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<EMAIL_ADDRESS>Togo Watanabe Nomoto Tanaka Arita

[a]Department of Materials Science and Engineering, Kyoto University, Sakyo,
Kyoto 606-8501, Japan [b]Center for Basic Research on Materials, National
Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan [c]Center
for Elements Strategy Initiative for Structural Materials, Kyoto University,
Sakyo, Kyoto 606-8501, Japan [d]Research Center for Advanced Science and
Technology, University of Tokyo, Meguro-ku, Tokyo 153-8904, Japan
[e]Nanostructures Research Laboratory, Japan Fine Ceramics Center, Nagoya
456-8587, Japan [f]RIKEN, Center for Emergent Matter Science, Saitama
351-0198, Japan

#  Algorithm for spin symmetry operation search

Atsushi  Hikaru  Takuya  Isao  Ryotaro

###### Abstract

A spin space group provides a suitable way to fully exploit 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 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 v0.1.1:
https://github.com/spglib/spinspg.

This paper presents an algorithm for determining the spin symmetry operations
of a given spin arrangement. Spin symmetry operations of a spin space group
act simultaneously on both the spatial and spin coordinates of the spin
arrangement.

##  1 Introduction

When the spin–orbit coupling (SOC) is negligible, a spin space group is an
appropriate concept to fully exploit the symmetry of a corresponding spin
arrangement [LITVIN1974538, Opechowski1986, PhysRevX.12.021016, 2105.12738].
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 [doi:10.1063/1.1708514, doi:10.1098/rspa.1966.0211] and has
recently been applied to the field of magnetism [PhysRevX.12.040501]: for
example, an analysis of a symmetry-adapted tensor of transport properties with
negligible SOC [PhysRevLett.119.187204, Zhang2018], symmetries of a spin
Hamiltonian [PhysRevB.106.144433], magnon band structure
[PhysRevB.105.064430], and classification of antiferromagnetism
[PhysRevX.12.031042].

When we consider spin space groups of given spin arrangements, we have to
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 an algorithm and its implementation
would benefit the 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 on only 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 spin symmetry
operations of a given commensurate spin arrangement, extending our magnetic
symmetry operation search algorithm [spglibv2]. 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 works by Litvin and Opechowski [Litvin:a09793, LITVIN1974538,
Litvin:a14103]. 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
[10.1093/acprof:oso/9780198510581.001.0001]. 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 [spglibv1]. The
implementation is distributed under the BSD 3-clause license in spinspg v0.1.1
on top of a crystal symmetry search algorithm [spglibv1]. For magnetic crystal
structures tabulated in magndata [Gallego:ks5532], the present algorithm and
implementation have been used to identify physical properties free from SOC
[spinspg-application].

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

##  2 Group structure of spin space group

We provide the definitions of spin symmetry operations and spin arrangements
in Sec. 2.1. We define the spin space group in Sec. 2.2. Then, we introduce
its derived groups, the spin-only group (Sec. 2.3) and spin translation group
(Sec. 2.4). Although these groups were already discussed in Litvin:a09793,
LITVIN1974538, and Litvin:a14103, we consider it beneficial to summarize these
results 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 translation parts of 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 $\mathrm{E}(3)$ and a spin rotation in the three-
dimensional orthogonal group $\mathrm{O}(3)$. The spin symmetry operation
$(g,\bm{W})\in\mathrm{E}(3)\times\mathrm{O}(3)$ acts on a pair of position
$\bm{r}$ and magnetic moments $\bm{m}$ as

$\displaystyle(g,\bm{W})(\bm{r},\bm{m})=(g\bm{r},\bm{Wm}).$ (1)

The product of two spin symmetry operations $(g,\bm{W})$ and
$(g^{\prime},\bm{W}^{\prime})$ is defined as

$\displaystyle(g,\bm{W})(g^{\prime},\bm{W}^{\prime})=(gg^{\prime},\bm{W}\bm{W}^{\prime}).$
(2)

Although uncommon in crystallography, we suppose both $g$ and $\bm{W}$ are
represented with Cartesian coordinates for later convenience. We denote basis
vectors

$\displaystyle\bm{A}=(\bm{a}_{1},\bm{a}_{2},\bm{a}_{3})=\begin{pmatrix}a_{1x}&a_{2x}&a_{3x}\\\
a_{1y}&a_{2y}&a_{3y}\\\ a_{1z}&a_{2z}&a_{3z}\\\ \end{pmatrix}.$ (3)

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

###  2.2 Spin space group

Let $\mathcal{G}$ be a subgroup of $\mathrm{E}(3)\times\mathrm{O}(3)$. When
the following $\mathcal{F}(\mathcal{G})$ and $\mathcal{D}(\mathcal{G})$ are
space groups, $\mathcal{G}$ is called a _spin space group_ [LITVIN1974538],

$\displaystyle\mathcal{F}(\mathcal{G})$
$\displaystyle=\left\\{g\in\mathrm{E}(3)\mathrel{}\middle|\mathrel{}\exists\bm{W}\in\mathrm{O}(3)\,s.t.\,(g,\bm{W})\in\mathcal{G}\right\\}$
(4) $\displaystyle\mathcal{D}(\mathcal{G})$
$\displaystyle=\left\\{g\in\mathrm{E}(3)\mathrel{}\middle|\mathrel{}(g,\bm{E})\in\mathcal{G}\right\\},$
(5)

where $\bm{E}$ stands for the identity matrix. For a spin space group
$\mathcal{G}$, we call $\mathcal{F}(\mathcal{G})$ a _family space group_ and
$\mathcal{D}(\mathcal{G})$ a _maximal space subgroup_. The maximal space
subgroup $\mathcal{D}(\mathcal{G})$ is a normal subgroup of $\mathcal{G}$. On
the contrary, the family space group $\mathcal{F}(\mathcal{G})$ is not a
subgroup of $\mathcal{G}\times\mathrm{O}(3)$ in general. Although
LITVIN1974538 did not impose the condition that $\mathcal{D}(\mathcal{G})$ is
crystallographic, we impose this condition to guarantee that a given spin
arrangement is commensurate. We confine our discussion and algorithms to
commensurate spin arrangements.

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

$\displaystyle(g,\bm{W})\mapsto\left((\bm{P},\bm{p})^{-1}g(\bm{P},\bm{p}),\bm{Q}^{-1}\bm{W}\bm{Q}\right).$
(6)

Two spin space groups $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ belong to the
same _spin-space-group type_ if they are transformed to the other by a pair of
an orientation-preserving transformation $(\bm{P},\bm{p})$111 A transformation
$(\bm{P},\bm{p})$ is called orientation-preserving if $\det\bm{P}>0$.  on the
spatial coordinates and a transformation matrix $\bm{Q}$ on the spin
coordinates.

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 [litvin2014magnetic]. 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 $\mathcal{G}$ is a set of spin
symmetry operations of $\mathcal{G}$ with identity spatial operations,

$\displaystyle\mathcal{P}_{\mathrm{so}}(\mathcal{G})=\left\\{\bm{W}\in\mathrm{O}(3)\mathrel{}\middle|\mathrel{}((\bm{E},\bm{0}),\bm{W})\in\mathcal{G}\right\\}.$
(7)

A direct product of an identity in spatial coordinates and
$\mathcal{P}_{\mathrm{so}}(\mathcal{G})$,
$1\times\mathcal{P}_{\mathrm{so}}(\mathcal{G})$, is a subgroup of
$\mathcal{G}$222 We denote a trivial group, consisting of a single element, as
$1$. .

###  2.4 Spin translation group

A _spin translation group_ of a spin space group $\mathcal{G}$ is a set of
spin symmetry operations with identity rotations in spatial coordinates,

$\displaystyle\mathcal{G}_{\mathrm{st}}(\mathcal{G})=\left\\{((\bm{E},\bm{v}),\bm{W})\mathrel{}\middle|\mathrel{}((\bm{E},\bm{v}),\bm{W})\in\mathcal{G}\right\\}.$
(8)

Litvin:a09793 classified the spin translation groups under the transformation
in Eq. (6).

