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Kohei Shinohara, [Atsushi Togo](https://orcid.org/0000-0001-8393-9766), Hikaru Watanabe, Takuya Nomoto, Isao Tanaka, Ryotaro Arita

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[Algorithm for spin symmetry operation search](https://mdr.nims.go.jp/datasets/a007de97-1578-4cff-8e13-bd1a26714f2e)

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1Algorithm for spin symmetry operation searchKohei Shinohara,a* Atsushi Togo,b Hikaru Watanabe,d Takuya Nomoto,dIsao Tanakaa,c,e and Ryotaro Aritad,faDepartment of Materials Science and Engineering, Kyoto University, Sakyo, Kyoto606-8501, Japan, bCenter for Basic Research on Materials, National Institute forMaterials Science, Tsukuba, Ibaraki 305-0047, Japan, cCenter for Elements StrategyInitiative for Structural Materials, Kyoto University, Sakyo, Kyoto 606-8501, Japan,dResearch Center for Advanced Science and Technology, University of Tokyo,Meguro-ku, Tokyo 153-8904, Japan, eNanostructures Research Laboratory, JapanFine Ceramics Center, Nagoya 456-8587, Japan, and fRIKEN, Center for EmergentMatter Science, Saitama 351-0198, Japan. E-mail: kshinohara0508@gmail.com(Received 0 XXXXXXX 0000; accepted 0 XXXXXXX 0000)AbstractA spin space group provides a suitable way to fully exploit the symmetry of a spinarrangement with a negligible spin–orbit coupling. There has been a growing interest inapplying spin symmetry analysis with the spin space group in the field of magnetism.However, there is no established algorithm to search for spin symmetry operationsof the spin space group. This paper presents an exhaustive algorithm for determiningspin symmetry operations of commensurate spin arrangements. The present algorithmsearches for spin symmetry operations from the symmetry operations of a correspond-ing nonmagnetic crystal structure and determines their spin-rotation parts by solvinga Procrustes problem. An implementation is distributed under a permissive free soft-ware license in spinspg v0.1.1: https://github.com/spglib/spinspg.PREPRINT: Acta Crystallographica Section A A Journal of the International Union of CrystallographyarXiv:2307.12228v2  [cond-mat.mtrl-sci]  22 Oct 2023https://github.com/spglib/spinspg21. IntroductionWhen the spin–orbit coupling (SOC) is negligible, a spin space group is an appropriateconcept to fully exploit the symmetry of a corresponding spin arrangement (Litvin &Opechowski, 1974; Opechowski, 1986; Liu et al., 2022; Yang et al., 2021). The spinarrangement comprises a crystal structure and magnetic moments. A spin symmetryoperation of the spin space group is assumed to act on the spatial and spin coor-dinates simultaneously, generalizing a magnetic symmetry operation of a magneticspace group. The spin space group was first introduced to analyze an extra symme-try 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 trans-port properties with negligible SOC (Železný et al., 2017; Zhang et al., 2018), symme-tries of a spin Hamiltonian (Zelenskiy et al., 2022), magnon band structure (Corticelliet al., 2022), and classification of antiferromagnetism (Šmejkal et al., 2022a).When we consider spin space groups of given spin arrangements, we have to firstexhaustively search for their spin symmetry operations. To the best of our knowledge,there is no rigorous algorithm to search for spin symmetry operations. Therefore, thedevelopment of an algorithm and its implementation would benefit the spin symmetryanalysis.There are a few differences between spin space groups and magnetic space groups interms of symmetry search algorithms. First, a spin space group may contain nontrivialoperations acting on only spin coordinates, called a spin-only group, which compli-cates the group structure of the spin space group. Second, although we only need toconsider at most double enlarged cells for magnetic space groups, the unit cell size mayarbitrarily change between a spin arrangement and its nonmagnetic correspondence.Lastly, spin rotations, which simultaneously act on magnetic moments, do not haveIUCr macros version 2.1.10: 2016/01/283to belong to a crystallographic point group.Here, we present a rigorous and robust algorithm for determining spin symme-try operations of a given commensurate spin arrangement, extending our magneticsymmetry operation search algorithm (Shinohara et al., to be published 2023). Thepresent algorithm fully exploits the group structure of the spin space groups andoutputs spin symmetry operations as a coset decomposition of the spin space group,based on the seminal works by Litvin and Opechowski (Litvin, 1973; Litvin & Ope-chowski, 1974; Litvin, 1977). We explicitly denote basis vectors of space groups andspin space groups and employ a lattice algorithm to deal with the