-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 , Southern University of Science and Technology, Shenzhen 518055, China 2School of Physics, Peking University, Beijing 100871, People’s Republic of China 3National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China and Collaborative Innovation 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 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 , 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

𝑍𝑍manifests2 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 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 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 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 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 (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 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 ( ) permits more discrete or continuous values under a spatial rotation ( ) ,

π‘ˆπ‘ˆwhereπ’Žπ’Ž πœ‘πœ‘ and denote the rotation axes, and the real scalars and are𝐢𝐢𝒏𝒏 πœƒπœƒthe rotationπ’Žπ’Ž angles. 𝒏𝒏This leads to much more symmetry operations πœ‘πœ‘and moreπœƒπœƒ 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 momenta𝑍𝑍2 protected by spin group symmetries.

II. General single- 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 𝒑𝒑� wavefunctions are labeled by the single-valued representations of𝐻𝐻 the 2π‘šπ‘š( )-𝑉𝑉determined𝒓𝒓�

𝑉𝑉 𝒓𝒓� 3 SG, of which the rotational elements contain solely the spatial rotations ( ). When

the general SOC term = ( ( ) Γ— ) is added, neither spatial𝐢𝐢𝒏𝒏 πœƒπœƒ rotation 1 𝑠𝑠𝑠𝑠𝑠𝑠 2 2 ( ) nor spin rotation𝐻𝐻 ( 2π‘šπ‘š) alone𝑐𝑐 πœ΅πœ΅π‘‰π‘‰, but𝒓𝒓� only𝒑𝒑� β‹…a 𝝈𝝈�locked combination ( ) ( )

𝐢𝐢can𝒏𝒏 πœƒπœƒkeep invariant. Tπ‘ˆπ‘ˆheπ’Žπ’Ž resultingπœ‘πœ‘ spinor wavefunctions furnish the doubleπ‘ˆπ‘ˆπ’π’ πœƒπœƒ-valued𝐢𝐢𝒏𝒏 πœƒπœƒ representations𝐻𝐻𝑠𝑠𝑠𝑠𝑠𝑠 of the ( )-determined double SG. To describe the magnetic systems, we apply = (𝑉𝑉 )𝒓𝒓� under the framework of single-particle mean-field approximationπ»π»π‘šπ‘šπ‘šπ‘šπ‘šπ‘š [39-41]𝑺𝑺, 𝒓𝒓�whereβ‹… 𝝈𝝈� ( ) stands for the r-dependent exchange field due to the distribution of local magnetic𝑺𝑺 moments𝒓𝒓� and is, strictly speaking, not the spin of electrons but the spin of quasiparticles arising𝝈𝝈� from exchange correlation among electrons. Note that resembles the form of double-exchange model widely used to describe the magneticπ»π»π‘šπ‘šπ‘šπ‘šπ‘šπ‘š phase transitions of manganites [42-44]. If + + is considered, the locking between spin and spatial rotations still holds,𝐻𝐻 while𝐻𝐻𝑠𝑠𝑠𝑠𝑠𝑠 the𝐻𝐻 π‘šπ‘šπ‘šπ‘šπ‘šπ‘šfull symmetry description requires double MSG with the inclusion of time-reversal operation .

If we 𝑇𝑇consider the symmetry operations of + , i.e., a magnetic system without SOC, it is straightforward to prove that the 𝐻𝐻completeπ»π»π‘šπ‘šπ‘šπ‘šπ‘šπ‘š locking of spatial and spin rotations is no longer required. Instead, partially locked rotations, i.e., ( ) ( ),

with ( ) keeping ( ) invariant, could keep invariant and π‘ˆπ‘ˆthusπ’Žπ’Ž πœ‘πœ‘ the𝐢𝐢𝒏𝒏 totalπœƒπœƒ Hamiltonian𝐢𝐢𝒏𝒏 πœƒπœƒ . Note that𝑉𝑉 𝒓𝒓�( ) presents the effects ofπ»π»π‘šπ‘šπ‘šπ‘šπ‘šπ‘š a spatially distributed magnetic

moments residing at the 𝑺𝑺atomic𝒓𝒓� 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, formingπ‘ˆπ‘ˆπ’Žπ’Ž πœ‘πœ‘ spin groups𝐢𝐢𝒏𝒏 πœƒπœƒ, 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 ( π‘ˆπ‘ˆ) π’Žπ’Žformingπœ‘πœ‘ 𝑇𝑇𝑇𝑇 π’Žπ’Žimproperπœ‘πœ‘ 𝐢𝐢𝒏𝒏 πœƒπœƒrotations𝐼𝐼𝐼𝐼𝒏𝒏 πœƒπœƒ and mirror reflections, time-reversal𝐼𝐼 symmetry 𝐢𝐢T𝑛𝑛 canπœƒπœƒ be viewed as the β€œinversion symmetry” in the spin space. For simplicity, we first consider proper rotations only, i.e., { ( )|| ( )} . The partial coupling

between spin and spatial operations implies π‘ˆπ‘ˆpureπ’Žπ’Ž πœ‘πœ‘ spin𝐢𝐢𝒏𝒏 rotationπœƒπœƒ { ( )|| } and coupled spin-spatial rotation, the latter of which forms nontrivial SPGsπ‘ˆπ‘ˆπ’Žπ’Ž πœ‘πœ‘ contain𝐸𝐸 ing elements of the form { ( )|| ( )} with ( ) ( is identity rotation)

except identity element {π‘ˆπ‘ˆπ’Žπ’Ž|| πœ‘πœ‘}. Ref.𝐢𝐢𝒏𝒏 [38]πœƒπœƒ constructed𝐢𝐢𝒏𝒏 πœƒπœƒ allβ‰  nontrivial𝐸𝐸 𝐸𝐸 SPGs in 3D crystals, by combining the factor groups𝐸𝐸 𝐸𝐸 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 moments𝐷𝐷6 on each site in general𝐢𝐢6𝑧𝑧 reduces𝐢𝐢2π‘₯π‘₯ the symmetry. Considering MPG symmetry (with SOC), the only spin

configuration𝐷𝐷6 that maintains symmetry is the in-plane spin arrangement shown in Fig. 2(c), while for SPG symmetry,𝐷𝐷6 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 magnitude𝑑𝑑𝑧𝑧 but different coplanar directions, = ( [ ], [ ], 0) (see Fig. 2(a)). The elements of the Hamiltonian

𝑺𝑺areπ’Šπ’Š written𝑆𝑆 𝐢𝐢𝐢𝐢𝐢𝐢 asπœ™πœ™ 𝑖𝑖 𝑆𝑆𝑆𝑆𝑆𝑆 πœ™πœ™π‘–π‘–

, = , = = , + , + , ( ) , (1)

𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗+1 𝑖𝑖 π‘—π‘—βˆ’1 𝑖𝑖 𝑗𝑗 𝑖𝑖 𝑗𝑗 𝑧𝑧 𝛼𝛼𝛼𝛼 whereοΏ½ 𝑖𝑖we𝑧𝑧̂ βˆ™chooseπœŽπœŽβƒ— 𝛼𝛼� 𝐻𝐻the�𝑗𝑗 basis𝑧𝑧̂ βˆ™ πœŽπœŽβƒ— functions𝛽𝛽� 𝑑𝑑� 𝛿𝛿with local𝛿𝛿 spinοΏ½ 𝑋𝑋 quantizationοΏ½πœ™πœ™ βˆ’ πœ™πœ™ �𝛼𝛼𝛼𝛼 axis𝛿𝛿 directing𝐽𝐽𝐽𝐽𝜎𝜎 along

5 local magnetic moments, i.e., = ( [ ], [ ], 0) , and [ ] is defined as

𝑧𝑧̂𝑖𝑖 𝐢𝐢𝐢𝐢𝐢𝐢 πœ™πœ™π‘–π‘– 𝑆𝑆𝑆𝑆𝑆𝑆 πœ™πœ™π‘–π‘– 𝑋𝑋 πœƒπœƒ [ ] πœƒπœƒ πœƒπœƒ . We then check all the possible { ( )|| } and 𝐢𝐢𝐢𝐢𝐢𝐢 οΏ½2οΏ½ 𝑖𝑖 𝑆𝑆𝑆𝑆𝑆𝑆 οΏ½2οΏ½ 𝑋𝑋 πœƒπœƒ ≑ οΏ½ πœƒπœƒ πœƒπœƒ οΏ½ π‘ˆπ‘ˆπ’Žπ’Ž πœ‘πœ‘ 𝐢𝐢6𝑧𝑧 { ( )||𝑖𝑖 𝑆𝑆𝑆𝑆𝑆𝑆}οΏ½ 2operationsοΏ½ 𝐢𝐢𝐢𝐢𝐢𝐢 οΏ½2οΏ½ that leave the Hamiltonian invariant, and find that a spatial

rotationπ‘ˆπ‘ˆπ’Žπ’Ž πœ‘πœ‘ 𝐢𝐢2π‘₯π‘₯ could couple a spin rotation with being the order of 2πœ‹πœ‹ 2𝑝𝑝𝑝𝑝 rotation and𝐢𝐢𝒏𝒏 οΏ½ 𝑑𝑑 οΏ½= 0, 1, … 1 (see Supplementaryπ‘ˆπ‘ˆπ’Žπ’Ž οΏ½ Section𝑑𝑑 οΏ½ IV)𝑑𝑑. Consequently, there

are 7 inequivalent𝑝𝑝 types 𝑑𝑑ofβˆ’ spin configurations containing and spatial rotations, including one collinear ferromagnetic (FM), one collinear𝐢𝐢6𝑧𝑧 antiferromagnetic𝐢𝐢2π‘₯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 β€œpartially” 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 serve𝐢𝐢𝒏𝒏 asπœƒπœƒ a unifiedπ‘ˆπ‘ˆπ’Žπ’Ž πœ‘πœ‘theory 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— 𝑇𝑇 𝑝𝑝0 2 ( = { , }, β€œΓ—β€ denotes the internal direct product,𝐺𝐺 while β€œ ” denotes𝑆𝑆𝑆𝑆 (external)𝑍𝑍 𝑇𝑇 direct𝑍𝑍2 product𝐸𝐸 𝑇𝑇 of two groups, see Supplementary Section I). WithβŠ— SOC, the symmetry degrades to Γ— by adding the constraint of the complete spin-space 𝑇𝑇 𝑝𝑝 2 coupling (type II MPG, 𝐺𝐺32 grey𝑍𝑍 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). πΊπΊπ‘šπ‘šπ‘šπ‘š 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

𝐺𝐺coplanar𝑠𝑠𝑠𝑠𝑠𝑠 moments, there exists a boson-like time-reversal group that forms the trivial SPG = { , ( ) = }, where denotes complex conjugation, rendering the 𝐾𝐾 full SPG𝑍𝑍2 𝐸𝐸 𝑇𝑇=π‘ˆπ‘ˆπ’Žπ’Ž πœ‹πœ‹ Γ— 𝐾𝐾 . For col𝐾𝐾linear moments, the full SPGs is = 𝐾𝐾 𝑠𝑠𝑠𝑠 𝑛𝑛𝑛𝑛𝑛𝑛 2 𝑠𝑠𝑠𝑠 Γ— ( (𝐺𝐺2) 𝐺𝐺 ) with𝑍𝑍 an additional (2) rotational symmetry group along𝐺𝐺 the 𝐾𝐾 𝑛𝑛𝑛𝑛𝑛𝑛 2 𝐺𝐺common𝑆𝑆𝑆𝑆 directionβ‹Š 𝑍𝑍 of the spins (see Appendix𝑆𝑆𝑆𝑆 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. Similarlyπ‘ˆπ‘ˆπ’Žπ’Ž πœ‘πœ‘ , the𝐢𝐢𝒏𝒏 analogousπœƒπœƒ 𝜏𝜏 symmetry𝜏𝜏 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) β€² † β€² † The first𝐻𝐻 termβˆ‘π›Όπ›Ό 𝛽𝛽isοΏ½ βˆ‘the<𝑅𝑅 nearest𝑖𝑖 𝑅𝑅 𝑗𝑗> 𝑑𝑑 neighborπ‘Žπ‘Žπ‘…π‘… 𝑖𝑖 𝛼𝛼𝛿𝛿𝛼𝛼 𝛽𝛽hoppingπ‘Žπ‘Žπ‘…π‘… 𝑗𝑗 𝛽𝛽 , and𝐽𝐽 βˆ‘ 𝑅𝑅the𝑖𝑖 π‘Žπ‘Žsecond𝑅𝑅 𝑖𝑖 𝛼𝛼 𝑺𝑺 π’Šπ’Štermβ‹… 𝝈𝝈 𝛼𝛼countsπ›½π›½π‘Žπ‘Žπ‘…π‘… 𝑖𝑖 the𝛽𝛽� effect of the local magnetic moment . Fig. 2(k) and 2(l) show the SOC-free band structures of such a 6-band model (includingπ‘Ίπ‘Ίπ’Šπ’Š 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 ’ ’ that only supports 1D irreducible corepresentations at the K point.Instead, the

𝐢𝐢𝐢𝐢𝐢𝐢band degeneracyπ‘šπ‘š 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 representations𝑑𝑑𝑧𝑧 (2) and (4) including spin (the number in parentheses denotes the degree of𝐾𝐾 1degeneracy𝐾𝐾).5 By choosing the eigenstates of spatial rotoinversion as the basis functions and adding spin degrees

of freedom, we operate the spin-group𝐢𝐢6𝑧𝑧𝐼𝐼 generators { || |0}, { || |0} and { || |0} on the Hamiltonian and find that (4) π‘ˆπ‘ˆsplits3𝑧𝑧 𝐢𝐢 to6𝑧𝑧 three𝐼𝐼 levelsπ‘ˆπ‘ˆ2π‘₯π‘₯ with𝐢𝐢2π‘₯π‘₯ two 1D irreducibleπ‘‡π‘‡π‘ˆπ‘ˆ2𝑧𝑧 𝐼𝐼 corepresentation (1), (1), 𝐾𝐾and5 a 2D irreducible corepresentation 𝑠𝑠 𝑠𝑠 (2), which is consistent with𝐾𝐾1 Fig. 2𝐾𝐾(l)2 calculated by TB model (see Supplementary 𝑠𝑠 Section𝐾𝐾3 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,𝑍𝑍2 which could 𝑇𝑇allow topological classification belonging to AII class in 2D subspaces of the 3D Brillouin𝑍𝑍2 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 β€œquasi-Kramers degeneracy”. 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 𝑧𝑧 𝑛𝑛 𝑧𝑧 1 2 excluded). We find that only two symmetry operations𝑇𝑇 𝐼𝐼 in magneticπ‘‡π‘‡π‘ˆπ‘ˆ πœ‹πœ‹ materials𝐢𝐢 πœ‹πœ‹ 𝜏𝜏also exist in MSG, corresponding to AFM TI (e.g., MnBi2Te4) [51-55] and topological semimetals with symmetry-protected𝑍𝑍2 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} ,

𝑧𝑧 { || [ ]|0}, { || [ ]| / } contain both pure spatial rotations, and𝑇𝑇 the𝐢𝐢 spinπœ‹πœ‹ -1/2 π‘₯π‘₯ 001 001 1 2 time𝑇𝑇 π‘šπ‘š-reversal 𝑇𝑇, andπ‘šπ‘š are 𝜏𝜏thus spin-group symmetries because of the decoupling

between spin and𝑇𝑇 lattice. The (001) surface bands with { || ( )|0} symmetry are all doubly degenerate, and thus do not protect gapless surface𝑇𝑇 states𝐢𝐢𝑧𝑧 πœ‹πœ‹ 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

𝑧𝑧 systems before.π‘šπ‘š Among them, { || [ ]|0} could lead to classification for a

001 2 system and the corresponding 3D quantum𝑇𝑇 π‘šπ‘š spin Hall phases, as will𝑍𝑍 be discussed below.

