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

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[Symmetry analysis with spin crystallographic groups: Disentangling effects free of spin-orbit coupling in emergent electromagnetism](https://mdr.nims.go.jp/datasets/e2312200-70b9-4747-96a8-664608072bc3)

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Symmetry analysis with spin crystallographic groups:Disentangling effects free of spin-orbit coupling in emergent electromagnetismHikaru Watanabe ,1, ∗ Kohei Shinohara ,2 Takuya Nomoto ,1 Atsushi Togo ,3 and Ryotaro Arita 1, 41Research Center for Advanced Science and Technology,University of Tokyo, Meguro-ku, Tokyo 153-8904, Japan2Department of Materials Science and Engineering,Kyoto University, Sakyo, Kyoto 606-8501, Japan3Center for Basic Research on Materials, National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan4RIKEN, Center for Emergent Matter Science, Saitama 351-0198, JapanRecent studies identified spin-order-driven phenomena such as spin-charge interconversion withoutrelying on the relativistic spin-orbit interaction. Those physical properties can be prominent insystems containing light magnetic atoms due to sizable exchange splitting and may pave the way forrealization of giant responses correlated with the spin degree of freedom. In this paper, we presenta systematic symmetry analysis based on the spin crystallographic groups and identify the physicalproperty of a vast number of magnetic materials up to 1500 in total. By decoupling the spin andorbital degrees of freedom, our analysis enables us to take a closer look into the relation betweenthe dimensionality of spin structures and the resultant physical properties and to identify the spinand orbital contributions separately. In stark contrast to the established analysis with magneticspace groups, the spin crystallographic group manifests richer symmetry including spin translationsymmetry and leads to emergent responses. For representative examples, we discuss geometricalnature of the anomalous Hall effect and magnetoelectric effect and classify the spin Hall effectarising from the nonrelativistic spin-charge coupling. Using the power of computational analysis,we apply our symmetry analysis to a wide range of magnets, encompassing complex magnets suchas those with noncoplanar spin structures as well as collinear and coplanar magnets. We identifyemergent multipoles relevant to physical responses and argue that our method provides a systematictool for exploring sizable electromagnetic responses driven by spin order.I. INTRODUCTIONSpintronics has experienced tremendous growth, and the concept has been discussed in various fields includingtopological electronic systems and superconductors. In recent years, spin-orbit coupling (SOC), a relativistic interac-tion between the charge and spin degrees of freedom, is particularly of matter due to its rich physical consequences.Search for candidate materials has taken place to maximize the physical responses associated with spin-orbit interac-tion in these decades. For example, giant spin-momentum splittings have been identified in systems with heavy atomshaving large SOC. Their strong spin-orbit entanglement has been demonstrated by spectroscopy [1, 2] and transportmeasurements [3]. The progress may let us consider the possibility of physics originating from SOC covering a broaderrange of materials other than what consists of heavy atoms, such as materials based on 3d transition metal elementswith negligible relativistic SOC.To this end, a concept of nonrelativistic spin-charge locking has been proposed in theories [4–7]. This coupling arisesfrom the spontaneous magnetic ordering without the help of SOC and thereby can exhibit exchange splitting energycomparable to that of the Coulomb interaction. The concept illuminates the potential impacts of light elements for thespintronic application and further identified advantageous properties compared to conventionally-studied materials;e.g., strong exchange splitting energy and large transition temperature. Notably, the magnetic order gives rise tocharacteristic spin-momentum-locking structure due to coupling between the order and structural property of crystalsas found in antiferromagnetic materials. These aspects are valuable for applications in the field of antiferromagneticspintronics gathering considerable interest as an emerging field in condensed matter physics [8–10]; for instance,various physical phenomena free from SOC have been clarified in the previous works such as the spin-polarizedcurrent induction [11–15], nonlinear response [16, 17], piezomagnetic effect [4, 18, 19], and magnetoresistance [20].Released from the SOC constraint, the array of localized spins does not have any favorable orientation described bythe crystal structure. The decoupling between spin and orbital degrees of freedom leads to a magnetic symmetry higherthan the conventional magnetic space group symmetry (Shubnikov group). Such magnetic symmetry without SOC iscovered by the spin crystallographic group such as spin space group and spin point group [21, 22] which includes a richer∗ hikaru-watanabe@g.ecc.u-tokyo.ac.jparXiv:2307.11560v2  [cond-mat.mtrl-sci]  22 Mar 2024https://orcid.org/0000-0001-7329-9638https://orcid.org/0000-0002-5907-2549https://orcid.org/0000-0002-4333-6773https://orcid.org/0000-0001-8393-9766https://orcid.org/0000-0001-5725-072Xmailto:hikaru-watanabe@g.ecc.u-tokyo.ac.jp2group structure due to the absence of SOC. The spin crystallographic groups have applied to analyzing the electronicstructure modified by the spontaneous spin order, particularly in the case of simple spin configurations [4, 5, 7, 23–25].For instance, recent studies identified a series of spin crystallographic symmetries providing the nonrelativistic spin-charge coupling by which the degeneracy at each crystal momentum is lifted. The materials manifesting such spincrystallographic symmetry are characterized by a collinear antiferromagnetic structure whose magnetic unit cell isthe same as the chemical one due to the zero propagation vector. Candidate materials such as MnTe for the so-calledaltermagnets belong to this class [10, 25–29].One can expect that there exist rich physical consequences of nonrelativistic spin-charge coupling in other kinds ofcollinear magnets as well as more complex spin-ordered systems (e.g., spin structure with nonzero propagation vectors),which have not been rarely investigated from the viewpoint of spin crystallographic group. The latter class encompassesintriguing systems such as noncollinear, noncoplanar, and multiferroic magnets [11]. A prototypical phenomenonunique to these materials is the geometrical Hall effect [30]. The effect occurs in magnets with noncoplanar spinstructure observed in materials such as systems with a triangular or net. The resultant time-reversal-symmetrybreaking resembles the orbital-flux order proposed in Ref. [31], and does not require the relativistic SOC effect [32].The SOC-free nature is in high contrast to the well-known anomalous Hall effect arising from the collinear and coplanarspin order [32–34] and may be responsible for the anomalous Hall responses of magnetic skyrmion crystals [35, 36].These prior studies indicate that the dimension of the spin structure is key to identifying the emergent physicalresponses induced by the spin order without the help of SOC. In this regard, the spin crystallographic group isadvantageous compared to the widely-used magnetic space group, because its group symmetry reflects a given spin-structure dimension.In this paper, we present the spin-crystallographic-group symmetry analysis covering not only the simple magneticmaterials having collinear and coplanar spin structures with the zero propagation vector but also complex spin-orderedsystems such as noncoplanar magnets and those with non-zero propagation vectors. The analysis incorporates thedimensionality of the spin structure and hence provides a convenient tool for identifying the emergent symmetrybreaking and associated phenomena which cannot be distinguished from the SOC-assisted contribution in terms ofthe SOC-accounted symmetry analysis based on magnetic space group and magnetic point group [37].Specifically, we demonstrate the following points of the present symmetry analysis; by separating the spin andorbital spaces, we can identify the contribution of each degree of freedom to physical responses. Our symmetryanalysis identified nontrivial spin crystallographic symmetry where the spin-space symmetry is kept highly symmetricto be unexpected by its crystal structure; e.g., the cubic spin-space symmetry despite the axial symmetry of thecrystal. Despite the inactive spin-related quantities, the symmetry does not forbid physical phenomena involving theorbital degree of freedom such as the geometrical Hall effect. Such nontrivial spin crystallographic symmetry enablesus to explore the spin-geometry-induced response so as to unambiguously distinguish it from the relativistic SOCeffect. The orbital-active but spin-inactive aspects highlight the significance of magnets with non-zero propagationvectors and are in sharp contrast to the previously identified SOC-free physical property, that is spin-active butorbital-inactive property [12]. Furthermore, our result suggests that emergent physical responses can be examinedin a semi-quantitative manner by combining the spin-crystallographic-group symmetry analysis with the physicalinsights into spin fluctuations correlated with the dimensionality of a spin structure. We explain these features bytaking several physical properties such as the anomalous Hall effect, magnetoelectric effect, and spin Hall effect. Thesymmetry analysis is computationally performed with the use of the algorithm for searching the spin space groupdeveloped by Shinohara et al. [38]. By classifying a vast number of magnetic materials (∼ 1500), we systematicallyclarify the SOC-free emergent properties of real magnetic materials.The organization of the paper is the following. In Sec. II, we overview the spin space group and introduce thesymmetry analysis with it. Based on the spin point group, the symmetry analysis is applied to some of ferromagneticand antiferromagnetic materials and their physical responses in Sec. III. Section IV is devoted to a high-throughputsymmetry analysis of magnetic materials listed in magndata [39, 40]. In light of emergent magnetic multipoles androtators for spin-charge coupling, we investigate the SOC-free physical properties and demonstrate that our symmetryanalysis clarifies the importance of the spin-structure dimension. We summarize the contents in Sec. V. The procedureof the symmetry analysis is sketched in Fig. 1.The symmetry analysis with given spin space group G and magnetic space group G are automatically computed onthe basis of spglib [41, 42] and spinspg [38]. Terms on group theory can be found in Appendix A.II. SPIN-GROUP OPERATIONS AND SPIN CRYSTALLOGRAPHIC GROUPWe introduce the transformation property of symmetry operations and overview the spin crystallographic symmetrysuch as the spin space group and spin point group. We not only explain the spin space group but also introducethe spin and orbital parts of the spin crystallographic group to disentangle the spin and orbital contributions to3FIG. 1. Procedure of symmetry analysis. (a) Input of crystal and spin structures. Dozens of data imported from magndata.