case when the unitcell size varies with and without magnetic moments. We search for the spin-rotationparts from three-dimensional orthogonal groups by solving a well-known optimiza-tion problem called a Procrustes problem (Gower & Dijksterhuis, 2004). Note thatwe restrict the present algorithm to commensurate spin arrangements in order to usesimilar inputs and outputs to an existing space group search implementation (Togo &Tanaka, 2018). The implementation is distributed under the BSD 3-clause license inspinspg v0.1.1 on top of a crystal symmetry search algorithm (Togo & Tanaka, 2018).For magnetic crystal structures tabulated in magndata (Gallego et al., 2016), thepresent algorithm and implementation have been used to identify physical propertiesfree from SOC (Watanabe et al., 2023).This paper is organized as follows. In Sec. 2, we give definitions of spin space groupsand 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 arrange-ment. In Sec. 4, we demonstrate the present spin symmetry operation search to a spinarrangement of a NiAs-type CrSe. The notations and terminology in this paper aresummarized in Table 1.IUCr macros version 2.1.10: 2016/01/2842. Group structure of spin space groupWe provide the definitions of spin symmetry operations and spin arrangements inSec. 2.1. We define the spin space group in Sec. 2.2. Then, we introduce its derivedgroups, the spin-only group (Sec. 2.3) and spin translation group (Sec. 2.4). Althoughthese groups were already discussed in Litvin (1973), Litvin & Opechowski (1974), andLitvin (1977), we consider it beneficial to summarize these results because we fullyexploit the group structure of the spin space group in searching for spin symmetryoperations.We note that spin-only groups and spin translation groups complicate the groupstructure of the spin space groups. For example, a spin point group, which ignorestranslation parts of spin symmetry operations of a spin space group, cannot be com-puted without going through the spin space group due to the existence of a nontrivialspin translation group in general. Although the analysis of spin point groups is notrequired to determine spin symmetry operations, we discuss the group structure ofspin point groups in Appendix B for completeness.2.1. Spin symmetry operation and spin arrangementA spin symmetry operation comprises a spatial operation in the three-dimensionalEuclidean group E(3) and a spin rotation in the three-dimensional orthogonal groupO(3). The spin symmetry operation (g,W ) ∈ E(3) × O(3) acts on a pair of positionr and magnetic moments m as(g,W )(r,m) = (gr,Wm). (1)The product of two spin symmetry operations (g,W ) and (g′,W ′) is defined as(g,W )(g′,W ′) = (gg′,WW ′). (2)IUCr macros version 2.1.10: 2016/01/285Although uncommon in crystallography, we suppose both g and W are representedwith Cartesian coordinates for later convenience. We denote basis vectorsA = (a1,a2,a3) =a1x a2x a3xa1y a2y a3ya1z a2z a3z . (3)When g with a matrix part R and a translation part v are represented with A, weexplicitly write g = (R,v)A, where R = ARA−1 and v = Av. A spin arrangementis a set of pairs of a crystal structure and magnetic moments.2.2. Spin space groupLet G be a subgroup of E(3)×O(3). When the following F(G) and D(G) are spacegroups, G is called a spin space group (Litvin & Opechowski, 1974),F(G) = {g ∈ E(3) | ∃W ∈ O(3) s.t. (g,W ) ∈ G} (4)D(G) = {g ∈ E(3) | (g,E) ∈ G} , (5)where E stands for the identity matrix. For a spin space group G, we call F(G) afamily space group and D(G) a maximal space subgroup. The maximal space subgroupD(G) is a normal subgroup of G. On the contrary, the family space group F(G) isnot a subgroup of G × O(3) in general. Although Litvin & Opechowski (1974) didnot impose the condition that D(G) is crystallographic, we impose this condition toguarantee that a given spin arrangement is commensurate. We confine our discussionand algorithms to commensurate spin arrangements.A spin symmetry operation (g,W ) is transformed by a transformation (P ,p) onthe spatial coordinates and a transformation matrix Q on the spin coordinates as(g,W ) 7→((P ,p)−1g(P ,p),Q−1WQ). (6)Two spin space groups G1 and G2 belong to the same spin-space-group type if they aretransformed to the other by a pair of an orientation-preserving transformation (P ,p)11A transformation (P ,p) is called orientation-preserving if detP > 0.IUCr macros version 2.1.10: 2016/01/286on the spatial coordinates and a transformation matrix Q on the spin coordinates.In the spin space group, we can consider rotating magnetic moments independentlywith spatial coordinates. On the other hand, we consider rotating magnetic momentsonly in association with spatial rotations and time-reversal operations in the magneticspace group (Litvin, 