The last symmetry { ( )|| | / } also supports magnetic TI, similar to 𝑧𝑧 π‘‡π‘‡π‘ˆπ‘ˆπ‘›π‘› πœ‹πœ‹ 𝐸𝐸 𝜏𝜏1 2 𝑍𝑍2 { || | / } , except that the surface Dirac point is located at (0, ) or ( , ) 𝑧𝑧 𝑇𝑇 𝐸𝐸 𝜏𝜏1 2 πœ‹πœ‹ πœ‹πœ‹ πœ‹πœ‹ 9 momenta in the momentum space.

We next take { || [ ]|0} and { ( )|| | / } as examples to illustrate the 𝑧𝑧 001 𝑛𝑛 1 2 new magnetic TI𝑇𝑇s andπ‘šπ‘š various unexpectedπ‘‡π‘‡π‘ˆπ‘ˆ πœ‹πœ‹ surface𝐸𝐸 𝜏𝜏 node structures. We start from a

3D Dirac𝑍𝑍2 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 insulator𝑺𝑺𝑧𝑧 phase at = /2 plane of the Brillion zone. Then, by building an AFM structure through cell-π‘˜π‘˜doubling𝑧𝑧 πœ‹πœ‹ (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, with𝑍𝑍 gapless Dirac surface states at 𝑧𝑧 001 𝑛𝑛 1 2 the𝑇𝑇 boundariesπ‘šπ‘š of all𝑇𝑇 2Dπ‘ˆπ‘ˆ planesπœ‹πœ‹ 𝐸𝐸 perpendicular𝜏𝜏 to axis. Consequently, it manifests surface Dirac node lines at = 0 or = lineπ‘˜π‘˜π‘§π‘§ for any surfaces perpendicular to

π‘₯π‘₯ π‘₯π‘₯ [ ] (Fig. 3(c) and 3(e)), whichπ‘˜π‘˜ is impossibleπ‘˜π‘˜ πœ‹πœ‹ in conventional TIs protected by or

π‘šπ‘š 001 𝑇𝑇 { || | / }. To examine the impact of each symmetry, we apply an in-plane FM canting 𝑧𝑧 𝑇𝑇 𝐸𝐸 𝜏𝜏1 2 to break { || [ ]|0} (Fig. 3(b)), then the surface node structure becomes a Dirac

001 point at (0𝑇𝑇, π‘šπ‘š) (Fig. 3(d) and 3(f)), which is consistent with the symmetry analysis shown in Tableπœ‹πœ‹ I. If { ( )|| | / } is broken by the dimerization of the two layers, 𝑧𝑧 𝑛𝑛 1 2 the Dirac point at (𝑇𝑇0π‘ˆπ‘ˆ, )πœ‹πœ‹ is 𝐸𝐸finally𝜏𝜏 gapped (Fig. 3(g) and 3(h)). Therefore, we demonstrate that unlike πœ‹πœ‹nonmagnetic materials and magnetic materials with SOC, magnetic materials with negligible SOC possess new topological classification

with unprecendented surface node structures protected by𝑍𝑍2 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 alongπ‘˜π‘˜ βˆ™ 𝑝𝑝the spin direction ( ), the βˆ’π‘–π‘–ΞΈπœŽπœŽπ‘§π‘§ π‘₯π‘₯ 𝑦𝑦 symmetry-𝑃𝑃allowed𝑃𝑃 𝜏𝜏 π‘–π‘–πœŽπœŽ 𝐾𝐾Hamiltonian𝑆𝑆𝑆𝑆 takes the form ( ) = ( ) + 𝑒𝑒( ) +

( ) + ( ) , where , , , and 𝐻𝐻, , π’Œπ’Œ, are𝑓𝑓 0Pauliπ’Œπ’Œ 𝜎𝜎 0matrices𝜏𝜏0 𝑓𝑓1 π’Œπ’Œacting𝜎𝜎0𝜏𝜏π‘₯π‘₯ on 𝑓𝑓spin2 π’Œπ’Œ and𝜎𝜎0𝜏𝜏 orbital𝑦𝑦 𝑓𝑓3 spacesπ’Œπ’Œ πœŽπœŽπ‘§π‘§πœπœ,𝑧𝑧 with the lastπœŽπœŽπ‘–π‘–= thre0 π‘₯π‘₯ 𝑦𝑦e𝑧𝑧 terms mutuallyπœπœπ‘–π‘–=0 π‘₯π‘₯ 𝑦𝑦 𝑧𝑧 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 SrMnBi𝑇𝑇 𝐼𝐼2 calculated𝑆𝑆𝑆𝑆 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 classification𝑇𝑇 withπ‘šπ‘š unprecedented surface nodal lines.

We next apply DFT𝑍𝑍2 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 β€² β€² of the two Dirac points at (0.280,0.280,0) and (0Ξ“οΏ½.293,0.293οΏ½ ,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 representativeπ‘šπ‘š 110 plane

(shown in green𝑍𝑍2 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,𝑍𝑍2 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

material𝑇𝑇 π‘šπ‘š choice110 here is to illustrate the new topological𝑇𝑇 phasesπ‘šπ‘š 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

𝑍𝑍2 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 𝑛𝑛 torque correlates the symmetry ofπœ’πœ’ 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 surfaceπ‘ˆπ‘ˆ 𝑛𝑛andπœ‹πœ‹ spin𝐸𝐸 𝜏𝜏 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) ( ) 𝑧𝑧 plane π‘˜π‘˜π‘§π‘§ 0,0 ( , 0) 𝑇𝑇 𝐸𝐸 𝜏𝜏1 2 ( , 0, ) and ( , , ) Dirac nodalπ‘œπ‘œπ‘œπ‘œ πœ‹πœ‹line at { ( )|| [ ]| / } (010) π‘₯π‘₯ πœ‹πœ‹ π‘˜π‘˜π‘§π‘§ lines πœ‹πœ‹ πœ‹πœ‹ π‘˜π‘˜π‘§π‘§ = π‘‡π‘‡π‘ˆπ‘ˆπ‘§π‘§ πœ‹πœ‹ π‘šπ‘š 001 𝜏𝜏1 2 π‘₯π‘₯ = 0 and = Possibleπ‘˜π‘˜ doubleπœ‹πœ‹ Dirac { || ( )|0} (001) π‘˜π‘˜π‘§π‘§ planesπ‘˜π‘˜ 𝑧𝑧 πœ‹πœ‹ point 𝑇𝑇 𝐢𝐢𝑧𝑧 πœ‹πœ‹ (0,0, ), (0, , ), Dirac nodal line at { || [ ]|0} ( , 0, ) 𝑧𝑧and ( , 𝑧𝑧, ) (xy0) π‘˜π‘˜ πœ‹πœ‹ π‘˜π‘˜ = 0 or = 𝑇𝑇 π‘šπ‘š 001 πœ‹πœ‹ π‘˜π‘˜π‘§π‘§ lines πœ‹πœ‹ πœ‹πœ‹ π‘˜π‘˜π‘§π‘§ π‘˜π‘˜π‘₯π‘₯ π‘˜π‘˜π‘₯π‘₯ πœ‹πœ‹ (0,0, ) (0, , ) Dirac nodal line at { || [ ]| / } (010) = 0 π‘₯π‘₯ π‘˜π‘˜π‘§π‘§ π‘Žπ‘Žlinesπ‘Žπ‘Žπ‘Žπ‘Ž πœ‹πœ‹ π‘˜π‘˜π‘§π‘§ 𝑇𝑇 π‘šπ‘š 001 𝜏𝜏1 2 π‘˜π‘˜π‘₯π‘₯ ( , 0, ) and ( , , ) { ( )|| [ ]| / } (010) Dirac nodal line at = π‘₯π‘₯ πœ‹πœ‹ π‘˜π‘˜π‘§π‘§ lines πœ‹πœ‹ πœ‹πœ‹ π‘˜π‘˜π‘§π‘§ π‘‡π‘‡π‘ˆπ‘ˆπ‘›π‘› πœ‹πœ‹ π‘šπ‘š 001 𝜏𝜏1 2 π‘˜π‘˜π‘₯π‘₯ πœ‹πœ‹

TRIM within = Dirac point at { ( )|| | / } (xy0) ( ) 𝑧𝑧 plane π‘˜π‘˜π‘§π‘§ πœ‹πœ‹ 0, ( , ) π‘‡π‘‡π‘ˆπ‘ˆπ‘›π‘› πœ‹πœ‹ 𝐸𝐸 𝜏𝜏1 2 πœ‹πœ‹ π‘œπ‘œπ‘œπ‘œ πœ‹πœ‹ πœ‹πœ‹

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,𝐢𝐢𝑛𝑛 spatialπœƒπœƒ and spin rotationsπ‘ˆπ‘ˆπ‘šπ‘š areπœ‘πœ‘ completely locked. For example, a spatial rotation by 2 /3 requires a simultaneous spin rotation by

2 /3 along the same axis. For the materialsπœ‹πœ‹ without SOC, the spin and spatial rotations areπœ‹πœ‹ 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 writtenπœ‹πœ‹ as { ( )|| ( )| } whereπœ‹πœ‹ the spatial and spin rotation axes and angles could be different.π‘ˆπ‘ˆπ‘šπ‘š πœ‘πœ‘ 𝐢𝐢𝑛𝑛 πœƒπœƒ 𝜏𝜏

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 generators𝐢𝐢6𝑧𝑧 are𝐢𝐢 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 complete𝐺𝐺𝑝𝑝0 locking between spin and spatial degrees of freedom; : MPG;𝐺𝐺𝑝𝑝 : full SPG; : nontrivial SPG. (j) a spinful kagome lattice with noncolinearπΊπΊπ‘šπ‘šπ‘šπ‘š AFM𝐺𝐺 spin𝑠𝑠𝑠𝑠 configuration𝐺𝐺𝑠𝑠𝑠𝑠 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 the𝐾𝐾 4parathesis represent the dimension of representations𝐾𝐾1 . The band eigenvalues of (l) at K include two two-fold degenerate points, belonging to 2D irreducible co-representation , and two 𝑠𝑠 nondegenerate points, belonging to and , respectively. 𝐾𝐾3 𝑠𝑠 𝑠𝑠 𝐾𝐾1 𝐾𝐾2

21

FIG. 3. magnetic topological insulators protected by SSG symmetries. (a-b) a

2 magnetic𝑍𝑍 system with (a) A-type AFM structure invariant under { || [ ]|0} and

𝑇𝑇 π‘šπ‘š 001 { ( )|| | / } SSG symmetry, (b) { ( )|| | / } SSG symmetry. (c-d) the 𝑧𝑧 𝑧𝑧 𝑛𝑛 1 2 𝑛𝑛 1 2 correspondingπ‘‡π‘‡π‘ˆπ‘ˆ πœ‹πœ‹ 𝐸𝐸 𝜏𝜏 [100] surface nodal structuresπ‘‡π‘‡π‘ˆπ‘ˆ in theπœ‹πœ‹ Brillouin𝐸𝐸 𝜏𝜏 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 𝑧𝑧 𝑛𝑛 1 2 layers. (h) the corresponding (100) surfaceπ‘‡π‘‡π‘ˆπ‘ˆ πœ‹πœ‹ band𝐸𝐸 𝜏𝜏dispersion with a gapped Dirac cone.

22

FIG. 4. topological phase protected by { || [ ]|0} symmetry in square-net

2 110 materials.𝑍𝑍 (a) crystal structure of AMnBi2 (A = Sr,𝑇𝑇 Caπ‘šπ‘š ). 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 symmetry𝑐𝑐 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 βˆ’1 line (denoted by cyan), represented by (0.οΏ½280,0οΏ½.280,0); and 4 Dirac points locatedΞ“ βˆ’ 𝑋𝑋 along line (denoted by blue), represented by (0.293,0.293,0.272) . (c)

Topological𝑀𝑀 βˆ’ nodal𝐺𝐺 -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 Brillouinπ‘šπ‘š zone110 parallel to [ ], as a 𝑍𝑍function2 of the momentum along the [110] direction. π‘šπ‘š 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 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) 𝑠𝑠 𝑇𝑇 𝑂𝑂 =𝑆𝑆𝑆𝑆{ , }, 𝑍𝑍 2 (A2) 𝑇𝑇 SO(3) = { ( )| = sin( )cos(𝑍𝑍2) +𝐸𝐸sin𝑇𝑇( )sin( ) + cos( ) , (0, ], π‘ˆπ‘ˆπ’π’ πœ”πœ” 𝒏𝒏 πœƒπœƒ (0,2 πœ‘πœ‘];𝒙𝒙� (0,2πœƒπœƒ ]}, πœ‘πœ‘ π’šπ’šοΏ½ πœƒπœƒ 𝒛𝒛 οΏ½ πœƒπœƒ ∈ πœ‹πœ‹ (A3)πœ‘πœ‘ ∈ where ( ) stands for spin rotationπœ‹πœ‹ πœ”πœ”with∈ rotationπœ‹πœ‹ axis and rotation angle . Such groupπ‘ˆπ‘ˆπ’π’ πœ”πœ” (3) is termed as the orthogonal group of spin𝒏𝒏 symmetries (OS). πœ”πœ” 𝑠𝑠 When considering𝑂𝑂 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) 𝑻𝑻 π‘₯π‘₯ 𝑦𝑦 𝑧𝑧 Then, spin group is𝑺𝑺 defined𝒓𝒓 𝑆𝑆 as 𝒓𝒓any𝑆𝑆 𝒓𝒓 𝑆𝑆 𝒓𝒓 of the external direct product of

a group with elements exerting𝐺𝐺𝑠𝑠 on three dimensional spatial coordinates (either PG or SG), denoted as , and the orthogonal groups of spin symmetries, i.e., (3): 𝑠𝑠 0 (3) = (SO(3) Γ— Z ). 𝑂𝑂 (A5) 𝑠𝑠 T 0⨂𝑂𝑂 0⨂ 2 24 For being PG , which is written as = { 0 ( ) | =𝑝𝑝0 ( ) ( ) + ( ) ( ) + ( ) , (0, ], π‘šπ‘š 𝑝𝑝0 𝐢𝐢𝒏𝒏 πœ”πœ” 𝐼𝐼 𝒏𝒏 𝑠𝑠𝑠𝑠𝑠𝑠 πœƒπœƒ(𝑐𝑐0𝑐𝑐𝑐𝑐,2 πœ‘πœ‘]; 𝒙𝒙� 𝑠𝑠𝑠𝑠𝑠𝑠(0,2πœƒπœƒ]𝑠𝑠𝑠𝑠𝑠𝑠, =πœ‘πœ‘0π’šπ’šοΏ½,1}, 𝑐𝑐 𝑐𝑐𝑐𝑐 πœƒπœƒ 𝒛𝒛� πœƒπœƒ ∈ πœ‹πœ‹(A6)

where ( ) stands for a spatialπœ‘πœ‘ ∈ rotationπœ‹πœ‹ πœ”πœ” operation∈ πœ‹πœ‹ withπ‘šπ‘š rotation axis and rotation- angle 𝐢𝐢,𝒏𝒏 whileπœ”πœ” stands for spatial inversion symmetry operation, every𝒏𝒏 subgroup of πœ”πœ” (3) 𝐼𝐼is called spin point group (SPG) , operations of which are 𝑠𝑠 denoted𝑝𝑝0⨂ as𝑂𝑂 𝐺𝐺𝑠𝑠𝑠𝑠 { || }, (A7) with (3) and . And 𝑔𝑔𝑠𝑠 𝑔𝑔 𝑠𝑠 𝑔𝑔s ∈ 𝑂𝑂 𝑔𝑔 ∈{ 𝑝𝑝||0 }{ || } = { || }, (A8)

