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Catalog of Altermagnetism in Magnetic Wallpaper/Space Groups and Nonsymmorphic Altermagnets

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Pith's one-line read Spinless time-reversal extends altermagnetism to every non-centrosymmetric crystal and, with nonsymmorphic symmetries, produces hourglass spin textures outside the six known wave types.

desk verdict Solid, usable catalog of all 17 wallpaper and 422 space groups that host conventional altermagnetism, with a clean nonsymmorphic hourglass extension that actually sits outside the six-wave taxonomy. read the letter →

arxiv 2607.10857 v1 pith:PUBPYCJJ submitted 2026-07-12 cond-mat.mtrl-sci cond-mat.str-el

Congcong Le , Fan Cui , Iao-Fai Io , Moritz Hirschmann , Xianxin Wu , Ching-Kai Chiu This is my paper
classification cond-mat.mtrl-scicond-mat.str-el
keywords altermagnetismmagneticwallpapergroupsspacespinlesstime-reversalnonsymmorphicsymmetryhourglassdispersionspin-momentumtexturespin-groupclassification
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

Altermagnets are collinear magnets with zero net magnetization that still split spin bands in momentum space. Earlier catalogs focused on centrosymmetric crystals and six wave types (d-, g-, i-wave in 2D and 3D). This paper shows that a single non-spatial symmetry—spinless time-reversal S—acts like inversion in momentum space, so the same spin-splitting patterns survive even when crystal inversion is absent. Applying two necessary conditions (a halving subgroup that guarantees compensated order, and the absence of inversion from the spin-flip coset so that PT is broken) yields a complete list: 17 magnetic wallpaper groups and 422 magnetic space groups. The paper maps the spin texture of every group in the primitive Brillouin zone and notes that nodal lines or planes can float free of high-symmetry planes. When S is further composed with a glide or screw, same-spin Kramers degeneracy appears on the zone boundary; compatibility with the usual opposite-spin nodes forces hourglass dispersions whose spin patterns lie outside the six established waves—for example an i-wave-like winding inside a tetragonal zone. The catalog therefore both enlarges the material search space and identifies a new mechanism that generates richer altermagnetic band structures.

What carries the argument

Spinless time-reversal S = U_π T, a non-spatial antiunitary that leaves each spin sector invariant while sending k → −k. It replaces inversion for Condition 5 and, when composed with a glide or screw, enforces same-spin Kramers degeneracy that forces hourglass band crossings.

What would settle it

Take any group listed as altermagnetic (e.g., P4′gm′ or Pna2₁ material Y₂Cu₂O₅) and measure its spin-resolved bands with finite SOC: if the predicted hourglass crossings or spin-sector partitions collapse or hybridize, the SOC-free catalog does not describe the real material.

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Extended reading notes

Core claim

The two necessary symmetry conditions identify exactly 17 altermagnetic wallpaper groups (12 centrosymmetric, 5 non-centrosymmetric) and 422 altermagnetic space groups (160 centrosymmetric, 262 non-centrosymmetric). Spinless time-reversal S makes non-centrosymmetric groups equivalent in momentum space to their centrosymmetric counterparts, while its composition with nonsymmorphic operations produces hourglass dispersions whose spin textures fall outside the six established wave types.

Load-bearing premise

The whole classification assumes zero spin-orbit coupling, so crystalline operations never mix the two spin blocks of the Hamiltonian.

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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

2 major / 6 minor

Summary. The manuscript constructs a complete symmetry catalog of conventional collinear altermagnetism for all 2D magnetic wallpaper groups and all 3D magnetic space groups. Using two necessary conditions—a index-2 halving subgroup that enforces compensated collinear order, and the requirement that spatial inversion (if present) not lie in the spin-flip coset, so that PT is broken—it identifies 17 altermagnetic wallpaper groups (12 centrosymmetric, 5 non-centrosymmetric) and 422 altermagnetic space groups (160 centrosymmetric, 262 non-centrosymmetric). A non-spatial spinless time-reversal symmetry S is used both to extend the catalog to non-centrosymmetric crystals (by acting as an effective inversion in momentum space) and, when composed with glide or screw operations, to enforce same-spin Kramers degeneracies that, together with opposite-spin altermagnetic nodes, produce hourglass dispersions and spin textures outside the six established d/g/i-wave types. Each group is assigned a wave type and a primitive-cell BZ spin map, with locked versus unlocked nodal lines/planes distinguished; the 422 count is cross-checked against an independent type-III enumeration and against prior literature.

