Pith. sign in

REVIEW 2 major objections 6 minor 45 references

Generalized Space Groups from Internal Configuration Spaces

T0 review · 2 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims a single construction generates all space-group frameworks and, in a dodecahedral case, a point node with topological charge |C|=12.

desk verdict Elegant Goursat unification of space-group generalizations, but the headline |C|=12 dodecahedral node depends on an unshown k·p block-diagonal form. read the letter →

arxiv 2608.03808 v2 pith:WV5TQKES submitted 2026-08-04 cond-mat.mtrl-sci

Zeying Zhang , Zhenye Li , Zhi-Ming Yu , Gui-Bin Liu , Yugui Yao This is my paper · ORCID
classification cond-mat.mtrl-sci
keywords generalizedspacegroupsinternaldegreesoffreedomGoursat'slemmadodecahedralgrouptopologicalchargepointnodek·pHamiltonianclassification
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 aims to give one construction that produces the symmetry group of any crystal whose sites carry extra internal structure—orientation, color, spin, multipole order, or anything else. Starting from the group of allowed internal transformations and the stabilizer of a reference configuration, it derives the maximal internal symmetry of a chosen set of configurations and couples that symmetry to spatial translations and rotations through Goursat's lemma. The framework includes ordinary, magnetic, spin, and color space groups as special cases. In the worked dodecahedral example, the construction yields a generalized space group whose band structure has a point node with topological charge |C|=12, a value the paper says has not appeared in prior crystallographic classifications. If correct, the result replaces several separate symmetry taxonomies with one recipe and opens internal degrees of freedom beyond the standard ones to the same kind of classification.

What carries the argument

The load-bearing object is the pair of normal subgroups and a quotient isomorphism supplied by Goursat's lemma. The construction first generates candidate configurations as orbits $X_H = N_{G_P}(H)\cdot x_0$ under the normalizer of a preset symmetry $H$; a chosen finite subset $C$ then defines the pointwise stabilizer $J = \bigcap_i \operatorname{Stab}(x_i)$ and the setwise stabilizer $K = \operatorname{Stab}(C)$, with $J\triangleleft K$. Goursat's lemma states that any subdirect product of a spatial group $G_S$ and an internal group with surjective projections is $G = \{(g,p)\in G_S\times K \mid \varphi(gN_S) = pJ\}$, where $N_S\triangleleft G_S$ and $\varphi$ is an isomorphism $G_S/N_S \to K/J$. This identity converts the physical choice of configuration set $C$ into an algebraic coupling between space and internal degrees of freedom; the quotient $K/J$ is the portion of internal symmetry that can permute configurations, and matching it to a spatial quotient is what creates a generalized space group.

What would settle it

Derive the full symmetry-allowed $k\cdot p$ Hamiltonian at $\Gamma$ from all 25 irreducible small representations, including every inter-block term allowed by the generalized space group, and compute the Chern number of each occupied band on a closed sphere around $\Gamma$; if the total topological charge is anything other than 12, or if the three $H_0$ blocks carry opposite chiralities, the claim collapses. A Wilson-loop or tight-binding calculation for any proposed material realization would also settle the node charge directly.