𝑔𝑔𝑠𝑠{ 𝑔𝑔|| 𝑔𝑔}𝑠𝑠′ 𝑔𝑔=β€² { 𝑔𝑔𝑠𝑠||𝑔𝑔𝑠𝑠′ }𝑔𝑔𝑔𝑔. β€² (A9) βˆ’1 βˆ’1 βˆ’1 We define and acting𝑔𝑔𝑠𝑠 on𝑔𝑔 the spin𝑔𝑔𝑠𝑠 space𝑔𝑔 and coordinate space, respectively: 𝑔𝑔𝑠𝑠 𝑔𝑔 : ( ) , (A10) 𝑔𝑔𝑠𝑠: 𝑺𝑺 β†’ 𝑅𝑅(𝑔𝑔)𝑠𝑠 𝑺𝑺, (A11) where ( ) and ( ) is the representation𝑔𝑔 𝒓𝒓 β†’ 𝑅𝑅 𝑔𝑔 matrix𝒓𝒓 of and in 3D Euclidian space, respectively.𝑅𝑅 𝑔𝑔𝑠𝑠 𝑅𝑅Then,𝑔𝑔 for the r-dependent spin arrangement𝑔𝑔𝑠𝑠 (𝑔𝑔), we have : ( ) ( ) ( ), 𝑺𝑺 𝒓𝒓 (A12) 𝑔𝑔:𝑠𝑠 𝑺𝑺( )𝒓𝒓 β†’ 𝑅𝑅( 𝑔𝑔( 𝑠𝑠 𝑺𝑺)𝒓𝒓). (A13) βˆ’1 Thus, it is natural to define𝑔𝑔 the𝑺𝑺 action𝒓𝒓 β†’ 𝑺𝑺 of𝑅𝑅 SPG𝑔𝑔 operations𝒓𝒓 on the spin arrangement as { || }: ( ) ( ) ( ( ) ). (A14) βˆ’1 If we are interested in𝑔𝑔 𝑠𝑠the𝑔𝑔 symmetry𝑺𝑺 𝒓𝒓 β†’ 𝑅𝑅groups𝑔𝑔𝑠𝑠 𝑺𝑺 of𝑅𝑅 𝑔𝑔realistic𝒓𝒓 molecule or crystal, we must consider another scalar valued function ( ) which stands for the electric

potential function contributed by atomic nuclei. Since𝑓𝑓 𝒓𝒓 spin rotation have no influence on electric potential, the action of spin point group is defined as: { || }: ( ) ( ( ) ). (A15) βˆ’1 Note: For = (𝑔𝑔𝑠𝑠)/𝑔𝑔 =𝑓𝑓 𝒓𝒓( β†’) 𝑓𝑓, 𝑅𝑅we𝑔𝑔 denoted𝒓𝒓 ( ( )) = ( ) / ( ( )) = ( 𝑔𝑔),𝑠𝑠 withπ‘ˆπ‘ˆ 𝒏𝒏 πœ”πœ”( 𝑔𝑔) being𝐢𝐢𝒏𝒏 πœ”πœ”the in 3D𝑅𝑅 Euclidianπ‘ˆπ‘ˆπ’π’ πœ”πœ” space𝑅𝑅𝒏𝒏 πœ”πœ”with 𝑅𝑅rotation𝐢𝐢𝒏𝒏 πœ”πœ” axis and𝑅𝑅𝒏𝒏 rotationπœ”πœ” direction𝑅𝑅𝒏𝒏 πœ”πœ” represented by and rotation angle represented by . For = / = , we denote ( ) = / (𝒏𝒏) = , with being the identity πœ”πœ” 𝑔𝑔𝑠𝑠 𝑇𝑇 𝑔𝑔 𝐼𝐼 𝑅𝑅 𝑇𝑇 25βˆ’ 𝕀𝕀 𝑅𝑅 𝐼𝐼 βˆ’π•€π•€ 𝕀𝕀 matrix. For being space group , then every subgroup of (3) is called a 𝑠𝑠 spin space𝐺𝐺 group (SSG) . We𝐺𝐺𝑠𝑠 0can further write SSG operations𝐺𝐺𝑠𝑠0⨂ as𝑂𝑂 𝐺𝐺𝑠𝑠𝑠𝑠 { || | }, (A16) where { | } with denoting PG𝑔𝑔𝑠𝑠 operation𝑔𝑔 𝒕𝒕 and is a three-component real vector denoting𝑔𝑔 𝒕𝒕 ∈ 𝐺𝐺translation𝑔𝑔 operation. We have 𝒕𝒕 { || | }{ || | } = { || | + }, (A17)

𝑔𝑔{𝑠𝑠 𝑔𝑔||𝒕𝒕 | 𝑔𝑔}𝑠𝑠′ =𝑔𝑔′{𝒕𝒕′ ||𝑔𝑔𝑠𝑠𝑔𝑔|𝑠𝑠′ 𝑔𝑔𝑔𝑔(β€² 𝑔𝑔𝒕𝒕′) }.𝒕𝒕 (A18) βˆ’1 βˆ’1 βˆ’1 βˆ’1 The actions of { 𝑔𝑔||𝑠𝑠 |𝑔𝑔}𝒕𝒕 on spin 𝑔𝑔arrangement𝑠𝑠 𝑔𝑔 βˆ’ and𝑅𝑅 𝑔𝑔 scalar𝒕𝒕 potential satisfy { 𝑔𝑔𝑠𝑠|| 𝑔𝑔| 𝒕𝒕}: ( ) ( ) ( ( )( )), (A19) βˆ’πŸπŸ 𝑔𝑔𝑠𝑠 {𝑔𝑔 |𝒕𝒕| |𝑺𝑺}𝒓𝒓: (β†’)𝑅𝑅 𝑔𝑔𝑠𝑠(𝑺𝑺(𝑅𝑅 𝑔𝑔)( 𝒓𝒓 βˆ’))𝒕𝒕. (A20) βˆ’πŸπŸ MSGs and MPGs include𝑔𝑔𝑠𝑠 𝑔𝑔 𝒕𝒕antisymmetry𝑓𝑓 𝒓𝒓 β†’ 𝑓𝑓 𝑅𝑅(or𝑔𝑔 time-𝒓𝒓reversal)βˆ’ 𝒕𝒕 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 trivial𝐺𝐺𝑠𝑠 spin group𝐺𝐺 𝑠𝑠𝑠𝑠 and a nontrivial𝐺𝐺𝑠𝑠𝑠𝑠

spin group . Trivial spin group stands for the group formed𝐺𝐺𝑑𝑑𝑑𝑑 by pure spin operations{ |𝐺𝐺|𝑛𝑛𝑛𝑛|0} (or { || }), while the nontrivial spin group stands for the group that contain𝑔𝑔 𝑠𝑠no𝐸𝐸 pure spin 𝑔𝑔operations,𝑠𝑠 𝐸𝐸 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 𝑔𝑔𝑠𝑠 𝐸𝐸 𝑔𝑔𝑠𝑠 𝐸𝐸 { || |0𝑺𝑺}: 𝒓𝒓 ( ) ( ) ( ). (B1) To analyze the trivial spin𝑔𝑔 𝑠𝑠groups𝐸𝐸 for𝑺𝑺 𝒓𝒓differentβ†’ 𝑅𝑅 𝑔𝑔 spin𝑠𝑠 𝑺𝑺 arrangements,𝒓𝒓 we divide all of the pure spin operations into 4 types, i.e., = ( ) ( 0) , = , =

( ) and = ( ) ( 0 𝑔𝑔𝑠𝑠), andπ‘ˆπ‘ˆπ’›π’› οΏ½analyzeπœ”πœ” πœ”πœ” theβ‰  conditions𝑔𝑔𝑠𝑠 for𝑇𝑇 𝑔𝑔𝑠𝑠( ) π‘ˆπ‘ˆto𝒙𝒙� beπœ‹πœ‹ invariants𝑇𝑇 𝑔𝑔𝑠𝑠 underπ‘ˆπ‘ˆπ’›π’›οΏ½ themπœ”πœ” 𝑇𝑇 separately.πœ”πœ” β‰  π‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž (Weπœ”πœ” β‰ chooseπœ‹πœ‹ specific axes of spin rotations𝑺𝑺 𝒓𝒓to simplify our analysis)

Type-1: = ( ) for 0

If 𝑔𝑔𝑠𝑠= π‘ˆπ‘ˆ(𝒛𝒛� πœ”πœ”) for πœ”πœ” β‰ 0 , and ( ) is invariant under such operation, i.e., ( ) = 𝑔𝑔𝑠𝑠( (π‘ˆπ‘ˆπ’›π’›οΏ½))πœ”πœ” ( ) . Then,πœ”πœ” β‰  there are𝑺𝑺 𝒓𝒓two types of spin arrangements that have 𝑺𝑺 𝒓𝒓( ) symmetry.𝑅𝑅 π‘ˆπ‘ˆπ’›π’›οΏ½ πœ”πœ” 𝑺𝑺 𝒓𝒓 π‘ˆπ‘ˆπ’π’ 1.πœ”πœ” If ( ) is zero for all , then ( ) obviously have ( ) symmetry. Spin arrangements𝑺𝑺 𝒓𝒓 having this𝒓𝒓 property𝑺𝑺 are𝒓𝒓 called nonmagneticπ‘ˆπ‘ˆπ’›π’›οΏ½ spinπœ”πœ” arrangements. 2. If ( ) does not belong to nonmagnetic spin arrangement, the spin

configuration𝑺𝑺 𝒓𝒓 ( ) with ( ) parallel the rotation direction for all have ( ) symmetry𝑺𝑺 𝒓𝒓. We define𝑺𝑺 𝒓𝒓 such spin arrangements as collinear𝒓𝒓 spin arrangementsπ‘ˆπ‘ˆπ’›π’›οΏ½ πœ”πœ” . Type-2: =

If 𝑔𝑔𝑠𝑠= 𝑇𝑇, then the spin arrangements invariant under { || |0} should satisfy ( ) = 𝑔𝑔𝑠𝑠 ( 𝑇𝑇), indicating nonmagnetic spin arrangements. 𝑇𝑇 𝐸𝐸 𝑺𝑺Type𝒓𝒓 -3:βˆ’ 𝑺𝑺 𝒓𝒓= ( ) For 𝑔𝑔𝑠𝑠 =π‘ˆπ‘ˆπ’™π’™οΏ½ (πœ‹πœ‹ )𝑇𝑇 , then the spin arrangements invariant under { ( ) || |0} 𝑔𝑔𝑠𝑠 π‘ˆπ‘ˆπ’™π’™οΏ½ πœ‹πœ‹ 𝑇𝑇 1 0 0 1 0 0 π‘ˆπ‘ˆπ’™π’™οΏ½ πœ‹πœ‹ 𝑇𝑇 𝐸𝐸 should satisfy ( ) = 0 1 0 ( ) = 0 1 0 ( ) . Then, the spin 0 0 1 βˆ’0 0 1 𝑺𝑺 𝒓𝒓 βˆ’ οΏ½ βˆ’ οΏ½ 𝑺𝑺 𝒓𝒓 οΏ½ οΏ½ 𝑺𝑺 𝒓𝒓 arrangements which have x componentβˆ’ being zero for all could have such symmetry. 27 𝒓𝒓 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.

Theπ‘ˆπ‘ˆ 𝒏𝒏spinπœ‹πœ‹ 𝑇𝑇arrangements𝐸𝐸 that are not coplanar, collinear or nonmagnetic are called noncoplanar spin arrangements. Type-4: = ( ) ( 0 )

A spin𝑔𝑔𝑠𝑠 arrangementπ‘ˆπ‘ˆπ’›π’›οΏ½ πœ”πœ” 𝑇𝑇 πœ”πœ” β‰ that π‘Žπ‘Žisπ‘Žπ‘Žπ‘Žπ‘Ž invariantπœ”πœ” β‰  πœ‹πœ‹ under { ( ) || |0} should satisfy 0 π‘ˆπ‘ˆπ‘§π‘§ πœ”πœ” 𝑇𝑇 0𝐸𝐸 ( ) = 0 ( ) = 0 ( ) , it is easy 𝑐𝑐𝑐𝑐𝑐𝑐0 πœ”πœ” βˆ’π‘ π‘ π‘ π‘ π‘ π‘ 0 πœ”πœ” 1 βˆ’π‘π‘π‘π‘π‘π‘0 πœ”πœ” 𝑠𝑠𝑠𝑠𝑠𝑠0 πœ”πœ” 1 𝑺𝑺 𝒓𝒓 βˆ’ �𝑠𝑠𝑠𝑠𝑠𝑠 πœ”πœ” 𝑐𝑐𝑐𝑐𝑐𝑐 πœ”πœ” οΏ½ 𝑺𝑺 𝒓𝒓 οΏ½βˆ’π‘ π‘ π‘ π‘ π‘ π‘  πœ”πœ” βˆ’π‘π‘π‘π‘π‘π‘ πœ”πœ” οΏ½ 𝑺𝑺 𝒓𝒓 ( ) ( ) = 0 to check that such equation has no solution for unless βˆ’ , indicating nonmagnetic spin arrangements. 𝑺𝑺 𝒓𝒓 𝑺𝑺 𝒓𝒓 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) 𝑠𝑠 2. Collinear spin arrangements 𝐺𝐺𝑑𝑑𝑑𝑑 𝑂𝑂 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 spin𝑆𝑆𝑆𝑆 arrangement≑ π‘ˆπ‘ˆπ’›π’›οΏ½ πœ”πœ”is πœ”πœ”also∈ invariantπœ‹πœ‹ under operation ( ) , where = + ( (0, ]) could be any direction perpendicularπ‘ˆπ‘ˆ 𝒏𝒏toπœ‹πœ‹ the𝑇𝑇 spin direction.𝒏𝒏 𝑐𝑐 𝑐𝑐𝑐𝑐𝑐𝑐We 𝒙𝒙�choose𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 a π’šπ’šοΏ½specificπœƒπœƒ ∈ directionπœ‹πœ‹ and then ( ) symmetry could generate a trivial spin group defined as 𝒙𝒙� 𝐾𝐾 { , ( ) }π‘ˆπ‘ˆ. 𝒙𝒙�Theπœ‹πœ‹ n𝑇𝑇 the full trivial spin group is the internal of 𝑍𝑍2(2≑) and𝐸𝐸 π‘ˆπ‘ˆ 𝒙𝒙� πœ‹πœ‹, i.e.,𝑇𝑇 𝑆𝑆𝑆𝑆 𝐾𝐾 𝑍𝑍2 = SO(2) . (B3) 𝐾𝐾 𝐺𝐺𝑑𝑑𝑑𝑑 28 β‹Š 𝑍𝑍2 We note that the internal semidirect product is because SO(2) is a of SO(2) while is not. 𝐾𝐾 𝐾𝐾 3. Coplanar spin arrangementsβ‹Š 𝑍𝑍2 𝑍𝑍 2 From the discussion above, the trivial spin group of this type of spin arrangement is = { , ( ) }, (B4) 𝐾𝐾 where denotes the direction𝐺𝐺 𝑑𝑑perpendicular𝑑𝑑 𝑍𝑍2 ≑ 𝐸𝐸 toπ‘ˆπ‘ˆπ’π’οΏ½ theπœ‹πœ‹ plane𝑇𝑇 of the spin arrangements. 4. Noncoplanar𝒏𝒏� spin arrangements The trivial spin group is this type has only identity element

= { }. (B5)

𝐺𝐺𝑑𝑑𝑑𝑑 𝐸𝐸 C. Full crystallographic spin point groups for collinear and coplanar

spin arrangements

Crystallographic SPGs are SPGs being subgroups of (3) where is 𝑠𝑠 one of the 32 crystallographic PGs. Construction of nontrivial𝑝𝑝0⨂ SPGs𝑂𝑂 from 32 PGs𝑝𝑝0 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 in spin space and affine group in physical space, i.e., (3) (3) , which are sufficient to describe noncoplanar spin arrangements according𝐺𝐺𝐺𝐺 βŠ— 𝐺𝐺to𝐺𝐺 𝐺𝐺the 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