Significance. If correct, the work supplies the definitive wallpaper- and space-group-level catalog of conventional collinear altermagnetism, including the previously under-treated non-centrosymmetric sector, and identifies a concrete nonsymmorphic mechanism (˜gS hourglass) that generates spin-splitting patterns beyond the six-wave taxonomy. The independent counting argument (674 type-III groups minus 252 with inversion in the coset equals 422) and agreement with Ref. [50] are strong internal checks. The generator-based BZ constructions, the effective four-band Hamiltonians near Γ and M for P4′gm′ (Eqs. 14–18), and the explicit hourglass criterion for 3D glides and screws are reproducible group-theoretic results that materials searches and model building can use directly. The SOC-free (spin-group) scope is stated clearly and matches the conventional definition of collinear altermagnetism.

major comments (2)
  1. Abstract and §7 emphasize that nonsymmorphic symmetries realize altermagnetic BZs beyond the six wave types, yet §4.2 and the conclusion leave a full classification of the spin-distribution BZs for the many 3D groups in Table 3 to future work. The 2D case (only P4′gm′) and two 3D prototypes (P4′bm′, P4′₂2₁2′) establish the mechanism, but the abstract’s claim of a catalog that “reveals” these new patterns overstates what is completed for 3D. Either provide at least a systematic assignment of same-spin manifold geometry (line vs plane) and hourglass connectivity for the Table 3 families, or narrow the abstract/conclusion language to “mechanism and candidate list, with full 3D BZ maps deferred.”
  2. §6 analyzes Y₂Cu₂O₅ (Pna2₁) in detail, but that material is the exceptional orthorhombic-glide case in which same-spin lines lie inside opposite-spin planes and fourfold degeneracy replaces hourglass crossings (explicitly noted in §4.2). The novel hourglass claim therefore lacks a single first-principles or tight-binding demonstration on a group that satisfies the hourglass criterion (e.g. P4′bm′ or P4′₂2₁2′ from Table 3). Adding one such calculation—or a clear statement that none of the MAGNDATA candidates in Table 3 currently realize the hourglass geometry—would make the materials section support rather than undercut the central nonsymmorphic advance.
minor comments (6)
  1. Fig. 1 caption and the main text use both “conventional-cell BZ” and “primitive-cell BZ”; a short sentence early in §3 stating which cell is used in Tables 2, 5, and 6 would reduce confusion.
  2. Table 2 marks P4′gm′ with an asterisk for the nonsymmorphic BZ, but the caption does not define the asterisk until the reader reaches §4; move the definition into the table caption.
  3. In §2.1 the identification S = U_π T and the statement that T and U_π “result in the same spin-flipping effect” are standard in the SOC-free limit, but a one-line reminder that this equivalence fails once SOC is restored would help non-specialist readers.
  4. Eqs. (8)–(10) and (25)–(27) use real spherical harmonics with different (ℓ,m) conventions in 2D versus 3D; a brief note that the 3D forms are chosen to match the nodal-plane counts of the six-wave taxonomy would clarify the correspondence.
  5. Several candidate materials are listed in Tables 5–6 with DFT deferred to the SM; the main text should at least cite the SM section numbers so that the claims are checkable without hunting.
  6. Typographical: “ani-wave-like” in the abstract should be “an i-wave-like”; “P4'gm'” vs “P4′gm′” notation is inconsistent across tables and figures.

Circularity Check

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No circularity: the 17/422 catalog and hourglass textures are a direct constructive enumeration from two stated symmetry conditions applied to the known magnetic groups, cross-checked independently.

full rationale

The paper's central results follow by applying two explicit selection rules (halving subgroup for compensated collinear order; inversion absent from the spin-flip coset so that PT is broken) to the finite, externally tabulated lists of type-III magnetic wallpaper and space groups. Sec. 2.5 counts 674 type-III MSGs, subtracts the 252 whose coset contains inversion, and obtains 422; the same number is recovered by the constructive point-group route of Secs. 3 and 5 and is stated to agree with the independent tabulation of Ref. [50]. The spinless time-reversal S is taken from the existing spin-group literature and used only as an effective inversion in momentum space; it is not defined in terms of the catalog it produces. Nonsymmorphic hourglass textures are derived from the compatibility of same-spin Kramers nodes of gS with the opposite-spin nodes of the altermagnetic order, again by direct construction (Sec. 4). No parameters are fitted to data and re-labeled as predictions, no uniqueness theorem is imported from the authors' own prior work, and no known empirical pattern is merely renamed. The SOC-free premise is an explicit scope limitation (Sec. 2.1), not a circular step. The derivation is therefore self-contained against external group tables and contains no circular reduction.

