Spin-group symmetry in magnetic materials with negligible spin-
orbit coupling
Pengfei Liu1,2, Jiayu Li1, Jingzhi Han2, Xiangang Wan3,* and Qihang Liu1,4,5,* 1Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China 2School of Physics, Peking University, Beijing 100871, Peopleβs Republic of China 3National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China 4Guangdong Provincial Key Laboratory of Computational Science and Material Design, Southern University of Science and Technology, Shenzhen 518055, China 5Shenzhen Key Laboratory of for Advanced Quantum Functional Materials and Devices, Southern University of Science and Technology, Shenzhen 518055, China *Emails: [email protected]; [email protected]
Abstract Symmetry formulated by group theory plays an essential role with respect to the laws of nature, from fundamental particles to condensed matter systems. Here, by combining symmetry analysis and tight-binding model calculations, we elucidate that the crystallographic symmetries of a vast number of magnetic materials with light elements, in which the neglect of relativistic spin-orbit coupling (SOC) is an appropriate approximation, are considerably larger than the conventional magnetic groups. Thus, a symmetry description that involves partially-decoupled spin and spatial rotations, dubbed as spin group, is required. Spin group permits more symmetry operations and thus more energy degeneracies that are disallowed by the magnetic groups. One consequence of the spin group is the new anti-unitary symmetries that protect SOC-free topological phases with unprecedented surface node structures. Our work not only
ππmanifests2 the physical reality of materials with weak SOC, but also shed light on the understanding of all solids with and without SOC by a unified group theory.
1 I. Introduction The study of symmetry has always been the kernel of condensed matter physics and materials chemistry, as it dictates the way in which wavefunctions of elementary excitations behave, including geometric phases, selection rules and degeneracies. The corresponding wavefunction properties thus reflect on the physical observables such as polarization, response susceptibility and band dispersions, etc. The symmetries of three- dimensional (3D) solids are believed to be described by a complete crystallographic group theory, including 32 point groups (PGs), 230 space groups (SGs), 122 magnetic PGs (MPGs), 1651 magnetic SGs (MSGs) and their double groups with spinor representations [1,2]. They apply to the nonmagnetic materials without and with spin- orbit coupling (SG and double SG) as well as the magnetic materials with spin-orbit coupling (double MSG). The recent prosperity of symmetry-protected topological phase has shed light on the electronic structures of condensed matter systems, which provide a fertile playground for a survey of various quasi-particles including Weyl, Dirac fermions and others beyond them [3-15]. Moreover, the theories of symmetry- based indicator and (magnetic) topological chemistry based on band representations allow a comprehensive classification of topological crystalline insulators and semimetals, leading to a dictionary of thousands of predicted topological materials [16- 31]. Topological phases without spin-orbit coupling (SOC) are also widely investigated in nonmagnetic materials [13,32-34]. However, the symmetry of the remaining quadrant of the magnetic materials with negligible relativistic SOC, which represents a vast number of compounds with light elements, is surprisingly seldom explored (see Fig. 1). The most striking characteristic of the materials in this quadrant is the nontrivial spin degrees of freedom yet decoupled with the orbital part. Thus, the corresponding symmetry is not fully described by any of the abovementioned SGs. Specifically, the symmetry operations of spin, e.g., spin rotations, and the symmetry operations of lattice can combine in the way disallowed by SOC, which form a composite symmetry group applied on both position space and spin space but not necessarily simultaneously. Such groups, dubbed as spin group including spin point group (SPG) and spin space group (SSG), was first derived in 1960-1970s to
2 account for the extra symmetries of Heisenberg Hamiltonian with the application on spin waves [35-38]. Here, in order to complete the last unexplored quadrant, we focus on describing spin-Β½ electrons in magnetic lattices by spin groups and their applications on the topological electronic structures. Starting from a general form of the time-independent Schrodinger equation, we first demonstrate that the description of the full symmetry group of a spin-orbit- decoupled system with on-site local magnetic moments naturally points to spin groups. Exemplified by a spinful hexagonal molecule and a kagome lattice, we conduct tight- binding models to elucidate that compared with the case with SOC where the spin and spatial rotations are completely locked to each other, without SOC the spin rotation ( ) permits more discrete or continuous values under a spatial rotation ( ) ,
ππwhereππ ππ and denote the rotation axes, and the real scalars and areπΆπΆππ ππthe rotationππ angles. ππThis leads to much more symmetry operations ππand moreππ energy degeneracies that cannot be understood by the conventional magnetic (double) groups. From the point of view of spin group, we construct a hierarchy for PG symmetries of both magnetic and nonmagnetic materials with or without SOC, providing alternative thinking of a unified theory for describing crystalline symmetries in solids. In addition, the decoupled spin rotation, combining with time-reversal and fractional translation, would lead to new topological phases such as SOC-free Z2 topological insulator (TI) with unprecedented surface node structures, further enriching the existing zoo of the topological materials. For materials realization, we show that square-net compounds
AMnBi2 (A = Sr, Ca) could realize such topological phases with surface nodal lines
as well as bulk Dirac points at generic momentaππ2 protected by spin group symmetries.
II. General single-electron Hamiltonians To begin with, we apply spatial and spin rotational operations on the general steady-state Hamiltonians to illustrate the requirements of symmetry operations in
= + ( ) various cases. For a nonmagnetic system without SOC [ 2 ], the πποΏ½ wavefunctions are labeled by the single-valued representations ofπ»π» the 2ππ( )-ππdeterminedπποΏ½
ππ πποΏ½ 3 SG, of which the rotational elements contain solely the spatial rotations ( ). When
the general SOC term = ( ( ) Γ ) is added, neither spatialπΆπΆππ ππ rotation 1 π π π π π π 2 2 ( ) nor spin rotationπ»π» ( 2ππ) aloneππ π΅π΅ππ, butπποΏ½ onlyπποΏ½ β a πποΏ½locked combination ( ) ( )
πΆπΆcanππ ππkeep invariant. Tππheππ resultingππ spinor wavefunctions furnish the doubleππππ ππ-valuedπΆπΆππ ππ representationsπ»π»π π π π π π of the ( )-determined double SG. To describe the magnetic systems, we apply = (ππ )πποΏ½ under the framework of single-particle mean-field approximationπ»π»ππππππ [39-41]πΊπΊ, πποΏ½whereβ πποΏ½ ( ) stands for the r-dependent exchange field due to the distribution of local magneticπΊπΊ momentsπποΏ½ and is, strictly speaking, not the spin of electrons but the spin of quasiparticles arisingπποΏ½ from exchange correlation among electrons. Note that resembles the form of double-exchange model widely used to describe the magneticπ»π»ππππππ phase transitions of manganites [42-44]. If + + is considered, the locking between spin and spatial rotations still holds,π»π» whileπ»π»π π π π π π theπ»π» ππππππfull symmetry description requires double MSG with the inclusion of time-reversal operation .
If we ππconsider the symmetry operations of + , i.e., a magnetic system without SOC, it is straightforward to prove that the π»π»completeπ»π»ππππππ locking of spatial and spin rotations is no longer required. Instead, partially locked rotations, i.e., ( ) ( ),
with ( ) keeping ( ) invariant, could keep invariant and ππthusππ ππ theπΆπΆππ totalππ HamiltonianπΆπΆππ ππ . Note thatππ πποΏ½( ) presents the effects ofπ»π»ππππππ a spatially distributed magnetic
moments residing at the πΊπΊatomicπποΏ½ sites with a given magnetization direction, coupling with spin through exchange-correlation interactions. Unlike SOC, such a spin-spatial coupling is nonrelativistic and constrain ( ) with ( ) in various ways
according to the specific spin arrangement, formingππππ ππ spin groupsπΆπΆππ ππ, as will be discussed below (see Appendix A-C). The Hamiltonians and symmetry groups describing different systems are summarized in Fig. 1, with the derivation of the constraints provided in Supplementary Section III.
III. Spin point group: partially decoupled spin and spatial rotation Spin group includes SPG and SSG. We first discuss SPG, whose elements are
4 denoted by { ( ), ( )|| ( ), ( )} . While spatial inversion can
combine ( ππ) ππformingππ ππππ ππimproperππ πΆπΆππ ππrotationsπΌπΌπΌπΌππ ππ and mirror reflections, time-reversalπΌπΌ symmetry πΆπΆTππ canππ be viewed as the βinversion symmetryβ in the spin space. For simplicity, we first consider proper rotations only, i.e., { ( )|| ( )} . The partial coupling
between spin and spatial operations implies ππpureππ ππ spinπΆπΆππ rotationππ { ( )|| } and coupled spin-spatial rotation, the latter of which forms nontrivial SPGsππππ ππ containπΈπΈ ing elements of the form { ( )|| ( )} with ( ) ( is identity rotation)
except identity element {ππππ|| ππ}. Ref.πΆπΆππ [38]ππ constructedπΆπΆππ ππ allβ nontrivialπΈπΈ πΈπΈ SPGs in 3D crystals, by combining the factor groupsπΈπΈ πΈπΈ of PGs and their exhaustive isomorphic groups as the spin part. Here, differ from mathematical construction, we focus on an exemplified structure with spin arrangements to illustrate how does the regime of spin group differentiate conventional magnetic group in permitting much more symmetry operations, and the resulting physical consequence in a magnetic material with negligible SOC in terms of band degeneracy and topological electronic structure. The introduction of the construction of nontrivial SPGs and its relationship with the construction of MPGs is provided in Appendix C. To illustrate that for certain configurations SPG generally possesses more symmetry operations than the conventional MPG, we consider a spinful hexagonal molecular structure with the spatial rotational group with generators and ,
as shown in Fig. 2(a)-2(h). Placing magnetic momentsπ·π·6 on each site in generalπΆπΆ6π§π§ reducesπΆπΆ2π₯π₯ the symmetry. Considering MPG symmetry (with SOC), the only spin
configurationπ·π·6 that maintains symmetry is the in-plane spin arrangement shown in Fig. 2(c), while for SPG symmetry,π·π·6 there are more possibilities. Without the loss of generality, we build a single-orbital (e.g., 2 ) tight-binding (TB) model with in-plane local magnetic moments having the same magnitudeπππ§π§ but different coplanar directions, = ( [ ], [ ], 0) (see Fig. 2(a)). The matrix elements of the Hamiltonian
πΊπΊareππ writtenππ πΆπΆπΆπΆπΆπΆ asππ ππ ππππππ ππππ
, = , = = , + , + , ( ) , (1)
ππ ππ ππ ππ+1 ππ ππβ1 ππ ππ ππ ππ π§π§ πΌπΌπΌπΌ whereοΏ½ ππweπ§π§Μ βchooseππβ πΌπΌοΏ½ π»π»theοΏ½ππ basisπ§π§Μ β ππβ functionsπ½π½οΏ½ π‘π‘οΏ½ πΏπΏwith localπΏπΏ spinοΏ½ ππ quantizationοΏ½ππ β ππ οΏ½πΌπΌπΌπΌ axisπΏπΏ directingπ½π½π½π½ππ along
5 local magnetic moments, i.e., = ( [ ], [ ], 0) , and [ ] is defined as
π§π§Μππ πΆπΆπΆπΆπΆπΆ ππππ ππππππ ππππ ππ ππ [ ] ππ ππ . We then check all the possible { ( )|| } and πΆπΆπΆπΆπΆπΆ οΏ½2οΏ½ ππ ππππππ οΏ½2οΏ½ ππ ππ β‘ οΏ½ ππ ππ οΏ½ ππππ ππ πΆπΆ6π§π§ { ( )||ππ ππππππ}οΏ½ 2operationsοΏ½ πΆπΆπΆπΆπΆπΆ οΏ½2οΏ½ that leave the Hamiltonian invariant, and find that a spatial
rotationππππ ππ πΆπΆ2π₯π₯ could couple a spin rotation with being the order of 2ππ 2ππππ rotation andπΆπΆππ οΏ½ ππ οΏ½= 0, 1, β¦ 1 (see Supplementaryππππ οΏ½ Sectionππ οΏ½ IV)ππ. Consequently, there
are 7 inequivalentππ types ππofβ spin configurations containing and spatial rotations, including one collinear ferromagnetic (FM), one collinearπΆπΆ6π§π§ antiferromagneticπΆπΆ2π₯π₯ (AFM), one noncollinear FM, and 4 coplanar AFM configurations. Their nontrivial SPG symbols and generators are shown in Fig. 2(b)-2(h). Furthermore, the matrix elements of the spin-group Hamiltonian are functions of the angles between the local moments of the neighboring sites; hence, rotating all the moments by a same angle leaves the eigenvalues of the Hamiltonian invariant, indicating the decoupling between spin space and real space. The abovementioned properties explicitly elucidates how do the spin and spatial rotation βpartiallyβ couple to each other in magnetic materials without SOC. By considering spatial and spin rotation separately, the SOC effect could be considered as a constraint to limit the relationship of ( ) and ( ) that reduce
symmetry. Consequently, spin group itself could serveπΆπΆππ asππ a unifiedππππ ππtheory of both nonmagnetic and magnetic groups, with and without SOC. We summarize the symmetry hierarchy in the context of SPG operations in Fig. 2(i). The specific hierarchy diagram for the spinful hexagonal molecule with various spin arrangements is shown in Supplementary Section IV. The nonmagnetic or paramagnetic phase without SOC, where spin rotation is fully independent on the spatial operations, has the highest symmetry, i.e., the direct product of the spatial part and spin part (3) Γ ππ ππ0 2 ( = { , }, βΓβ denotes the internal direct product,πΊπΊ while β β denotesππππ (external)ππ ππ directππ2 productπΈπΈ ππ of two groups, see Supplementary Section I). Withβ SOC, the symmetry degrades to subgroups Γ by adding the constraint of the complete spin-space ππ ππ 2 coupling (type II MPG, πΊπΊ32 greyππ groups). The further addition of magnetic orders leads
6 to conventional magnetic group , including type I (32 colorless MPGs) and type
III (58 black-white MPGs). πΊπΊππππ The symmetry hierarchy also has another branch by adding magnetic order first and then SOC, leading to SPGs and MPGs, respectively. There are 598 nontrivial SPGs [38], which could describe noncoplanar spin arrangements. In addition, for
πΊπΊcoplanarπ π π π π π moments, there exists a boson-like time-reversal group that forms the trivial SPG = { , ( ) = }, where denotes complex conjugation, rendering the πΎπΎ full SPGππ2 πΈπΈ ππ=ππππ ππ Γ πΎπΎ . For colπΎπΎlinear moments, the full SPGs is = πΎπΎ π π π π ππππππ 2 π π π π Γ ( (πΊπΊ2) πΊπΊ ) withππ an additional (2) rotational symmetry group alongπΊπΊ the πΎπΎ ππππππ 2 πΊπΊcommonππππ directionβ ππ of the spins (see Appendixππππ B). After a comprehensive classification, we obtain 252 and 90 SPGs for describing the symmetries of coplanar and collinear magnetic structures without SOC, respectively (see Appendix C).