𝐺𝐺𝑠𝑠𝑠𝑠 𝐺𝐺 𝑛𝑛b𝑛𝑛𝑛𝑛ecause𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 is always a normal subgroup of . This group 𝐺𝐺could𝑛𝑛𝑛𝑛𝑛𝑛 ⋉ also𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 be written as𝐺𝐺 direct𝑑𝑑𝑑𝑑𝑑𝑑 product Γ— by proper selection𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 of such that is also a normal subgroup of 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑. Thus, to classify the point𝐺𝐺 groups𝑛𝑛𝑛𝑛𝑛𝑛 of the form𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 Γ— , we have to find all𝐺𝐺 the𝑛𝑛𝑛𝑛𝑛𝑛 types𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 of nontrivial SPG (denoted as ) that could𝐺𝐺 perform𝑛𝑛𝑛𝑛𝑛𝑛 𝐺𝐺 𝑑𝑑internal𝑑𝑑𝑑𝑑 direct product with trivial spin groups . Since 598 𝐺𝐺types𝑛𝑛𝑛𝑛𝑛𝑛 of nontrivial SPG are complete and any SPGs can be written as 𝐺𝐺the𝑑𝑑𝑑𝑑𝑑𝑑 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 in𝐺𝐺0 theβŠ— 𝑠𝑠 𝑂𝑂following 2-step𝐺𝐺0 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 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 βˆ’1 𝐺𝐺and𝑑𝑑𝑑𝑑𝑑𝑑 = for all , and then𝑔𝑔 𝐺𝐺 𝑛𝑛determine𝑛𝑛𝑛𝑛𝑔𝑔 𝐺𝐺𝑛𝑛 𝑛𝑛𝑛𝑛the correspond𝑔𝑔 ∈ 𝐺𝐺ing𝑑𝑑𝑑𝑑𝑑𝑑 βˆ’1 β„Ž ’s𝐺𝐺 𝑑𝑑(oneπ‘‘π‘‘π‘‘π‘‘β„Ž nontrivial𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 SPG mightβ„Ž ∈ 𝐺𝐺 𝑛𝑛corresponds𝑛𝑛𝑛𝑛 to several full SPGs). Since both 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛𝐺𝐺 𝑑𝑑and𝑑𝑑𝑑𝑑 are subgroups of (3) , and = {{ || }} , The 𝑠𝑠 𝐺𝐺condition𝑛𝑛𝑛𝑛𝑛𝑛 𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 = for all 𝑝𝑝0⨂𝑂𝑂 and 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 ∩ 𝐺𝐺𝑑𝑑=𝑑𝑑𝑑𝑑 𝐸𝐸for𝐸𝐸 all βˆ’1 βˆ’1 implies𝑔𝑔 that𝐺𝐺𝑛𝑛 𝑛𝑛𝑛𝑛𝑔𝑔 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 is a group 𝑔𝑔and∈ 𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 =β„Ž πΊπΊπ‘‘π‘‘π‘‘π‘‘π‘‘π‘‘Γ—β„Ž 𝐺𝐺. 𝑑𝑑Then,𝑑𝑑𝑑𝑑 we getβ„Ž all∈ types𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 of full SPGs represented𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 by Γ— 𝐺𝐺𝑛𝑛.𝑛𝑛𝑛𝑛 𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 Step 2: We consider full SPGs𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 obtained𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 from Step 1, represented as groups Γ— , Γ— , …, Γ— , with , , … and 1 2 𝑛𝑛 1 2 𝑛𝑛 𝐺𝐺belonging𝑛𝑛𝑛𝑛𝑛𝑛 𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 to different𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 types𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 of nontrivial𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 SPGs,𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 but with𝐺𝐺 𝑛𝑛𝑛𝑛𝑛𝑛 ×𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛, Γ— 𝐺𝐺𝑛𝑛,…𝑛𝑛𝑛𝑛 1 2 and Γ— actually belonging to the same type of 𝐺𝐺full𝑛𝑛𝑛𝑛𝑛𝑛 SPGs.𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 Then,𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 we 𝐺𝐺choose𝑑𝑑𝑑𝑑𝑑𝑑 𝑛𝑛 𝑛𝑛𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑𝑑𝑑 one 𝐺𝐺of the 𝐺𝐺 Γ— to represent this full SPG. Or in other word, eliminate multiple 𝑖𝑖 𝑛𝑛𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑𝑑𝑑 counting of𝐺𝐺 the equivalent𝐺𝐺 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 𝐾𝐾 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 𝑍𝑍2 30 described by Γ— ( (2)) . Next, we separately classify full SPGs for 𝐾𝐾 𝑛𝑛𝑛𝑛𝑛𝑛 2 coplanar spin arrangements𝐺𝐺 𝑍𝑍 ⋉ and𝑆𝑆𝑆𝑆 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 𝐾𝐾 𝑑𝑑𝑑𝑑𝑑𝑑 2 consider the relative directions of spin rotation axis𝐺𝐺 and space𝑍𝑍 rotation axis in the following derivation because he definition of β€œtype of spin group” 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 {{ || }, { ( )|| }}, 𝐾𝐾 the rotation axis should be either parallel or perpendicular𝑍𝑍2 ≑ to every𝐸𝐸 𝐸𝐸 spin𝑇𝑇 rotationπ‘ˆπ‘ˆπ’π’ πœ‹πœ‹ 𝐸𝐸axis of the spin rotation𝒏𝒏 part of , i.e., . Otherwise, the condition that = 𝑠𝑠 𝐾𝐾 should be invariant under 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 will 𝐺𝐺not𝑛𝑛𝑛𝑛𝑛𝑛 be satisfied. Thus, this step excludes𝐺𝐺𝑑𝑑𝑑𝑑𝑑𝑑 the𝑍𝑍2

nontrivial SPGs whose 𝐺𝐺spin𝑛𝑛𝑛𝑛𝑛𝑛 part are polyhedral group, including 𝑠𝑠 , , , , because we cannot find𝐺𝐺𝑝𝑝 a direction that is either parallel or

𝑇𝑇perpendiπ‘‡π‘‡β„Ž 𝑇𝑇𝑑𝑑 cular𝑂𝑂 π‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž to 𝑂𝑂allβ„Ž of the rotation axis. Thus, the options left for the spin part are 27 axial groups: ( = 𝑠𝑠 1,2,3,4,6) , ( = 2,3,4,6) , , ( = 2,3,4𝐺𝐺,6𝑛𝑛)𝑛𝑛𝑛𝑛 , ( = 2, 3, 4, 6), , 𝐢𝐢𝑛𝑛(𝑛𝑛 =

2, 3, 4, 6), 𝐷𝐷,𝑛𝑛 𝑛𝑛, 𝐢𝐢 𝑖𝑖 πΆπΆπ‘›π‘›β„Ž 𝑛𝑛 π·π·π‘›π‘›β„Ž 𝑛𝑛 𝐢𝐢𝑠𝑠 𝐢𝐢𝑛𝑛𝑛𝑛 𝑛𝑛 For 𝑆𝑆4 𝐢𝐢being3𝑖𝑖 𝑆𝑆6 π‘Žπ‘Ždifferentπ‘Žπ‘Žπ‘Žπ‘Ž 𝐷𝐷3𝑑𝑑 groups, the direction is constrained differently in 𝑠𝑠 𝑛𝑛𝑛𝑛𝑛𝑛 order that 𝐺𝐺the condition that = for all 𝒏𝒏 is satisfied. We separate βˆ’1 𝐾𝐾 𝐾𝐾 2 2 𝑛𝑛𝑛𝑛𝑛𝑛 the ways is constrained into𝑔𝑔 5𝑍𝑍 cases.𝑔𝑔 𝑍𝑍 𝑔𝑔 ∈ 𝐺𝐺

𝒏𝒏 Case1: For being or , is not constrained. 𝑠𝑠 Case2: For𝐺𝐺 𝑛𝑛𝑛𝑛𝑛𝑛 𝐢𝐢1 being𝐢𝐢𝑖𝑖 𝒏𝒏 = {{ || }, { ( )| }} or = 𝑠𝑠 {{ || }, { ( )| }} 𝐺𝐺or𝑛𝑛𝑛𝑛𝑛𝑛 = {{ || }, { 𝐢𝐢(𝑠𝑠 )| }𝐸𝐸, { 𝐸𝐸 (𝑇𝑇)π‘ˆπ‘ˆ|𝒛𝒛� }πœ‹πœ‹, { 𝐸𝐸| }} there is 𝐢𝐢only2

one𝐸𝐸 two𝐸𝐸 -foldπ‘ˆπ‘ˆπ’›π’›οΏ½ πœ‹πœ‹spin𝐸𝐸 rotation𝐢𝐢2β„Ž ( 𝐸𝐸) in𝐸𝐸 π‘ˆπ‘ˆπ’›π’›οΏ½ . πœ‹πœ‹Thus𝐸𝐸 𝑇𝑇 π‘ˆπ‘ˆwhich𝒛𝒛� πœ‹πœ‹ 𝐸𝐸could𝑇𝑇 be𝐸𝐸 either parallel or 𝑠𝑠 π‘ˆπ‘ˆπ’›π’›οΏ½ πœ‹πœ‹ 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 𝒏𝒏 31 perpendicular to . (When is perpendicular to , then groups corresponding to

different belongs𝒛𝒛� to the same𝒏𝒏 types up to conjugate𝒛𝒛� transformations ) Case3: For𝒏𝒏 being the groups which have rotations of order larger than 2, 𝑠𝑠 should be parallel𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 to the principal axis of . 𝒏𝒏 𝑠𝑠 Case4: For being , should𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 be parallel to one of the 2-fold spin 𝑠𝑠 rotation axis.𝐺𝐺 𝑛𝑛𝑛𝑛𝑛𝑛 𝐷𝐷2 π‘œπ‘œπ‘œπ‘œ 𝐷𝐷2β„Ž 𝒏𝒏 Case5: For being , should either be perpendicular to one mirror or 𝑠𝑠 parallel to the𝐺𝐺 two𝑛𝑛𝑛𝑛𝑛𝑛-fold rotation𝐢𝐢2𝑣𝑣 axis.𝒏𝒏 It is easy to see that for the 5 cases, the condition that = for all βˆ’1 is also satisfied. Then, we get all of the types of SPGs𝑔𝑔 that𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 can𝑔𝑔 be𝐺𝐺 𝑛𝑛written𝑛𝑛𝑛𝑛 as 𝐾𝐾 𝑔𝑔 ∈ 𝑍𝑍×2 with some types possibly being identical. 𝐾𝐾 𝑛𝑛𝑛𝑛𝑛𝑛 2 Step𝐺𝐺 2 𝑍𝑍 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

𝑛𝑛𝑛𝑛𝑛𝑛 of { ( )|| } with those operations,𝑇𝑇 𝐺𝐺 for the full group being Γ— . That is 𝐾𝐾 𝒏𝒏 𝑛𝑛𝑛𝑛𝑛𝑛 2 to say,𝑇𝑇 ifπ‘ˆπ‘ˆ weπœ‹πœ‹ have𝐸𝐸 a spin point group Γ— with been a nontrivial𝐺𝐺 𝑍𝑍 SPG that 1 𝐾𝐾 1 𝑛𝑛𝑛𝑛𝑛𝑛 2 𝑛𝑛𝑛𝑛𝑛𝑛 could be written as the form 𝐺𝐺 = +𝑍𝑍 { || } 𝐺𝐺 with being the 1 1 subgroup of of order 2 that𝐺𝐺𝑛𝑛 does𝑛𝑛𝑛𝑛 not𝐻𝐻 contain𝑇𝑇 𝐸𝐸 �𝐺𝐺, 𝑛𝑛then,𝑛𝑛𝑛𝑛 βˆ’ we𝐻𝐻 οΏ½have: 𝐻𝐻 1 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 Γ— 𝑇𝑇 1 𝐾𝐾 𝑛𝑛𝑛𝑛𝑛𝑛 2 =𝐺𝐺( + {𝑍𝑍 || } ) Γ— {{ || }, { ( )|| }} 1 𝑛𝑛𝑛𝑛𝑛𝑛 𝒏𝒏 = (𝐻𝐻 + {𝑇𝑇 (𝐸𝐸 )οΏ½|𝐺𝐺| } βˆ’ 𝐻𝐻 οΏ½ )𝐸𝐸× {𝐸𝐸{ ||π‘‡π‘‡π‘ˆπ‘ˆ}, { πœ‹πœ‹ (𝐸𝐸 )|| }} 1 𝒏𝒏 𝑛𝑛𝑛𝑛𝑛𝑛 𝒏𝒏 = 𝐻𝐻 Γ— π‘ˆπ‘ˆ , πœ‹πœ‹ 𝐸𝐸 οΏ½ 𝐺𝐺 βˆ’ 𝐻𝐻 οΏ½ 𝐸𝐸 𝐸𝐸 𝑇𝑇 π‘ˆπ‘ˆ πœ‹πœ‹ 𝐸𝐸 (C1) 2 𝐾𝐾 with containing𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 no𝑍𝑍2 . This implies that in all of the SPGs of the form 2 ×𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 can be chosen 𝑇𝑇such that the corresponding𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 to is formed by 𝐾𝐾 𝑠𝑠 𝑛𝑛𝑛𝑛𝑛𝑛 2 𝑛𝑛𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛𝑛𝑛 pure𝐺𝐺 spin𝑍𝑍 rotations. 𝐺𝐺 𝐺𝐺

Furthermore, it is obvious that two SPGs, Γ— and Γ— , with 𝐴𝐴 𝐾𝐾 𝐡𝐡 𝐾𝐾 𝐴𝐴 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 𝑍𝑍2 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 𝑍𝑍2 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 32 and being different types and containing no , should be different types of SPG. 𝐡𝐡 Thus, 𝐺𝐺we𝑛𝑛𝑛𝑛𝑛𝑛 can use all of the nontrivial SPGs 𝑇𝑇that have spin part being 9 𝑠𝑠 axial PGs of pure spin rotation ( = 1,2,3,4𝐺𝐺,𝑛𝑛6𝑛𝑛𝑛𝑛) and ( = 2,3,4,6)𝐺𝐺 𝑛𝑛to𝑛𝑛𝑛𝑛 construct all types of SPGs of the form 𝐢𝐢𝑛𝑛 𝑛𝑛× to avoid multiple𝐷𝐷𝑛𝑛 counting𝑛𝑛 of the same type 𝐾𝐾 𝑛𝑛𝑛𝑛𝑛𝑛 2 of group. Then there are 4 cases𝐺𝐺 left: 𝑍𝑍

Case1: For = , could be random directions which corresponds the same 𝑠𝑠 types of SPG𝐺𝐺. 𝑛𝑛𝑛𝑛𝑛𝑛 𝐢𝐢1 𝒏𝒏 Case2: For = = {{ || }, { ( )| }} , which could be either parallel or 𝑠𝑠 perpendicular𝐺𝐺 to𝑛𝑛𝑛𝑛𝑛𝑛 axis.𝐢𝐢2 𝐸𝐸 𝐸𝐸 π‘ˆπ‘ˆπ’›π’›οΏ½ πœ‹πœ‹ 𝐸𝐸 𝒏𝒏

Case3: For 𝑧𝑧= = {{ || }, { ( )| }, { ( )| }, { ( )| }} , could be 𝑠𝑠 𝑛𝑛𝑛𝑛𝑛𝑛 2 𝒙𝒙� π’šπ’šοΏ½ 𝒛𝒛� parallel to one𝐺𝐺 of the𝐷𝐷 3 two-𝐸𝐸fold𝐸𝐸 rotationπ‘ˆπ‘ˆ πœ‹πœ‹ axes.𝐸𝐸 π‘ˆπ‘ˆ Theseπœ‹πœ‹ 𝐸𝐸 3 twoπ‘ˆπ‘ˆ -foldπœ‹πœ‹ 𝐸𝐸 rotation𝒏𝒏 axes are equivalent for . But they are not necessarily equivalent for the whole nontrivial 𝑠𝑠 SPG . Thus,𝐺𝐺 𝑛𝑛the𝑛𝑛𝑛𝑛 groups which separately have parallel to , and axis could 𝐺𝐺belong𝑛𝑛𝑛𝑛𝑛𝑛 to the same types of SPG or different types𝒏𝒏 of SPG. π‘₯π‘₯ 𝑦𝑦 𝑧𝑧 Case4: For being one of the left 6 groups ( = 3,4,6) and ( = 3,4,6), 𝑠𝑠 has to be parallel𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 to the principal axis . 𝐢𝐢𝑛𝑛 𝑛𝑛 𝐷𝐷𝑛𝑛 𝑛𝑛