2014). Thus, the spin space group can be regarded as a super-group of the magnetic space group under an appropriate correspondence between spinsymmetry operations and magnetic symmetry operations as discussed in Appendix A.2.3. Spin-only groupA spin-only group of a spin space group G is a set of spin symmetry operations ofG with identity spatial operations,Pso(G) = {W ∈ O(3) | ((E,0),W ) ∈ G} . (7)A direct product of an identity in spatial coordinates and Pso(G), 1 × Pso(G), is asubgroup of G2.2.4. Spin translation groupA spin translation group of a spin space group G is a set of spin symmetry operationswith identity rotations in spatial coordinates,Gst(G) = {((E,v),W ) | ((E,v),W ) ∈ G} . (8)Litvin (1973) classified the spin translation groups under the transformation in Eq. (6).We denote a translation subgroup and a point group of space group R as T (R) andP(R), respectively,T (R) = {(E, t) | (E, t) ∈ R} (9)P(R) = {W | ∃v s.t. (W ,v) ∈ R} . (10)2We denote a trivial group, consisting of a single element, as 1.IUCr macros version 2.1.10: 2016/01/287Then, the spin-only group Pso(G) and translation subgroup of D(G) are a normalsubgroup of Gst(G). Because (T (D(G)) × 1) ∩ (1 × Pso(G)) = 1, their direct productT (D(G))×Pso(G) is also a normal subgroup of Gst(G). Thus, we can consider a factorgroup Gst(G)/(T (D(G))×Pso(G)), which is finite for commensurate spin arrangements.Finally, Gst(G) is a normal subgroup of G. The factor group G/Gst(G) is isomorphic toP(F(G)) and thus finite.3. Spin symmetry operation searchWe provide an algorithm to search for spin symmetry operations from a given spinarrangement represented by the following four objects: (1) basis vectors of its latticeA = (a1,a2,a3), (2) an array of point coordinates of sites in its unit cell X =(x1, · · · ,xN ), (3) an array of atomic types of sites in its unit cell T = (t1, · · · , tN ),and (4) an array of magnetic moments of sites in its unit cell M = (m1, · · · ,mN ),where N is the number of sites in the unit cell.To search for spin symmetry operations robustly, comparisons of point coordinatesand magnetic moments should be performed within tolerances in practice. We adoptthe same absolute tolerance parameter for point coordinates as Togo & Tanaka (2018).For magnetic moments, we use another tolerance parameter ϵmag to identify that twomagnetic moments Wmi and mσg(i) are equal to the other if∥∥Wmi −mσg(i)∥∥2< ϵmag. (11)The present algorithm extends the authors’ previous work on detecting magneticsymmetry operations (Shinohara et al., to be published 2023). We first consider aspace group of nonmagnetic crystal structure (A,X,T ) in Sec. 3.1. Next, we search fornormal subgroups of the spin space group: the spin-only group Pso(G) in Sec. 3.2 andthe translation subgroup T (D(G)) in Sec. 3.3. With these normal subgroups of Gst(G),IUCr macros version 2.1.10: 2016/01/288we search for coset representatives of the spin translation group Gst(G)/(T (D(G)) ×Pso(G)) in Sec. 3.4. We search for coset representatives of the spin space group G/Gst(G)in Sec. 3.5. Because the notation in this section is abstract to unambiguously deal withseveral basis vectors, it may be helpful to read it alongside the examples in Sec. 4.3.1. Space group of nonmagnetic crystal structureA candidate for spatial operations of G can be derived from symmetry operationsof a crystal structure (A,X,T ) ignoring the magnetic moments. We also employ thespace group S given as a stabilizer of E(3) preserving (A,X,T ):S =g = (R,v)A ∈ E(3)∣∣∣∣∣∣∃σg ∈ SN , ∀i = 1, · · ·N,Rxi + v ≡ xσg(i) (mod 1)ti = tσg(i) , (12)where SN is a symmetric group of degree N . Note that g = (R,v)A maps point coor-dinates xi to Rxi+v. The mapped point coordinates coincide with point coordinatesin X up to modulo one, inducing permutation σg.The existing crystal symmetry search algorithm (Togo & Tanaka, 2018) can findprimitive basis vectors AS of T (S) and coset decomposition of S over TA, where wewrite a translation subgroup formed by A asTA ={(E,n)A∣∣ n ∈ Z3}. (13)The input basis vector A can be represented as an integer linear combination ofAS . Thus, an integer matrix U ∈ Z3×3 exists such that A = ASU . The translationsubgroup TAS is decomposed asTAS =⊔tS(E, tS)ASTA, (14)where tSis a centering vector in a unit cell spanned by A. The coset decompositionof S is written asS =⊔RS(RS,vRS)ASTAS . (15)IUCr macros version 2.1.10: 2016/01/289Here, vRS is a translation part of a symmetry operation with a matrix part RS.For the spin arrangement (A,X,T ,M), the spin space group can be expressed asa stabilizer of S ×O(3) that preserves (A,X,T ,M),G ={(g,W ) ∈ S ×O(3)∣∣Wmi = mσg(i) (i = 1, . . . , N)}. (16)This expression serves as the starting point for the spin symmetry operation search.For notation simplicity, we