Watch

Extended reading notes

Core claim

The central claim is that all decorated-crystal symmetry groups, old and new, fit into a single construction. Given a group $G_P$ acting on an internal configuration space $X$ and a reference configuration $x_0$ with stabilizer $P$, each subgroup $H\le P$ generates a candidate orbit $X_H = N_{G_P}(H)\cdot x_0$ of configurations sharing $H$. A physically selected subset $C\subseteq X_H$ has pointwise stabilizer $J$ and setwise stabilizer $K$ ($J\triangleleft K$), and Goursat's lemma couples the quotient $K/J$ to the quotient $G_S/N_S$ of a spatial space group through an isomorphism $\varphi: G_S/N_S \to K/J$. In the dodecahedral example $G_P=SO(3)$, $P=I\simeq A_5$, and the two-configuration set gives $J=T$, $K=O$, so $K/J\simeq\mathbb{Z}_2$; coupling to the body-centered spatial quotient $I432/P432$ produces a generalized space group isomorphic to $T\rtimes_\alpha I432$. At $\Gamma$ the group's 25 irreducible representations include a six-dimensional one whose symmetry-allowed $k\cdot p$ Hamiltonian has the form $H_0\oplus H_0\oplus H_0$, with each two-band block carrying topological charge $|C_0|=4$ on a closed surface around $\Gamma$; the three blocks share chirality, so the total charge is $|C|=12$. The paper presents this as a symmetry-enforced point node beyond anything previously reported in ordinary, magnetic, or spin space groups.

Load-bearing premise

The $|C|=12$ result assumes the symmetry-allowed $k\cdot p$ Hamiltonian is exactly three identical two-band chiral blocks with the same chirality and no inter-block coupling; the paper presents this form but does not derive it from the 25 irreducible representations, so an allowed coupling between blocks could change the total charge.

Editorial extensions

If this is right

  • Any crystal whose local objects live in an internal configuration space with a reference stabilizer can be assigned a generalized space group by choosing a finite configuration subset and matching its $K/J$ quotient to a spatial quotient.
  • Ordinary, magnetic, spin, and color space groups are recovered as special cases of the same construction, so results in those frameworks can be viewed as instances of one classification.
  • The dodecahedral pair $T\triangleleft O$, coupled to the index-two spatial quotient $I432/P432$, gives a generalized space group that is a semidirect product $T\rtimes_\alpha I432$, not a direct product, so the internal symmetry is nontrivially twisted by translations.
  • The six-dimensional representation at $\Gamma$ supports a symmetry-enforced chiral point node with total topological charge $|C|=12$, exceeding the charges available in previously classified ordinary, magnetic, and spin space groups.
  • The construction handles both discrete configuration sets, such as two orientations, and continuous families, such as orientations sharing a $C_n$ axis, so the same formalism applies to ordered crystals and to orientational or cluster order with continuous degeneracy.

Reading between the lines

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

  • A systematic enumeration over choices of $(G_P, P, H, C)$ would produce a catalogue of generalized space groups whose band representations and allowed topological charges could be computed; the dodecahedral example is one entry, not the whole table.
  • The $|C|=12$ record is tied to the uncoupled form $H_0\oplus H_0\oplus H_0$; testing whether symmetry-allowed inter-block coupling can be switched on without changing the total charge would decide whether the record survives beyond the stated model.
  • Because the construction lets $G_P$ be any group, not just subgroups of $O(3)$, it suggests that internal spaces such as skyrmion textures, molecular orientations, and multipolar configurations may host symmetry-enforced band crossings that no $O(3)$-based classification can predict.
  • The semidirect structure $T\rtimes_\alpha I432$ means band representations of these generalized space groups will be governed by a twisted internal group action, so existing band-representation databases for ordinary and magnetic space groups may not capture the new phases.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 paper develops a general construction of 'generalized space groups' (GSGs) for crystals whose sites carry internal degrees of freedom beyond position and chemical identity. Starting from a group GP of allowed internal transformations and the stabilizer P of a reference object, the authors define candidate internal configurations via normalizer orbits, compute the pointwise (J) and setwise (K) stabilizers of a selected configuration set C, and then couple the internal quotient K/J to a spatial quotient GS/NS using Goursat's lemma. They argue that ordinary, magnetic, spin, and color space groups appear as special cases. As a central example, they consider a dodecahedral internal object with P=I≃A5, obtain the normal pair T◁O with O/T≃Z2, and construct a GSG based on the spatial group I432 with an index-two spatial subgroup. They report that a six-band k·p Hamiltonian for this group takes the form H=H0⊕H0⊕H0 and hosts a point node with total topological charge |C|=12, which they state exceeds any previously reported symmetry-enforced point node in the established crystallographic frameworks.