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

$\displaystyle\mathcal{T}(\mathcal{R})$
$\displaystyle=\left\\{(\bm{E},\bm{t})\mathrel{}\middle|\mathrel{}(\bm{E},\bm{t})\in\mathcal{R}\right\\}$
(9) $\displaystyle\mathcal{P}(\mathcal{R})$
$\displaystyle=\left\\{\bm{W}\mathrel{}\middle|\mathrel{}\exists\bm{v}\,s.t.\,(\bm{W},\bm{v})\in\mathcal{R}\right\\}.$
(10)

Then, the spin-only group $\mathcal{P}_{\mathrm{so}}(\mathcal{G})$ and
translation subgroup of $\mathcal{D}(\mathcal{G})$ are a normal subgroup of
$\mathcal{G}_{\mathrm{st}}(\mathcal{G})$. Because
$(\mathcal{T}(\mathcal{D}(\mathcal{G}))\times
1)\cap(1\times\mathcal{P}_{\mathrm{so}}(\mathcal{G}))=1$, their direct product
$\mathcal{T}(\mathcal{D}(\mathcal{G}))\times\mathcal{P}_{\mathrm{so}}(\mathcal{G})$
is also a normal subgroup of $\mathcal{G}_{\mathrm{st}}(\mathcal{G})$. Thus,
we can consider a factor group
$\mathcal{G}_{\mathrm{st}}(\mathcal{G})/(\mathcal{T}(\mathcal{D}(\mathcal{G}))\times\mathcal{P}_{\mathrm{so}}(\mathcal{G}))$,
which is finite for commensurate spin arrangements. Finally,
$\mathcal{G}_{\mathrm{st}}(\mathcal{G})$ is a normal subgroup of
$\mathcal{G}$. The factor group
$\mathcal{G}/\mathcal{G}_{\mathrm{st}}(\mathcal{G})$ is isomorphic to
$\mathcal{P}(\mathcal{F}(\mathcal{G}))$ 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: (1) basis vectors
of its lattice $\bm{A}=(\bm{a}_{1},\bm{a}_{2},\bm{a}_{3})$, (2) an array of
point coordinates of sites in its unit cell
$\bm{X}=(\bm{x}_{1},\cdots,\bm{x}_{N})$, (3) an array of atomic types of sites
in its unit cell $\bm{T}=(t_{1},\cdots,t_{N})$, and (4) an array of magnetic
moments of sites in its unit cell $\bm{M}=(\bm{m}_{1},\cdots,\bm{m}_{N})$,
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 spglibv1. For magnetic moments, we use another tolerance parameter
$\epsilon_{\mathrm{mag}}$ to identify that two magnetic moments
$\bm{W}\bm{m}_{i}$ and $\bm{m}_{\sigma_{g}(i)}$ are equal to the other if

$\displaystyle\left\lVert\bm{W}\bm{m}_{i}-\bm{m}_{\sigma_{g}(i)}\right\rVert_{2}<\epsilon_{\mathrm{mag}}.$
(11)

The present algorithm extends the authors’ previous work on detecting magnetic
symmetry operations [spglibv2]. We first consider a space group of nonmagnetic
crystal structure $(\bm{A},\bm{X},\bm{T})$ in Sec. 3.1. Next, we search for
normal subgroups of the spin space group: the spin-only group
$\mathcal{P}_{\mathrm{so}}(\mathcal{G})$ in Sec. 3.2 and the translation
subgroup $\mathcal{T}(\mathcal{D}(\mathcal{G}))$ in Sec. 3.3. With these
normal subgroups of $\mathcal{G}_{\mathrm{st}}(\mathcal{G})$, we search for
coset representatives of the spin translation group
$\mathcal{G}_{\mathrm{st}}(\mathcal{G})/(\mathcal{T}(\mathcal{D}(\mathcal{G}))\times\mathcal{P}_{\mathrm{so}}(\mathcal{G}))$
in Sec. 3.4. We search for coset representatives of the spin space group
$\mathcal{G}/\mathcal{G}_{\mathrm{st}}(\mathcal{G})$ in Sec. 3.5. Because the
notation in this section is abstract to unambiguously deal with several basis
vectors, it may be helpful to read it alongside the examples in Sec. 4.

###  3.1 Space group of nonmagnetic crystal structure

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

$\displaystyle\mathcal{S}$
$\displaystyle=\left\\{g=(\overline{\bm{R}},\overline{\bm{v}})_{\bm{A}}\in\mathrm{E}(3)\mathrel{}\middle|\mathrel{}\begin{array}[]{l}\exists\sigma_{g}\in\mathfrak{S}_{N},\forall
i=1,\cdots N,\\\
\overline{\bm{R}}\bm{x}_{i}+\overline{\bm{v}}\equiv\bm{x}_{\sigma_{g}(i)}\,(\mathrm{mod}\,1)\\\
t_{i}=t_{\sigma_{g}(i)}\end{array}\right\\},$ (15)

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

The existing crystal symmetry search algorithm [spglibv1] can find primitive
basis vectors $\bm{A}_{\mathcal{S}}$ of $\mathcal{T}(\mathcal{S})$ and coset
decomposition of $\mathcal{S}$ over $\mathcal{T}_{\bm{A}}$, where we write a
translation subgroup formed by $\bm{A}$ as

$\displaystyle\mathcal{T}_{\bm{A}}=\left\\{(\bm{E},\bm{n})_{\bm{A}}\mathrel{}\middle|\mathrel{}\bm{n}\in\mathbb{Z}^{3}\right\\}.$
(16)

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

$\displaystyle\mathcal{T}_{\bm{A}_{\mathcal{S}}}=\bigsqcup_{\overline{\bm{t}}^{\mathcal{S}}}\left(\bm{E},\overline{\bm{t}}^{\mathcal{S}}\right)_{\bm{A}_{\mathcal{S}}}\mathcal{T}_{\bm{A}},$
(17)

where $\overline{\bm{t}}^{\mathcal{S}}$ is a centering vector in a unit cell
spanned by $\bm{A}$. The coset decomposition of $\mathcal{S}$ is written as

$\displaystyle\mathcal{S}$
$\displaystyle=\bigsqcup_{\overline{\bm{R}}^{\mathcal{S}}}\left(\overline{\bm{R}}^{\mathcal{S}},\overline{\bm{v}}_{\overline{\bm{R}}^{\mathcal{S}}}\right)_{\bm{A}_{\mathcal{S}}}\mathcal{T}_{\bm{A}_{\mathcal{S}}}.$
(18)

Here, $\overline{\bm{v}}_{\overline{\bm{R}}^{\mathcal{S}}}$ is a translation
part of a symmetry operation with a matrix part
$\overline{\bm{R}}^{\mathcal{S}}$.

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

$\displaystyle\mathcal{G}$
$\displaystyle=\left\\{(g,\bm{W})\in\mathcal{S}\times\mathrm{O}(3)\mathrel{}\middle|\mathrel{}\bm{W}\bm{m}_{i}=\bm{m}_{\sigma_{g}(i)}\,(i=1,\dots,N)\right\\}.$
(19)

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

###  3.2 Spin-only group search

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

$\displaystyle\mathcal{P}_{\mathrm{so}}$
$\displaystyle=\left\\{\bm{W}\in\mathrm{O}(3)\mathrel{}\middle|\mathrel{}\bm{W}\bm{m}_{i}=\bm{m}_{i}\quad(i=1,\dots,N)\right\\}.$
(20)

As shown in Table 2, when a spin space group $\mathcal{G}$ is a stabilizer of
a spin arrangement, spin-only groups are classified into four types up to
transformations [LITVIN1974538, PhysRevX.12.021016]: _nonmagnetic_ ,
_collinear_ , _coplanar_ , and _noncoplanar_ spin arrangements.

In Sec. 3.2.1, we provide an algorithm for detecting
$\mathcal{P}_{\mathrm{so}}$ using the eigenvalue decomposition of
$\bm{M}\bm{M}^{\top}$. The spin-only group search should be performed within
tolerances for magnetic moments in practice. In fact, selecting appropriate
tolerances is challenging. In Sec. 3.2.2, we propose a robust algorithm,
building on the previous section, to alleviate the difficulty.

