Algorithm for spin symmetry operation search

Shinohara, K.; Togo, A.; Watanabe, H.; Nomoto, T.; Tanaka, I.; Arita, R.

doi:10.1107/S2053273323009257

research papers

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ISSN: 2053-2733

Volume 80| Part 1| January 2024| Pages 94-103

https://doi.org/10.1107/S2053273323009257

Algorithm for spin symmetry operation search

Kohei Shinohara,^a ^* Atsushi Togo,^b Hikaru Watanabe,^c Takuya Nomoto,^c Isao Tanaka ^a,^d,^e and Ryotaro Arita ^c,^f

^aDepartment of Materials Science and Engineering, Kyoto University, Sakyo, Kyoto 606-8501, Japan, ^bCenter for Basic Research on Materials, National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan, ^cResearch Center for Advanced Science and Technology, University of Tokyo, Meguro-ku, Tokyo 153-8904, Japan, ^dCenter for Elements Strategy Initiative for Structural Materials, Kyoto University, Sakyo, Kyoto 606-8501, Japan, ^eNanostructures Research Laboratory, Japan Fine Ceramics Center, Nagoya 456-8587, Japan, and ^fRIKEN, Center for Emergent Matter Science, Saitama 351-0198, Japan
^*Correspondence e-mail: [email protected]

Edited by S. J. L. Billinge, Columbia University, USA (Received 23 July 2023; accepted 21 October 2023)

A spin space group provides a suitable way of fully exploiting the symmetry of a spin arrangement with a negligible spin–orbit coupling. There has been a growing interest in applying spin symmetry analysis with the spin space group in the field of magnetism. However, there is no established algorithm to search for spin symmetry operations of the spin space group. This paper presents an exhaustive algorithm for determining the spin symmetry operations of commensurate spin arrangements. The present algorithm searches for spin symmetry operations from the symmetry operations of a corresponding nonmagnetic crystal structure and determines their spin-rotation parts by solving a Procrustes problem. An implementation is distributed under a permissive free software license in spinspg Version 0.1.1, available at https://github.com/spglib/spinspg.

Keywords: spin space groups; spin symmetry operations; spin arrangements; Procrustes problems; Hermite normal forms.

1. Introduction

When the spin–orbit coupling (SOC) is negligible, a spin space group is an appropriate concept for full exploitation of the symmetry of a corresponding spin arrangement (Litvin & Opechowski, 1974 ; Opechowski, 1986 ; Yang et al., 2021 ; Liu et al., 2022 ). The spin arrangement comprises a crystal structure and magnetic moments. A spin symmetry operation of the spin space group is assumed to act on the spatial and spin coordinates simultaneously, generalizing a magnetic symmetry operation of a magnetic space group. The spin space group was first introduced to analyze an extra symmetry of a spin Hamiltonian for neutron scattering experiments (Brinkman & Elliott, 1966 ; Brinkman et al., 1966 ) and has recently been applied to the field of magnetism (Šmejkal et al., 2022b ); for example, an analysis of a symmetry-adapted tensor of transport properties with negligible SOC (Železný et al., 2017 ; Zhang et al., 2018 ), symmetries of a spin Hamiltonian (Zelenskiy et al., 2022 ), magnon band structure (Corticelli et al., 2022 ) and classification of antiferromagnetism (Šmejkal et al., 2022a ).

When we consider spin space groups of given spin arrangements, we must first exhaustively search for their spin symmetry operations. To the best of our knowledge, there is no rigorous algorithm to search for spin symmetry operations. Therefore, the development of such an algorithm and its implementation would benefit spin symmetry analysis.

There are a few differences between spin space groups and magnetic space groups in terms of symmetry search algorithms. First, a spin space group may contain nontrivial operations acting only on spin coordinates, called a spin-only group, which complicates the group structure of the spin space group. Second, although we only need to consider at most double enlarged cells for magnetic space groups, the unit-cell size may arbitrarily change between a spin arrangement and its nonmagnetic correspondence. Lastly, spin rotations, which simultaneously act on magnetic moments, do not have to belong to a crystallographic point group.

Here, we present a rigorous and robust algorithm for determining the spin symmetry operations of a given commensurate spin arrangement, extending our magnetic symmetry operation search algorithm (Shinohara et al., 2023 ). The present algorithm fully exploits the group structure of the spin space groups and outputs spin symmetry operations as a coset decomposition of the spin space group, based on the seminal work by Litvin and Opechowski (Litvin, 1973 , 1977 ; Litvin & Opechowski, 1974). We explicitly denote basis vectors of space groups and spin space groups, and employ a lattice algorithm to deal with the case when the unit-cell size varies with and without magnetic moments. We search for the spin-rotation parts from three-dimensional orthogonal groups by solving a well known optimization problem called a Procrustes problem (Gower & Dijksterhuis, 2004 ). Note that we restrict the present algorithm to commensurate spin arrangements in order to use similar inputs and outputs to an existing space-group search implementation (Togo & Tanaka, 2018 ). The implementation is distributed under the BSD 3-clause license in spinspg Version 0.1.1 on top of a crystal symmetry search algorithm (Togo & Tanaka, 2018). For magnetic crystal structures tabulated in MAGNDATA (Gallego et al., 2016 ), the present algorithm and implementation have been used to identify physical properties free from SOC (Watanabe et al., 2023 ).

This paper is organized as follows. In Section 2 we give definitions of spin space groups and their derived groups. In Section 3 we provide an algorithm for determining a spin-only group, spin translation group and spin space group of a given spin arrangement. In Section 4 we demonstrate the present spin symmetry operation search applied to the spin arrangement of an NiAs-type CrSe. The notations and terminology in this paper are summarized in Table 1.

Table 1
Notation and terminology in this paper

Symbol	Meaning
$[A \sqcup B]$	Disjoint union of set A and B with A ∩ B = ∅
E(3)	Three-dimensional Euclidean group
O(3)	Three-dimensional orthogonal group
E	Identity matrix
$[\specialfonts {\frak S}_{N}]$	Symmetric group of degree N
∥y∥₂ = $[\sqrt{ {\bf y}^{\rm T} {\bf y} }]$	l² norm of vector y
$[\\|{\bf B} \\|_{\rm F} = \sqrt { {\rm Tr} \, {\bf B}^{\rm T} {\bf B}}]$	Frobenius norm of matrix B
1	Trivial group

r	Position in Cartesian coordinates
m	Magnetic moments
Spin symmetry operation (g, W)	Pair of spatial operation g and spin rotation W
$[({\overline {\bf R}}, {\overline {\bf v}})_{\bf A} = ({\bf A} {\overline {\bf R}} {\bf A}^{-1}, {\bf A} {\overline {\bf v}})]$	Spatial operation with basis vectors A

A = (a₁, a₂, a₃)	Basis vectors
$[{\bf X} = ({\bf x}_{1}, \cdots, {\bf x}_{N})]$	Array of point coordinates
$[{\bf T} = (t_{1}, \cdots, t_{N})]$	Array of atomic types
$[{\bf M} = ({\bf m}_{1}, \cdots, {\bf m}_{N})]$	Array of magnetic moments
Spin arrangement (A, X, T, M)	Pair of crystal structure and magnetic moments
(A, X, T)	Crystal structure ignoring magnetic moments
ε_mag	Absolute tolerance to compare magnetic moments

$[{\cal S}, {\cal R}]$	Space group
Translation subgroup $[{\cal T} ({\cal R})]$	Translation parts of $[{\cal R}]$ with identity rotations
Point group $[{\cal P} ({\cal R})]$	Group obtained from the rotation parts of $[{\cal R}/{\cal T} ({\cal R})]$
$[{\cal T}_{\bf A}]$	Translation group spanned by A
σ_g	Permutation of sites induced by g
$[{\bf A}_{\cal S} = {\bf A} {\bf U}^{-1}]$	Primitive basis vectors of $[{\cal T}({\cal S})]$

Spin-only group $[{\cal P}_{\rm so} = {\cal P}_{\rm so}({\cal G})]$	Spin-rotation parts with identity spatial operations
$[\sigma_{i}, {\hat {\bf n}}_{i}]$	Eigenvalue and eigenvector of MM^T
$[{\hat {\bf n}}_{\parallel}]$	Parallel direction for collinear spin arrangement
$[{\hat {\bf n}}_{\perp}]$	Perpendicular direction for coplanar spin arrangement

$[{\bf A}_{\cal D} = {\bf A}_{\cal S} {\bf V}]$	Basis vectors of $[{\cal T} ({\cal D} ({\cal G}))]$

Spin translation group $[{\cal G}_{\rm st} = {\cal G}_{\rm st}({\cal G})]$	Subgroup of $[{\cal G}]$ with identity rotations
$[{\bf M}_{g} = ({\bf m}_{\sigma_{g}(1)} \cdots {\bf m}_{\sigma_{g}(N)})]$	Array of magnetic moments permuted by σ_g

Spin space group $[{\cal G}]$	See Section 2.2
Family space group $[{\cal F} ({\cal G})]$	Space group composed of spatial parts of $[{\cal G}]$
Maximal space subgroup $[{\cal D} = {\cal D}({\cal G})]$	Subgroup of $[{\cal G}]$ with identity spin-rotation parts
Spin space-group type	See Section 2.2
(P, p)	Transformation on spatial coordinates
Q	Transformation matrix on spin coordinates

Family spin point group $[{\cal B}({\cal G})]$	Spin-rotation parts of $[{\cal G}]$
Spin point group $[{\cal U}({\cal G})]$	Pairs of spatial and spin rotations of $[{\cal G}]$

$[{\cal M}]$	Magnetic space group

2. Group structure of a spin space group

We provide the definitions of spin symmetry operations and spin arrangements in Section 2.1. We define the spin space group in Section 2.2 and then introduce its derived groups, the spin-only group (Section 2.3) and spin translation group (Section 2.4). Although these groups were already discussed by Litvin (1973, 1977) and Litvin & Opechowski (1974), we consider it beneficial to summarize these results here because we fully exploit the group structure of the spin space group in searching for spin symmetry operations.