(b) Spin space group G and magnetic space group G computationally identified. The effect of spin-orbit coupling (SOC) isnot considered in G but is taken into account in G. The spin-space-group symmetry is given by the spin rotation (W ) andby orbital-space operations such as rotation (R) and translation (t) operations. The magnetic space group is comprised of theorbital-space operations (R, t) with or without the time-reversal operation θ such as R′ = θR. For the magnetic space group G,the orbital-space operations R,R′ also act on spins due to the SOC constraint. (c) The space group is reduced to its point groupby omitting the translation operation t. Spin (P) and magnetic (P) point groups are obtained from G and G, respectively. Thespin point group is further divided into (Psp,Porb) consisting of either spin- or orbital-space symmetry. The spin crystallographicgroups (G,P) are reduced to the spin-orbital-coupled groups (G,P) by SOC. (d) The symmetry-adapted form of a given tensorχ̂ obtained by the spin or magnetic point group symmetry. The red-colored components (χsp, χorb) originates from the spin-order-induced symmetry breaking without SOC, and the origin of each component is further attributed to the spin (χsp)and orbital (χorb) degree of freedom. SOC entangles the spin contribution with the orbital (χ1, χ4) and induces additionalcomponents (χ2, χ3).physical phenomena. We concisely introduce the notations of spin crystallographic groups we adopt, while theterminology related to the group theory and its mathematical aspect are summarized in Appendix A. Concerning thecrystallographic property, interested readers can refer to Refs. [22, 43].Owing to the absence of generic spin-orbital coupling, the rotation operation separately acts on the spin and orbitalspace in terms of the spin group symmetry. Let g = (R,W ) be the combination of symmetry operations R and Wacting on the orbital and spin space, respectively.Firstly, we consider the point group symmetry with rotation operations (R,W ) and raise some examples of the basictransformation property such as position r, momentum p, and spin s. For instance, let us take the orbital-space-only operation gorb = (R, 1), the time-reversal operation θ = (1,−1), and spin-space proper rotation gsp = (1,W )(detW = +1). Note that the time-reversal operation is denoted by the space-inversion operation in the spin spaceby following Refs. [38, 44]. The operators are transformed under each symmetry operation as shown in Fig. 2. Whenthe improper property holds for the spin-space operation as detW = −1, the operation g includes the time-reversaloperation θ such as that with the spin-space space-inversion (W = −1) and mirror operation (W = m). It follows that4g with detW = +1 (detW = −1) is unitary (anti-unitary). Owing to the time-reversal operation θ, the operationg with detW = −1 can act on the orbital space as it flips the time-reversal-odd quantities, e.g., p [Fig. 2(b)] andorbital magnetization. On the other hand, the proper rotation in the spin space does affect only the spin degree offreedom [Fig. 2(c)]. For a spin-group operation g = (R,W ), the operators are transformed asg ra g−1 = rbTba(R), (1)g pa g−1 = detW · pbTba(R), (2)g sa g−1 = sbTba(W ), (3)where we introduced the three-dimensional orthogonal matrices T̂ (u) (u ∈ O(3)) in accordance with the vectorialsymmetry of each object such as T̂ (−1) = −1 for the inversion operation (1 is the identity matrix in three-dimensionalsystem) 1.We generalize the symmetry argument to the case of the tensor quantity Oabc···. The transformation is written byg Oabc··· g−1 = Oa′b′c′···D(A)a′a (g)D(B)b′b (g)D(C)c′c (g) · · · , (4)where we introduced the representation matrices for physical quantities A,B,C, · · · labeled by the indices a, b, c, · · · ,respectively. For the aforementioned three quantities, the representation matrices are explicitly given byD̂(r)(g) = T̂ (R), D̂(p)(g) = detW · T̂ (R), D̂(s)(g) = T̂ (W ). (5)Taking the operations depicted in Fig. 2, the representation matrices are explicitly given byD̂(r)(gorb) = T̂ (R), D̂(p)(gorb) = T̂ (R), D̂(s)(gorb) = 1, (6)D̂(r)(θ) = 1, D̂(p)(θ) = −1, D̂(s)(θ) = T̂ (W ), (7)D̂(r)(gsp) = 1, D̂(p)(gsp) = 1, D̂(s)(gsp) = T̂ (W ), (8)by which, for instance, the spin-space mirror operation (R = 1,W = m) is obtained by combining the representationmatrices of θ with that of gsp. Then, in the absence of SOC, the transformation property of spins underW can be givensimilarly to that of a polar vector. Owing to the irrelevant role of proper spin-space rotations in the transformationof r and p [Eq. (5)], the system has the orbital time-reversal symmetry if there exists a symmetry operation given byg = (1,W ) (detW = −1).FIG. 2. Spin-group transformation of electron depicted by position r, momentum p, and spin s. The spin-group operation g =(R,W ) (R is orbital-space rotation, W is spin-space rotation). (a) Orbital-space operation acting on position and momentumwhile leading to no action on spins (W = 1). (b) Spin-space inversion operation same as the time-reversal operation (R =1, W = −1) flipping the time-reversal-odd quantities such as r and s. (c) Spin-space operation satisfying R = 1 and the properrotation condition (detW = +1). It gives no transformation related to the orbital-space objects.Next, we consider the structure of the spin crystallographic group. In terms of the space group symmetry, the orbital-space operation h is comprised of the point group operation R and the translation operation t as h = (R, t). The1 Note that the representation matrices are in the Cartesian coordinates though they are usually in the basis spanned by the Bravaisvectors in the field of crystallography. This is because the spin-space operations do not necessarily belong to the Bravais class same asthat for a given crystal structure in the framework of spin crystallographic group.5spin space group G is a set of symmetry operations g = (h,W ) under which crystal structures and spin configurationdwelling on each magnetic atom are invariant. In stark contrast to the well-known magnetic space group (Shubnikovgroup) G [45], we can take symmetry operations acting on objects in the orbital and spin space independently [43].The difference can be inferred from the adopted Hamiltonian as follows.Let us consider Hamiltonian manifesting the spin-space-group symmetry. The Hamiltonian not only consists ofkinetic and potential Hamiltonians H0 for paramagnetic states but also takes into account the spontaneous spinordering by the molecular field. The total Hamiltonian is given byHSG = H0 +Hmag, (9)where the molecular-field term isHmag =∑iB(i) · s(i), (10)with indices for the sites i. The exchange-splitting field B(i) is defined at each magnetic site (B(i) = 0 for nonmagneticatoms). The Hamiltonian [Eq. (9)] is invariant under the associated spin space group asg ∈ G, g HSG g−1 = HSG. (11)The paramagnetic part satisfies the following relation for g = (h,W ) ∈ G asH0 = (h,W )H0 (h,W )−1= (h, 1)H0 (h, 1)−1, (12)where the spin rotation W is irrelevant. The orbital-space part h is therefore restricted by the atomic configurationand generated by the space group of a given crystal structure. On the other hand, the exchange Hamiltonian istransformed under both h and W asg(∑iB(i)a s(i)a)g−1 =∑iB(i)a ·(g s(i)a g−1), (13)=∑iB(i)a · s(ih)b Tba(W ). (14)The orbital-space operation permutes the sites as h : ri 7→ rih = h−1 ri h. Owing to Eq. (11), the spin-groupoperations should satisfyT̂ (W )B(i) = B(ih), (15)for every site, and the coupling arises between the orbital and spin degrees of freedom. As the result, the spin-spaceoperations are determined by Hmag, whereas orbital-space operations are by the total Hamiltonian. Importantly, theoperations h and W are taken independently as long as they satisfy Eq. (15). The property clearly distinguishes thespin space group from the magnetic space group. We note that the spin-space-group symmetry holds in general if oneproperly takes into account the spin order in a SOC-free manner, while the spin-ordering effect is simply taken as themolecular field for illustrative purposes.For the case of the magnetic space group, we similarly treat the magnetic order as the molecular fields and addspin-orbit interaction HSOC to the Hamiltonian. The Hamiltonian reads asHMG = H0 +Hmag +HSOC, (16)where the SOC Hamiltonian is given byHSOC = λL · s. (17)L denotes the atomic orbital angular momentum and λ is the strength of SOC. The additional symmetry constraintby the SOC Hamiltonian leads to the group-subgroup relation G (G < G). In sharp contrast to the spin space groupG, the SOC Hamiltonian imposes the following constraint on (h,W ) = ((R, t) ,W ) ∈ G,detR ·R = detW ·W. (18)That is, the proper rotation parts of h and W should be the same as each other [38]. Eq. (18), along with Eq. (15),ties the orbital space with the spin space.6Then, let us overview the group structure of spin space group G which has been investigated in Refs. [43, 46]. Gcontains the spin-only group Gso as a normal subgroup (G ▷ Gso). The spin-only group solely consists of the spinsymmetry operations such asGso = ((1,0),Pso) = {((1,0),W ) | W ∈ Pso}. (19)The group is determined by the dimension of a given spin configuration, which we call the spin-structure dimensionDsp [22]. For one-dimensional magnets (collinear magnets) denoted by Dsp = 1, the magnetic moments are parallelor anti-parallel to the axis n in the spin space. The spin-only group is given by an internal semidirect product,Pso = SO(2)⋊ {1,m∥}, (20)by which the vector n is invariant. We can take rotation operations along n with an arbitrary angle and mirrorreflection m∥ whose mirror plane contains the axis n. For the two-dimensional case (Dsp = 2, noncollinear butcoplanar), the spin-only group isPso = {1,m⊥}. (21)The mirror operation m⊥ shares its plane with the spins spanning the two-dimensional plane. Lastly, in the three-dimensional case (Dsp = 3, noncoplanar), the spin-only group trivially consists of only the identity operation;Pso = {1}. (22)Note that the orbital time-reversal symmetry is preserved in collinear magnets as well as coplanar magnets due tothe spin-space mirror operation. By using the spin-only group, the spin space group is decomposed as [46]G = Gso × G. (23)As a result, we obtain the nontrivial spin space group G whose spin-space operation W is intimately coupled to theorbital-space operation h = (R, t) such as the combined operation of spin rotation and translation g = ((1, t),W );i.e., the symmetry operation ((R, t),W ) with W ̸= 1 satisfies (R, t) ̸= (1,0).The set of spin translation operations {((1, t),W )} in G forms the group which we denote the nontrivial spintranslation group Gst. When the spin order does not modify the paramagnetic unit cell, the nontrivial spin translationgroup is reduced to the translation group T = {((1, t), 1)}. The nontrivial spin translation group is a normal subgroupof the nontrivial spin space group (G ▷ Gst). Thus, we decompose G by Gst asG =⋃igi Gst. (24)The representative gi = ((R, t) ,W ) indicates that the spin-space operation is coupled to the orbital-space pointgroup operations otherwise it is the identity (W = 1); in other words, the orbital-space point group operation in giis nontrivial (R ̸= 1) except for the identity operation ((1,0), 1). Note that one can obtain another decomposition ofthe spin space group by the spin translation group defined by Gst = Gso × Gst (see also Appendix A). To corroboratethe effect of the spin-structure dimension Dsp on emergent responses, we here utilize the decomposition of Eq. (23)in the following. The spin space group G for given crystal and spin structures can be computationally obtained [38].For instance, we consider the body-centered-cubic Fe (space group Im3̄m, No. 229) whose ferromagnetic spinpolarization is along the [001] axis. The magnetic space group G = I4/mm′m′ indicates the spontaneous crystallinesymmetry reduction from the cubic to tetragonal under the SOC effect. On the other hand, the spin space group Gretains high symmetry in the ordered state. The spin space group comprises the spin-only group Gso given by Eq. (20)with the spin-space axis n = [001]. By dividing G by the spin-only group, we obtain the nontrivial spin space groupG [Eq. (23)]. Since the ferromagnetic order does not modify the unit cell, the spin translation group is the translationgroup Tbcc for the bcc centering. Then, the nontrivial spin space group isG =⋃igi Gst =⋃igiTbcc, (25)where {gi} forms the cubic point group same as that for the paramagnetic state (m3̄m). As a result, the nontrivialspin space group is the same as paramagnetic one G = Im3̄m (each spin-space operation is the identify operation andhence omitted), and the overall spin space group symmetry is given byG ={(h,W ) | h ∈ Im3̄m, W ∈ SO(2)⋊ {1,m∥}}. (26)7The orbital-space cubic symmetry is intact in its spin space group. The symmetry restoration results from releasingthe system from the SOC constraint of Eq. (18). The restoration is observed for many magnetic materials as well asthe ferromagnet.The macroscopic physical properties are of our interest, and thus it is enough to take into account the spin pointgroup given by ignoring the translation operations asP(G) = {(R,W ) | ((R, t) ,W ) ∈ G}. (27)According to Eq. (23), the spin point group is similarly decomposed asP(G) = Pso × P. (28)In the right-hand side, P is derived from the nontrivial spin space group G similarly to Eq. (27).Following the convention in Ref. [44], the