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

The paper is a pure symmetry classification. It imports standard crystallographic and magnetic-group facts, the conventional definition of collinear altermagnetism (no SOC, compensated order, identical spins at ±k), and the known spinless time-reversal operator S. No free parameters are fitted; no new particles or forces are postulated. The only domain-level assumptions are the five conditions of Sec. 2 and the restriction to type-III magnetic groups.

assumptions (4)
  • domain assumption Conventional altermagnetism requires five conditions: momentum-dependent spin splitting, negligible SOC, collinear magnetization, compensated order, and identical spin at opposite momenta.
    Stated as the selection principles in Sec. 2; the entire catalog is built only for systems obeying them.
  • domain assumption Spinless time-reversal S = U_π T acts within each spin block as an effective time-reversal and maps k → −k, thereby enforcing Condition 5 even without crystal inversion.
    Introduced in Sec. 2.1 and used for all non-centrosymmetric and nonsymmorphic constructions; taken from the spin-group literature.
  • domain assumption Only type-III magnetic groups can host altermagnetism; type-I are ferromagnetic and type-IV enforce full spin degeneracy via T_a.
    Derived in Sec. 2.3 from the action of the composite translation-plus-spin-flip operator.
  • standard math Standard Opechowski–Guccione notation and the known lists of 17 wallpaper groups and 230 space groups (and their magnetic extensions) are complete and correctly labeled.
    Background crystallographic fact used throughout the tables.

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Pith. "Pith review of Catalog of Altermagnetism in Magnetic Wallpaper/Space Groups and Nonsymmorphic Altermagnets." pith.science (2026). https://pith.science/paper/PUBPYCJJ

@misc{pith2026260710857,
  author       = {Pith},
  title        = {Pith review of: Catalog of Altermagnetism in Magnetic Wallpaper/Space Groups and Nonsymmorphic Altermagnets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PUBPYCJJ}},
  note         = {Machine review of arXiv:2607.10857}
}
abstract

Conventional altermagnetism, characterized by compensated collinear spin alignment and spin splitting, exhibits identical spin states at opposite momenta. In this work, we employ a non-spatial global symmetry $S$, the spinless time-reversal symmetry, which effectively replaces inversion symmetry in preserving the spin-state equivalence; hence, we systematically extend the classification of altermagnetism to all possible non-centrosymmetric crystals. By analyzing the necessary symmetry conditions, we provide a complete catalog of altermagnetic orders for all 2D magnetic wallpaper groups and all 3D magnetic space groups, identifying 17 altermagnetic wallpaper groups (12 centrosymmetric and 5 non-centrosymmetric) and 422 altermagnetic space groups (160 centrosymmetric and 262 non-centrosymmetric). This catalog assigns each altermagnetic wallpaper and space group to one of the six altermagnetic wave types established in the literature and presents its distinct spin distribution in the Brillouin zone (BZ); notably, the low-energy wave-type description does not necessarily extend throughout the full BZ, since the spin-degenerate nodal lines and planes can be unpinned from the high-symmetry planes. Beyond the catalog, nonsymmorphic symmetries further bring new patterns of the altermagnetic BZs through the emergence of hourglass dispersions, which arise from the compatibility relations between two symmetry-protected degenerate manifolds: same-spin and opposite-spin degeneracies. In both the non-centrosymmetric altermagnetism and the emergence of the hourglass dispersion, the spinless time-reversal symmetry plays the key role. Our work extends the symmetry catalog of altermagnetism and reveals that nonsymmorphic symmetries are essential for realizing altermagnetic band structures beyond the six established wave types, such as an $i$-wave-like spin winding in a tetragonal BZ.

