Symmetry-breaking induced transition among net-zero-magnetization magnets

Guangzhao Wang; San-Dong Guo; Xiao-Shu Guo

REVIEW 2 major objections 6 minor 49 references

Symmetry-breaking induced transition among net-zero-magnetization magnets

T0 review · 2 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Symmetry breaking in a single monolayer family produces the full sequence of net-zero-magnetization magnets, from PT-antiferromagnet through altermagnet to fully-compensated ferrimagnet, with the expected spin-splitting symmetries.

desk verdict A clean symmetry-based demonstration of PT-AFM to altermagnet to fully-compensated ferrimagnet transitions in one 2D family, but the material realization rests on a single Hubbard U value with no sensitivity test. read the letter →

arxiv 2501.06829 v1 pith:TOOKJU5L submitted 2025-01-12 cond-mat.mtrl-sci

San-Dong Guo , Xiao-Shu Guo , Guangzhao Wang This is my paper · ORCID

classification cond-mat.mtrl-sci

keywords net-zero-magnetizationmagnetsaltermagnetismfully-compensatedferrimagnetPT-antiferromagnetJanusengineeringisovalentalloyingspinsplittinganomalousvalleyHalleffect

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 claims that the three kinds of net-zero-magnetization magnets — PT-antiferromagnets, altermagnets, and fully-compensated ferrimagnets — are connected by symmetry breaking, and that one can walk through the whole sequence in a single two-dimensional material family. Starting from a PT-antiferromagnetic CrC2S6 monolayer with both inversion and mirror symmetry, the authors show by density functional theory that replacing one sulfur layer with selenium (Janus engineering) breaks inversion while keeping the mirror, yielding an altermagnet with i-wave spin splitting, while substituting one chromium by molybdenum (isovalent alloying) breaks both symmetries, yielding fully-compensated ferrimagnets with s-wave splitting. The same altermagnetic splitting is reproduced by an external electric field on the parent compound, making the symmetry-breaking route reversible and voltage-tunable. If correct, this gives a clear material playground for comparing and exploiting the distinct spin-splitting symmetry classes and for anomalous valley Hall transport in zero-moment magnets.

What carries the argument

The organizing device is the symmetric-connection classification of the two oppositely polarized magnetic sublattices: [C2∥P] (inversion-connected, PT-antiferromagnet), [C2∥C/M] (rotation/mirror-connected, altermagnet), and [C2∥Null] (asymmetrically connected, fully-compensated ferrimagnet). The paper's specific strategy is to start from a PT-antiferromagnet that carries both P and a mirror (C/M) symmetry, so that breaking just one symmetry toggles the class: P-breaking with C/M intact gives momentum-dependent (d/g/i-wave) splitting, and breaking both gives s-wave splitting. In the CrC2S6 example the symmetry operations are concrete: parent P-31m with P and Mxy; Janus CrC2S3Se3 in P31m without P but with Mxy; alloyed CrMoC2S6/CrMoC2S3Se3 in P312/P3 without either. The machinery is what makes the transitions predictable and the spin-splitting symmetry diagnosable from the space group alone.

What would settle it

Compute the CrC2S6 family with a different Hubbard U (or a hybrid functional) and with noncollinear spin arrangements; if the AFM1 order ceases to be the ground state, or the band structures lose the i-wave/s-wave spin splitting, the paper's central claim is refuted. Experimentally, growing the monolayer and performing spin-resolved ARPES or magnetic circular dichroism on Cr2C2S3Se3 and CrMoC2S6 would directly test whether the splitting symmetries match the predictions.

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Extended reading notes

Core claim

Starting from a collinear PT-antiferromagnet whose opposite-spin sublattices are connected simultaneously by inversion P and a mirror Mxy, breaking only P (keeping Mxy) converts the material into an altermagnet with momentum-dependent, i-wave spin splitting; breaking both P and Mxy converts it into a fully-compensated ferrimagnet with global s-wave splitting. In the CrC2S6 monolayer, these two breakings are realized chemically: Janus substitution of one S layer by Se gives CrC2S3Se3 (altermagnet), and isovalent Cr-to-Mo substitution gives CrMoC2S6 and CrMoC2S3Se3 (fully-compensated ferrimagnets with out-of-plane easy axis and valley polarization of opposite sign at the two valleys). The parent remains a 2.31 eV bandgap semiconductor with strictly zero total moment, and the derived phases retain zero net moment while showing the predicted spin-splitting symmetries and, in the ferrimagnets, a large orbital-mismatch-driven splitting near the Fermi level and an anomalous valley Hall effect. An external electric field of 0.30 V/Å on the parent reproduces the altermagnetic splitting, with the splitting order reversing when the field reverses.