𝒏𝒏 Thus, we can classify the types of full𝑧𝑧 SPGs of the form Γ— into 11 types, 𝐾𝐾 𝑛𝑛𝑛𝑛𝑛𝑛 2 shown in the Table C1. Finally, we get 252 types of crystallographic𝐺𝐺 𝑍𝑍SPGs of the form

Γ— . These SPGs are listed in Table C2-C12. Note that the operation 𝐾𝐾 𝑛𝑛𝑛𝑛𝑛𝑛 2 {𝐺𝐺 ( 𝑍𝑍)|| } is actually a mirror operation in spin space, and thus is denoted by usingπ‘‡π‘‡π‘ˆπ‘ˆπ’π’ Hermannπœ‹πœ‹ 𝐸𝐸 –Mauguin notation. π‘šπ‘š

TABLE C1. Classifications of the SPGs of the form Γ— , with 𝐾𝐾 𝐾𝐾 {{ || }, { ( )|| }} and the corresponding being𝐺𝐺𝑛𝑛 𝑛𝑛𝑛𝑛one 𝑍𝑍of2 the 9 groups𝑍𝑍2 ≑ 𝑠𝑠 𝐸𝐸( 𝐸𝐸= 1𝑇𝑇,2π‘ˆπ‘ˆ,3𝒏𝒏,4πœ‹πœ‹,6) , 𝐸𝐸 ( = 2,3,4,6) . And the principal𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 axis of ( if exists) are 𝑠𝑠 𝐢𝐢assumed𝑛𝑛 𝑛𝑛 to be along𝐷𝐷 z𝑛𝑛 direction.𝑛𝑛 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 Direction of Number of types 𝑠𝑠 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 𝒏𝒏 33 Type-I random 32 Type-II parallel to z axis 58 𝐢𝐢1 perpendicular to x Type-III 𝐢𝐢2 58 axis 2 Type-IV 𝐢𝐢 parallel to z axis 5 Type-V parallel to z axis 4 𝐢𝐢3 Type-VI parallel to z axis 7 𝐢𝐢4 parallel to , or 𝐢𝐢6 axis(when the Type-VII π‘₯π‘₯ 𝑦𝑦 2 three directions lead 𝑧𝑧 𝐷𝐷2 to identical types) parallel to x y or z axis (when the three Type-VIII 63 directions lead to 𝐷𝐷2 different types) Type-IX parallel to z axis 10 Type-X parallel to z axis 5 𝐷𝐷3 Type-XI parallel to z axis 8 𝐷𝐷4 Total number of classes: 252 𝐷𝐷6

TABLE C2. The types of type-I SPGs of the form Γ— , i.e., the types the spin 𝐾𝐾 part group of which could be written as = is𝐺𝐺 𝑛𝑛shown𝑛𝑛𝑛𝑛 𝑍𝑍 2in this table. There are 32 𝑠𝑠 types in total. 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 𝐢𝐢1

= Γ—

𝐾𝐾 1 1 1 𝐺𝐺𝑠𝑠𝑠𝑠 𝐺𝐺𝑛𝑛1𝑛𝑛𝑛𝑛 𝑍𝑍2 𝐺𝐺𝑝𝑝0 𝑁𝑁𝑝𝑝0 𝐺𝐺𝑛𝑛 𝑛𝑛𝑛𝑛 1 π‘šπ‘š 1 1 1 1 1 2 2 12 12π‘šπ‘š1 οΏ½ οΏ½ οΏ½ οΏ½ 1 1 π‘šπ‘š 1 2/ 2/ 21/ 21/ π‘šπ‘š 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 2 2 1 1 2 1 1 2π‘šπ‘š 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 222 222 1 212 12 1 212 12 π‘šπ‘š1 π‘šπ‘šπ‘šπ‘š π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 1 1 1 1 1 π‘šπ‘š 1 4 4 1 1 4 1 1 1 4 1 1 π‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 1 π‘šπ‘š 4 4 4 4 1 4/ 4/ 41/ 41/ π‘šπ‘š 1 οΏ½ οΏ½ οΏ½ οΏ½ 1 1 1 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 34 422 422 4 2 2 4 2 2 1 1 1 1 1 1 1 π‘šπ‘š 4 4 4 4 1 42 42 1 41 2 1 1 41 2 1 π‘šπ‘š1 π‘šπ‘šπ‘šπ‘š π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 4/ 4/1 1 1 4/1 1 1 π‘šπ‘š 1 οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 3 3 1 1 31 1 1 1 31 11 π‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 1 π‘šπ‘š 3 3 3 3 1 32 32 31 2 31 2π‘šπ‘š 1 οΏ½ οΏ½ οΏ½ οΏ½ 1 1 1 1 π‘šπ‘š 3 3 3 3 1 3 3 131 131 π‘šπ‘š1 π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6 6 1 16 1 16 π‘šπ‘š1 οΏ½π‘šπ‘š οΏ½π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 6 6 16 16π‘šπ‘š1 οΏ½ οΏ½ οΏ½ οΏ½ 1 1 π‘šπ‘š 622 622 6 2 2 6 2 2 1 1 1 1 1 1 1 π‘šπ‘š 6/ 6/ 6/ 6/ 1 6 6 61 1 16 1 π‘šπ‘š 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6 2 6 2 1 61 1 2 1 61 1 2 π‘šπ‘š1 π‘šπ‘šπ‘šπ‘š π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6/ 6/ 6/1 1 1 6/1 1 1 π‘šπ‘š 1 οΏ½π‘šπ‘š οΏ½π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 23 23 1 1 2 13 1 1 1 2 13 11 π‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 1 1 1 π‘šπ‘š 2/ 3 2/ 3 2/ 3 2/ 3 1 43 43 1 4 13 1 1 4 13 1 π‘šπ‘š1 π‘šπ‘šοΏ½ π‘šπ‘šοΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ 432 432 1 41 31 2 1 41 31 2 π‘šπ‘š1 οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 1 1 1 1 1 1 π‘šπ‘š 4/ 32 4/ 3 2 4/ 32/ / 4/ 3 2/ / 1 11 1 1 π‘šπ‘šοΏ½ π‘šπ‘š οΏ½ 1 1 1 1 1 1 π‘šπ‘š π‘šπ‘šοΏ½ π‘šπ‘š π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š TABLE C3. The types of type-II SPGs of the form Γ— , i.e., the types the spin 𝐾𝐾 part group of which could be written as = 𝐺𝐺and𝑛𝑛𝑛𝑛𝑛𝑛 the𝑍𝑍 2direction of which is 𝑠𝑠 parallel to z axis, is shown in this table. There𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 are 𝐢𝐢582 types in total. 𝒏𝒏

= Γ— 𝐾𝐾 𝑝𝑝0 𝑝𝑝0 𝑛𝑛𝑛𝑛𝑛𝑛 𝑠𝑠𝑠𝑠 𝑛𝑛𝑛𝑛𝑛𝑛 2 𝐺𝐺1 𝑁𝑁1 𝐺𝐺 1 𝐺𝐺 𝐺𝐺1 1 𝑍𝑍 2 1 22 2𝑧𝑧2π‘šπ‘šπ‘§π‘§1 οΏ½ οΏ½ οΏ½ 2 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 1 1 2/ 2 22/ 2𝑧𝑧/ π‘šπ‘šπ‘§π‘§ 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2/ 12/2 2𝑧𝑧2/2 π‘šπ‘šπ‘§π‘§1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2/ 1 22/1 2𝑧𝑧2/1 π‘šπ‘šπ‘§π‘§1 π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 2 2 2 2 2 2𝑧𝑧 2 2π‘šπ‘šπ‘§π‘§ 1 π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š 2 2 2 12 2𝑧𝑧 2𝑧𝑧 12π‘šπ‘šπ‘§π‘§1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 2 2 1 2𝑧𝑧 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 35 222 2 2 2 2 2 2 2 1 1 2 2 1 2𝑧𝑧 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 2/ 1 2 1 2 2 1 2𝑧𝑧 2𝑧𝑧 π‘šπ‘šπ‘§π‘§1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 222 1 1 2 1 1 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 4 2 2 2 4 2 2𝑧𝑧 2𝑧𝑧 4 2𝑧𝑧 1 π‘šπ‘šπ‘§π‘§ π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 2 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 4 2 4 4 1 4/ 2/ 42/ 42𝑧𝑧/ π‘šπ‘šπ‘§π‘§ 1 οΏ½ οΏ½ οΏ½ 4/ 4 24/1 2𝑧𝑧4/ 1 π‘šπ‘šπ‘§π‘§ 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 4 24/2 2𝑧𝑧4/ 2𝑧𝑧 π‘šπ‘šπ‘§π‘§1 π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š 422 4 14 22 2 14 2𝑧𝑧 2π‘šπ‘šπ‘§π‘§ 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 2 2 1 2𝑧𝑧 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 422 222 4 2 2 4 2 2 1 2 1 2 2𝑧𝑧 1 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 4 4 4 4 1 4 2 142 2 1 42𝑧𝑧 2𝑧𝑧 π‘šπ‘šπ‘§π‘§1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 42 4 2 41 2 2 2𝑧𝑧4 1 2 2𝑧𝑧 π‘šπ‘šπ‘§π‘§1 π‘šπ‘šπ‘šπ‘š π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 42 2 14222 1 42𝑧𝑧 22𝑧𝑧 π‘šπ‘šπ‘§π‘§1 οΏ½ π‘šπ‘š οΏ½ οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 42 222 24221 2𝑧𝑧422𝑧𝑧 1 π‘šπ‘šπ‘§π‘§1 οΏ½ π‘šπ‘š π‘šπ‘šπ‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 4/ 42 4/2 1 2 4/2𝑧𝑧 1 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 1 οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 4/ 4 24/2 2 1 2 4/2𝑧𝑧 2𝑧𝑧 1 π‘šπ‘šπ‘§π‘§1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 14/2 1 1 1 4/2𝑧𝑧 1 1 π‘šπ‘šπ‘§π‘§ 1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 4/ 24/1 1 2 24𝑧𝑧 / 1 1 2𝑧𝑧 π‘šπ‘šπ‘§π‘§1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 422 14/1 2 2 14/ 1 2𝑧𝑧 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 3 3 1 2 32 2 1 2𝑧𝑧 23𝑧𝑧 21𝑧𝑧 π‘šπ‘šπ‘§π‘§ π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 32 3 32 2 32𝑧𝑧 2π‘šπ‘šπ‘§π‘§ 1 οΏ½ οΏ½ οΏ½ 1 2 1 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 3 3 3 3 1 3 3 132 132𝑧𝑧 π‘šπ‘šπ‘§π‘§1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 3 3 132 1 32𝑧𝑧 π‘šπ‘šπ‘§π‘§1 οΏ½π‘šπ‘š οΏ½ οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 3 32 231 2𝑧𝑧3 1 π‘šπ‘šπ‘§π‘§ 1 οΏ½π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 6 3 2 26 2𝑧𝑧 26𝑧𝑧 π‘šπ‘š1𝑧𝑧 οΏ½π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 6 3 26 2𝑧𝑧6π‘šπ‘šπ‘§π‘§1 οΏ½ οΏ½ οΏ½ 2 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 622 6 6 2 2 6 2 2 1 1 2 2 1 2𝑧𝑧 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 622 32 6 2 2 6 2 2 1 2 1 2 2𝑧𝑧 1 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 6/ 3 6/ 6/ 1 6/ 6 26/2 2𝑧𝑧 6/2𝑧𝑧 π‘šπ‘šπ‘§π‘§1 π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š 6/ 6 26/1 26𝑧𝑧 / 1 π‘šπ‘šπ‘§π‘§1 π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š 6 6 61 2 61 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 6 3 162 2 1 62𝑧𝑧 2𝑧𝑧 π‘šπ‘šπ‘§π‘§1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6 2 6 2 61 2 2 2𝑧𝑧6 1 2𝑧𝑧 2 π‘šπ‘šπ‘§π‘§1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 2 2 1 2𝑧𝑧 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ οΏ½π‘šπ‘š οΏ½ οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 36 6 2 3 6 2 6 2 1 6 2 32 261 22 2𝑧𝑧61 2𝑧𝑧2π‘šπ‘šπ‘§π‘§1 οΏ½π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 6/ 3 6/2 2 1 62/𝑧𝑧 2𝑧𝑧 1 π‘šπ‘šπ‘§π‘§ 1 οΏ½π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 6/ 6 2 26/2 1 2 2𝑧𝑧 6/2𝑧𝑧 1 2𝑧𝑧 π‘šπ‘šπ‘§π‘§1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š οΏ½π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6/ 6/ 26/1 2 1 26𝑧𝑧 / 1 2𝑧𝑧 1 π‘šπ‘šπ‘§π‘§1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š οΏ½π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6/ 6 16/1 2 2 1 6/1 2𝑧𝑧 2𝑧𝑧 π‘šπ‘šπ‘§π‘§1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6/ 622 16/2 1 1 16/ 2𝑧𝑧 1 1 π‘šπ‘šπ‘§π‘§ 1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 2/ 3 23 1 22/ 2 32 1 22𝑧𝑧/ 2𝑧𝑧 23𝑧𝑧 π‘šπ‘š1 𝑧𝑧 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 43 23 1 4 23 2 1 42𝑧𝑧3 2𝑧𝑧 π‘šπ‘šπ‘§π‘§1 π‘šπ‘šοΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ 432 23 2 41 32 2 2𝑧𝑧 41 32𝑧𝑧 2 π‘šπ‘šπ‘§π‘§1 οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 2 1 2 2𝑧𝑧 1 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 4/ 32/ 2/ 3 4/ 3 2/ 4/ 3 2/ 1 4/ 32/ 43 24/1 1322/2 2𝑧𝑧4/ 1 1 32𝑧𝑧 2/2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 1 π‘šπ‘šοΏ½ π‘šπ‘š π‘šπ‘šοΏ½ π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š 4/ 32/ 432 24/2 2322/1 2𝑧𝑧4/ 2𝑧𝑧 2𝑧𝑧3 22𝑧𝑧/ 1 π‘šπ‘šπ‘§π‘§1 π‘šπ‘šοΏ½ π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š 1 2 2 1 2 1 2𝑧𝑧 2𝑧𝑧 1 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ π‘šπ‘šοΏ½ π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š TABLE C4. The types of type-III SPGs of the form Γ— , i.e., the types the spin 𝐾𝐾 part of which could be written as = and the𝐺𝐺 direction𝑛𝑛𝑛𝑛𝑛𝑛 𝑍𝑍2 of which is parallel 𝑠𝑠 to x axis, is shown in this table. There𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 are 58𝐢𝐢2 types in total. 𝒏𝒏

