Altermagnetic Spin-Point Groups

• Altermagnetic spin-point groups are finite symmetry groups that act independently on spin and real space in weak spin–orbit-coupling regimes, defining magnetic orders with zero net dipole.
• They decouple spin and spatial rotations, leading to nonrelativistic momentum-dependent spin splitting that differentiates them from ferromagnetic and Rashba–Dresselhaus behaviors.
• Their tensorial classifications predict macroscopic responses such as piezomagnetism, Kerr effects, and quantized photocurrents, as evidenced in materials like RuO₂ and MnTiO₃.

Altermagnetic spin-point groups are the finite symmetry groups that act separately on spin space and real space in the weak-spin–orbit-coupling regime and classify magnetic orders whose electronic bands exhibit nonrelativistic, momentum-dependent spin splitting despite vanishing net magnetization. In the altermagnetic case, the defining symmetry content is broken time reversal, absence of an allowed net dipole, and the presence of higher-rank time-reversal-odd, parity-even invariants that enforce $\mathbf B(\mathbf k)=-\mathbf B(-\mathbf k)$ rather than conventional ferromagnetic or Rashba–Dresselhaus behavior. The subject is developed through complementary formalisms: spin-point groups and spin-space groups, tensorial classifications in magnetic point groups, and reduced magnetic-group descriptions in which electron spin is treated as a pseudoscalar (Radaelli, 2024, Jiang et al., 2023, Schiff et al., 2023, Turek, 2022).

1. Formal definition and group-theoretic setting

A spin-point group $X$ is a finite subgroup of $O(3)\times SU(2)$ whose elements act on both real space and spin space. In the notation $[S\!\parallel\!R]$, $R$ acts on $\mathbf r\in\mathbb R^3$ while $S$ acts on the magnetic moment or spin degrees of freedom; the group law is

$$[\,S_1\!\parallel\!R_1\,]\cdot[\,S_2\!\parallel\!R_2\,]=[\,S_1S_2\!\parallel\!R_1R_2\,].$$

If spin and spatial operations are locked, one recovers ordinary magnetic point groups; if they are independent, one obtains the crystallographic spin-point groups appropriate to weak or intermediate spin–orbit coupling (Schiff et al., 2023).

The same structure arises as the point-group quotient of a spin-space group. In that language, a spin-space group $\mathcal G^s$ contains operations $\{U\!\parallel\!R|\tau\}$ acting on the pseudovector field $X$0 by

$X$1

and the corresponding spin-point group is obtained by modding out translations: $X$2 For collinear orders, the spin-only subgroup is continuous, $X$3, so the nontrivial content lies in how discrete spatial operations are paired with spin flips or spin rotations (Jiang et al., 2023).

This decoupling is the essential distinction from conventional magnetic point groups. In the nonrelativistic collinear limit, magnetic point groups do not untangle spin and spatial rotations, whereas spin-point groups can contain operations that exchange spin-up and spin-down sublattices through a nontrivial spatial operation without requiring the same rotation in real and spin space. This is precisely the symmetry mechanism that distinguishes altermagnets from both collinear ferromagnets and Kramers-degenerate antiferromagnets (Šmejkal et al., 2021).

2. Relation to magnetic point groups and reduced magnetic-group formulations

An altermagnetic magnetic point group $X$4 is defined by three conditions: $X$5, the spin splitting obeys $X$6, and the group forbids a net dipole while admitting higher-rank, time-odd, parity-even tensors. In Radaelli’s tensorial formulation, this means that the lowest nonzero time-odd invariant is of odd rank $X$7: rank $X$8 corresponds to ferromagnetism, while rank $X$9 corresponds to Rashba–Dresselhaus families (Radaelli, 2024).

This tensorial viewpoint was introduced partly to clarify a recurrent contrast in the literature between spin groups and magnetic point groups. The tensor approach shows that both frameworks can be discussed within a common invariant-theory language. A direct implication is that altermagnetism is not a category external to magnetic crystallography; rather, it is a specific symmetry sector in which the lowest allowed time-odd invariant has altermagnetic parity and rank (Radaelli, 2024).