Load-bearing premise

The central assumption is that the density-functional calculations with the chosen Hubbard U correction (3 eV on chromium and molybdenum) and the four collinear spin arrangements tested actually find the true magnetic ground state; if a noncollinear ordering or a different U value changes which magnetic order wins, the predicted altermagnet and fully-compensated ferrimagnet phases in these specific compounds would not form as described.

Editorial extensions

If this is right

The same symmetry recipe — start from a PT-antiferromagnet with P plus rotation/mirror, then break P, or P and mirror — should generate altermagnets and fully-compensated ferrimagnets from other honeycomb PT-AFM parents, not only CrC2S6.
Reversing the external electric field reverses the order of spin-splitting in the altermagnetic phase, which the paper identifies as a handle for tuning spin currents in spintronic devices.
The fully-compensated ferrimagnets exhibit spontaneous valley polarization with opposite Berry curvature at K and -K, so shifting the Fermi level between valleys produces an anomalous valley Hall effect in a zero-moment magnet.
A single material family now allows direct comparison of properties that otherwise live in unrelated compounds: PT-AFM spin degeneracy, altermagnetic d/i-wave splitting, and ferrimagnetic s-wave splitting under the same lattice and magnetic ordering.

Reading between the lines

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

The symmetry criterion is parameter-free: any collinear PT-antiferromagnet with the same P+C/M group constraints should show the same transitions, so the specific Hubbard U value matters for quantitative band gaps and moments but not for the existence of the splitting symmetries.
If the altermagnetic CrC2S3Se3 and ferrimagnetic CrMoC2S6 monolayers can be exfoliated or grown, spin-resolved angle-resolved photoemission should directly image the contrasting i-wave vs s-wave spin splitting, and a Hall-bar measurement should detect the predicted transverse voltage.
The external-field route on the parent suggests a reversible, non-volatile two-state switch between spin-degenerate and spin-split regimes, and combining Janus and alloying could yield lateral heterostructures with a built-in interface between altermagnet and ferrimagnet.
Because only four collinear magnetic configurations were checked, the authors' identification of AFM1 as the ground state could be tested by noncollinear spin-spiral calculations; if a noncollinear order wins, the predicted net-zero phases in these exact compounds would not form, though the symmetry rationale would survive.

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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

Net-zero-magnetization magnets are an active area, and this paper does a useful thing: it takes one parent PT-AFM (CrC2S6) and shows, with symmetry arguments plus DFT+U, how Janus engineering and isovalent alloying produce an altermagnet (Cr2C2S3Se3) and two fully-compensated ferrimagnets (CrMoC2S6, CrMoC2S3Se3). The specific compounds are new predictions, and the band-structure symmetries from the text match what the symmetry classification expects: i-wave splitting in the Janus case, s-wave splitting in the alloyed cases. The phonon, AIMD, and elastic-constant checks are appropriate. The Berry curvature and valley splitting numbers (7.4 and 3.3 meV) are reasonable and consistent with the MAE values.

The soft spot, as you flagged, is the single Hubbard U. All four ground states come from U=3.00 eV on both Cr and Mo, taken from a prior paper. There is no U sweep, and the energy differences in Figure 3(b) are reported without absolute scale, so we can't see how close the competing orders are. It's entirely possible that a different U (or a noncollinear configuration) changes the ground state, which would undo the specific material claims. This is not a fatal flaw in the symmetry story — the point-group argument is independent of U — but it is load-bearing for the material realization. A referee should ask for a U sensitivity test (2–4 eV) and at least a check of noncollinear orders before publication.

Two smaller points: the external-field calculation is described only briefly, so I'd like to see how the field is applied in the slab and whether the 0.30 V/Å result is a proper response or just an order-of-magnitude estimate. And the paper leans heavily on the authors' own prior definitions of fully-compensated ferrimagnets; that's not a problem in itself, but it makes the paper less self-contained for someone new to the terminology.

Overall, I think the paper is a solid, modest contribution. It doesn't redefine the field, but it gives a clean concrete example of transitions among three classes of net-zero-magnetization magnets, with plausible (not proven) material candidates. I would send it to review, with the U-sensitivity request as a condition. For a reading group, it's a reasonable way to get into the altermagnet/fully-compensated ferrimagnet discussion, but I wouldn't call it essential.