I V. Spin space group and band degeneracy By considering the translational symmetry of the lattice, one can easily generalize the operations of SPG to SSG, { ( )|| ( )| } , where denotes the spatial
translation within a primitive cell. Similarlyππππ ππ , theπΆπΆππ analogousππ ππ symmetryππ hierarchy for SPG in Fig. 2(i) could be generalized to SSG by involving color Bravais lattices, which certainly includes the current 1651 MSGs (also known as Shubnikov groups). Because of the complexity of color Bravais lattices, the exhaustive construction of SSGs is complicated, with infinite number of possible types. We briefly discuss the issue in Supplementary Section II. We next perform a case study to illustrate the additional band degeneracies induced by SSG symmetry. We consider a transition-metal layer with kagome lattice and noncollinear AFM spin configuration shown in Fig. 2(e), which is similar to the spin
arrangement of the noncollinear antiferromagnets Mn3Ge and Mn3Sn [45-48], as shown in Fig. 2(j). By constructing a simple single-orbital TB lattice model, we show that the electronic structure of such a magnetic lattice system should be described by SSG rather than MSG. The lattice Hamiltonian is written as follows:
7 = , , ; , , , , , , + , , , ( ) , , , . (2) β² β β² β The firstπ»π» termβπΌπΌ π½π½isοΏ½ βthe<π π nearestππ π π ππ> π‘π‘ neighborπππ π ππ πΌπΌπΏπΏπΌπΌ π½π½hoppingπππ π ππ π½π½ , andπ½π½ β π π theππ ππsecondπ π ππ πΌπΌ πΊπΊ ππtermβ ππ πΌπΌcountsπ½π½πππ π ππ theπ½π½οΏ½ effect of the local magnetic moment . Fig. 2(k) and 2(l) show the SOC-free band structures of such a 6-band model (includingπΊπΊππ spin) with zero moments and magnetic moments shown in Fig. 2(j), respectively. The nonmagnetic kagome structure with P6/mmm symmetry exhibits its prototypical band structure, including a flat band and a Dirac cone at K. By adding the noncolinear AFM order, the spin-degenerate bands in Fig. 2(k) splits to two sets of Dirac cones without opening a gap. While for colinear AFM order the Hamiltonian is block-diagonal for spin-up and spin-down components [49], the noncolinear Hamiltonian such as Eq. (2) cannot be decomposed straightforwardly. Hence, we apply the group representation theory that fully describes the system to analyze the band degeneracies at the high-symmetry momenta. Apparently, such two-fold degeneracy certainly cannot be interpreted with a MSG β β that only supports 1D irreducible corepresentations at the K point.Instead, the
πΆπΆπΆπΆπΆπΆband degeneracyππ and compatibility relation at the K valley of the band structure shown in Fig. 2(l) can be successfully explained by spin group symmetry with a higher symmetry. As shown in Fig. 2(k), the 2 orbitals on the three kagome sites (Wyckoff
position 3g) give rise to two kinds of representationsπππ§π§ (2) and (4) including spin (the number in parentheses denotes the degree ofπΎπΎ 1degeneracyπΎπΎ).5 By choosing the eigenstates of spatial rotoinversion as the basis functions and adding spin degrees
of freedom, we operate the spin-groupπΆπΆ6π§π§πΌπΌ generators { || |0}, { || |0} and { || |0} on the Hamiltonian and find that (4) ππsplits3π§π§ πΆπΆ to6π§π§ threeπΌπΌ levelsππ2π₯π₯ withπΆπΆ2π₯π₯ two 1D irreducibleππππ2π§π§ πΌπΌ corepresentation (1), (1), πΎπΎand5 a 2D irreducible corepresentation π π π π (2), which is consistent withπΎπΎ1 Fig. 2πΎπΎ(l)2 calculated by TB model (see Supplementary π π SectionπΎπΎ3 V for details).
V. Quasi-Kramers degeneracy and Z2 magnetic topological insulator Spin-group operations, including enormous number of combinations of pure spatial operations, time-reversal and pure spin rotations, significantly enhance the
8 symmetries of magnetic materials. As a result, there have to be various extra topological phases unexpected before, protected by spin-group symmetries. Analogous to TIs
protected by , we next consider anti-unitary operations squared into -1 in spin groups,ππ2 which could ππallow topological classification belonging to AII class in 2D subspaces of the 3D Brillouinππ2 zone [50]. We list all such symmetry operations in the regime of SSG in Table 1, and define the resulting degeneracy at certain high-symmetry k-points as βquasi-Kramers degeneracyβ. Furthermore, we only consider the symmetries that still persist in certain cleaved surfaces, i.e., possibly having symmetry- protected topological surface states (e.g., { || |0} and { ( )|| ( )| / } are π§π§ ππ π§π§ 1 2 excluded). We find that only two symmetry operationsππ πΌπΌ in magneticππππ ππ materialsπΆπΆ ππ ππalso exist in MSG, corresponding to AFM TI (e.g., MnBi2Te4) [51-55] and topological semimetals with symmetry-protectedππ2 double helicoid surface states predicted in nonmagnetic systems [56,57], respectively. We confirm that such topological phases could still exist without SOC. On the other hand, the other five symmetries supporting quasi-Kramers degeneracy exist solely in SSGs, without any analogues in nonmagnetic materials or magnetic materials with large SOC. Among them, { || ( )|0} ,
π§π§ { || [ ]|0}, { || [ ]| / } contain both pure spatial rotations, andππ theπΆπΆ spinππ -1/2 π₯π₯ 001 001 1 2 timeππ ππ-reversal ππ, andππ are ππthus spin-group symmetries because of the decoupling
between spin andππ lattice. The (001) surface bands with { || ( )|0} symmetry are all doubly degenerate, and thus do not protect gapless surfaceππ statesπΆπΆπ§π§ ππ in general. However, if the z axis is the axis of high rotational symmetry or there are additional spin rotational symmetries, the surface states may manifest double Dirac point, which is predicted only in bulk bands before [10,58]. The three spin-group symmetries containing spatial mirror reflection supports surface Dirac nodal line, which is not reported in magnetic
π§π§ systems before.ππ Among them, { || [ ]|0} could lead to classification for a
001 2 system and the corresponding 3D quantumππ ππ spin Hall phases, as willππ be discussed below.
The last symmetry { ( )|| | / } also supports magnetic TI, similar to π§π§ ππππππ ππ πΈπΈ ππ1 2 ππ2 { || | / } , except that the surface Dirac point is located at (0, ) or ( , ) π§π§ ππ πΈπΈ ππ1 2 ππ ππ ππ 9 momenta in the momentum space.
We next take { || [ ]|0} and { ( )|| | / } as examples to illustrate the π§π§ 001 ππ 1 2 new magnetic TIππs andππ various unexpectedππππ ππ surfaceπΈπΈ ππ node structures. We start from a
3D Diracππ2 semimetal model without SOC [59], which is analogous to a 3D version of graphene. Such a phase can easily transform to a Weyl semimetal under local magnetic moments along the z direction. We can thus tune the hopping parameters to realize a Chern insulatorπΊπΊπ§π§ phase at = /2 plane of the Brillion zone. Then, by building an AFM structure through cell-ππdoublingπ§π§ ππ (Fig. 3(a)), we can annihilate the Weyl points with opposite chirality and create a gapped insulator. By constructing an 8-band model (see Supplementary Section VI for details), we realize a magnetic TI protected by both
2 { || [ ]|0} and { ( )|| | / } symmetries, withππ gapless Dirac surface states at π§π§ 001 ππ 1 2 theππ boundariesππ of allππ 2Dππ planesππ πΈπΈ perpendicularππ to axis. Consequently, it manifests surface Dirac node lines at = 0 or = lineπππ§π§ for any surfaces perpendicular to
π₯π₯ π₯π₯ [ ] (Fig. 3(c) and 3(e)), whichππ is impossibleππ ππ in conventional TIs protected by or
ππ 001 ππ { || | / }. To examine the impact of each symmetry, we apply an in-plane FM canting π§π§ ππ πΈπΈ ππ1 2 to break { || [ ]|0} (Fig. 3(b)), then the surface node structure becomes a Dirac
001 point at (0ππ, ππ) (Fig. 3(d) and 3(f)), which is consistent with the symmetry analysis shown in Tableππ I. If { ( )|| | / } is broken by the dimerization of the two layers, π§π§ ππ 1 2 the Dirac point at (ππ0ππ, )ππ is πΈπΈfinallyππ gapped (Fig. 3(g) and 3(h)). Therefore, we demonstrate that unlike ππnonmagnetic materials and magnetic materials with SOC, magnetic materials with negligible SOC possess new topological classification
with unprecendented surface node structures protected byππ2 SSG symmetries. We note that the previous studies about SOC-free TIs focused on spinless system protected by pure crystalline symmetry without considering spin rotation or time-reversal symmetry, which differs from the situation discussed here [32].
VI. Materials realization of nodal-line semimetals A remarkable consequence of symmetry and topology in the electronic structure
10 of materials is the existence of protected degeneracies, leading to various topological semimetals such as Dirac, Weyl, nodal-line and nodal-surface semimetals. Thus, the corresponding symmetry design principles could be established to conduct a comprehensive material search. Here we take two widely studied magnetic topological semimetals, e.g., SrMnBi2 and CaMnBi2, to illustrate their unrevealed bulk and surface nodes that only exist under the regime of spin-group symmetries, including Dirac points at arbitrary k-points and surface nodal lines. Note that we turn off SOC in the calculation of these well-studied large-SOC materials to illustrate the distinct topological phases in a semi-realistic setup. The identification of more suitable material candidates described by spin-group symmetries is left for future works.
AMnBi2 (A = Sr, Ca) are layered materials with anisotropic Dirac fermions (see Fig. 4(a)), inspiring the study of square-net materials as topological semimetals such as ZrSiS [60-62]. Despite a checkerboard-type AFM configuration, the nodal properties in such compounds are typically treated via nonmagnetic models [63]. Without SOC, the diagnosis for nonmagnetic topological semimetals with inversion symmetry
indicates that AMnBi2 are nodal-line semimetals [34](see Supplementary Section VII). The magnetic order brings an inhomogeneous effective exchange field resembling the effect of SOC [64], turning the nodal lines to discrete Dirac points with four-fold degeneracy. Compared with the case with SOC where the Dirac points only occur at high-order rotational axes or Brillouin zone boundary, the Dirac points protected by spin-group symmetry could occur even at arbitrary k-points, like chiral Weyl points. Such peculiar property could be understood by the low-energy Hamiltonian. By
applying ( )and (2) spin rotation alongππ β ππthe spin direction ( ), the βππΞΈπππ§π§ π₯π₯ π¦π¦ symmetry-ππallowedππ ππ ππππ πΎπΎHamiltonianππππ takes the form ( ) = ( ) + ππ( ) +
( ) + ( ) , where , , , and π»π», , ππ, areππ 0Pauliππ ππ 0matricesππ0 ππ1 ππactingππ0πππ₯π₯ on ππspin2 ππ andππ0ππ orbitalπ¦π¦ ππ3 spacesππ πππ§π§ππ,π§π§ with the lastππππ= thre0 π₯π₯ π¦π¦eπ§π§ terms mutuallyππππ=0 π₯π₯ π¦π¦ π§π§ anti-commute with each other, leading to stable Dirac nodes that could appear at generic momenta and cannot be gapped by any perturbation that maintain { || |0} , (2) spin rotation and translation symmetries. The Dirac points of SrMnBiππ πΌπΌ2 calculatedππππ by density functional
11 theory (DFT) are shown in Fig. 4(b).
According to the above discussion, { || [ ]|0} spin-group symmetry in
110 AMnBi2 protects topological classificationππ withππ unprecedented surface nodal lines.
We next apply DFTππ2 calculations on SrMnBi2 under uniaxial pressure (the lattice constant along z is reduced by 10%) to verify this. Fig. 4(c) plots the surface states on the (001) surface, showing gapless Dirac cone at both and M point. The existence β² β² of the two Dirac points at (0.280,0.280,0) and (0ΞοΏ½.293,0.293οΏ½ ,0.272) , protects a
region in which any vertical planes in the Brillouin zone parallel to [ ] yield a
nontrivial 2D = 1 phase, as indicated by the Wilson loop of a representativeππ 110 plane
(shown in greenππ2 in Fig. 4(b)) and the transition of as a function of the momentum along the [110] direction (see Figs. 4(d) and 4(e)). Furthermore,ππ2 the surface nodes form a line between the surface projections of the two Dirac points. The four-fold rotation symmetry in this system transforms the nodal line into 4 nodal lines, with two protected by { || [ ]|0} , and the other two protected by { || [ ]|0} . We note that the
materialππ ππ choice110 here is to illustrate the new topologicalππ phasesππ 11οΏ½ 0protected by spin-group symmetries by well-known topological materials, mostly with large SOC. Considering the current topological materials with heavy elements suffered by their defective nature and chemical instability, the power of spin group naturally guides us to a vast group of stable materials with light elements for the candidates of topological systems.
VII. Discussion Although established decades ago, the concept of spin group is not widely explored or applied due to the lack of suitable condensed matter scenarios, especially in spin-Β½ electronic systems. However, the recent progress of modern condensed matter physics, in which the geometric phase, topological matter and emergent quasiparticles play an essential role, paves an avenue for the application of such symmetry groups in describing complicated magnetic materials. The main purpose of our work is to establish the connection between the powerful but previously overlooked symmetry group and the frontier of quantum material studies. The abovementioned symmetry- protected degeneracy and topological classification are merely the tip of the
ππ2 12 iceberg for the application of the spin group, leaving fruitful diversity of topological phases and emergent fermions induced by such an enhanced symmetry group to be further explored. Such nodal structures in both bulk and surface states could also shed light on the non-Abelian band topology in magnetic metals [13]. Furthermore, symmetry indicators based on spin group would also give rise to more possibilities of topological crystalline insulators and semimetals. While SOC is an intrinsic relativistic property for all materials depending on the atomic mass of the constituting elements, the theory of spin-group, which describes the symmetry of a magnetic ground state, acts as the very starting point to understand the behavior of magnetic materials with SOC. For most materials even with strong SOC, e.g., 10-100 meV, its influence on the electronic structure is still small compared with those caused by hopping, exchange splitting, and crystal field, etc (typically in the order of eV). Consequently, one can construct the zero-order Hamiltonian of a magnetic ground state based on spin group and add SOC as high-order perturbation terms. Such approach also provides an alternative paradigm to accurately understand the role of SOC by differentiating the contribution of SOC and the contribution of magnetic moments and crystal lattice. Apart from the topological phase of matter, spin-group theory also sheds lights on other physical entities. These include ground-state properties such as spin/orbit polarization, Berry curvature, and linear responses such as anomalous/spin Hall conductance, (inverse) spin Galvanic effect, (inverse) Faraday effect. For instance, each element of the response tensors ( ) in Kubo formalism for observables like spin-orbit ππ torque correlates the symmetry ofππ crystals [65], determining its zero/nonzero value but not the magnitude. Therefore, the conventional MSG cannot tell if a symmetry- permitted element is tiny or large even if the neglect of SOC is an appropriate approximation. Such an element could turn out to be zero under the regime of spin group, providing rational guiding principles for experiments. For example, the spin-
conductivity tensor was calculated in noncolinear antiferromagnets Mn3Sn without SOC, showing more symmetry restrictions compared to the case with SOC [66,67]. Another example of the possible application of spin group is the AFM-induced spin
13 splitting, which has caught great interest recently [68-72]. Such spin splitting and emergent effects could be explained in the framework of spin group theory. AFM- induced spin splitting occurs when certain spin group symmetry, e.g., { ( )|| | } is
broken; and the emergent effects like particular shapes of fermi surfaceππ ππandππ spinπΈπΈ ππ Hall conductance are restricted by the point group part of SSG when the SOC is turned off [73,74]. To achieve a complete survey of AFM-induced spin splitting, the theory of a full spin group provided above is required. In a word, spin-group description both manifests the reality of materials with weak SOC and clarifies the consequences of SOC in measurable quantities for materials with strong SOC. Last but not the least, since the symmetries of spin and space degree of freedom are considered separately, spin group could provide a unified group theory for describing materials in all the four quadrants of Fig. 1. Recall that the diagnosis of degeneracy and topological phases with and without SOC has been very different because of the applications of single-valued and double-valued representations for the same symmetry operations, leading to distinct commutation relations and eigenvalues in different contexts. In the regime of spin group, the little co-group representations in the momentum space of different quadrants are naturally connected with each other by decomposing the subduced projective representation of the anti-unitary parent group, as shown in the hierarchy relationship of Fig. 2(i). To conclude, spin group serves as a bridge to connect the seemingly independent descriptions based on nonmagnetic groups and magnetic groups and paves a new avenue for understanding the emergent properties of magnetic quantum materials.
Acknowledgements We thank Chen Fang, Zhida Song, Zhi Wang and Alex Zunger for helpful discussions. This work was supported by National Key R&D Program of China under Grant No. 2020YFA0308900, the National Natural Science Foundation of China under Grant No. 11874195 and 11834006, Guangdong Innovative and Entrepreneurial Research Team Program under Grant No. 2017ZT07C062, Guangdong Provincial Key Laboratory for Computational Science and Material Design under Grant No. 2019B030301001, the
14 Shenzhen Science and Technology Program (Grant No.KQTD20190929173815000) and Center for Computational Science and Engineering of Southern University of Science and Technology. X.W. also acknowledges the support from the Tencent Foundation through the XPLORER PRIZE.
References [1] M. S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group theory: application to the physics of condensed matter (Springer Science & Business Media, 2007). [2] C. Bradley and A. Cracknell, The mathematical theory of symmetry in solids: representation theory for point groups and space groups (Oxford University Press, 2009). [3] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. Grigorieva, S. Dubonos, Firsov, and Aa, Nature 438, 197 (2005). [4] X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B 83, 205101 (2011). [5] A. A. Burkov, M. D. Hook, and L. Balents, Phys. Rev. B 84, 235126 (2011). [6] Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu, H. Weng, X. Dai, and Z. Fang, Phys. Rev. B 85, 195320 (2012). [7] S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, and A. M. Rappe, Phys. Rev. Lett. 108, 140405 (2012). [8] C. Fang, M. J. Gilbert, X. Dai, and B. A. Bernevig, Phys. Rev. Lett. 108, 266802 (2012). [9] S. M. Young and C. L. Kane, Phys. Rev. Lett. 115, 126803 (2015). [10] B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, C. Felser, R. J. Cava, and B. A. Bernevig, Science 353 (2016). [11] F. Tang, X. Luo, Y. Du, Y. Yu, and X. Wan, arXiv preprint arXiv:1612.05938 (2016). [12] Q. Liu and A. Zunger, Phys. Rev. X 7, 021019 (2017). [13] Q. Wu, A. A. Soluyanov, and T. BzduΕ‘ek, Science 365, 1273 (2019). [14] F. Tang and X. Wan, arXiv preprint arXiv:2103.08477 (2021).