####  3.2.1 Spin-only group search by eigenvalue decomposition

We consider a moment tensor of $\bm{M}\in\mathbb{R}^{3\times N}$,

$\displaystyle\bm{M}\bm{M}^{\top}=\sum_{i=1}^{N}\bm{m}_{i}\otimes\bm{m}_{i}.$
(21)

Because $\bm{M}\bm{M}^{\top}$ is a symmetric semi-definite matrix, we can
consider its eigenvalue decomposition,

$\displaystyle\bm{M}\bm{M}^{\top}=\sum_{r=1}^{3}\sigma_{r}\hat{\bm{n}}_{r}\otimes\hat{\bm{n}}_{r},$
(22)

where $\sigma_{1}\geq\sigma_{2}\geq\sigma_{3}\geq 0$ and
$\\{\hat{\bm{n}}_{r}\\}_{r=1}^{3}$ 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, $\sigma_{1}=\sigma_{2}=\sigma_{3}=0$. In a collinear spin arrangement,
all magnetic moments align parallel or antiparallel to a direction
$\hat{\bm{n}}_{\parallel}$. The eigenvector $\hat{\bm{n}}_{1}$ with the
largest eigenvalue should be parallel or antiparallel to
$\hat{\bm{n}}_{\parallel}$ and the other eigenvalues $\sigma_{2}$ and
$\sigma_{3}$ should be zero. In a coplanar spin arrangement, all magnetic
moments align perpendicular to a direction $\hat{\bm{n}}_{\perp}$. In this
case, the eigenvector $\hat{\bm{n}}_{3}$ with the smallest eigenvalue should
be parallel or antiparallel to $\hat{\bm{n}}_{\perp}$ and the smallest
eigenvalue $\sigma_{3}$ should be zero. In other cases, the spin arrangement
is noncoplanar. Note that the site order of magnetic moments $\bm{M}$ does not
affect the classification because $\bm{M}\bm{M}^{\top}$ in Eq. (21) 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 judging
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
$\epsilon_{\mathrm{mag}}$ in Eq. (11) as follows.

  1. 1.

We compute the eigenvalues $\sigma_{i}$ and eigenvectors $\hat{\bm{n}}_{i}$ in
Eq. (22).

  2. 2.

If all magnetic moments are close to zero within $\epsilon_{\mathrm{mag}}$,

$\displaystyle\left\lVert\bm{m}_{i}\right\rVert_{2}<\epsilon_{\mathrm{mag}}\quad(i=1,\dots,N),$
(23)

the spin arrangement is nonmagnetic.

  3. 3.

If not, we check if the eigenvector $\hat{\bm{n}}_{1}$ is a parallel or
antiparallel direction for all magnetic moments,

$\displaystyle
2\left\lVert\bm{m}_{i}-(\bm{m}_{i}\cdot\hat{\bm{n}}_{1})\hat{\bm{n}}_{1}\right\rVert_{2}<\epsilon_{\mathrm{mag}}\quad(\forall
i=1,\dots,N).$ (24)

When spin symmetry operations in a collinear spin-only group prescribed by a
direction $\hat{\bm{n}}_{1}$ act on $\bm{m}_{i}$, the acted 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 magnetic moments, which
corresponds to the diameter of the cone with direction $\hat{\bm{n}}_{1}$. If
the inequality holds for all magnetic moments, the spin arrangement is
collinear.

  4. 4.

If not, we check if the eigenvector $\hat{\bm{n}}_{3}$ is a perpendicular
direction for all magnetic moments,

$\displaystyle
2\left|(\bm{m}_{i}\cdot\hat{\bm{n}}_{3})\right|<\epsilon_{\mathrm{mag}}\quad(\forall
i=1,\dots,N).$ (25)

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

  5. 5.

Otherwise, the spin arrangement is noncoplanar.

###  3.3 Translation subgroup search

We search for primitive basis vectors $\bm{A}_{\mathcal{D}}$ of the
translation subgroup
$\mathcal{T}(\mathcal{D}(\mathcal{G}))=\left\\{(\bm{E},\bm{v})\mathrel{}\middle|\mathrel{}((\bm{E},\bm{v}),\bm{E})\in\mathcal{G}\right\\}$.
The group–subgroup relationships of translation subgroups are shown in Fig. 2.
Because $\mathcal{T}_{\bm{A}_{\mathcal{D}}}$ is between
$\mathcal{T}_{\bm{A}_{\mathcal{S}}}$ and $\mathcal{T}_{\bm{A}}$,
$\mathcal{T}_{\bm{A}}\trianglelefteq\mathcal{T}_{\bm{A}_{\mathcal{D}}}\trianglelefteq\mathcal{T}_{\bm{A}_{\mathcal{S}}}$,
we only need to examine finite coset representatives of
$\mathcal{T}_{\bm{A}_{\mathcal{S}}}/\mathcal{T}_{\bm{A}}$ for candidates of
$\mathcal{T}_{\bm{A}_{\mathcal{S}}}/\mathcal{T}_{\bm{A}_{\mathcal{D}}}$. For
every coset
$\left(\bm{E},\overline{\bm{t}}^{\mathcal{S}}\right)_{\bm{A}_{\mathcal{S}}}\mathcal{T}_{\bm{A}}$
in Eq. (17), we check if
$\left(\bm{E},\overline{\bm{t}}^{\mathcal{S}}\right)_{\bm{A}_{\mathcal{S}}}$
preserves magnetic moments $\bm{M}$,

$\displaystyle\mathcal{T}_{\bm{A}_{\mathcal{D}}}$
$\displaystyle=\left\\{g=\left(\bm{E},\overline{\bm{t}}^{\mathcal{S}}\right)_{\bm{A}_{\mathcal{S}}}\mathrel{}\middle|\mathrel{}\begin{array}[]{l}g\mathcal{T}_{\bm{A}}\in\mathcal{T}_{\bm{A}_{\mathcal{S}}}/\mathcal{T}_{\bm{A}},\\\
\bm{m}_{i}=\bm{m}_{\sigma_{g}(i)}\,(i=1,\dots,N)\end{array}\right\\}\mathcal{T}_{\bm{A}}.$
(28)

An integer matrix $\bm{V}\in\mathbb{Z}^{3\times 3}$ exists such that
$\bm{A}_{\mathcal{D}}=\bm{A}_{\mathcal{S}}\bm{V}$. A lattice algorithm to
generate $\bm{V}$ from centering vectors $\overline{\bm{t}}^{\mathcal{S}}$ in
Eq. (28) is presented in Appendix C.

###  3.4 Spin translation group search

We can find a spin rotation $\bm{W}$ for a given symmetry operation
$g\in\mathcal{S}$ by solving a Procrustes problem
[10.1093/acprof:oso/9780198510581.001.0001] as follows. We write magnetic
moments permuted by $g$ as

$\displaystyle\bm{M}_{g}=\left(\bm{m}_{\sigma_{g}(1)},\cdots,\bm{m}_{\sigma_{g}(N)}\right),$
(29)

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

$\displaystyle\tilde{\bm{W}}$
$\displaystyle=\operatorname*{arg\,min}_{\bm{W}\in\mathrm{O}(3)}\left\lVert\bm{M}_{g}-\bm{W}\bm{M}\right\rVert_{F}.$
(30)

Here the displacement between the magnetic moments and operated ones by
$(g,\bm{W})$ is measured by the Frobenius norm
$\left\lVert\cdot\right\rVert_{F}$. The solution to Eq. (30) can be explicitly
written as

$\displaystyle\tilde{\bm{W}}=\bm{Y}\bm{Z}^{\top},$ (31)

where $\bm{Y}$ and $\bm{Z}$ are orthogonal matrices of the singular value
decomposition of $\bm{M}_{g}\bm{M}^{\top}$,

$\displaystyle\bm{M}_{g}\bm{M}^{\top}=\bm{Y}\bm{\Sigma}\bm{Z}^{\top}.$ (32)

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

$\displaystyle\left\lVert\bm{m}_{i}-\tilde{\bm{W}}\bm{m}_{\sigma_{g}(i)}\right\rVert_{2}<\epsilon_{\mathrm{mag}}$
(33)

holds for every site $i$. If Eq. (33) does not hold for some site $i$, we
reject $(g,\tilde{\bm{W}})$ as a spin symmetry operation.