Referee Report

2 major / 6 minor

Summary. The paper proposes a symmetry-based framework for transitions among three classes of net-zero-magnetization magnets—PT-antiferromagnets, altermagnets, and fully-compensated ferrimagnets—and claims first-principles verification in a single 2D materials family: PT-antiferromagnetic Cr2C2S6, altermagnetic Cr2C2S3Se3 (via Janus engineering), and fully-compensated ferrimagnetic CrMoC2S6 and CrMoC2S3Se3 (via isovalent alloying). DFT+U calculations are used to establish the AFM1 ground state, stability (phonons, AIMD, elastic constants), spin-splitting symmetries, and Berry-curvature-based anomalous valley Hall effect. The central conceptual message is that breaking P (and C/M) symmetries of a parent PT-antiferromagnet transforms the spin-splitting symmetry from degenerate to i-wave (altermagnet) or s-wave (fully-compensated ferrimagnet).

Significance. If the material realization is robust, the paper provides a clear and conceptually useful demonstration of how symmetry-breaking operations connect three important classes of net-zero-magnetization magnets within one 2D family. The proposed Cr2C2S6-derived compounds would constitute a promising platform for exploring momentum-dependent versus global spin splitting, and for potential spintronic and valleytronic applications. The paper includes several commendable elements: symmetry arguments tied to magnetic space groups, explicit collinear magnetic configuration searches, phonon/AIMD/elastic stability checks, and Berry-curvature illustrations of the anomalous valley Hall effect. However, the material-specific predictions rest on a single DFT+U parametrization, and the paper would be significantly strengthened by demonstrating robustness of the magnetic ground state and spin-splitting with respect to the Hubbard U and to noncollinear magnetic order.

major comments (2)

[Computational detail and Figure 3] The central material-realization claim depends on AFM1 being the magnetic ground state for all four compounds, yet this is established only at a single PBE+U value, U=3.00 eV for both Cr and Mo, taken from Ref. [40]. No U-dependence is reported. Since U directly controls the relative energies of FM, AFM2, AFM3, and AFM1, and can also alter the d-orbital ordering near the gap, the paper should provide a U-sweep (for example, U = 2.0 to 4.0 eV) for the energy differences in Figure 3(b), and ideally verify that the AFM1 ground state and the reported spin-splitting patterns survive across this range. In addition, only four collinear configurations are considered; noncollinear spin arrangements are not tested, leaving an unexamined possibility for an even lower-energy state. These checks are load-bearing because the paper's mode of validation is explicitly first-principles verification in these specific compounds.
[Material realization, Figure 4(e)] The external-electric-field route is used to argue that a PT-antiferromagnet can be turned into an altermagnet, with Cr2C2S6 at E = 0.30 V/Å shown as the example. The computational implementation of the field is not described: the paper does not state whether a sawtooth potential, dipole-correction scheme, or other method is used in the periodic slab, nor whether the field strength is large enough to be realistic while remaining below dielectric breakdown. Because the external field is one of the three proposed symmetry-breaking methods in the abstract and introduction, these details matter for reproducibility and for assessing whether the reported spin-splitting is a genuine field effect rather than a numerical artifact. Please specify the implementation and add a field-strength dependence (e.g., E = 0.1–0.5 V/Å) for the band structure.

minor comments (6)

[Abstract and main text] The chemical formulas are inconsistent: the abstract uses CrC2S6, CrC2S3Se3, CrMoC2S6, and CrMoC2S3Se3, while the main text uses Cr2C2S6, Cr2C2S3Se3, CrMoC2S6, and CrMoC2S3Se3. Please unify the notation, as the two-Cr formula is the one supported by the structure and stoichiometry.
[Introduction] There is a typo in 'anomalous Halll/Nernst effect'; 'Halll' should be 'Hall'.
[Stability paragraph] The phrase 'confirming thier mechanical stabilities' contains a typo ('thier' should be 'their').
[Figure 1 caption] The symbols C2, C/M, and P are used in the caption but not defined there; define them (two-fold rotation in spin space, rotation/mirror symmetry in lattice space, inversion) so the figure is self-contained.
[Computational detail] The abbreviations 'V ASP' and 'PA W' contain spacing artifacts; they should read 'VASP' and 'PAW'.
[Figure 3(b)] The energy differences are described only as positive; providing numerical values or a table would make the ground-state assignment more quantitative and would strengthen the paper.