A complementary reduction is possible for nonrelativistic collinear magnets when electron spin is treated as a pseudoscalar. In that setting, one replaces the full spin group by a modified magnetic group with elements $O(3)\times SU(2)$0, $O(3)\times SU(2)$1, and the spin operator transforms as

$O(3)\times SU(2)$2

The resulting invariant tensors near $O(3)\times SU(2)$3 are classified by magnetic Laue groups. Turek showed that exactly ten nontrivial magnetic Laue groups of category (c) admit momentum-dependent spin splitting, and that these are in one-to-one correspondence with the ten nontrivial spin-Laue classes in the Šmejkal–Sinova–Jungwirth classification. Among them, the four classes with $O(3)\times SU(2)$4 — $O(3)\times SU(2)$5, $O(3)\times SU(2)$6, $O(3)\times SU(2)$7, and $O(3)\times SU(2)$8 — are precisely those compatible with a nonzero spin-conductivity tensor (Turek, 2022).

This multiplicity of descriptions is therefore not contradictory. It reflects different quotients and different invariant objects: full spin-point groups, magnetic point groups, and reduced magnetic Laue groups encode the same nonrelativistic symmetry breaking at different levels of resolution.

3. Classification landscape

Several classification schemes are simultaneously in use, each counting a different object. The following summary captures the numerical structure reported across the literature.

Framework	Object counted	Result
Crystallographic spin-point groups	Nontrivial spin-point groups	598
Enumerated spin-space groups up to supercell index $O(3)\times SU(2)$9	Collinear SSGs	1,421
Tensorial magnetic-point-group classification	Altermagnetic MPGs	69 in 18 classes
Two-dimensional spin-group classification	Altermagnetic spin-layer groups	7
Polar altermagnetic spin groups	Polar collinear altermagnetic spin-point groups	11
Noncentrosymmetric SPG response classification	SPGs with pure-spin CPGE	10

The tensorial classification of the 69 altermagnetic magnetic point groups is especially explicit. For a given band $[S\!\parallel\!R]$0 and wavevector $[S\!\parallel\!R]$1, the effective Zeeman field is expanded as

$[S\!\parallel\!R]$2

with $[S\!\parallel\!R]$3 fully symmetric in the Greek indices. Even-rank tensors are $[S\!\parallel\!R]$4-odd and $[S\!\parallel\!R]$5-even, corresponding to Rashba–Dresselhaus families; odd-rank tensors are $[S\!\parallel\!R]$6-even and $[S\!\parallel\!R]$7-odd, corresponding to altermagnetic families. The lowest odd rank $[S\!\parallel\!R]$8 allowed by symmetry defines the altermagnetic class (Radaelli, 2024).

In this scheme, Classes I–XVII have $[S\!\parallel\!R]$9, whereas Class XVIII is the single $R$0 case in which all $R$1 tensors vanish and the lowest nonzero invariant is a rank-5 hexadecapolar form. The cubic Class XVIII, with magnetic point groups $R$2, $R$3, and $R$4, has

$R$5

and is distinguished by the absence of quadratic altermagnetic invariants (Radaelli, 2024).

A second taxonomy, developed in spin-group language, organizes three-dimensional altermagnets into six characteristic spin-momentum-locking types: three planar and three bulk, with even winding numbers $R$6 and lowest harmonics denoted $R$7, $R$8, and $R$9. The associated nontrivial spin-Laue groups are all of the form

$\mathbf r\in\mathbb R^3$0

where $\mathbf r\in\mathbb R^3$1 preserves same-spin sublattices and $\mathbf r\in\mathbb R^3$2 exchanges opposite-spin sublattices (Šmejkal et al., 2021).

In two dimensions, spin-group analysis yields seven nontrivial altermagnetic spin-layer groups rather than the five spin-Laue groups familiar from earlier three-dimensional discussions. This two-dimensional extension produces the symmetry basis used to classify monolayer materials such as MnTeMoO$\mathbf r\in\mathbb R^3$3 and VP$\mathbf r\in\mathbb R^3$4H$\mathbf r\in\mathbb R^3$5(NO$\mathbf r\in\mathbb R^3$6)$\mathbf r\in\mathbb R^3$7 (Zeng et al., 2024).

4. Representation theory, band degeneracies, and canonical examples

The representation theory of crystallographic spin-point groups shows that nontrivial spin-point groups have co-representations corresponding exactly to the irreducible representations or co-representations of regular or black-and-white groups, while total spin groups with continuous spin-only factors can generate genuinely new co-representations. In the collinear case with non-unitary $\mathbf r\in\mathbb R^3$8, these new co-representations can have dimensions up to $\mathbf r\in\mathbb R^3$9, and their construction is controlled by the Dimmock test

$S$0

with the standard cases $S$1, $S$2, and $S$3 (Schiff et al., 2023).