spin point group is denoted by the paired operations WR for g = (R,W ).For example, when the spin point group is obtained such as22/−1m, (29)it is generated by a set of operations(2, 2) , (m,−1) . (30)The orbital point group symmetry is given by 2/m whose two-fold rotation and mirror reflection are connected withthe spin-space two-fold rotation and space-inversion, respectively.Bearing in mind that the time-reversal operation is related to improper rotations in the spin space, we reduce agiven spin crystallographic point group to the point groups consisting of either spin-space or orbital-space operations.The spin-space part is defined byPsp(P) = {W | (R,W ) ∈ P}, (31)where the orbital-space operations (R) are irrelevant. The orbital-space part corresponds to a well-known magneticpoint group and similarly reads asPorb(P) = {RW | (R,W ) ∈ P}, (32)where RW = R for detW = +1 and RW = R′ ≡ θR for detW = −1; e.g., RW = 1′ = θ is the orbital time-reversaloperation. We again note that the improperness of the spin-space operation W should be incorporated into theorbital-space symmetry to respect the effect of the time-reversal operation. Similarly to the case with SOC, we referto the orbital part as either colorless, gray, or black-white in terms of the orbital time-reversal operation RW = 1′(see Appendix A). For the example of Eq. (29), the spin-space and orbital-space point groups are respectively givenbyPsp(P) = 2/m, (33)andPorb(P) = 2/m′. (34)The obtained orbital part is a black-white group.To demonstrate the role of spin-group symmetry analysis, it is better to make a comparison to the conventionalanalysis based on magnetic point groups with the SOC effect. The spin-orbital-coupled (SO-coupled) magnetic pointgroup P is derived from P by respecting the SOC constraint of Eq. (18). Corresponding to Eq. (30), we obtain thecolorless magnetic point groupP = 2, (35)where no operation involving the time-reversal operation exists in contrast to the black-white point group of Eq. (34).The series of magnetic symmetry is summarized in Fig. 1(b,c).We aim to identify the physical phenomena emerging from the spin order without relying on SOC, and the seriesof point groups (P,Psp,Porb,P) are convenient; The orbital (spin) part of the spin point group suffices to analyze thesymmetry of the object in the orbital (spin) space, while that of the SO-entangled object is determined by the overallspin point group. For instance, recalling the transformation property of objects depicted in Fig. 2, each transformationin Eq. (5) is sufficiently described by the group of Eq. (31) for spin-space objects and of Eq. (32) for orbital-spaceobjects, respectively. On the other hand, for an example of SO-coupled operator, the spin current (Jsba ∼ {pa, sb}/2)manifests the transformation property given by the direct product of representation matrices as D̂(p)(g) × D̂(s)(g).Thus, the spin point group [Eq. (27)] is indispensable to describe the representation matrix for Jsba . In the followingparts, we raise examples of spin crystallographic groups. The group symmetry is identified by the computationalmethods proposed in Refs. [38, 41, 42], and hence we do not show explicit derivations.8III. SPIN-GROUP CLASSIFICATION OF PHYSICAL PROPERTYWe consider the physical properties of the magnetic materials with or without SOC on the basis of Sec. II. Firstly,we present the spin-point-group symmetry analysis of the linear response function as well as that with the SO-coupledmagnetic point group [37]. We classify the response into T-even and T-odd contributions, which are allowed withoutand with the time-reversal-symmetry breaking, respectively [47–50]. By generalizing the previous classification of dcresponses [47, 48], we present the classification taking account of the frequency dependence (Sec. IIIA).In particular, we identify two aspects of the spin-group symmetry analysis through the comparative study with themagnetic point group; (1) intact symmetry in the orbital space leads to vanishing responses irrelevant to the spindegree of freedom, while it is not the case for spin-related phenomena. (2) the nontrivial spin translation symmetrymakes the spin space highly symmetric even in the presence of the spontaneous spin order and hence severely forbidsspin-related phenomena such as the spin magnetoelectric effect and spin Hall effect. These contrasting circumstancesare facilitated to identify through the identification of the spin space group which can be comprised of the nontrivialspin translation group while preceding symmetry analysis is for the magnetic materials with the zero propagationvector [11, 12]. We also introduce multipolar degrees of freedom relevant to those responses. The identification of agiven physical property is based on the developed computational method.A. Response function and T-even/T-odd decompositionWe consider the linear response formula to illustrate the symmetry of the transport phenomena. The formula iswritten byXi(ω) = χXYij (ω)F(Y )j (ω), (36)where the physical quantities Xi, Yj and the force F(Y )j conjugate to Yj are in the frequency (ω) domain. In theframework of the linear response theory [51], we can derive the constraint on the response coefficient from thepreserved symmetry in a quantum-mechanical manner [52]. Applying the symmetry operation g of a given pointgroup G, we obtain the symmetry constraintχXYij (ω) = χXYkl (ω)D(X)ki (g)D(Y )lj (g), (37)for the unitary operation andχXYij (ω) = χY Xkl (ω)(D(Y )kj (g))∗ (D(X)li (g))∗, (38)for the anti-unitary operation. We introduced the representation matrices for Xi and Yj as in Eq. (4). The anti-unitary symmetry relates the response function χXYij with χY Xji for the inverse response Yj = χY Xji F(X)i , in whichthe force F (X) is required to be conjugate to X. When the current participates in the response such as the electricconductivity Ji = σijEj , it is convenient to rewrite the response function by the canonical correlation function. Thesymmetry argument is similarly described for the canonical correlation (see Appendix B).Furthermore, one can decompose the response into the symmetric and antisymmetric parts [48]. The Lehmannrepresentation of the response function isχXYij (ω) =∑abρa − ρbω + iη + ϵa − ϵb⟨a |Xi | b⟩ ⟨b |Yj | a⟩ , (39)≡∑abρabω + iη + ϵabXiabYjba, (40)with the adiabaticity parameter η = +0 and with ρab = ρa − ρb. The indices a, b are for the eigenstates of themany-body Hamiltonian in equilibrium, ϵa is the eigen-energy, and ρa is the Boltzmann factor parametrized by ϵa.The symmetric (s) and antisymmetric (a) parts are defined by dividing the prefactor intoρabω + iη + ϵab=ω + iη(ω + iη)2 − ϵ2abρab −ϵab(ω + iη)2 − ϵ2abρab, (41)= κaab + κsab. (42)9These terms show the odd or even parity under the permutation of indices (a, b), and thereby we obtain the decom-position as χXYij = χXY,sij + χXY,aij . Similarly partitioning the product of matrix elements of Xi and Yj ,XiabYjba =12(XiabYjba +XibaYjab)+12(XiabYjba −XibaYjab), (43)≡ {Xi, Yj}ab + [Xi, Yj ]ab . (44)After the summation over (a, b), the surviving terms are κsab {Xi, Yj}ab and κaab [Xi, Yj ]ab. It indicates that thesymmetric and antisymmetric parts of the indices (a, b) are respectively the symmetric and antisymmetric terms withrespect to the permutation of the response and field (Xi, Yj). As a result, the symmetric and antisymmetric parts ofχXYij are recast asχXY,sij (ω) =12(χXYij (ω) + χY Xji (ω)), (45)χXY,aij (ω) =12(χXYij (ω)− χY Xji (ω)). (46)When the time-reversal symmetry is intact, we obtainXiabYjba = θXθY Xib̄āYjāb̄, (47)where ā is the time-reversal partner for the eigenstate a and θX is the parity of X under the time-reversal operation.Since the paired states have the same energy (ϵa = ϵā), it is shown that only the symmetric part survives for the caseof θXθY = +1 while the antisymmetric part does when θXθY = −1. Once the time-reversal symmetry is lost, weobtain the other contributions, that is, the symmetric term for θXθY = −1 and the antisymmetric for θXθY = +1.We label the contributions allowed in time-reversal-symmetric systems by T-even contributions and those arisingfrom the time-reversal-symmetry breaking by T-odd contributions. Note that one can utilize the orbital time-reversalsymmetry g = (1,W ) (detW = −1) instead of the time-reversal symmetry g = (1,−1), when X and Y are in theorbital space. Keeping the symmetric and antisymmetric decomposition of Eqs. (45) and (46) in mind, the T-evencontribution gives rise to χXY,sij for θXθY = +1 and χXY,aij for θXθY = −1. On the other hand, the T-odd contributionis complementary to the T-even, that is, χXY,sij for θXθY = −1 and χXY,aij for θXθY = +1.The symmetric-antisymmetric partition and even-odd classification with respect to the time-reversal operationimply the frequency dependence of the response. To be more specific, each term is even- or odd-order in the frequencyω as χ̂a ∼ ω2n+1 and χ̂s ∼ ω2n in the limit of η → 0 [Eq. (42)]. Considering the static limit (ω → 0), two partssimilarly indicate the dependence on the relaxation time τ . This can be intuitively understood by replacing theadiabaticity parameter η with the phenomenological scattering rate as η → τ−1. For instance, the antisymmetric partis recast asχXY,aij →∑a,b−iτ−1τ−2 + ϵ2ab[Xi, Yj ]ab , (48)whose equi-energy matrix elements give rise to contribution ∼ τ1 such as the Drude term of electric conductivity. Theantisymmetric term leads to the term O(τ2n+1) which may be characteristic of the transport phenomena in metals,and the symmetric term corresponds to the contributions as large as O(τ2n) including what may appear in insulatorssuch as anomalous Hall conductivity. In some cases, the dc antisymmetric term is labeled by an extrinsic (dissipative,absorptive) effect, while the symmetric is intrinsic (dissipation-less, reactive) [47, 48].It is noteworthy that the symmetric responses are related to equilibrium properties of materials, that is physicalquantities one can observe without dissipation, in some cases. When we assume equilibrium conditions for Eq. (36),that is, the dc limit and zero antisymmetric contribution, the remaining term is solely symmetric and satisfiesχXYij = χY Xji , (49)The symmetry of (Xi, Yj) implies the phenomenological free energy given byFXY = −χXYij FXi FYj . (50)The relation of Eq. (49) is reproduced by the free energy because Xi = −∂FXY /∂FXi and Yi = −∂FXY /∂FYj .The discussion can be applied to various equilibrium properties such as piezoelectric, piezomagnetic, magnetoelectriceffects, and so on [37].10In Table I, we summarize the classification in terms of the symmetric and antisymmetric parts. We also list someexamples of the T-even/T-odd classification by taking (X,Y ) = (J ,E) (electric conductivity), (M ,E) (magneto-electric effect [53], magneto-galvanic effect [54–56]), (ε̂,E) with strain εij (piezoelectric and magneto-piezoelectriceffect), and (ε̂,H) (piezomagnetic and kinetically-piezomagnetic effect). The T-odd contribution for (ε̂,E), calledmagnetopiezoelectric effect, has recently been proposed by theories [48, 57] and demonstrated in experiments [58–60].TABLE I. Classification of the response function by symmetric and antisymmetric parts. The frequency dependence for theac response χ̂(ω) and relaxation-time dependence for the dc response χ̂dc are listed. The symmetric and antisymmetric termsare further categorized by the T-even and T-odd contributions in the light of the total time-reversal parity θtot = θXθY . Wetabulate some specific classifications for (X,Y ) denoted by the pair of response X and field Y .antisymmetric symmetricχ̂(ω) ω2n+1 ω2nχ̂dc τ2n+1 τ2nθtot +1 -1 +1 -1T-odd T-even T-even T-odd(X,Y )(J ,E) Drude Hall(M ,E) magnetogalvanic magnetoelectric(ε̂,E) magnetopiezoelectric piezoelectric(ε̂,H) kinetically-piezomagnetic piezomagneticWe introduce the symmetry of response functions based on the unitary and anti-unitary properties of symmetryoperations without specifying the group. Thus, the symmetry argument works in the case with and without SOC. Wealso note that the decomposition plays a powerful role in the nonlinear response as well [61–63]. In the same spirit ofthe T-even/T-odd decomposition, theoretical studies have been presented in a diagrammatic fashion [64, 65].B. Geometrical Hall effect and spin/orbital magnetizationWe revisit the relation between the Hall response and the magnetization from the viewpoint of the spin crystallo-graphic group. The Hall response reads asJi = ϵijkσHk Ej , (51)where the Hall