= Γ— 𝐾𝐾 𝑝𝑝0 𝑝𝑝0 𝑛𝑛𝑛𝑛𝑛𝑛 𝑠𝑠𝑠𝑠 𝑛𝑛𝑛𝑛𝑛𝑛 2 𝐺𝐺1 𝑁𝑁1 𝐺𝐺 1 𝐺𝐺 𝐺𝐺1 1 𝑍𝑍 2 1 22 2𝑧𝑧2π‘šπ‘šπ‘₯π‘₯1 οΏ½ οΏ½ οΏ½ 2 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ 1 1 2/ 2 22/ 2𝑧𝑧/ π‘šπ‘šπ‘₯π‘₯ 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2/ 12/2 2𝑧𝑧2/2 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2/ 1 22/1 2𝑧𝑧2/1 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 2 2 2 2 2 2𝑧𝑧 2 2π‘šπ‘šπ‘₯π‘₯ 1 π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š 2 2 2 12 2𝑧𝑧 2𝑧𝑧 12π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 222 2 1 222 22 1 22𝑧𝑧2 2𝑧𝑧2 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 2 2 1 2𝑧𝑧 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ 2/ 1 2 1 2 2 1 2𝑧𝑧 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 222 1 1 2 1 1 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ 1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 4 2 2 2 4 2 2𝑧𝑧 2𝑧𝑧 4 2𝑧𝑧 1 π‘šπ‘šπ‘₯π‘₯ π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 2 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ 4 2 4 4 1 4/ 2/ 42/ 42𝑧𝑧/ π‘šπ‘šπ‘₯π‘₯ 1 οΏ½ οΏ½ οΏ½ 4/ 4 24/1 2𝑧𝑧4/ 1 π‘šπ‘šπ‘₯π‘₯ 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 4 24/2 2𝑧𝑧4/ 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š 1 2 1 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ π‘šπ‘š π‘šπ‘š π‘šπ‘š 37 422 4 4 2 2 4 2 2 1 1 2 2 1 2𝑧𝑧 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ 422 222 4 2 2 4 2 2 1 2 1 2 2𝑧𝑧 1 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ 4 4 4 4 1 4 2 142 2 1 42𝑧𝑧 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 42 4 2 41 2 2 2𝑧𝑧4 1 2 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘šπ‘šπ‘š π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 42 2 14222 1 42𝑧𝑧 22𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 οΏ½ π‘šπ‘š οΏ½ οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 42 222 24221 2𝑧𝑧422𝑧𝑧 1 π‘šπ‘šπ‘₯π‘₯1 οΏ½ π‘šπ‘š π‘šπ‘šπ‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 4/ 42 4/2 1 2 4/2𝑧𝑧 1 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ 1 οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 4/ 4 24/2 2 1 2 4/2𝑧𝑧 2𝑧𝑧 1 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 14/2 1 1 1 4/2𝑧𝑧 1 1 π‘šπ‘šπ‘₯π‘₯ 1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 4/ 24/1 1 2 24𝑧𝑧 / 1 1 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 422 14/1 2 2 14/ 1 2𝑧𝑧 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ 1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 3 3 1 2 32 2 1 2𝑧𝑧 23𝑧𝑧 21𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 32 3 32 2 32𝑧𝑧 2π‘šπ‘šπ‘₯π‘₯ 1 οΏ½ οΏ½ οΏ½ 1 2 1 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ 3 3 3 3 1 3 3 132 132𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 3 3 132 1 32𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 οΏ½π‘šπ‘š οΏ½ οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 3 32 231 2𝑧𝑧3 1 π‘šπ‘šπ‘₯π‘₯ 1 οΏ½π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 6 3 2 26 2𝑧𝑧 26𝑧𝑧 π‘šπ‘š1π‘₯π‘₯ οΏ½π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 6 3 26 2𝑧𝑧6π‘šπ‘šπ‘₯π‘₯1 οΏ½ οΏ½ οΏ½ 2 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ 622 6 6 2 2 6 2 2 1 1 2 2 1 2𝑧𝑧 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ 622 32 6 2 2 6 2 2 1 2 1 2 2𝑧𝑧 1 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ 6/ 3 6/ 6/ 1 6/ 6 26/2 2𝑧𝑧 6/2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š 6/ 6 26/1 26𝑧𝑧 / 1 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š 6 6 61 2 61 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 6 3 162 2 1 62𝑧𝑧 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6 2 6 2 61 2 2 2𝑧𝑧6 1 2𝑧𝑧 2 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6 2 3 162 22 1 62𝑧𝑧 2𝑧𝑧2π‘šπ‘šπ‘₯π‘₯1 οΏ½π‘šπ‘š οΏ½ οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 6 2 32 261 22 2𝑧𝑧61 2𝑧𝑧2π‘šπ‘šπ‘₯π‘₯1 οΏ½π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 6/ 3 6/2 2 1 62/𝑧𝑧 2𝑧𝑧 1 π‘šπ‘šπ‘₯π‘₯ 1 οΏ½π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 6/ 6 2 26/2 1 2 2𝑧𝑧 6/2𝑧𝑧 1 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š οΏ½π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6/ 6/ 26/1 2 1 26𝑧𝑧 / 1 2𝑧𝑧 1 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š οΏ½π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6/ 6 16/1 2 2 1 6/1 2𝑧𝑧 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6/ 622 16/2 1 1 16/ 2𝑧𝑧 1 1 π‘šπ‘šπ‘₯π‘₯ 1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 2/ 3 23 1 22/ 2 32 1 22𝑧𝑧/ 2𝑧𝑧 23𝑧𝑧 π‘šπ‘š1 π‘₯π‘₯ π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 43 23 1 4 23 2 1 42𝑧𝑧3 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘šοΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ 2 1 2 2𝑧𝑧 1 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 38 432 23 4 3 2 4 3 2 1 2 1 2 2𝑧𝑧 1 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ 4/ 32/ 2/ 3 4/ 3 2/ 4/ 3 2/ 1 4/ 32/ 43 24/1 1322/2 2𝑧𝑧4/ 1 1 32𝑧𝑧 22/𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ 1 π‘šπ‘šοΏ½ π‘šπ‘š π‘šπ‘šοΏ½ π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š 4/ 32/ 432 24/2 2322/1 2𝑧𝑧4/ 2𝑧𝑧 2𝑧𝑧3 22𝑧𝑧/ 1 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘šοΏ½ π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š 1 2 2 1 2 1 2𝑧𝑧 2𝑧𝑧 1 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯ π‘šπ‘šοΏ½ π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š TABLE C5. The types of type-IV SPGs of the form Γ— , i.e., the types the spin 𝐾𝐾 part of which could be written as = and the𝐺𝐺 direction𝑛𝑛𝑛𝑛𝑛𝑛 𝑍𝑍2 of which is parallel 𝑠𝑠 to z axis, is shown in this table. There𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 are 5𝐢𝐢 3types in total. 𝒏𝒏

= Γ—

𝑁𝑁𝑝𝑝0 𝐾𝐾 3 1 3 𝐺𝐺𝑠𝑠𝑠𝑠 𝐺𝐺3𝑛𝑛𝑛𝑛𝑛𝑛1 𝑍𝑍2 𝐺𝐺𝑝𝑝0 𝐺𝐺𝑛𝑛 𝑛𝑛𝑛𝑛 6 36 3𝑧𝑧6π‘šπ‘šπ‘§π‘§1 οΏ½ οΏ½ οΏ½ οΏ½ 6 2 36 3𝑧𝑧6π‘šπ‘šπ‘§π‘§1 οΏ½ π‘šπ‘š οΏ½ οΏ½ 3 3𝑧𝑧 π‘šπ‘šπ‘§π‘§ 6/ 2/ 6/ 6/ 1 23 222 3 2 13 3𝑧𝑧2 13 π‘šπ‘š1𝑧𝑧 π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 3 1 3𝑧𝑧 π‘šπ‘šπ‘§π‘§

TABLE C6. The types of type-V SPGs of the form Γ— , i.e., the types the spin 𝐾𝐾 part of which could be written as = and the𝐺𝐺 𝑛𝑛direction𝑛𝑛𝑛𝑛 𝑍𝑍2 of which is parallel 𝑠𝑠 to z axis, is shown in this table. There𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 are 4𝐢𝐢 4types in total. 𝒏𝒏

= Γ—

𝐾𝐾 4 1 4 𝐺𝐺𝑠𝑠𝑠𝑠 𝐺𝐺4𝑛𝑛𝑛𝑛𝑛𝑛1 𝑍𝑍2 𝐺𝐺𝑝𝑝0 𝑁𝑁𝑝𝑝0 𝐺𝐺𝑛𝑛 𝑛𝑛𝑛𝑛 4 4𝑧𝑧 π‘šπ‘šπ‘§π‘§ 4 1 4 4 1 4/ 1 44/ 44/𝑧𝑧 π‘šπ‘šπ‘§π‘§ 1 οΏ½ οΏ½ οΏ½ οΏ½ 4/ 44/2 4𝑧𝑧 4/2𝑧𝑧 π‘šπ‘šπ‘§π‘§1 π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š 4 1 4𝑧𝑧 1 π‘šπ‘šπ‘§π‘§ π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š TABLE C7. The types of type-VI SPGs of the form Γ— , i.e., the types the spin 𝐾𝐾 part of which could be written as = and the𝐺𝐺 direction𝑛𝑛𝑛𝑛𝑛𝑛 𝑍𝑍2 of which is parallel 𝑠𝑠 to z axis, is shown in this table. There𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 are 7𝐢𝐢 6classes in total. 𝒏𝒏

𝑝𝑝0 = ×𝑠𝑠𝑠𝑠 𝑁𝑁 𝐺𝐺 𝐾𝐾 3 1 3 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛3 𝑍𝑍12 𝐺𝐺𝑝𝑝0 𝐺𝐺𝑛𝑛 𝑛𝑛𝑛𝑛 6 1 66 6𝑧𝑧6π‘šπ‘šπ‘§π‘§1 οΏ½ οΏ½ οΏ½ 6 6𝑧𝑧 π‘šπ‘šπ‘§π‘§ οΏ½ οΏ½ οΏ½ 39 6 1 6 6 1 6 6𝑧𝑧 π‘šπ‘šπ‘§π‘§ 6/ 2 6/ 6/ 1 6/ 36/2 3𝑧𝑧 6/2𝑧𝑧 π‘šπ‘šπ‘§π‘§1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 6/ 1 66/1 6𝑧𝑧6/ 1 π‘šπ‘šπ‘§π‘§ 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 2/ 3 222 26/ 2 3 26𝑧𝑧/ 2𝑧𝑧 3π‘šπ‘šπ‘§π‘§ 1 π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š 1 2 6 1 2𝑧𝑧 6𝑧𝑧 π‘šπ‘šπ‘§π‘§ π‘šπ‘šοΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ TABLE C8. The classes of type-VII SPGs of the form Γ— , i.e., the types the 𝐾𝐾 spin part of which could be written as = and 𝐺𝐺the𝑛𝑛𝑛𝑛𝑛𝑛 direction𝑍𝑍2 of which is 𝑠𝑠 parallel to x, y or z axis when the three directions𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 𝐷𝐷 lead2 to identical classes,𝒏𝒏 is shown in this table. There are 2 types in total.

222 1 2 2 2 2 2 2 1 𝐺𝐺𝑝𝑝0 𝑁𝑁𝑝𝑝0 𝐺𝐺𝑛𝑛 𝑛𝑛𝑛𝑛 𝐺𝐺𝑠𝑠𝑠𝑠 2π‘₯π‘₯ 2𝑦𝑦 2𝑧𝑧 2π‘₯π‘₯ 2𝑦𝑦 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 1 1 2π‘₯π‘₯ 2𝑦𝑦 2𝑧𝑧 2π‘₯π‘₯ 2𝑦𝑦 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ π‘šπ‘šπ‘šπ‘šπ‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š TABLE C9. The types of type-VIII SPGs of the form Γ— , i.e., the types the spin 𝐾𝐾 part of which could be written as = and the𝐺𝐺 direction𝑛𝑛𝑛𝑛𝑛𝑛 𝑍𝑍2 of which is parallel 𝑠𝑠 to x, y or z axis when the three directions𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 lead𝐷𝐷2 to different types, 𝒏𝒏is shown in this table. There are 63 types in total.