RuO$S$4 is the paradigmatic worked example. Starting from the nonmagnetic space group $S$5 (#136), the quotient $S$6 admits a real representation sending the coset generator $S$7 to a spin-space mirror $S$8. After dropping translations, the spin-point group is

$S$9

Within the spin-space-group framework, this extra unlocked symmetry predicts protected double degeneracies along $$[\,S_1\!\parallel\!R_1\,]\cdot[\,S_2\!\parallel\!R_2\,]=[\,S_1S_2\!\parallel\!R_1R_2\,].$$0 in the absence of SOC, whereas elsewhere in the Brillouin zone the same symmetry permits the momentum-odd spin splitting that characterizes altermagnetism (Jiang et al., 2023, Chen et al., 2023).

The low-energy form of the RuO$$[\,S_1\!\parallel\!R_1\,]\cdot[\,S_2\!\parallel\!R_2\,]=[\,S_1S_2\!\parallel\!R_1R_2\,].$$1 splitting is also a canonical tensor example. In the tetragonal Class X, the rank-3 invariant yields

$$[\,S_1\!\parallel\!R_1\,]\cdot[\,S_2\!\parallel\!R_2\,]=[\,S_1S_2\!\parallel\!R_1R_2\,].$$2

Radaelli identifies RuO$$[\,S_1\!\parallel\!R_1\,]\cdot[\,S_2\!\parallel\!R_2\,]=[\,S_1S_2\!\parallel\!R_1R_2\,].$$3 as a “poster child” altermagnet and reports a giant $$[\,S_1\!\parallel\!R_1\,]\cdot[\,S_2\!\parallel\!R_2\,]=[\,S_1S_2\!\parallel\!R_1R_2\,].$$4 splitting along $$[\,S_1\!\parallel\!R_1\,]\cdot[\,S_2\!\parallel\!R_2\,]=[\,S_1S_2\!\parallel\!R_1R_2\,].$$5; Pb$$[\,S_1\!\parallel\!R_1\,]\cdot[\,S_2\!\parallel\!R_2\,]=[\,S_1S_2\!\parallel\!R_1R_2\,].$$6MnO$$[\,S_1\!\parallel\!R_1\,]\cdot[\,S_2\!\parallel\!R_2\,]=[\,S_1S_2\!\parallel\!R_1R_2\,].$$7 is given as a collinear antiferromagnetic insulator with observed piezomagnetism and Kerr response in the same class (Radaelli, 2024).

A second canonical example is the trigonal Class XIV, whose generators can be taken as $$[\,S_1\!\parallel\!R_1\,]\cdot[\,S_2\!\parallel\!R_2\,]=[\,S_1S_2\!\parallel\!R_1R_2\,].$$8, $$[\,S_1\!\parallel\!R_1\,]\cdot[\,S_2\!\parallel\!R_2\,]=[\,S_1S_2\!\parallel\!R_1R_2\,].$$9, and $\mathcal G^s$0. The rank-3 tensor has the nonzero entries $\mathcal G^s$1, $\mathcal G^s$2, and $\mathcal G^s$3, giving

$\mathcal G^s$4

This is a $\mathcal G^s$5-wave texture in the $\mathcal G^s$6 plane. CrSb and HoMnO$\mathcal G^s$7 are listed as material realizations consistent with this tensor form (Radaelli, 2024).

In two dimensions, the symmetry logic is analogous but the admissible groups differ. The seven spin-layer groups exhaust the nondegenerate altermagnetic possibilities compatible with the two-dimensional restriction that $\mathcal G^s$8 remain in the spin-identity half of the index-2 splitting. Monolayer MnTeMoO$\mathcal G^s$9 is assigned to $\{U\!\parallel\!R|\tau\}$0, and monolayer VP$\{U\!\parallel\!R|\tau\}$1H$\{U\!\parallel\!R|\tau\}$2(NO$\{U\!\parallel\!R|\tau\}$3)$\{U\!\parallel\!R|\tau\}$4 to $\{U\!\parallel\!R|\tau\}$5, with the resulting nodal lines and angular anisotropies of spin splitting determined directly by the spin-flip subset $\{U\!\parallel\!R|\tau\}$6 (Zeng et al., 2024).