Significance. If the construction and the dodecahedral example are fully validated, the paper provides a genuinely unifying algebraic framework for symmetry classifications of crystals with arbitrary internal degrees of freedom. The use of Goursat's lemma to couple internal and spatial quotients is clean and natural, and the explicit reduction of the known cases (SG, MSG, SSG, CG) to a single construction is valuable. The machine-assisted computations with GAP and MagneticKP are a strength, as is the concrete prediction of a topological charge |C|=12, which would be a notable result. However, the headline claim rests on the asserted form H=H0⊕H0⊕H0 and on the equal chirality of the three blocks, neither of which is demonstrated in the manuscript. The significance of the paper therefore depends on closing this verification gap; the abstract construction itself appears sound.

major comments (2)
  1. [Dodecahedral group as an example (H0⊕H0⊕H0 and |C|=12)] The statement that the symmetry-allowed k·p Hamiltonian up to third order is exactly H=H0⊕H0⊕H0, and that the three blocks carry the same chirality, is asserted without derivation. The six-dimensional representation of the generalized little group is not specified, and no representation matrices are given that would allow the reader to verify that all inter-block couplings vanish and that the three 2×2 blocks are identical rather than merely three independent blocks. Because any symmetry-allowed inter-block coupling could change the total Chern number on a closed surface enclosing Γ, the headline claim of a 'symmetry-enforced point node with |C|=12' is not yet established. Please provide the explicit basis and representation matrices (or a reproducible MagneticKP input and output that yields the full set of allowed terms), and show independently that the three blocks have identical chirality.
  2. [Dodecahedral group as an example (third-order truncation and Wilson loop)] The topological charge |C|=12 is computed from a Hamiltonian truncated at third order in momentum. The paper should justify that this charge is a well-defined invariant of the full symmetry-constrained model: on a sufficiently small sphere around Γ the leading-order spectrum should remain gapped, so that higher-order terms cannot change the total Chern number. In addition, the Wilson-loop spectrum in Fig. 2(d) would be reproducible only if the loop radius/path and the values of c1 and c2 used in the numerical computation are stated, or if it is explicitly noted that the charge is independent of these parameters.
minor comments (6)
  1. [Dodecahedral group as an example] There is a typo: 'On the internal side, On the Tis an index-two normal subgroup' should read 'On the internal side, T is an index-two normal subgroup.'
  2. [Conclusion] The conclusion contains 'an full group GP'; this should be 'a full group GP'.
  3. [Table I caption] The caption phrase 'in each case' is dangling, and the sentence beginning 'For DGSGs, SO(3)/I is the configuration space...' would be clearer if it were split into separate statements for the candidate space and the selected subset C.
  4. [Dodecahedral group as an example (use of MSG PI432)] The text says the MSG PI432 is used only to identify an index-two subgroup and that its antiunitary operation is not retained. This is potentially confusing because the resulting GSG is not a magnetic space group; please clarify in one or two sentences why the MSG machinery is being invoked and what exactly is taken from it.
  5. [Dodecahedral group as an example (notation for p0)] The notation p0=C_{20\bar{1}1} is not defined; please specify the dodecahedral coordinate convention and the rotation axis so that the reader can identify the operation in Fig. 2(b).
  6. [Reference [39]] Reference [39] is an unpublished preprint and the paper relies on its method for computing irreducible small representations. Please provide a short summary of the method in the main text or in a supplement, or include the necessary definitions so that the present paper is self-contained on this point.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the generalized space-group construction and the dodecahedral |C|=12 charge are computed from stabilizer data, Goursat's lemma, and representation-theoretic input, not built into the inputs.

