Antiferromagnetic SPGs: Symmetry & Applications

• Antiferromagnetic SPGs are symmetry groups that act simultaneously on real and spin space, defining the parent structure for nonrelativistic antiferromagnets.
• They determine when momentum-dependent spin splitting and strain-induced Zeeman responses occur, with validations from first-principles studies in materials like CoF₂.
• SPGs connect magnetic order to electronic, optical, and transport phenomena, offering a symmetry-based framework for material design and response prediction.

Searching arXiv for recent and foundational papers on antiferromagnetic spin point groups and closely related spin-group symmetry. Antiferromagnetic spin point groups (SPGs) are symmetry groups for magnetic structures whose elements act jointly on real space and spin space, and they become the natural point-group-level language for antiferromagnets when spin and lattice operations are correlated but not rigidly locked, especially in weak- and intermediate-spin-orbit-coupling regimes (Schiff et al., 2023). In this framework, a symmetry element is written as $[S\|R]$ or equivalently $\{O_s\Vert O_l\}$, with $R$ or $O_l$ acting in real space and $S$ or $O_s$ acting in spin space; magnetic point groups are recovered as the special case in which the real-space and spin-space operations are locked to one another (Zhai et al., 9 Jun 2025). For antiferromagnets, SPGs organize the symmetry of compensated order, encode when momentum-dependent spin splitting is allowed or forbidden without spin-orbit coupling, clarify how collinear and noncollinear orders differ, and provide a parent nonrelativistic symmetry that descends to orientation-dependent magnetic point groups once spin-orbit interaction is included (Zhang et al., 11 Dec 2025). They also supply a route from magnetic structure to concrete electronic, optical, and transport consequences, including altermagnetic spin splitting, strain-induced Zeeman-type responses, high-harmonic spin-current selection rules, and topology controlled by antiunitary magnetic symmetries (Zhai et al., 9 Jun 2025, Mizuno, 30 Jun 2026, Bouhon et al., 2020).

1. Definition, notation, and relation to magnetic crystallography

The defining feature of an SPG is that its operations act on both real space and spin space. In the notation used for collinear antiferromagnets and altermagnets, an element is written as $[g_s \parallel g_l]$, where $g_s$ is an operation in spin space and $g_l$ is an operation in real space (Zhai et al., 9 Jun 2025). A closely related notation used for collinear magnets writes an element as $\{O_s\Vert O_l\}$, with the same division of labor between spin-space and lattice-space actions (Zhang et al., 11 Dec 2025). The representation-theoretic treatment of crystallographic SPGs formulates them as finite groups whose elements act simultaneously on a spin texture $\{O_s\Vert O_l\}$0, with a generic action $\{O_s\Vert O_l\}$1 (Schiff et al., 2023).

This is more general than an ordinary crystallographic point group, which acts only in real space, and more general than a magnetic point group, which allows time reversal but in the strong-spin-orbit-coupling setting effectively ties spin and lattice transformations together (Schiff et al., 2023). In weak spin-orbit coupling, that locking is absent, so SPGs rather than magnetic point groups become the appropriate nonrelativistic symmetry objects (Mizuno, 30 Jun 2026). The same conceptual distinction motivates the full spin-space-group formalism, where a symmetry operation is written $\{O_s\Vert O_l\}$2 and the point-group reduction yields SPG-level data such as spin-space point groups, spin site-symmetry groups, and spin little co-groups (Yu et al., 23 Apr 2026).

A recurring structural decomposition is the separation between a spin-only subgroup and a nontrivial spin point-group part. For collinear magnets, one paper writes the SPG as

$\{O_s\Vert O_l\}$3

where $\{O_s\Vert O_l\}$4 is the spin-only part and $\{O_s\Vert O_l\}$5 is the nontrivial spin point group (Zhai et al., 9 Jun 2025). In the treatment of collinear magnetic structures, the spin-only part is

$\{O_s\Vert O_l\}$6

with $\{O_s\Vert O_l\}$7 a spin rotation about the ordered-moment axis, $\{O_s\Vert O_l\}$8 a $\{O_s\Vert O_l\}$9-rotation about an axis perpendicular to it, and $R$0 time reversal (Zhai et al., 9 Jun 2025). The representation-theoretic account of crystallographic SPGs states the full spin point group as a product $R$1, with $R$2 the spin-only group and $R$3 the nontrivial spin point group; for collinear systems the relevant spin-only class is

$R$4

(Schiff et al., 2023).

