Altermagnetic Spin Laue Groups

• Altermagnetic spin Laue groups are group-theoretical structures that define the combined spatial and spin symmetries in nonrelativistic, time-reversal‐breaking materials with zero net magnetization.
• They enforce an even-in-momentum, odd-in-spin splitting of electronic bands without relying on spin–orbit coupling, leading to unique multiferroic and magnetoelectric properties.
• The classification uses symmetry selection rules, tensor invariants, and spin-point group analysis to guide effective Hamiltonian construction in compounds like BiFeO₃, BaCuF₄, and Ca₃Mn₂O₇.

Altermagnetic spin Laue groups are group-theoretical structures that classify the combined spatial and spin symmetries underpinning nonrelativistic, time-reversal-breaking electronic states with compensated (zero net) magnetization. Unlike in conventional ferro- or antiferromagnets, altermagnetic ordering manifests as a symmetry-enforced, even-in-momentum, odd-in-spin splitting (typically d-, g-, or i-wave) of electronic bands which is not reliant on spin-orbit coupling. These groups provide the rigorous setting for understanding symmetry selection rules, response tensors, and the topological protection of nodal features in altermagnetic materials, and form the symmetry backbone for new classes of multiferroics, magnetoelectric effects, and spin-dependent transport phenomena.

1. Formal Definition and Structural Elements

In non-relativistic crystals, the conventional Laue group, $\mathcal L = \{ R \} \cup \{ iR \}$, is composed of all point-group rotations $R$ and inversion $i$. For general collinear magnets, one must track not only spatial operations but also time-reversal $\Theta$ and spin-space rotations. A spin Laue group $\mathcal L_s$ is thus constructed as the set of combined operations: $$[R_c \parallel R_s],\quad [R_c \parallel R_s]\,i,\quad [R_c \parallel R_s]\,\Theta,\quad [R_c \parallel R_s]\,i\Theta,$$ where $R_c$ acts in real space and $R_s \in SO(3)$ is a spin rotation aligned to preserve the particular antiferromagnetic or altermagnetic order parameter.

In altermagnets, time-reversal symmetry $\Theta$ is broken, but a subset of spatial operations only survives when paired with appropriate spin-space operations, often 180° rotations $C_2^s$. The minimal “spin-point group” is

$$\mathcal G_s = \{ [R_c \parallel R_s] \} \cup \{ [R_c \parallel R_s]\,\Theta \},$$

and the full spin Laue group becomes

$\mathcal L_s = \mathcal G_s \cup i\mathcal G_s.$

Practical computations often ignore explicit $\Theta$ in the generators, noting that half the elements imply time-reversal on the anti-symmetry operations.

2. Enumeration and Classification of Polar Altermagnetic Spin Laue Groups

Starting from ten crystallographic polar point groups $\{1,\,2,\,m,\,mm2,\,4,\,4mm,\,3,\,3m,\,6,\,6mm\}$, analysis shows that only eight admit collinear altermagnetism. Three of these permit two inequivalent time-reversal embeddings, yielding eleven distinct polar altermagnetic spin-Laue groups. Each group is specified by a Hermann–Mauguin-like spin symbol, its group order, generators, and the momentum-space structure of the leading altermagnetic mode.

The defining property across all altermagnetic spin Laue groups is that a subset of spatial operations, when paired with proper spin rotations, enforce an alternating sign of spin polarization between symmetry-related sublattices, leading to a distinct, symmetry-enforced nonrelativistic spin splitting.