We note that spin-only groups and spin translation groups complicate the group structure of the spin space groups. For example, a spin point group, which ignores the translation parts of the spin symmetry operations of a spin space group, cannot be computed without going through the spin space group due to the existence of a nontrivial spin translation group in general. Although the analysis of spin point groups is not required to determine spin symmetry operations, we discuss the group structure of spin point groups in Appendix B for completeness.

2.1. Spin symmetry operation and spin arrangement

A spin symmetry operation comprises a spatial operation in the three-dimensional Euclidean group E(3) and a spin rotation in the three-dimensional orthogonal group O(3). The spin symmetry operation (g, W) ∈ E(3) × O(3) acts on a pair of position r and magnetic moments m as

$[(g, {\bf W}) \, ({\bf r}, {\bf m}) = (g {\bf r}, {\bf Wm}). \eqno(1)]$

The product of two spin symmetry operations (g, W) and (g′, W′) is defined as

$[(g, {\bf W}) \, (g^{\prime}, {\bf W}^{\prime}) = (g g^{\prime}, {\bf W} {\bf W}^{\prime}). \eqno(2)]$

Although uncommon in crystallography, we suppose both g and W to be represented with Cartesian coordinates for later convenience. We denote the basis vectors

$[{\bf A} = ({\bf a}_{1}, {\bf a}_{2}, {\bf a}_{3}) = \left (\matrix{ a_{1x} & a _{2x} & a_{3x} \cr a_{1y} & a_{2y} & a_{3y} \cr a_{1z} & a_{2z} & a_{3z} \cr } \right). \eqno(3)]$

When g with a matrix part R and a translation part v are represented by A, we explicitly write $[g = ({\overline {\bf R}}, {\overline {\bf v}})_{\bf A}]$ , where $[{\bf R} = {\bf A} {\overline {\bf R}} {\bf A}^{-1}]$ and $[{\bf v} = {\bf A} {\overline {\bf v}}]$ . A spin arrangement is a set of pairs of a crystal structure and magnetic moments.

2.2. Spin space group

Let $[{\cal G}]$ be a subgroup of E(3) × O(3). When the following $[{\cal F} ({\cal G})]$ and $[{\cal D} ({\cal G})]$ are space groups, $[{\cal G}]$ is called a spin space group (Litvin & Opechowski, 1974),

$[{\cal F} ({\cal G}) = \left \{ g \in {\rm E}(3) \ | \ \exists {\bf W} \in {\rm O}(3) \ {\rm s.t.} \ (g, {\bf W}) \in {\cal G } \right \}, \eqno(4)]$

$[{\cal D} ({\cal G}) = \left \{ g \in {\rm E}(3) \ | \ (g, {\bf E}) \in {\cal G} \right \}, \eqno(5)]$

where E stands for the identity matrix. For a spin space group $[{\cal G}]$ , we call $[{\cal F} ({\cal G})]$ a family space group and $[{\cal D} ({\cal G})]$ a maximal space subgroup. The maximal space subgroup $[{\cal D} ({\cal G})]$ is a normal subgroup of $[{\cal G}]$ . In contrast, the family space group $[{\cal F} ({\cal G})]$ is not a subgroup of $[{\cal G} \times {\rm O}(3)]$ in general. Although Litvin & Opechowski (1974) did not impose the condition that $[{\cal D} ({\cal G})]$ be crystallographic, we do impose this condition here to guarantee that a given spin arrangement is commensurate. We confine our discussion and algorithms to commensurate spin arrangements.

A spin symmetry operation (g, W) is transformed by a transformation (P, p) on the spatial coordinates and a transformation matrix Q on the spin coordinates as

$[(g, {\bf W}) \ \mapsto \ \left [({\bf P}, {\bf p})^{-1} g ({\bf P}, {\bf p}), {\bf Q}^{-1} {\bf W} {\bf Q} \right] . \eqno(6)]$

Two spin space groups $[{\cal G}_{1}]$ and $[{\cal G}_{2}]$ belong to the same spin space-group type if they are transformed into each other by a pair of an orientation-preserving transformation (P, p) on the spatial coordinates and a transformation matrix Q on the spin coordinates; a transformation (P, p) is called orientation preserving if $[\det {\bf P} \, \gt \, 0]$ .

In the spin space group, we can consider rotating magnetic moments independently with spatial coordinates. On the other hand, we consider rotating magnetic moments only in association with spatial rotations and time-reversal operations in the magnetic space group (Litvin, 2014 ). Thus, the spin space group can be regarded as a supergroup of the magnetic space group under an appropriate correspondence between spin symmetry operations and magnetic symmetry operations, as discussed in Appendix A.

2.3. Spin-only group

A spin-only group of a spin space group $[{\cal G}]$ is a set of spin symmetry operations of $[{\cal G}]$ with identity spatial operations,

$[{\cal P}_{\rm so} ({\cal G}) = \left \{ {\bf W} \in {\rm O} (3) \ | \ (({\bf E}, {\bf 0}), {\bf W}) \in {\cal G} \right \}. \eqno(7)]$

Denoting a trivial group, consisting of a single element, as 1, the direct product of an identity in spatial coordinates and $[{\cal P}_{\rm so} ({\cal G})]$ , $[1 \times {\cal P}_{\rm so} ({\cal G})]$ , is a subgroup of $[{\cal G}]$ .

2.4. Spin translation group

A spin translation group of a spin space group $[{\cal G}]$ is a set of spin symmetry operations with identity rotations in spatial coordinates,

$[{\cal G}_{\rm st} ({\cal G}) = \left \{ (({\bf E}, {\bf v}), {\bf W}) \ | \ (({\bf E}, {\bf v}), {\bf W}) \in {\cal G} \right \}. \eqno(8)]$

Litvin (1973) classified the spin translation groups under the transformation in equation (6).

We denote a translation subgroup and a point group of space group $[{\cal R}]$ as $[{\cal T} ({\cal R})]$ and $[{\cal P} ({\cal R})]$ , respectively,

$[{\cal T} ({\cal R}) = \left \{ ({\bf E}, {\bf t}) \ | \ ({\bf E}, {\bf t}) \in {\cal R} \right \}, \eqno(9)]$

$[{\cal P} ({\cal R}) = \left \{ {\bf W} \ | \ \exists {\bf v} \ {\rm s.t.} \ ({\bf W}, {\bf v}) \in {\cal R} \right \}. \eqno(10)]$

Then, the spin-only group $[{\cal P}_{\rm so} ({\cal G})]$ and translation subgroup of $[{\cal D} ({\cal G})]$ are a normal subgroup of $[{\cal G}_{\rm st} ({\cal G})]$ . Because $[({\cal T} ({\cal D} ({\cal G})) \times 1) \cap]$ $[(1 \times {\cal P}_{\rm so} ({\cal G})) = 1]$ , their direct product $[{\cal T} ({\cal D} ({\cal G})) \times {\cal P}_{\rm so} ({\cal G})]$ is also a normal subgroup of $[{\cal G}_{\rm st} ({\cal G})]$ . Thus, we can consider a factor group $[{\cal G}_{\rm st} ({\cal G}) / ({\cal T} ({\cal D} ({\cal G})) \times {\cal P}_{\rm so} ({\cal G}))]$ , which is finite for commensurate spin arrangements. Finally, $[{\cal G}_{\rm st} ({\cal G})]$ is a normal subgroup of $[{\cal G}]$ . The factor group $[{\cal G} / {\cal G}_{\rm st} ({\cal G})]$ is isomorphic to $[{\cal P} ({\cal F} ({\cal G}))]$ and thus finite.

3. Spin symmetry operation search

We provide an algorithm to search for spin symmetry operations from a given spin arrangement represented by the following four objects: (i) basis vectors of its lattice A = (a₁, a₂, a₃), (ii) an array of point coordinates of sites in its unit cell X = $[({\bf x}_{1}, \cdots, {\bf x}_{N})]$ , (iii) an array of atomic types of sites in its unit cell T = $[(t_{1}, \cdots, t_{N})]$ , and (iv) an array of magnetic moments of sites in its unit cell M = $[({\bf m}_{1}, \cdots, {\bf m}_{N})]$ , where N is the number of sites in the unit cell.

To search for spin symmetry operations robustly, comparisons of point coordinates and magnetic moments should be performed within tolerances in practice. We adopt the same absolute tolerance parameter for point coordinates as Togo & Tanaka (2018). For magnetic moments, we use another tolerance parameter ε_mag to identify that two magnetic moments Wm_i and $[{\bf m}_{\sigma_{g}(i)}]$ are equal to each other if

$[\| {\bf W} {\bf m}_{i} - {\bf m}_{\sigma_{g}(i)} \|_{2} \, \lt \, \epsilon_{\rm mag}. \eqno(11)]$

The present algorithm extends our previous work on detecting magnetic symmetry operations (Shinohara et al., 2023). We first consider a space group of a nonmagnetic crystal structure (A, X, T) in Section 3.1. Next, we search for normal subgroups of the spin space group: the spin-only group $[{\cal P}_{\rm so} ({\cal G})]$ in Section 3.2 and the translation subgroup $[{\cal T} ({\cal D} ({\cal G}))]$ in Section 3.3. With these normal subgroups of $[{\cal G}_{\rm st} ({\cal G})]$ , we search for coset representatives of the spin translation group $[{\cal G}_{\rm st} ({\cal G}) /]$ $[({\cal T} ({\cal D} ({\cal G})) \times {\cal P}_{\rm so} ({\cal G}))]$ in Section 3.4. We search for coset representatives of the spin space group $[{\cal G} / {\cal G}_{\rm st} ({\cal G})]$ in Section 3.5. Because the notation in this section is abstract to deal unambiguously with several basis vectors, it may be helpful to read it alongside the examples in Section 4.