A central conceptual distinction follows. In nonrelativistic collinear magnets, the same scalar magnetic pattern can support many vector magnetic structures because the spin-alignment direction is not yet locked to the lattice. The magnetic structure is then the direct product of a scalar magnetic structure and an orientation vector $R$5, with local moments written as $R$6 or $R$7 (Zhang et al., 11 Dec 2025). In this sense, the SPG is the parent symmetry of the collinear antiferromagnet before spin-orbit coupling selects a particular magnetic point group.

2. Collinear antiferromagnets and the parent-symmetry role of SPGs

For collinear antiferromagnets, SPGs provide a nonrelativistic parent description that is independent of the chosen Néel-vector orientation. The paper on spin-orientation-driven polarization states that a collinear magnetic structure should be separated into a scalar magnetic structure and an orientation vector $R$8, and that the corresponding SPG is the appropriate symmetry group before spin-orbit interaction couples spin and lattice (Zhang et al., 11 Dec 2025). Once spin-orbit interaction is included, the symmetry is lowered to an orientation-dependent magnetic point group: $R$9 The paper states this in words as “the MPG is a subgroup of the SPG, and the operations of the MPG should be compatible with the orientation vector” (Zhang et al., 11 Dec 2025).

This parent-to-descendant relation is especially important for antiferromagnets because a single scalar antiferromagnetic pattern can produce distinct vector states, such as

$O_l$0

all sharing the same parent SPG but generally reducing to different magnetic point groups once spin-orbit interaction is active (Zhang et al., 11 Dec 2025). The practical implication is that polarity, tensor responses, and multiferroic behavior can depend on spin orientation even though the underlying antiferromagnetic scalar order is unchanged.

The CuFeS$O_l$1 example provides an explicit antiferromagnetic realization. Its parent SPG is

$O_l$2

while the orientation $O_l$3 yields the nonpolar magnetic point group

$O_l$4

$O_l$5 yields

$O_l$6

also nonpolar, and $O_l$7 yields the polar magnetic point group

$O_l$8

which allows

$O_l$9

(Zhang et al., 11 Dec 2025). The same parent antiferromagnetic SPG therefore contains both nonpolar and polar descendants, depending only on the Néel-vector orientation. The paper reports that first-principles calculations on this room-temperature antiferromagnet with $S$0 confirm a sinusoidal orientation dependence of polarization with period $S$1 (Zhang et al., 11 Dec 2025).

The formal collinear classification also interfaces with older symmetry analyses that did not use explicit SPG terminology. The work on momentum-dependent spin splitting in collinear antiferromagnets without atomic spin-orbit coupling relies on ordinary crystallographic point groups, multipoles, SU(2) spin symmetry, and the antiunitary symmetry $S$2, where $S$3 is a spin-space rotation reversing the spin quantization axis (Hayami et al., 2019). That paper shows that in the absence of atomic spin-orbit coupling, $S$4 enforces

$S$5

so spin splitting induced by collinear antiferromagnetic order must be even in $S$6 (Hayami et al., 2019). The symmetry logic is SPG-like even though the formal SPG language is not used.

A related reformulation shows that, for nonrelativistic collinear magnetic systems with perfect translation invariance, the full spin-group description can be reduced to modified magnetic point groups if the electron spin is treated as an effective pseudoscalar under spatial operations and sign-reversed only by antisymmetry (Turek, 2022). In that formulation, the spin-point-group content survives as a reduced “skeleton” magnetic group $S$7, where $S$8 tracks whether the operation preserves or interchanges the spin channels (Turek, 2022). This suggests that, in the strictly collinear nonrelativistic limit, SPG data can sometimes be compressed without loss of the symmetry classification relevant for spin splitting near $S$9.