Table: Summary of the 11 Polar Altermagnetic Spin Laue Groups

Spin Laue Symbol	Order	Generators (in $[R_c \parallel R_s]$)	Leading Spin-Splitting Term
$^{2}2$	4	$[E\parallel E]$, $[C_2\parallel C_2]$	$k_x^2\,\sigma_x$
$^{2}m$	4	$[E\parallel E]$, $[E\parallel M]$	$k_y^2\,\sigma_y$
$^{2}m^{1}m^{2}2$	8	$$[E\parallel E], [E\parallel M_x], [C_2\parallel M_y], [C_2\parallel C_{2z}]$$	$k_y k_z\,\sigma_y$
$^{2}m^{2}m^{1}2$	8	$$[E\parallel E], [C_2\parallel M_x], [C_2\parallel M_y], [E\parallel C_{2z}]$$	$k_x k_y\,\sigma_z$
$^{2}4$	4	$[E\parallel E]$, $[C_4\parallel C_4]$	$k_x k_y\,\sigma_z$
$^{2}m^{1}m^{2}4$	8	$$[E\parallel E], [E\parallel M_x], [C_4\parallel M_y], [C_2\parallel C_{2z}]$$	$(k_x k_z \pm k_y k_z)\,\sigma_y$
$^{2}m^{2}m^{1}4$	8	$$[E\parallel E], [C_4\parallel M_x], [C_2\parallel C_{2y}], [E\parallel C_{4z}]$$	...
$^{2}3m$	8	$$[E\parallel E], [C_3\parallel C_3], [E\parallel M], [C_2\parallel C_2]$$	$k_x(k_x^2 - 3k_y^2)\,\sigma_z$ (i-wave)
$^{2}6$	4	$[E\parallel E]$, $[C_6\parallel C_6]$	$k_x k_y\,\sigma_z$
$^{2}m^{1}m^{2}6$	8	$$[E\parallel E], [E\parallel M_x], [C_6\parallel M_y], [C_3\parallel C_{3z}]$$	$(k_x^2 - k_y^2)k_z\,\sigma_x$ (g-wave)
$^{2}m^{2}m^{1}6$	8	$$[E\parallel E], [C_6\parallel M_x], [E\parallel C_{3z}], [C_3\parallel M_y]$$	...

In every group, the leading spin splitting is even in $\mathbf{k}$ and odd in spin, and the overall sign reverses under spatial inversion—a direct reflection of the underlying magnetoelectric point group symmetry.

3. Symmetry-Based Selection Rules and Group-Theoretical Procedures

The group-theoretical classification proceeds via:

• Parent group selection: Only those polar point groups that enable two-sublattice collinear antiferromagnetism with time-reversal breaking and partial retention of parent symmetry via combined spatial–spin operations are eligible.
• Extension with $Z_2^T$ and $SO(3)$: Group extension includes time-reversal (as a generator, but physically broken) and spin rotation symmetry, focusing on whether each spatial symmetry must be paired with a spin rotation (usually a 180° flip).
• Spin-group analysis: Factorizes out global $SO(2)$ rotations about the collinear axis and retains only nontrivial, symmetry-enforcing operations that map $M(\mathbf r)\sim +1$ on one sublattice to $M(\mathbf r)\sim -1$ on the other.
• Doubling by inversion: The spin point group is doubled by spatial inversion to form the full spin Laue group.

This procedure establishes all allowed embeddings and identifies the corresponding invariants that characterize the altermagnetic order.

4. Concrete Material Examples: Mapping and Spin-Splitting Forms

BaCuF₄

• Parent group: $mm2$ ($Cmc2_1$).
• Spin-point group: $SO(2)\times Z_2\times\,^{2}m^{1}m^{2}2$.
• Generators: $$[E\parallel E], [E\parallel M_x], [C_2\parallel M_y], [C_2\parallel C_{2z}]$$.
• Spin-splitting: Real-space $d$-wave, $D_{yz}\,\sigma_y$; in $\mathbf{k}$-space: $\Delta H(\mathbf k)\propto k_y k_z\,\sigma_y$.

Ca₃Mn₂O₇

• Parent group: $mm2$.
• Spin-point group: $SO(2)\times Z_2\times\,^{2}m^{2}m^{1}2$.
• Generators: $$[E\parallel E], [C_2\parallel M_x], [C_2\parallel M_y], [E\parallel C_{2z}]$$.
• Spin-splitting: $d$-wave $D_{xy}\,\sigma_z$; in $\mathbf{k}$-space: $k_x k_y\,\sigma_z$.

BiFeO₃ (collinear phase)

• Parent group: $3m$ ($R3c$).
• Spin-point group: $^{2}3m$.
• Generators: $$[E\parallel E], [C_3\parallel C_3], [E\parallel M_{(1\bar10)}], [C_2\parallel C_2]$$.
• Spin-splitting: $i$-wave, sixfold nodal: $$\Delta H(\mathbf k)\propto (k_x^3-3k_x k_y^2)\,\sigma_z$$.