= Γ— 𝐾𝐾 𝑝𝑝0 𝑝𝑝0 𝑛𝑛𝑛𝑛𝑛𝑛 𝑠𝑠𝑠𝑠 𝑛𝑛𝑛𝑛𝑛𝑛 2 𝐺𝐺 𝑁𝑁 𝐺𝐺 𝐺𝐺 2/𝐺𝐺 1𝑍𝑍 2/ 1 2/ 2𝑧𝑧2/2π‘₯π‘₯ π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘š 2𝑧𝑧 2π‘₯π‘₯ 2𝑧𝑧2/2π‘₯π‘₯ π‘šπ‘šπ‘¦π‘¦1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2𝑧𝑧 2π‘₯π‘₯ π‘šπ‘šπ‘§π‘§ 2 1 2 1 2 π‘šπ‘š 2π‘₯π‘₯ 2𝑦𝑦 2𝑧𝑧2π‘šπ‘šπ‘₯π‘₯1 2π‘₯π‘₯ 2𝑦𝑦 2𝑧𝑧 π‘šπ‘š π‘šπ‘š π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š 2π‘₯π‘₯ 2𝑦𝑦 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 1 π‘šπ‘š π‘šπ‘š 2 2𝑧𝑧 2𝑧𝑧 2π‘₯π‘₯ π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2𝑧𝑧 2𝑧𝑧 2π‘₯π‘₯ 2𝑧𝑧 2𝑧𝑧 2π‘₯π‘₯ π‘šπ‘šπ‘¦π‘¦1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 2𝑧𝑧 2𝑧𝑧 2π‘₯π‘₯ π‘šπ‘šπ‘§π‘§1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2π‘₯π‘₯ 2𝑦𝑦 1 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2π‘₯π‘₯ 2𝑦𝑦 1 2π‘₯π‘₯ 2𝑦𝑦 1 π‘šπ‘šπ‘¦π‘¦1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 2π‘₯π‘₯ 42/𝑦𝑦 1 π‘šπ‘š1𝑧𝑧 π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 2 4/ 2𝑧𝑧4/2π‘₯π‘₯ π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘š 2𝑧𝑧 2π‘₯π‘₯ 2𝑧𝑧4/2π‘₯π‘₯ π‘šπ‘šπ‘¦π‘¦1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2𝑧𝑧 2π‘₯π‘₯ π‘šπ‘šπ‘§π‘§ π‘šπ‘š 40 4 2 2 1 422 2 4 2 2 2π‘₯π‘₯42𝑦𝑦22𝑧𝑧2π‘šπ‘šπ‘₯π‘₯1 2π‘₯π‘₯ 2𝑦𝑦 2𝑧𝑧 2π‘₯π‘₯ 2𝑦𝑦 2𝑧𝑧 π‘šπ‘šπ‘¦π‘¦ 4 1 4 2 4 2π‘₯π‘₯42𝑦𝑦 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 2π‘₯π‘₯ 2𝑦𝑦 2𝑧𝑧 π‘šπ‘š π‘šπ‘š π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š 2π‘₯π‘₯ 42𝑦𝑦 2 2𝑧𝑧 π‘šπ‘šπ‘¦π‘¦1 π‘šπ‘š π‘šπ‘š 42 2 4 2 2π‘₯π‘₯42𝑦𝑦22𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 οΏ½ π‘šπ‘š 2π‘₯π‘₯ 2𝑦𝑦 2𝑧𝑧 2π‘₯π‘₯42𝑦𝑦22𝑧𝑧 π‘šπ‘šπ‘¦π‘¦1 οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 42/π‘₯π‘₯ 2𝑦𝑦 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 1 οΏ½ 4/ 4 4/ π‘šπ‘š 2𝑧𝑧4/2𝑧𝑧 2π‘₯π‘₯ 2𝑦𝑦 π‘šπ‘šπ‘₯π‘₯1 2𝑧𝑧 2𝑧𝑧 2π‘₯π‘₯ 2𝑦𝑦 π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š π‘šπ‘š 2𝑧𝑧4/ 2𝑧𝑧 2π‘₯π‘₯ 2𝑦𝑦 π‘šπ‘šπ‘§π‘§1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 4 4/ 14/2π‘₯π‘₯ 2𝑦𝑦 2𝑦𝑦 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 2π‘₯π‘₯ 2𝑦𝑦 2𝑦𝑦 14/2π‘₯π‘₯ 2𝑦𝑦 2𝑦𝑦 π‘šπ‘šπ‘¦π‘¦1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 2π‘₯π‘₯ 2𝑦𝑦 2𝑦𝑦 π‘šπ‘šπ‘§π‘§ 4/ 1 4/ 2/ 4/ π‘šπ‘š π‘šπ‘š π‘šπ‘š 2𝑧𝑧4/1 2π‘₯π‘₯ 2𝑦𝑦 π‘šπ‘šπ‘₯π‘₯1 2𝑧𝑧 1 2π‘₯π‘₯ 2𝑦𝑦 π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 2𝑧𝑧4/1 2π‘₯π‘₯ 2𝑦𝑦 π‘šπ‘šπ‘§π‘§1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 2 4/ 2𝑧𝑧4/2π‘₯π‘₯ 1 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2𝑧𝑧 2π‘₯π‘₯ 1 2𝑧𝑧 2𝑧𝑧4/2π‘₯π‘₯ 1 2𝑧𝑧 π‘šπ‘šπ‘¦π‘¦1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 2𝑧𝑧4/ 2π‘₯π‘₯ 1 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 222 4/ 2𝑧𝑧4/2π‘₯π‘₯ 2π‘₯π‘₯ 2𝑦𝑦 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2𝑧𝑧 2π‘₯π‘₯ 2π‘₯π‘₯ 2𝑦𝑦 2𝑧𝑧4/2π‘₯π‘₯ 2π‘₯π‘₯ 2𝑦𝑦 π‘šπ‘šπ‘¦π‘¦1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 2𝑧𝑧 2π‘₯π‘₯ 3 2π‘₯π‘₯ 2𝑦𝑦1 π‘šπ‘šπ‘§π‘§ π‘šπ‘š π‘šπ‘š π‘šπ‘š 3 3 3 2π‘₯π‘₯32𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 οΏ½ π‘šπ‘š 2π‘₯π‘₯ 2𝑧𝑧 2π‘₯π‘₯32𝑧𝑧 π‘šπ‘šπ‘¦π‘¦1 οΏ½π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 2π‘₯π‘₯ 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 6 2 2 1 622 3 6 2 2 οΏ½ π‘šπ‘š 2π‘₯π‘₯62𝑦𝑦22𝑧𝑧2π‘šπ‘šπ‘₯π‘₯1 2π‘₯π‘₯ 2𝑦𝑦 2𝑧𝑧 2π‘₯π‘₯ 2𝑦𝑦 2𝑧𝑧 π‘šπ‘šπ‘¦π‘¦ 6/ 1 6/ 3 6/ 2𝑧𝑧6/2π‘₯π‘₯ π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘š 2𝑧𝑧 2π‘₯π‘₯ 2𝑧𝑧6/2π‘₯π‘₯ π‘šπ‘šπ‘¦π‘¦1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2𝑧𝑧 2π‘₯π‘₯ π‘šπ‘šπ‘§π‘§ 6 1 6 3 6 π‘šπ‘š 2π‘₯π‘₯62𝑦𝑦 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 2π‘₯π‘₯ 2𝑦𝑦 2𝑧𝑧 π‘šπ‘š π‘šπ‘š π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š 2π‘₯π‘₯ 62𝑦𝑦 2𝑧𝑧 2 π‘šπ‘šπ‘¦π‘¦1 π‘šπ‘š π‘šπ‘š 6 2 3 6 2 2𝑧𝑧62π‘₯π‘₯ 2𝑦𝑦2π‘šπ‘šπ‘₯π‘₯1 οΏ½ π‘šπ‘š 2𝑧𝑧 2π‘₯π‘₯ 2𝑦𝑦 2𝑧𝑧62π‘₯π‘₯ 2𝑦𝑦2π‘šπ‘šπ‘¦π‘¦1 οΏ½π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 62/𝑧𝑧 2π‘₯π‘₯ 2𝑦𝑦 π‘šπ‘šπ‘§π‘§ 1 οΏ½ 6/ 3 6/ π‘šπ‘š 2𝑧𝑧6/2𝑧𝑧 2π‘₯π‘₯ 2𝑦𝑦 π‘šπ‘šπ‘₯π‘₯1 2𝑧𝑧 2𝑧𝑧 2π‘₯π‘₯ 2𝑦𝑦 π‘šπ‘š π‘šπ‘š π‘šπ‘š οΏ½ 2𝑧𝑧 2𝑧𝑧 2π‘₯π‘₯ 2𝑦𝑦 π‘šπ‘šπ‘§π‘§ π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6/ 1 6/ 6 6/ π‘šπ‘š π‘šπ‘š π‘šπ‘š 2𝑧𝑧6/1 2π‘₯π‘₯ 2𝑦𝑦 π‘šπ‘šπ‘₯π‘₯1 2𝑧𝑧 1 2π‘₯π‘₯ 2𝑦𝑦 π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š π‘šπ‘š 2𝑧𝑧 1 2π‘₯π‘₯ 2𝑦𝑦 π‘šπ‘šπ‘§π‘§ π‘šπ‘š π‘šπ‘š π‘šπ‘š 41 6/ 1 6/ 6 6/ 16/2𝑧𝑧 2π‘₯π‘₯ 2π‘₯π‘₯ π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 2𝑧𝑧 2π‘₯π‘₯ 2π‘₯π‘₯ 16/2𝑧𝑧 2π‘₯π‘₯ 2π‘₯π‘₯ π‘šπ‘šπ‘¦π‘¦1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 6/2𝑧𝑧 2π‘₯π‘₯ 2π‘₯π‘₯ π‘šπ‘šπ‘§π‘§1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 6/ 3 6/ 2𝑧𝑧6/2π‘₯π‘₯ 1 2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2𝑧𝑧 2π‘₯π‘₯ 1 2𝑧𝑧 2𝑧𝑧6/2π‘₯π‘₯ 1 2𝑧𝑧 π‘šπ‘šπ‘¦π‘¦1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 2𝑧𝑧6/ 2π‘₯π‘₯ 1 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 6/ 32 6/ 2𝑧𝑧6/2π‘₯π‘₯ 2𝑦𝑦 2π‘₯π‘₯ π‘šπ‘šπ‘₯π‘₯1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2𝑧𝑧 2π‘₯π‘₯ 2𝑦𝑦 2π‘₯π‘₯ 2𝑧𝑧6/2π‘₯π‘₯ 2𝑦𝑦 2π‘₯π‘₯ π‘šπ‘šπ‘¦π‘¦1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 2𝑧𝑧 2π‘₯π‘₯ 2𝑦𝑦 2π‘₯π‘₯ π‘šπ‘šπ‘§π‘§ 4/ 3 2/ 1 4/ 3 2 π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 32/ 23 2π‘₯π‘₯4/2𝑦𝑦 2𝑦𝑦32π‘₯π‘₯2/2𝑧𝑧 π‘šπ‘šπ‘₯π‘₯1 2π‘₯π‘₯ 2𝑦𝑦 2𝑦𝑦 2π‘₯π‘₯ οΏ½ / π‘šπ‘š π‘šπ‘š π‘šπ‘š οΏ½ 2π‘₯π‘₯4/2𝑦𝑦 2𝑦𝑦32π‘₯π‘₯2/2𝑧𝑧 π‘šπ‘šπ‘¦π‘¦1 π‘šπ‘šοΏ½ π‘šπ‘š 2𝑧𝑧 π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š 2π‘₯π‘₯ 2𝑦𝑦 2𝑦𝑦 2π‘₯π‘₯ 2𝑧𝑧 π‘šπ‘šπ‘§π‘§ π‘šπ‘š οΏ½ π‘šπ‘š TABLE C10. The classes of type-IX SPGs of the form Γ— , i.e., the types the 𝐾𝐾 spin part of which could be written as = and 𝐺𝐺the𝑛𝑛𝑛𝑛𝑛𝑛 direction𝑍𝑍2 of which is 𝑠𝑠 parallel to z axis, is shown in this table. There𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 are𝐷𝐷 103 types in total. 𝒏𝒏

= Γ—

𝑁𝑁𝑝𝑝0 𝐾𝐾 32 1 3 2 𝐺𝐺𝑠𝑠𝑠𝑠 3𝐺𝐺𝑛𝑛2𝑛𝑛𝑛𝑛 1 𝑍𝑍2 𝐺𝐺𝑝𝑝0 𝐺𝐺 𝑛𝑛𝑛𝑛𝑛𝑛 3𝑧𝑧 2π‘₯π‘₯ 3𝑧𝑧 2π‘₯π‘₯ π‘šπ‘šπ‘§π‘§ 3 1 3 3 1 3 1 3𝑧𝑧32π‘₯π‘₯ 3𝑧𝑧32π‘₯π‘₯ π‘šπ‘šπ‘§π‘§1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 622 2 36𝑧𝑧 2π‘₯π‘₯ 2 36𝑧𝑧 2π‘₯π‘₯ π‘šπ‘š2𝑧𝑧 1 οΏ½π‘šπ‘š οΏ½ οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 3𝑧𝑧 2π‘₯π‘₯ 2π‘₯π‘₯π‘₯π‘₯ 3𝑧𝑧 2π‘₯π‘₯ 2π‘₯π‘₯π‘₯π‘₯ π‘šπ‘šπ‘§π‘§ 6 2 6 6 1 6 2 3𝑧𝑧 62π‘₯π‘₯ 2π‘₯π‘₯π‘₯π‘₯ 2 3𝑧𝑧 62π‘₯π‘₯ 2π‘₯π‘₯π‘₯π‘₯ 2 π‘šπ‘šπ‘§π‘§1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 2/ 3𝑧𝑧 2π‘₯π‘₯ 2π‘₯π‘₯π‘₯π‘₯ 3𝑧𝑧 2π‘₯π‘₯ 26π‘₯π‘₯π‘₯π‘₯ π‘šπ‘šπ‘§π‘§ οΏ½π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 6/ 6/ / 3𝑧𝑧 1 π‘šπ‘š 43 222 3𝑧𝑧 14 23π‘₯π‘₯ 2π‘₯π‘₯π‘₯π‘₯ 1 24π‘₯π‘₯ 32π‘₯π‘₯π‘₯π‘₯ π‘šπ‘šπ‘§π‘§1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 432 222 2π‘₯π‘₯ 43𝑧𝑧 32π‘₯π‘₯ 2 2π‘₯π‘₯ 43𝑧𝑧 32π‘₯π‘₯ 2 π‘šπ‘šπ‘§π‘§1 οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 2π‘₯π‘₯ 3𝑧𝑧 2π‘₯π‘₯ 2π‘₯π‘₯ 3𝑧𝑧 2π‘₯π‘₯ π‘šπ‘šπ‘§π‘§ 4/ 3 2 4/ 3 2 4/ 32/ / 2π‘₯π‘₯ 1 3𝑧𝑧 2π‘₯π‘₯ / 2π‘₯π‘₯ 11 3𝑧𝑧 2π‘₯π‘₯ π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ 2π‘₯π‘₯ 2π‘₯π‘₯ π‘šπ‘šπ‘§π‘§ π‘šπ‘šοΏ½ π‘šπ‘š π‘šπ‘š π‘šπ‘š TABLE C11. The types of type-X SPGs of the form Γ— , i.e., the types the spin 𝐾𝐾 part of which could be written as = and the𝐺𝐺 direction𝑛𝑛𝑛𝑛𝑛𝑛 𝑍𝑍2 of which is parallel 𝑠𝑠 to z axis, is shown in this table. There𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 are 5𝐷𝐷 types4 in total. 𝒏𝒏

42

= Γ—

𝑁𝑁𝑝𝑝0 𝐾𝐾 422 1 4 2 2 𝐺𝐺𝑠𝑠𝑠𝑠4 2𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛2 𝑍𝑍12 𝐺𝐺𝑝𝑝0 𝐺𝐺𝑛𝑛 𝑛𝑛𝑛𝑛 4𝑧𝑧 2π‘₯π‘₯ 2π‘₯π‘₯π‘₯π‘₯ 4𝑧𝑧 2π‘₯π‘₯ 2π‘₯π‘₯π‘₯π‘₯ π‘šπ‘šπ‘§π‘§ 4 1 4 2 2 4 2 2 1 42 1 4𝑧𝑧4 2π‘₯π‘₯2 2π‘₯π‘₯π‘₯π‘₯ 4𝑧𝑧4 2π‘₯π‘₯2 2π‘₯π‘₯π‘₯π‘₯ π‘šπ‘šπ‘§π‘§ 1 π‘šπ‘šπ‘šπ‘š 4𝑧𝑧 2π‘₯π‘₯ 2π‘₯π‘₯π‘₯π‘₯ 4𝑧𝑧 2π‘₯π‘₯ 2π‘₯π‘₯4π‘₯π‘₯ π‘šπ‘šπ‘§π‘§ οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 4/ 4/ / 4𝑧𝑧 1 π‘šπ‘š 1 4𝑧𝑧 1 24π‘₯π‘₯ 2π‘₯π‘₯π‘₯π‘₯ 1 2π‘₯π‘₯ 2π‘₯π‘₯4π‘₯π‘₯ π‘šπ‘šπ‘§π‘§ π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ / 4𝑧𝑧 / 4𝑧𝑧 1 οΏ½ 2𝑧𝑧 2π‘₯π‘₯ 2π‘₯π‘₯π‘₯π‘₯ 2𝑧𝑧 2π‘₯π‘₯ 2π‘₯π‘₯π‘₯π‘₯ π‘šπ‘šπ‘§π‘§ π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š TABLE C12. The types of type-XI SPGs of the form Γ— , i.e., the types the spin 𝐾𝐾 part of which could be written as = and the𝐺𝐺 direction𝑛𝑛𝑛𝑛𝑛𝑛 𝑍𝑍2 of which is parallel 𝑠𝑠 to z axis, is shown in this table. There𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 are 8𝐷𝐷 types4 in total. 𝒏𝒏

= Γ—

𝑁𝑁𝑝𝑝0 𝐾𝐾 3 1 3 𝐺𝐺𝑠𝑠𝑠𝑠 3 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 1𝑍𝑍 2 𝐺𝐺𝑝𝑝0 𝐺𝐺 𝑛𝑛𝑛𝑛𝑛𝑛 622 1 66𝑧𝑧 22π‘₯π‘₯ 2 66𝑧𝑧 22π‘₯π‘₯ 2π‘šπ‘šπ‘§π‘§ 1 οΏ½π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 6𝑧𝑧 2π‘₯π‘₯ 2𝑙𝑙 6𝑧𝑧 2π‘₯π‘₯ 2𝑙𝑙 π‘šπ‘šπ‘§π‘§ 6 1 6 6 1 6 2 1 6𝑧𝑧 62π‘₯π‘₯ 2𝑙𝑙 2 6𝑧𝑧 62π‘₯π‘₯ 2𝑙𝑙 2 π‘šπ‘šπ‘§π‘§1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6/ 1 66/𝑧𝑧 2π‘₯π‘₯ 2𝑙𝑙 66/𝑧𝑧 2π‘₯π‘₯ 2𝑙𝑙 π‘šπ‘šπ‘§π‘§ 1 οΏ½π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 6/ 6𝑧𝑧 6/2𝑧𝑧 2π‘₯π‘₯ 2𝑙𝑙 6𝑧𝑧 6/2𝑧𝑧 2π‘₯π‘₯ 2𝑙𝑙 π‘šπ‘šπ‘§π‘§1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 2 6𝑧𝑧 1 26π‘₯π‘₯ 2𝑙𝑙 6𝑧𝑧 1 2π‘₯π‘₯ 6 2𝑙𝑙 π‘šπ‘šπ‘§π‘§ π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 3𝑧𝑧 3𝑧𝑧 6/ / / 1 222 2𝑧𝑧4/2π‘₯π‘₯ 2π‘₯π‘₯π‘₯π‘₯3 2 2𝑧𝑧 42/π‘₯π‘₯ 2π‘₯π‘₯π‘₯π‘₯ 3π‘šπ‘šπ‘§π‘§2 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 32/ /22 2𝑧𝑧 6𝑧𝑧 22 / 22 21𝑧𝑧 6𝑧𝑧 22 π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ 2π‘₯π‘₯ 2π‘₯π‘₯ π‘šπ‘šπ‘§π‘§ π‘šπ‘šοΏ½ π‘šπ‘š π‘šπ‘š π‘šπ‘š C2. Classification of full SPGs for collinear spin arrangements