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

$\displaystyle\mathcal{G}_{\mathrm{st}}$
$\displaystyle=\left\\{g=\left((\bm{E},\overline{\bm{t}}^{\mathcal{D}})_{\bm{A}_{\mathcal{D}}},\bm{W}_{\overline{\bm{t}}^{\mathcal{D}}}\right)\mathrel{}\middle|\mathrel{}\begin{array}[]{l}(\bm{E},\overline{\bm{t}}^{\mathcal{D}})_{\bm{A}_{\mathcal{D}}}\mathcal{T}_{\bm{A}_{\mathcal{D}}}\in\mathcal{T}_{\bm{A}_{\mathcal{S}}}/\mathcal{T}_{\bm{A}_{\mathcal{D}}}\\\
\bm{W}_{\overline{\bm{t}}^{\mathcal{D}}}\bm{m}_{i}=\bm{m}_{\sigma_{g}(i)}\,(\forall
i=1,\cdots,N)\end{array}\right\\}\left(\mathcal{T}_{\bm{A}_{\mathcal{D}}}\times\mathcal{P}_{\mathrm{so}}\right),$
(36)

where $\overline{\bm{t}}^{\mathcal{D}}$ takes a distinct translation up to
$\mathcal{T}_{\bm{A}_{\mathcal{D}}}$ and we choose
$\bm{W}_{\overline{\bm{t}}^{\mathcal{D}}=\bm{0}}=\bm{E}$.

###  3.5 Spin space group search

Because the translation subgroup of $\mathcal{F}(\mathcal{G})$ is
$\mathcal{T}_{\bm{A}_{\mathcal{D}}}$, spatial operation parts of $\mathcal{G}$
should belong to a maximal subgroup $\mathcal{S}_{\mathcal{D}}$ of
$\mathcal{S}$ with its translation subgroup
$\mathcal{T}_{\bm{A}_{\mathcal{D}}}$. Figure 2 shows the group–subgroup
relationship between $\mathcal{S}$ and $\mathcal{S}_{\mathcal{D}}$. A rotation
$\bm{A}_{\mathcal{S}}\overline{\bm{R}}^{\mathcal{S}}\bm{A}_{\mathcal{S}}^{-1}\in\mathcal{P}(\mathcal{S})$
is compatible with $\mathcal{T}_{\bm{A}_{\mathcal{D}}}$ if
$\bm{V}^{-1}\overline{\bm{R}}^{\mathcal{S}}\bm{V}$ is an integer matrix
[PhysRevB.77.224115]. Thus, the compatible subgroup
$\mathcal{S}_{\mathcal{D}}$ is written as

$\displaystyle\mathcal{S}_{\mathcal{D}}$
$\displaystyle=\left\\{\left(\overline{\bm{R}}^{\mathcal{S}},\overline{\bm{v}}_{\overline{\bm{R}}^{\mathcal{S}}}+\overline{\bm{t}}_{\overline{\bm{R}}^{\mathcal{S}}}\right)_{\bm{A}_{\mathcal{S}}}\mathrel{}\middle|\mathrel{}\begin{array}[]{l}\left(\overline{\bm{R}}^{\mathcal{S}},\overline{\bm{v}}_{\overline{\bm{R}}^{\mathcal{S}}}\right)\mathcal{T}_{\bm{A}_{\mathcal{S}}}\in\mathcal{S}/\mathcal{T}_{\bm{A}_{\mathcal{S}}}\\\
\bm{V}^{-1}\overline{\bm{R}}^{\mathcal{S}}\bm{V}\in\mathbb{Z}^{3\times 3}\\\
\left(\bm{E},\overline{\bm{t}}_{\overline{\bm{R}}^{\mathcal{S}}}\right)_{\bm{A}_{\mathcal{S}}}\mathcal{T}_{\bm{A}}\in\mathcal{T}_{\bm{A}_{\mathcal{S}}}/\mathcal{T}_{\bm{A}}\end{array}\right\\}\mathcal{T}_{\bm{A}_{\mathcal{D}}},$
(40)

where $\overline{\bm{t}}_{\overline{\bm{R}}^{\mathcal{S}}}$ is a centering
vector in $\mathcal{T}_{\bm{A}_{\mathcal{S}}}/\mathcal{T}_{\bm{A}}$. The
additional centering vector
$\overline{\bm{t}}_{\overline{\bm{R}}^{\mathcal{S}}}$ is necessary for
$\mathcal{S}_{\mathcal{D}}$ to form a group [nebe20061, Stokes:vk5013].

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

$\displaystyle\mathcal{G}=\left\\{\left(\left(\overline{\bm{R}}^{\mathcal{S}},\overline{\bm{v}}_{\overline{\bm{R}}^{\mathcal{S}}}+\overline{\bm{t}}_{\overline{\bm{R}}^{\mathcal{S}}}\right)_{\bm{A}_{\mathcal{S}}},\bm{W}_{\overline{\bm{R}}^{\mathcal{S}}}\right)\mathrel{}\middle|\mathrel{}\begin{array}[]{l}\left(\overline{\bm{R}}^{\mathcal{S}},\overline{\bm{v}}_{\overline{\bm{R}}^{\mathcal{S}}}+\overline{\bm{t}}_{\overline{\bm{R}}^{\mathcal{S}}}\right)_{\bm{A}_{\mathcal{S}}}\mathcal{T}_{\bm{A}_{\mathcal{D}}}\in\mathcal{S}_{\mathcal{D}}/\mathcal{T}_{\bm{A}_{\mathcal{D}}}\\\
\bm{W}_{\overline{\bm{R}}^{\mathcal{S}}}\bm{m}_{i}=\bm{m}_{\sigma_{g}(i)}\,(\forall
i=1,\cdots,N)\end{array}\right\\}\mathcal{G}_{\mathrm{st}}.$ (43)

##  4 Examples of spin symmetry operation search

We consider a spin arrangement of the NiAs-type CrSe [PhysRev.122.1402,
Litvin:a09793] (#2.35 of magndata [Gallego:ks5532]) as an example of spin
symmetry operation search, illustrated in Fig. 3. The hexagonal lattice of
CrSe has the following basis vectors

$\displaystyle\bm{A}=\begin{pmatrix}a&-\frac{1}{2}a&0\\\
0&\frac{\sqrt{3}}{2}a&0\\\ 0&0&c\\\ \end{pmatrix}.$

The fractional coordinates and magnetic moments of atoms are as follows.