3. Spin splitting, altermagnetism, and SPG selection rules in antiferromagnets

One of the central uses of antiferromagnetic SPGs is the classification of momentum-dependent spin splitting without spin-orbit coupling. In the magnetic-space-group language, a necessary condition for generic-$O_s$0 spin splitting is the absence of both

$O_s$1

because if either symmetry is present then

$O_s$2

and each $O_s$3 point is spin-degenerate (Yuan et al., 2020). The paper emphasizes that this is fundamentally a symmetry classification of when antiferromagnets can or cannot exhibit momentum-dependent spin splitting without spin-orbit coupling, and that the logic is directly portable to SPG language once translational details are stripped away (Yuan et al., 2020).

At the point-group level, the classification of collinear antiferromagnetic and altermagnetic spin splitting is formulated directly in terms of SPGs. For collinear antiferromagnets with $O_s$4-wave altermagnetic splitting, the effective momentum-space term is

$O_s$5

where $O_s$6 refers to the collinear spin axis (Zhai et al., 9 Jun 2025). The crucial symmetry observation is that $O_s$7 transforms identically to the strain tensor component $O_s$8, leading to the strain analogue

$O_s$9

(Zhai et al., 9 Jun 2025). In this treatment, the decisive symmetry object is the nontrivial SPG part $[g_s \parallel g_l]$0, because the spin-only group leaves both the strain tensor and $[g_s \parallel g_l]$1 invariant (Zhai et al., 9 Jun 2025).

A full scan of the 58 collinear antiferromagnetic SPGs identifies 15 SPGs that host strain-induced nonrelativistic Zeeman-type spin splittings, and the paper states that these 15 SPGs coincide exactly with the 15 collinear SPGs previously identified as hosting $[g_s \parallel g_l]$2-wave altermagnetic spin splittings (Zhai et al., 9 Jun 2025). The result is both a group-theoretic classification and a material-design rule. For example, the SPG

$[g_s \parallel g_l]$3

allows

$[g_s \parallel g_l]$4

with the corresponding $[g_s \parallel g_l]$5-space invariant

$[g_s \parallel g_l]$6

(Zhai et al., 9 Jun 2025).

First-principles calculations without spin-orbit coupling corroborate this SPG classification in CoF$[g_s \parallel g_l]$7, LiFe$[g_s \parallel g_l]$8F$[g_s \parallel g_l]$9, and La$g_s$0O$g_s$1Mn$g_s$2Se$g_s$3. A shear strain of $g_s$4 produces spin splittings of up to $g_s$5, $g_s$6, and $g_s$7 meV, respectively, with a linear dependence $g_s$8 (Zhai et al., 9 Jun 2025). The important SPG-level conclusion is that spin-group symmetry alone, without spin-orbit coupling, can predict both momentum-dependent and strain-induced spin splitting in compensated collinear antiferromagnets.

A broader algorithmic catalogue of antiferromagnetic spin-momentum locking is formulated with magnetic point groups, little co-groups, and coset representatives, and the paper states that its method can be extended straightforwardly to spin groups and spin space groups (Hu et al., 2024). In that framework, the symmetry data determining the locking are: the little co-group $g_s$9, which fixes spin orientation at a seed momentum, and coset representatives $g_l$0, which generate all symmetry-related valleys and their spin textures through

$g_l$1

(Hu et al., 2024). This suggests that SPG-based classifications can be organized by the same stabilizer-orbit logic.

The 2D adsorption framework makes the SPG role fully explicit. By classifying all point operations derived from 80 layer groups, that work derives 63 antiferromagnetic SPGs for 2D materials, of which 26 are intrinsically altermagnetic and 37 host spin degeneracy; from those 37, it further isolates 15 SPGs in which surface adsorption can break the degeneracy-protecting operation while preserving a sublattice-connecting rotation or mirror, thereby inducing altermagnetism (Liu et al., 13 Jul 2025). The abstract states the symmetry-engineering rule directly: selectively breaking the operations protecting spin degeneracy enables nonrelativistic spin-split states, while preserving rotation or mirror symmetries connecting opposite sublattices ensures zero net magnetization (Liu et al., 13 Jul 2025). In monolayer VPS$g_l$2 and MnPSe$g_l$3, adsorption breaks $g_l$4-type spin-degeneracy protection while retaining a $g_l$5-type symmetry, and first-principles calculations confirm pronounced spin splitting without spin-orbit coupling (Liu et al., 13 Jul 2025).