denote the maximal space subgroup of G as D = D(G).3.2. Spin-only group searchBecause a symmetry operation in the spin-only group of G does not change theorder of point coordinates, the spin-only group of (A,X,T ,M) is expressed asPso = {W ∈ O(3) | Wmi = mi (i = 1, . . . , N)} . (17)As shown in Table 2, when a spin space group G is a stabilizer of a spin arrangement,spin-only groups are classified into four types up to transformations (Litvin & Ope-chowski, 1974; Liu et al., 2022): nonmagnetic, collinear, coplanar, and noncoplanarspin arrangements.In Sec. 3.2.1, we provide an algorithm for detecting Pso using the eigenvalue decom-position of MM⊤. The spin-only group search should be performed within tolerancesfor magnetic moments in practice. In fact, selecting appropriate tolerances is challeng-ing. In Sec. 3.2.2, we propose a robust algorithm, building on the previous section, toalleviate the difficulty.3.2.1. Spin-only group search by eigenvalue decomposition We consider a momenttensor of M ∈ R3×N ,MM⊤ =N∑i=1mi ⊗mi. (18)IUCr macros version 2.1.10: 2016/01/2810Because MM⊤ is a symmetric semi-definite matrix, we can consider its eigenvaluedecomposition,MM⊤ =3∑r=1σrn̂r ⊗ n̂r, (19)where σ1 ≥ σ2 ≥ σ3 ≥ 0 and {n̂r}3r=1 are orthonormal.The spin-only group is classified based on the eigenvalues {σi}3i=1, as summarizedin Table 2. In a nonmagnetic spin arrangement, all magnetic moments are zero. Inthis case, all eigenvalues are zero, σ1 = σ2 = σ3 = 0. In a collinear spin arrangement,all magnetic moments align parallel or antiparallel to a direction n̂∥. The eigenvectorn̂1 with the largest eigenvalue should be parallel or antiparallel to n̂∥ and the othereigenvalues σ2 and σ3 should be zero. In a coplanar spin arrangement, all magneticmoments align perpendicular to a direction n̂⊥. In this case, the eigenvector n̂3 withthe smallest eigenvalue should be parallel or antiparallel to n̂⊥ and the smallest eigen-value σ3 should be zero. In other cases, the spin arrangement is noncoplanar. Notethat the site order of magnetic moments M does not affect the classification becauseMM⊤ in Eq. (18) remains invariant under a permutation of the site order.3.2.2. Numerically robust spin-only group search The spin-group search algorithm inthe previous section requires judging whether the eigenvalues are zero or positive,which needs an additional tolerance parameter. To reduce the number of toleranceparameters for usability, we modify the spin-group search algorithm solely with thetolerance ϵmag in Eq. (11) as follows.1. We compute the eigenvalues σi and eigenvectors n̂i in Eq. (19).2. If all magnetic moments are close to zero within ϵmag,∥mi∥2 < ϵmag (i = 1, . . . , N), (20)the spin arrangement is nonmagnetic.IUCr macros version 2.1.10: 2016/01/28113. If not, we check if the eigenvector n̂1 is a parallel or antiparallel direction for allmagnetic moments,2 ∥mi − (mi · n̂1)n̂1∥2 < ϵmag (∀i = 1, . . . , N). (21)When spin symmetry operations in a collinear spin-only group prescribed by adirection n̂1 act on mi, the acted magnetic moments draw a cone as shown inFig. 1 (a). The left-hand side of the above inequality is the largest displacementbetween the acted magnetic moments, which corresponds to the diameter of thecone with direction n̂1. If the inequality holds for all magnetic moments, thespin arrangement is collinear.4. If not, we check if the eigenvector n̂3 is a perpendicular direction for all magneticmoments,2 |(mi · n̂3)| < ϵmag (∀i = 1, . . . , N). (22)The left-hand side of the above inequality is the largest displacement betweenmagnetic moments acted by a coplanar spin-only group along n̂3 as shown inFig. 1 (b). If the inequality holds for all magnetic moments, the spin arrangementis coplanar.5. Otherwise, the spin arrangement is noncoplanar.3.3. Translation subgroup searchWe search for primitive basis vectors AD of the translation subgroup T (D(G)) ={(E,v) | ((E,v),E) ∈ G}. The group–subgroup relationships of translation subgroupsare shown in Fig. 2. Because TAD is between TAS and TA, TA ⊴ TAD ⊴ TAS , we onlyneed to examine finite coset representatives of TAS/TA for candidates of TAS/TAD .For every coset(E, tS)ASTA in Eq. (14), we check if(E, tS)ASpreserves magneticIUCr macros version 2.1.10: 2016/01/2812moments M ,TAD ={g =(E, tS)AS∣∣∣∣ gTA ∈ TAS/TA,mi = mσg(i) (i = 1, . . . , N)}TA. (23)An integer matrix V ∈ Z3×3 exists such that AD = ASV . A lattice algorithm togenerate V from centering vectors tSin Eq. (23) is presented in Appendix C.3.4. Spin translation group searchWe can find a spin rotation W for a given symmetry operation g ∈ S by solvinga Procrustes problem (Gower & Dijksterhuis, 2004) as follows. We write magneticmoments permuted by g asMg =(mσg(1), · · · ,mσg(N)), (24)where σg is a permutation of N