15 [15] Z.-M. Yu, Z. Zhang, G.-B. Liu, W. Wu, X.-P. Li, R.-W. Zhang, S. A. Yang, and Y. Yao, arXiv preprint arXiv:2102.01517 (2021). [16] B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Nature 547, 298 (2017). [17] H. C. Po, A. Vishwanath, and H. Watanabe, Nat. Commun. 8, 1 (2017). [18] J. Kruthoff, J. de Boer, J. van Wezel, C. L. Kane, and R.-J. Slager, Phys. Rev. X 7, 041069 (2017). [19] Z. Song, T. Zhang, Z. Fang, and C. Fang, Nat. Commun. 9, 1 (2018). [20] H. Watanabe, H. C. Po, and A. Vishwanath, Sci. Adv. 4, eaat8685 (2018). [21] F. Tang, H. C. Po, A. Vishwanath, and X. Wan, Nat. Phys. 15, 470 (2019). [22] F. Tang, H. C. Po, A. Vishwanath, and X. Wan, Nature 566, 486 (2019). [23] M. G. Vergniory, L. Elcoro, C. Felser, N. Regnault, B. A. Bernevig, and Z. Wang, Nature 566, 480 (2019). [24] T. Zhang, Y. Jiang, Z. Song, H. Huang, Y. He, Z. Fang, H. Weng, and C. Fang, Nature 566, 475 (2019). [25] A. Bouhon, G. F. Lange, and R.-J. Slager, arXiv preprint arXiv:2010.10536 (2020). [26] L. Elcoro, B. J. Wieder, Z. Song, Y. Xu, B. Bradlyn, and B. A. Bernevig, arXiv preprint arXiv:2010.00598 (2020). [27] S. Ono, H. C. Po, and H. Watanabe, Sci. Adv. 6, eaaz8367 (2020). [28] S. Ono, H. C. Po, and K. Shiozaki, arXiv preprint arXiv:2008.05499 (2020). [29] Y. Xu, L. Elcoro, Z.-D. Song, B. J. Wieder, M. G. Vergniory, N. Regnault, Y. Chen, C. Felser, and B. A. Bernevig, Nature 586, 702 (2020). [30] J. Yang, Z.-X. Liu, and C. Fang, arXiv preprint arXiv:2009.07864 (2020). [31] B. Peng, Y. Jiang, Z. Fang, H. Weng, and C. Fang, arXiv preprint arXiv:2102.12645 (2021). [32] A. Alexandradinata, C. Fang, M. J. Gilbert, and B. A. Bernevig, Phys. Rev. Lett. 113, 116403 (2014). [33] C. Fang, Y. Chen, H.-Y. Kee, and L. Fu, Phys. Rev. B 92, 081201 (2015). [34] Z. Song, T. Zhang, and C. Fang, Phys. Rev. X 8, 031069 (2018).
16 [35] W. F. Brinkman and R. J. Elliott, J. Appl. Phys. 37, 1457 (1966). [36] W. F. Brinkman and R. J. Elliott, Proc. R. Soc. A 294, 343 (1966). [37] D. B. Litvin and W. Opechowski, Physica 76, 538 (1974). [38] D. B. Litvin, Acta Crystallogr. A 33, 279 (1977). [39] J. Kubler, K. H. Hock, J. Sticht, and A. R. Williams, Journal of Physics F: Metal Physics 18, 469 (1988). [40] D. Hobbs, G. Kresse, and J. Hafner, Phys. Rev. B 62, 11556 (2000). [41] U. Von Barth and L. Hedin, J. Phys. C: Solid State Phys. 5, 1629 (1972). [42] C. Zener, Phys. Rev. 82, 403 (1951). [43] S. Yunoki, J. Hu, A. L. Malvezzi, A. Moreo, N. Furukawa, and E. Dagotto, Phys. Rev. Lett. 80, 845 (1998). [44] E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep. 344, 1 (2001). [45] T. Nagamiya, S. Tomiyoshi, and Y. Yamaguchi, Solid State Commun. 42, 385 (1982). [46] S. Tomiyoshi and Y. Yamaguchi, J. Phys. Soc. Jpn. 51, 2478 (1982). [47] P. J. Brown, V. Nunez, F. Tasset, J. B. Forsyth, and P. Radhakrishna, J. Phys. Condens. Matter 2, 9409 (1990). [48] J. Liu and L. Balents, Phys. Rev. Lett. 119, 087202 (2017). [49] W. Brzezicki and M. Cuoco, Phys. Rev. B 95, 155108 (2017). [50] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, New J. Phys. 12, 065010 (2010). [51] R. S. K. Mong, A. M. Essin, and J. E. Moore, Phys. Rev. B 81, 245209 (2010). [52] Y. Gong et al., Chin. Phys. Lett. 36, 076801 (2019). [53] J. Li, Y. Li, S. Du, Z. Wang, B.-L. Gu, S.-C. Zhang, K. He, W. Duan, and Y. Xu, Sci. Adv. 5, eaaw5685 (2019). [54] M. M. Otrokov et al., Nature 576, 416 (2019). [55] D. Zhang, M. Shi, T. Zhu, D. Xing, H. Zhang, and J. Wang, Phys. Rev. Lett. 122, 206401 (2019). [56] C. Fang, L. Lu, J. Liu, and L. Fu, Nat. Phys. 12, 936 (2016). [57] H. Cheng, Y. Sha, R. Liu, C. Fang, and L. Lu, Phys. Rev. Lett. 124, 104301 (2020).
17 [58] B. J. Wieder, Y. Kim, A. M. Rappe, and C. L. Kane, Phys. Rev. Lett. 116, 186402 (2016). [59] P. Delplace, J. Li, and D. Carpentier, EPL 97, 67004 (2012). [60] J. Park et al., Phys. Rev. Lett. 107, 126402 (2011). [61] L. M. Schoop et al., Nat. Commun. 7, 1 (2016). [62] S. Klemenz, S. Lei, and L. M. Schoop, Annu. Rev. Mater. Res. 49, 185 (2019). [63] G. Lee, M. A. Farhan, J. S. Kim, and J. H. Shim, Phys. Rev. B 87, 245104 (2013). [64] S. I. Pekar and E. I. Rashba, Zh. Eksperim. i Teor. Fiz. 47 (1964). [65] J. Ε½eleznΓ½, H. Gao, A. Manchon, F. Freimuth, Y. Mokrousov, J. Zemen, J. MaΕ‘ek, J. Sinova, and T. Jungwirth, Phys. Rev. B 95, 014403 (2017). [66] J. Ε½eleznΓ½, Y. Zhang, C. Felser, and B. Yan, Phys. Rev. Lett. 119, 187204 (2017). [67] Y. Zhang, J. Ε½eleznΓ½, Y. Sun, J. Van Den Brink, and B. Yan, New J. Phys. 20, 073028 (2018). [68] S. Hayami, Y. Yanagi, and H. Kusunose, J. Phys. Soc. Jpn. 88, 123702 (2019). [69] S. Hayami, Y. Yanagi, and H. Kusunose, Phys. Rev. B 101, 220403 (2020). [70] L.-D. Yuan, Z. Wang, J.-W. Luo, E. I. Rashba, and A. Zunger, Phys. Rev. B 102, 014422 (2020). [71] L.-D. Yuan, Z. Wang, J.-W. Luo, and A. Zunger, Phys. Rev. Mater. 5, 014409 (2021). [72]L. Ε mejkal, J. Sinova, and T. Jungwirth, arXiv preprint arXiv:2105.05820 (2021). [73] R. GonzΓ‘lez-HernΓ‘ndez, L. Ε mejkal, K. VΓ½bornΓ½, Y. Yahagi, J. Sinova, T. Jungwirth, and J. Ε½eleznΓ½, Phys. Rev. Lett. 126, 127701 (2021). [74] H.-Y. Ma, M. Hu, N. Li, J. Liu, W. Yao, J.-F. Jia, and J. Liu, Nat. Commun. 12, 2846 (2021). [75] W. Opechowski, Crystallographic and Metacrystallographic Groups (North- Holland, 1986).
TABLE I. Spin space group (SSG) symmetries supporting quasi-Kramers degeneracy. The SSG symmetries, the momenta with protected 2-fold degeneracy, the surfaces that
18 maintain the corresponding symmetry and the possible surface states with various nodal structures, are listed.
Momenta with SSG Surface with protected 2-fold Possible surface states Symmetry the symmetry degeneracy
TRIM within = 0 Dirac point at { || | / } (xy0) ( ) π§π§ plane πππ§π§ 0,0 ( , 0) ππ πΈπΈ ππ1 2 ( , 0, ) and ( , , ) Dirac nodalππππ ππline at { ( )|| [ ]| / } (010) π₯π₯ ππ πππ§π§ lines ππ ππ πππ§π§ = πππππ§π§ ππ ππ 001 ππ1 2 π₯π₯ = 0 and = Possibleππ doubleππ Dirac { || ( )|0} (001) πππ§π§ planesππ π§π§ ππ point ππ πΆπΆπ§π§ ππ (0,0, ), (0, , ), Dirac nodal line at { || [ ]|0} ( , 0, ) π§π§and ( , π§π§, ) (xy0) ππ ππ ππ = 0 or = ππ ππ 001 ππ πππ§π§ lines ππ ππ πππ§π§ πππ₯π₯ πππ₯π₯ ππ (0,0, ) (0, , ) Dirac nodal line at { || [ ]| / } (010) = 0 π₯π₯ πππ§π§ ππlinesππππ ππ πππ§π§ ππ ππ 001 ππ1 2 πππ₯π₯ ( , 0, ) and ( , , ) { ( )|| [ ]| / } (010) Dirac nodal line at = π₯π₯ ππ πππ§π§ lines ππ ππ πππ§π§ ππππππ ππ ππ 001 ππ1 2 πππ₯π₯ ππ
TRIM within = Dirac point at { ( )|| | / } (xy0) ( ) π§π§ plane πππ§π§ ππ 0, ( , ) ππππππ ππ πΈπΈ ππ1 2 ππ ππππ ππ ππ
19
FIG. 1. Four-quadrant diagram describing the symmetry of solids with/without magnetic order and/or SOC. The general steady-state Hamiltonians, space groups and their representative group elements are shown for each quadrant. Compared with the conventional crystallographic groups, the key characteristic of spin group is the partial decoupling between spatial rotation ( ) and spin rotation ( ). For the materials with SOC, i.e., the quadrant III and IV,πΆπΆππ spatialππ and spin rotationsππππ areππ completely locked. For example, a spatial rotation by 2 /3 requires a simultaneous spin rotation by
2 /3 along the same axis. For the materialsππ without SOC, the spin and spatial rotations areππ completely or partially decoupled, which implies that one symmetry operation could be composed of a spin and a spatial rotation with different rotation axes and angles. For the nonmagnetic case (quadrant II), we could either consider spatial rotation only, or add a totally unconstrained spin rotation, which constitutes a SO(3) group for spin. For magnetic cases (quadrant I), spin rotation is constrained by the magnetic orders of the system, which allows more operations that are disallowed by SOC but less than the full SO(3) group. The schematic plot in quadrant I shows that, for a specific magnetic order, we can have a symmetry operation that is composed of a spatial rotation by 2 /3 and a spin rotation of 4 /3 along the same axis. Such operations are writtenππ as { ( )|| ( )| } whereππ the spatial and spin rotation axes and angles could be different.ππππ ππ πΆπΆππ ππ ππ
20
FIG. 2. Symmetries of various spin configurations within the regime of spin point group.
(a) a spinful hexagonal molecule with D6 spatial symmetry. (b-h) seven inequivalent spin configurations of the hexagonal molecule containing and spatial rotations. The corresponding nontrivial SPG symbols and generatorsπΆπΆ6π§π§ areπΆπΆ 2shown.π₯π₯ (i) symmetry hierarchy from SPG to MPG for various spin arrangements. SOC: spin-orbit coupling; MO: magnetic order. : PG for spinless system (containing spatial operations only); : PG with completeπΊπΊππ0 locking between spin and spatial degrees of freedom; : MPG;πΊπΊππ : full SPG; : nontrivial SPG. (j) a spinful kagome lattice with noncolinearπΊπΊππππ AFMπΊπΊ spinπ π π π configurationπΊπΊπ π π π shown in panel (e). (k-l) SOC-free band structure of the kagome lattice without (k) and with (l) magnetic order. The band eigenvalues at K include a 4D irreducible representation and a 2D irreducible representation if considering spin. The numbers in theπΎπΎ 4parathesis represent the dimension of representationsπΎπΎ1 . The band eigenvalues of (l) at K include two two-fold degenerate points, belonging to 2D irreducible co-representation , and two π π nondegenerate points, belonging to and , respectively. πΎπΎ3 π π π π πΎπΎ1 πΎπΎ2
21
FIG. 3. magnetic topological insulators protected by SSG symmetries. (a-b) a
2 magneticππ system with (a) A-type AFM structure invariant under { || [ ]|0} and
ππ ππ 001 { ( )|| | / } SSG symmetry, (b) { ( )|| | / } SSG symmetry. (c-d) the π§π§ π§π§ ππ 1 2 ππ 1 2 correspondingππππ ππ πΈπΈ ππ [100] surface nodal structuresππππ in theππ BrillouinπΈπΈ ππ zone, including (c) surface nodal line and (d) surface Dirac cone. (e-f) the corresponding surface band dispersion.
(g) the configuration with broken { ( )|| | / } by the dimerization of the two π§π§ ππ 1 2 layers. (h) the corresponding (100) surfaceππππ ππ bandπΈπΈ ππdispersion with a gapped Dirac cone.