4. Noncollinear antiferromagnetic SPGs and high-order spin splitting

The SPG framework becomes substantially richer for coplanar and noncoplanar antiferromagnets. A recent classification based on spin-group theory tabulates 1249 nonequivalent spin point groups, including time-reversal-containing cases overlooked in earlier enumerations (Elcoro et al., 17 Jun 2026). The classification is organized into collinear, coplanar, and noncoplanar classes, with and without $g_l$6, and is implemented in the SPGENPOS database on the Bilbao Crystallographic Server (Elcoro et al., 17 Jun 2026).

For antiferromagnets, the decisive statement is that, except for SpPGs containing

$g_l$7

all other coplanar and noncoplanar SpPGs allow non-relativistic spin splitting at some order in the expansion of the crystal momentum (Elcoro et al., 17 Jun 2026). This is a major departure from the collinear case, where time reversal forbids any nonrelativistic spin splitting if the collinear spin-only symmetry is preserved (Elcoro et al., 17 Jun 2026). The effective Zeeman field is expanded as

$g_l$8

and the paper finds lowest-order monomials ranging from $g_l$9 to $\{O_s\Vert O_l\}$0, with the exception of $\{O_s\Vert O_l\}$1 (Elcoro et al., 17 Jun 2026).

The coplanar case has particularly clear SPG constraints. If the spin plane is perpendicular to $\{O_s\Vert O_l\}$2, the intrinsic coplanar spin-only symmetry implies that the in-plane components $\{O_s\Vert O_l\}$3 are even in $\{O_s\Vert O_l\}$4 while the out-of-plane component $\{O_s\Vert O_l\}$5 is odd in $\{O_s\Vert O_l\}$6 (Elcoro et al., 17 Jun 2026). If time reversal $\{O_s\Vert O_l\}$7 is also present, then $\{O_s\Vert O_l\}$8, so the polarization is forced perpendicular to the spin plane (Elcoro et al., 17 Jun 2026). For noncoplanar structures, no such intrinsic planar restriction exists.

The classification identifies new high-order spin textures in some noncentrosymmetric SpPGs. For coplanar $\{O_s\Vert O_l\}$9-wave ($\{O_s\Vert O_l\}$00) splitting, the allowed form is

$\{O_s\Vert O_l\}$01

for coplanar $\{O_s\Vert O_l\}$02-wave ($\{O_s\Vert O_l\}$03)

$\{O_s\Vert O_l\}$04

and for coplanar $\{O_s\Vert O_l\}$05-wave ($\{O_s\Vert O_l\}$06)

$\{O_s\Vert O_l\}$07

(Elcoro et al., 17 Jun 2026). The paper identifies LaMnAu$\{O_s\Vert O_l\}$08 as a real coplanar magnetic structure showing $\{O_s\Vert O_l\}$09 spin splitting (Elcoro et al., 17 Jun 2026).

This noncollinear classification also sharpens the relation between SPGs and magnetic point groups. The paper states that the magnetic point group of a magnetic structure is always a subgroup of its SpPG, but that noncollinear antiferromagnets often possess symmetry visible only in the spin-space action $\{O_s\Vert O_l\}$10 (Elcoro et al., 17 Jun 2026). This suggests that for noncollinear antiferromagnets, SPGs are not merely a refinement but often the minimal symmetry language that captures the allowed nonrelativistic band splitting.