conductivity may be classified into three parts σH = σn+σKL+σg [66]; Normal Hall effect σn allowedunder the external magnetic field, Karplus-Luttinger (KL) Hall effect σKL, and geometrical (spontaneous, topological)Hall effect σg. The latter two contributions result from the magnetic ordering and are therefore summarized to theanomalous Hall effect, while they differ with respect to the role of SOC [10, 30]. The KL Hall effect can appear in thepresence of SOC as investigated in diverse magnetic materials such as ferromagnets, those with weak ferromagnetism,and compensated collinear and coplanar antiferromagnets [67–72]. The contribution may be appreciable in systemshaving large uniform spin magnetization (Msp) as observed that the SOC-assisted anomalous Hall conductivity istypically proportional to their uniform magnetization [73]. We note that the empirical rule is not applicable to someseries of antiferromagnetic materials such as Mn3Sn. For instance, the magnetic multipolar fields offer the anomalousHall response with the help of SOC but without the net magnetization [74]. In contrast, the geometrical Hall effect ischaracteristic of noncoplanar magnets whose geometrical texture of spins allows quasiparticles to be deflected withoutthe help of SOC. We identify the anomalous Hall effect driven by the spin order without SOC as the geometricalHall effect by referring to the nontrivial geometrical texture irrespective of intrinsic or extrinsic cause, which is anoncoplanar spin structure. The definition covers the known mechanism for the SOC-free anomalous Hall effect suchas that induced by the fictitious magnetic flux arising from the spin order [32].The Hall conductivity σH is an axial and time-reversal-odd vector defined in the orbital space, coinciding withthe symmetry of the orbital magnetization Morb. Then, we can verify the anomalous Hall effect by Morb of a givenmagnetic material. Beyond the symmetry, the correlation between orbital magnetization and anomalous Hall effectcan be found as clarified by the well-known Středa formula [75]. The orbital magnetization may cover a broad range ofmaterials hosting the anomalous Hall effect such as systems with orbital flux [31] and Graphene-based ferromagneticsystems [76]. The symmetry-adapted form of Morb is computationally identified by Eq. (4) with a given magneticsymmetry. We note that the orbital magnetization exists while the localized magnetic moments are attributed to thespin degree of freedom [Eq. (10)] with quenched atomic orbital angular momentum.11It is of paramount interest how the KL and geometrical terms are distinguished since the distinction clarifies theSOC effect on emergent physical responses. For instance, the geometrical effect has been intensively studied in earlyworks; e.g., those with the pyrochlore ferromagnets such as Nd2Mo2O7 [32, 34, 66, 77]. Nd2Mo2O7 undergoes theferromagnetic order of Mo atoms and subsequently the noncoplanar magnetic order of Nd atoms as temperaturedecreases. The two magnetic states with different spin-structure dimensions are labeled by the same magnetic pointgroup in the SO-coupled case. Two types of anomalous Hall responses therefore cannot be distinguished by thesymmetry in the conventional context. This is, however, not the case in the framework of the spin crystallographicgroup.Considering the ferromagnetic Fe of Eq. (26), we derive the orbital part of the spin point group Porb by Eq. (32)to identify the symmetry of orbital magnetization Morb dwelling on the orbital space. The obtained Porb is a graygroup written byPorb = m3̄m1′, (52)in which the orbital time-reversal symmetry (1′) comes from the spin-space mirror symmetry in the spin-only groupof Eq. (20). It indicates the zero orbital magnetization and vanishing anomalous Hall effect which are odd-parityunder the orbital time-reversal operation, consistent with the symmetry analysis presented in Ref. [12]. On the otherhand, once the SOC is switched on, the ferroic spin magnetization is admixed with the orbital magnetization andmanifests the favorable direction with respect to the crystal axes. The resultant point group symmetry is reduced tothe tetragonal magnetic point group P = 4/mm′m′ allowing for the spin-orbital-entangled magnetization along thefour-fold rotation axis as derived in the established symmetry analysis [37].As a result, the Hall response of Fe is attributed to the KL contribution (σKL ̸= 0, σg = 0). This argument canbe applied to spin space groups for all the one- and two-dimensional spin configurations [Eqs. (20),(21)]. Then, ifthe magnetic moments spanning low-dimensional structure are supposed to originate from the spin magnetization, itcan be said that the ordered state preserves the orbital time-reversal symmetry. It follows that the correspondingorbital-space point group is gray in terms of the magnetic point group. It similarly indicates the absence of physicalphenomena originating from the violation of the orbital time reversal symmetry such as the orbital piezomagneticeffect (Sec. III) and orbital magnetoelectric effect (Sec. III C).FIG. 3. (a) Crystal (left panel) and spin (right panel) structures of CoTa3S6. (b) The tetrahedron spanned by four spins inthe magnetic structure. The black lines denote the two-fold rotation axes relevant to the spin translation group.The anomalous Hall response does not suffer from such a severe symmetry constraint in the case of the noncoplanarmagnets because of the trivial spin-only group [Eq. (22)]. It is noteworthy that the nontrivial spin translationsymmetry realizes the orbital magnetization not admixed with the spin counterpart. Let us consider a layeredmaterial CoTa3S6 for an example [78, 79] (Fig. 3). After the computational search for the magnetic symmetry [42],the magnetic space group symmetry is identified toG = P32′. (53)The associated SO-coupled magnetic point group P = 32′ leads to the conclusion that the spin (Msp) and orbitalmagnetization (Morb) can concurrently show up along the three-fold rotation axis. Thus, we cannot distinguish theKL and geometrical contributions to the Hall effect within the conventional magnetic point group analysis.12Next, We consider the spin-crystallographic-group symmetry identified by the method proposed in Ref. [38]. Owingto the noncoplanar spin structure, the spin-only group is trivial (Pso = {1}), and thereby the spin space group iscoincident with the nontrivial spin space group, G = G in Eq. (23). For the (nontrivial) spin translation group, wesimilarly obtain Gst = Gst. The spin space group is written by the internal semidirect product of the spin translationgroup Gst and the remaining part H;G = Gst ⋊H. (54)Since we are interested in the macroscopic physical property, it is sufficient to take into account the point groupsymmetry. The point group is obtained from Eq. (54) asP = Pst ⋊ PH. (55)The point groups Pst and PH are respectively derived from Gst and H as in Eq. (27). The spin-translation part Pstgives rise to the spin-only-group symmetry written byPst = {(1,W ) |W ∈ {1, 2X , 2Y , 2Z}} . (56)The point group 222 = {1, 2X , 2Y , 2Z} is given by the mutually-orthogonal two-fold rotation axes (X,Y, Z) by whichfour orientations of Co spins are interchanged [Fig. 3(b)]. The remaining part isPH =3 6 m(100)2 m(010)2. (57)We notice that the six-fold symmetry of the crystal (space group P6322, No. 182) is intact in the ordered phasewithout the SOC effect.We are interested in the spin and orbital magnetization which are related to the KL and geometrical Hall effects,respectively. Then, it is enough to consider the spin and orbital parts projected from P as in Eqs. (31) and (32). Thespin part isPsp(P) = 222⋊ 3m = 4̄3m, (58)and the orbital part isPorb(P) = 62′2′. (59)The spin translation symmetry of Eq. (56) leads to the cubic symmetry Psp = 4̄3m despite the hexagonal crystalstructure of CoTa3S6. The spin point group symmetry enhanced by the spin translation symmetry has not beenaddressed in previous studies of emergent responses. On the other hand, the orbital part manifests the axial symmetrywhose rotation axis is [001] similar to the SO-coupled case of Eq. (53). Using the spin and orbital parts in the spinpoint group, we identify the allowed spin and orbital magnetization,Msp = 0, Morb ∥ [001]. (60)As a result, the anomalous Hall conductivity vector σH ∥ [001] can appear without the help of SOC (σg ̸= 0).Furthermore, the zero spin magnetization follows from the cubic symmetry in the spin space. These propertiesindicate that the anomalous Hall effect of CoTa3S6 can be attributed to the geometrical Hall effect (σg) and that theKL contribution (σKL) may be suppressed due to vanishing spin polarization since it is empirically expected to beproportional to the uniform magnetization.We have clarified two characteristic aspects of spin group symmetry in this section. The spin-only group associatedwith low-dimensional spin configuration gives rise to strong constraints on the orbital-space objects, forbidding physicalresponses arising from the violation of the orbital time-reversal symmetry. On the other hand, the noncoplanar spinstructure may lead to such emergent responses from orbital degrees of freedom, while the responses relevant to the spindegree of freedom may vanish due to the high symmetry of the spin space originating from the spin translation group.These contrasting situations are systematically understood by the spin-crystallographic-group symmetry analysisincorporating more details of spin structures such as the spin-structure dimension Dsp and spin-translation symmetrybeyond the magnetic space group.C. Magnetoelectric effect and spin/orbital magnetic quadrupole momentsThe spin-crystallographic-group symmetry analysis distinguishes the role of spin and orbital degrees of freedom inthe magnetoelectric effect because it separately identifies the spin and orbital magnetization as in Eq. (60). We hereconsider the correlation between magnetization and electric polarization written byPi = χPMij Hj , Mi = χMPij Ej . (61)13The frequency dependence is suppressed. In particular, the symmetric term is called magnetoelectric effect and theantisymmetric is the inverse magneto-galvanic effect [53, 55] (see also Table I). In the following, we focus on theDC magnetoelectric effect αij = χMP,sij (ω = 0), which appears even in systems with no electric conductivity, that is,insulators at the zero temperature. The magnetoelectric effect is further divided into the spin and orbital parts asαij = αspij + αorbij . (62)where the spin and orbital magnetization participate in the response, respectively. In the light of the spin crystal-lographic group, the symmetry of spin and orbital magnetoelectric effect is respectively determined by the whole ofand orbital part of the spin point group, since the former is a spin-orbital-coupled response and the latter consists ofonly the orbital degree of freedom.Firstly, we consider a prototypical magnetoelectric material Cr2O3 [80, 81] [Fig. 4(a)]. Its collinear antiferromagneticorder does not break translation symmetry due to the zero propagation vector of the spin configuration, and thenontrivial spin translation group is equal to the translation group for the paramagnetic state as Gst = T . Supposingthat the spins are aligned to the [001] axis in the spin space, the spin crystallographic point group isP = Pso × P, (63)where the spin-only group Pso is for the one-dimensional spin configuration [Eq. (20)]Pso = SO(2)⋊{1,m∥}. (64)The remaining part isP =3̄ 3̄mm. (65)The spin part associated with P manifests centrosymmetric point group symmetry asPsp(P) = O(2)⋊ {1,m∥}, (66)and the orbital part is given by a centrosymmetric gray groupPorb(P) = 3̄m1′, (67)which differs from the noncentrosymmetric black-white point group P = 3̄′m′ for the SO-coupled case.The absence of orbital contribution (αorbij = 0) follows from either of the space-inversion or time-reversal symmetryin the orbital part of the spin point group [Eq. (67)] because of the odd parity under those operations [53]. On the otherhand, when taking into account the spin-space proper rotations, the spin point group of Eq. (63) does not preservethe time-reversal or space-inversion symmetry as −11,1 −1 ̸∈ P and may allow for finite spin magnetoelectric effect.The symmetry of the spin contribution is obtained as follows. In the SO-coupled case with the magnetic point groupP = 3̄′m′, the allowed SO-entangled magnetoelectric effect is given by all the diagonal components