$\displaystyle\mathrm{Cr}:$
$\displaystyle\bm{x}_{1}=\left(0,0,0\right)^{\top},\bm{m}_{1}=\left(-\frac{1}{2}m_{x},-\frac{\sqrt{3}}{2}m_{x},-m_{z}\right)^{\top}$
$\displaystyle\mathrm{Cr}:$
$\displaystyle\bm{x}_{2}=\left(\frac{1}{3},\frac{2}{3},0\right)^{\top},\bm{m}_{2}=\left(m_{x},0,-m_{z}\right)^{\top}$
$\displaystyle\mathrm{Cr}:$
$\displaystyle\bm{x}_{3}=\left(\frac{2}{3},\frac{1}{3},0\right)^{\top},\bm{m}_{3}=\left(-\frac{1}{2}m_{x},\frac{\sqrt{3}}{2}m_{x},-m_{z}\right)^{\top}$
$\displaystyle\mathrm{Cr}:$
$\displaystyle\bm{x}_{4}=\left(0,0,\frac{1}{2}\right)^{\top},\bm{m}_{4}=\left(\frac{1}{2}m_{x},\frac{\sqrt{3}}{2}m_{x},m_{z}\right)^{\top}$
$\displaystyle\mathrm{Cr}:$
$\displaystyle\bm{x}_{5}=\left(\frac{1}{3},\frac{2}{3},\frac{1}{2}\right)^{\top},\bm{m}_{5}=\left(-m_{x},0,m_{z}\right)^{\top}$
$\displaystyle\mathrm{Cr}:$
$\displaystyle\bm{x}_{6}=\left(\frac{2}{3},\frac{1}{3},\frac{1}{2}\right)^{\top},\bm{m}_{6}=\left(\frac{1}{2}m_{x},-\frac{\sqrt{3}}{2}m_{x},m_{z}\right)^{\top}$
$\displaystyle\mathrm{Se}:$
$\displaystyle\bm{x}_{7}=\left(0,\frac{1}{3},\frac{1}{4}\right)^{\top},\bm{m}_{7}=\left(0,0,0\right)^{\top}$
$\displaystyle\mathrm{Se}:$
$\displaystyle\bm{x}_{8}=\left(\frac{1}{3},0,\frac{1}{4}\right)^{\top},\bm{m}_{8}=\left(0,0,0\right)^{\top}$
$\displaystyle\mathrm{Se}:$
$\displaystyle\bm{x}_{9}=\left(\frac{1}{3},\frac{1}{3},\frac{3}{4}\right)^{\top},\bm{m}_{9}=\left(0,0,0\right)^{\top}$
$\displaystyle\mathrm{Se}:$
$\displaystyle\bm{x}_{10}=\left(0,\frac{2}{3},\frac{3}{4}\right)^{\top},\bm{m}_{10}=\left(0,0,0\right)^{\top}$
$\displaystyle\mathrm{Se}:$
$\displaystyle\bm{x}_{11}=\left(\frac{2}{3},0,\frac{3}{4}\right)^{\top},\bm{m}_{11}=\left(0,0,0\right)^{\top}$
$\displaystyle\mathrm{Se}:$
$\displaystyle\bm{x}_{12}=\left(\frac{2}{3},\frac{2}{3},\frac{1}{4}\right)^{\top},\bm{m}_{12}=\left(0,0,0\right)^{\top}$

Here, the magnetic moments are represented with Cartesian coordinates.

###  4.1 Space group of nonmagnetic crystal structure

The space-group type of $\mathcal{S}$ for the crystal structure ignoring
magnetic moments is $P6_{3}/mmc$ (No. 194). One of the primitive basis vectors
for $\mathcal{S}$ and the transformation matrix are

$\displaystyle\bm{A}_{\mathcal{S}}$
$\displaystyle=\begin{pmatrix}0&-\frac{1}{2}a&0\\\
-\frac{\sqrt{3}}{3}a&\frac{\sqrt{3}}{6}a&0\\\ 0&0&-c\\\ \end{pmatrix}$
$\displaystyle\bm{U}$ $\displaystyle=\begin{pmatrix}-1&-1&0\\\ -2&1&0\\\
0&0&-1\\\ \end{pmatrix}$

with $\bm{A}=\bm{A}_{\mathcal{S}}\bm{U}$ as defined in Sec. 3.1. There are
three coset representatives for
$\mathcal{T}_{\bm{A}_{\mathcal{S}}}/\mathcal{T}_{\bm{A}}$,

$\displaystyle\mathcal{T}_{\bm{A}_{\mathcal{S}}}$
$\displaystyle=g_{1}\mathcal{T}_{\bm{A}}\sqcup g_{2}\mathcal{T}_{\bm{A}}\sqcup
g_{3}\mathcal{T}_{\bm{A}}$ (44) $\displaystyle g_{1}$
$\displaystyle=\left(\bm{E},(0,0,0)^{\top}\right)_{\bm{A}_{\mathcal{S}}}$
$\displaystyle g_{2}$
$\displaystyle=\left(\bm{E},(-1,-1,0)^{\top}\right)_{\bm{A}_{\mathcal{S}}}$
$\displaystyle g_{3}$
$\displaystyle=\left(\bm{E},(-1,0,0)^{\top}\right)_{\bm{A}_{\mathcal{S}}}.$

###  4.2 Spin-only group

The moment tensor of $\bm{M}$ is given by

$\displaystyle\bm{M}\bm{M}^{\top}$
$\displaystyle=\begin{pmatrix}3m_{x}^{2}&0&0\\\ 0&3m_{x}^{2}&0\\\
0&0&6m_{z}^{2}\\\ \end{pmatrix}.$

Its eigenvalues $3m_{x}^{2}$, $3m_{x}^{2}$, and $6m_{z}^{2}$ are all positive.
Consequently, the spin arrangement is noncoplanar with
$\mathcal{P}_{\mathrm{so}}=1$.

###  4.3 Translation subgroup of maximal space subgroup

The translation $g_{1}$ in Eq. (44) belongs to
$\mathcal{T}_{\mathcal{A}_{\mathcal{D}}}$. On the other hand, $g_{2}$ does not
belong to $\mathcal{T}_{\mathcal{A}_{\mathcal{D}}}$ because it maps
$\bm{x}_{1}$ to $\bm{x}_{3}$, whereas $\bm{m}_{1}\neq\bm{m}_{3}$. Similarly,
$g_{3}$ does not belong to $\mathcal{T}_{\mathcal{A}_{\mathcal{D}}}$.
Therefore, $\mathcal{T}_{\bm{A}_{\mathcal{D}}}$ is identical to
$\mathcal{T}_{\bm{A}_{\mathcal{S}}}$ with
$\bm{A}_{\mathcal{D}}=\bm{A}_{\mathcal{S}}\bm{V}$ as defined in Sec. 3.3 and
we can choose $\bm{V}=\bm{E}$.

###  4.4 Coset representatives of spin translation group

There are three candidates for the spatial operation parts of coset
representatives of $\mathcal{G}_{\mathrm{st}}$: $g_{1}$, $g_{2}$, and $g_{3}$.
For translation $g_{1}$ in Eq. (44), we obtain

$\displaystyle\bm{M}_{g_{1}}\bm{M}^{\top}=\begin{pmatrix}3m_{x}^{2}&0&0\\\
0&3m_{x}^{2}&0\\\ 0&0&6m_{z}^{2}\\\ \end{pmatrix}.$

From the singular value decomposition of $\bm{M}_{g_{1}}\bm{M}^{\top}$, we
calculate the spin-rotation part $\tilde{\bm{W}}_{1}$ for $g_{1}$ as
$\tilde{\bm{W}}_{1}=\bm{E}$. For translation $g_{2}$, the singular value
decomposition of $\bm{M}_{g_{2}}\bm{M}^{\top}$ is

$\displaystyle\bm{M}_{g_{2}}\bm{M}^{\top}=\begin{pmatrix}-\frac{1}{2}&\frac{\sqrt{3}}{2}&0\\\
-\frac{\sqrt{3}}{2}&-\frac{1}{2}&0\\\ 0&0&1\\\
\end{pmatrix}\begin{pmatrix}3m_{x}^{2}&0&0\\\ 0&3m_{x}^{2}&0\\\
0&0&6m_{x}^{2}\\\ \end{pmatrix}\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\\\
\end{pmatrix}^{\top}.$

Hence, a candidate for a spin-rotation part with $g_{2}$ is

$\displaystyle\tilde{\bm{W}}_{2}=\begin{pmatrix}-\frac{1}{2}&\frac{\sqrt{3}}{2}&0\\\
-\frac{\sqrt{3}}{2}&-\frac{1}{2}&0\\\ 0&0&1\\\
\end{pmatrix}\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\\\
\end{pmatrix}^{\top}=\begin{pmatrix}-\frac{1}{2}&\frac{\sqrt{3}}{2}&0\\\
-\frac{\sqrt{3}}{2}&-\frac{1}{2}&0\\\ 0&0&1\\\ \end{pmatrix}.$

Similarly, a candidate for a spin-rotation part with $g_{3}$ is

$\displaystyle\tilde{\bm{W}}_{3}=\begin{pmatrix}-\frac{1}{2}&-\frac{\sqrt{3}}{2}&0\\\
\frac{\sqrt{3}}{2}&-\frac{1}{2}&0\\\ 0&0&1\\\ \end{pmatrix}.$

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

$\displaystyle\mathcal{G}_{\mathrm{st}}=(g_{1},\tilde{\bm{W}}_{1})(\mathcal{T}_{\bm{A}_{\mathcal{D}}}\times
1)\sqcup(g_{2},\tilde{\bm{W}}_{2})(\mathcal{T}_{\bm{A}_{\mathcal{D}}}\times
1)\sqcup(g_{3},\tilde{\bm{W}}_{3})(\mathcal{T}_{\bm{A}_{\mathcal{D}}}\times
1).$