3.1. Space group of a nonmagnetic crystal structure

A candidate for spatial operations of $[{\cal G}]$ can be derived from symmetry operations of a crystal structure (A, X, T) ignoring the magnetic moments. We also employ the space group $[{\cal S}]$ given as a stabilizer of E(3) preserving (A, X, T):

$[\specialfonts {\cal S} = \left \{ g = ({\overline {\bf R}}, {\overline {\bf v}})_{\bf A} \in {\rm E} ({\rm 3}) \ \left | \matrix{\exists \sigma_{g} \in {\frak S}_{N}, \forall i = {\rm 1}, \cdots N, \cr {\overline {\bf R}} {\bf x}_{i} + {\overline {\bf v}} \equiv {\bf x}_{\sigma_{g}(i)} \, (\bmod \, {\rm 1}) \hfill \cr t_{i} = t_{\sigma_{g}(i)} \hfill } \right. \right \}, \eqno(12)]$

where $[\specialfonts {\frak S}_{N}]$ is a symmetric group of degree N. Note that $[g = ({\overline {\bf R}}, {\overline {\bf v}})_{\bf A}]$ maps point coordinates x_i to $[{\overline {\bf R}} {\bf x}_{i} + {\overline {\bf v}}]$ . The mapped point coordinates coincide with point coordinates in X up to modulo one, inducing permutation σ_g.

The existing crystal symmetry search algorithm (Togo & Tanaka, 2018) can find primitive basis vectors $[{\bf A}_{\cal S}]$ of $[{\cal T} ({\cal S})]$ and coset decomposition of $[{\cal S}]$ over $[{\cal T}_{\bf A}]$ , where we write a translation subgroup formed by A as

$[{\cal T}_{\bf A} = \left \{ ({\bf E}, {\bf n})_{\bf A} \ | \ {\bf n} \in {\bb Z}^{3} \right \}. \eqno(13)]$

The input basis vector A can be represented as an integer linear combination of $[{\bf A}_{\cal S}]$ . Thus, an integer matrix $[{\bf U} \in {\bb Z}^{3\times 3}]$ exists such that $[{\bf A} = {\bf A}_{\cal S} {\bf U}]$ . The translation subgroup $[{\cal T}_{{\bf A}_{\cal S}}]$ is decomposed as

$[{\cal T}_{{\bf A}_{\cal S}} = \bigsqcup_{{\overline {\bf t}}^{\,{\cal S}}} \left ({\bf E}, {\overline {\bf t}}^{\,{\cal S}} \right)_{{\bf A}_{\cal S}} {\cal T}_{\bf A}, \eqno(14)]$

where $[{\overline {\bf t}}^{\,{\cal S}}]$ is a centering vector in a unit cell spanned by A. The coset decomposition of $[{\cal S}]$ is written as

$[{\cal S} = \bigsqcup_{{\overline {\bf R}}^{\,{\cal S}}} \left ({\overline {\bf R}}^{\,{\cal S}}, {\overline {\bf v}}_{{\overline {\bf R}}^{\,{\cal S}}} \right)_{{\bf A}_{\cal S}} {\cal T}_{{\bf A}_{\cal S}}. \eqno(15)]$

Here, $[{\overline {\bf v}}_{{\overline {\bf R}}^{\,{\cal S}}}]$ is a translation part of a symmetry operation with a matrix part $[{\overline {\bf R}}^{\,{\cal S}}]$ .

For the spin arrangement (A, X, T, M), the spin space group can be expressed as a stabilizer of $[{\cal S} \times {\rm O}(3)]$ that preserves (A, X, T, M),

$[{\cal G} = \left \{ (g, {\bf W}) \in {\cal S} \times {\rm O} (3) \ | \ {\bf W} {\bf m}_{i} = {\bf m}_{\sigma_{g}(i)} \quad (i = 1, \ldots, N) \right \}. \eqno(16)]$

This expression serves as the starting point for the spin symmetry operation search. For notation simplicity, we denote the maximal space subgroup of $[{\cal G}]$ as $[{\cal D} = {\cal D} ({\cal G})]$ .

3.2. Spin-only group search

Because a symmetry operation in the spin-only group of $[{\cal G}]$ does not change the order of point coordinates, the spin-only group of (A, X, T, M) is expressed as

$[{\cal P}_{\rm so} = \left \{ {\bf W} \in {\rm O} (3) \ | \ {\bf W} {\bf m}_{i} = {\bf m}_{i} \quad (i = 1, \ldots, N) \right \}. \eqno(17)]$

As shown in Table 2, when a spin space group $[{\cal G}]$ is a stabilizer of a spin arrangement, spin-only groups are classified into four types up to transformations (Litvin & Opechowski, 1974; Liu et al., 2022): nonmagnetic, collinear, coplanar and non-coplanar spin arrangements.

Table 2
Classification of spin-only groups up to transformations

The spin-only groups are represented in Hermann–Mauguin symbols (Hahn et al., 2016 ).

Spin arrangement	Spin-only group	Eigenvalues of MM^T
Nonmagnetic	∞∞m ≅ O(3)	σ₁ = σ₂ = σ₃ = 0
Collinear	$[\infty m \cong {\rm SO}(2) {\times \hskip -1.7pt \hbox{\vrule height 4.6pt depth -0.1pt}} \, {\bb Z}_{2}]$	σ₁ > σ₂ = σ₃ = 0
Coplanar	m	σ₁ ≥ σ₂ > σ₃ = 0
Non-coplanar	1	σ₁ ≥ σ₂ ≥ σ₃ > 0

In Section 3.2.1 we provide an algorithm for detecting $[{\cal P}_{\rm so}]$ using the eigenvalue decomposition of MM^T. The spin-only group search should be performed within tolerances for magnetic moments in practice. In fact, selecting appropriate tolerances is challenging. In Section 3.2.2 we propose a robust algorithm, building on the previous section, to alleviate this difficulty.

3.2.1. Spin-only group search by eigenvalue decomposition

We consider a moment tensor of $[{\bf M} \in {\bb R}^{3\times N}]$ ,

$[{\bf M} {\bf M}^{\rm T} = \sum_{i = 1}^{N} {\bf m}_{i} \otimes {\bf m}_{i}. \eqno(18)]$

Because MM^T is a symmetric semi-definite matrix, we can consider its eigenvalue decomposition,

$[{\bf M} {\bf M}^{\rm T} = \sum_{r = 1}^{3} \sigma_{r} {\hat {\bf n}}_{r} \otimes {\hat {\bf n}}_{r}, \eqno(19)]$

where σ₁ ≥ σ₂ ≥ σ₃ ≥ 0 and $[\{ {\hat {\bf n}}_{r} \}_{r = 1}^{3}]$ are orthonormal.

The spin-only group is classified based on the eigenvalues $[\{ \sigma_{i} \}_{i = 1}^{3}]$ , as summarized in Table 2. In a nonmagnetic spin arrangement, all magnetic moments are zero. In this case, all eigenvalues are zero, σ₁ = σ₂ = σ₃ = 0. In a collinear spin arrangement, all magnetic moments align parallel or antiparallel to a direction $[{\hat {\bf n}}_{\parallel}]$ . The eigenvector $[{\hat {\bf n}}_{1}]$ with the largest eigenvalue should be parallel or antiparallel to $[{\hat {\bf n}}_{\parallel}]$ , and the other eigenvalues σ₂ and σ₃ should be zero. In a coplanar spin arrangement, all magnetic moments align perpendicular to a direction $[{\hat {\bf n}}_{\perp}]$ . In this case, the eigenvector $[{\hat {\bf n}}_{3}]$ with the smallest eigenvalue should be parallel or antiparallel to $[{\hat {\bf n}}_{\perp}]$ and the smallest eigenvalue σ₃ should be zero. In other cases, the spin arrangement is non-coplanar. Note that the site order of magnetic moments M does not affect the classification because MM^T in equation (18) remains invariant under a permutation of the site order.

3.2.2. Numerically robust spin-only group search

The spin-group search algorithm in the previous section requires a judgment on whether the eigenvalues are zero or positive, which needs an additional tolerance parameter. To reduce the number of tolerance parameters for usability, we modify the spin-group search algorithm solely with the tolerance ε_mag in equation (11) as follows.

(i) We compute the eigenvalues σ_i and eigenvectors $[{\hat {\bf n}}_{i}]$ in equation (19).

(ii) If all magnetic moments are close to zero within ε_mag,

$[\| {\bf m}_{i} \|_{2} \, \lt \, \epsilon_{\rm mag} \quad (i = 1, \ldots, N), \eqno(20)]$

the spin arrangement is nonmagnetic.