22
FIG. 4. topological phase protected by { || [ ]|0} symmetry in square-net
2 110 materials.ππ (a) crystal structure of AMnBi2 (A = Sr,ππ Caππ ). The A atoms are not plotted for clarification. (b) Locations of 16 Dirac points and the corresponding surface nodal lines of pressured SrMnBi2 (with axis reduced by 10%). The equivalent Dirac points that are connected by symmetryππ are denoted by the same color. There are 3 types of nonequivalent Dirac points, with 8 Dirac points located at generic momenta (denoted by purple), represented by (0.690,0.066,0) Γ ; 4 Dirac points located along β1 line (denoted by cyan), represented by (0.οΏ½280,0οΏ½.280,0); and 4 Dirac points locatedΞ β ππ along line (denoted by blue), represented by (0.293,0.293,0.272) . (c)
Topologicalππ β nodalπΊπΊ -line surface states of pressured SrMnBi2. (d) Wilson loop calculation
in on a plane (colored by green in panel b) parallel to [ ]. (e) The transition of 2D
value defined at the vertical planes in the Brillouinππ zone110 parallel to [ ], as a ππfunction2 of the momentum along the [110] direction. ππ 110
23 APPENDIX
A. Definition of spin group and its classifications
32 crystallographic point groups (PGs) and 230 space groups (SGs) are groups that aim to describe three-dimensional (3D) nonmagnetic crystals. For magnetic crystals, the ordered spin arrangements in periodic lattices are generally described by 122 magnetic point groups (MPGs) and 1651 magnetic space groups (MSGs). Such magnetic groups introduce the antisymmetric time-reversal operation T that flips double-valued properties like spin, and thus enlarges the number of types of crystallographic group. However, spin, as a vector in 3D Euclidian vector space, could have more than two values in realistic spin arrangements, the symmetry operations of which include spin rotations and spin inversion forming an orthogonal group O(3) that keeps dot product of any two vectors in 3D vector space invariant. The inversion of spin is realized through T, for which we write the orthogonal group acting on the spin space as (3) = (3) Γ , (A1) π π ππ ππ =ππππ{ , }, ππ 2 (A2) ππ SO(3) = { ( )| = sin( )cos(ππ2) +πΈπΈsinππ( )sin( ) + cos( ) , (0, ], ππππ ππ ππ ππ (0,2 ππ];πποΏ½ (0,2ππ ]}, ππ πποΏ½ ππ ππ οΏ½ ππ β ππ (A3)ππ β where ( ) stands for spin rotationππ ππwithβ rotationππ axis and rotation angle . Such groupππππ ππ (3) is termed as the orthogonal group of spinππ symmetries (OS). ππ π π When consideringππ a spin arrangement in real space, we must include spin operations and spatial operations at the same time. One spin arrangement could be represented by a three-component vector valued function:
( ) = ( ( ), ( ), ( )) . (A4) π»π» π₯π₯ π¦π¦ π§π§ Then, spin group isπΊπΊ definedππ ππ as ππanyππ subgroupππ ππ ππ of the external direct product of
a group with elements exertingπΊπΊπ π on three dimensional spatial coordinates (either PG or SG), denoted as , and the orthogonal groups of spin symmetries, i.e., (3): π π 0 (3) = (SO(3) Γ Z ). ππ (A5) π π T 0β¨ππ 0β¨ 2 24 For being PG , which is written as = { 0 ( ) | =ππ0 ( ) ( ) + ( ) ( ) + ( ) , (0, ], ππ ππ0 πΆπΆππ ππ πΌπΌ ππ π π π π π π ππ(ππ0ππππ,2 ππ]; πποΏ½ π π π π π π (0,2ππ]π π π π π π , =ππ0πποΏ½,1}, ππ ππππ ππ πποΏ½ ππ β ππ(A6)
where ( ) stands for a spatialππ β rotationππ ππ operationβ ππ withππ rotation axis and rotation- angle πΆπΆ,ππ whileππ stands for spatial inversion symmetry operation, everyππ subgroup of ππ (3) πΌπΌis called spin point group (SPG) , operations of which are π π denotedππ0β¨ asππ πΊπΊπ π π π { || }, (A7) with (3) and . And πππ π ππ π π ππs β ππ ππ β{ ππ||0 }{ || } = { || }, (A8)
πππ π { ππ|| ππ}π π β² ππ=β² { πππ π ||πππ π β² }ππππ. β² (A9) β1 β1 β1 We define and actingπππ π onππ the spinπππ π spaceππ and coordinate space, respectively: πππ π ππ : ( ) , (A10) πππ π : πΊπΊ β π π (ππ)π π πΊπΊ, (A11) where ( ) and ( ) is the representationππ ππ β π π ππ matrixππ of and in 3D Euclidian space, respectively.π π πππ π π π Then,ππ for the r-dependent spin arrangementπππ π (ππ), we have : ( ) ( ) ( ), πΊπΊ ππ (A12) ππ:π π πΊπΊ( )ππ β π π ( ππ( π π πΊπΊ)ππ). (A13) β1 Thus, it is natural to defineππ theπΊπΊ actionππ β πΊπΊ ofπ π SPGππ operationsππ on the spin arrangement as { || }: ( ) ( ) ( ( ) ). (A14) β1 If we are interested inππ π π theππ symmetryπΊπΊ ππ β π π groupsπππ π πΊπΊ ofπ π ππrealisticππ molecule or crystal, we must consider another scalar valued function ( ) which stands for the electric
potential function contributed by atomic nuclei. Sinceππ ππ spin rotation have no influence on electric potential, the action of spin point group is defined as: { || }: ( ) ( ( ) ). (A15) β1 Note: For = (πππ π )/ππ =ππ ππ( β) ππ, π π weππ denotedππ ( ( )) = ( ) / ( ( )) = ( ππ),π π withππ ππ ππ( ππ) beingπΆπΆππ ππthe rotation matrix in 3Dπ π Euclidianππππ ππ spaceπ π ππ ππwith π π rotationπΆπΆππ ππ axis andπ π ππ rotationππ directionπ π ππ ππ represented by and rotation angle represented by . For = / = , we denote ( ) = / (ππ) = , with being the identity ππ πππ π ππ ππ πΌπΌ π π ππ 25β ππ π π πΌπΌ βππ ππ matrix. For being space group , then every subgroup of (3) is called a π π spin spaceπΊπΊ group (SSG) . WeπΊπΊπ π 0can further write SSG operationsπΊπΊπ π 0β¨ asππ πΊπΊπ π π π { || | }, (A16) where { | } with denoting PGπππ π operationππ ππ and is a three-component real vector denotingππ ππ β πΊπΊtranslationππ operation. We have ππ { || | }{ || | } = { || | + }, (A17)
ππ{π π ππ||ππ | ππ}π π β² =ππβ²{ππβ² ||πππ π ππ|π π β² ππππ(β² ππππβ²) }.ππ (A18) β1 β1 β1 β1 The actions of { ππ||π π |ππ}ππ on spin ππarrangementπ π ππ β andπ π ππ scalarππ potential satisfy { πππ π || ππ| ππ}: ( ) ( ) ( ( )( )), (A19) βππ πππ π {ππ |ππ| |πΊπΊ}ππ: (β)π π πππ π (πΊπΊ(π π ππ)( ππ β))ππ. (A20) βππ MSGs and MPGs includeπππ π ππ ππantisymmetryππ ππ β ππ π π (orππ time-ππreversal)β ππ operation in addition to spatial symmetry operations, while neglecting spin rotation operations. Thus, they are incomplete in the sense for describing the full symmetry of a general spin arrangement. When spin-orbit coupling is included, MSGs and MPGs are accurate for describing symmetry of Hamiltonian in physics because spin and lattice degrees of freedom must rotate synchronously, which binds the spin rotations in spatial rotations. Hence, including only spatial rotation is enough for describing full symmetry. However, when relativistic spin-orbit coupling is negligible, spin rotations and spatial rotations have to be considered separately, for which spin groups should be applied. Ref. [75] shows that every spin group (either SPG or SSG ) can
be written as a direct product of a so-called trivialπΊπΊπ π spin groupπΊπΊ π π π π and a nontrivialπΊπΊπ π π π
spin group . Trivial spin group stands for the group formedπΊπΊπ‘π‘π‘π‘ by pure spin operations{ |πΊπΊ|ππππ|0} (or { || }), while the nontrivial spin group stands for the group that containππ π π noπΈπΈ pure spin ππoperations,π π πΈπΈ i.e., all of the group elements contain spatial operations except the identity. In Appendix B, we analyze all possible trivial spin groups for different types of spin arrangement. In Appendix C, we comprehensively develop all the possible combinations between nontrivial SPGs and trivial SPGs to provide a full description of the possible SPGs.
26
B. Groups consisting of pure spin operations
A trivial spin group consists of elements of the form { || |0} (or { || } ),
which act on the spin configuration ( ) as πππ π πΈπΈ πππ π πΈπΈ { || |0πΊπΊ}: ππ ( ) ( ) ( ). (B1) To analyze the trivial spinππ π π groupsπΈπΈ forπΊπΊ ππdifferentβ π π ππ spinπ π πΊπΊ arrangements,ππ we divide all of the pure spin operations into 4 types, i.e., = ( ) ( 0) , = , =
( ) and = ( ) ( 0 πππ π ), andππππ οΏ½analyzeππ ππ theβ conditionsπππ π forππ πππ π ( ) ππtoπποΏ½ beππ invariantsππ πππ π underπππποΏ½ themππ ππ separately.ππ β ππππππ (Weππ β chooseππ specific axes of spin rotationsπΊπΊ ππto simplify our analysis)
Type-1: = ( ) for 0
If πππ π = ππ(πποΏ½ ππ) for ππ β 0 , and ( ) is invariant under such operation, i.e., ( ) = πππ π ( (πππποΏ½))ππ ( ) . Then,ππ β there areπΊπΊ ππtwo types of spin arrangements that have πΊπΊ ππ( ) symmetry.π π πππποΏ½ ππ πΊπΊ ππ ππππ 1.ππ If ( ) is zero for all , then ( ) obviously have ( ) symmetry. Spin arrangementsπΊπΊ ππ having thisππ propertyπΊπΊ areππ called nonmagneticπππποΏ½ spinππ arrangements. 2. If ( ) does not belong to nonmagnetic spin arrangement, the spin
configurationπΊπΊ ππ ( ) with ( ) parallel the rotation direction for all have ( ) symmetryπΊπΊ ππ. We defineπΊπΊ ππ such spin arrangements as collinearππ spin arrangementsπππποΏ½ ππ . Type-2: =
If πππ π = ππ, then the spin arrangements invariant under { || |0} should satisfy ( ) = πππ π ( ππ), indicating nonmagnetic spin arrangements. ππ πΈπΈ πΊπΊTypeππ -3:β πΊπΊ ππ= ( ) For πππ π =πππποΏ½ (ππ )ππ , then the spin arrangements invariant under { ( ) || |0} πππ π πππποΏ½ ππ ππ 1 0 0 1 0 0 πππποΏ½ ππ ππ πΈπΈ should satisfy ( ) = 0 1 0 ( ) = 0 1 0 ( ) . Then, the spin 0 0 1 β0 0 1 πΊπΊ ππ β οΏ½ β οΏ½ πΊπΊ ππ οΏ½ οΏ½ πΊπΊ ππ arrangements which have x componentβ being zero for all could have such symmetry. 27 ππ We define all spin arrangements satisfying this condition that are not nonmagnetic and collinear spin arrangements as coplanar spin arrangements. Therefore, every spin arrangement that is nonmagnetic, collinear and coplanar spin arrangement are invariant under { ( ) || |0} symmetry operation.
Theππ ππspinππ ππarrangementsπΈπΈ that are not coplanar, collinear or nonmagnetic are called noncoplanar spin arrangements. Type-4: = ( ) ( 0 )
A spinπππ π arrangementπππποΏ½ ππ ππ ππ β that ππisππππ invariantππ β ππ under { ( ) || |0} should satisfy 0 πππ§π§ ππ ππ 0πΈπΈ ( ) = 0 ( ) = 0 ( ) , it is easy ππππππ0 ππ βπ π π π π π 0 ππ 1 βππππππ0 ππ π π π π π π 0 ππ 1 πΊπΊ ππ β οΏ½π π π π π π ππ ππππππ ππ οΏ½ πΊπΊ ππ οΏ½βπ π π π π π ππ βππππππ ππ οΏ½ πΊπΊ ππ ( ) ( ) = 0 to check that such equation has no solution for unless β , indicating nonmagnetic spin arrangements. πΊπΊ ππ πΊπΊ ππ In conclusion, there are 4 types of spin arrangements that have different trivial spin groups that separately belongs to 4 types. (types of spin group is defined in Appendix C). Since the spatial part is always identity in trivial spin groups, we only write down the part for simplicity. 1. Nonmagnetic spin arrangements The trivial spin group for this type of spin arrangement is invariant under all the pure spin operations: = (3). (B2) π π 2. Collinear spin arrangements πΊπΊπ‘π‘π‘π‘ ππ Any spin rotation along the direction of spin arrangement could leave this type of spin arrangement invariant, thus, such spin arrangement have (2) { ( )|
(0,2 ]} spin rotation group. Furthermore, this type of spinππππ arrangementβ‘ πππποΏ½ ππis ππalsoβ invariantππ under operation ( ) , where = + ( (0, ]) could be any direction perpendicularππ ππtoππ theππ spin direction.ππ ππ ππππππWe πποΏ½chooseπ π π π π π π π a πποΏ½specificππ β directionππ and then ( ) symmetry could generate a trivial spin group defined as πποΏ½ πΎπΎ { , ( ) }ππ. πποΏ½Theππ nππ the full trivial spin group is the internal semidirect product of ππ2(2β‘) andπΈπΈ ππ πποΏ½ ππ, i.e.,ππ ππππ πΎπΎ ππ2 = SO(2) . (B3) πΎπΎ πΊπΊπ‘π‘π‘π‘ 28 β ππ2 We note that the internal semidirect product is because SO(2) is a normal subgroup of SO(2) while is not. πΎπΎ πΎπΎ 3. Coplanar spin arrangementsβ ππ2 ππ 2 From the discussion above, the trivial spin group of this type of spin arrangement is = { , ( ) }, (B4) πΎπΎ where denotes the directionπΊπΊ π‘π‘perpendicularπ‘π‘ ππ2 β‘ πΈπΈ toπππποΏ½ theππ planeππ of the spin arrangements. 4. NoncoplanarπποΏ½ spin arrangements The trivial spin group is this type has only identity element
= { }. (B5)
πΊπΊπ‘π‘π‘π‘ πΈπΈ C. Full crystallographic spin point groups for collinear and coplanar
spin arrangements
Crystallographic SPGs are SPGs being subgroups of (3) where is π π one of the 32 crystallographic PGs. Construction of nontrivialππ0β¨ SPGsππ from 32 PGsππ0 is similar to obtaining the MPGs. However, the orthogonal group of spin symmetries, SO(3) Γ Z , has infinite number of operations, introduction of which into T crystallographic2 PGs requires us to find all of the normal subgroups of the 32 PGs, and find all of the groups that are subgroups of SO(3) Γ Z and isomorphic to the T corresponding quotient groups. Ref. [37] shows that all the2 nontrivial SPGs can be obtained in the above approach if we find all normal subgroups of the 32 PGs, construct all quotient groups from these normal subgroups, and find all subgroups of SO(3) Γ Z T that are isomorphic to the quotient groups through all possible isomorphic relations. By2 applying such procedure, Ref. [38] obtains 598 types of nontrivial SPGs, with two groups defined to belong to the same type if they are conjugate subgroups of the direct product of general linear group in spin space and affine group in physical space, i.e., (3) (3) , which are sufficient to describe noncoplanar spin arrangements accordingπΊπΊπΊπΊ β πΊπΊtoπΊπΊ πΊπΊthe discussion in Appendix B. However, for collinear and coplanar spin arrangements, there are pure spin
29 operations, i.e., we have to consider SPGs that are product of nontrivial and trivial SPGs, i.e., = . Such product could always be written as semidirect product
πΊπΊπ π π π πΊπΊ ππbππππecauseπΊπΊπ‘π‘π‘π‘π‘π‘ is always a normal subgroup of . This group πΊπΊcouldππππππ β alsoπΊπΊπ‘π‘π‘π‘π‘π‘ be written asπΊπΊ directπ‘π‘π‘π‘π‘π‘ product Γ by proper selectionπΊπΊπππππππΊπΊπ‘π‘π‘π‘π‘π‘ of such that is also a normal subgroup of πΊπΊππππππ πΊπΊπ‘π‘π‘π‘π‘π‘. Thus, to classify the pointπΊπΊ groupsππππππ of the formπΊπΊππππππ Γ , we have to find allπΊπΊ theππππππ typesπΊπΊπ‘π‘π‘π‘π‘π‘ of nontrivial SPG (denoted as ) that couldπΊπΊ performππππππ πΊπΊ π‘π‘internalπ‘π‘π‘π‘ direct product with trivial spin groups . Since 598 πΊπΊtypesππππππ of nontrivial SPG are complete and any SPGs can be written as πΊπΊtheπ‘π‘π‘π‘π‘π‘ direct product of nontrivial SPGs and trivial SPGs, such classification should lead to a complete set of SPGs for describing coplanar and collinear spin arrangements. (We neglect nonmagnetic spin arrangements because they obviously have symmetry group
(3) with being one of the PGs or SGs). We conduct such classification inπΊπΊ0 theβ π π ππfollowing 2-stepπΊπΊ0 procedure: Step 1: We find all of the types of nontrivial SPG that allow that both and
are invariant under each other, i.e., satisfy = for all πΊπΊππππππ β1 πΊπΊandπ‘π‘π‘π‘π‘π‘ = for all , and thenππ πΊπΊ ππdetermineππππππ πΊπΊππ ππππthe correspondππ β πΊπΊingπ‘π‘π‘π‘π‘π‘ β1 β βsπΊπΊ π‘π‘(oneπ‘π‘π‘π‘β nontrivialπΊπΊπ‘π‘π‘π‘π‘π‘ SPG mightβ β πΊπΊ ππcorrespondsππππ to several full SPGs). Since both πΊπΊπππππππΊπΊ π‘π‘andπ‘π‘π‘π‘ are subgroups of (3) , and = {{ || }} , The π π πΊπΊconditionππππππ πΊπΊπ‘π‘π‘π‘π‘π‘ = for all ππ0β¨ππ and πΊπΊππππππ β© πΊπΊπ‘π‘=π‘π‘π‘π‘ πΈπΈforπΈπΈ all β1 β1 impliesππ thatπΊπΊππ ππππππ πΊπΊππππππ is a group ππandβ πΊπΊπ‘π‘π‘π‘π‘π‘ =β πΊπΊπ‘π‘π‘π‘π‘π‘Γβ πΊπΊ. π‘π‘Then,π‘π‘π‘π‘ we getβ allβ typesπΊπΊππππππ of full SPGs representedπΊπΊπππππππΊπΊπ‘π‘π‘π‘π‘π‘ by Γ πΊπΊππ.ππππ πΊπΊπ‘π‘π‘π‘π‘π‘ πΊπΊππππππ πΊπΊπ‘π‘π‘π‘π‘π‘ Step 2: We consider full SPGsπΊπΊππππππ obtainedπΊπΊπ‘π‘π‘π‘π‘π‘ from Step 1, represented as groups Γ , Γ , β¦, Γ , with , , β¦ and 1 2 ππ 1 2 ππ πΊπΊbelongingππππππ πΊπΊπ‘π‘π‘π‘π‘π‘ to differentπΊπΊππππππ typesπΊπΊπ‘π‘π‘π‘π‘π‘ of nontrivialπΊπΊππππππ SPGs,πΊπΊπ‘π‘π‘π‘π‘π‘ but withπΊπΊ ππππππ ΓπΊπΊππππππ, Γ πΊπΊππ,β¦ππππ 1 2 and Γ actually belonging to the same type of πΊπΊfullππππππ SPGs.πΊπΊπ‘π‘π‘π‘π‘π‘ Then,πΊπΊππππππ we πΊπΊchooseπ‘π‘π‘π‘π‘π‘ ππ ππππππ π‘π‘π‘π‘π‘π‘ one πΊπΊof the πΊπΊ Γ to represent this full SPG. Or in other word, eliminate multiple ππ ππππππ π‘π‘π‘π‘π‘π‘ counting ofπΊπΊ the equivalentπΊπΊ types of full SPGs. As discussed in Appendix B, the full SPGs for coplanar spin arrangements can be
written as Γ , while the full SPGs for collinear spin arrangements could be πΎπΎ πΊπΊππππππ ππ2 30 described by Γ ( (2)) . Next, we separately classify full SPGs for πΎπΎ ππππππ 2 coplanar spin arrangementsπΊπΊ ππ β andππππ for collinear spin arrangements.