###  4.5 Coset representatives of spin space group

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

$\displaystyle g=\left(\begin{pmatrix}-1&1&0\\\ -1&0&0\\\ 0&0&-1\\\
\end{pmatrix},\bm{0}\right)_{\bm{A}_{\mathcal{D}}}\in\mathcal{S}_{\mathcal{D}}$

gives a permutation of sites
$\sigma_{g}=\left(4,5,6,1,2,3,12,7,11,9,10,8\right)$ and the singular value
decomposition of $\bm{M}_{g}\bm{M}$

$\displaystyle\bm{M}_{g}\bm{M}=\begin{pmatrix}-1&0&0\\\ 0&-1&0\\\ 0&0&-1\\\
\end{pmatrix}\begin{pmatrix}3m_{x}^{2}&0&0\\\ 0&3m_{x}^{2}&0\\\
0&0&6m_{x}^{2}\\\ \end{pmatrix}\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\\\
\end{pmatrix}^{\top},$

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

A magnetic space group $\mathcal{M}$ of the NiAs-type CrSe is $P31m^{\prime}$
(BNS number 157.55), which has
$|\mathcal{M}/\mathcal{T}_{\bm{A}_{\mathcal{D}}}|=6$ coset representatives.
Because the spin translation group $\mathcal{G}_{\mathrm{st}}$ contains three-
fold spin-rotation parts, these spin symmetry operations
($(g_{2},\tilde{\bm{W}}_{2})$ and $(g_{3},\tilde{\bm{W}}_{3})$ in Sec. 4.4)
are not captured in $\mathcal{M}$. Similarly, the spin symmetry operations
with six-fold spatial rotations or rotoinversions are not preserved in
$\mathcal{M}$.

##  5 Conclusion

We have presented the algorithm for determining 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,
$\bm{M}\bm{M}^{\top}$. 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
$\mathcal{T}_{\bm{A}}$, one by the primitive basis vector
$\mathcal{T}_{\bm{A}_{\mathcal{S}}}$, and one by the primitive basis vectors
for the spin space group $\mathcal{T}_{\bm{A}_{\mathcal{D}}}$. 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. 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.

##### Note added

After this work was completed, we became aware of recent preprints on the
classification and enumeration of spin space-group types [2307.10364,
2307.10369, 2307.10371]. Two of these works, 2307.10364 and 2307.10371, 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 $\bm{W}$. On the other hand, 2307.10364 directly determines
$\bm{W}$ for a symmetry operation $g$ such that selected three magnetic
moments $\bm{m}_{i}\quad(i=1,2,3)$ are transformed into
$\bm{m}_{\sigma_{g}(i)}$, which may not be robust against numerical noises and
experimental uncertainty of magnetic moments.

2307.10371 identifies spin symmetry operations on top of spglib [spglibv1]
similar to ours. However, they do not incorporate translation subgroups
$\mathcal{T}_{\bm{A}_{\mathcal{D}}}$ and $\mathcal{T}_{\bm{A}}$. 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 $((\bm{R},\bm{v}),\bm{W})$ for a
Hamiltonian without the spin–orbit coupling (SOC), we derive the condition
that $((\bm{R},\bm{v}),\bm{W})$ has a corresponding magnetic symmetry
operation for the Hamiltonian with SOC. The SOC introduces an additional term
to the Hamiltonian proportional to $\hat{\bm{L}}\cdot\hat{\bm{\sigma}}$, with
the angular momentum operator $\hat{\bm{L}}$ and the Pauli matrices
$\hat{\bm{\sigma}}=(\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z})$. A
spin symmetry operation $((\bm{R},\bm{v}),\bm{W})$ acts on $\hat{\bm{L}}$ and
$\hat{\bm{\sigma}}$ as

$\displaystyle\hat{\bm{L}}$
$\displaystyle\mapsto(\det\bm{R})(\det\bm{W})\left(\sum_{\nu=x,y,z}R_{\mu\nu}\hat{L}_{\nu}\right)_{\mu=x,y,z}$
(45) $\displaystyle\hat{\bm{\sigma}}$
$\displaystyle\mapsto\left(\sum_{\nu=x,y,z}W_{\mu\nu}\hat{\sigma}_{\nu}\right)_{\mu=x,y,z},$
(46)

where $\bm{R}$ and $\bm{W}$ are represented with Cartesian coordinates. Note
that $\det\bm{W}$ in Eq. (45) reflects a time reversal operation. Thus,
$((\bm{R},\bm{v}),\bm{W})$ acts on the SOC term as

$\displaystyle\hat{\bm{L}}\cdot\hat{\bm{\sigma}}$
$\displaystyle\mapsto(\det\bm{R})(\det\bm{W})\sum_{\mu,\nu=x,y,z}\left[\bm{R}^{\top}\bm{W}\right]_{\mu\nu}\hat{L}_{\mu}\hat{\sigma}_{\nu}.$
(47)

To preserve the SOC term, $(\det\bm{R})(\det\bm{W})\bm{R}^{\top}\bm{W}$ should
be identity. Because $(\det\bm{R})\bm{R}$ and $(\det\bm{W})\bm{W}$ belong to
$\mathrm{SO}(3)$, this condition is equivalent to

$\displaystyle(\det\bm{R})\bm{R}=(\det\bm{W})\bm{W}.$ (48)

Therefore, if $((\bm{R},\bm{v}),\bm{W})$ satisfies Eq. (48), it has a
corresponding magnetic symmetry operation $((\bm{R},\bm{v}),\det\bm{W})$ with
symmetry operation part $(\bm{R},\bm{v})$ and time-reversal part $\det\bm{W}$.
We identify $\det\bm{W}=1$ with an identity operation and $\det\bm{W}=-1$ with
a time-reversal operation. Here, we assume that a magnetic symmetry operation
$((\bm{R},\bm{v}),\theta)$ acts on a magnetic moments $\bm{m}$ as
$\theta(\det\bm{R})\bm{R}\bm{m}$.

Conversely, a magnetic symmetry operation $((\bm{R},\bm{v}),\theta)$ is mapped
to a spin symmetry operation $((\bm{R},\bm{v}),\theta\bm{R})$. We can confirm
this mapping satisfies Eq. (48) by
$(\det\theta\bm{R})\theta\bm{R}=(\det\bm{R})\bm{R}$. Because this mapping is
injective, the index of a magnetic space group $\mathcal{M}$ in its
translation subgroup $\mathcal{T}_{\bm{A}_{\mathcal{D}}}$ is a divisor of the
index of a corresponding spin space group $\mathcal{G}$ in
$\mathcal{T}_{\bm{A}_{\mathcal{D}}}$,
$|\mathcal{G}/\mathcal{T}_{\bm{A}_{\mathcal{D}}}|\equiv
0\,(\mathrm{mod}\,|\mathcal{M}/\mathcal{T}_{\bm{A}_{\mathcal{D}}}|)$.