αxx, αyy, αzz [37].In the absence of SOC, one should consider additional constraints due to the spin-only group [Eq. (64)]. The SO(2)symmetry in the spin-only group forbids the spin polarization response transverse to the collinear axis (αspyj , αspzj = 0)but allows the longitudinal as αspzj ̸= 0. The symmetry analysis is summarized asαspzz ̸= 0 otherwise αspij = 0, αorbij = 0. (68)As a result, only the longitudinal magnetization can respond to the applied electric field in a SOC-free manner andis purely ascribed to the spin origin.Although we assumed the DC case, the symmetry analysis similarly holds for the AC responses. The AC mag-netoelectric effect denoted by χMP,hij (ω) triggers the nonreciprocal optical activity [82–84]. Owing to the effectivespace-inversion (W 1̄ with detW = 1) and orbital time-reversal symmetry (W 1 with detW = −1) in the spin pointgroup of Eq. (63), the optical activity arises solely from the spin magnetic-dipole transition but does not include theorbital magnetic-dipole or electric-quadrupole effects in the absence of SOC.Next, we again consider CoTa3S6 to demonstrate the role of spin translation symmetry in the magnetoelectric effect.Similarly to zero spin magnetization, the cubic spin-space symmetry [Eq. (58)] forbids the spin magnetoelectric effect,αspij = 0. On the other hand, the orbital part of Eq. (59) leads to finite orbital magnetoelectric effects αorbxy = −αorbyx .Then, the magnetoelectric effect of CoTa3S6 is summarized asαspij = 0, αorbxy = −αorbyx ̸= 0. (69)14In the SO-coupled point group symmetry of Eq. (53), αxy = −αyx is similarly allowed while the spin effect is admixed.The symmetry analysis shows the possibility of the SOC-free magnetoelectric effect dominated by the orbitalcontribution. Among known mechanisms for the magnetoelectricity [85], the identified response may originate fromthe exchange striction mechanism [86, 87] and the dynamical phase [88] which do not require SOC. Note that we herediscussed the orbital magnetoelectric effect induced by the noncoplanar spin order [89] rather than that what arisesfrom the orbital-current order [90].We took the overview of the relation between the anomalous Hall effect and orbital magnetization in Sec. III B.A similar discussion can be found in the case of the magnetoelectricity; the response may be correlated with higher-order anisotropy of magnetic charge, that is, the magnetic quadrupole moment Qij [91, 92]. The symmetry of Qij isschematically given by the tensor product of the magnetization and position as Qij ∼ Mirj . According to the space-time symmetry, we can find the correspondence between the magnetic quadrupole moments and the magnetoelectriceffect given byQij ↔ αij . (70)The magnetization Mi can be classified into the spin and orbital contributions in terms of the spin crystallographicgroup. Then, the symmetry analysis of the allowed magnetoelectric effect can be reproduced by identifying therelevant spin/orbital magnetic quadrupole moments with the use of Eq. (4).For instance, let us consider the multipolar degree of freedom corresponding to the magnetoelectric effect of CoTa3S6.The allowed multipole moment is the orbital toroidal moment Qorbxy −Qorbyx ∼ (Morb × r)z where the toroidal momentis a time-reversal-odd polar vector. The symmetry of the toroidal moment is consistent with that of the orbital mag-netoelectric effect in Eq. (69). The toroidal moment polarized along the [001] direction can be intuitively understoodby its orbital magnetization and crystal structure. The space group symmetry of CoTa3S6 (No. 182, P6322) does nothave any improper rotation symmetry in the orbital space [93], and hence every axially symmetrical quantity can becoupled to the corresponding polar-symmetry quantity with preserving the time-reversal parity. In the present case,the orbital magnetization (time-reversal-odd axial vector) is coupled to the orbital toroidal moment ( time-reversal-odd vector) as in the case of magnetochiral anisotropy [94] [Fig. 4(b,c)]. In the present case, the orbital magnetization(time-reversal-odd axial vector) is coupled to the orbital toroidal moment (time-reversal-odd polar vector) as in thecase of magnetochiral anisotropy [94] [Fig. 4(b,c)]. The coupling between the orbital magnetization and toroidalmoment is a consequence of its chiral crystal structure, and the ligands surrounding magnetic Co atoms play essen-tial roles. We checked that the noncoplanar spin structure of Co atoms does give rise to orbital magnetization butmanifests no orbital toroidal moment without Ta and S atoms. Interestingly, beyond the symmetry analysis, recenttheoretical studies identified that the magnetic quadrupole moment [95–98] covers not only the magnetoelectric effectbut also other cross-correlated responses [99] when the system is insulating.FIG. 4. (a) Crystal and spin structures of Cr2O3 where spins are collinear along the [001] direction. (b,c) Chirality of thecrystal structure of CoTa3S6 implied by TaS6 prism with the twisted coordination of Co atoms. The twisting arrangementdetermines the chirality χ = ±1. When orbital magnetization (blue-colored vector) is formed along the [001] direction due toits noncoplanar spin ordering, (b) the toroidal moment (red-colored vector) is anti-parallel to it, (c) while it is parallel with theopposite chirality.15D. Spin Hall response and rotatorsLet us consider another spin-related response, the electric-field (Ej) induction of spin-polarized current Jski . Thespin-polarized current may be given by Jski = {Ji, sk}/2 which is comprised of the spin- and orbital-space objects.The response formula is given byJski = σkijEj . (71)The parity under the time-reversal symmetry is θJsθE = +1, and the T-even contribution is symmetric while theT-odd is antisymmetric according to the classification in Sec. III A. For the Hall response (ϵijpσkij) in the DC limit, theT-even effect includes the well-known spin Hall effect prominent in the spin-orbit-coupled semiconductors [100, 101],while the T-odd called the magnetic spin Hall effect is unique to magnetic metals [11–13, 102–104]. The absence ofSOC and spin order, indicating the isotropic spin-space symmetry, leads to the vanishing response.We refer to the symmetry analysis of Refs. [11, 12] and decompose σkij into the T-even and T-odd components. Thetarget material, a noncollinear but coplanar magnet Mn3Sn, is attracting a lot of attention because of its potentialapplication for spintronic and magneto-optical components [Fig. 5(a)] [10, 73]. The spin crystallographic point groupis given byP = Pso × P. (72)The spin-only group isPso ={(1,W ) |W ∈ {1,m(001)}}, (73)for a two-dimensional spin structure. The remaining part is [24]P =3 6/1mm(120)mm(110)m, (74)where the spin configuration is taken to preserve the spin point group symmetry for g =(2[100], 2[100]). The spin partassociated with P isPsp(P) = m(001) × 3m = 6̄2m, (75)and the orbital part isPorb(P) = 6/mmm1′, (76)coinciding with the SO-coupled magnetic point group for the paramagnetic state, while the SO-coupled magnetic pointgroup for the magnetic state shows the crystal-class reduction from hexagonal to orthorhombic as P = mm′m′. Thus,as far as only the orbital degrees of freedom are concerned, the antiferromagnetic state of Mn3Sn does not show anyphysical phenomena arising from the symmetry breaking. On the other hand, owing to the spontaneously-emergedanisotropy in the spin space, the electric field can stimulate the spin-polarized current. The response is explicitlygiven byσzxy = −σzyx, (77)for the T-even contributions andσxxx = σyxy = σyyx = −σxyy, (78)for the T-odd components. These spin-orbit-free components may overwhelm those requiring the SOC effect [11, 12].Following the discussion parallel to those in the previous sections III B and III C, the symmetry analysis is applicableto more complex spin structures and can be extended to cover the orbital counterpart such as the orbital-current Halleffect, which is denoted with the current whose magnetic polarization is attributed to the orbital origin [105–107]. Forinstance, CoTa3S6 with the cubic spin-space symmetry [Eq. (58)] leads to the zero spin-current response σk,spij = 0.The orbital counterpart, however, is allowed even without SOC such as the T-odd contributions σz,orbxx , σz,orbyy , andσz,orbzz .We further consider the quantities relevant to the T-even/T-odd spin Hall responses by the analogy of the SOCHamiltonian [50]. Recalling the expression for the atomic SOC of Eq. (17), one can replace the orbital angularmomentum L ∼ r × p with the cross product of the electric field and current E × J , since the space-time symmetry16FIG. 5. (a) Spin structure of Mn3Sn. (b) Transverse conversion between the charge and spin currents denoted by the rotatorRij . The Hall plane perpendicular to the xi direction is for the current whose spin is polarized along the xj direction in thespin space.is same in pairs of vectors (r,E) and (p,J). Then, the relativistic spin-orbit interaction may correspond to the spinHall response asL · s ∼ (J ×E) · s = ϵijkEjJisk ↔ σkij = ϵijkσ0, (79)where ϵijkσ0 indicates the spin Hall effect whose spin polarization is perpendicular to the Hall plane defined by theE and J [Fig. 5(b)]. That is why the spin Hall effect generically exists under the SOC effect. The correspondencebetween the T-even spin Hall effect and the product of Li and sj may be generalized to that in the framework of thespin crystallographic group. Then, we here introduce the T-even rotator Reij giving the transverse correlation betweencharge and spin currents. The symmetry of Reij coincides with the product of the time-reversal-odd axial vectors Liand sj which are defined in the orbital and spin space, respectively. The T-even rotator may be attributed to thespin-resolved Berry curvature playing a crucial role in the intrinsic spin Hall effect. The T-even rotator denotes theHall response for the xj-polarized spin current denoted by the Hall plane perpendicular to the i-direction [Fig. 5(b)],Reij ↔ ϵiabσjab (80)Specifically, the trace∑i Reii corresponds to Eq. (17). For instance, referring to the spin crystallographic point groupof Mn3Sn [Eq. (72)], we identify the T-even rotator Rezz ̸= 0 corresponding to the spin Hall response of Eq. (77) wherethe z = (001) Hall plane is obtained as (E × J)z and the spin current is polarized along the z-direction. The T-evenspin Hall effect is a response characteristic of noncollinear spin systems under no SOC effect [12] as we corroboratein Sec. IV.It is straightforward to derive the similar quantity relevant to the magnetic spin Hall effect, that is, T-odd rotatorRoij ∼ Lisj with the time-reversal-even axial vector Li defined in the orbital space [50]. The symmetry of the T-oddrotator agrees with the spin-current vorticity clarified in a recent theoretical study [108]. In the case of Mn3Sn, theT-odd rotator is absent without the SOC effect (Roij = 0). This is consistent with the symmetry analysis of Eq. (78)whose field and response can be longitudinal to each other.The symmetry analysis based on rotators can be applied to spin-current responses to another stimulus such as thetemperature gradient, as long as the replaced field shares the same symmetry as the electric field, that is a time-reversal-odd polar vector. Then, the high-throughput symmetry analysis presented in the following section allows usto identify SOC-free spincaloritronic responses such as anomalous spin Nernst effect.IV. HIGH-THROUGHPUT SYMMETRY ANALYSIS OF SPIN GROUP SYMMETRYThe computational search for the spin space group allows us to identify physical properties free from the SOCeffect [38, 42]. We present symmetry analysis with dozens of observed spin configurations obtained from magn-17data [39, 40]. We have performed the symmetry analysis of 1512 magnetic materials which have no site disorder. Forthe spin-structure dimension, 914 collinear, 403 coplanar, and 195 noncoplanar spin systems are studied. The magneticmaterials are numbered by following the identification number provided in magndata such as Cr2O3 (# 0.59).In this section, we discuss the physical quantities such as spin/orbital magnetization and magnetic quadrupolemoments, and T-even/T-odd rotators introduced in Sec. III to investigate emergent physical phenomena. Providingsome examples of spin space group (G) with comparison to the analysis with magnetic space group (G), we investigatecharacteristic physical properties in the viewpoint of symmetry. Although the electromagnetic responses relying onnonrelativistic spin-charge coupling has been mainly discussed for spin structures with the zero propagation vector,our high-throughput symmetry analysis further identifies candidate materials offering intriguing physical phenomenaarising from a complex spin structure such as purely-orbital magnetoelectric effect and motivates us to revisit knownmaterials from the perspective of SOC-free responses.A. Spin crystallographic symmetryLet us classify the magnetic materials in terms of the spin crystallographic or magnetic space group symmetry. Sincethe spin space group comprises its corresponding magnetic space group as a subgroup, the orders of groups satisfythe relation |G|/|G| ∈ N = {1, 2, 3, · · · } where we consider the nontrivial spin space group G instead of G. Figure 6illustrates how many symmetry operations are restored by neglecting SOC. For instance, the maximal symmetryrestoration occurs in the case of a noncoplanar magnet CrSe (#2.35) [109]. The hexagonal crystalline symmetry(space group No. 194, P63/mmc) is intact for the nontrivial spin space group G, while the crystal class is reducedto the trigonal for the magnetic space group G. The nontrivial spin translation group as large as |Gst| = 3 alsocontributes to the higher symmetry of G [38].0 100 200 300 400|G|024681012|G|/|G|CollinearCoplanarNoncoplanarFIG. 