C1. Classification of full SPGs for coplanar spin arrangements
For coplanar type spin arrangements, we have = . Note that we do not πΎπΎ π‘π‘π‘π‘π‘π‘ 2 consider the relative directions of spin rotation axisπΊπΊ and spaceππ rotation axis in the following derivation because he definition of βtype of spin groupβ implies that variation of relative direction of spin rotation axis and space rotation axis will not give rise to the different types of spin group. Step 1 The step 1 outlined above implies that, if we write {{ || }, { ( )|| }}, πΎπΎ the rotation axis should be either parallel or perpendicularππ2 β‘ to everyπΈπΈ πΈπΈ spinππ rotationππππ ππ πΈπΈaxis of the spin rotationππ part of , i.e., . Otherwise, the condition that = π π πΎπΎ should be invariant under πΊπΊππππππ will πΊπΊnotππππππ be satisfied. Thus, this step excludesπΊπΊπ‘π‘π‘π‘π‘π‘ theππ2
nontrivial SPGs whose πΊπΊspinππππππ part are polyhedral group, including π π , , , , because we cannot findπΊπΊππ a direction that is either parallel or
ππperpendiππβ ππππ cularππ ππππππ to ππallβ of the rotation axis. Thus, the options left for the spin part are 27 axial groups: ( = π π 1,2,3,4,6) , ( = 2,3,4,6) , , ( = 2,3,4πΊπΊ,6ππ)ππππ , ( = 2, 3, 4, 6), , πΆπΆππ(ππ =
2, 3, 4, 6), π·π·,ππ ππ, πΆπΆ ππ πΆπΆππβ ππ π·π·ππβ ππ πΆπΆπ π πΆπΆππππ ππ For ππ4 πΆπΆbeing3ππ ππ6 ππdifferentππππ π·π·3ππ groups, the direction is constrained differently in π π ππππππ order that πΊπΊthe condition that = for all ππ is satisfied. We separate β1 πΎπΎ πΎπΎ 2 2 ππππππ the ways is constrained intoππ 5ππ cases.ππ ππ ππ β πΊπΊ
ππ Case1: For being or , is not constrained. π π Case2: ForπΊπΊ ππππππ πΆπΆ1 beingπΆπΆππ ππ = {{ || }, { ( )| }} or = π π {{ || }, { ( )| }} πΊπΊorππππππ = {{ || }, { πΆπΆ(π π )| }πΈπΈ, { πΈπΈ (ππ)ππ|πποΏ½ }ππ, { πΈπΈ| }} there is πΆπΆonly2
oneπΈπΈ twoπΈπΈ -foldπππποΏ½ ππspinπΈπΈ rotationπΆπΆ2β ( πΈπΈ) inπΈπΈ πππποΏ½ . ππThusπΈπΈ ππ ππwhichπποΏ½ ππ πΈπΈcouldππ beπΈπΈ either parallel or π π πππποΏ½ ππ πΊπΊππππππ ππ 31 perpendicular to . (When is perpendicular to , then groups corresponding to
different belongsπποΏ½ to the sameππ types up to conjugateπποΏ½ transformations ) Case3: Forππ being the groups which have rotations of order larger than 2, π π should be parallelπΊπΊππππππ to the principal axis of . ππ π π Case4: For being , shouldπΊπΊππππππ be parallel to one of the 2-fold spin π π rotation axis.πΊπΊ ππππππ π·π·2 ππππ π·π·2β ππ Case5: For being , should either be perpendicular to one mirror or π π parallel to theπΊπΊ twoππππππ-fold rotationπΆπΆ2π£π£ axis.ππ It is easy to see that for the 5 cases, the condition that = for all β1 is also satisfied. Then, we get all of the types of SPGsππ thatπΊπΊππππππ canππ beπΊπΊ ππwrittenππππ as πΎπΎ ππ β ππΓ2 with some types possibly being identical. πΎπΎ ππππππ 2 StepπΊπΊ 2 ππ There are some types gotten in step 1 that are identical. This is because the operations that implicitly contain in could always be changed to the product
ππππππ of { ( )|| } with those operations,ππ πΊπΊ for the full group being Γ . That is πΎπΎ ππ ππππππ 2 to say,ππ ifππ weππ haveπΈπΈ a spin point group Γ with been a nontrivialπΊπΊ ππ SPG that 1 πΎπΎ 1 ππππππ 2 ππππππ could be written as the form πΊπΊ = +ππ { || } πΊπΊ with being the 1 1 subgroup of of order 2 thatπΊπΊππ doesππππ notπ»π» containππ πΈπΈ οΏ½πΊπΊ, ππthen,ππππ β weπ»π» οΏ½have: π»π» 1 πΊπΊππππππ Γ ππ 1 πΎπΎ ππππππ 2 =πΊπΊ( + {ππ || } ) Γ {{ || }, { ( )|| }} 1 ππππππ ππ = (π»π» + {ππ (πΈπΈ )οΏ½|πΊπΊ| } β π»π» οΏ½ )πΈπΈΓ {πΈπΈ{ ||ππππ}, { ππ (πΈπΈ )|| }} 1 ππ ππππππ ππ = π»π» Γ ππ , ππ πΈπΈ οΏ½ πΊπΊ β π»π» οΏ½ πΈπΈ πΈπΈ ππ ππ ππ πΈπΈ (C1) 2 πΎπΎ with containingπΊπΊππππππ noππ2 . This implies that in all of the SPGs of the form 2 ΓπΊπΊππππππ can be chosen ππsuch that the correspondingπΊπΊππππππ to is formed by πΎπΎ π π ππππππ 2 ππππππ ππππππ pureπΊπΊ spinππ rotations. πΊπΊ πΊπΊ
Furthermore, it is obvious that two SPGs, Γ and Γ , with π΄π΄ πΎπΎ π΅π΅ πΎπΎ π΄π΄ πΊπΊππππππ ππ2 πΊπΊππππππ ππ2 πΊπΊππππππ 32 and being different types and containing no , should be different types of SPG. π΅π΅ Thus, πΊπΊweππππππ can use all of the nontrivial SPGs ππthat have spin part being 9 π π axial PGs of pure spin rotation ( = 1,2,3,4πΊπΊ,ππ6ππππ) and ( = 2,3,4,6)πΊπΊ ππtoππππ construct all types of SPGs of the form πΆπΆππ ππΓ to avoid multipleπ·π·ππ countingππ of the same type πΎπΎ ππππππ 2 of group. Then there are 4 casesπΊπΊ left: ππ
Case1: For = , could be random directions which corresponds the same π π types of SPGπΊπΊ. ππππππ πΆπΆ1 ππ Case2: For = = {{ || }, { ( )| }} , which could be either parallel or π π perpendicularπΊπΊ toππππππ axis.πΆπΆ2 πΈπΈ πΈπΈ πππποΏ½ ππ πΈπΈ ππ
Case3: For π§π§= = {{ || }, { ( )| }, { ( )| }, { ( )| }} , could be π π ππππππ 2 πποΏ½ πποΏ½ πποΏ½ parallel to oneπΊπΊ of theπ·π· 3 two-πΈπΈfoldπΈπΈ rotationππ ππ axes.πΈπΈ ππ Theseππ πΈπΈ 3 twoππ -foldππ πΈπΈ rotationππ axes are equivalent for . But they are not necessarily equivalent for the whole nontrivial π π SPG . Thus,πΊπΊ ππtheππππ groups which separately have parallel to , and axis could πΊπΊbelongππππππ to the same types of SPG or different typesππ of SPG. π₯π₯ π¦π¦ π§π§ Case4: For being one of the left 6 groups ( = 3,4,6) and ( = 3,4,6), π π has to be parallelπΊπΊππππππ to the principal axis . πΆπΆππ ππ π·π·ππ ππ
ππ Thus, we can classify the types of fullπ§π§ SPGs of the form Γ into 11 types, πΎπΎ ππππππ 2 shown in the Table C1. Finally, we get 252 types of crystallographicπΊπΊ ππSPGs of the form
Γ . These SPGs are listed in Table C2-C12. Note that the operation πΎπΎ ππππππ 2 {πΊπΊ ( ππ)|| } is actually a mirror operation in spin space, and thus is denoted by usingππππππ Hermannππ πΈπΈ βMauguin notation. ππ
TABLE C1. Classifications of the SPGs of the form Γ , with πΎπΎ πΎπΎ {{ || }, { ( )|| }} and the corresponding beingπΊπΊππ ππππone ππof2 the 9 groupsππ2 β‘ π π πΈπΈ( πΈπΈ= 1ππ,2ππ,3ππ,4ππ,6) , πΈπΈ ( = 2,3,4,6) . And the principalπΊπΊππππππ axis of ( if exists) are π π πΆπΆassumedππ ππ to be alongπ·π· zππ direction.ππ πΊπΊππππππ Direction of Number of types π π πΊπΊππππππ ππ 33 Type-I random 32 Type-II parallel to z axis 58 πΆπΆ1 perpendicular to x Type-III πΆπΆ2 58 axis 2 Type-IV πΆπΆ parallel to z axis 5 Type-V parallel to z axis 4 πΆπΆ3 Type-VI parallel to z axis 7 πΆπΆ4 parallel to , or πΆπΆ6 axis(when the Type-VII π₯π₯ π¦π¦ 2 three directions lead π§π§ π·π·2 to identical types) parallel to x y or z axis (when the three Type-VIII 63 directions lead to π·π·2 different types) Type-IX parallel to z axis 10 Type-X parallel to z axis 5 π·π·3 Type-XI parallel to z axis 8 π·π·4 Total number of classes: 252 π·π·6
TABLE C2. The types of type-I SPGs of the form Γ , i.e., the types the spin πΎπΎ part group of which could be written as = isπΊπΊ ππshownππππ ππ 2in this table. There are 32 π π types in total. πΊπΊππππππ πΆπΆ1
= Γ
πΎπΎ 1 1 1 πΊπΊπ π π π πΊπΊππ1ππππ ππ2 πΊπΊππ0 ππππ0 πΊπΊππ ππππ 1 ππ 1 1 1 1 1 2 2 12 12ππ1 οΏ½ οΏ½ οΏ½ οΏ½ 1 1 ππ 1 2/ 2/ 21/ 21/ ππ 1 ππ ππ ππ ππ 2 2 1 1 2 1 1 2ππ 1 ππ ππ ππ ππ 222 222 1 212 12 1 212 12 ππ1 ππππ ππππ ππ ππ ππ ππ 1 1 1 1 1 1 ππ 1 4 4 1 1 4 1 1 1 4 1 1 ππ ππππππ ππππππ ππ ππ ππ ππ ππ ππ 1 1 ππ 4 4 4 4 1 4/ 4/ 41/ 41/ ππ 1 οΏ½ οΏ½ οΏ½ οΏ½ 1 1 1 1 ππ ππ ππ ππ ππ 34 422 422 4 2 2 4 2 2 1 1 1 1 1 1 1 ππ 4 4 4 4 1 42 42 1 41 2 1 1 41 2 1 ππ1 ππππ ππππ ππ ππ ππ ππ 4/ 4/ 4/1 1 1 4/1 1 1 ππ 1 οΏ½ ππ οΏ½ ππ οΏ½ ππ οΏ½ ππ 3 3 1 1 31 1 1 1 31 11 ππ ππππππ ππππππ ππ ππ ππ ππ ππ ππ 1 1 ππ 3 3 3 3 1 32 32 31 2 31 2ππ 1 οΏ½ οΏ½ οΏ½ οΏ½ 1 1 1 1 ππ 3 3 3 3 1 3 3 131 131 ππ1 ππ ππ ππ ππ 6 6 1 16 1 16 ππ1 οΏ½ππ οΏ½ππ οΏ½ ππ οΏ½ ππ 6 6 16 16ππ1 οΏ½ οΏ½ οΏ½ οΏ½ 1 1 ππ 622 622 6 2 2 6 2 2 1 1 1 1 1 1 1 ππ 6/ 6/ 6/ 6/ 1 6 6 61 1 16 1 ππ 1 ππ ππ ππ ππ 6 2 6 2 1 61 1 2 1 61 1 2 ππ1 ππππ ππππ ππ ππ ππ ππ 6/ 6/ 6/1 1 1 6/1 1 1 ππ 1 οΏ½ππ οΏ½ππ οΏ½ ππ οΏ½ ππ 23 23 1 1 2 13 1 1 1 2 13 11 ππ ππππππ ππππππ ππ ππ ππ ππ ππ ππ 1 1 1 1 ππ 2/ 3 2/ 3 2/ 3 2/ 3 1 43 43 1 4 13 1 1 4 13 1 ππ1 πποΏ½ πποΏ½ ππ οΏ½ ππ οΏ½ 432 432 1 41 31 2 1 41 31 2 ππ1 οΏ½ ππ οΏ½ ππ οΏ½ ππ οΏ½ ππ 1 1 1 1 1 1 ππ 4/ 32 4/ 3 2 4/ 32/ / 4/ 3 2/ / 1 11 1 1 πποΏ½ ππ οΏ½ 1 1 1 1 1 1 ππ πποΏ½ ππ ππ ππ οΏ½ ππ ππ TABLE C3. The types of type-II SPGs of the form Γ , i.e., the types the spin πΎπΎ part group of which could be written as = πΊπΊandππππππ theππ 2direction of which is π π parallel to z axis, is shown in this table. ThereπΊπΊππππππ are πΆπΆ582 types in total. ππ
