,-sigma} except on special momentum loci or when combined with another operation. In addition, the no-spin-orbit-coupling scope is not explicitly stated, and the numerical phase diagram lacks error estimates and sufficient reproducibility details.

major comments (4)
  1. [Abstract and Symmetry analysis (paragraphs around Eq. (1) and the discussion of [C2||M])] The abstract's criterion (iii) is not sufficient as stated for the odd-parity relation E_{k,sigma}=E_{-k,-sigma} defined after Eq. (1). A symmetry [C2||M] with M a mirror plane enforces only E_{k,sigma}=E_{Mk,-sigma}; it yields E_{k,sigma}=E_{-k,-sigma} only on the mirror-invariant locus or when an additional operation, such as [C2||Ebar_{BL}tau], relates the remaining momentum components. The text itself indicates this: near Fig. 2(a) the p-wave dalternagnet is described by [C2||Ebar_{BL}tau], not by [C2||M] alone. The criterion should therefore be reformulated as '[C2||Ebar], or [C2||M] together with an additional operation that connects -k to Mk', or the definition of odd-parity ALM should be generalized to include spin splitting antisymmetric under a mirror operation rather than only under inversion. This is a load-bearing point because the central claim in the abstract and conclusion is built on this criterion.
  2. [Symmetry analysis, paragraph after Eq. (2)] The spin-group decomposition into a direct product of a spin-only group and a nontrivial spin group assumes that spin and real-space degrees of freedom are decoupled, i.e., that spin-orbit coupling is absent. This no-SOC scope is not stated in the abstract or conclusion, and the criteria are presented as general conditions for odd-parity ALM. Since in real materials with SOC the operation [C2||M] acting independently on spin and lattice is not a symmetry, the sufficient conditions may fail or need modification. The manuscript should explicitly state that all derivations and criteria apply in the nonrelativistic limit (negligible spin-orbit coupling) and discuss the implications when this limit is violated.
  3. [Haldane-Hubbard model, Eq. (8) and Fig. 3(c)] The numerical demonstration of odd-parity ALM relies on the cluster slave-spin approximation, but the paper gives no error bars, no cluster-size convergence analysis, and no explicit statement of the cluster size used for the phase diagram. The phase diagram in Fig. 3(c) is therefore not reproducible from the text alone, particularly because the text states that U_AFM ≈ 3.1 in the cluster slave-spin method while the large-scale QMC value is ≈ 3.8, yet no systematic extrapolation is shown. Since the phase boundaries are a supporting illustration rather than the core symmetry claim, this issue does not invalidate the central derivation, but it should be addressed by providing the missing numerical details and a discussion of the approximation's reliability.
  4. [Haldane-Hubbard model, discussion of the square-lattice system] The claim that the square-lattice system exhibits a time-reversal-symmetric energy band, E_{k,sigma}=E_{-k,-sigma}, is attributed to the symmetry [C2||Ebar_{BL}tau], but the action of the translation tau in this symmetry is not explained. The text states that tau is the minimal vector connecting opposite-spin sublattices, but it does not specify how the symmetry acts on the Hamiltonian, which is essential to verify that the combination actually produces the odd-parity relation. This needs a concise clarification in the main text or a reference to a specific equation in the supplemental material.
minor comments (6)
  1. [Throughout] There are several typographical errors, including 'Bellow' for 'Below' in the Haldane-Hubbard section and 'in spit of' for 'in spite of' in the symmetry analysis.
  2. [Eq. (1) and surrounding text] The notation E_{k,sigma}=E_{-k+(-)sigma} is ambiguous; it should be written clearly as E_{k,sigma}=E_{-k,-sigma} for odd parity and E_{k,sigma}=E_{-k,sigma} for even parity.
  3. [Symmetry analysis, paragraph defining nonmagnetic TRS] The phrase 'nonmagnetic time reversal symmetry' and its identification with 'real-space TRS' could be defined more explicitly. The connection between the spin-only symmetry [Cbar2||E] and the absence of [Cbar2||T] is central to the argument, so it would help to state precisely that the nonmagnetic TRS operation acts only on real space and not on spin.
  4. [Fig. 3(c)] The phase diagram is described in the text as having regions of different colors, but the figure is not reproduced in color in the text, and the legend is not described in sufficient detail. Please ensure the figure is legible in grayscale or add a clear description of the phase boundaries.
  5. [Reference list, Ref. [34]] The author list of Ref. [34] appears garbled ('S.-W. Gedik, Nuh Cheong' and 'B. Ilyas, E. Erge¸cen' in particular); please check and correct the reference.
  6. [Abstract] The phrase 'standard odd-parity ALM' is not defined; it would be clearer to say 'the usual odd-parity spin splitting E_{k,sigma}=E_{-k,-sigma}' or to define the term at first use.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the symmetry criteria are derived from the spin-group action, and the only self-citation (cluster slave spin method) is not load-bearing.