6. Distribution of magnetic materials parametrized by the order of the magnetic space group (|G|) and by the ratiobetween the order of the nontrivial spin space group (|G|) and |G|. The distributions are categorized based on whether thespin-structure dimension is collinear, coplanar (but noncollinear), or noncoplanar.For a more detailed comparison, we classify the orbital symmetry of the spin point group (Porb) and the SO-coupledmagnetic point group (P) in terms of the time-reversal symmetry such as colorless, gray, and black-white groups (seeAppendix A). Owing to the spin-only group, the low-dimensional spin structure (Dsp = 1, 2) makes the orbital-spacesymmetry gray irrespective of its SO-coupled magnetic point group symmetry [see examples of Eq. (26) for Dsp = 1and Eq. (72) for Dsp = 2]. On the other hand, the noncoplanar system (Dsp = 3) can be characterized by any of threedifferent types of magnetic point groups. Table II shows the classification result. The absence of the SOC constraint[Eq. (18)] allows for the additional symmetry related to the time-reversal operation and hence some of the colorlessgroups among P are turned into black-white with respect to Porb.18TABLE II. Classification table of the SO-coupled magnetic point group (P) and the orbital part of the spin point group (Porb)for the noncoplanar magnets in light of the time-reversal symmetry. Note that all the Porb is gray for the collinear and coplanarmagnets irrespective of the type of P.Colorless Gray Black-WhiteP 46 46 103Porb 42 46 107We take some examples to compare the magnetic symmetry with and without SOC. For an example of low-dimensional spin structures, we consider a coplanar magnet Ba3MnSb2O9 (# 1.0.46). The material crystallizes in thestructure denoted by the centrosymmetric space group C2/m (No. 15) [110], and its coplanar spin structure (Dsp = 2)is formed by magnetic moments at Mn atoms [Fig. 7(a)]. No symmetry including the time-reversal operation existsin the magnetic space group G = C2 (type I) allowing for both of ferroelectric and ferromagnetic polarizations. Onthe other hand, such multiferroic property is missing if there exists no SOC effect. The spin space group reads asG = Gso × G. (81)The two-dimensional spin structure corresponds to the spin-only groupGso = {((1,0), 1) ,((1,0),m(001))}, (82)and the nontrivial part G is isomorphic to the space group C2/m. G is generated by((1, t), 1) ,((−1,0),m(010)),((2[010], t̃),m(100)), (83)in addition to the trivial translation operations associated with the monoclinic crystal structure. We here introducedthe translations t = (0.5, 0.5, 0) and t̃ = (0, 0, 0.5). When G is reduced to the spin point group P, the orbital partPorb is centrosymmetric and gray asPorb(P) = 2/m1′, (84)in contrast to the SO-coupled (P = 2). Consequently, the spin group symmetry forbids various physical phenomenaactivated by the time-reversal or space-inversion symmetry breaking.Next, we consider a noncoplanar magnet Mn3CuN (#2.5) [111]. The complex spin structure consists of magneticmoments at Mn sites having two different moduli [Fig. 7(b)]. The magnetic space group isG = P4/m, (85)from which the magnetic point group is P = 4/m. Thus, owing to the spin order and SOC, the cubic crystallinesymmetry (space group No. 221, Pm3̄m) is reduced to the tetragonal. The magnetic symmetry allows for themagnetization along the [001] axis.Then, let us consider its spin space group symmetry. With the non-zero propagation vector of the spin configuration,the spin space group includes the spin translation group Gst generated by((1, t), 2z) , (86)with t = (0.5, 0.5, 0) and by trivial translation operations without any spin-space rotations. Then, we obtain the cosetdecomposition of the spin space group asG =⋃igi Gst, (87)where the representatives gi are given by the identity and((−1,0), 1) ,((4+[001],0), 4+[001]),((m(010),0),mα),((m(100),0),mβ). (88)The mirror operation W = mα is depicted in Fig. 7(c). Importantly, the spin-space-group operations related tomα,mβ are preserved if without the SOC condition of Eq. (18) and makes the orbital part Porb black-white. Thespin point group is obtained asP =2[001] 14[001]4/1mmβmmγm, (89)19with the spin-space mirror operation W = mγ associated with the orbital-space mirror reflection R = m(11̄0). Ac-cordingly, we obtainPsp(P) = 4mm, (90)for the spin part, andPorb(P) = 4/mm′m′, (91)for the orbital part. The resulting black-white symmetry of Porb differs from the colorless magnetic point groupP = 4/m for the SO-coupled case.FIG. 7. Spin configurations of (a) Ba3MnSb2O9 with only Mn atoms and of (b,c) Mn3CuN. In (c), the spin space operation((m(010),0),mα)is depicted. The mirror operations W = mβ ,mγ can be similarly obtained.B. Emergent physical properties1. MagnetizationHere we consider the spin and orbital magnetization arising from the spin ordering. Although the uniform spinmagnetization trivially appears in the ferromagnetic materials, the orbital counterpart is severely forbidden due to thespin-only group in simple spin structures such as collinear and coplanar configurations (see Sec. III B). Then, we focuson the 195 noncoplanar magnets which may possess orbital magnetization. We show the classification concerningmagnetization in Table III. The classification also covers the magnetization identified by the magnetic symmetry(G,P) including the SOC effect, that is, the spin-orbital-entangled magnetization denoted by MSOC. When eithernonzero spin or orbital magnetization exists in a given spin space group, MSOC is similarly allowed due to the group-subgroup relation of G < G. Then, we classify the noncoplanar magnets into four classes in terms of magnetization;(M1) nonzero spin and orbital magnetization even without SOC, (M2) nonzero spin but zero orbital magnetizationwithout SOC, (M3) zero spin but nonzero orbital magnetization without SOC, and (M4) zero magnetization withoutSOC, but nonzero with SOC. We do not discuss the case of zero magnetization with and without SOC.The system with orbital-free spin magnetization (Class M2 of Table III) is trivial since such type of magnetizationcan be found in typical ferromagnetic materials as well. On the other hand, the spin-free orbital magnetization (ClassM3) indicates a nontrivial spin group symmetry hosting the orbital magnetization not to be concomitant with spinmagnetization. The materials of Class M3 are as follows; DyCrWO6 (# 0.316), CuB2O4 (# 0.431), Fe3F8(H2O)2(# 2.61), TbCrO3 (# 2.62), DyCrO3 (# 2.63, # 2.64), and MgCr2O4 (# 3.4). The candidate materials are mostlyinsulators in contrast to the noncoplanar magnetic metal CoTa3S6 discussed in Sec. III B. Thus, they may not bepromising candidates offering the geometrical Hall effect, whereas the orbital magnetization should participate insimilar phenomena for quasiparticles conductive in electrically-insulating materials such as phonon and magnon [112].The orbital magnetization also plays an important role in various magneto-optical phenomena such as Faraday ro-tations [113]. Interestingly, the optical response may be tolerant to extrinsic effects such as skew scattering yieldinganomalous Hall effect [30] and hence it may be a good test bed for investigating the intrinsic role of the orbitalmagnetization in emergent responses.We observe that CrSe (#2.35) shows the magnetization if and only if SOC is taken into account (Class M4). Thisis because the cubic symmetry of spin space group G is reduced to the trigonal magnetic point group under SOC.20We also notice that the spin and orbital contributions to the equilibrium properties may be distinguished even whenboth are allowed such as in Class M1 of Table III. As an example for Class M1 of Table III, the spin group symmetryof Mn3O4 (# 2.52) leads to the spin and orbital magnetizations given byMsp ∥ [010], Morb ∥ [001]. (92)The two perpendicular magnetizations get entangled with each other under the SOC effect, and the magnetizationcan be in the (100) plane. Note that we cannot determine the relative orientation of spin-space axes with respectto the orbital-space coordinate system without SOC. We, however, determined the spin-space axes of Eq. (92) byreferring to the spin configuration observed in experiments, and thus the peculiar relation between spin and orbitalmagnetization may give an implication; e.g., the anomalous Hall effect σxy may be larger than another transversecomponent σzx, because the former may be related to the orbital magnetization Morb ∥ [001] while the latter resultsfrom SOC.TABLE III. Classification of noncoplanar magnetic materials in terms of spin and orbital magnetization Msp,Morb. MSOCdenotes the magnetization under the SOC effect. “Num.” denotes the number of data of magnetic materials belonging to eachclass. The materials without any magnetization are not shown.Msp Morb MSOC Num.M1 ✓ ✓ ✓ 69M2 ✓ ✓ 2M3 ✓ ✓ 7M4 ✓ 1Total 79Finally, we comment on the relation between Class M2 and the spin scalar chirality. The noncoplanar nature canbe quantified by the spin scalar chirality vector C, which shares the same spin crystallographic symmetry as that ofthe orbital magnetization and geometrical Hall effect. The quantity is written byCi =∫drϵijks(r) ·[∂∂rjs(r)× ∂∂rks(r)]. (93)The importance of the spin scalar chirality has been explored in various materials such as Kagomé lattice [32, 34, 66,114, 115] and magnetic skyrmion crystals [35, 36].The spin scalar chirality may be defined in a lattice system asCi =∑△Sα · (Sβ × Sγ) ({α, β, γ} ∈ △) , (94)where three spins spanning the triangle (△) are in the same plane normal to the xi axis [116]. The triangular unit maybe hard to identify in general except for known examples such as layered material [78] and pyrochlore magnet [66, 77].On the other hand, our symmetry analysis allows us to identify the orbital magnetization Morb as well as the spinscalar chirality C in continuum space. It implies that our symmetry analysis unambiguously identifies the geometricalHall effect for complex magnetic materials where the discretely defined spin chirality may be hard to identify.2. Magnetic quadrupole momentWe consider the spin and orbital quadrupole moments Qspij and Qorbij . While the orbital contribution similarly doesnot show up without a noncoplanar spin structure, the spin contribution is of interest in the low-dimensional spinstructures. Furthermore, we can gain insight into relativistic effects on the quadrupole-mediated physical responsessuch as the magnetoelectric effect from the comparison between the magnetic quadrupole moments with and withoutSOC. Thus, we categorize the magnetic materials by the spin/orbital/SO-coupled (QSOCij ) quadrupole moments andby the spin-structure dimension (Table IV).Being consistent with the spin-only-group symmetry for Dsp = 1 and Dsp = 2, the quadrupole moments originatefrom only the spin degree of freedom (Class Q2) without SOC for low-dimensional spin structures, while both spinand orbital contributions are admixed (Class Q1) in noncoplanar case (Dsp = 3). Interestingly, only the orbital partis allowed in the SOC-free manner (Class Q3) for U3As4 (# 0.169), U3P4 (# 0.170), CrSe (#2.35), MgCr2O4 (# 3.4).21TABLE IV. Classification of magnetic quadrupole moments. The spin (Qsp), orbital (Qorb), and spin-orbital-coupled contri-butions (QSOC) are classified by the spin-structure dimension (Dsp = 1, 2, 3). The materials without any quadrupole momentsare not shown.Qsp Qorb QSOCDsp1 2 3Q1 ✓ ✓ ✓ 0 0 42Q2 ✓ ✓ 142 94 0Q3 ✓ ✓ 0 0 4Q4 ✓ 93 24 1Total 235 212 47It is noteworthy that the noncoplanar magnet CrSe possesses the orbital quadrupole moment without any uniformmagnetization (see Appendix C). The Class Q4 of Table IV indicates the system whose magnetic quadrupole momentappears if and only if SOC is included. The magnetoelectric property is inactive without the SOC effect in somecollinear antiferromagnets because of strong constraints from the spin-only group.The magnetoelectric effect related to magnetic quadrupole moment has been intensively studied mainly withcollinear magnets [53] and with incommensurate magnetic systems [85, 117]. It, however, has been rarely ex-plored for commensurate but noncollinear magnetic materials such as what manifests the orbital magnetoelectriceffect [89, 118, 119]. Thus, the present classification may lead us to a deep comprehension of the relation between thespin-structure dimension and the magnetoelectric effect.3. Rotators for the spin-polarized current responsesFinally, let us consider the T-even and T-odd spin-current rotators Re,oij . These quantities are comprised of thespin degree of freedom in j-th component of Rij and vanishes in the paramagnetic state without SOC. Under theSOC effect, the T-even rotator Reij is allowed in every system as in Eq. (79), while the T-odd rotator Roij requiresthe time-reversal symmetry breaking 2. Then, we identify candidate materials possessing the T-even rotator withoutSOC (Reij), T-odd rotator without SOC (Roij), and T-odd rotator under the SOC effect (RoSij ) for each spin-structuredimension (Table V).TABLE V. Classification of the spin-current rotators. Magnetic material data with the spin-structure dimension Dsp = 1, 2, 3are classified in terms of the T-even (Re), T-odd (Rorb), and SO-coupled T-odd rotators (RoS). The materials without anyrotators are not shown.Re Ro RoSDsp1 2 3R1 ✓ ✓ ✓ 0 115 100R2 ✓ ✓ 130 19 0R3 ✓ 123 42 5R4 ✓ 0 164 79R5 ✓ ✓ 0 5 4Total 253 345 188For the collinear case (Dsp = 1), the T-even rotator vanishes due to the spin-only group symmetry (Reij = 0), butthe T-odd contribution can be finite (Class R2 and R3). As a result, the spin Hall effect of collinear magnets isgenerically attributed to the magnetic origin and hence is unique to magnetic metals. It is noteworthy that the spincurrent induced by the T-odd spin Hall effect is not accompanied by the charge current in the collinear and coplanarmagnets due to the absence of the anomalous Hall effect without SOC. This situation is distinct from the Hall effectof SO-coupled systems where the magnetic spin Hall current is admixed with the charge Hall current [14].For coplanar and noncoplanar magnets (Dsp = 2, 3), both the electric and T-odd rotators are not forbidden ingeneral. In particular, the candidates for Class R5 may show a sizable T-even spin Hall effect without the help of2 To be more precise, the T-odd rotator requires the axial symmetry and violation of the combined symmetry of the space-inversion andtime-reversal symmetry in addition to the time-reversal-symmetry breaking.22SOC as demonstrated in the first-principles study such as that for Mn3Sn [12]. On the other hand, to be different fromthe collinear case, the T-odd rotator rarely appears without being admixed with the T-even rotator in noncoplanarmagnets (Class R2 and R3), because a complex spin structure providing the T-odd rotator secondarily induces thetime-reversal-symmetric spin-charge anisotropy as well.We note that the symmetry analysis refers to the spin structures reported in experiments, that is, the spin con-figurations including the SOC effect such as small canting by the Dzyaloshinskii-Moriya interaction. To obtain aproper insight into the SOC-free responses, we have to remove the SOC corrections by performing the comparativeand first-principles study with and without the SOC effect [120, 121]. The noncollinear spin configuration does notnecessarily require the SOC effect because the collinear configuration is not favorable in some cases such as in frus-trated systems [122, 123]. More exploration of the SOC-free magnet and its emergent responses is a future work tobe addressed.V. DISCUSSION AND SUMMARYWe mainly focused on linear responses such as anomalous Hall, magnetoelectric, and spin Hall responses. Thepowerful features, such as T-even/T-odd decomposition and classification in terms of spin and orbital degrees offreedom, work in analyzing nonlinear responses as well. We exemplify it by nonreciprocal DC current induction inCoTa3S6. The response reads asJi(ω = 0) = σi;jkEj(ω0)Ek(−ω0). (95)It is called photocurrent response for ω0 ̸= 0 [124] and nonreciprocal conductivity for ω0 = 0 [125]. Performing theT-even and T-odd decompositions, we obtain the allowed componentsσz;xy = −σz;yx, σy;zx = −σy;xz, σx;yz = −σx;zy, (96)for the T-even contribution andσz;xx = σz;xx, σx;zx = σy;zy, σx;xz = σy;yz, σz;zz, (97)for the T-odd. Although the T-even components are due to the noncentrosymmetric crystal structure of CoTa3S6,the T-odd component is correlated with emergence of the orbital magnetic toroidal moment [see Sec. III C andFig. 4(b)] [94, 126–128]. This implies the giant nonlinear response driven by the nonrelativistic spin-charge cou-pling [17]. Supporting this argument, it has been identified that the giant exchange splitting leads to the sizablenonreciprocal conductivity in Mn-based antiferromagnetic metals [61].Furthermore, we can separate the spin and orbital contributions for the magnetization-related nonlinear responses.By replacing the dc electric current in Eq. (95) with the spin-/orbital-polarized DC Jsai /JLai , the symmetry analysiscan be applied to the nonreciprocal spin/orbital current induction σa,spi;jk /σa,orbi;jk . Owing to the high symmetric spinspace, the spin part vanishes (σa,spi;jk = 0) but the orbital contribution exists (σa,orbi;jk ̸= 0) in the case of CoTa3S6. Thesituation is different from that considered in the previous study on the spin contribution [129].Our work provides a systematic tool for investigating SOC-free responses, whereas it does not address quantitativeaspects of responses due to the limitation of symmetry analysis. Previous studies reported that the SOC-free physicalresponse can be sizable such as spin-polarized current induction [12, 15] and piezomagnetic effect [18], but criteria foridentifying giant spin-driven phenomena remain elusive. These problems are expected to be addressed in future worksas guided by our symmetry analysis. For instance, the classification of the magnetic quadrupole moments (Table IV)may motivate us to revisit the magnetoelectric effect. Historically, the effect has been mainly explored with simple andcollinear magnets such as Cr2O3. For collinear magnets without SOC, the one-dimensional spin-only group allows foronly the longitudinal magnetoelectric effect where induced magnetization is collinear to the spins as in Eq. (68). Giventhat small longitudinal spin fluctuations suppress the magnetoelectric effect at the low temperature [82], the SOC-freemagnetoelectric effect is typically small for the collinear magnets without thermal fluctuations. On the other hand,coplanar and noncoplanar magnetic materials may host significant spin magnetoelectric responses due to remainingspin fluctuations. Although the importance of the noncollinear property has been highlighted by prior theoreticalstudies [89], candidate materials are not fully explored. The developed spin-group symmetry analysis incorporatedinto the computational design of magnetic materials [120] may facilitate further investigations into complex spinstructures enhancing magnetoelectric responses.Previous theoretical studies investigated the spin-momentum coupling originating from the spin order without theSOC effect [5–7, 130–132]. The purpose of this paper is to clarify the macroscopic physical properties, and the spin-space-group analysis of the spin-splitting structure is out of the present scope. The systematic classification, however,23can be similarly obtained by combining the spin-space-group symmetry with the classification of spin momentumlocking in the light of multipolar degrees of freedom [48–50]. By properly taking the characteristics of the spin spacegroup, we can get accurate criteria for spontaneous spin-momentum splitting. Interestingly, various emergent physicalresponses can occur even without the nonrelativistic spin-momentum locking such as exemplified by the geometricalHall effect of CoTa3S6 because of the spin-translation degree of freedom.To summarize, we established the spin-crystallographic-group symmetry analysis applicable to complex spin struc-tures such as that with a nontrivial spin translation group. Our symmetry analysis is powerful enough to cultivatefurther understandings of spin-order-induced emergent responses and relativistic corrections to them in various mag-netic materials. In stark contrast to the widely adopted magnetic space group, the spin space group takes intoaccount the spin-structure dimension and conveniently allows us to identify the geometric contributions to physicalresponses such as the geometrical Hall effect and orbital magnetoelectric effect. The spin-space-group symmetry andcharacteristic physical responses are automatically identified by our computational methods developed in Ref. [38]and in this work. Performing the computational classification of dozens of magnetic materials, we systematicallyidentified intriguing systems such as what hosts purely-orbital magnetization, purely-orbital magnetic quadrupole po-larization, and the spin-order-induced T-even/T-odd spin Hall effect. The developed symmetry analysis will deepenour understanding of spin-orbit-free phenomena by combining first-principles material design.The program 3 of searching the symmetry-adapted tensors with a given spin space group is based on spglib [41, 42]and spinspg [38], and it is distributed under the BSD 3-clause license. The program allows for the T-even/T-odddecomposition and supports various physical properties such as equilibrium property, linear response, and nonlinearresponses.Note added—Recently, we noticed Refs. [133–135] relevant to our work. These works worked on the identification and classificationof spin space groups.ACKNOWLEDGEMENTThe authors are grateful Rikuto Oiwa and Susumu Minami for fruitful comments and discussions. This work issupported by JSPS-KAKENHI (No. JP23K13058, JP22H00290, JP21H04437, JP21H04990, JP21J00453, JP21J10712,and JP19H05825), JST-CREST Program (No. JPMJCR18T3, JPMJCR23O4), and JST-PRESTO (No. JPMJPR20L7).The crystal structure and its spin configuration are visualized by a useful software vesta [136].Appendix A: Notes on group theory1. General properties of group theoryThe terminology used in the paper is briefly mentioned to make the paper self-contained. Let g = (h,W ) , g′ =(h′,W ′) be operations of the spin space group G, the multiplication law is defined asg · g′ = (h · h′,W ·W ′) , (A1)where the multiplications of the spin- and orbital-space operations are similarly defined in that for the O(3) group andfor the space group, respectively. Accordingly, the spin space group satisfies the group axioms, that is, associativity(g · g′ ∈ G), the existence of identity (id.) operation (for id. ∈ G, g · id. = id. · g = g), and the existence of the inverseoperation (for g ∈ G, there uniquely exists the operation g−1 ∈ G such that g · g−1 = g−1 · g = id.). We denote |G| asthe order of a group G (the number of operations in G).Let H be a group whose all the operations are in G, and the group-subgroup relation holds as G > H. Thegroup-subgroup relation indicates that the group G can be decomposed by its subgroup H asG = g1H ∪ g2H ∪ · · · ∪ gnH, (A2)=⋃igiH. (A3)3 https://github.com/Hi-Wat/scg-symmetry-search.githttps://github.com/Hi-Wat/scg-symmetry-search.git24Representatives gi ∈ G are chosen to satisfy the relationgiH ∩ gjH = ϕ (i ̸= j), (A4)indicating that every intersection is empty (ϕ). The decomposition is similarly performed from the right-hand side asG =⋃iH gi. (A5)Let us consider the subgroup H commuting with every operation of its supergroup G asg ∈ G, gH = Hg, (A6)then H is a normal subgroup of G denoted asH ◁ G. (A7)A trivial normal subgroup is G itself (G ◁ G). If H ◁ G holds, two-fold coset decompositions in Eqs. (A3) and (A5)are equivalent. In that case, the cosets {giH} form the factor group G/H whose multiplication law isgiH · gjH = (gigj)H. (A8)While the representatives of the factor group {gi} themselves do not form a group in general, there may exist thegroup K = {gi} satisfying the group axiom such as gi · gj ∈ K. Accordingly, G is recast as the internal semidirectproduct of H and K,G = H ⋊K, (A9)where H,K < G , H ◁ G, and there exists the trivial intersection between H and K as K ∩ H = {id.