Circularity Check

0 steps flagged · score 2.0 of 10

No substantive circularity: the DFT material predictions are computed, not fitted, and the symmetry-breaking 'transitions' are a classification restatement rather than a derived result.

full rationale

The paper's load-bearing material claims are first-principles results: the AFM1 ground state, spin-splitting symmetries, zero total moments, band gaps, MAE, and Berry curvatures of Cr2C2S6, Cr2C2S3Se3, CrMoC2S6, and CrMoC2S3Se3 are obtained from DFT+U calculations with a fixed U=3.00 eV taken from an external reference. No parameter is fitted to the target spin-splitting or to the predicted transition, and no fitted quantity is renamed as a prediction. The general statement that breaking P while preserving C/M turns a PT-antiferromagnet into an altermagnet is a direct consequence of the symmetry classification adopted from refs. [5,6] and is used as an organizing framework rather than as evidence for the numerically computed electronic structures; this is definitional taxonomy, not circular derivation. Self-citations are frequent, especially for fully-compensated ferrimagnets (refs. [7,29-33]) and for the claim that CrMoC2S6 is a fully-compensated ferrimagnet, but the present paper independently recomputes the relevant bands, moments, MAE, and Berry curvatures for CrMoC2S6 and CrMoC2S3Se3, so those citations are not the load-bearing evidence. The absence of a U-sensitivity test and the restricted search over four collinear magnetic configurations are correctness risks, not circularity, and therefore do not increase the circularity score.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The predictions rest on DFT+U with a literature U value, a collinear magnetic order assumption, and a prior symmetry classification. No new particles, forces, or conserved quantities are introduced.

free parameters (2)

Hubbard U for Cr and Mo d orbitals = 3.00 eV
Chosen from Ref [40]; enters all PBE+U calculations and affects band gaps, spin-splitting, MAE, and valley splittings, but is not fitted in this paper.
External electric field strength E = 0.30 V/Å
Hand-chosen value used to demonstrate field-induced spin-splitting in Cr2C2S6; no sweep or threshold analysis is reported.

assumptions (4)

domain assumption Kohn-Sham DFT with GGA-PBE+U accurately describes the electronic and magnetic properties of these 2D magnets.
All quantitative results in Material realization and Figures 4 and 5 come from this.
domain assumption The four collinear magnetic orders (FM, AFM1, AFM2, AFM3) considered are sufficient to identify the magnetic ground state.
Figure 3 compares only these orders; noncollinear and spin-spiral states are not tested.
domain assumption The U value from prior work on CrMoA2S6 transfers to CrC2S6 and its Janus and alloy derivatives.
Computational detail cites Ref [40]; no U-sweep is reported.
standard math The classification of net-zero-magnetization magnets by connection operators [C2||P], [C2||C/M], and [C2||Null] is correct.
Used in the transition section to map symmetry breaking to altermagnet and fully-compensated ferrimagnet; based on refs [5,6].

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Cite this review

Pith. "Pith review of Symmetry-breaking induced transition among net-zero-magnetization magnets." pith.science (2026). https://pith.science/paper/TOOKJU5L

@misc{pith2026250106829,
  author       = {Pith},
  title        = {Pith review of: Symmetry-breaking induced transition among net-zero-magnetization magnets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TOOKJU5L}},
  note         = {Machine review of arXiv:2501.06829}
}

abstract

Net-zero-magnetization magnets have garnered intensive research attention due to their ultradense and ultrafast potential. In terms of the symmetric classification of connecting magnetic atoms with opposite spin polarization, the net-zero-magnetization magnets mainly include $PT$-antiferromagnet (the joint symmetry ($PT$) of space inversion symmetry ($P$) and time-reversal symmetry ($T$)), altermagnet and fully-compensated ferrimagnet. Studying transitions among net-zero-magnetization magnets is essentially the research on symmetry breaking, which can also clearly reveal the transformation of spin-splitting symmetry. Symmetry breaking can be achieved through methods such as Janus engineering, isovalent alloying, and external electric field. Here, we start from a parent $PT$-antiferromagnet that simultaneously possesses both $P$ and rotational/mirror symmetries to induce altermagnet and fully-compensated ferrimagnet. Based on first-principles calculations, the proposed transitions can be verified in $PT$-antiferromagnet $\mathrm{CrC_2S_6}$ monolayer. By Janus engineering and isovalent alloying, $\mathrm{CrC_2S_6}$ can change into altermagnetic $\mathrm{CrC_2S_3Se_3}$ and fully-compensated ferrimagnetic $\mathrm{CrMoC_2S_6}$. The $\mathrm{CrC_2S_3Se_3}$ can also become fully-compensated ferrimagnetic $\mathrm{CrMoC_2S_3Se_3}$ by isovalent alloying. Our work provides a clear and intuitive example to explain the transitions among net-zero-magnetization magnets, which can inspire more research on net-zero-magnetization magnets.

Figures

Figures reproduced from arXiv: 2501.06829 by the authors.

**Figure 2.** FIG. 2. (Color online) Transitions among crystal structures: the Cr [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3. (Color online) (a): four magnetic configurations, [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. (Color online) The spin-polarized energy band structures of Cr [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: FIG. 5. (Color online) The enlarged energy band struc [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

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