These mappings provide explicit guidance for symmetry analysis and effective Hamiltonian construction in real systems.

5. Order Parameter Invariants and Tensor Structure

The altermagnetic order parameter transforms as an even-in-momentum, odd-in-spin tensor. The general invariant takes the form: $$H_{\rm alterm} = \sum_{j,k,\alpha} \eta_{\alpha\,jk} k_j k_k \sigma_\alpha + \dots$$ with the relevant $\eta_{\alpha\,jk}$ dictated by the symmetry of the spin Laue group. For example:

• $d$-wave: only $\eta_{y\,yz}=\eta_{y\,zy}\ne0$;
• $g$-wave: nonzero $\eta$'s involve $k_xk_z$ or $k_yk_z$ pairs;
• $i$-wave: needs cubic combinations $k_x(k_x^2-3k_y^2)\sigma_z$.

This formalism prescribes the lowest order in momentum/spin harmonics allowed by the symmetry, crucial for modeling the physics of altermagnetic phases.

6. Physical Implications and the Altermagnetoelectric Effect

Every altermagnetic spin Laue group is also a magnetoelectric point group: the symmetry requires that the characteristic spin splitting is odd under time reversal and real-space inversion, while reversal of the polar structure by an electric field also reverses the sign of the spin splitting (the “altermagnetoelectric effect”). This nonrelativistic, symmetry-enforced coupling between spin order and polar order, mediated in part by lattice distortions (as in BiFeO₃, BaCuF₄, and Ca₃Mn₂O₇), opens a pathway to electric-field control of altermagnetic order parameters, distinct from mechanisms in conventional multiferroics.

7. Compact Table: Generators and Leading Modes in the 11 Polar Altermagnetic Spin Laue Groups

Symbol	Order	Generators	Leading Mode
$^{2}2$	4	$[E\parallel E], [C_2\parallel C_2]$	$k_x^2\,\sigma_x$
$^{2}m$	4	$[E\parallel E], [E\parallel M]$	$k_y^2\,\sigma_y$
$^{2}m^{1}m^{2}2$	8	$$[E\parallel E], [E\parallel M_x], [C_2\parallel M_y], [C_2\parallel C_{2z}]$$	$k_y k_z\,\sigma_y$
$^{2}m^{2}m^{1}2$	8	$$[E\parallel E], [C_2\parallel M_x], [C_2\parallel M_y], [E\parallel C_{2z}]$$	$k_x k_y\,\sigma_z$
$^{2}4$	4	$[E\parallel E], [C_4\parallel C_4]$	$k_x k_y\,\sigma_z$
$^{2}m^{1}m^{2}4$	8	$$[E\parallel E], [E\parallel M_x], [C_4\parallel M_y], [C_2\parallel C_{2z}]$$	$(k_x k_z\pm k_y k_z)\,\sigma_y$
$^{2}m^{2}m^{1}4$	8	$$[E\parallel E], [C_4\parallel M_x], [C_2\parallel C_{2y}], [E\parallel C_{4z}]$$	...
$^{2}3m$	8	$$[E\parallel E], [C_3\parallel C_3], [E\parallel M], [C_2\parallel C_2]$$	$i$-wave: $k_x(k_x^2-3k_y^2)\sigma_z$
$^{2}6$	4	$[E\parallel E], [C_6\parallel C_6]$	$k_x k_y\,\sigma_z$
$^{2}m^{1}m^{2}6$	8	$$[E\parallel E], [E\parallel M_x], [C_6\parallel M_y], [C_3\parallel C_{3z}]$$	$g$-wave: $(k_x^2 - k_y^2)k_z\,\sigma_x$
$^{2}m^{2}m^{1}6$	8	$$[E\parallel E], [C_6\parallel M_x], [E\parallel C_{3z}], [C_3\parallel M_y]$$	...

These results comprehensively structure the landscape of collinear polar altermagnetism, providing essential symmetry constraints for both theoretical construction and experimental identification in complex oxide, fluoride, and multiferroic materials.

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References:

(Šmejkal, 29 Nov 2024, Šmejkal et al., 2021, Turek, 2022, Zhai et al., 9 Jun 2025, Dale et al., 27 Nov 2024, Chen et al., 2023, Radaelli, 18 Jul 2024)