##  Appendix B Spin point group

We define a _family spin point group_ of a spin space group $\mathcal{G}$ as

$\displaystyle\mathcal{B}(\mathcal{G})$
$\displaystyle=\left\\{\bm{W}\in\mathrm{O}(3)\mathrel{}\middle|\mathrel{}\exists
g\,s.t.\,(g,\bm{W})\in\mathcal{G}\right\\}.$ (49)

When we write a coset decomposition of $\mathcal{G}$ by its spin translation
group $\mathcal{G}_{\mathrm{st}}(\mathcal{G})$ as

$\displaystyle\mathcal{G}=\bigsqcup_{\bm{R}}\left((\bm{R},\bm{v}_{\bm{R}}),\bm{W}_{\bm{R}}\right)\mathcal{G}_{\mathrm{st}}(\mathcal{G}),$
(50)

A _spin point group_ of $\mathcal{G}$ is

$\displaystyle\mathcal{U}(\mathcal{G})=\left\\{\left(\bm{R},\bm{W}_{\bm{R}}\mathcal{B}(\mathcal{G}_{\mathrm{st}}(\mathcal{G}))\right)\right\\}_{\bm{R}\in\mathcal{P}(\mathcal{F}(\mathcal{G}))}.$
(51)

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

In general, we cannot choose $\bm{W}_{\bm{R}}$ so that
$\\{(\bm{R},\bm{W}_{\bm{R}})\\}$ is a group under the multiplication in
$\mathrm{O}(3)\times\mathrm{O}(3)$. We show one of the counterexamples in
which $\\{(\bm{R},\bm{W}_{\bm{R}})\\}$ is not closed as a group, with a spin
arrangement

$\displaystyle\bm{A}$ $\displaystyle=\begin{pmatrix}4&0&0\\\ 0&6&0\\\ 0&0&8\\\
\end{pmatrix}$ $\displaystyle\bm{X}$
$\displaystyle=\begin{pmatrix}0.4&0.6&0.4&0.4\\\ 0.4&0.6&0.4&0.4\\\
0&0.25&0.5&0.75\\\ \end{pmatrix}$ $\displaystyle\bm{M}$
$\displaystyle=\begin{pmatrix}1&0&-1&0\\\ 0&1&0&-1\\\ 0.4&0.4&0.4&0.4\\\
\end{pmatrix},$

illustrated in Fig. 4. This spin arrangement is noncoplanar with
$\mathcal{P}_{\mathrm{so}}(\mathcal{G})=1$. The spin space group $\mathcal{G}$
is obtained as

$\displaystyle\mathcal{G}$
$\displaystyle=\left\\{g_{1},g_{2},g_{3},g_{4},g_{5},g_{6},g_{7},g_{8}\right\\}\left(\mathcal{T}(\mathcal{D}(\mathcal{G}))\times\mathcal{P}_{\mathrm{so}}(\mathcal{G})\right)$
$\displaystyle g_{1}$
$\displaystyle=\left(\left(1,\bm{0}\right)_{\bm{A}},1\right)$ $\displaystyle
g_{2}$
$\displaystyle=\left(\left(1,\frac{1}{2}\bm{c}\right)_{\bm{A}},2_{z}\right)$
$\displaystyle g_{3}$
$\displaystyle=\left(\left(\overline{1},\frac{3}{4}\bm{c}\right)_{\bm{A}},m_{xy}\right)$
$\displaystyle g_{4}$
$\displaystyle=\left(\left(\overline{1},\frac{1}{4}\bm{c}\right)_{\bm{A}},m_{x\overline{y}}\right)$
$\displaystyle g_{5}$
$\displaystyle=\left(\left(2,\frac{3}{4}\bm{c}\right)_{\bm{A}},4^{-}_{z}\right)$
$\displaystyle g_{6}$
$\displaystyle=\left(\left(2,\frac{1}{4}\bm{c}\right)_{\bm{A}},4^{+}_{z}\right)$
$\displaystyle g_{7}$
$\displaystyle=\left(\left(m,\frac{1}{2}\bm{c}\right)_{\bm{A}},m_{x}\right)$
$\displaystyle g_{8}$
$\displaystyle=\left(\left(m,\bm{0}\right)_{\bm{A}},m_{y}\right),$

where we denote a mirror operation along $xy$ axis as $m_{xy}$. The spin point
group of $\mathcal{G}$ is

$\displaystyle\mathcal{U}(\mathcal{G})$
$\displaystyle=\left\\{(1,1),(\overline{1},m_{xy}),(2,4^{-}_{z}),(m,m_{x})\right\\}\left(1\times\mathcal{B}(\mathcal{G}_{\mathrm{st}}(\mathcal{G}))\right)$
(52) $\displaystyle\mathcal{B}(\mathcal{G}_{\mathrm{st}}(\mathcal{G}))$
$\displaystyle=\left\\{1,2_{z}\right\\}$

The coset representatives in Eq. (52) are not closed as a subgroup of
$\mathrm{O}(3)\times\mathrm{O}(3)$ 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 $\mathrm{O}(3)\times\mathrm{O}(3)$ in
this example because $\mathcal{U}(\mathcal{G})\cong\mathbb{Z}_{4}$,
$\mathcal{B}(\mathcal{G}_{\mathrm{st}}(\mathcal{G}))\cong\mathbb{Z}_{2}$, and
$\mathbb{Z}_{4}$ cannot be written as an internal semidirect product
$\mathbb{Z}_{2}\rtimes\mathbb{Z}_{2}$.

##  Appendix C Generating transformation matrix from centering vectors

To demonstrate how to generate a transformation matrix $\bm{V}$ in Sec. 3.3,
we consider the following coplanar fcc structure with the conventional basis,

$\displaystyle\bm{A}$ $\displaystyle=\begin{pmatrix}a&0&0\\\ 0&a&0\\\ 0&0&a\\\
\end{pmatrix}$ $\displaystyle\bm{X}$
$\displaystyle=\begin{pmatrix}0&0&\frac{1}{2}&\frac{1}{2}\\\
0&\frac{1}{2}&0&\frac{1}{2}\\\ 0&\frac{1}{2}&\frac{1}{2}&0\\\ \end{pmatrix}$
$\displaystyle\bm{M}$ $\displaystyle=\begin{pmatrix}0&0&m&m\\\ 0&0&0&0\\\
m&m&0&0\\\ \end{pmatrix}.$

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

$\displaystyle\bm{A}_{\mathcal{S}}$
$\displaystyle=\bm{A}\bm{U}^{-1}=\frac{a}{2}\begin{pmatrix}0&1&1\\\ 1&0&1\\\
1&1&0\\\ \end{pmatrix}$ $\displaystyle\bm{U}$
$\displaystyle=\begin{pmatrix}-1&1&1\\\ 1&-1&1\\\ 1&1&-1\\\ \end{pmatrix}.$

There are four centering vectors in
$\mathcal{T}_{\bm{A}_{\mathcal{S}}}/\mathcal{T}_{\bm{A}}$,

$\displaystyle\mathcal{T}_{\bm{A}_{\mathcal{S}}}$
$\displaystyle=g_{1}\mathcal{T}_{\bm{A}}\sqcup g_{2}\mathcal{T}_{\bm{A}}\sqcup
g_{3}\mathcal{T}_{\bm{A}}\sqcup g_{4}\mathcal{T}_{\bm{A}}$ $\displaystyle
g_{1}$
$\displaystyle=\left(\bm{E},(0,0,0)^{\top}\right)_{\bm{A}_{\mathcal{S}}}$
$\displaystyle g_{2}$
$\displaystyle=\left(\bm{E},(1,0,0)^{\top}\right)_{\bm{A}_{\mathcal{S}}}$
$\displaystyle g_{3}$
$\displaystyle=\left(\bm{E},(0,1,0)^{\top}\right)_{\bm{A}_{\mathcal{S}}}$
$\displaystyle g_{4}$
$\displaystyle=\left(\bm{E},(0,0,1)^{\top}\right)_{\bm{A}_{\mathcal{S}}}.$

The half of centering vectors form the translation subgroup of
$\mathcal{D}(\mathcal{G})$,

$\displaystyle\mathcal{T}(\mathcal{D}(\mathcal{G}))=g_{1}\mathcal{T}_{\bm{A}}\sqcup
g_{2}\mathcal{T}_{\bm{A}}.$

Because $\mathcal{T}(\mathcal{D}(\mathcal{G}))$ is spanned by column vectors
of $\bm{U}$ and centering vectors of $g_{1}$ and $g_{2}$, we can rewrite
$\mathcal{T}(\mathcal{D}(\mathcal{G}))$ as

$\displaystyle\mathcal{T}(\mathcal{D}(\mathcal{G}))$
$\displaystyle=\bm{A}_{\mathcal{S}}\tilde{\bm{U}}\mathbb{Z}^{5}$
$\displaystyle\tilde{\bm{U}}$ $\displaystyle=\begin{pmatrix}-1&1&1&0&1\\\
1&-1&1&0&0\\\ 1&1&-1&0&0\\\ \end{pmatrix}.$

We need to find a transformation matrix $\bm{V}\in\mathbb{Z}^{3\times 3}$ such
that