Figures

Figures reproduced from arXiv: 2607.10857 by the authors.

Figure 1
Figure 1. (color online) Classification of 2D and 3D altermagnetic systems, with and without inversion symmetry, into [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. (color online) Stepwise construction of spin-momentum textures in the Brillouin zone for representative AWGs: [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. (color online) Crystal lattice and band structure for AWG P1m [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: (color online) Nonsymmorphic altermagnetism in the magnetic wallpaper group P4 ′ gm′ . (a) Schematic of the hourglass dispersion along a path connecting the BZ boundary (ky = π) to the diagonal symmetry lines (kx = ±ky). A symmetry-protected crossing (nodal line) occur…
Figure 5
Figure 5. Figure 5: (color online) Nonsymmorphic altermagnetism in the altermagnetic space groups P4′bm′ and P4 ′ 2212 ′ . (a) and (b) BZs of altermagnetic space groups P4′bm′ and P4 ′ 2212 ′ : Magenta lines and planes indicate same￾spin twofold degeneracies, while the green planes repres…
Figure 6
Figure 6. Figure 6: (color online)Generation process of the altermagnetic types in centrosymmetric ASGs (a) P2 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: (color online) Non-centrosymmetric altermagnetic metal Y [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]

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Works this paper leans on

77 extracted references · 14 canonical work pages

  1. [50]

    npj Quantum Materials , author =

    Altermagnetism classification , volume =. npj Quantum Materials , author =. 2025 , pages =. doi:10.1038/s41535-025-00756-5 , number =

    doi:10.1038/s41535-025-00756-5 2025

  2. [1]

    Beyond Conventional Ferromagnetism and Antiferromagnetism: A Phase with Nonrelativistic Spin and Crystal Rotation Symmetry , author =. Phys. Rev. X , volume =. 2022 , month =. doi:10.1103/PhysRevX.12.031042 , url =

    doi:10.1103/physrevx.12.031042 2022

  3. [2]

    Emerging Research Landscape of Altermagnetism , author =. Phys. Rev. X , volume =. 2022 , month =. doi:10.1103/PhysRevX.12.040501 , url =

    doi:10.1103/physrevx.12.040501 2022

  4. [3]

    Altermagnetic Routes to Majorana Modes in Zero Net Magnetization , author =. Phys. Rev. Lett. , volume =. 2024 , month =. doi:10.1103/PhysRevLett.133.106601 , url =

    doi:10.1103/physrevlett.133.106601 2024

  5. [4]

    Topological superconductivity in two-dimensional altermagnetic metals , author =. Phys. Rev. B , volume =. 2023 , month =. doi:10.1103/PhysRevB.108.184505 , url =

    doi:10.1103/physrevb.108.184505 2023

  6. [5]

    Two-dimensional altermagnets: Superconductivity in a minimal microscopic model , author =. Phys. Rev. B , volume =. 2023 , month =. doi:10.1103/PhysRevB.108.224421 , url =

    doi:10.1103/physrevb.108.224421 2023

  7. [6]

    Nature Communications , author =

    Finite-momentum. Nature Communications , author =. 2024 , pages =. doi:10.1038/s41467-024-45951-3 , number =

    doi:10.1038/s41467-024-45951-3 2024

  8. [7]

    dc Josephson Effect in Altermagnets , author =. Phys. Rev. Lett. , volume =. 2023 , month =. doi:10.1103/PhysRevLett.131.076003 , url =

    doi:10.1103/physrevlett.131.076003 2023

Show all 77 references
  1. [8]

    Phase-shifted Andreev levels in an altermagnet Josephson junction , author =. Phys. Rev. B , volume =. 2023 , month =. doi:10.1103/PhysRevB.108.075425 , url =

    2023 doi

  2. [9]

    arXiv , url=:2309.14427 , year=

    Zero-field finite-momentum and field-induced superconductivity in altermagnets , author=. arXiv , url=:2309.14427 , year=

    arXiv

  3. [10]

    Majorana corner modes and tunable patterns in an altermagnet heterostructure , author =. Phys. Rev. B , volume =. 2023 , month =. doi:10.1103/PhysRevB.108.205410 , url =

    2023 doi

  4. [11]