= Γ πΎπΎ ππ0 ππ0 ππππππ π π π π ππππππ 2 πΊπΊ1 ππ1 πΊπΊ 1 πΊπΊ πΊπΊ1 1 ππ 2 1 22 2π§π§2πππ§π§1 οΏ½ οΏ½ οΏ½ 2 2π§π§ πππ§π§ 1 1 2/ 2 22/ 2π§π§/ πππ§π§ 1 ππ ππ ππ 2/ 12/2 2π§π§2/2 πππ§π§1 ππ ππ ππ 2/ 1 22/1 2π§π§2/1 πππ§π§1 ππ ππ ππ ππ 2 2 2 2 2 2π§π§ 2 2πππ§π§ 1 ππ οΏ½ ππ ππ 2 2 2 12 2π§π§ 2π§π§ 12πππ§π§1 ππππ ππ ππ ππ ππ 1 2 2 1 2π§π§ 2π§π§ πππ§π§ ππππ ππ ππ ππ ππ ππ 35 222 2 2 2 2 2 2 2 1 1 2 2 1 2π§π§ 2π§π§ πππ§π§ 2/ 1 2 1 2 2 1 2π§π§ 2π§π§ πππ§π§1 ππππππ ππ ππ ππ ππ ππ ππ ππ 222 1 1 2 1 1 2π§π§ πππ§π§ 1 ππππππ ππππ ππ ππ ππ ππ ππ ππ 4 2 2 2 4 2 2π§π§ 2π§π§ 4 2π§π§ 1 πππ§π§ ππππππ ππ ππ ππ ππ ππ ππ 2 2π§π§ πππ§π§ 4 2 4 4 1 4/ 2/ 42/ 42π§π§/ πππ§π§ 1 οΏ½ οΏ½ οΏ½ 4/ 4 24/1 2π§π§4/ 1 πππ§π§ 1 ππ ππ ππ ππ 4/ 4 24/2 2π§π§4/ 2π§π§ πππ§π§1 ππ οΏ½ ππ ππ 422 4 14 22 2 14 2π§π§ 2πππ§π§ 1 ππ ππ ππ 1 2 2 1 2π§π§ 2π§π§ πππ§π§ 422 222 4 2 2 4 2 2 1 2 1 2 2π§π§ 1 2π§π§ πππ§π§ 4 4 4 4 1 4 2 142 2 1 42π§π§ 2π§π§ πππ§π§1 ππππ ππ ππ ππ ππ 42 4 2 41 2 2 2π§π§4 1 2 2π§π§ πππ§π§1 ππππ ππππ ππ ππ ππ ππ 42 2 14222 1 42π§π§ 22π§π§ πππ§π§1 οΏ½ ππ οΏ½ οΏ½ ππ οΏ½ ππ 42 222 24221 2π§π§422π§π§ 1 πππ§π§1 οΏ½ ππ ππππ οΏ½ ππ οΏ½ ππ 4/ 42 4/2 1 2 4/2π§π§ 1 2π§π§ πππ§π§ 1 οΏ½ ππ οΏ½ ππ οΏ½ ππ 4/ 4 24/2 2 1 2 4/2π§π§ 2π§π§ 1 πππ§π§1 ππππππ οΏ½ ππ ππ ππ ππ ππ ππ ππ 4/ 14/2 1 1 1 4/2π§π§ 1 1 πππ§π§ 1 ππππππ ππππ ππ ππ ππ ππ ππ ππ 4/ 4/ 24/1 1 2 24π§π§ / 1 1 2π§π§ πππ§π§1 ππππππ ππππππ ππ ππ ππ ππ ππ ππ 4/ 422 14/1 2 2 14/ 1 2π§π§ 2π§π§ πππ§π§ 1 ππππππ ππ ππ ππ ππ ππ ππ ππ 3 3 1 2 32 2 1 2π§π§ 23π§π§ 21π§π§ πππ§π§ ππππππ ππ ππ ππ ππ ππ ππ 32 3 32 2 32π§π§ 2πππ§π§ 1 οΏ½ οΏ½ οΏ½ 1 2 1 2π§π§ πππ§π§ 3 3 3 3 1 3 3 132 132π§π§ πππ§π§1 ππ ππ ππ 3 3 132 1 32π§π§ πππ§π§1 οΏ½ππ οΏ½ οΏ½ ππ οΏ½ ππ 3 32 231 2π§π§3 1 πππ§π§ 1 οΏ½ππ ππ οΏ½ ππ οΏ½ ππ 6 3 2 26 2π§π§ 26π§π§ ππ1π§π§ οΏ½ππ οΏ½ ππ οΏ½ ππ 6 3 26 2π§π§6πππ§π§1 οΏ½ οΏ½ οΏ½ 2 2π§π§ πππ§π§ 622 6 6 2 2 6 2 2 1 1 2 2 1 2π§π§ 2π§π§ πππ§π§ 622 32 6 2 2 6 2 2 1 2 1 2 2π§π§ 1 2π§π§ πππ§π§ 6/ 3 6/ 6/ 1 6/ 6 26/2 2π§π§ 6/2π§π§ πππ§π§1 ππ οΏ½ ππ ππ 6/ 6 26/1 26π§π§ / 1 πππ§π§1 ππ οΏ½ ππ ππ 6 6 61 2 61 2π§π§ πππ§π§ 1 ππ ππ ππ 6 3 162 2 1 62π§π§ 2π§π§ πππ§π§1 ππππ ππ ππ ππ ππ 6 2 6 2 61 2 2 2π§π§6 1 2π§π§ 2 πππ§π§1 ππππ ππ ππ ππ ππ ππ 1 2 2 1 2π§π§ 2π§π§ πππ§π§ οΏ½ππ οΏ½ οΏ½ ππ οΏ½ ππ 36 6 2 3 6 2 6 2 1 6 2 32 261 22 2π§π§61 2π§π§2πππ§π§1 οΏ½ππ ππ οΏ½ ππ οΏ½ ππ 6/ 3 6/2 2 1 62/π§π§ 2π§π§ 1 πππ§π§ 1 οΏ½ππ οΏ½ ππ οΏ½ ππ 6/ 6 2 26/2 1 2 2π§π§ 6/2π§π§ 1 2π§π§ πππ§π§1 ππππππ οΏ½ππ ππ ππ ππ ππ ππ ππ 6/ 6/ 26/1 2 1 26π§π§ / 1 2π§π§ 1 πππ§π§1 ππππππ οΏ½ππ ππ ππ ππ ππ ππ ππ 6/ 6 16/1 2 2 1 6/1 2π§π§ 2π§π§ πππ§π§1 ππππππ ππ ππ ππ ππ ππ ππ ππ 6/ 622 16/2 1 1 16/ 2π§π§ 1 1 πππ§π§ 1 ππππππ ππππ ππ ππ ππ ππ ππ ππ 2/ 3 23 1 22/ 2 32 1 22π§π§/ 2π§π§ 23π§π§ ππ1 π§π§ ππππππ ππ ππ ππ ππ ππ ππ 43 23 1 4 23 2 1 42π§π§3 2π§π§ πππ§π§1 πποΏ½ ππ οΏ½ ππ οΏ½ 432 23 2 41 32 2 2π§π§ 41 32π§π§ 2 πππ§π§1 οΏ½ ππ οΏ½ ππ οΏ½ ππ 2 1 2 2π§π§ 1 2π§π§ πππ§π§ 4/ 32/ 2/ 3 4/ 3 2/ 4/ 3 2/ 1 4/ 32/ 43 24/1 1322/2 2π§π§4/ 1 1 32π§π§ 2/2π§π§ πππ§π§ 1 πποΏ½ ππ πποΏ½ ππ οΏ½ ππ ππ οΏ½ ππ 4/ 32/ 432 24/2 2322/1 2π§π§4/ 2π§π§ 2π§π§3 22π§π§/ 1 πππ§π§1 πποΏ½ ππ οΏ½ ππ ππ οΏ½ ππ ππ οΏ½ ππ 1 2 2 1 2 1 2π§π§ 2π§π§ 1 2π§π§ πππ§π§ πποΏ½ ππ ππ οΏ½ ππ ππ οΏ½ ππ TABLE C4. The types of type-III SPGs of the form Γ , i.e., the types the spin πΎπΎ part of which could be written as = and theπΊπΊ directionππππππ ππ2 of which is parallel π π to x axis, is shown in this table. ThereπΊπΊππππππ are 58πΆπΆ2 types in total. ππ
= Γ πΎπΎ ππ0 ππ0 ππππππ π π π π ππππππ 2 πΊπΊ1 ππ1 πΊπΊ 1 πΊπΊ πΊπΊ1 1 ππ 2 1 22 2π§π§2πππ₯π₯1 οΏ½ οΏ½ οΏ½ 2 2π§π§ πππ₯π₯ 1 1 2/ 2 22/ 2π§π§/ πππ₯π₯ 1 ππ ππ ππ 2/ 12/2 2π§π§2/2 πππ₯π₯1 ππ ππ ππ 2/ 1 22/1 2π§π§2/1 πππ₯π₯1 ππ ππ ππ ππ 2 2 2 2 2 2π§π§ 2 2πππ₯π₯ 1 ππ οΏ½ ππ ππ 2 2 2 12 2π§π§ 2π§π§ 12πππ₯π₯1 ππππ ππ ππ ππ ππ 222 2 1 222 22 1 22π§π§2 2π§π§2 πππ₯π₯1 ππππ ππ ππ ππ ππ ππ 1 2 2 1 2π§π§ 2π§π§ πππ₯π₯ 2/ 1 2 1 2 2 1 2π§π§ 2π§π§ πππ₯π₯1 ππππππ ππ ππ ππ ππ ππ ππ ππ 222 1 1 2 1 1 2π§π§ πππ₯π₯ 1 ππππππ ππππ ππ ππ ππ ππ ππ ππ 4 2 2 2 4 2 2π§π§ 2π§π§ 4 2π§π§ 1 πππ₯π₯ ππππππ ππ ππ ππ ππ ππ ππ 2 2π§π§ πππ₯π₯ 4 2 4 4 1 4/ 2/ 42/ 42π§π§/ πππ₯π₯ 1 οΏ½ οΏ½ οΏ½ 4/ 4 24/1 2π§π§4/ 1 πππ₯π₯ 1 ππ ππ ππ ππ 4/ 4 24/2 2π§π§4/ 2π§π§ πππ₯π₯1 ππ οΏ½ ππ ππ 1 2 1 2π§π§ πππ₯π₯ ππ ππ ππ 37 422 4 4 2 2 4 2 2 1 1 2 2 1 2π§π§ 2π§π§ πππ₯π₯ 422 222 4 2 2 4 2 2 1 2 1 2 2π§π§ 1 2π§π§ πππ₯π₯ 4 4 4 4 1 4 2 142 2 1 42π§π§ 2π§π§ πππ₯π₯1 ππππ ππ ππ ππ ππ 42 4 2 41 2 2 2π§π§4 1 2 2π§π§ πππ₯π₯1 ππππ ππππ ππ ππ ππ ππ 42 2 14222 1 42π§π§ 22π§π§ πππ₯π₯1 οΏ½ ππ οΏ½ οΏ½ ππ οΏ½ ππ 42 222 24221 2π§π§422π§π§ 1 πππ₯π₯1 οΏ½ ππ ππππ οΏ½ ππ οΏ½ ππ 4/ 42 4/2 1 2 4/2π§π§ 1 2π§π§ πππ₯π₯ 1 οΏ½ ππ οΏ½ ππ οΏ½ ππ 4/ 4 24/2 2 1 2 4/2π§π§ 2π§π§ 1 πππ₯π₯1 ππππππ οΏ½ ππ ππ ππ ππ ππ ππ ππ 4/ 14/2 1 1 1 4/2π§π§ 1 1 πππ₯π₯ 1 ππππππ ππππ ππ ππ ππ ππ ππ ππ 4/ 4/ 24/1 1 2 24π§π§ / 1 1 2π§π§ πππ₯π₯1 ππππππ ππππππ ππ ππ ππ ππ ππ ππ 4/ 422 14/1 2 2 14/ 1 2π§π§ 2π§π§ πππ₯π₯ 1 ππππππ ππ ππ ππ ππ ππ ππ ππ 3 3 1 2 32 2 1 2π§π§ 23π§π§ 21π§π§ πππ₯π₯ ππππππ ππ ππ ππ ππ ππ ππ 32 3 32 2 32π§π§ 2πππ₯π₯ 1 οΏ½ οΏ½ οΏ½ 1 2 1 2π§π§ πππ₯π₯ 3 3 3 3 1 3 3 132 132π§π§ πππ₯π₯1 ππ ππ ππ 3 3 132 1 32π§π§ πππ₯π₯1 οΏ½ππ οΏ½ οΏ½ ππ οΏ½ ππ 3 32 231 2π§π§3 1 πππ₯π₯ 1 οΏ½ππ ππ οΏ½ ππ οΏ½ ππ 6 3 2 26 2π§π§ 26π§π§ ππ1π₯π₯ οΏ½ππ οΏ½ ππ οΏ½ ππ 6 3 26 2π§π§6πππ₯π₯1 οΏ½ οΏ½ οΏ½ 2 2π§π§ πππ₯π₯ 622 6 6 2 2 6 2 2 1 1 2 2 1 2π§π§ 2π§π§ πππ₯π₯ 622 32 6 2 2 6 2 2 1 2 1 2 2π§π§ 1 2π§π§ πππ₯π₯ 6/ 3 6/ 6/ 1 6/ 6 26/2 2π§π§ 6/2π§π§ πππ₯π₯1 ππ οΏ½ ππ ππ 6/ 6 26/1 26π§π§ / 1 πππ₯π₯1 ππ οΏ½ ππ ππ 6 6 61 2 61 2π§π§ πππ₯π₯ 1 ππ ππ ππ 6 3 162 2 1 62π§π§ 2π§π§ πππ₯π₯1 ππππ ππ ππ ππ ππ 6 2 6 2 61 2 2 2π§π§6 1 2π§π§ 2 πππ₯π₯1 ππππ ππ ππ ππ ππ ππ 6 2 3 162 22 1 62π§π§ 2π§π§2πππ₯π₯1 οΏ½ππ οΏ½ οΏ½ ππ οΏ½ ππ 6 2 32 261 22 2π§π§61 2π§π§2πππ₯π₯1 οΏ½ππ ππ οΏ½ ππ οΏ½ ππ 6/ 3 6/2 2 1 62/π§π§ 2π§π§ 1 πππ₯π₯ 1 οΏ½ππ οΏ½ ππ οΏ½ ππ 6/ 6 2 26/2 1 2 2π§π§ 6/2π§π§ 1 2π§π§ πππ₯π₯1 ππππππ οΏ½ππ ππ ππ ππ ππ ππ ππ 6/ 6/ 26/1 2 1 26π§π§ / 1 2π§π§ 1 πππ₯π₯1 ππππππ οΏ½ππ ππ ππ ππ ππ ππ ππ 6/ 6 16/1 2 2 1 6/1 2π§π§ 2π§π§ πππ₯π₯1 ππππππ ππ ππ ππ ππ ππ ππ ππ 6/ 622 16/2 1 1 16/ 2π§π§ 1 1 πππ₯π₯ 1 ππππππ ππππ ππ ππ ππ ππ ππ ππ 2/ 3 23 1 22/ 2 32 1 22π§π§/ 2π§π§ 23π§π§ ππ1 π₯π₯ ππππππ ππ ππ ππ ππ ππ ππ 43 23 1 4 23 2 1 42π§π§3 2π§π§ πππ₯π₯1 πποΏ½ ππ οΏ½ ππ οΏ½ 2 1 2 2π§π§ 1 2π§π§ πππ₯π₯ οΏ½ ππ οΏ½ ππ οΏ½ ππ 38 432 23 4 3 2 4 3 2 1 2 1 2 2π§π§ 1 2π§π§ πππ₯π₯ 4/ 32/ 2/ 3 4/ 3 2/ 4/ 3 2/ 1 4/ 32/ 43 24/1 1322/2 2π§π§4/ 1 1 32π§π§ 22/π§π§ πππ₯π₯ 1 πποΏ½ ππ πποΏ½ ππ οΏ½ ππ ππ οΏ½ ππ 4/ 32/ 432 24/2 2322/1 2π§π§4/ 2π§π§ 2π§π§3 22π§π§/ 1 πππ₯π₯1 πποΏ½ ππ οΏ½ ππ ππ οΏ½ ππ ππ οΏ½ ππ 1 2 2 1 2 1 2π§π§ 2π§π§ 1 2π§π§ πππ₯π₯ πποΏ½ ππ ππ οΏ½ ππ ππ οΏ½ ππ TABLE C5. The types of type-IV SPGs of the form Γ , i.e., the types the spin πΎπΎ part of which could be written as = and theπΊπΊ directionππππππ ππ2 of which is parallel π π to z axis, is shown in this table. ThereπΊπΊππππππ are 5πΆπΆ 3types in total. ππ
= Γ
ππππ0 πΎπΎ 3 1 3 πΊπΊπ π π π πΊπΊ3ππππππ1 ππ2 πΊπΊππ0 πΊπΊππ ππππ 6 36 3π§π§6πππ§π§1 οΏ½ οΏ½ οΏ½ οΏ½ 6 2 36 3π§π§6πππ§π§1 οΏ½ ππ οΏ½ οΏ½ 3 3π§π§ πππ§π§ 6/ 2/ 6/ 6/ 1 23 222 3 2 13 3π§π§2 13 ππ1π§π§ ππ ππ ππ ππ 1 3 1 3π§π§ πππ§π§
TABLE C6. The types of type-V SPGs of the form Γ , i.e., the types the spin πΎπΎ part of which could be written as = and theπΊπΊ ππdirectionππππ ππ2 of which is parallel π π to z axis, is shown in this table. ThereπΊπΊππππππ are 4πΆπΆ 4types in total. ππ
= Γ
πΎπΎ 4 1 4 πΊπΊπ π π π πΊπΊ4ππππππ1 ππ2 πΊπΊππ0 ππππ0 πΊπΊππ ππππ 4 4π§π§ πππ§π§ 4 1 4 4 1 4/ 1 44/ 44/π§π§ πππ§π§ 1 οΏ½ οΏ½ οΏ½ οΏ½ 4/ 44/2 4π§π§ 4/2π§π§ πππ§π§1 ππ οΏ½ ππ ππ 4 1 4π§π§ 1 πππ§π§ ππ ππ ππ ππ TABLE C7. The types of type-VI SPGs of the form Γ , i.e., the types the spin πΎπΎ part of which could be written as = and theπΊπΊ directionππππππ ππ2 of which is parallel π π to z axis, is shown in this table. ThereπΊπΊππππππ are 7πΆπΆ 6classes in total. ππ
ππ0 = Γπ π π π ππ πΊπΊ πΎπΎ 3 1 3 πΊπΊππππππ3 ππ12 πΊπΊππ0 πΊπΊππ ππππ 6 1 66 6π§π§6πππ§π§1 οΏ½ οΏ½ οΏ½ 6 6π§π§ πππ§π§ οΏ½ οΏ½ οΏ½ 39 6 1 6 6 1 6 6π§π§ πππ§π§ 6/ 2 6/ 6/ 1 6/ 36/2 3π§π§ 6/2π§π§ πππ§π§1 ππ ππ ππ 6/ 1 66/1 6π§π§6/ 1 πππ§π§ 1 ππ ππ ππ ππ 2/ 3 222 26/ 2 3 26π§π§/ 2π§π§ 3πππ§π§ 1 ππ οΏ½ ππ ππ 1 2 6 1 2π§π§ 6π§π§ πππ§π§ πποΏ½ ππ οΏ½ ππ οΏ½ TABLE C8. The classes of type-VII SPGs of the form Γ , i.e., the types the πΎπΎ spin part of which could be written as = and πΊπΊtheππππππ directionππ2 of which is π π parallel to x, y or z axis when the three directionsπΊπΊππππππ π·π· lead2 to identical classes,ππ is shown in this table. There are 2 types in total.