sites induced by g. Rather than directly searchingfor W ∈ O(3) such that Wmi = mσg(i), we choose a candidate W̃ by solving thefollowing Procrustes problem:W̃ = argminW∈O(3)∥Mg −WM∥F . (25)Here the displacement between the magnetic moments and operated ones by (g,W )is measured by the Frobenius norm ∥·∥F . The solution to Eq. (25) can be explicitlywritten asW̃ = Y Z⊤, (26)where Y andZ are orthogonal matrices of the singular value decomposition ofMgM⊤,MgM⊤ = Y ΣZ⊤. (27)After we obtain W̃ , (g, W̃ ) is taken as a spin symmetry operation if the condition∥∥∥mi − W̃mσg(i)∥∥∥2< ϵmag (28)IUCr macros version 2.1.10: 2016/01/2813holds for every site i. If Eq. (28) does not hold for some site i, we reject (g, W̃ ) as aspin symmetry operation.For coset representatives of (E, tD)ADTAD ∈ TAS/TAD , we search for a correspond-ing spin rotation WtD using the above algorithm if it exists. The coset decompositionof Gst can be written asGst ={g =((E, tD)AD ,WtD) ∣∣∣∣∣ (E, tD)ADTAD ∈ TAS/TADWtDmi = mσg(i) (∀i = 1, · · · , N)}(TAD × Pso) ,(29)where tDtakes a distinct translation up to TAD and we choose WtD=0= E.3.5. Spin space group searchBecause the translation subgroup of F(G) is TAD , spatial operation parts of G shouldbelong to a maximal subgroup SD of S with its translation subgroup TAD . Figure 2shows the group–subgroup relationship between S and SD. A rotation ASRSA−1S ∈P(S) is compatible with TAD if V −1RSV is an integer matrix (Hart & Forcade, 2008).Thus, the compatible subgroup SD is written asSD =(RS,vRS + tRS)AS∣∣∣∣∣∣∣∣(RS,vRS)TAS ∈ S/TASV −1RSV ∈ Z3×3(E, tRS)ASTA ∈ TAS/TA TAD , (30)where tRS is a centering vector in TAS/TA. The additional centering vector tRS isnecessary for SD to form a group (Nebe, 2011; Stokes & Campbell, 2017).For a coset representative of SD/TAD , a corresponding spin-rotation part WRS canalso be determined by solving the Procrustes problem as presented in Sec. 3.4. Finally,we obtain all spin symmetry operations in a coset decompositionG ={((RS,vRS + tRS)AS,WRS) ∣∣∣∣∣(RS,vRS + tRS)ASTAD ∈ SD/TADWRSmi = mσg(i) (∀i = 1, · · · , N)}Gst.(31)IUCr macros version 2.1.10: 2016/01/28144. Examples of spin symmetry operation searchWe consider a spin arrangement of the NiAs-type CrSe (Corliss et al., 1961; Litvin,1973) (#2.35 of magndata (Gallego et al., 2016)) as an example of spin symmetryoperation search, illustrated in Fig. 3. The hexagonal lattice of CrSe has the followingbasis vectorsA =a −12a 00√32 a 00 0 c .The fractional coordinates and magnetic moments of atoms are as follows.Cr :x1 = (0, 0, 0)⊤ ,m1 =(−12mx,−√32mx,−mz)⊤Cr :x2 =(13,23, 0)⊤,m2 = (mx, 0,−mz)⊤Cr :x3 =(23,13, 0)⊤,m3 =(−12mx,√32mx,−mz)⊤Cr :x4 =(0, 0,12)⊤,m4 =(12mx,√32mx,mz)⊤Cr :x5 =(13,23,12)⊤,m5 = (−mx, 0,mz)⊤Cr :x6 =(23,13,12)⊤,m6 =(12mx,−√32mx,mz)⊤Se :x7 =(0,13,14)⊤,m7 = (0, 0, 0)⊤Se :x8 =(13, 0,14)⊤,m8 = (0, 0, 0)⊤Se :x9 =(13,13,34)⊤,m9 = (0, 0, 0)⊤Se :x10 =(0,23,34)⊤,m10 = (0, 0, 0)⊤Se :x11 =(23, 0,34)⊤,m11 = (0, 0, 0)⊤Se :x12 =(23,23,14)⊤,m12 = (0, 0, 0)⊤Here, the magnetic moments are represented with Cartesian coordinates.IUCr macros version 2.1.10: 2016/01/28154.1. Space group of nonmagnetic crystal structureThe space-group type of S for the crystal structure ignoring magnetic moments isP63/mmc (No. 194). One of the primitive basis vectors for S and the transformationmatrix areAS = 0 −12a 0−√33 a√36 a 00 0 −cU =−1 −1 0−2 1 00 0 −1with A = ASU as defined in Sec. 3.1. There are three coset representatives forTAS/TA,TAS = g1TA ⊔ g2TA ⊔ g3TA (32)g1 =(E, (0, 0, 0)⊤)ASg2 =(E, (−1,−1, 0)⊤)ASg3 =(E, (−1, 0, 0)⊤)AS.4.2. Spin-only groupThe moment tensor of M is given byMM⊤ =3m2x 0 00 3m2x 00 0 6m2z .Its eigenvalues 3m2x, 3m2x, and 6m2z are all positive. Consequently, the spin arrangementis noncoplanar with Pso = 1.4.3. Translation subgroup of maximal space subgroupThe translation g1 in Eq. (32) belongs to TAD . On the other hand, g2 does notbelong to TAD because it maps x1 to x3, whereas m1 ̸= m3. Similarly, g3 does notIUCr macros version 2.1.10: 2016/01/2816belong to TAD . Therefore, TAD is identical to TAS with AD = ASV as defined inSec. 