}. WhenG = H ⋊K holds, the operations in G can be written asG = {h · k |h ∈ H, k ∈ K} . (A10)If we further demand the relation K ◁ G, the group G is given by the (internal) direct productG = H ×K, (A11)by which the operations h ∈ H and k ∈ K commute with each other in Eq. (A10).2. Group structure of spin space groupThe spin space group G has a hierarchical structure given as follows [46]. The entire group can be recast as thecoset decomposition with the spin translation group (Gst ◁ G)G =⋃igi Gst. (A12)The spin translation group is given byGst = {((1, t),W ) ∈ G} , (A13)containing spin-only operations and combinations of the spin rotation and translation such as ((1,0),W ) , ((1, t),W ).The spin translation group is further divided as [46]Gst = Gso × Gst. (A14)The spin only group Gso is comprised of only the spin rotation operationGso = {((1,0),W ) ∈ G} , (A15)in which the point group operations {W} form the spin-only group Pso. A group Gst in Eq. (A14) denotes thenontrivial spin translation group whose spin rotation should be coupled to the translation.25We can perform the decomposition of the spin space group in a different manner from that in Eq. (A12) asG = Gso × G, (A16)by which the nontrivial spin space group G is defined. The nontrivial spin space group contains the nontrivial spintranslation group as its normal subgroup (Gst ◁ G). Thus, the coset decomposition is obtained asG =⋃igi Gst. (A17)In the main text, we mainly discuss the spin space group by using the decomposition of Eq. (A16).The nontrivial spin space group can be given by the internal semidirect product in some cases; e.g., CoTa3S6 ofEq. (54). It is, however, not always the case. For instance, G in Eq. (87) cannot be given that way. See also Ref. [38].The representatives of the factor group G/Gst are g = ((R, t),W ) such that R ̸= 1 otherwise g is the identity. Thefactor group{gi Gst}is therefore isomorphic to a nontrivial spin point group PH [24, 44] defined byPH = {(R,W ) | (R,W ) ̸= (1,W ) for W ̸= 1}. (A18)3. Conventional classification of magnetic symmetryThe magnetic space and point groups account for the magnetic symmetry of nonmagnetic and magnetic sys-tems [137]. In the following, we consider widely-used magnetic symmetry respecting the SOC constraint. The groupconsists of symmetry operations with and without the time-reversal operation which are respectively unitary and anti-unitary, while it may contain only unitary operations in some cases. In particular, when there exists an anti-unitaryoperation in group G, G has a normal subgroup H consisting of only unitary operations whose order is half that ofG (|H| = |G|/2). Then, we obtain the coset decompositionG = H ∪ aH, (A19)where a ̸∈ H is an anti-unitary operation including the time-reversal operation θ ≡ 1′. Otherwise, the group is formedby only the unitary operations. In terms of anti-unitary symmetry, magnetic symmetry is classified as follows.Magnetic point groupThe magnetic point group is comprised of orbital-space unitary (R) and anti-unitary (R′ = θR) operations whereR belongs to the O(3) group. The magnetic point groups P are classified into three types;Colorless group: no element including the time-reversal operation in P,Gray group: the time-reversal symmetry trivially holds as 1′ ∈ P,Black-White group: otherwise, i.e., a = R′ with R ̸= 1 in Eq. (A19).Magnetic space groupThe magnetic space group G is formed by the point-group operation R (R′) without (with) the time-reversaloperation and by the translation operation t. These two operations are frequently summarized to the Seitz notationsuch as (R, t), (R′, t̃).Similarly to the magnetic point group, the magnetic space groups are classified into four types with respect to theanti-unitary symmetry [44].Type I: no element including the time-reversal operation in G,Type II: the time-reversal symmetry trivially holds as (1′,0) ∈ G,Type III: the time-reversal operation is combined with the point-group operation as a = (R′, t) with R ̸= 1 inEq. (A19),26Type IV: the time-reversal operation is combined with the translational operation as a = (1′, t̃).Note that the translation t̃ belongs to the translational group T0 = {(1, t)} for the paramagnetic phase but is notincluded in that for the magnetic state.Magnetic materials do not show the trivial time-reversal symmetry related to g = (1′,0). Thus, we obtain theone-to-one correspondence between the types of the magnetic point group and magnetic space group as (P,G) =(Colorless, I), (Gray, IV), (Black-White, III).Appendix B: Canonical correlation for linear response theory and its symmetry constraintLet us introduce the linear response function of Xi = χXYij F(Y )j defined in Eq. (36) in the frequency domain asχXYij (ω) =∫ ∞0dteiωt−ηtχXYij (t), (B1)with the infinitesimal positive parameter η = +0 building the causality into the response. The response function inthe time domain can be written as the form of canonical correlation as [51]χXYij (t) ≡ ΓXẎij =∫ 1/T0dτTr[ρeqẎj(−iτ)Xi(t)]. (B2)The operators are in the Heisenberg representation, X(t) = eiHtXe−iHt with the unperturbed Hamiltonian H.We also introduced the temperature T and the density operator for the (unperturbed) equilibrium state ρeq =e−H/T /Tr[e−H/T]. Following Ref. [52], the transformation property of the canonical correlation function (in thefrequency domain) isΓXẎij (ω) = ΓXẎkl (ω)D(X)ki (g)D(Y )lj (g), (B3)for a preserving unitary operation g andΓXẎij (ω) = −ΓẎ Xkl (ω)[D(X)ki (g)]∗ [D(Y )lj (g)]∗, (B4)for an anti-unitary operation. Let us consider the electric conductivity by adopting the electric current X = J andthe electric polarization Y = P . When the orbital time-reversal symmetry g = (1,W ) (detW = −1) is preserved ina given magnetic group such as spin point group P and SO-coupled magnetic point group P, we can relate differentcomponents of the electric conductivity σij = χJPij with each other asσij = χJPij = κJJij = −κJJkl (−1)kj (1)li = κJJji = χJPji = σji, (B5)by using Ṗ = J . This means the Onsager reciprocity.Appendix C: Classification of noncoplanar magnetsWe summarize the classification of noncoplanar magnets shown in Sec. IV. For details of the definitions of eachquantity, please refer to the corresponding tables (Tables III, IV, V).TABLE VI: Classification of noncoplanar magnets in terms of magneti-zation ( Msp, Morb, MSOC), magnetic quadrupole moment ( Qsp, Qorb,QSOC), and rotators ( Re, Ro, RoS). The names of each material referto magndata.# Msp Morb MSOC Qsp Qorb QSOC Re Ro RoS0.102_Mn2GeO4.mcif ✓ ✓ ✓0.103_Mn2GeO4.mcif ✓ ✓ ✓ ✓ ✓ ✓0.106_DyVO3.mcif ✓ ✓ ✓ ✓ ✓ ✓0.127_Dy3Al5O12.mcif ✓0.135_Ni3B7O13Br.mcif ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓27TABLE VI (cont.)# Msp Morb MSOC Qsp Qorb QSOC Re Ro RoS0.136_Co3B7O13Br.mcif ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓0.141_Tb5Ge4.mcif ✓ ✓ ✓ ✓0.145_Co3TeO6.mcif ✓ ✓ ✓ ✓0.150_NiS2.mcif ✓ ✓ ✓0.151_Tm2Mn2O7.mcif ✓ ✓ ✓ ✓ ✓ ✓0.157_Yb2Sn2O7.mcif ✓ ✓ ✓ ✓ ✓ ✓0.158_Yb2Ti2O7.mcif ✓ ✓ ✓ ✓ ✓ ✓0.167_Nd3Sb3Mg2O14.mcif ✓ ✓ ✓ ✓ ✓ ✓0.168_NH4Fe2F6.mcif ✓ ✓ ✓0.169_U3As4.mcif ✓ ✓ ✓ ✓ ✓0.170_U3P4.mcif ✓ ✓ ✓ ✓ ✓0.184_Nd5Si4.mcif ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓0.185_Nd5Ge4.mcif ✓ ✓ ✓ ✓ ✓ ✓0.203_Mn3Ge.mcif ✓ ✓ ✓ ✓ ✓0.204_Ca2MnReO6.mcif ✓ ✓ ✓ ✓ ✓ ✓0.20_MnTe2.mcif ✓ ✓ ✓0.218_Co2SiO4.mcif ✓ ✓ ✓0.219_Co2SiO4.mcif ✓ ✓ ✓0.220_Mn2SiO4.mcif ✓ ✓ ✓ ✓ ✓ ✓0.221_Fe2SiO4.mcif ✓ ✓ ✓0.236_CaFe4Al8.mcif ✓ ✓ ✓0.240_Er2Cu2O5.mcif ✓ ✓ ✓ ✓ ✓ ✓0.250_(NH2(CH3)2)(FeCo(HCOO)6).mcif ✓ ✓ ✓ ✓ ✓ ✓0.251_(NH2(CH3)2)(FeMn(HCOO)6).mcif ✓ ✓ ✓ ✓ ✓ ✓0.268_Tb2MnNiO6.mcif ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓0.269_Tb2MnNiO6.mcif ✓ ✓ ✓ ✓ ✓ ✓0.281_Co2V2O7.mcif ✓ ✓ ✓ ✓0.292_NiTe2O5.mcif ✓ ✓ ✓0.294_Cu4(OD)6FBr.mcif ✓ ✓ ✓ ✓ ✓ ✓0.29_Er2Ti2O7.mcif ✓ ✓ ✓0.2_Cd2Os2O7.mcif ✓0.311_CoGeO3.mcif ✓ ✓ ✓ ✓0.316_DyCrWO6.mcif ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓0.318_Tm2CoMnO6.mcif ✓ ✓ ✓ ✓ ✓ ✓0.326_Nd2Sn2O7.mcif ✓0.339_Nd2Hf2O7.mcif ✓0.33_HoMnO3.mcif ✓ ✓ ✓ ✓ ✓ ✓0.340_Nd2Zr2O7.mcif ✓0.342_Tb3Ge5.mcif ✓ ✓ ✓ ✓ ✓ ✓0.347_Er2ReC2.mcif ✓ ✓ ✓ ✓0.349_Nd2NiO4.mcif ✓ ✓ ✓ ✓ ✓ ✓0.352_TbFeO3.mcif ✓ ✓ ✓ ✓ ✓ ✓0.357_CaFe5O7.mcif ✓ ✓ ✓ ✓ ✓ ✓0.368_(CH3NH3)(Co(COOH)3.mcif ✓ ✓ ✓ ✓ ✓ ✓0.369_(CH3NH3)(Co(COOH)3.mcif ✓ ✓ ✓ ✓ ✓ ✓0.388_Co3Al2Si3O12.mcif ✓ ✓ ✓ ✓0.394_Cu2CdB2O6.mcif ✓ ✓ ✓ ✓0.39_Nd2NaRuO6.mcif ✓ ✓ ✓ ✓ ✓ ✓0.411_Tb5Ge4.mcif ✓ ✓ ✓ ✓0.412_Tb5Ge4.mcif ✓ ✓ ✓ ✓0.419_ErGe2O7.mcif ✓ ✓ ✓ ✓ ✓ ✓0.42_HoMnO3.mcif ✓0.430_Yb3Pt4.mcif ✓ ✓ ✓ ✓0.431_CuB2O4.mcif ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓0.43_HoMnO3.mcif ✓0.440_SrCuTe2O6.mcif ✓ ✓ ✓ ✓ ✓ ✓0.450_Nd5Ge4.mcif ✓ ✓ ✓ ✓ ✓ ✓0.478_SmCrO3.mcif ✓ ✓ ✓0.479_SmCrO3.mcif ✓ ✓ ✓ ✓ ✓ ✓0.488_YbMnO3.mcif ✓0.489_YbMnO3.mcif ✓28TABLE VI (cont.)# Msp Morb MSOC Qsp Qorb QSOC Re Ro RoS0.48_Tb2Sn2O7.mcif ✓ ✓ ✓ ✓ ✓ ✓0.490_YbMnO3.mcif ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓0.49_Ho2Ru2O7.mcif ✓ ✓ ✓ ✓ ✓ ✓0.51_Ho2Ru2O7.mcif ✓ ✓ ✓ ✓ ✓ ✓0.530_SrCuTe2O6.mcif ✓ ✓ ✓ ✓ ✓ ✓0.544_Mn2FeReO6.mcif ✓ ✓ ✓ ✓ ✓ ✓0.545_Mn2FeReO6.mcif ✓ ✓ ✓ ✓ ✓ ✓0.571_CoSO4.mcif ✓ ✓ ✓0.572_Na2NiCrF7.mcif ✓ ✓ ✓ ✓ ✓ ✓0.573_Na2NiCrF7.mcif ✓ ✓ ✓ ✓ ✓ ✓0.574_MnFeF5(H2O)2.mcif ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓0.576_Cr2F5.mcif ✓ ✓ ✓ ✓ ✓ ✓0.578_NaBaFe2F9.mcif ✓ ✓ ✓ ✓ ✓ ✓0.584_Fe2F5(H2O)2.mcif ✓ ✓ ✓ ✓ ✓ ✓0.60_[NH2(CH3)2]n[FeIIIFeII(HCOO)6]n.mcif ✓ ✓ ✓ ✓ ✓ ✓0.64_MnV2O4.mcif ✓ ✓ ✓ ✓ ✓ ✓0.652_HoMnO3.mcif ✓0.658_BaCuTe2O6.mcif ✓0.696_SmCrO3.mcif ✓ ✓ ✓ ✓ ✓ ✓0.697_SmCrO3.mcif ✓ ✓ ✓ ✓ ✓ ✓0.70_Na3Co(CO3)2Cl.mcif ✓ ✓ ✓0.715_HoCrWO6.mcif ✓ ✓ ✓ ✓ ✓ ✓0.726_CsMn2F6.mcif ✓ ✓ ✓ ✓ ✓ ✓0.727_CsMn2F6.mcif ✓ ✓ ✓ ✓ ✓ ✓0.740_Dy3Ga5O12.mcif ✓0.741_Er3Ga5O12.mcif ✓0.743_Ho3Al5O12.mcif ✓0.744_Tb3Al5O12.mcif ✓0.745_Ho3Ga5O12.mcif ✓0.746_Tb3Ga5O12.mcif ✓0.756_GaV4S8.mcif ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓0.763_Mn5(PO4)2(PO3(OH))2(HOH)4.mcif ✓ ✓ ✓ ✓ ✓ ✓0.764_Mn5(PO4)2(PO3(OH))2(HOH)4.mcif ✓ ✓ ✓ ✓ ✓ ✓0.765_Mn5(PO4)2(PO3(OH))2(HOH)4.mcif ✓ ✓ ✓ ✓ ✓ ✓0.77_Tb2Ti2O7.mcif ✓ ✓ ✓ ✓ ✓ ✓0.78_NiN2O6.mcif ✓ ✓ ✓ ✓ ✓ ✓0.806_Fe2Se2O7.mcif ✓ ✓ ✓ ✓0.807_Fe2Se2O7.mcif ✓ ✓ ✓ ✓0.808_Fe2Se2O7.mcif ✓ ✓ ✓ ✓0.809_Fe2WO6.mcif ✓ ✓ ✓ ✓0.851_C7H14NFeCl4.mcif ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓0.862_Eu2Ir2O7.mcif ✓0.870_Pr2NiIrO6.mcif ✓ ✓ ✓ ✓ ✓ ✓0.874_Nd2NiIrO6.mcif ✓ ✓ ✓ ✓ ✓ ✓0.875_Nd2NiIrO6.mcif ✓ ✓ ✓ ✓ ✓ ✓0.877_Nd2ZnIrO6.mcif ✓0.878_Nd2ZnIrO6.mcif ✓0.879_Nd2ZnIrO6.mcif ✓0.883_NaCo2(SeO3)2(OH).mcif ✓ ✓ ✓ ✓ ✓ ✓0.898_Mn3IrSi.mcif ✓ ✓ ✓ ✓ ✓ ✓0.899_Mn3IrGe.mcif ✓ ✓ ✓ ✓ ✓ ✓0.900_Mn3CoGe.mcif ✓ ✓ ✓ ✓ ✓ ✓0.90_Rb2Fe2O(AsO4)2.mcif ✓ ✓ ✓0.916_Cd2Os2O7.mcif ✓0.91_Rb2Fe2O(AsO4)2.mcif ✓ ✓ ✓ ✓ ✓ ✓0.941_Er2O3.mcif ✓ ✓ ✓0.942_Er2Ge2O7.mcif ✓ ✓ ✓ ✓ ✓ ✓0.943_Yb2Ge2O7.mcif ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓0.944_Yb2Ir2O7.mcif ✓ ✓ ✓ ✓ ✓ ✓0.945_Yb2Ir2O7.mcif ✓0.948_CaNi3P4O14.mcif ✓ ✓ ✓ ✓ ✓ ✓29TABLE VI (cont.)# Msp Morb MSOC Qsp Qorb QSOC Re Ro RoS0.950_LaErO3.mcif ✓ ✓ ✓0.954_Nd2Ir2O7.mcif ✓0.958_Mn3Si2Te6.mcif ✓ ✓ ✓ ✓ ✓ ✓0.96_CoSO4.mcif ✓ ✓ ✓0.97_FeSb2O4.mcif ✓ ✓1.0.23_Dy3Ru4Al12.mcif ✓ ✓ ✓ ✓ ✓ ✓1.0.52_Tb14Ag51.mcif ✓ ✓ ✓ ✓1.102_U2Ni2In.mcif ✓1.115_Dy3Ru4Al12.mcif ✓1.135_C8H10Co2O11.mcif ✓1.138_MgV2O4.mcif ✓1.161_PrFe3(BO3)4.mcif ✓1.167_NiS2.mcif ✓1.201_Cr2ReO6.mcif ✓1.207_U2Rh2Sn.mcif ✓1.235_Ba(TiO)Cu4(PO4)4.mcif ✓1.267_Dy2Co3Al9.mcif ✓1.274_DyFeWO6.mcif ✓1.279_Ho2Cu2O5.mcif ✓1.299_GdMn2O5.mcif ✓1.300_GdMn2O5.mcif ✓1.303_Dy3Ru4Al12.mcif ✓1.307_Mn5Si3.mcif1.326_PrMn2O5.mcif ✓1.327_LaMn2O5.mcif ✓1.342_Co3(PO4)2.mcif ✓1.498_Cu6(SiO3)6(H2O)6.mcif ✓1.595_CaCoSO.mcif ✓1.680_Nd2NiIrO6.mcif ✓1.710_BaFe2Se3.mcif ✓1.720_Yb2O3.mcif ✓1.73_CaV2O4.mcif ✓1.75_BiMn2O5.mcif ✓1.85_alpha-Mn.mcif ✓1.89_DyFe3(BO3)4.mcif ✓1.92_HoFe3(BO3)4.mcif ✓2.18_Sc2NiMnO6.mcif ✓2.19_Mn3ZnC.mcif ✓ ✓ ✓ ✓ ✓2.32_Dy3Ru4Al12.mcif ✓ ✓ ✓ ✓ ✓ ✓2.33_Na2Mn3Se4.mcif ✓2.35_CrSe.mcif ✓ ✓ ✓ ✓2.37_La8Cu7O19.mcif ✓2.38_Pb2MnWO6.mcif ✓ ✓ ✓ ✓ ✓ ✓2.3_HoNiO3.mcif ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓2.52_Mn3O4.mcif ✓ ✓ ✓ 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