$\displaystyle\mathcal{T}(\mathcal{D}(\mathcal{G}))=\bm{A}_{\mathcal{S}}\bm{V}\mathbb{Z}^{3},$
(53)

and it is known to be achieved by the Hermite normal form of $\tilde{\bm{U}}$
[Cohen1993],

$\displaystyle\tilde{\bm{U}}=\begin{pmatrix}1&0&0&0&0\\\ 0&1&0&0&0\\\
0&1&2&0&0\\\ \end{pmatrix}\begin{pmatrix}-1&1&1&0&1\\\ 1&-1&1&0&0\\\
0&1&-1&0&0\\\ 0&0&0&1&0\\\ 0&1&0&0&0\\\ \end{pmatrix},$

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

$\displaystyle\mathcal{T}(\mathcal{D}(\mathcal{G}))$
$\displaystyle=\bm{A}_{\mathcal{S}}\begin{pmatrix}1&0&0&0&0\\\ 0&1&0&0&0\\\
0&1&2&0&0\\\ \end{pmatrix}\mathbb{Z}^{5}$
$\displaystyle=\bm{A}_{\mathcal{S}}\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&1&2\\\
\end{pmatrix}\mathbb{Z}^{3}.$

Consequently, we generate the integer matrix

$\displaystyle\bm{V}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&1&2\end{pmatrix}.$

[references]

Table 1: Notation and terminology in this paper. Symbol | Meaning
---|---
$A\sqcup B$ |  Disjoint union of set $A$ and $B$ with $A\cap B=\varnothing$
$\mathrm{E}(3)$ | Three-dimensional Euclidean group
$\mathrm{O}(3)$ | Three-dimensional orthogonal group
$\bm{E}$ | Identity matrix
$\mathfrak{S}_{N}$ |  Symmetric group of degree $N$
$\left\lVert\bm{y}\right\rVert_{2}=\sqrt{\bm{y}^{\top}\bm{y}}$ |  $l^{2}$ norm of vector $\bm{y}$
$\left\lVert\bm{B}\right\rVert_{F}=\sqrt{\operatorname{Tr}\bm{B}^{\top}\bm{B}}$ |  Frobenius norm of matrix $\bm{B}$
$1$ | Trivial group
$\bm{r}$ | Position in Cartesian coordinates
$\bm{m}$ | Magnetic moments
Spin symmetry operation $(g,\bm{W})$ |  Pair of spatial operation $g$ and spin rotation $\bm{W}$
$(\overline{\bm{R}},\overline{\bm{v}})_{\bm{A}}=(\bm{A}\overline{\bm{R}}\bm{A}^{-1},\bm{A}\overline{\bm{v}})$ |  Spatial operation with basis vectors $\bm{A}$
$\bm{A}=(\bm{a}_{1},\bm{a}_{2},\bm{a}_{3})$ | Basis vectors
$\bm{X}=(\bm{x}_{1},\cdots,\bm{x}_{N})$ | Array of point coordinates
$\bm{T}=(t_{1},\cdots,t_{N})$ | Array of atomic types
$\bm{M}=(\bm{m}_{1},\cdots,\bm{m}_{N})$ | Array of magnetic moments
Spin arrangement $(\bm{A},\bm{X},\bm{T},\bm{M})$ | Pair of crystal structure and magnetic moments
$(\bm{A},\bm{X},\bm{T})$ | Crystal structure ignoring magnetic moments
$\epsilon_{\mathrm{mag}}$ | Absolute tolerance to compare magnetic moments
$\mathcal{S},\mathcal{R}$ | Space group
Translation subgroup $\mathcal{T}(\mathcal{R})$ |  Translation parts of $\mathcal{R}$ with identity rotations
Point group $\mathcal{P}(\mathcal{R})$ |  Group obtained from the rotation parts of $\mathcal{R}/\mathcal{T}(\mathcal{R})$
$\mathcal{T}_{\bm{A}}$ |  Translation group spanned by $\bm{A}$
$\sigma_{g}$ |  Permutation of sites induced by $g$
$\bm{A}_{\mathcal{S}}=\bm{A}\bm{U}^{-1}$ |  Primitive basis vectors of $\mathcal{T}(\mathcal{S})$
Spin-only group $\mathcal{P}_{\mathrm{so}}=\mathcal{P}_{\mathrm{so}}(\mathcal{G})$ | Spin-rotation parts with identity spatial operations
$\sigma_{i},\hat{\bm{n}_{i}}$ |  Eigenvalue and eigenvector of $\bm{M}\bm{M}^{\top}$
$\hat{\bm{n}}_{\parallel}$ | Parallel direction for collinear spin arrangement
$\hat{\bm{n}_{\perp}}$ | Perpendicular direction for coplanar spin arrangement
$\bm{A}_{\mathcal{D}}=\bm{A}_{\mathcal{S}}\bm{V}$ |  Basis vectors of $\mathcal{T}(\mathcal{D}(\mathcal{G}))$
Spin translation group $\mathcal{G}_{\mathrm{st}}=\mathcal{G}_{\mathrm{st}}(\mathcal{G})$ |  Subgroup of $\mathcal{G}$ with identity rotations
$\bm{M}_{g}=\left(\bm{m}_{\sigma_{g}(1)}\cdots\bm{m}_{\sigma_{g}(N)}\right)$ |  Array of magnetic moments permuted by $\sigma_{g}$
Spin space group $\mathcal{G}$ |  See Sec. 2.2
Family space group $\mathcal{F}(\mathcal{G})$ |  Space group composed of spatial parts of $\mathcal{G}$
Maximal space subgroup $\mathcal{D}=\mathcal{D}(\mathcal{G})$ |  Subgroup of $\mathcal{G}$ with identity spin-rotation parts
Spin-space-group type |  See Sec. 2.2
$(\bm{P},\bm{p})$ | Transformation on spatial coordinates
$\bm{Q}$ | Transformation matrix on spin coordinates
Family spin point group $\mathcal{B}(\mathcal{G})$ |  Spin-rotation parts of $\mathcal{G}$
Spin point group $\mathcal{U}(\mathcal{G})$ |  Pairs of spatial and spin rotations of $\mathcal{G}$
$\mathcal{M}$ | Magnetic space group
Table 2:  Classification of spin-only groups up to transformations. The spin-only groups are represented in Hellmann–Mauguin symbols [hahn2016point].  Spin arrangement |  Spin-only group |  Eigenvalues of $\bm{M}\bm{M}^{\top}$
---|---|---
Nonmagnetic |  $\infty\infty m\cong\mathrm{O}(3)$ |  $\sigma_{1}=\sigma_{2}=\sigma_{3}=0$
Collinear |  $\infty m\cong\mathrm{SO}(2)\rtimes\mathbb{Z}_{2}$ |  $\sigma_{1}>\sigma_{2}=\sigma_{3}=0$
Coplanar |  $m$ |  $\sigma_{1}\geq\sigma_{2}>\sigma_{3}=0$
Noncoplanar |  $1$ |  $\sigma_{1}\geq\sigma_{2}\geq\sigma_{3}>0$
Figure 1:  Magnetic moments acted by (a) collinear and (b) coplanar spin-only
groups. (a) A collinear spin-only group along axis $\hat{\bm{n}}_{1}$ is
generated from rotations along $\hat{\bm{n}}_{1}$ and mirror operations
preserving $\hat{\bm{n}}_{1}$. The red dotted line indicates the largest
displacement between magnetic moments acted the collinear spin-only group. (b)
A coplanar spin-only group along axis $\hat{\bm{n}}_{3}$ is generated from a
mirror operation perpendicular to $\hat{\bm{n}}_{3}$. The red dotted line
indicates the largest displacement between magnetic moments acted the coplanar
spin-only group.  Figure 2:  Group–subgroup relationship of translation
subgroups and space groups derived from 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 a diagram. Note that
$\mathcal{S}_{\mathcal{D}}\mathcal{T}_{\bm{A}_{\mathcal{S}}}$ could be a
proper subgroup of $\mathcal{S}$.  Figure 3:  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.  Figure 4:  Spin arrangement example in Appendix B. Its spin point
group is not closed under $\mathrm{O}(3)\times\mathrm{O}(3)$ due to the spin
translation group.