(iii) If not, we check whether the eigenvector $[{\hat {\bf n}}_{1}]$ is a parallel or antiparallel direction for all magnetic moments,

$[2 \| {\bf m}_{i} - ({\bf m}_{i} \cdot {\hat {\bf n}}_{1}) \, {\hat {\bf n}}_{1} \|_{2} \, \lt \, \epsilon_{\rm mag} \quad (\forall i = 1, \ldots, N). \eqno(21)]$

When spin symmetry operations in a collinear spin-only group prescribed by a direction $[{\hat {\bf n}}_{1}]$ act on m_i, the acted-on magnetic moments draw a cone, as shown in Fig. 1(a). The left-hand side of the above inequality is the largest displacement between the acted-on magnetic moments, which corresponds to the diameter of the cone with direction $[{\hat {\bf n}}_{1}]$ . If the inequality holds for all magnetic moments, the spin arrangement is collinear.

Figure 1
Magnetic moments acted on by (a) collinear and (b) coplanar spin-only groups. (a) A collinear spin-only group along the axis $[{\hat {\bf n}}_{1}]$ is generated from rotations along $[{\hat {\bf n}}_{1}]$ and mirror operations preserving $[{\hat {\bf n}}_{1}]$ . The red dotted line indicates the largest displacement between magnetic moments acting on the collinear spin-only group. (b) A coplanar spin-only group along the axis $[{\hat {\bf n}}_{3}]$ is generated from a mirror operation perpendicular to $[{\hat {\bf n}}_{3}]$ . The red dotted line indicates the largest displacement between magnetic moments acting on the coplanar spin-only group.

(iv) If not, we check whether the eigenvector $[{\hat {\bf n}}_{3}]$ is a perpendicular direction for all magnetic moments,

$[2 \left | ({\bf m}_{i} \cdot {\hat {\bf n}}_{3}) \right | \, \lt \, \epsilon_{\rm mag} \quad (\forall i = 1, \ldots, N). \eqno(22)]$

The left-hand side of the above inequality is the largest displacement between magnetic moments acted on by a coplanar spin-only group along $[{\hat {\bf n}}_{3}]$ , as shown in Fig. 1(b). If the inequality holds for all magnetic moments, the spin arrangement is coplanar.

(v) Otherwise, the spin arrangement is non-coplanar.

3.3. Translation subgroup search

We search for primitive basis vectors $[{\bf A}_{\cal D}]$ of the translation subgroup $[{\cal T} ({\cal D} ({\cal G})) = \{ ({\bf E}, {\bf v}) \ | \ (({\bf E}, {\bf v}), {\bf E}) \in {\cal G} \}]$ . The group–subgroup relationships of translation subgroups are shown in Fig. 2. Because $[{\cal T}_{{\bf A}_{\cal D}}]$ is between $[{\cal T}_{{\bf A}_{\cal S}}]$ and $[{\cal T}_{\bf A}]$ , $[{\cal T}_{\bf A} \, {\underline {\triangleleft}} \, {\cal T}_{{\bf A}_{\cal D}} \, {\underline {\triangleleft}} \, {\cal T}_{{\bf A}_{\cal S}}]$ , we only need to examine finite coset representatives of $[{\cal T}_{{\bf A}_{\cal S}} / {\cal T}_{\bf A}]$ for candidates of $[{\cal T}_{{\bf A}_{\cal S}} / {\cal T}_{{\bf A}_{\cal D}}]$ . For every coset $[({\bf E}, {\overline {\bf t}}^{\,{\cal S}})_{{\bf A}_{\cal S}} {\cal T}_{\bf A}]$ in equation (14), we check whether $[({\bf E}, {\overline {\bf t}}^{\,{\cal S}} )_{{\bf A}_{\cal S}}]$ preserves magnetic moments M,

$[{\cal T}_{{\bf A}_{\cal D}} = \left \{ g = \left ( {\bf E}, {\overline {\bf t}}^{\,{\cal S}} \right )_{{\bf A}_{\cal S}} \ \left | \ \matrix{ g{\cal T}_{{\bf A}} \in {\cal T}_{{\bf A }_{\cal S}} / {\cal T}_{\bf A}, \hfill \cr {\bf m}_{i} = {\bf m}_{\sigma_{g}(i)} \ (i = 1, \ldots, N)} \right . \right \} {\cal T}_{\bf A}. \eqno(23)]$

Figure 2
The group–subgroup relationship of translation subgroups and space groups derived from a spin space group. The nodes represent translation subgroups and space groups. Each edge indicates that a lower group is a subgroup of an upper group in the diagram. Note that $[{\cal S}_{\cal D} {\cal T}_{{\bf A}_{\cal S}}]$ could be a proper subgroup of $[{\cal S}]$ .

An integer matrix $[{\bf V} \in {\bb Z}^{3\times 3}]$ exists such that $[{\bf A}_{\cal D} = {\bf A}_{\cal S} {\bf V}]$ . A lattice algorithm to generate V from centering vectors $[{\overline {\bf t}}^{\,{\cal S}}]$ in equation (23) is presented in Appendix C.

3.4. Spin translation group search

We can find a spin rotation W for a given symmetry operation $[g \in {\cal S}]$ by solving a Procrustes problem (Gower & Dijksterhuis, 2004) as follows. We write magnetic moments permuted by g as

$[{\bf M}_{g} = \left ( {\bf m}_{\sigma_{g}(1)}, \cdots, {\bf m}_{\sigma_{g}(N)} \right ), \eqno(24)]$

where σ_g is a permutation of N sites induced by g. Rather than directly searching for W ∈ O(3) such that $[{\bf W} {\bf m}_{i}]$ = $[{\bf m}_{\sigma_{g}(i)}]$ , we choose a candidate $[{\tilde {\bf W}}]$ by solving the following Procrustes problem:

$[{\tilde {\bf W}} = {\mathop{\rm argmin}\limits_{{\bf W} \in {\rm O}(3)}} \ \| {\bf M}_{g} - {\bf W} {\bf M} \|_{\rm F}. \eqno(25)]$

Here the displacement between the magnetic moments and those operated on by (g, W) is measured by the Frobenius norm ∥·∥_F. The solution to equation (25) can be explicitly written as

$[{\tilde {\bf W}} = {\bf Y} {\bf Z}^{\rm T}, \eqno(26)]$

where Y and Z are orthogonal matrices of the singular value decomposition of M_gM^T,

$[{\bf M}_{g} {\bf M}^{\rm T} = {\bf Y} {\bf \Sigma} {\bf Z}^{\rm T}. \eqno(27)]$

After we obtain $[{\tilde {\bf W}}]$ , $[(g,{\tilde {\bf W}})]$ is taken as a spin symmetry operation if the condition

$[\| {\bf m}_{i} - {\tilde {\bf W}} {\bf m}_{\sigma_{g}(i)} \|_{2} \, \lt \, \epsilon_{\rm mag} \eqno(28)]$

holds for every site i. If equation (28) does not hold for some site i, we reject $[(g,{\tilde {\bf W}})]$ as a spin symmetry operation.

For coset representatives of $[({\bf E}, {\overline {\bf t}}^{\,{\cal D}})_{{\bf A}_{\cal D}} {\cal T}_{{\bf A}_{\cal D}} \in {\cal T}_{{\bf A}_{\cal S}} / {\cal T}_{{\bf A}_{\cal D}}]$ , we search for a corresponding spin rotation $[{\bf W}_{{\overline {\bf t}}^{\,{\cal D}}}]$ using the above algorithm if it exists. The coset decomposition of $[{\cal G}_{\rm st}]$ can be written as

$[\eqalignno{ & {\cal G}_{\rm st} = \cr & \left \{ g = \left ( ({\bf E}, {\overline {\bf t}}^{\,{\cal D}})_{{\bf A}_{\cal D}}, {\bf W}_{{\overline {\bf t}}^{\,{\cal D}}} \right ) \ \left | \ \matrix{ ({\bf E}, {\overline {\bf t}}^{\,{\cal D}})_{{\bf A}_{\cal D}} {\cal T}_{{\bf A}_{\cal D}} \in {\cal T}_{{\bf A}_{\cal S}} / {\cal T}_{{\bf A}_{\cal D}} \hfill \cr {\bf W}_{{\overline {\bf t}}^{\,{\cal D}}} {\bf m}_{i} = {\bf m}_{\sigma_{g}(i)} \, (\forall i = 1, \cdots, N)} \right. \right \} \cr & \times \left ( {\cal T}_{{\bf A}_{\cal D}} \times {\cal P}_{\rm so} \right ) , &(29)}]$

where $[{\overline {\bf t}}^{\,{\cal D}}]$ takes a distinct translation up to $[{\cal T}_{{\bf A}_{\cal D}}]$ and we choose $[{\bf W}_{{\overline {\bf t}}^{\,{\cal D}} = {\bf 0}} = {\bf E}]$ .