full rationale

The paper's derivation chain is self-contained in the relevant sense. It starts from explicit inputs (G_P, the stabilizer P of a reference configuration, a preset subgroup H, and a selected configuration set C) and computes the pointwise and setwise stabilizers J and K directly from the defining equations J = intersection of Stab(x_i) and K = Stab(C). Goursat's lemma is cited to a standard textbook (Hall, Theorem 5.5.1), so the coupling between K/J and a spatial quotient is an external mathematical fact rather than an assumption tailored to the results. For the dodecahedral example, the pair T ◁ O with O/T ≅ Z_2 is derived from normalizer-quotient computations for H = D_3, D_5, and T; it is not assumed as a target. The headline charge |C|=12 is neither an input nor a fitted parameter: after the group G is fixed, the paper computes the 25 irreducible small representations at Γ, identifies one six-dimensional representation, and uses MagneticKP to generate the symmetry-allowed k·p Hamiltonian H = H0 ⊕ H0 ⊕ H0 up to third order. The topological charge is then obtained by evaluating the Wilson-loop spectrum of that Hamiltonian. Nothing in the construction is adjusted to reproduce |C|=12, and the charge is not used to select C, H, or the quotient data. The self-citations to GAP, MagneticKP, and the companion representation method are citations to computational tools and methods, not to an unverified theorem that already contains the dodecahedral result; these outputs are code-reproducible and independently checkable. The lack of a fully written-out derivation of H0 ⊕ H0 ⊕ H0 from the 25 irreps is a transparency or verification gap, but it is not circularity: an incorrect or incomplete symmetry analysis would be a wrong derivation, not an equation that is equal to its input by construction. Therefore no circular step meets the evidentiary standard required here.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

The framework uses standard group theory (Goursat's lemma) and the explicit assumption of full spatial projection. The dodecahedral example introduces one new predicted object, the charge-12 node, which lacks independent experimental evidence. The effective Hamiltonian contains two real coefficients that do not affect the topological charge.

free parameters (2)
  • c1 = not fitted (arbitrary real coefficient)
    Real coefficient of the kx ky kz term in H0. The topological charge does not depend on its value as long as it is nonzero.
  • c2 = not fitted (arbitrary real coefficient)
    Real coefficient of the quadratic terms in H0. The charge is independent of its value; if it vanishes the node gap closes and the charge is not defined.
assumptions (4)
  • standard math Goursat's lemma: for groups A and B, subgroups of A×B with surjective projections are classified by normal subgroups N≤A, M≤B and an isomorphism φ:A/N→B/M.
    Used to parametrize all possible generalized space groups; cited at Eq. (1) with Ref. [36].
  • domain assumption The spatial projection of the generalized space group equals the full parent space group G_S.
    Stated explicitly at the start of the Construction principle; excludes structures where the spatial symmetry is a proper subgroup.
  • domain assumption The physically selected configuration set C is finite, ensuring K/J is finite and can be mapped to a finite quotient of a space group.
    Justified by periodicity: 'a periodic ordered crystal selects only a finite configuration set C'. Used throughout the example.
  • ad hoc to paper Internal operations in the dodecahedral example act on electron wavefunctions, so the generalized little group has the stated representation content.
    The k·p model treats G as a symmetry group acting on electronic states; the physical coupling between molecular orientation and electron orbitals is not specified. This underpins the charge-12 prediction.
invented entities (1)
  • Point node with topological charge |C|=12
    purpose: Emergent quasiparticle predicted in the dodecahedral generalized space group; demonstrates that GSGs can support topological charges beyond established frameworks.
    No concrete material realization or experimental signature is proposed; the node exists only in a hypothetical symmetry group.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Generalized Space Groups from Internal Configuration Spaces." pith.science (2026). https://pith.science/paper/WV5TQKES

@misc{pith2026260803808,
  author       = {Pith},
  title        = {Pith review of: Generalized Space Groups from Internal Configuration Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WV5TQKES}},
  note         = {Machine review of arXiv:2608.03808}
}
abstract