    Altermagnetic superconducting diode effect , author =. Phys. Rev. B , volume =. 2024 , month =. doi:10.1103/PhysRevB.110.024503 , url =

    2024 doi

  5. [12]

    Andreev reflection at the altermagnet-superconductor interface , author =. Phys. Rev. B , volume =. 2023 , month =. doi:10.1103/PhysRevB.108.L060508 , url =

    2023 doi

  6. [13]

    Nature Reviews Materials , author =

    Anomalous. Nature Reviews Materials , author =. 2022 , pages =. doi:10.1038/s41578-022-00430-3 , number =

    2022 doi

  7. [14]

    Nature Electronics , author =

    An anomalous. Nature Electronics , author =. 2022 , pages =. doi:10.1038/s41928-022-00866-z , number =

    2022 doi

  8. [15]

    Spontaneous Anomalous Hall Effect Arising from an Unconventional Compensated Magnetic Phase in a Semiconductor , author =. Phys. Rev. Lett. , volume =. 2023 , month =. doi:10.1103/PhysRevLett.130.036702 , url =

    2023 doi

  9. [16]

    and Zhang, Y

    Bai, H. and Zhang, Y. C. and Zhou, Y. J. and Chen, P. and Wan, C. H. and Han, L. and Zhu, W. X. and Liang, S. X. and Su, Y. C. and Han, X. F. and Pan, F. and Song, C. , journal =. Efficient Spin-to-Charge Conversion via Altermagnetic Spin Splitting Effect in Antiferromagnet. 2...

    2023 doi

  10. [17]

    Crystal Thermal Transport in Altermagnetic

    Zhou, Xiaodong and Feng, Wanxiang and Zhang, Run-Wu and. Crystal Thermal Transport in Altermagnetic. Phys. Rev. Lett. , volume =. 2024 , month =. doi:10.1103/PhysRevLett.132.056701 , url =

    2024 doi

  11. [18]

    Nature Electronics , author =

    Tilted spin current generated by the collinear antiferromagnet ruthenium dioxide , volume =. Nature Electronics , author =. 2022 , pages =. doi:10.1038/s41928-022-00744-8 , number =

    2022 doi

  12. [19]

    and Han, L

    Bai, H. and Han, L. and Feng, X. Y. and Zhou, Y. J. and Su, R. X. and Wang, Q. and Liao, L. Y. and Zhu, W. X. and Chen, X. Z. and Pan, F. and Fan, X. L. and Song, C. , journal =. Observation of Spin Splitting Torque in a Collinear Antiferromagnet. 2022 , month =. doi:10.1103/P...

    2022 doi

  13. [20]

    Observation of Spin-Splitter Torque in Collinear Antiferromagnetic

    Karube, Shutaro and Tanaka, Takahiro and Sugawara, Daichi and Kadoguchi, Naohiro and Kohda, Makoto and Nitta, Junsaku , journal =. Observation of Spin-Splitter Torque in Collinear Antiferromagnetic. 2022 , month =. doi:10.1103/PhysRevLett.129.137201 , url =

    2022 doi

  14. [21]

    Nature Communications , author =

    Spin-neutral currents for spintronics , volume =. Nature Communications , author =. 2021 , pages =. doi:10.1038/s41467-021-26915-3 , number =

    2021 doi

  15. [22]

    Giant and Tunneling Magnetoresistance in Unconventional Collinear Antiferromagnets with Nonrelativistic Spin-Momentum Coupling , author =. Phys. Rev. X , volume =. 2022 , month =. doi:10.1103/PhysRevX.12.011028 , url =

    2022 doi

  16. [23]

    Efficient Electrical Spin Splitter Based on Nonrelativistic Collinear Antiferromagnetism , author =. Phys. Rev. Lett. , volume =. 2021 , month =. doi:10.1103/PhysRevLett.126.127701 , url =

    2021 doi

  17. [24]

    Journal of the Physical Society of Japan , volume =

    Hayami ,Satoru and Yanagi ,Yuki and Kusunose ,Hiroaki , title =. Journal of the Physical Society of Japan , volume =. 2019 , doi =

    2019

  18. [25]

    2024 , eprint=

    AI-accelerated Discovery of Altermagnetic Materials , author=. 2024 , eprint=

    2024

  19. [26]