full rationale

The symmetry analysis is self-contained. Equation (2) defines the spin-group action, and Equation (1) is obtained by applying powers of [C2||C_{nz}^1] to E_{kσ}; the criteria (i)-(iii) then follow from requiring the absence of [Cbar2||T] and the presence of [C2||Ebar] or [C2||M] in the nontrivial spin group. This is a derivation from the group action, not a restatement of a fitted quantity: no parameter is fitted to the claim it explains, and the Haldane-Hubbard calculation is explicitly chosen as an illustrative realization rather than a test with a circular coin-flip structure. The only self-citation is to the cluster slave spin method [45-47]; the method originates in the independent Ref. [45] (Lee and Lee), so the same-group applications [46,47] are not load-bearing. The skeptic's objection that [C2||M] alone enforces only E_{kσ}=E_{Mk,-σ} rather than E_{kσ}=E_{-k,-σ} is a correctness concern about the stated sufficient condition, not a circularity concern; the paper itself uses [C2||Ebar_BL tau] to obtain the odd-parity relation in both examples. No step reduces to its input by construction.

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

The central symmetry claim rests on the standard spin-group classification from the cited literature plus three physics assumptions: collinear compensated order, no spin-orbit coupling, and the specific effect of breaking nonmagnetic TRS. No parameters are fitted to data; self-consistent mean-field quantities are internal to the method. No new physical entities are postulated.

assumptions (5)
  • standard math Spin groups of collinear magnets can be decomposed as r_s x R_s with spin-only group {Cinfinity, Cbar2}, using the classification of nontrivial spin groups into types I, II, and III.
    Adopted from Litvin and Opechowski and Smejkal et al.; the paper uses these results without reproving them.
  • domain assumption The magnetic state is collinear, long-range, and compensated, so the spin-only group remains {Cinfinity, Cbar2}.
    The sufficient conditions are framed for this class of magnets; noncollinear or uncompensated states are outside the derivation.
  • domain assumption Spin and real-space transformations are independent, i.e., no spin-orbit coupling is present.
    The spin group element [Ri||Rjv] acts separately on spin and spatial coordinates in Eq. 2; this decoupling fails with spin-orbit coupling, a scope the paper does not state.
  • ad hoc to paper Breaking nonmagnetic TRS removes [Cbar2||T] but leaves [Cbar2||E] and the collinear spin-only group intact.
    The paper assumes the only effect of sublattice currents or light on the spin group is to delete the [Cbar2||T] element; this is plausible but not proven.
  • domain assumption The cluster slave-spin method with the given cluster size approximates the ground state well enough to locate phase boundaries.
    The method is cited from refs 45-47; the paper notes accuracy improves with cluster size.

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Pith. "Pith review of The odd-parity altermagnetism: A spin group study." pith.science (2026). https://pith.science/paper/MPMAKFRN

@misc{pith2026250709906,
  author       = {Pith},
  title        = {Pith review of: The odd-parity altermagnetism: A spin group study},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MPMAKFRN}},
  note         = {Machine review of arXiv:2507.09906}
}
abstract

Following recent intensive studies on altermagnetism(ALM) characterized by non-relativistic even-parity spin splitting, realizing unconventional odd-parity magnetism has also attracted increasing interest. Here, using symmetry arguments based on spin-group analyses, we elucidate sufficient conditions for the emergence of odd-parity spin splitting in collinear antiferromagnetic systems, which is further established as the standard odd-parity ALM. It is derived that the odd-parity ALM arises from the following criteria: (i)the breaking nonmagnetic time reversal symmetry(TRS), i.e., the breaking real-space TRS; (ii)the long-range collinear compensated magnetism; (iii)the symmetry $[C_{2}||\bar{E}]$ or $[C_{2}||M]$ connecting opposite-spin sublattices, where $C_{2}$, $\bar{E}$, and $M$ respectively represent a $180^{\circ}$ rotation around the axis perpendicular to spins, the inversion, and the mirror reflection separating opposite-spin sublattices, directly reflecting the high-order harmonic($l\ge3$) and the $p$-wave($l=1$) odd-parity ALM, respectively. Moreover, we utilize the well-known Haldane-Hubbard model to identify odd-parity spin splitting in the collinear ALM ground state, where (i)the nonmagnetic TRS is broken by opposite sublattice currents coming from the Haldane hopping; (ii)the symmetry $[C_{2}||\bar{E}]$ is ensured because the currents flowing on opposite-spin sublattices are reversed.

Figures

Figures reproduced from arXiv: 2507.09906 by the authors.

Figure 1
Figure 1. FIG. 1. (Color online) The schematic cartoon of the spin [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (Color online) The schematic illustration of the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (Color online) The electron spectrum function [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

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  1. Stripe-Order Altermagnetism: Nematic Spin Splitting beyond the $l$-Wave Classification

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  2. Nonrelativistic Spin-Orbit-Coupling Effects in Odd-Parity Coplanar Magnets

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    Bilayer odd-parity coplanar magnets constructed from altermagnets realize tunable nonrelativistic SOC spin textures equivalent to relativistic counterparts.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.