3.5. Spin space-group search

Because the translation subgroup of $[{\cal F} ({\cal G})]$ is $[{\cal T}_{{\bf A}_{\cal D}}]$ , the spatial operation parts of $[{\cal G}]$ should belong to a maximal subgroup $[{\cal S}_{\cal D}]$ of $[{\cal S}]$ with its translation subgroup $[{\cal T}_{{\bf A}_{\cal D}}]$ . Fig. 2 shows the group–subgroup relationship between $[{\cal S}]$ and $[{\cal S}_{\cal D}]$ . A rotation $[{\bf A}_{\cal S} {\overline {\bf R}}^{\,{\cal S}} {\bf A}_{\cal S}^{-1} \in {\cal P} ({\cal S})]$ is compatible with $[{\cal T}_{{\bf A}_{\cal D}}]$ if $[{\bf V}^{-1} {\overline {\bf R}}^{\,{\cal S}} {\bf V}]$ is an integer matrix (Hart & Forcade, 2008 ). Thus, the compatible subgroup $[{\cal S}_{\cal D}]$ is written as

$[\eqalignno{ & {\cal S}_{\cal D} = \cr & \left \{ \left ( {\overline {\bf R}}^{\,{\cal S}}, {\overline {\bf v}}_{{\overline {\bf R}}^{\,{\cal S}}} + {\overline {\bf t}}_{{\overline {\bf R}}^{\,{\cal S}}} \right )_{{\bf A}_{\cal S}} \ \left | \ \matrix{ \left ( {\overline {\bf R}}^{\,{\cal S}}, {\overline {\bf v}}_{{\overline {\bf R}}^{\,{\cal S}}} \right ) {\cal T}_{{\bf A}_{\cal S}} \in {\cal S} / {\cal T}_{{\bf A}_{\cal S}} \hfill \cr {\bf V}^{-1} {\overline {\bf R}}^{\,{\cal S}} {\bf V} \in {\bb Z}^{3\times 3} \hfill \cr \left ( {\bf E}, {\overline {\bf t}}_{{\overline {\bf R}}^{\,{\cal S}}} \right )_{{\bf A}_{\cal S}} {\cal T}_{\bf A} \in {\cal T}_{{\bf A}_{\cal S}} / {\cal T}_{\bf A}} \right. \right \} {\cal T}_{{\bf A}_{\cal D}} , \cr &&(30)}]$

where $[{\overline {\bf t}}_{{\overline {\bf R}}^{\,{\cal S}}}]$ is a centering vector in $[{\cal T}_{{\bf A}_{\cal S}} / {\cal T}_{\bf A}]$ . The additional centering vector $[{\overline {\bf t}}_{{\overline {\bf R}}^{\,{\cal S}}}]$ is necessary for $[{\cal S}_{\cal D}]$ to form a group (Nebe, 2011 ; Stokes & Campbell, 2017 ).

For a coset representative of $[{\cal S}_{\cal D} / {\cal T}_{{\bf A}_{\cal D}}]$ , a corresponding spin-rotation part $[{\bf W}_{{\overline {\bf R}}^{\,{\cal S}}}]$ can also be determined by solving the Procrustes problem as presented in Section 3.4. Finally, we obtain all spin symmetry operations in a coset decomposition,

$[\eqalignno{ & {\cal G} = \cr & \quad \Bigg \{ \left ( \left ( {\overline {\bf R}}^{\,{\cal S}}, {\overline {\bf v}}_{{\overline {\bf R}}^{\,{\cal S}}} + {\overline {\bf t}}_{{\overline {\bf R}}^{\,{\cal S}}} \right )_{{\bf A}_{\cal S}}, {\bf W}_{{\overline {\bf R}}^{\,{\cal S}}} \right ) \cr & \quad \left | \ \matrix{ \left ( {\overline {\bf R}}^{\,{\cal S}}, {\overline {\bf v}}_{{\overline {\bf R}}^{\,{\cal S}}} + {\overline {\bf t}}_{{\overline {\bf R}}^{\,{\cal S}}} \right )_{{\bf A}_{\cal S}} {\cal T}_{{\bf A}_{\cal D}} \in {\cal S}_{\cal D} / {\cal T}_{{\bf A}_{\cal D}} \cr {\bf W}_{{\overline {\bf R}}^{\,{\cal S}}} {\bf m}_{i} = {\bf m}_{\sigma_{g}(i)} \, (\forall i = 1, \cdots, N) \hfill } \right \} {\cal G}_{\rm st}. &(31)}]$

4. Examples of spin symmetry operation search

We consider a spin arrangement of the NiAs-type CrSe (Corliss et al., 1961 ; Litvin, 1973) [entry No. 2.35 of MAGNDATA (Gallego et al., 2016)] as an example of a spin symmetry operation search, illustrated in Fig. 3. The hexagonal lattice of CrSe has the following basis vectors:

$[{\bf A} = \left ( \matrix{ a & -{{1} \over {2}}a & 0 \cr 0 & {{\sqrt 3} \over {2}}a & 0 \cr 0 & 0 & c } \right ) .]$

The fractional coordinates and magnetic moments of the atoms are as follows:

$[{\rm Cr}{:} \ {\bf x}_{1} = (0, 0, 0)^{\rm T}, \ {\bf m}_{1} = \left ( -{{1} \over {2}}m_{x}, -{{\sqrt 3} \over {2}}m_{x}, -m_{z} \right )^{\rm T} , ]$

$[{\rm Cr}{:} \ {\bf x}_{2} = \left ( {{1} \over {3}}, {{2} \over {3}}, 0 \right )^{\rm T}, \ {\bf m}_{2} = \left ( m_{x}, 0, -m_{z} \right )^{\rm T} ,]$

$[{\rm Cr}{:} \ {\bf x}_{3} = \left ( {{2} \over {3}}, {{1} \over {3}}, 0 \right )^{\rm T}, \ {\bf m}_{3} = \left ( -{{1} \over {2}} m_{x}, {{\sqrt 3} \over {2}} m_{x}, -m_{z} \right ) ^{\rm T} , ]$

$[{\rm Cr}{:} \ {\bf x}_{4} = \left ( 0, 0, {{1} \over {2}} \right )^{\rm T}, \ {\bf m} _{4} = \left ( {{1} \over {2}} m_{x}, {{\sqrt 3} \over {2}} m_{x}, m_{z} \right )^{\rm T} ,]$

$[{\rm Cr}{:} \ {\bf x}_{5} = \left ( {{1} \over {3}}, {{2} \over {3}}, {{1} \over {2}} \right )^{\rm T}, \ {\bf m}_{5} = \left ( -m_{x}, 0, m_{z} \right )^{\rm T} ,]$

$[{\rm Cr}{:} \ {\bf x}_{6} = \left ( {{2} \over {3}}, {{1} \over {3}}, {{1} \over {2}} \right )^{\rm T}, \ {\bf m}_{6} = \left ( {{1} \over {2}} m_{x}, -{{\sqrt 3} \over {2}} m_{x}, m_{z} \right )^{\rm T} , ]$

$[{\rm Se}{:} \ {\bf x}_{7} = \left (0, {{1} \over {3}}, {{1} \over {4}} \right )^{\rm T}, \ {\bf m}_{7} = (0, 0, 0)^{\rm T} ,]$

$[{\rm Se}{:} \ {\bf x}_{8} = \left ( {{1} \over {3}}, 0, {{1} \over {4}} \right )^{\rm T}, \ {\bf m}_{8} = (0, 0, 0)^{\rm T} ,]$

$[{\rm Se}{:} \ {\bf x}_{9} = \left ( {{1} \over {3}}, {{1} \over {3}}, {{3} \over {4}} \right )^{\rm T}, \ {\bf m}_{9} = (0, 0, 0)^{\rm T} ,]$

$[{\rm Se}{:} \ {\bf x}_{10} = \left ( 0, {{2} \over {3}}, {{3} \over {4}} \right )^{\rm T}, \ {\bf m}_{10} = (0, 0, 0)^{\rm T} ,]$

$[{\rm Se}{:} \ {\bf x}_{11} = \left ( {{2} \over {3}}, 0, {{3} \over {4}} \right)^{\rm T}, \ {\bf m}_{11} = (0, 0, 0)^{\rm T} ,]$

$[{\rm Se}{:} \ {\bf x}_{12} = \left ( {{2} \over {3}}, {{2} \over {3}}, {{1} \over {4}} \right )^{\rm T}, \ {\bf m}_{12} = (0, 0, 0)^{\rm T} .]$

Here, the magnetic moments are represented with Cartesian coordinates.

Figure 3
A spin arrangement example for NiAs-type CrSe. The gray and orange balls represent Cr and Se atoms, respectively. The red arrows denote magnetic moments of Cr atoms with equal magnitudes.

4.1. Space group of a nonmagnetic crystal structure

The space-group type of $[{\cal S}]$ for the crystal structure ignoring magnetic moments is P6₃/mmc (No. 194). One of the primitive basis vectors A for $[{\cal S}]$ and the transformation matrix U are

$[{\bf A}_{\cal S} = \left ( \matrix{ 0 & -{{1} \over {2}}a & 0 \cr -{{\sqrt 3} \over {3}}a & {{\sqrt 3} \over {6}}a & 0 \cr 0 & 0 & -c } \right ) , ]$

$[{\bf U} = \left ( \matrix{ -1 & -1 & 0 \cr -2 & 1 & 0 \cr 0 & 0 & -1} \right ) , ]$

with $[{\bf A} = {\bf A}_{\cal S} {\bf U}]$ as defined in Section 3.1. There are three coset representatives for $[{\cal T}_{{\bf A}_{\cal S}} / {\cal T}_{\bf A}]$ ,

$[\eqalignno{{\cal T}_{{\bf A}_{\cal S}} = & \, g_{1} {\cal T}_{\bf A} \sqcup g_{2} {\cal T}_{\bf A} \sqcup g_{3} {\cal T}_{\bf A}, &(32) \cr g_{1} = & \, \left ( {\bf E}, (0, 0, 0)^{\rm T} \right )_{{\bf A}_{\cal S}}, \cr g_{2} = & \, \left ( {\bf E}, (-1, -1, 0)^{\rm T} \right )_{{\bf A}_{\cal S}}, \cr g_{3} = & \, \left ( {\bf E}, (-1, 0, 0)^{\rm T} \right )_{{\bf A}_{\cal S}} .}]$

4.2. Spin-only group

The moment tensor of M is given by

$[{\bf M} {\bf M}^{\rm T} = \left ( \matrix{ 3m_{x}^{2} & 0 & 0 \cr 0 & 3m_{x}^{2} & 0 \cr 0 & 0 & 6m_{z}^{2} } \right ) . ]$

Its eigenvalues 3m_x², 3m_x² and 6m_z² are all positive. Consequently, the spin arrangement is non-coplanar with $[{\cal P}_{\rm so}]$ = 1.