We develop a unified construction of generalized space groups for crystals with unconventional internal degrees of freedom. Starting from the full group $G_P$ of allowed internal transformations and the stabilizer $P$ of a reference object, we determine the pointwise and setwise symmetries, $J$ and $K$, of the allowed configuration set. Goursat's lemma then couples the internal quotient $K/J$ to a spatial quotient. The framework includes ordinary, magnetic, spin, and color space groups as special cases. As an example, we consider a dodecahedral object with $P=I\simeq A_5$, for which we obtain the nontrivial pair $T\triangleleft O$ with $O/T\simeq\mathbb Z_2$. The resulting generalized space group hosts a point node with topological charge $|C|=12$.

Figures

Figures reproduced from arXiv: 2608.03808 by the authors.

Figure 1
Figure 1. FIG. 1. Construction of a generalized space group. The pre [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Two dodecahedral orientations sharing a [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Review history (3 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

45 extracted references · 2 canonical work pages

  1. [1]

    Bradley and A

    C. Bradley and A. Cracknell,Mathematical theory of symmetry in solids: representation theory for point groups and space groups(OUP, Oxford, 2009)

    2009

  2. [2]

    Lax,Symmetry Principles in Solid State and Molec- ular Physics(Courier Corporation, 2001)

    M. Lax,Symmetry Principles in Solid State and Molec- ular Physics(Courier Corporation, 2001)

    2001

  3. [3]

    M. S. Dresselhaus, G. Dresselhaus, and A. Jorio,Group theory: application to the physics of condensed matter (Springer-Verlag, Berlin, 2008)

    2008

  4. [4]

    Bansil, H

    A. Bansil, H. Lin, and T. Das, Colloquium: Topological band theory, Rev. Mod. Phys.88, 021004 (2016)

    2016

  5. [5]

    C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with sym- metries, Rev. Mod. Phys.88, 035005 (2016)

    2016

  6. [6]

    Cano and B

    J. Cano and B. Bradlyn, Band Representations and Topological Quantum Chemistry, Annu. Rev. Condens. Matter Phys.12, 225 (2021)

    2021

  7. [7]

    Z.-M. Yu, Z. Zhang, G.-B. Liu, W. Wu, X.-P. Li, R.-W. Zhang, S.A.Yang,andY.Yao,Encyclopediaofemergent particles in three-dimensional crystals, Sci. Bull.67, 375 (2022)

    2022

  8. [8]

    Opechowski and R

    W. Opechowski and R. Guccione,Magnetic symmetry (New York: Academic Press, 1965) publisher: Academic Press

    1965

Show all 45 references
  1. [9]

    C. J. Bradley and B. L. Davies, Magnetic Groups and Their Corepresentations, Rev. Mod. Phys.40, 359 (1968)

    1968

  2. [10]

    D. B. Litvin,Magnetic Subperiodic Groups and Magnetic Space Groups(Wiley Online Library, 2016)

    2016

  3. [11]

    W. F. Brinkman and R. J. Elliott, Theory of spin-space groups, Proc. R. Soc. Lond. A294, 343 (1966)

    1966

  4. [12]

    Litvin and W

    D. Litvin and W. Opechowski, Spin Groups, Physica76, 538 (1974)

    1974

  5. [13]

    D. B. Litvin, Spin point groups, Acta Crystallogr. A33, 279 (1977)

    1977

  6. [14]

    X. Chen, J. Ren, Y. Zhu, Y. Yu, A. Zhang, P. Liu, J. Li, Y. Liu, C. Li, and Q. Liu, Enumeration and Represen- tation Theory of Spin Space Groups, Phys. Rev. X14, 031038 (2024)

    2024

  7. [15]

    Jiang, Z

    Y. Jiang, Z. Song, T. Zhu, Z. Fang, H. Weng, Z.-X. Liu, J. Yang, and C. Fang, Enumeration of Spin-Space Groups: Toward a Complete Description of Symmetries of Magnetic Orders, Phys. Rev. X14, 031039 (2024)

