GoKawiilFurther classification of magnetic geometries based on the FM/AFM dichotomy
On the basis of the FM/AFM dichotomy, the SSG framework enables further classification of magnetic geometry. Here we focus on the SSG-based classification of various antiferromagnetic geometries, especially for noncollinear magnets, which were also phenomenologically described previously such as Néel-type, spiral and multi-q AFM. Experimentally, the spin distribution across crystallographic primitive cells is typically described by the propagation vector q. However, q alone cannot capture the complexity of the magnetic geometry within a single primitive cell. Moreover, when the lattice periodicity and the propagation vector period are mismatched, q fails to reflect the modulation of the crystal field on the magnetic configuration. Furthermore, even the propagation of spiral magnetic order is hardly captured by MSGs, necessitating the application of SSGs.
As mentioned in the main text, we introduce spin translational group T spin , which consists of the combination of pure spin-space operation and fractional translation {g s ||1|τ}. Because the components of T spin , g s and τ act in different spaces and their multiplicative actions commute, the group T spin follows the group structure of its τ component and is, thus, Abelian.
The classification constitutes four distinct categories, as shown in Extended Data Fig. 1. For i k = 1, T spin only consists of the identity operation and the complexity of magnetic geometry is only included in the magnetic primary cell. A typical example is CuMnAs with antiparallel spin arrangement for the two Mn atoms within a primary cell. Therefore, such a type of AFM is classified as primary AFM. In the case of i k = 2, T spin has an order 2 spin translational operation, whose spin-space part can be −1, 2 or m. Examples include the intrinsic magnetic topological insulator MnBi 2 Te 4 (refs. 42,43), which has two magnetic atoms with antiparallel spin connected by {U 2 ||1|τ 1/2 } (\({\tau }_{1/2}=0,0,\frac{1}{2}\)). Owing to the correspondence between the collinear SSG and MSG in the group structure, its group symbol can be simplified as R I 1−31m∞m1. Such a category aligns with the pedagogical one-dimensional AFM chain, referred to as bicolour AFM.
The case of i k > 2 could be further divided into two categories based on whether T spin is cyclic. If T spin is a cyclic group, such as n, −n (n > 2), the magnetic geometry aligns with a high-order spin rotation associated with translation. We select EuIn 2 As 2 as an example in which the magnetic moments are connected by {U 3 ||1|τ 1/3 }, forming a so-called spiral AFM8,44. Finally, if T spin is a non-cyclic Abelian group, the spin rotations with different axes must be mapped to translations in different directions. Such mappings result in a more intricate multi-q magnetic geometry, as observed in antiferromagnetic [111]-strained cubic γ-FeMn (ref. 26) and CoNb 3 S 6 (ref. 27), referred to as multiaxial AFM. Apparently, both spiral and multiaxial AFM cannot be described by MSGs, in which the corresponding T spin only allows {−1||1|τ} operation. Furthermore, FM can also be classified into the four categories in the same way. For example, a helimagnet with AFM geometries and a FM magnetic canting can be directly described by combining a T spin with i k > 2 and a polar P spin .
Extended Data Fig. 2 summarizes the quantities and proportions of materials exhibiting each type of AFM geometry in the MAGNDATA database obtained by our online program FINDSPINGROUP. On the basis of T spin , AFM geometries are further classified into primary (660, 32.0%), bicolour (857, 41.5%), spiral (24, 1.2%) and multiaxial (45, 2.2%) categories. In Supplementary Information sections 2.1 and 2.2, we provide an exhaustive list of all materials and their oriented SSG including the dichotomy of FM/AFM and further geometries classification based on T spin .
SOC tensor
To describe the transformation of SOC under SSG operations, we reformulate it in a form that explicitly allows for independent coordinate systems in real space and spin space:
$$\begin{array}{c}\begin{array}{c}\begin{array}{c}{\hat{H}}_{{\rm{SOC}}}={\lambda }{\hat{{\bf{L}}}}^{{\rm{T}}}{\boldsymbol{\chi }}\hat{{\boldsymbol{\sigma }}}=\lambda \sum _{i,j}{\chi }_{{ij}}{\hat{L}}_{i}{\hat{\sigma }}_{j}\\ =\,\lambda ({\hat{L}}_{1}\,{\hat{L}}_{2}\,{\hat{L}}_{3})\left(\begin{array}{ccc}{{\bf{r}}}_{1}\cdot {{\bf{s}}}_{1} & {{\bf{r}}}_{1}\cdot {{\bf{s}}}_{2} & {{\bf{r}}}_{1}\cdot {{\bf{s}}}_{3}\\ {{\bf{r}}}_{2}\cdot {{\bf{s}}}_{1} & {{\bf{r}}}_{2}\cdot {{\bf{s}}}_{2} & {{\bf{r}}}_{2}\cdot {{\bf{s}}}_{3}\\ {{\bf{r}}}_{3}\cdot {{\bf{s}}}_{1} & {{\bf{r}}}_{3}\cdot {{\bf{s}}}_{2} & {{\bf{r}}}_{3}\cdot {{\bf{s}}}_{3}\end{array}\right)\,\left(\begin{array}{c}{\hat{\sigma }}_{1}\\ {\hat{\sigma }}_{2}\\ {\hat{\sigma }}_{3}\end{array}\right),\end{array}\end{array}\end{array}$$ (1)
in which λ, \(\hat{{\bf{L}}}\) and \(\hat{{\boldsymbol{\sigma }}}\) represent the SOC coefficient, effective orbital angular momentum operator and spin operator, respectively; r i and s j are the unit base vectors with i = 1, 2, 3 and j = 1, 2, 3 for real space and spin space, respectively; χ represents a 3 × 3 SOC tensor matrix, defined as χ = {χ ij = r i · s j |i = 1, 2, 3; j = 1, 2, 3}. For a general SSG operation {g s ||g l }, the transformation of χ can be expressed as:
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