3.3 and we can choose V = E.4.4. Coset representatives of spin translation groupThere are three candidates for the spatial operation parts of coset representativesof Gst: g1, g2, and g3. For translation g1 in Eq. (32), we obtainMg1M⊤ =3m2x 0 00 3m2x 00 0 6m2z .From the singular value decomposition of Mg1M⊤, we calculate the spin-rotationpart W̃1 for g1 as W̃1 = E. For translation g2, the singular value decomposition ofMg2M⊤ isMg2M⊤ = −12√32 0−√32 −12 00 0 13m2x 0 00 3m2x 00 0 6m2x1 0 00 1 00 0 1⊤.Hence, a candidate for a spin-rotation part with g2 isW̃2 = −12√32 0−√32 −12 00 0 11 0 00 1 00 0 1⊤= −12√32 0−√32 −12 00 0 1 .Similarly, a candidate for a spin-rotation part with g3 isW̃3 =−12 −√32 0√32 −12 00 0 1 .The spin symmetry operations (gi, W̃i) (i = 1, 2, 3) all preserve magnetic moments.Consequently, the spin translation group is obtained asGst = (g1, W̃1)(TAD × 1) ⊔ (g2, W̃2)(TAD × 1) ⊔ (g3, W̃3)(TAD × 1).4.5. Coset representatives of spin space groupBecause TAD and TAS coincide in this example, S is fully compatible with TAD ,SD = S. For 24 coset representatives of SD/TAD , we search for their spin-rotationIUCr macros version 2.1.10: 2016/01/2817parts by solving the Procrustes problems. For example, a six-fold spatial rotoinversiong =−1 1 0−1 0 00 0 −1 ,0AD∈ SDgives a permutation of sites σg = (4, 5, 6, 1, 2, 3, 12, 7, 11, 9, 10, 8) and the singular valuedecomposition of MgMMgM =−1 0 00 −1 00 0 −13m2x 0 00 3m2x 00 0 6m2x1 0 00 1 00 0 1⊤,which results in W̃ = −E. Similarly, all of the 24 coset representatives of SD/TADgive spin symmetry operations preserving M .A magnetic space group M of the NiAs-type CrSe is P31m′ (BNS number 157.55),which has |M/TAD | = 6 coset representatives. Because the spin translation group Gstcontains three-fold spin-rotation parts, these spin symmetry operations ((g2, W̃2) and(g3, W̃3) in Sec. 4.4) are not captured in M. Similarly, the spin symmetry operationswith six-fold spatial rotations or rotoinversions are not preserved in M.5. ConclusionWe have presented the algorithm for determining spin symmetry operations of a givenspin arrangement. The spin-only group is robustly determined from the eigenvaluedecomposition of the moment tensor of magnetic moments, MM⊤. We have explicitlyconsidered the three translation subgroups to address the enlargement of the unitcell due to the spin translation group: the translation subgroup spanned by the inputbasis vectors TA, one by the primitive basis vector TAS , and one by the primitive basisvectors for the spin space group TAD . Spin-rotation parts of the coset representatives ofthe spin translation group and spin space group are found by solving the Procrustesproblem to match the original magnetic moments and permuted ones. The presentalgorithm is implemented in spinspg under a permissive license. In future work, itIUCr macros version 2.1.10: 2016/01/2818will be beneficial to identify the spin-space-group type and a suitable transformationfrom the spin symmetry operations. Our presented algorithm and implementation willadvance spin symmetry analysis in crystallography and condensed matter physics.Note added After this work was completed, we became aware of recent preprints onthe classification and enumeration of spin space-group types (Xiao et al., 2023; Renet al., 2023; Jiang et al., 2023). Two of these works, Xiao et al. (2023) and Jianget al. (2023), appear to use a workflow similar to ours to determine spin symmetryoperations although they only provide a brief description and their implementationsare not available to the public.Our presented algorithm uses the Procrustes problem to determine a spin-rotationpart W . On the other hand, Xiao et al. (2023) directly determines W for a sym-metry operation g such that selected three magnetic moments mi (i = 1, 2, 3) aretransformed into mσg(i), which may not be robust against numerical noises and exper-imental uncertainty of magnetic moments.Jiang et al. (2023) identifies spin symmetry operations on top of spglib (Togo &Tanaka, 2018) similar to ours. However, they do not incorporate translation subgroupsTAD and TA. This omission makes their algorithm complicated and non-exhaustive.Appendix ACorrespondence between spin symmetry operation and magneticsymmetry operationWhen we consider a spin symmetry operation ((R,v),W ) for a Hamiltonian with-out the spin–orbit coupling (SOC), we derive the condition that ((R,v),W ) has acorresponding magnetic symmetry operation for the Hamiltonian with SOC. The SOCintroduces an additional term to the Hamiltonian proportional to L̂·σ̂, with the angu-IUCr macros version 2.1.10: 2016/01/2819lar momentum operator L̂ and the Pauli matrices σ̂ = (σ̂x, σ̂y, σ̂z). A spin symmetryoperation ((R,v),W ) acts on L̂ and σ̂ asL̂ 7→ (detR)(detW )( ∑ν=x,y,zRµνL̂ν)µ=x,y,z(33)σ̂ 7→( ∑ν=x,y,zWµν σ̂ν)µ=x,y,z, (34)where R and W are represented with Cartesian coordinates. Note that detW inEq. (33) reflects a time reversal operation. Thus, ((R,v),W ) acts on the SOC termasL̂ · σ̂ 7→ (detR)(detW )∑µ,ν=x,y,z[R⊤W]µνL̂µσ̂ν . (35)To preserve the SOC term, (detR)(detW )R⊤W should be identity. Because (detR)Rand (detW )W belong to SO(3), this condition is equivalent to(detR)R = (detW )W . (36)Therefore, if ((R,v),W ) satisfies Eq. (36), it has a corresponding magnetic symme-try operation ((R,v),detW ) with symmetry operation part (R,v) and time-reversalpart detW . We identify detW = 1 with an identity operation and detW = −1with a time-reversal operation. Here, we