    2024

  8. [16]

    Z. Xiao, J. Zhao, Y. Li, R. Shindou, and Z.-D. Song, Spin Space Groups: Full Classification and Applications, Phys. Rev. X14, 031037 (2024)

    2024

  9. [17]

    van der Waerden and J

    B. van der Waerden and J. J. Burkhardt, Farbgruppen, Z. Kristallogr.115, 231 (1961)

    1961

  10. [18]

    Shubnikov and N.V

    A.V. Shubnikov and N.V. Belov,Colored Symmetry (THE MACMILLAN COMPANY, 1964)

    1964

  11. [19]

    R. L. E. Schwarzenberger, Colour Symmetry: PART ONE: SYMMETRY GROUPS, Bull. Lond. Math. Soc. 16, 209 (1984)

    1984

  12. [20]

    Lifshitz, Theory of color symmetry for periodic and quasiperiodic crystals, Rev

    R. Lifshitz, Theory of color symmetry for periodic and quasiperiodic crystals, Rev. Mod. Phys.69, 1181 (1997)

    1997

  13. [21]

    P. A. Heiney, J. E. Fischer, A. R. McGhie, W. J. Ro- manow, A. M. Denenstein, J. P. McCauley Jr., A. B. Smith, and D. E. Cox, Orientational ordering transition in solid C60, Phys. Rev. Lett.66, 2911 (1991)

    1991

  14. [22]

    H. L. B. Boström, M. S. Senn, and A. L. Goodwin, Recipes for improper ferroelectricity in molecular per- ovskites, Nat. Commun.9, 2380 (2018)

    2018

  15. [23]

    Z. Liu, X. Zhang, Y. Mao, Y. Y. Zhu, Z. Yang, C. T. Chan, and P. Sheng, Locally Resonant Sonic Materials, Science289, 1734 (2000)

    2000

  16. [24]

    Eichenfield, J

    M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, Optomechanical crystals, Nature462, 78 (2009)

    2009

  17. [25]

    L. M. Nash, D. Kleckner, A. Read, V. Vitelli, A. M. Turner, and W. T. M. Irvine, Topological mechanics of gyroscopic metamaterials, Proc. Natl. Acad. Sci. U.S.A. 112, 14495 (2015)

    2015

  18. [26]

    S.-Z.Lin, C.D.Batista,andA.Saxena,Internalmodesof a skyrmion in the ferromagnetic state of chiral magnets, Phys. Rev. B89, 024415 (2014)

    2014

  19. [27]

    Zhang, J

    X. Zhang, J. Xia, Y. Zhou, X. Liu, H. Zhang, and M. Ezawa, Skyrmion dynamics in a frustrated ferromag- netic film and current-induced helicity locking-unlocking transition, Nat. Commun.8, 1717 (2017)

    2017

  20. [28]

    Murakami, J

    Y. Murakami, J. P. Hill, D. Gibbs, M. Blume, I. Koyama, M. Tanaka, H. Kawata, T. Arima, Y. Tokura, K. Hirota, and Y. Endoh, Resonant x-ray scattering from orbital ordering in LaMnO3, Phys. Rev. Lett.81, 582 (1998)

    1998

  21. [29]

    Watanabe and Y

    H. Watanabe and Y. Yanase, Group-theoretical classifi- cation of multipole order: Emergent responses and can- didate materials, Phys. Rev. B98, 245129 (2018)

    2018

  22. [30]

    D. A. Lenz, R. Blaak, C. N. Likos, and B. M. Mladek, Mi- croscopically Resolved Simulations Prove the Existence of Soft Cluster Crystals, Phys. Rev. Lett.109, 228301 (2012)

    2012

  23. [31]