    Antiferromagnetism in

    Ahn, Kyo-Hoon and Hariki, Atsushi and Lee, Kwan-Woo and Kune. Antiferromagnetism in. Phys. Rev. B , volume =. 2019 , month =. doi:10.1103/PhysRevB.99.184432 , url =

    2019 doi

  20. [27]

    Giant momentum-dependent spin splitting in centrosymmetric low- Z antiferromagnets , author =. Phys. Rev. B , volume =. 2020 , month =. doi:10.1103/PhysRevB.102.014422 , url =

    2020 doi

  21. [28]

    Mazin and Klaus Koepernik and Michelle D

    Igor I. Mazin and Klaus Koepernik and Michelle D. Johannes and Rafael González-Hernández and Libor Šmejkal , title =. Proceedings of the National Academy of Sciences , volume =. 2021 , doi =

    2021

  22. [29]

    Broken Kramers Degeneracy in Altermagnetic MnTe , author =. Phys. Rev. Lett. , volume =. 2024 , month =. doi:10.1103/PhysRevLett.132.036702 , url =

    2024 doi

  23. [30]

    Nature Communications , author =

    Direct observation of altermagnetic band splitting in. Nature Communications , author =. 2024 , pages =. doi:10.1038/s41467-024-46476-5 , number =

    2024 doi

  24. [31]

    Nature , author =

    Altermagnetic lifting of. Nature , author =. 2024 , pages =. doi:10.1038/s41586-023-06907-7 , number =

    2024 doi

  25. [32]

    Nature , author =

    Observation of plaid-like spin splitting in a noncoplanar antiferromagnet , volume =. Nature , author =. 2024 , pages =. doi:10.1038/s41586-024-07023-w , number =

    2024 doi

  26. [33]

    Nature Communications , author =

    Three-dimensional mapping of the altermagnetic spin splitting in. Nature Communications , author =. 2025 , pages =. doi:10.1038/s41467-025-56647-7 , number =

    2025 doi

  27. [34]

    Topological responses from gapped Weyl points in 2D altermagnets , author=. Phys. Rev. B , volume=. 2025 , doi=. 2403.09520 , archivePrefix=

    2025 arXiv

  28. [35]

    Minimal models for altermagnetism , author =. Phys. Rev. B , volume =. 2024 , month =. doi:10.1103/PhysRevB.110.144412 , url =

    2024 doi

  29. [36]

    1974 , issn =

    Spin groups , journal =. 1974 , issn =. doi:https://doi.org/10.1016/0031-8914(74)90157-8 , url =

    1974 doi

  30. [37]

    Litvin, D. B. Spin point groups. Acta Crystallographica Section A. 1977. doi:10.1107/S0567739477000709 , url =

    1977 doi

  31. [38]

    Spin Space Groups: Full Classification and Applications , author =. Phys. Rev. X , volume =. 2024 , month =. doi:10.1103/PhysRevX.14.031037 , url =

    2024 doi

  32. [39]

    Enumeration of Spin-Space Groups: Toward a Complete Description of Symmetries of Magnetic Orders , author =. Phys. Rev. X , volume =. 2024 , month =. doi:10.1103/PhysRevX.14.031039 , url =

    2024 doi

  33. [40]

    Enumeration and Representation Theory of Spin Space Groups , author =. Phys. Rev. X , volume =. 2024 , month =. doi:10.1103/PhysRevX.14.031038 , url =

    2024 doi

  34. [41]

    and Perez-Mato, J

    Gallego, Samuel V. and Perez-Mato, J. Manuel and Elcoro, Luis and Tasci, Emre S. and Hanson, Robert M. and Momma, Koichi and Aroyo, Mois I. and Madariaga, Gotzon. MAGNDATA : towards a database of magnetic structures. I.The commensurate case. Journal of Applied Crystallography....

    2016 doi

  35. [42]

    and Perez-Mato, J

    Gallego, Samuel V. and Perez-Mato, J. Manuel and Elcoro, Luis and Tasci, Emre S. and Hanson, Robert M. and Aroyo, Mois I. and Madariaga, Gotzon. MAGNDATA : towards a database of magnetic structures. II. The incommensurate case. Journal of Applied Crystallography. 2016. doi:10....