A conceptually related but lower-dimensional example appears in the spin-$\{O_s\Vert O_l\}$11 Kitaev–Gamma chain near the antiferromagnetic Kitaev point. There the parent discrete spin-orbital symmetry is $\{O_s\Vert O_l\}$12, and ordered phases are classified by reductions $\{O_s\Vert O_l\}$13, $\{O_s\Vert O_l\}$14, and $\{O_s\Vert O_l\}$15 (Yang et al., 2020). Read through an SPG lens, the $\{O_s\Vert O_l\}$16 phase is a rank-1 magnetic phase with residual trigonal symmetry, while the intermediate $\{O_s\Vert O_l\}$17 regime is a rank-2 spin-nematic phase that preserves time reversal through the factor $\{O_s\Vert O_l\}$18 (Yang et al., 2020). This illustrates how antiferromagnetic spin-point symmetry can organize not only band splitting but also multipolar order and scale-dependent symmetry breaking.

5. Responses and emergent phenomena constrained by antiferromagnetic SPGs

Antiferromagnetic SPGs constrain not only static band structure but also dynamical and response properties. In high-harmonic generation, a weak-spin-orbit-coupling SPG framework distinguishes ferromagnetic, antiferromagnetic, and altermagnetic phases by whether the spin-flip branch appears globally, partially, or not at all (Mizuno, 30 Jun 2026). In the minimal $\{O_s\Vert O_l\}$19 example, the type-II antiferromagnetic SPG is

$\{O_s\Vert O_l\}$20

whose defining feature is a global spin-flip branch in addition to the ordinary spatial branch (Mizuno, 30 Jun 2026). This branch leaves charge-current selection rules largely conventional but imposes strong additional constraints on spin-current harmonics. Under axis-aligned linearly polarized driving, the antiferromagnetic phase is identified by the absence of the corresponding longitudinal spin-current harmonics, while ferromagnetic and altermagnetic phases are not separated by charge-current response alone (Mizuno, 30 Jun 2026). Under single-helicity circular driving, the spin-flip branch forbids $\{O_s\Vert O_l\}$21-polarized spin-current harmonics and shifts the allowed $\{O_s\Vert O_l\}$22-polarized harmonics by $\{O_s\Vert O_l\}$23 (Mizuno, 30 Jun 2026).

A different response problem is light-induced spin torque. A survey over all 122 magnetic point groups finds that Kramers-degenerate collinear antiferromagnets protected by $\{O_s\Vert O_l\}$24 or $\{O_s\Vert O_l\}$25 can support only Néel-vector torque $\{O_s\Vert O_l\}$26 and not total induced magnetization $\{O_s\Vert O_l\}$27 under linearly polarized light (Zhou et al., 11 Apr 2025). The relevant object is a subgroup $\{O_s\Vert O_l\}$28 governing sublattice-odd Néel responses, which is the nearest analogue in that work to an AFM-specific spin-point-group construction (Zhou et al., 11 Apr 2025). Although the language is MPG-based, the SPG connection is direct: the response decomposes into sublattice-even and sublattice-odd channels, and only the latter survives in compensated antiferromagnets.

Pure piezospintronics in insulating antiferromagnets provides another symmetry-controlled example. That work classifies magnetic point groups allowing a pure piezospintronic effect, defines the spin dipole moment

$\{O_s\Vert O_l\}$29

and the strain response

$\{O_s\Vert O_l\}$30

(Chen et al., 1 Jul 2025). The authors explicitly note that in negligible spin-orbit coupling, magnetic point-group analysis may overpredict allowed responses because “a magnetic group may allow a non-zero spin dipole moment, but the corresponding spin group may forbid it” (Chen et al., 1 Jul 2025). This is a direct SPG caveat: tensors carrying both spin and spatial indices can be further restricted by spin-space symmetry even when the magnetic point group allows them.