4.3. Translation subgroup of a maximal space subgroup

The translation g₁ in equation (32) belongs to $[{\cal T}_{{\bf A}_{\cal D}}]$ . On the other hand, g₂ does not belong to $[{\cal T}_{{\bf A}_{\cal D}}]$ because it maps x₁ to x₃, whereas m₁ ≠ m₃. Similarly, g₃ does not belong to $[{\cal T}_{{\bf A}_{\cal D}}]$ . Therefore, $[{\cal T}_{{\bf A}_{\cal D}}]$ is identical to $[{\cal T}_{{\bf A}_{\cal S}}]$ with $[{\bf A}_{\cal D} = {\bf A}_{\cal S} {\bf V}]$ as defined in Section 3.3 and we can choose V = E.

4.4. Coset representatives of a spin translation group

There are three candidates for the spatial operation parts of coset representatives of $[{\cal G}_{\rm st}]$ : g₁, g₂ and g₃. For translation g₁ in equation (32), we obtain

$[{\bf M}_{g_{1}} {\bf M}^{\rm T} = \left ( \matrix{ 3m_{x}^{2} & 0 & 0 \cr 0 & 3m_{x}^{2} & 0 \cr 0 & 0 & 6m_{z}^{2} } \right ). ]$

From the singular value decomposition of $[{\bf M}_{g_{1}} {\bf M}^{\rm T}]$ , we calculate the spin-rotation part $[{\tilde {\bf W}}_{1}]$ for g₁ as $[{\tilde {\bf W}}_{1} = {\bf E}]$ . For translation g₂, the singular value decomposition of $[{\bf M}_{g_{2}} {\bf M}^{\rm T}]$ is

$[\eqalign{ & {\bf M}_{g_{2}} {\bf M}^{\rm T} = \cr & \quad \left ( \matrix{ -{{1} \over {2}} & {{\sqrt 3} \over {2}} & 0 \cr -{{\sqrt 3} \over {2}} & -{{1} \over {2}} & 0 \cr 0 & 0 & 1 } \right ) \left ( \matrix{ 3m_{x}^{2} & 0 & 0 \cr 0 & 3m_{x}^{2} & 0 \cr 0 & 0 & 6m_{x}^{2} } \right ) \left ( \matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 } \right) ^{\rm T} .}]$

Hence, a candidate for a spin-rotation part with g₂ is

$[\eqalign{{\tilde {\bf W}}_{2} = & \, \left ( \matrix{ -{{1} \over {2}} & {{\sqrt 3} \over {2}} & 0 \cr -{{\sqrt 3} \over {2}} & -{{1} \over {2}} & 0 \cr 0 & 0 & 1 } \right ) \left ( \matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 } \right )^{\rm T} \cr = & \, \left ( \matrix{ -{{1} \over {2}} & {{\sqrt 3} \over {2}} & 0 \cr -{{\sqrt 3} \over {2}} & -{{1} \over {2}} & 0 \cr 0 & 0 & 1 } \right) .}]$

Similarly, a candidate for a spin-rotation part with g₃ is

$[{\tilde {\bf W}}_{3} = \left ( \matrix{ -{{1} \over {2}} & -{{\sqrt 3} \over {2}} & 0 \cr {{\sqrt 3} \over {2}} & -{{1} \over {2}} & 0 \cr 0 & 0 & 1 } \right ) . ]$

The spin symmetry operations $[(g_{i}, {\tilde {\bf W}}_{i})]$ (i = 1, 2, 3) all preserve magnetic moments. Consequently, the spin translation group is obtained as

$[\eqalign{ {\cal G}_{\rm st} = & \, (g_{1}, {\tilde {\bf W}}_{1}) \, ({\cal T}_{{\bf A}_{\cal D}} \times 1) \sqcup (g_{2}, {\tilde {\bf W}}_{2}) \, ({\cal T}_{{\bf A}_{\cal D}} \times 1) \cr & \, \sqcup (g_{3}, {\tilde {\bf W}}_{3}) \, ({\cal T}_{{\bf A}_{\cal D}} \times 1) .}]$

4.5. Coset representatives of a spin space group

Because $[{\cal T}_{{\bf A}_{\cal D}}]$ and $[{\cal T}_{{\bf A}_{\cal S}}]$ coincide in this example, $[{\cal S}]$ is fully compatible with $[{\cal T}_{{\bf A}_{\cal D}}]$ , $[{\cal S}_{\cal D} = {\cal S}]$ . For the 24 coset representatives of $[{\cal S}_{\cal D} / {\cal T}_{{\bf A}_{\cal D}}]$ , we search for their spin-rotation parts by solving the Procrustes problems. For example, a sixfold spatial rotoinversion

$[g = \left (\left ( \matrix{ -1 & 1 & 0 \cr -1 & 0 & 0 \cr 0 & 0 & -1 } \right ) , {\bf 0} \right )_{{\bf A}_{\cal D}} \in {\cal S}_{\cal D} ]$

gives a permutation of sites σ_g = (4, 5, 6, 1, 2, 3, 12, 7, 11, 9, 10, 8) and the singular value decomposition of M_gM^T,

$[\eqalign{ & {\bf M}_{g} {\bf M}^{\rm T} = \cr & \quad \left ( \matrix{ -1 & 0 & 0 \cr 0 & -1 & 0 \cr 0 & 0 & -1 } \right ) \left ( \matrix{ 3m_{x}^{2} & 0 & 0 \cr 0 & 3m_{x}^{2} & 0 \cr 0 & 0 & 6m_{x}^{2} } \right ) \left ( \matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 } \right )^{\rm T} ,}]$

which results in $[{\tilde {\bf W}} = -{\bf E}]$ . Similarly, all of the 24 coset representatives of $[{\cal S}_{\cal D}/{\cal T}_{{\bf A}_{\cal D}}]$ give spin symmetry operations preserving M.

A magnetic space group $[{\cal M}]$ of the NiAs-type CrSe is P31m′ (BNS number 157.55), which has $[|{\cal M}/{\cal T}_{{\bf A}_{\cal D}}| = 6]$ coset representatives. Because the spin translation group $[{\cal G}_{\rm st}]$ contains threefold spin-rotation parts, these spin symmetry operations [ $[(g_{2}, {\tilde {\bf W}}_{2})]$ and $[(g_{3}, {\tilde {\bf W}}_{3})]$ in Section 4.4] are not captured in $[{\cal M}]$ . Similarly, the spin symmetry operations with sixfold spatial rotations or rotoinversions are not preserved in $[{\cal M}]$ .

5. Conclusions

We have presented an algorithm for determining the spin symmetry operations of a given spin arrangement. The spin-only group is robustly determined from the eigenvalue decomposition of the moment tensor of magnetic moments, MM^T. We have explicitly considered the three translation subgroups to address the enlargement of the unit cell due to the spin translation group: the translation subgroup spanned by the input basis vectors $[{\cal T}_{\bf A}]$ , one by the primitive basis vector $[{\cal T}_{{\bf A}_{\cal S}}]$ and one by the primitive basis vectors for the spin space group $[{\cal T}_{{\bf A}_{\cal D}}]$ . Spin-rotation parts of the coset representatives of the spin translation group and spin space group are found by solving the Procrustes problem to match the original magnetic moments and permuted ones. The present algorithm is implemented in spinspg under a permissive license, available at https://github.com/spglib/spinspg. In future work, it will be beneficial to identify the spin space-group type and a suitable transformation from the spin symmetry operations. Our presented algorithm and implementation will advance spin symmetry analysis in crystallography and condensed matter physics.

Notes added during review. After this work was completed, we became aware of recent preprints on the classification and enumeration of spin space-group types (Xiao et al., 2023 ; Ren et al., 2023 ; Jiang et al., 2023 ). Two of these reports, Xiao et al. (2023) and Jiang et al. (2023), appear to use a workflow similar to ours to determine spin symmetry operations, although they only provide a brief description and their implementations are not available to the public.

Our presented algorithm uses the Procrustes problem to determine a spin-rotation part W. On the other hand, Xiao et al. (2023) directly determine W for a symmetry operation g such that three selected magnetic moments $[{\bf m}_{i}]$ (i = 1, 2, 3) are transformed into $[{\bf m}_{\sigma_{g}(i)}]$ , which may not be robust against numerical noise and the experimental uncertainty of magnetic moments.

Jiang et al. (2023) identify spin symmetry operations on top of spglib (Togo & Tanaka, 2018) similar to ours. However, they do not incorporate translation subgroups $[{\cal T}_{{\bf A}_{\cal D}}]$ and $[{\cal T}_{\bf A}]$ . This omission makes their algorithm complicated and non-exhaustive.