    2016 doi

  36. [43]

    and Garlea, Vasile O

    Sanjeewa, Liurukara D. and Garlea, Vasile O. and McGuire, Michael A. and McMillen, Colin D. and Cao, Huibo and Kolis, Joseph W. , journal =. Structural and magnetic characterization of the one-dimensional S=5/2 antiferromagnetic chain system. 2016 , month =. doi:10.1103/PhysRe...

    2016 doi

  37. [44]

    Garca-Muoz, J. L. and Rodrguez-Carvajal, J. and Obradors, X. and Vallet-Reg, M. and Gonz\'alez-Calbet, J. and Parras, M. , journal =. Complex magnetic structures of the rare-earth cuprates. 1991 , month =. doi:10.1103/PhysRevB.44.4716 , url =

    1991 doi

  38. [45]

    Morozkin and O

    A.V. Morozkin and O. Isnard and R. Nirmala and S.K. Malik , keywords =. Magnetic structure of the rare earth intermetallic compound Ce4Ge3 , journal =. 2009 , issn =. doi:https://doi.org/10.1016/j.jallcom.2009.06.191 , url =

    2009 doi

  39. [46]

    Nirmala and A.V

    R. Nirmala and A.V. Morozkin and O. Isnard and A.K. Nigam , keywords =. Understanding the magnetic ground state of rare-earth intermetallic compound Ce4Sb3: Magnetization and neutron diffraction studies , journal =. 2009 , issn =. doi:https://doi.org/10.1016/j.jmmm.2008.08.102 , url =

    2009 doi

  40. [47]

    Magnetic Structure of α‐Tb2C3 , volume =

    Atoji, Masao , date =. Magnetic Structure of α‐Tb2C3 , volume =. doi:10.1063/1.1675373 , pages =

    doi

  41. [48]

    A metallic room-temperature d-wave altermagnet , issn =

    Jiang, Bei and Hu, Mingzhe and Bai, Jianli and Song, Ziyin and Mu, Chao and Qu, Gexing and Li, Wan and Zhu, Wenliang and Pi, Hanqi and Wei, Zhongxu and Sun, Yu-Jie and Huang, Yaobo and Zheng, Xiquan and Peng, Yingying and He, Lunhua and Li, Shiliang and Luo, Jianlin and Li, Zh...

    doi

  42. [49]

    Spontaneous antisymmetric spin splitting in noncollinear antiferromagnets without spin-orbit coupling , author =. Phys. Rev. B , volume =. 2020 , month =. doi:10.1103/PhysRevB.101.220403 , url =

    2020 doi

  43. [51]

    npj Quantum Materials , author =

    Altermagnetism with non-collinear spins , volume =. npj Quantum Materials , author =. 2024 , pages =. doi:10.1038/s41535-024-00626-6 , number =

    2024 doi

  44. [52]

    Nature , author =

    Hourglass fermions , volume =. Nature , author =. 2016 , pages =. doi:10.1038/nature17410 , abstract =

    2016 doi

  45. [53]

    Interplay between altermagnetism and nonsymmorphic symmetries generating large anomalous Hall conductivity by semi-Dirac points induced anticrossings , author =. Phys. Rev. B , volume =. 2023 , month =. doi:10.1103/PhysRevB.108.115138 , url =

    2023 doi

  46. [54]

    2015 , month =

    Shiozaki, Ken and Sato, Masatoshi and Gomi, Kiyonori , journal =. 2015 , month =. doi:10.1103/PhysRevB.91.155120 , url =

    2015 doi

  47. [55]

    Wieder and Barry Bradlyn and Zhijun Wang and Jennifer Cano and Youngkuk Kim and Hyeong-Seok D

    Benjamin J. Wieder and Barry Bradlyn and Zhijun Wang and Jennifer Cano and Youngkuk Kim and Hyeong-Seok D. Kim and Andrew M. Rappe and C. L. Kane and B. Andrei Bernevig , title =. Science , volume =. 2018 , doi =

    2018

  48. [56]

    Antonenko and Jörn W

    Qian Song and Srdjan Stavrić and Paolo Barone and Andrea Droghetti and Daniil S. Antonenko and Jörn W. F. Venderbos and Connor A. Occhialini and Batyr Ilyas and Emre Ergeçen and Nuh Gedik and Sang-Wook Cheong and Rafael M. Fernandes and Silvia Picozzi and Riccardo Comin , titl...