assume that a magnetic symmetry operation((R,v), θ) acts on a magnetic moments m as θ(detR)Rm.Conversely, a magnetic symmetry operation ((R,v), θ) is mapped to a spin sym-metry operation ((R,v), θR). We can confirm this mapping satisfies Eq. (36) by(det θR)θR = (detR)R. Because this mapping is injective, the index of a magneticspace group M in its translation subgroup TAD is a divisor of the index of a corre-sponding spin space group G in TAD , |G/TAD | ≡ 0 (mod |M/TAD |).Appendix BSpin point groupIUCr macros version 2.1.10: 2016/01/2820We define a family spin point group of a spin space group G asB(G) = {W ∈ O(3) | ∃g s.t. (g,W ) ∈ G} . (37)When we write a coset decomposition of G by its spin translation group Gst(G) asG =⊔R((R,vR),WR)Gst(G), (38)A spin point group of G isU(G) = {(R,WRB(Gst(G)))}R∈P(F(G)) . (39)The spin point group U(G) is a subgroup of O(3)×(O(3)/B(Gst(G))) and is isomorphicto G/Gst(G).In general, we cannot choose WR so that {(R,WR)} is a group under the multi-plication in O(3)×O(3). We show one of the counterexamples in which {(R,WR)} isnot closed as a group, with a spin arrangementA =4 0 00 6 00 0 8X =0.4 0.6 0.4 0.40.4 0.6 0.4 0.40 0.25 0.5 0.75M = 1 0 −1 00 1 0 −10.4 0.4 0.4 0.4 ,illustrated in Fig. 4. This spin arrangement is noncoplanar with Pso(G) = 1. The spinIUCr macros version 2.1.10: 2016/01/2821space group G is obtained asG = {g1, g2, g3, g4, g5, g6, g7, g8} (T (D(G))× Pso(G))g1 = ((1,0)A , 1)g2 =((1,12c)A, 2z)g3 =((1,34c)A,mxy)g4 =((1,14c)A,mxy)g5 =((2,34c)A, 4−z)g6 =((2,14c)A, 4+z)g7 =((m,12c)A,mx)g8 = ((m,0)A ,my) ,where we denote a mirror operation along xy axis as mxy. The spin point group of GisU(G) ={(1, 1), (1,mxy), (2, 4−z ), (m,mx)}(1× B(Gst(G))) (40)B(Gst(G)) = {1, 2z}The coset representatives in Eq. (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 thisexample because U(G) ∼= Z4, B(Gst(G)) ∼= Z2, and Z4 cannot be written as an internalsemidirect product Z2 ⋊ Z2.Appendix CGenerating transformation matrix from centering vectorsTo demonstrate how to generate a transformation matrix V in Sec. 3.3, we considerIUCr macros version 2.1.10: 2016/01/2822the following coplanar fcc structure with the conventional basis,A =a 0 00 a 00 0 aX =0 0 12120 12 0 120 1212 0M = 0 0 m m0 0 0 0m m 0 0 .One of the primitive basis vectors for the nonmagnetic crystal structure isAS = AU−1 =a20 1 11 0 11 1 0U =−1 1 11 −1 11 1 −1 .There are four centering vectors in TAS/TA,TAS = g1TA ⊔ g2TA ⊔ g3TA ⊔ g4TAg1 =(E, (0, 0, 0)⊤)ASg2 =(E, (1, 0, 0)⊤)ASg3 =(E, (0, 1, 0)⊤)ASg4 =(E, (0, 0, 1)⊤)AS.The half of centering vectors form the translation subgroup of D(G),T (D(G)) = g1TA ⊔ g2TA.Because T (D(G)) is spanned by column vectors of U and centering vectors of g1 andg2, we can rewrite T (D(G)) asT (D(G)) = ASŨZ5Ũ =−1 1 1 0 11 −1 1 0 01 1 −1 0 0 .IUCr macros version 2.1.10: 2016/01/2823We need to find a transformation matrix V ∈ Z3×3 such thatT (D(G)) = ASV Z3, (41)and it is known to be achieved by the Hermite normal form of Ũ (Cohen, 1993),Ũ =1 0 0 0 00 1 0 0 00 1 2 0 0−1 1 1 0 11 −1 1 0 00 1 −1 0 00 0 0 1 00 1 0 0 0 ,where Ũ is decomposed into a product of a lower triangular integer matrix and aunimodular matrix. Because the above 5× 5 integer matrix is unimodular, we obtainT (D(G)) = AS1 0 0 0 00 1 0 0 00 1 2 0 0Z5= AS1 0 00 1 00 1 2Z3.Consequently, we generate the integer matrixV =1 0 00 1 00 1 2 .ReferencesBrinkman, W. & Elliott, R. J. (1966). J. Appl. Phys. 37(3), 1457–1459.Brinkman, W. F., Elliott, R. J. & Peierls, R. E. (1966). Proc. Math. Phys. Eng. Sci. 294(1438),343–358.Cohen, H. (1993). A Course in Computational Algebraic Number Theory. Springer BerlinHeidelberg.Corliss, L. M., Elliott, N., Hastings, J. M. & Sass, R. L. (1961). Phys. Rev. 122, 1402–1406.Corticelli, A., Moessner, R. & McClarty, P. A. (2022). Phys. Rev. B, 105, 064430.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(5), 1750–1776.Gower, J. C. & Dijksterhuis, G. B. (2004). Procrustes Problems. Oxford University Press.Hahn, T., Klapper, H., Müller, U. & Aroyo, M. (2016). 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Spin space groups: Full classifi-cation and applications. https://arxiv.org/abs/2307.10364.Yang, J., Liu, Z.