5. Extensions: noncollinear, polar, and non-crystalline altermagnetic spin groups

Although the earliest spin-point-group treatments focused on collinear magnets, the formalism has been extended in several directions. One extension uses colour symmetry for noncollinear antiferromagnets. In that framework, each distinct spin direction is treated as a colour, and the colour point group $\{U\!\parallel\!R|\tau\}$7 organizes the momentum-space spin texture. The full texture is decomposed as

$\{U\!\parallel\!R|\tau\}$8

where $\{U\!\parallel\!R|\tau\}$9 is the lowest-order SOC-independent component invariant under global spin rotations encoded by the colour symmetry: $X$00 The explicitly altermagnetic component is constructed as

$X$01

Examples include $X$02 for Mn$X$03Ir(Ge,Si), $X$04 for Pb$X$05MnO$X$06, and $X$07 for Mn$X$08GaN (Radaelli et al., 6 Jan 2025).

A second extension concerns polar altermagnetic spin groups. Starting from the ten polar nonmagnetic point groups, one asks which collinear compensated orders preserve a combined $X$09 symmetry with $X$10. Out of the 37 possible collinear altermagnetic spin-point groups enumerated in the cited work, exactly 11 are polar. Representative cases are $X$11 for BaCuF$X$12, $X$13 for Ca$X$14Mn$X$15O$X$16, and a collinear BiFeO$X$17 phase assigned to a polar altermagnetic spin group. These systems were proposed as altermagnetic multiferroics and as the setting for an altermagnetoelectric effect (Šmejkal, 2024).

A third extension removes the assumption of global crystal symmetry. In an amorphous or otherwise non-crystalline setting with two directional orbitals per site, the order parameter

$X$18

can preserve a local altermagnetic spin-point group

$X$19

with the general sequence

$X$20

This establishes that altermagnetic spin-point-group structure can be anchored in local orbital-shape symmetry rather than global crystallographic rotations (d'Ornellas et al., 11 Apr 2025).

These generalizations support a broader interpretation of altermagnetic spin-point groups: they are not limited to crystalline, two-sublattice, collinear antiferromagnets, even though that case remains the best-developed and experimentally most visible realization.

6. Macroscopic responses, topology, and dynamical probes

The tensorial formulation makes the connection between symmetry and bulk response unusually direct. The quadratic altermagnetic tensor has Jahn symbol $X$21, identical to the tensor forms of the linear piezomagnetic tensor $X$22 and the linear magneto-optical Kerr tensor $X$23. By Neumann’s principle, every one of the 66 magnetic point groups in Classes I–XVII must therefore allow both

$X$24

and a nonzero polar Kerr effect. The single $X$25 cubic class forbids piezomagnetism and MOKE while allowing quartic altermagnetism through the pure hexadecapole (Radaelli, 2024).

Nonlinear optical responses can also be organized directly by spin-point groups. For the second-order bulk photovoltaic effect,

$X$26

the SPG invariance conditions determine which tensor components survive. In the classification of noncentrosymmetric altermagnetic SPGs, 27 groups are partitioned according to whether $X$27 and $X$28 vanish. Ten SPGs admit a pure-spin CPGE, meaning $X$29 while $X$30. In altermagnetic Weyl semimetals this yields a quantized pure-spin CPGE,

$X$31

The paper constructs a tight-binding model for the spin-space group $X$32 and identifies MnTiO$X$33 as a first-principles candidate (Yoshida et al., 6 Sep 2025).

Dynamical symmetry extends the same logic to high-harmonic generation. In a minimal altermagnetic model, the characteristic generator is $X$34, and harmonic selection rules follow from the condition

$X$35

For single-helicity circular driving in the $X$36 case, the $X$37-polarized spin-current harmonics satisfy a $X$38 alternation. The cited work emphasizes that this fourfold alternation is a unique dynamical fingerprint of the altermagnetic spin-point group and is not reproduced by ferromagnetic or simple antiferromagnetic phases (Mizuno, 30 Jun 2026).

Light-induced spin torque provides a further diagnostic. For linearly polarized light, the second-order susceptibility

$X$39

transforms as a rank-3 polar–pseudotensor. The complete enumeration over the 122 magnetic point groups identifies 11 centrosymmetric magnetic point groups with vanishing equilibrium moment but nonzero light-induced total spin torque in altermagnets. The contrast with PT-invariant antiferromagnets is explicit in the case studies: FeSe monolayer has $X$40 strictly, whereas the altermagnetic V$X$41Se$X$42O monolayer supports finite $X$43 and a global $X$44 under linearly polarized light (Zhou et al., 11 Apr 2025).

Taken together, these results place altermagnetic spin-point groups at the junction of magnetic crystallography, invariant theory, and band-topological response theory. They classify not only static spin splitting, but also protected degeneracies, permitted tensor components, multiferroic couplings, nonlinear photocurrents, and dynamical spin-current harmonics.