APPENDIX A

Correspondence between spin symmetry operation and magnetic symmetry operation

When we consider a spin symmetry operation ((R, v), W) for a Hamiltonian without spin–orbit coupling (SOC), we derive the condition that ((R, v), W) has a corresponding magnetic symmetry operation for the Hamiltonian with SOC. The SOC introduces an additional term to the Hamiltonian proportional to $[{\hat {\bf L}} \cdot {\hat {\bf \sigma}}]$ , with the angular momentum operator $[{\hat {\bf L}}]$ and the Pauli matrices $[{\hat {\bf \sigma}} = ({\hat \sigma}_{x}, {\hat \sigma}_{y}, {\hat \sigma}_{z})]$ . A spin symmetry operation ((R, v), W) acts on $[{\hat {\bf L}}]$ and $[{\hat {\bf \sigma}}]$ as

$[{\hat {\bf L}} \ \mapsto \ (\det{\bf R}) \, (\det{\bf W}) \left ( \sum_{\nu = x,y,z} R_{\mu\nu} {\hat L}_{\nu} \right )_{\mu = x,y,z} , \eqno(33)]$

$[{\hat {\bf \sigma}} \ \mapsto \ \left ( \sum_{\nu = x,y,z} W_{\mu\nu} {\hat \sigma}_{\nu} \right )_{\mu = x,y,z} , \eqno(34)]$

where R and W are represented with Cartesian coordinates. Note that $[\det{\bf W}]$ in equation (33) reflects a time-reversal operation. Thus, ((R, v), W) acts on the SOC term as

$[{\hat {\bf L}} \cdot {\hat {\bf \sigma}} \ \mapsto \ (\det{\bf R}) \, (\det{\bf W}) \sum_{\mu,\nu = x,y,z} \left [ {\bf R}^{\rm T} {\bf W} \right ]_{\mu\nu} {\hat L}_{\mu} {\hat \sigma}_{\nu} . \eqno(35)]$

To preserve the SOC term, $[(\det{\bf R}) \, (\det{\bf W}) \, {\bf R}^{\rm T} {\bf W}]$ should be identity. Because $[(\det{\bf R}) \, {\bf R}]$ and $[(\det{\bf W}) \, {\bf W}]$ belong to SO(3), this condition is equivalent to

$[(\det{\bf R}) \, {\bf R} = (\det{\bf W}) \, {\bf W}. \eqno(36)]$

Therefore, if ((R, v), W) satisfies equation (36), it has a corresponding magnetic symmetry operation $[(({\bf R}, {\bf v}), \det{\bf W})]$ with symmetry operation part (R, v) and time-reversal part $[\det{\bf W}]$ . We identify $[\det{\bf W} = 1]$ with an identity operation and $[\det{\bf W} = -1]$ with a time-reversal operation. Here, we assume that a magnetic symmetry operation ((R, v), θ) acts on a magnetic moment m as $[\theta (\det{\bf R}) \, {\bf R} {\bf m}]$ .

Conversely, a magnetic symmetry operation ((R, v), θ) is mapped to a spin symmetry operation ((R, v), θR). We can confirm this mapping satisfies equation (36) by $[(\det\theta{\bf R}) \, \theta {\bf R}]$ = $[(\det{\bf R}) \, {\bf R}]$ . Because this mapping is injective, the index of a magnetic space group $[{\cal M}]$ in its translation subgroup $[{\cal T}_{{\bf A}_{\cal D}}]$ is a divisor of the index of a corresponding spin space group $[{\cal G}]$ in $[{\cal T}_{{\bf A}_{\cal D}}]$ , $[|{\cal G}/{\cal T}_{{\bf A}_{\cal D}}| \equiv 0 \, ({\rm mod} \, | \, {\cal M}/{\cal T}_{{\bf A}_{\cal D}}|)]$ .

APPENDIX B

Spin point group

We define a family spin point group of a spin space group $[{\cal G}]$ as

$[{\cal B} ({\cal G}) = \left \{ {\bf W} \in {\rm O}(3) \ | \ \exists g \ {\rm s.t.} \ (g, {\bf W}) \in {\cal G} \right \} . \eqno(37)]$

We write a coset decomposition of $[{\cal G}]$ by its spin translation group $[{\cal G}_{\rm st} ({\cal G})]$ as

$[{\cal G} = \bigsqcup_{\bf R} \left ( ({\bf R}, {\bf v}_{\bf R}), {\bf W}_{\bf R} \right ) \, {\cal G}_{\rm st} ({\cal G}) . \eqno(38)]$

A spin point group of $[{\cal G}]$ is

$[{\cal U} ({\cal G}) = \left \{ \left ( {\bf R}, {\bf W}_{\bf R} {\cal B} ({\cal G}_{\rm st} ({\cal G})) \right ) \right \}_{{\bf R} \in {\cal P}({\cal F}({\cal G}))} . \eqno(39)]$

The spin point group $[{\cal U} ({\cal G})]$ is a subgroup of $[{\rm O}(3) \times]$ $[({\rm O}(3)/{\cal B}({\cal G}_{\rm st}({\cal G})))]$ and is isomorphic to $[{\cal G}/{\cal G}_{\rm st}({\cal G})]$ .

In general, we cannot choose W_R so that {(R, W_R)} is a group under the multiplication in O(3) × O(3). We show here one of the counterexamples in which {(R, W_R)} is not closed as a group, with a spin arrangement

$[{\bf A} = \left ( \matrix{ 4 & 0 & 0 \cr 0 & 6 & 0 \cr 0 & 0 & 8 } \right )]$

$[{\bf X} = \left ( \matrix{ 0.4 & 0.6 & 0.4 & 0.4 \cr 0.4 & 0.6 & 0.4 & 0.4 \cr 0 & 0.25 & 0.5 & 0.75 } \right )]$

$[{\bf M} = \left ( \matrix{ 1 & 0 & -1 & 0 \cr 0 & 1 & 0 & -1 \cr 0.4 & 0.4 & 0.4 & 0.4 } \right ) , ]$

illustrated in Fig. 4. This spin arrangement is non-coplanar with $[{\cal P}_{\rm so}({\cal G}) = 1]$ . The spin space group $[{\cal G}]$ is obtained as

$[{\cal G} = \left \{ g_{1}, g_{2}, g_{3}, g_{4}, g_{5}, g_{6}, g_{7}, g_{8} \right \} \, \left ( {\cal T} ({\cal D} ({\cal G})) \times {\cal P}_{\rm so} ({\cal G}) \right ) , ]$

$[g_{1} = \left ( \left ( 1, {\bf 0} \right )_{\bf A}, 1 \right )]$

$[g_{2} = \left ( \left ( 1, {\scriptstyle {{1} \over {2}}} {\bf c} \right )_{\bf A}, 2_{z} \right )]$

$[g_{3} = \left ( \left ( {\overline 1}, {\scriptstyle {{3} \over {4}}} {\bf c} \right )_{\bf A}, m_{xy} \right )]$

$[g_{4} = \left ( \left ( {\overline 1}, {\scriptstyle {{1} \over {4}}} {\bf c} \right )_{\bf A} , m_{x {\overline y}} \right )]$

$[g_{5} = \left ( \left ( 2, {\scriptstyle {{3} \over {4}}} {\bf c} \right )_{\bf A}, 4^{-}_{z} \right )]$

$[g_{6} = \left ( \left ( 2, {\scriptstyle {{1} \over {4}}} {\bf c} \right )_{\bf A}, 4^{+}_{z} \right )]$

$[g_{7} = \left ( \left ( m, {\scriptstyle {{1} \over {2}}} {\bf c} \right )_{\bf A}, m_{x} \right )]$

$[g_{8} = \left ( \left ( m, {\bf 0} \right )_{\bf A}, m_{y} \right ) ,]$

where we denote a mirror operation along the xy axis as m_xy. The spin point group of $[{\cal G}]$ is

$[\eqalignno{{\cal U} ({\cal G}) = & \, \left \{ (1, 1), ({\overline 1}, m_{xy}), (2, 4^{-}_{z}), (m, m_{x}) \right \} \left ( 1 \times {\cal B} ({\cal G}_{\rm st} ({\cal G})) \right ), \cr &&(40) \cr {\cal B} ({\cal G}_{\rm st} ({\cal G})) = & \, \left \{ 1, 2_{z} \right \} .}]$

The coset representatives in equation (40) are not closed as a subgroup of O(3) × O(3) because $[(2, 4^{-}_{z})^{-1}]$ = $[(2, 4^{+}_{z})]$ does not belong to the coset representatives. In fact, we cannot choose coset representatives to form a subgroup of O(3) × O(3) in this example because $[{\cal U} ({\cal G}) \cong {\bb Z}_{4}]$ , $[{\cal B}({\cal G}_{\rm st} ({\cal G})) \cong {\bb Z}_{2}]$ , and $[{\bb Z}_{4}]$ cannot be written as an internal semidirect product $[{\bb Z}_{2} {\times \hskip -1.7pt \hbox{\vrule height 4.6pt depth -0.1pt}} \, {\bb Z}_{2}]$ .

Figure 4
The spin arrangement example described in Appendix B

. Its spin point group is not closed under O(3) × O(3) due to the spin translation group.