Topology likewise exposes the boundary between SPG intuition and full space-group necessity. The correspondence between antiferromagnetic and ferro/ferrimagnetic topological phases in type-IV magnetic space groups relies on nonsymmorphic time-reversal operations such as $\{O_s\Vert O_l\}$31 and $\{O_s\Vert O_l\}$32, so the decisive enforcement is usually magnetic-space-group-level rather than purely SPG-level (Bouhon et al., 2020). Still, the SPG viewpoint captures the qualitative mechanism: $\{O_s\Vert O_l\}$33 or $\{O_s\Vert O_l\}$34 rotational sectors together with a $\{O_s\Vert O_l\}$35-type antiunitary operation can force nontrivial topology, while the sign of $\{O_s\Vert O_l\}$36 distinguishes real-band topology from Kramers-degenerate structure (Bouhon et al., 2020). The paper explicitly notes that SPG captures the “style” of the mechanism, whereas the actual diagnosis requires little-group representations and translation phases (Bouhon et al., 2020).

6. Databases, computational identification, and current scope

A notable recent development is the transition from theoretical SPG classification to computational infrastructure. The representation-theoretic classification of crystallographic spin point groups establishes 598 nontrivial SPGs and shows that their co-irreps correspond exactly to the irreps or co-irreps of regular or black-and-white point groups, while total spin groups can acquire genuinely new co-irreps when continuous rotational freedom is present (Schiff et al., 2023). That work provides explicit co-irrep tables and shows their usefulness on electronic bands of altermagnets and on magnon spectra (Schiff et al., 2023).

At the space-group level, the SSG framework enumerates over 100000 SSGs under a four-index nomenclature, identifies inequivalent SSGs applicable to collinear, coplanar, and noncoplanar magnetic configurations, and provides the online program findspingroup.com for determining SSG symmetries of magnetic crystals (Chen et al., 2023). The relation to antiferromagnetic SPGs is indirect but explicit: the finite spin-space point group $\{O_s\Vert O_l\}$37 arises from quotient data $\{O_s\Vert O_l\}$38, and spin site-symmetry groups are stated to be isomorphic to one of the 598 nontrivial SPGs (Chen et al., 2023).

The oriented-spin-space-group framework extends this further by introducing oriented SSGs (OSSGs), whose purpose is to fix the relative orientation of spin and lattice spaces and thereby connect the nonrelativistic SSG description to the relativistic MSG descendant (Yu et al., 23 Apr 2026). The paper states that spin site-symmetry groups of an OSSG are always isomorphic to a subset of the 941 SPGs, and that only 375 SPGs allow local magnetic moments (Yu et al., 23 Apr 2026). In this framework, the useful point-group-level objects are no longer only a single global SPG but also spin-space point groups, spin site-symmetry groups, and spin little co-groups, which together determine whether an antiferromagnet allows net spin magnetization, anomalous Hall effect, momentum-dependent spin splitting, or spin-induced ferroelectricity (Yu et al., 23 Apr 2026).

Several limitations remain explicit. Many results are restricted to collinear magnets, negligible spin-orbit coupling, or 2D layer groups (Zhai et al., 9 Jun 2025, Zhang et al., 11 Dec 2025, Liu et al., 13 Jul 2025). Other work shows that topology often requires the full magnetic-space-group structure, especially when fractional translations determine the square of antiunitary symmetries (Bouhon et al., 2020). The “pseudoscalar electron spin” reduction applies only to nonrelativistic collinear magnets with perfect translational invariance (Turek, 2022). Conversely, the noncollinear SpPG classification shows that the inclusion of time reversal at the spin-point-group level is essential for many type-IV magnetic structures and that earlier tabulations omitting it were incomplete (Elcoro et al., 17 Jun 2026). These restrictions do not diminish the role of antiferromagnetic SPGs; they delimit the circumstances under which SPGs suffice on their own and the circumstances under which they must be embedded in a broader SSG or MSG framework.

Taken together, the modern literature places antiferromagnetic SPGs at the center of a layered symmetry hierarchy. They are the natural point-group objects for weak-spin-orbit antiferromagnets, the parent symmetries of collinear order before spin-orbit locking, the direct symmetry classifiers of nonrelativistic spin splitting in collinear and noncollinear magnets, and increasingly the point-level data structures used inside full SSG/OSSG workflows for materials discovery and response prediction (Schiff et al., 2023, Zhai et al., 9 Jun 2025, Liu et al., 13 Jul 2025, Elcoro et al., 17 Jun 2026, Yu et al., 23 Apr 2026).