222 1 2 2 2 2 2 2 1 πΊπΊππ0 ππππ0 πΊπΊππ ππππ πΊπΊπ π π π 2π₯π₯ 2π¦π¦ 2π§π§ 2π₯π₯ 2π¦π¦ 2π§π§ πππ§π§ 1 1 2π₯π₯ 2π¦π¦ 2π§π§ 2π₯π₯ 2π¦π¦ 2π§π§ πππ§π§ ππππππ οΏ½ ππ ππ ππ ππ ππ ππ TABLE C9. The types of type-VIII SPGs of the form Γ , i.e., the types the spin πΎπΎ part of which could be written as = and theπΊπΊ directionππππππ ππ2 of which is parallel π π to x, y or z axis when the three directionsπΊπΊππππππ leadπ·π·2 to different types, ππis shown in this table. There are 63 types in total.
= Γ πΎπΎ ππ0 ππ0 ππππππ π π π π ππππππ 2 πΊπΊ ππ πΊπΊ πΊπΊ 2/πΊπΊ 1ππ 2/ 1 2/ 2π§π§2/2π₯π₯ πππ₯π₯1 ππ 2π§π§ 2π₯π₯ 2π§π§2/2π₯π₯ πππ¦π¦1 ππ ππ ππ 2π§π§ 2π₯π₯ πππ§π§ 2 1 2 1 2 ππ 2π₯π₯ 2π¦π¦ 2π§π§2πππ₯π₯1 2π₯π₯ 2π¦π¦ 2π§π§ ππ ππ ππππ ππ ππ 2π₯π₯ 2π¦π¦ 2π§π§ πππ§π§ 1 ππ ππ 2 2π§π§ 2π§π§ 2π₯π₯ πππ₯π₯1 ππ ππ ππ 2π§π§ 2π§π§ 2π₯π₯ 2π§π§ 2π§π§ 2π₯π₯ πππ¦π¦1 ππππππ ππ ππ ππ ππ ππ ππ 2π§π§ 2π§π§ 2π₯π₯ πππ§π§1 ππ ππ ππ 2π₯π₯ 2π¦π¦ 1 πππ₯π₯1 ππ ππ ππ 2π₯π₯ 2π¦π¦ 1 2π₯π₯ 2π¦π¦ 1 πππ¦π¦1 ππππππ ππ ππ ππ ππ ππ ππ ππ 2π₯π₯ 42/π¦π¦ 1 ππ1π§π§ ππ ππ ππ 4/ 2 4/ 2π§π§4/2π₯π₯ πππ₯π₯1 ππ 2π§π§ 2π₯π₯ 2π§π§4/2π₯π₯ πππ¦π¦1 ππ ππ ππ 2π§π§ 2π₯π₯ πππ§π§ ππ 40 4 2 2 1 422 2 4 2 2 2π₯π₯42π¦π¦22π§π§2πππ₯π₯1 2π₯π₯ 2π¦π¦ 2π§π§ 2π₯π₯ 2π¦π¦ 2π§π§ πππ¦π¦ 4 1 4 2 4 2π₯π₯42π¦π¦ 2π§π§ πππ₯π₯1 2π₯π₯ 2π¦π¦ 2π§π§ ππ ππ ππππ ππ ππ 2π₯π₯ 42π¦π¦ 2 2π§π§ πππ¦π¦1 ππ ππ 42 2 4 2 2π₯π₯42π¦π¦22π§π§ πππ₯π₯1 οΏ½ ππ 2π₯π₯ 2π¦π¦ 2π§π§ 2π₯π₯42π¦π¦22π§π§ πππ¦π¦1 οΏ½ ππ οΏ½ ππ οΏ½ ππ 42/π₯π₯ 2π¦π¦ 2π§π§ πππ§π§ 1 οΏ½ 4/ 4 4/ ππ 2π§π§4/2π§π§ 2π₯π₯ 2π¦π¦ πππ₯π₯1 2π§π§ 2π§π§ 2π₯π₯ 2π¦π¦ ππ ππ ππ ππππππ οΏ½ ππ ππ ππ 2π§π§4/ 2π§π§ 2π₯π₯ 2π¦π¦ πππ§π§1 ππ ππ ππ 4/ 4 4/ 14/2π₯π₯ 2π¦π¦ 2π¦π¦ πππ₯π₯1 ππ ππ ππ 1 2π₯π₯ 2π¦π¦ 2π¦π¦ 14/2π₯π₯ 2π¦π¦ 2π¦π¦ πππ¦π¦1 ππππππ ππ ππ ππ ππ ππ ππ 1 2π₯π₯ 2π¦π¦ 2π¦π¦ πππ§π§ 4/ 1 4/ 2/ 4/ ππ ππ ππ 2π§π§4/1 2π₯π₯ 2π¦π¦ πππ₯π₯1 2π§π§ 1 2π₯π₯ 2π¦π¦ ππ ππ ππ ππππππ ππ ππ ππ ππ 2π§π§4/1 2π₯π₯ 2π¦π¦ πππ§π§1 ππ ππ ππ 4/ 2 4/ 2π§π§4/2π₯π₯ 1 2π§π§ πππ₯π₯1 ππ ππ ππ 2π§π§ 2π₯π₯ 1 2π§π§ 2π§π§4/2π₯π₯ 1 2π§π§ πππ¦π¦1 ππππππ ππππ ππ ππ ππ ππ ππ ππ 2π§π§4/ 2π₯π₯ 1 2π§π§ πππ§π§ 1 ππ ππ ππ 4/ 222 4/ 2π§π§4/2π₯π₯ 2π₯π₯ 2π¦π¦ πππ₯π₯1 ππ ππ ππ 2π§π§ 2π₯π₯ 2π₯π₯ 2π¦π¦ 2π§π§4/2π₯π₯ 2π₯π₯ 2π¦π¦ πππ¦π¦1 ππππππ ππ ππ ππ ππ ππ ππ 2π§π§ 2π₯π₯ 3 2π₯π₯ 2π¦π¦1 πππ§π§ ππ ππ ππ 3 3 3 2π₯π₯32π§π§ πππ₯π₯1 οΏ½ ππ 2π₯π₯ 2π§π§ 2π₯π₯32π§π§ πππ¦π¦1 οΏ½ππ οΏ½ ππ οΏ½ ππ 2π₯π₯ 2π§π§ πππ§π§ 6 2 2 1 622 3 6 2 2 οΏ½ ππ 2π₯π₯62π¦π¦22π§π§2πππ₯π₯1 2π₯π₯ 2π¦π¦ 2π§π§ 2π₯π₯ 2π¦π¦ 2π§π§ πππ¦π¦ 6/ 1 6/ 3 6/ 2π§π§6/2π₯π₯ πππ₯π₯1 ππ 2π§π§ 2π₯π₯ 2π§π§6/2π₯π₯ πππ¦π¦1 ππ ππ ππ 2π§π§ 2π₯π₯ πππ§π§ 6 1 6 3 6 ππ 2π₯π₯62π¦π¦ 2π§π§ πππ₯π₯1 2π₯π₯ 2π¦π¦ 2π§π§ ππ ππ ππππ ππ ππ 2π₯π₯ 62π¦π¦ 2π§π§ 2 πππ¦π¦1 ππ ππ 6 2 3 6 2 2π§π§62π₯π₯ 2π¦π¦2πππ₯π₯1 οΏ½ ππ 2π§π§ 2π₯π₯ 2π¦π¦ 2π§π§62π₯π₯ 2π¦π¦2πππ¦π¦1 οΏ½ππ οΏ½ ππ οΏ½ ππ 62/π§π§ 2π₯π₯ 2π¦π¦ πππ§π§ 1 οΏ½ 6/ 3 6/ ππ 2π§π§6/2π§π§ 2π₯π₯ 2π¦π¦ πππ₯π₯1 2π§π§ 2π§π§ 2π₯π₯ 2π¦π¦ ππ ππ ππ οΏ½ 2π§π§ 2π§π§ 2π₯π₯ 2π¦π¦ πππ§π§ ππππππ ππ ππ ππ 6/ 1 6/ 6 6/ ππ ππ ππ 2π§π§6/1 2π₯π₯ 2π¦π¦ πππ₯π₯1 2π§π§ 1 2π₯π₯ 2π¦π¦ ππ ππ ππ ππππππ οΏ½ ππ ππ ππ 2π§π§ 1 2π₯π₯ 2π¦π¦ πππ§π§ ππ ππ ππ 41 6/ 1 6/ 6 6/ 16/2π§π§ 2π₯π₯ 2π₯π₯ πππ₯π₯1 ππ ππ ππ 1 2π§π§ 2π₯π₯ 2π₯π₯ 16/2π§π§ 2π₯π₯ 2π₯π₯ πππ¦π¦1 ππππππ ππ ππ ππ ππ ππ ππ 1 6/2π§π§ 2π₯π₯ 2π₯π₯ πππ§π§1 ππ ππ ππ 6/ 3 6/ 2π§π§6/2π₯π₯ 1 2π§π§ πππ₯π₯1 ππ ππ ππ 2π§π§ 2π₯π₯ 1 2π§π§ 2π§π§6/2π₯π₯ 1 2π§π§ πππ¦π¦1 ππππππ ππ ππ ππ ππ ππ ππ ππ 2π§π§6/ 2π₯π₯ 1 2π§π§ πππ§π§ 1 ππ ππ ππ 6/ 32 6/ 2π§π§6/2π₯π₯ 2π¦π¦ 2π₯π₯ πππ₯π₯1 ππ ππ ππ 2π§π§ 2π₯π₯ 2π¦π¦ 2π₯π₯ 2π§π§6/2π₯π₯ 2π¦π¦ 2π₯π₯ πππ¦π¦1 ππππππ ππ ππ ππ ππ ππ ππ 2π§π§ 2π₯π₯ 2π¦π¦ 2π₯π₯ πππ§π§ 4/ 3 2/ 1 4/ 3 2 ππ ππ ππ 4/ 32/ 23 2π₯π₯4/2π¦π¦ 2π¦π¦32π₯π₯2/2π§π§ πππ₯π₯1 2π₯π₯ 2π¦π¦ 2π¦π¦ 2π₯π₯ οΏ½ / ππ ππ ππ οΏ½ 2π₯π₯4/2π¦π¦ 2π¦π¦32π₯π₯2/2π§π§ πππ¦π¦1 πποΏ½ ππ 2π§π§ ππ οΏ½ ππ ππ 2π₯π₯ 2π¦π¦ 2π¦π¦ 2π₯π₯ 2π§π§ πππ§π§ ππ οΏ½ ππ TABLE C10. The classes of type-IX SPGs of the form Γ , i.e., the types the πΎπΎ spin part of which could be written as = and πΊπΊtheππππππ directionππ2 of which is π π parallel to z axis, is shown in this table. ThereπΊπΊππππππ areπ·π· 103 types in total. ππ
= Γ
ππππ0 πΎπΎ 32 1 3 2 πΊπΊπ π π π 3πΊπΊππ2ππππ 1 ππ2 πΊπΊππ0 πΊπΊ ππππππ 3π§π§ 2π₯π₯ 3π§π§ 2π₯π₯ πππ§π§ 3 1 3 3 1 3 1 3π§π§32π₯π₯ 3π§π§32π₯π₯ πππ§π§1 ππ ππ ππ 622 2 36π§π§ 2π₯π₯ 2 36π§π§ 2π₯π₯ ππ2π§π§ 1 οΏ½ππ οΏ½ οΏ½ ππ οΏ½ ππ 3π§π§ 2π₯π₯ 2π₯π₯π₯π₯ 3π§π§ 2π₯π₯ 2π₯π₯π₯π₯ πππ§π§ 6 2 6 6 1 6 2 3π§π§ 62π₯π₯ 2π₯π₯π₯π₯ 2 3π§π§ 62π₯π₯ 2π₯π₯π₯π₯ 2 πππ§π§1 ππππ ππ ππ ππ ππ 2/ 3π§π§ 2π₯π₯ 2π₯π₯π₯π₯ 3π§π§ 2π₯π₯ 26π₯π₯π₯π₯ πππ§π§ οΏ½ππ ππ οΏ½ ππ οΏ½ ππ 6/ 6/ / 3π§π§ 1 ππ 43 222 3π§π§ 14 23π₯π₯ 2π₯π₯π₯π₯ 1 24π₯π₯ 32π₯π₯π₯π₯ πππ§π§1 ππππππ ππ ππ ππ ππ ππ ππ 432 222 2π₯π₯ 43π§π§ 32π₯π₯ 2 2π₯π₯ 43π§π§ 32π₯π₯ 2 πππ§π§1 οΏ½ ππ οΏ½ ππ οΏ½ ππ 2π₯π₯ 3π§π§ 2π₯π₯ 2π₯π₯ 3π§π§ 2π₯π₯ πππ§π§ 4/ 3 2 4/ 3 2 4/ 32/ / 2π₯π₯ 1 3π§π§ 2π₯π₯ / 2π₯π₯ 11 3π§π§ 2π₯π₯ ππππππ ππ οΏ½ ππ οΏ½ 2π₯π₯ 2π₯π₯ πππ§π§ πποΏ½ ππ ππ ππ TABLE C11. The types of type-X SPGs of the form Γ , i.e., the types the spin πΎπΎ part of which could be written as = and theπΊπΊ directionππππππ ππ2 of which is parallel π π to z axis, is shown in this table. ThereπΊπΊππππππ are 5π·π· types4 in total. ππ
42
= Γ
ππππ0 πΎπΎ 422 1 4 2 2 πΊπΊπ π π π 4 2πΊπΊππππππ2 ππ12 πΊπΊππ0 πΊπΊππ ππππ 4π§π§ 2π₯π₯ 2π₯π₯π₯π₯ 4π§π§ 2π₯π₯ 2π₯π₯π₯π₯ πππ§π§ 4 1 4 2 2 4 2 2 1 42 1 4π§π§4 2π₯π₯2 2π₯π₯π₯π₯ 4π§π§4 2π₯π₯2 2π₯π₯π₯π₯ πππ§π§ 1 ππππ 4π§π§ 2π₯π₯ 2π₯π₯π₯π₯ 4π§π§ 2π₯π₯ 2π₯π₯4π₯π₯ πππ§π§ οΏ½ ππ οΏ½ ππ οΏ½ ππ 4/ 4/ / 4π§π§ 1 ππ 1 4π§π§ 1 24π₯π₯ 2π₯π₯π₯π₯ 1 2π₯π₯ 2π₯π₯4π₯π₯ πππ§π§ ππππππ ππ ππ ππ ππ ππ ππ 4/ / 4π§π§ / 4π§π§ 1 οΏ½ 2π§π§ 2π₯π₯ 2π₯π₯π₯π₯ 2π§π§ 2π₯π₯ 2π₯π₯π₯π₯ πππ§π§ ππππππ ππ ππ ππ ππ ππ ππ TABLE C12. The types of type-XI SPGs of the form Γ , i.e., the types the spin πΎπΎ part of which could be written as = and theπΊπΊ directionππππππ ππ2 of which is parallel π π to z axis, is shown in this table. ThereπΊπΊππππππ are 8π·π· types4 in total. ππ
= Γ
ππππ0 πΎπΎ 3 1 3 πΊπΊπ π π π 3 πΊπΊππππππ 1ππ 2 πΊπΊππ0 πΊπΊ ππππππ 622 1 66π§π§ 22π₯π₯ 2 66π§π§ 22π₯π₯ 2πππ§π§ 1 οΏ½ππ οΏ½ ππ οΏ½ ππ 6π§π§ 2π₯π₯ 2ππ 6π§π§ 2π₯π₯ 2ππ πππ§π§ 6 1 6 6 1 6 2 1 6π§π§ 62π₯π₯ 2ππ 2 6π§π§ 62π₯π₯ 2ππ 2 πππ§π§1 ππππ ππ ππ ππ ππ 6/ 1 66/π§π§ 2π₯π₯ 2ππ 66/π§π§ 2π₯π₯ 2ππ πππ§π§ 1 οΏ½ππ οΏ½ ππ οΏ½ ππ 6/ 6π§π§ 6/2π§π§ 2π₯π₯ 2ππ 6π§π§ 6/2π§π§ 2π₯π₯ 2ππ πππ§π§1 ππππππ οΏ½ ππ ππ ππ ππ ππ ππ 2 6π§π§ 1 26π₯π₯ 2ππ 6π§π§ 1 2π₯π₯ 6 2ππ πππ§π§ ππππππ ππ ππ ππ ππ ππ ππ ππ 3π§π§ 3π§π§ 6/ / / 1 222 2π§π§4/2π₯π₯ 2π₯π₯π₯π₯3 2 2π§π§ 42/π₯π₯ 2π₯π₯π₯π₯ 3πππ§π§2 ππππππ ππ ππ ππ ππ ππ ππ 4/ 32/ /22 2π§π§ 6π§π§ 22 / 22 21π§π§ 6π§π§ 22 ππ οΏ½ ππ οΏ½ 2π₯π₯ 2π₯π₯ πππ§π§ πποΏ½ ππ ππ ππ C2. Classification of full SPGs for collinear spin arrangements
For collinear spin arrangements, we should consider groups of the form Γ πΎπΎ ππππππ 2 (2) . πΊπΊ οΏ½ππ β
ππππStep 1οΏ½ We firstly write (2) as (2) = {{ ( ) || |0}| (0,2 ]}. Because of
the presence of ππππ(2) = {{ππππ( ) || ππ|0πποΏ½}| ππ ππ(0πΈπΈ,2 ]}ππ , βall ofππ the rotations contained in shouldππππ have rotationπππποΏ½ ππ ππaxisπΈπΈ parallelππ β to ππ, so that the condition that π π πΊπΊππππππ πποΏ½ 43 = for all (2) is satisfied. Thus, should have no β1 πΎπΎ π π moreβ πΊπΊ ππthanππππβ 1 πΊπΊrotationππππππ axis.β βThen,ππ2 β theππππ options left for areπΊπΊππππππ 14 PGs: ( = π π 1,2,3,4,6) , , ( = 2,3,4,6) , , , , . For πΊπΊππππππ being these PGs,πΆπΆππ ππthe π π ππ ππβ π π 4 3ππ 6 ππππππ condition thatπΆπΆ πΆπΆ ππ (2) = πΆπΆ ππ πΆπΆ(2)ππ for all πΊπΊ is also satisfied. β1 πΎπΎ πΎπΎ 2 2 ππππππ Step 2 ππ ππ β ππππ ππ ππ β ππππ ππ β πΊπΊ Since is parallel to the axis of rotation of , and (2) contains spin π π rotations withπποΏ½ arbitrary rotation angle along . ThenπΊπΊ ππifππππ there isππππ a spin group of the
form Γ (2) , then couldπποΏ½ be chosen such that there is no spin πΎπΎ ππππππ 2 ππππππ rotationπΊπΊ at allοΏ½ inππ β ππππ, similarοΏ½ to theπΊπΊ chosen of in Γ such that πΎπΎ ππππππ ππππππ ππππππ 2 ππππππ contains no , describedπΊπΊ in section C1. πΊπΊ πΊπΊ ππ πΊπΊ
Thus, theππ options left for are or = so that there is no multiple π π ππ ππππππ 1 ππ 2 counting of the same types of fullπΊπΊ SPGs representedπΆπΆ πΆπΆ by ππ Γ (2) . And the πΎπΎ remaining βs to be considered are actually 90 typesπΊπΊππππππ includingοΏ½ππ2 β ππππ32 typeοΏ½-I MPGs
and 58 typeπΊπΊ-IIππππππ MPGs.