    doi

  49. [57]

    Birch and Priya R

    Rinsuke Yamada and Max T. Birch and Priya R. Baral and Shun Okumura and Ryota Nakano and Shang Gao and Motohiko Ezawa and Takuya Nomoto and Jan Masell and Yuki Ishihara and Kamil K. Kolincio and Ilya Belopolski and Hajime Sagayama and Hironori Nakao and Kazuki Ohishi and Takas...

    doi

  50. [58]

    Topology of nonsymmorphic crystalline insulators and superconductors , author =. Phys. Rev. B , volume =. 2016 , month =. doi:10.1103/PhysRevB.93.195413 , url =

    2016 doi

  51. [59]

    , title =

    Peierls, R. , title =. Zeitschrift f. 1933 , doi =

    1933

  52. [60]

    , title =

    Guo, Yaqian and Liu, Hui and Janson, Oleg and Fulga, Ion Cosma and van den Brink, Jeroen and Facio, Jorge I. , title =. Materials Today Physics , volume =. 2023 , doi =

    2023

  53. [61]

    and Rau, Jeffrey G

    McClarty, Paul A. and Rau, Jeffrey G. , title =. Phys. Rev. Lett. , volume =. 2024 , doi =

    2024

  54. [62]

    and Romh

    Schiff, Hana and McClarty, Paul and Rau, Jeffrey G. and Romh. Collinear altermagnets and their Landau theories , journal =. 2025 , eprint =

    2025

  55. [63]

    , title =

    Radaelli, Paolo G. , title =. Phys. Rev. B , volume =. 2024 , doi =

    2024

  56. [64]

    and Zhao, Y.-J

    Zeng, S. and Zhao, Y.-J. , title =. Phys. Rev. B , volume =. 2024 , doi =

    2024

  57. [65]

    and Chen, X

    Liu, Y. and Chen, X. and Yu, Y. and Etxebarria, J. and Perez-Mato, J. M. and Liu, Q. , title =. Nature , volume =. 2026 , eprint =

    2026

  58. [66]

    2023 , eprint =

    P-wave magnets , author =. 2023 , eprint =

    2023

  59. [67]

    Rooj, Suman and Saxena, Sugandha and Ganguli, Nirmal , title =. Phys. Rev. B , volume =. 2025 , doi =

    2025

  60. [68]

    Physical Review B , volume =

    Nodal surface semimetals: Theory and material realization , author =. Physical Review B , volume =. 2018 , doi =

    2018

  61. [69]

    Nature Communications , volume =

    Kramers nodal line metals , author =. Nature Communications , volume =. 2021 , doi =

    2021

  62. [70]

    Physical Review , volume =

    Effect of Time-Reversal Symmetry on Energy Bands of Crystals , author =. Physical Review , volume =. 1937 , doi =

    1937

  63. [71]

    Physical Review , volume =

    Spin-Orbit Interaction in Graphite , author =. Physical Review , volume =. 1965 , doi =

    1965

  64. [72]

    Physical Review Letters , volume =

    Dirac Semimetals in Two Dimensions , author =. Physical Review Letters , volume =. 2015 , doi =

    2015

  65. [73]

    Nature , volume =

    Symmetry-enforced topological nodal planes at the Fermi surface of a chiral magnet , author =. Nature , volume =. 2021 , doi =

    2021

  66. [74]

    Physical Review Materials , volume =

    Prediction of low-Z collinear and noncollinear antiferromagnetic compounds having momentum-dependent spin splitting even without spin-orbit coupling , author =. Physical Review Materials , volume =. 2021 , doi =

    2021

  67. [75]

    Computational Materials Science , volume =

    High-throughput electronic band structure calculations: Challenges and tools , author =. Computational Materials Science , volume =. 2010 , doi =

    2010

  68. [76]

    Frontiers of Physics , volume =

    Classification and design of two-dimensional altermagnets , author =. Frontiers of Physics , volume =. 2026 , doi =

    2026

  69. [77]

    2013 , doi =

    Magnetic Group Tables: 1-, 2- and 3-Dimensional Magnetic Subperiodic Groups and Magnetic Space Groups , author =. 2013 , doi =

    2013

Pith tools

Reviewed July 14, 2026 · model on record in the stance chip above.