    Stiakakis, N

    E. Stiakakis, N. Jung, N. Adžić, T. Balandin, E. Kentzinger, U. Rücker, R. Biehl, J. K. G. Dhont, U.Jonas,andC.N.Likos,Self-assemblingclustercrystals from DNA-based dendritic nanostructures, Nat. Com- mun.12, 7167 (2021)

    2021

  24. [32]

    O. M. Yaghi, M. O’Keeffe, N. W. Ockwig, H. K. Chae, M. Eddaoudi, and J. Kim, Reticular synthesis and the design of new materials, Nature423, 705 (2003)

    2003

  25. [33]

    M. Li, D. Li, M. O’Keeffe, and O. M. Yaghi, Topologi- cal analysis of metal–organic frameworks with polytopic linkers and/or multiple building units and the minimal transitivity principle, Chem. Rev.114, 1343 (2014)

    2014

  26. [34]

    L. T. Glasby, J. L. Cordiner, J. C. Cole, and P. Z. Moghadam, Topological characterization of metal– organic frameworks: A perspective, Chem. Mater.36, 9013 (2024)

    2024

  27. [35]

    Mariette, E

    C. Mariette, E. Trzop, J.-Y. Mevellec, A. Boucekkine, A. Ghoufi, G. Maurin, E. Collet, M. C. Muñoz, J. A. Real, and B. Toudic, Symmetry breakings in a metal or- 6 ganicframeworkwithaconfinedguest,Phys.Rev.B101, 134103 (2020)

    2020

  28. [36]

    Hall,The theory of groups(The Macmillan Company,

    M. Hall,The theory of groups(The Macmillan Company,

  29. [37]

    Šmejkal, J

    L. Šmejkal, J. Sinova, and T. Jungwirth, Beyond Con- ventional Ferromagnetism and Antiferromagnetism: A Phase with Nonrelativistic Spin and Crystal Rotation Symmetry, Phys. Rev. X12, 031042 (2022)

    2022

  30. [38]

    The GAP Group,GAP – Groups, Algorithms, and Pro- gramming, Version 4.14.0(The GAP Group, 2024)

    2024

  31. [39]

    Zhang, G.-B

    Z. Zhang, G.-B. Liu, M. Tian, R.-W. Zhang, Z.-M. Yu, andY.Yao,Spinlayergroupsandtheircorepresentations (2026), arXiv:2605.25484 [cond-mat.mtrl-sci]

    2026 arXiv

  32. [40]

    Zhang, Z.-M

    Z. Zhang, Z.-M. Yu, G.-B. Liu, Z. Li, S. A. Yang, and Y. Yao, MagneticKP: A package for quickly construct- ingk·pmodels of magnetic and nonmagnetic crystals, Comput. Phys. Commun.290, 108784 (2023)

    2023

  33. [41]

    Tang and X

    F. Tang and X. Wan, Exhaustive construction of effective models in 1651 magnetic space groups, Phys. Rev. B104, 085137 (2021)

    2021

  34. [42]

    Tang and X

    F. Tang and X. Wan, Complete classification of band nodal structures and massless excitations, Phys. Rev. B 105, 155156 (2022)

    2022

  35. [43]

    G.-B. Liu, Z. Zhang, Z.-M. Yu, S. A. Yang, and Y. Yao, Systematic investigation of emergent particles in type-III magnetic space groups, Phys. Rev. B105, 085117 (2022)

    2022

  36. [44]

    Zhang, G.-B

    Z. Zhang, G.-B. Liu, Z.-M. Yu, S. A. Yang, and Y. Yao, Encyclopedia of emergent particles in type-IV magnetic space groups, Phys. Rev. B105, 104426 (2022)

    2022

  37. [45]

    Yang, Z.-X

    J. Yang, Z.-X. Liu, and C. Fang, Symmetry invariants and classes of quasiparticles in magnetically ordered sys- tems having weak spin-orbit coupling, Nat. Commun.15, 10203 (2024)

    2024

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.