-X. & Fang, C., (2021). Symmetry invariants in magnetically ordered systemshaving weak spin-orbit coupling. https://arxiv.org/abs/2105.12738.Zelenskiy, A., Monchesky, T. L., Plumer, M. L. & Southern, B. W. (2022). Phys. Rev. B, 106,144433.Zhang, Y., Železný, J., Sun, Y., van den Brink, J. & Yan, B. (2018). New J. Phys. 20(7),073028.IUCr macros version 2.1.10: 2016/01/28https://www.iucr.org/publ/978-0-9553602-2-0https://www.iucr.org/publ/978-0-9553602-2-0https://arxiv.org/abs/2307.10369https://github.com/spglib/spglibhttps://github.com/spglib/spglibhttps://arxiv.org/abs/2307.10364https://arxiv.org/abs/2105.1273825Table 1. Notation and terminology in this paper.Symbol MeaningA ⊔B Disjoint union of set A and B with A ∩B = ∅E(3) Three-dimensional Euclidean groupO(3) Three-dimensional orthogonal groupE Identity matrixSN Symmetric group of degree N∥y∥2 =√y⊤y l2 norm of vector y∥B∥F =√TrB⊤B Frobenius norm of matrix B1 Trivial groupr Position in Cartesian coordinatesm Magnetic momentsSpin symmetry operation (g,W ) Pair of spatial operation g and spin rotation W(R,v)A = (ARA−1,Av) Spatial operation with basis vectors AA = (a1,a2,a3) Basis vectorsX = (x1, · · · ,xN ) Array of point coordinatesT = (t1, · · · , tN ) Array of atomic typesM = (m1, · · · ,mN ) Array of magnetic momentsSpin arrangement (A,X,T ,M) Pair of crystal structure and magnetic moments(A,X,T ) Crystal structure ignoring magnetic momentsϵmag Absolute tolerance to compare magnetic momentsS,R Space groupTranslation subgroup T (R) Translation parts of R with identity rotationsPoint group P(R) Group obtained from the rotation parts of R/T (R)TA Translation group spanned by Aσg Permutation of sites induced by gAS = AU−1 Primitive basis vectors of T (S)Spin-only group Pso = Pso(G) Spin-rotation parts with identity spatial operationsσi, n̂i Eigenvalue and eigenvector of MM⊤n̂∥ Parallel direction for collinear spin arrangementn̂⊥ Perpendicular direction for coplanar spin arrangementAD = ASV Basis vectors of T (D(G))Spin translation group Gst = Gst(G) Subgroup of G with identity rotationsMg =(mσg(1) · · ·mσg(N))Array of magnetic moments permuted by σgSpin space group G See Sec. 2.2Family space group F(G) Space group composed of spatial parts of GMaximal space subgroup D = D(G) Subgroup of G with identity spin-rotation partsSpin-space-group type See Sec. 2.2(P ,p) Transformation on spatial coordinatesQ Transformation matrix on spin coordinatesFamily spin point group B(G) Spin-rotation parts of GSpin point group U(G) Pairs of spatial and spin rotations of GM Magnetic space groupTable 2. Classification of spin-only groups up to transformations. The spin-only groups arerepresented in Hellmann–Mauguin symbols (Hahn et al., 2016).Spin arrangement Spin-only group Eigenvalues of MM⊤Nonmagnetic ∞∞m ∼= O(3) σ1 = σ2 = σ3 = 0Collinear ∞m ∼= SO(2)⋊ Z2 σ1 > σ2 = σ3 = 0Coplanar m σ1 ≥ σ2 > σ3 = 0Noncoplanar 1 σ1 ≥ σ2 ≥ σ3 > 0IUCr macros version 2.1.10: 2016/01/2826Fig. 1. Magnetic moments acted by (a) collinear and (b) coplanar spin-only groups.(a) A collinear spin-only group along axis n̂1 is generated from rotations alongn̂1 and mirror operations preserving n̂1. The red dotted line indicates the largestdisplacement between magnetic moments acted the collinear spin-only group. (b)A coplanar spin-only group along axis n̂3 is generated from a mirror operationperpendicular to n̂3. The red dotted line indicates the largest displacement betweenmagnetic moments acted the coplanar spin-only group.IUCr macros version 2.1.10: 2016/01/2827TATADTASSDSDTASSFig. 2. Group–subgroup relationship of translation subgroups and space groupsderived from spin space group. The nodes represent translation subgroups and spacegroups. Each edge indicates that a lower group is a subgroup of an upper group ina diagram. Note that SDTAS could be a proper subgroup of S.IUCr macros version 2.1.10: 2016/01/2828Fig. 3. Spin arrangement example for NiAs-type CrSe. The gray and orange ballsrepresent Cr and Se atoms, respectively. The red arrows denote magnetic momentsof Cr atoms with equal magnitudes.IUCr macros version 2.1.10: 2016/01/2829Fig. 4. Spin arrangement example in Appendix B. Its spin point group is not closedunder O(3)×O(3) due to the spin translation group.SynopsisThis paper presents an algorithm for determining the spin symmetry operations of a givenspin arrangement. Spin symmetry operations of a spin space group act simultaneously on boththe spatial and spin coordinates of the spin arrangement.IUCr macros version 2.1.10: 2016/01/28 Introduction Group structure of spin space group Spin symmetry operation and spin arrangement Spin space group Spin-only group Spin translation group Spin symmetry operation search Space group of nonmagnetic crystal structure Spin-only group search Spin-only group search by eigenvalue decomposition Numerically robust spin-only group search Translation subgroup search Spin translation group search Spin space group search Examples of spin symmetry operation search Space group of nonmagnetic crystal structure Spin-only group Translation subgroup of maximal space subgroup Coset representatives of spin translation group Coset representatives of spin space group Conclusion A Correspondence between spin symmetry operation and magnetic symmetry operation B Spin point group C Generating transformation matrix from centering vectors