In conclusion, there are 90 crystallographic SPGs of the form Γ πΎπΎ ππππππ 2 (2) , shown in Table C13-C14. (2) is actually the πΊπΊ group,οΏ½ππ inβ πΎπΎ ππππSchoenfliesοΏ½ notation, acting on spin space.ππ2 βHermannππππ βMauguin notationπΆπΆβ isπ£π£ used in the following table, which write as .
πΆπΆβπ£π£ βππ
TABLE C13. 32 SPGs of the form Γ (2) with the corresponding πΎπΎ being . The first column listsπΊπΊ theππππππ οΏ½ππ 2correspondingβ ππππ οΏ½ to shown in the π π πΊπΊsecondππππππ column.πΆπΆ1 The third column lists πΊπΊallππ0 of the types of SPGsπΊπΊππππππ of the form
Γ (2) having shown in the second column. πΎπΎ πΊπΊ ππππππ οΏ½ππ2 β ππππ οΏ½ πΊπΊππππππ
= Γ (2)
πΎπΎ π π π π ππππππ 2 1ππ 0 ππ1ππππ πΊπΊ πΊπΊ οΏ½ ππ1 β ππππ οΏ½ πΊπΊ πΊπΊ 1 βππ 44 1 1 1 1 2 12 12βππ1 οΏ½ οΏ½ οΏ½ 1 1 βππ 1 2/ 21/ 21/ βππ 1 ππ ππ ππ 2 1 1 2 1 1 2βππ 1 ππ ππ ππ 222 1 212 12 1 212 12 βππ1 ππππ ππ ππ ππ ππ 1 1 1 1 1 1 βππ 1 4 1 1 4 1 1 1 4 1 1β ππ ππππππ ππ ππ ππ ππ ππ ππ 1 1 βππ 4 4 4 1 4/ 41/ 41/ βππ 1 οΏ½ οΏ½ οΏ½ 422 14 21 2 14 21 2βππ 1 ππ ππ ππ 1 1 1 1 1 1 βππ 4 4 4 1 42 1 41 2 1 1 41 2 1 βππ1 ππππ ππ ππ ππ ππ 4/ 4/1 1 1 4/1 1 1 βππ 1 οΏ½ ππ οΏ½ ππ οΏ½ ππ 3 1 1 31 1 1 1 31 11 βππ ππππππ ππ ππ ππ ππ ππ ππ 1 1 βππ 3 3 3 1 32 31 2 31 2βππ 1 οΏ½ οΏ½ οΏ½ 1 1 1 1 βππ 3 3 3 1 3 131 131 βππ1 ππ ππ ππ 6 1 16 1 16 βππ1 οΏ½ππ οΏ½ ππ οΏ½ ππ 6 16 16βππ1 οΏ½ οΏ½ οΏ½ 1 1 βππ 622 6 2 2 6 2 2 1 1 1 1 1 1 1 βππ 6/ 6/ 6/ 1 6 61 1 16 1 βππ 1 ππ ππ ππ 6 2 1 61 1 2 1 61 1 2 βππ1 ππππ ππ ππ ππ ππ 6/ 6/1 1 1 6/1 1 1 βππ 1 οΏ½ππ οΏ½ ππ οΏ½ ππ 23 1 1 2 13 1 1 1 2 13 1 1β ππ ππππππ ππ ππ ππ ππ ππ ππ 1 1 1 1 βππ 2/ 3 2/ 3 2/ 3 1 43 1 4 13 1 1 4 13 1 βππ1 πποΏ½ ππ οΏ½ ππ οΏ½ 432 1 41 31 2 1 41 31 2 βππ1 οΏ½ ππ οΏ½ ππ οΏ½ ππ 1 1 1 1 1 1 βππ 4/ 32/ 4/ 3 2/ 4/ 3 2/ 1 1 1 1 1 1 1 1 1 1 1 βππ πποΏ½ ππ ππ οΏ½ ππ ππ οΏ½ ππ
TABLE C14. 58 types of SPGs of the form Γ (2) with being πΎπΎ π π ππππππ ππππππ ( ). πΊπΊ οΏ½ππ2 β ππππ οΏ½ πΊπΊ ππ πΆπΆ ππ ππ2
45 = Γ (2)
πΎπΎ π π π π ππππππ 2 1ππ 0 ππ1ππππ πΊπΊ πΊπΊ 1 οΏ½ππ1 β ππππ οΏ½ πΊπΊ πΊπΊ 2 1οΏ½2 1οΏ½2βππ1 οΏ½ οΏ½ οΏ½ 1οΏ½ 1οΏ½ βππ 1 2/ 21οΏ½/ 21οΏ½/ βππ 1 ππ ππ ππ 2/ 12/1οΏ½ 12/1οΏ½ βππ1 ππ ππ ππ 2/ 1οΏ½2/1 1οΏ½2/1 βππ1 ππ ππ ππ 2 1οΏ½ 1οΏ½ 2 1οΏ½ 1οΏ½ 2βππ 1 ππ ππ ππ 2 1οΏ½ 1οΏ½ 12 1οΏ½ 1οΏ½ 12βππ1 ππππ ππ ππ ππ ππ 222 1 21οΏ½2 1οΏ½2 1 21οΏ½2 1οΏ½2 βππ1 ππππ ππ ππ ππ ππ 1 1οΏ½ 1οΏ½ 1 1οΏ½ 1οΏ½ βππ 1 1 1οΏ½ 1οΏ½ 1 1οΏ½ 1οΏ½ βππ1 ππππππ ππ ππ ππ ππ ππ ππ 1 1 1οΏ½ 1 1 1οΏ½ βππ1 ππππππ ππ ππ ππ ππ ππ ππ 4 1οΏ½ 1οΏ½ 4 1οΏ½ 1οΏ½ 1οΏ½ 4 1οΏ½ 1β ππ ππππππ ππ ππ ππ ππ ππ ππ 1οΏ½ 1οΏ½ βππ 4 4 4 1 4/ 41οΏ½/ 41οΏ½/ βππ 1 οΏ½ οΏ½ οΏ½ 4/ 1οΏ½4/1 1οΏ½4/1 βππ1 ππ ππ ππ 4/ 1οΏ½4/1οΏ½ 1οΏ½4/1οΏ½ βππ1 ππ ππ ππ 422 14 21οΏ½ 2 14 21οΏ½ 2βππ 1 ππ ππ ππ 1 1οΏ½ 1οΏ½ 1 1οΏ½ 1οΏ½ βππ 422 4 2 2 4 2 2 1 1οΏ½ 1 1οΏ½ 1οΏ½ 1 1οΏ½ βππ 4 4 4 1 4 141οΏ½ 1οΏ½ 141οΏ½ 1οΏ½ βππ1 ππππ ππ ππ ππ ππ 42 1οΏ½ 41 2 1οΏ½ 1οΏ½ 41 2 1οΏ½ βππ1 ππππ ππ ππ ππ ππ 42 141οΏ½21οΏ½ 141οΏ½21οΏ½ βππ1 οΏ½ ππ οΏ½ ππ οΏ½ ππ 42 1οΏ½41οΏ½21 1οΏ½41οΏ½21 βππ1 οΏ½ ππ οΏ½ ππ οΏ½ ππ 4/ 4/1οΏ½ 1 1οΏ½ 4/1οΏ½ 1 1οΏ½ βππ 1 οΏ½ ππ οΏ½ ππ οΏ½ ππ 4/ 1οΏ½4/1οΏ½ 1οΏ½ 1 1οΏ½4/1οΏ½ 1οΏ½ 1 βππ1 ππππππ ππ ππ ππ ππ ππ ππ 4/ 14/1οΏ½ 1 1 14/1οΏ½ 1 1 βππ1 ππππππ ππ ππ ππ ππ ππ ππ 4/ 1οΏ½4/1 1 1οΏ½ 1οΏ½4/1 1 1οΏ½ βππ1 ππππππ ππ ππ ππ ππ ππ ππ 4/ 14/1 1οΏ½ 1οΏ½ 14/1 1οΏ½ 1οΏ½ βππ1 ππππππ ππ ππ ππ ππ ππ ππ 3 1 1οΏ½ 31οΏ½ 1οΏ½ 1 1οΏ½ 31οΏ½ 1οΏ½1 βππ ππππππ ππ ππ ππ ππ ππ ππ 32 31οΏ½ 2 31οΏ½ 2βππ 1 οΏ½ οΏ½ οΏ½ 1 1οΏ½ 1 1οΏ½ βππ 3 3 3 1 3 131οΏ½ 131οΏ½ βππ1 ππ ππ ππ 3 131οΏ½ 131οΏ½ βππ1 οΏ½ππ οΏ½ ππ οΏ½ ππ 3 1οΏ½31 1οΏ½31 βππ1 οΏ½ππ οΏ½ ππ οΏ½ ππ 6 1οΏ½ 1οΏ½6 1οΏ½ 1οΏ½6 βππ1 οΏ½ππ οΏ½ ππ οΏ½ ππ 1οΏ½ 1οΏ½ βππ οΏ½ οΏ½ 46 οΏ½ 6 6 6 1 1οΏ½ 1οΏ½ βππ 622 6 2 2 6 2 2 1 1 1οΏ½ 1οΏ½ 1 1οΏ½ 1οΏ½ βππ 622 6 2 2 6 2 2 1 1οΏ½ 1 1οΏ½ 1οΏ½ 1 1οΏ½ βππ 6/ 6/ 6/ 1 6/ 1οΏ½6/1οΏ½ 1οΏ½6/1οΏ½ βππ1 ππ ππ ππ 6/ 1οΏ½6/1 1οΏ½6/1 βππ1 ππ ππ ππ 6 61 1οΏ½ 61 1οΏ½ βππ 1 ππ ππ ππ 6 161οΏ½ 1οΏ½ 161οΏ½ 1οΏ½ βππ1 ππππ ππ ππ ππ ππ 6 2 1οΏ½ 61 1οΏ½ 2 1οΏ½ 61 1οΏ½ 2 βππ1 ππππ ππ ππ ππ ππ 6 2 161οΏ½ 1οΏ½2 161οΏ½ 1οΏ½2βππ1 οΏ½ππ οΏ½ ππ οΏ½ ππ 6 2 1οΏ½61 1οΏ½2 1οΏ½61 1οΏ½2βππ1 οΏ½ππ οΏ½ ππ οΏ½ ππ 6/ 6/1οΏ½ 1οΏ½ 1 6/1οΏ½ 1οΏ½ 1 βππ 1 οΏ½ππ οΏ½ ππ οΏ½ ππ 6/ 1οΏ½6/1οΏ½ 1 1οΏ½ 1οΏ½6/1οΏ½ 1 1οΏ½ βππ1 ππππππ ππ ππ ππ ππ ππ ππ 6/ 1οΏ½6/1 1οΏ½ 1 1οΏ½6/1 1οΏ½ 1 βππ1 ππππππ ππ ππ ππ ππ ππ ππ 6/ 16/1 1οΏ½ 1οΏ½ 16/1 1οΏ½ 1οΏ½ βππ1 ππππππ ππ ππ ππ ππ ππ ππ 6/ 16/1οΏ½ 1 1 16/1οΏ½ 1 1 βππ1 ππππππ ππ ππ ππ ππ ππ ππ 2/ 3 1 21οΏ½/ 1οΏ½ 31οΏ½ 1 21οΏ½/ 1οΏ½ 31οΏ½ β1ππ ππππππ ππ ππ ππ ππ ππ ππ 43 1 4 1οΏ½3 1οΏ½ 1 4 1οΏ½3 1οΏ½ βππ1 πποΏ½ ππ οΏ½ ππ οΏ½ 432 1οΏ½ 41 31οΏ½ 2 1οΏ½ 41 31οΏ½ 2 βππ1 οΏ½ ππ οΏ½ ππ οΏ½ ππ 1οΏ½ 1 1οΏ½ 1οΏ½ 1 1οΏ½ βππ 4/ 32/ 4/ 3 2/ 4/ 3 2/ 1 4/ 32/ 1οΏ½4/1 131οΏ½2/1οΏ½ 1οΏ½4/1 131οΏ½2/1οΏ½ βππ1 πποΏ½ ππ ππ οΏ½ ππ ππ οΏ½ ππ 4/ 32/ 1οΏ½4/1οΏ½ 1οΏ½31οΏ½2/1 1οΏ½4/1οΏ½ 1οΏ½31οΏ½2/1 βππ1 πποΏ½ ππ ππ οΏ½ ππ ππ οΏ½ ππ 1 1οΏ½ 1οΏ½ 1 1οΏ½ 1 1οΏ½ 1οΏ½ 1 1οΏ½ βππ πποΏ½ ππ ππ οΏ½ ππ ππ οΏ½ ππ
47