APPENDIX C

Generating a transformation matrix from centering vectors

To demonstrate how to generate a transformation matrix V in Section 3.3, we consider the following coplanar face-centered cubic structure with the conventional basis,

$[{\bf A} = \left ( \matrix{ a & 0 & 0 \cr 0 & a & 0 \cr 0 & 0 & a } \right )]$

$[{\bf X} = \left ( \matrix{ 0 & 0 & {{1} \over {2}} & {{1} \over {2}} \cr 0 & {{1} \over {2}} & 0 & {{1} \over {2}} \cr 0 & {{1} \over {2}} & {{1} \over {2}} & 0 } \right )]$

$[{\bf M} = \left ( \matrix{ 0 & 0 & m & m \cr 0 & 0 & 0 & 0 \cr m & m & 0 & 0 } \right ) . ]$

One of the primitive basis vectors for a nonmagnetic crystal structure is

$[{\bf A}_{\cal S} = {\bf A} {\bf U}^{-1} = {{a} \over {2}} \left ( \matrix{ 0 & 1 & 1 \cr 1 & 0 & 1 \cr 1 & 1 & 0 } \right )]$

$[{\bf U} = \left ( \matrix{ -1 & 1 & 1 \cr 1 & -1 & 1 \cr 1 & 1 & -1 } \right ) . ]$

There are four centering vectors in $[{\cal T}_{{\bf A}_{\cal S}}/{\cal T}_{\bf A}]$ ,

$[{\cal T}_{{\bf A}_{\cal S}} = g_{1} {\cal T}_{\bf A} \sqcup g_{2} {\cal T}_{\bf A} \sqcup g_{3} {\cal T}_{\bf A} \sqcup g_{4} {\cal T}_{\bf A} , ]$

$[g_{1} = \left ( {\bf E}, (0, 0, 0)^{\rm T} \right )_{{\bf A}_{\cal S}}]$

$[g_{2} = \left ( {\bf E}, (1, 0, 0)^{\rm T} \right )_{{\bf A}_{\cal S}}]$

$[g_{3} = \left ( {\bf E}, (0, 1, 0)^{\rm T} \right )_{{\bf A}_{\cal S}}]$

$[g_{4} = \left ( {\bf E}, (0, 0, 1)^{\rm T} \right )_{{\bf A}_{\cal S}} .]$

Half of the centering vectors form the translation subgroup of $[{\cal D}({\cal G})]$ ,

$[{\cal T} ({\cal D} ({\cal G})) = g_{1} {\cal T}_{\bf A} \sqcup g_{2} {\cal T}_{\bf A} .]$

Because $[{\cal T} ({\cal D} ({\cal G}))]$ is spanned by column vectors of U and centering vectors of g₁ and g₂, we can rewrite $[{\cal T} ({\cal D} ({\cal G}))]$ as

$[{\cal T} ({\cal D} ({\cal G})) = {\bf A}_{\cal S} {\tilde {\bf U}} {\bb Z}^{5} , ]$

$[{\tilde {\bf U}} = \left ( \matrix{ -1 & 1 & 1 & 0 & 1 \cr 1 & -1 & 1 & 0 & 0 \cr 1 & 1 & -1 & 0 & 0 } \right ) .]$

We need to find a transformation matrix $[{\bf V} \in {\bb Z}^{3\times 3}]$ such that

$[{\cal T} ({\cal D} ({\cal G})) = {\bf A}_{\cal S} {\bf V} {\bb Z}^{3} , \eqno(41)]$

and this is known to be achieved by the Hermite normal form of $[{\tilde {\bf U}}]$ (Cohen, 1993 ),

$[{\tilde {\bf U}} = \left ( \matrix{ 1 & 0 & 0 & 0 & 0 \cr 0 & 1 & 0 & 0 & 0 \cr 0 & 1 & 2 & 0 & 0 } \right ) \left ( \matrix{ -1 & 1 & 1 & 0 & 1 \cr 1 & -1 & 1 & 0 & 0 \cr 0 & 1 & -1 & 0 & 0 \cr 0 & 0 & 0 & 1 & 0 \cr 0 & 1 & 0 & 0 & 0 } \right ) , ]$

where $[{\tilde {\bf U}}]$ is decomposed into a product of a lower triangular integer matrix and a unimodular matrix. Because the above 5×5 integer matrix is unimodular, we obtain

$[\eqalign{ {\cal T} ({\cal D} ({\cal G})) = & \, {\bf A}_{\cal S} \left ( \matrix{ 1 & 0 & 0 & 0 & 0 \cr 0 & 1 & 0 & 0 & 0 \cr 0 & 1 & 2 & 0 & 0 } \right ) {\bb Z}^{5} \cr = & \, {\bf A}_{\cal S} \left ( \matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 1 & 2 } \right ) {\bb Z}^{3} .}]$

Consequently, we generate the integer matrix

$[{\bf V} = \left ( \matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 1 & 2 } \right ) .]$

Funding information

The following funding is acknowledged: Japan Society for the Promotion of Science (grant No. 21J10712).

References

Brinkman, W. & Elliott, R. J. (1966). J. Appl. Phys. 37, 1457–1459. CrossRef CAS Web of Science Google Scholar
Brinkman, W. F., Elliott, R. J. & Peierls, R. E. (1966). Proc. Math. Phys. Eng. Sci. 294, 343–358. CAS Google Scholar
Cohen, H. (1993). A Course in Computational Algebraic Number Theory. Berlin/Heidelberg: Springer. Google Scholar
Corliss, L. M., Elliott, N., Hastings, J. M. & Sass, R. L. (1961). Phys. Rev. 122, 1402–1406. CrossRef CAS Web of Science Google Scholar
Corticelli, A., Moessner, R. & McClarty, P. A. (2022). Phys. Rev. B, 105, 064430. Web of Science CrossRef Google Scholar
Gallego, S. V., Perez-Mato, J. M., Elcoro, L., Tasci, E. S., Hanson, R. M., Momma, K., Aroyo, M. I. & Madariaga, G. (2016). J. Appl. Cryst. 49, 1750–1776. Web of Science CrossRef CAS IUCr Journals Google Scholar
Gower, J. C. & Dijksterhuis, G. B. (2004). Procrustes Problems. Oxford University Press. Google Scholar
Hahn, T., Klapper, H., Müller, U. & Aroyo, M. (2016). International Tables for Crystallography, Vol A, 6th ed., edited by M. I. Aroyo, ch. 3.2, pp. 720–776. Chichester: Wiley. Google Scholar
Hart, G. L. W. & Forcade, R. W. (2008). Phys. Rev. B, 77, 224115. Web of Science CrossRef Google Scholar
Jiang, Y., Song, Z., Zhu, T., Fang, Z., Weng, H., Liu, Z.-X., Yang, J. & Fang, C. (2023). arXiv:2307.10371. Google Scholar
Litvin, D. & Opechowski, W. (1974). Physica, 76, 538–554. CrossRef CAS Web of Science Google Scholar
Litvin, D. B. (1973). Acta Cryst. A29, 651–660. CrossRef CAS IUCr Journals Web of Science Google Scholar
Litvin, D. B. (1977). Acta Cryst. A33, 279–287. CrossRef IUCr Journals Web of Science Google Scholar
Litvin, D. B. (2014). Magnetic Group Tables. https://www.iucr.org/publ/978-0-9553602-2-0. Chester: IUCr. Google Scholar
Liu, P., Li, J., Han, J., Wan, X. & Liu, Q. (2022). Phys. Rev. X, 12, 021016. Google Scholar
Nebe, G. (2011). International Tables for Crystallography, Vol. A1, 2nd ed., edited by H. Wondratschek & U. Müller, ch. 1.4, pp. 27–40. Heidelberg: Springer. Google Scholar
Opechowski, W. (1986). Crystallographic and Metacrystallographic Groups. Amsterdam: North-Holland. Google Scholar
Ren, J., Chen, X., Zhu, Y., Yu, Y., Zhang, A., Li, J., Li, C. & Liu, Q. (2023). arXiv:2307.10369. Google Scholar
Shinohara, K., Togo, A. & Tanaka, I. (2023). Acta Cryst. A79, 390–398. Web of Science CrossRef IUCr Journals Google Scholar
Šmejkal, L., Sinova, J. & Jungwirth, T. (2022a). Phys. Rev. X, 12, 031042. Google Scholar
Šmejkal, L., Sinova, J. & Jungwirth, T. (2022b). Phys. Rev. X, 12, 040501. Google Scholar
Stokes, H. T. & Campbell, B. J. (2017). Acta Cryst. A73, 4–13. Web of Science CrossRef IUCr Journals Google Scholar
Togo, A. & Tanaka, I. (2018). Spglib: a Software Library for Crystal Symmetry Search. https://github.com/spglib/spglib. Google Scholar
Watanabe, H., Shinohara, K., Nomoto, T., Togo, A. & Arita, R. (2023). arXiv:2307.11560. Google Scholar
Xiao, Z., Zhao, J., Li, Y., Shindou, R. & Song, Z.-D. (2023). arXiv:2307.10364. Google Scholar
Yang, J., Liu, Z.-X. & Fang, C. (2021). arXiv:2105.12738. Google Scholar
Zelenskiy, A., Monchesky, T. L., Plumer, M. L. & Southern, B. W. (2022). Phys. Rev. B, 106, 144433. Web of Science CrossRef Google Scholar
Železný, J., Zhang, Y., Felser, C. & Yan, B. (2017). Phys. Rev. Lett. 119, 187204. Web of Science PubMed Google Scholar
Zhang, Y., Železný, J., Sun, Y., van den Brink, J. & Yan, B. (2018). New J. Phys. 20, 073028. Web of Science CrossRef Google Scholar

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Volume 80| Part 1| January 2024| Pages 94-103

https://doi.org/10.1107/S2053273323009257

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Algorithm for spin symmetry operation search

1. Introduction

2. Group structure of a spin space group

2.1. Spin symmetry operation and spin arrangement

2.2. Spin space group

2.3. Spin-only group

2.4. Spin translation group

3. Spin symmetry operation search

3.1. Space group of a nonmagnetic crystal structure

3.2. Spin-only group search

3.2.1. Spin-only group search by eigenvalue decomposition

3.2.2. Numerically robust spin-only group search

3.3. Translation subgroup search

3.4. Spin translation group search

3.5. Spin space-group search

4. Examples of spin symmetry operation search

4.1. Space group of a nonmagnetic crystal structure

4.2. Spin-only group

4.3. Translation subgroup of a maximal space subgroup

4.4. Coset representatives of a spin translation group

4.5. Coset representatives of a spin space group

5. Conclusions

APPENDIX A

Correspondence between spin symmetry operation and magnetic symmetry operation

APPENDIX B

Spin point group

APPENDIX C

Generating a transformation matrix from centering vectors

Funding information

References

research papers