For collinear spin arrangements, we should consider groups of the form Γ— 𝐾𝐾 𝑛𝑛𝑛𝑛𝑛𝑛 2 (2) . 𝐺𝐺 �𝑍𝑍 ⋉

𝑆𝑆𝑆𝑆Step 1οΏ½ We firstly write (2) as (2) = {{ ( ) || |0}| (0,2 ]}. Because of

the presence of 𝑆𝑆𝑆𝑆(2) = {{𝑆𝑆𝑆𝑆( ) || π‘ˆπ‘ˆ|0π’Žπ’ŽοΏ½}| πœ”πœ” 𝑇𝑇(0𝐸𝐸,2 ]}πœ”πœ” , ∈all ofπœ‹πœ‹ the rotations contained in should𝑆𝑆𝑆𝑆 have rotationπ‘ˆπ‘ˆπ’Žπ’ŽοΏ½ πœ”πœ” 𝑇𝑇axis𝐸𝐸 parallelπœ”πœ” ∈ to πœ‹πœ‹, so that the condition that 𝑠𝑠 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 π’Žπ’ŽοΏ½ 43 = for all (2) is satisfied. Thus, should have no βˆ’1 𝐾𝐾 𝑠𝑠 moreβ„Ž 𝐺𝐺 𝑛𝑛thanπ‘›π‘›π‘›π‘›β„Ž 1 𝐺𝐺rotation𝑛𝑛𝑛𝑛𝑛𝑛 axis.β„Ž ∈Then,𝑍𝑍2 ⋉ the𝑆𝑆𝑆𝑆 options left for are𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 14 PGs: ( = 𝑠𝑠 1,2,3,4,6) , , ( = 2,3,4,6) , , , , . For 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 being these PGs,𝐢𝐢𝑛𝑛 𝑛𝑛the 𝑠𝑠 𝑖𝑖 π‘›π‘›β„Ž 𝑠𝑠 4 3𝑖𝑖 6 𝑛𝑛𝑛𝑛𝑛𝑛 condition that𝐢𝐢 𝐢𝐢 𝑛𝑛 (2) = 𝐢𝐢 𝑆𝑆 𝐢𝐢(2)𝑆𝑆 for all 𝐺𝐺 is also satisfied. βˆ’1 𝐾𝐾 𝐾𝐾 2 2 𝑛𝑛𝑛𝑛𝑛𝑛 Step 2 𝑔𝑔 𝑍𝑍 ⋉ 𝑆𝑆𝑆𝑆 𝑔𝑔 𝑍𝑍 ⋉ 𝑆𝑆𝑆𝑆 𝑔𝑔 ∈ 𝐺𝐺 Since is parallel to the axis of rotation of , and (2) contains spin 𝑠𝑠 rotations withπ’Žπ’ŽοΏ½ arbitrary rotation angle along . Then𝐺𝐺 𝑛𝑛if𝑛𝑛𝑛𝑛 there is𝑆𝑆𝑆𝑆 a spin group of the

form Γ— (2) , then couldπ’Žπ’ŽοΏ½ be chosen such that there is no spin 𝐾𝐾 𝑛𝑛𝑛𝑛𝑛𝑛 2 𝑛𝑛𝑛𝑛𝑛𝑛 rotation𝐺𝐺 at allοΏ½ in𝑍𝑍 ⋉ 𝑆𝑆𝑆𝑆, similarοΏ½ to the𝐺𝐺 chosen of in Γ— such that 𝐾𝐾 𝑛𝑛𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛𝑛𝑛 2 𝑛𝑛𝑛𝑛𝑛𝑛 contains no , described𝐺𝐺 in section C1. 𝐺𝐺 𝐺𝐺 𝑍𝑍 𝐺𝐺

Thus, the𝑇𝑇 options left for are or = so that there is no multiple 𝑠𝑠 𝑇𝑇 𝑛𝑛𝑛𝑛𝑛𝑛 1 𝑖𝑖 2 counting of the same types of full𝐺𝐺 SPGs represented𝐢𝐢 𝐢𝐢 by 𝑍𝑍 Γ— (2) . And the 𝐾𝐾 remaining ’s to be considered are actually 90 types𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 including�𝑍𝑍2 ⋉ 𝑆𝑆𝑆𝑆32 typeοΏ½-I MPGs

and 58 type𝐺𝐺-II𝑛𝑛𝑛𝑛𝑛𝑛 MPGs.

In conclusion, there are 90 crystallographic SPGs of the form Γ— 𝐾𝐾 𝑛𝑛𝑛𝑛𝑛𝑛 2 (2) , shown in Table C13-C14. (2) is actually the 𝐺𝐺 group,�𝑍𝑍 in⋉ 𝐾𝐾 𝑆𝑆𝑆𝑆SchoenfliesοΏ½ notation, acting on spin space.𝑍𝑍2 ⋉Hermann𝑆𝑆𝑆𝑆 –Mauguin notation𝐢𝐢∞ is𝑣𝑣 used in the following table, which write as .

πΆπΆβˆžπ‘£π‘£ βˆžπ‘šπ‘š

TABLE C13. 32 SPGs of the form Γ— (2) with the corresponding 𝐾𝐾 being . The first column lists𝐺𝐺 the𝑛𝑛𝑛𝑛𝑛𝑛 �𝑍𝑍 2corresponding⋉ 𝑆𝑆𝑆𝑆 οΏ½ to shown in the 𝑠𝑠 𝐺𝐺second𝑛𝑛𝑛𝑛𝑛𝑛 column.𝐢𝐢1 The third column lists 𝐺𝐺all𝑝𝑝0 of the types of SPGs𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛 of the form

Γ— (2) having shown in the second column. 𝐾𝐾 𝐺𝐺 𝑛𝑛𝑛𝑛𝑛𝑛 �𝑍𝑍2 ⋉ 𝑆𝑆𝑆𝑆 οΏ½ 𝐺𝐺𝑛𝑛𝑛𝑛𝑛𝑛

= Γ— (2)

𝐾𝐾 𝑠𝑠𝑠𝑠 𝑛𝑛𝑛𝑛𝑛𝑛 2 1𝑝𝑝 0 𝑛𝑛1𝑛𝑛𝑛𝑛 𝐺𝐺 𝐺𝐺 οΏ½ 𝑍𝑍1 ⋉ 𝑆𝑆𝑆𝑆 οΏ½ 𝐺𝐺 𝐺𝐺 1 βˆžπ‘šπ‘š 44 1 1 1 1 2 12 12βˆžπ‘šπ‘š1 οΏ½ οΏ½ οΏ½ 1 1 βˆžπ‘šπ‘š 1 2/ 21/ 21/ βˆžπ‘šπ‘š 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2 1 1 2 1 1 2βˆžπ‘šπ‘š 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 222 1 212 12 1 212 12 βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 1 1 1 1 1 βˆžπ‘šπ‘š 1 4 1 1 4 1 1 1 4 1 1∞ π‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 1 βˆžπ‘šπ‘š 4 4 4 1 4/ 41/ 41/ βˆžπ‘šπ‘š 1 οΏ½ οΏ½ οΏ½ 422 14 21 2 14 21 2βˆžπ‘šπ‘š 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 1 1 1 1 1 βˆžπ‘šπ‘š 4 4 4 1 42 1 41 2 1 1 41 2 1 βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 4/1 1 1 4/1 1 1 βˆžπ‘šπ‘š 1 οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 3 1 1 31 1 1 1 31 11 βˆžπ‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 1 βˆžπ‘šπ‘š 3 3 3 1 32 31 2 31 2βˆžπ‘šπ‘š 1 οΏ½ οΏ½ οΏ½ 1 1 1 1 βˆžπ‘šπ‘š 3 3 3 1 3 131 131 βˆžπ‘šπ‘š1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 6 1 16 1 16 βˆžπ‘šπ‘š1 οΏ½π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 6 16 16βˆžπ‘šπ‘š1 οΏ½ οΏ½ οΏ½ 1 1 βˆžπ‘šπ‘š 622 6 2 2 6 2 2 1 1 1 1 1 1 1 βˆžπ‘šπ‘š 6/ 6/ 6/ 1 6 61 1 16 1 βˆžπ‘šπ‘š 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 6 2 1 61 1 2 1 61 1 2 βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6/ 6/1 1 1 6/1 1 1 βˆžπ‘šπ‘š 1 οΏ½π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 23 1 1 2 13 1 1 1 2 13 1 1∞ π‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 1 1 1 βˆžπ‘šπ‘š 2/ 3 2/ 3 2/ 3 1 43 1 4 13 1 1 4 13 1 βˆžπ‘šπ‘š1 π‘šπ‘šοΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ 432 1 41 31 2 1 41 31 2 βˆžπ‘šπ‘š1 οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 1 1 1 1 1 1 βˆžπ‘šπ‘š 4/ 32/ 4/ 3 2/ 4/ 3 2/ 1 1 1 1 1 1 1 1 1 1 1 βˆžπ‘šπ‘š π‘šπ‘šοΏ½ π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š

TABLE C14. 58 types of SPGs of the form Γ— (2) with being 𝐾𝐾 𝑠𝑠 𝑛𝑛𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛𝑛𝑛 ( ). 𝐺𝐺 �𝑍𝑍2 ⋉ 𝑆𝑆𝑆𝑆 οΏ½ 𝐺𝐺 𝑇𝑇 𝐢𝐢 𝑖𝑖 𝑍𝑍2

45 = Γ— (2)

𝐾𝐾 𝑠𝑠𝑠𝑠 𝑛𝑛𝑛𝑛𝑛𝑛 2 1𝑝𝑝 0 𝑛𝑛1𝑛𝑛𝑛𝑛 𝐺𝐺 𝐺𝐺 1 �𝑍𝑍1 ⋉ 𝑆𝑆𝑆𝑆 οΏ½ 𝐺𝐺 𝐺𝐺 2 1οΏ½2 1οΏ½2βˆžπ‘šπ‘š1 οΏ½ οΏ½ οΏ½ 1οΏ½ 1οΏ½ βˆžπ‘šπ‘š 1 2/ 21οΏ½/ 21οΏ½/ βˆžπ‘šπ‘š 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2/ 12/1οΏ½ 12/1οΏ½ βˆžπ‘šπ‘š1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2/ 1οΏ½2/1 1οΏ½2/1 βˆžπ‘šπ‘š1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2 1οΏ½ 1οΏ½ 2 1οΏ½ 1οΏ½ 2βˆžπ‘šπ‘š 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 2 1οΏ½ 1οΏ½ 12 1οΏ½ 1οΏ½ 12βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 222 1 21οΏ½2 1οΏ½2 1 21οΏ½2 1οΏ½2 βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 1οΏ½ 1οΏ½ 1 1οΏ½ 1οΏ½ βˆžπ‘šπ‘š 1 1 1οΏ½ 1οΏ½ 1 1οΏ½ 1οΏ½ βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 1 1οΏ½ 1 1 1οΏ½ βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 4 1οΏ½ 1οΏ½ 4 1οΏ½ 1οΏ½ 1οΏ½ 4 1οΏ½ 1∞ π‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 1οΏ½ 1οΏ½ βˆžπ‘šπ‘š 4 4 4 1 4/ 41οΏ½/ 41οΏ½/ βˆžπ‘šπ‘š 1 οΏ½ οΏ½ οΏ½ 4/ 1οΏ½4/1 1οΏ½4/1 βˆžπ‘šπ‘š1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 1οΏ½4/1οΏ½ 1οΏ½4/1οΏ½ βˆžπ‘šπ‘š1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 422 14 21οΏ½ 2 14 21οΏ½ 2βˆžπ‘šπ‘š 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 1 1οΏ½ 1οΏ½ 1 1οΏ½ 1οΏ½ βˆžπ‘šπ‘š 422 4 2 2 4 2 2 1 1οΏ½ 1 1οΏ½ 1οΏ½ 1 1οΏ½ βˆžπ‘šπ‘š 4 4 4 1 4 141οΏ½ 1οΏ½ 141οΏ½ 1οΏ½ βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 42 1οΏ½ 41 2 1οΏ½ 1οΏ½ 41 2 1οΏ½ βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 42 141οΏ½21οΏ½ 141οΏ½21οΏ½ βˆžπ‘šπ‘š1 οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 42 1οΏ½41οΏ½21 1οΏ½41οΏ½21 βˆžπ‘šπ‘š1 οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 4/ 4/1οΏ½ 1 1οΏ½ 4/1οΏ½ 1 1οΏ½ βˆžπ‘šπ‘š 1 οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 4/ 1οΏ½4/1οΏ½ 1οΏ½ 1 1οΏ½4/1οΏ½ 1οΏ½ 1 βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 14/1οΏ½ 1 1 14/1οΏ½ 1 1 βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 1οΏ½4/1 1 1οΏ½ 1οΏ½4/1 1 1οΏ½ βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 4/ 14/1 1οΏ½ 1οΏ½ 14/1 1οΏ½ 1οΏ½ βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 3 1 1οΏ½ 31οΏ½ 1οΏ½ 1 1οΏ½ 31οΏ½ 1οΏ½1 βˆžπ‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 32 31οΏ½ 2 31οΏ½ 2βˆžπ‘šπ‘š 1 οΏ½ οΏ½ οΏ½ 1 1οΏ½ 1 1οΏ½ βˆžπ‘šπ‘š 3 3 3 1 3 131οΏ½ 131οΏ½ βˆžπ‘šπ‘š1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 3 131οΏ½ 131οΏ½ βˆžπ‘šπ‘š1 οΏ½π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 3 1οΏ½31 1οΏ½31 βˆžπ‘šπ‘š1 οΏ½π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 6 1οΏ½ 1οΏ½6 1οΏ½ 1οΏ½6 βˆžπ‘šπ‘š1 οΏ½π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 1οΏ½ 1οΏ½ βˆžπ‘šπ‘š οΏ½ οΏ½ 46 οΏ½ 6 6 6 1 1οΏ½ 1οΏ½ βˆžπ‘šπ‘š 622 6 2 2 6 2 2 1 1 1οΏ½ 1οΏ½ 1 1οΏ½ 1οΏ½ βˆžπ‘šπ‘š 622 6 2 2 6 2 2 1 1οΏ½ 1 1οΏ½ 1οΏ½ 1 1οΏ½ βˆžπ‘šπ‘š 6/ 6/ 6/ 1 6/ 1οΏ½6/1οΏ½ 1οΏ½6/1οΏ½ βˆžπ‘šπ‘š1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 6/ 1οΏ½6/1 1οΏ½6/1 βˆžπ‘šπ‘š1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 6 61 1οΏ½ 61 1οΏ½ βˆžπ‘šπ‘š 1 π‘šπ‘š π‘šπ‘š π‘šπ‘š 6 161οΏ½ 1οΏ½ 161οΏ½ 1οΏ½ βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6 2 1οΏ½ 61 1οΏ½ 2 1οΏ½ 61 1οΏ½ 2 βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6 2 161οΏ½ 1οΏ½2 161οΏ½ 1οΏ½2βˆžπ‘šπ‘š1 οΏ½π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 6 2 1οΏ½61 1οΏ½2 1οΏ½61 1οΏ½2βˆžπ‘šπ‘š1 οΏ½π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 6/ 6/1οΏ½ 1οΏ½ 1 6/1οΏ½ 1οΏ½ 1 βˆžπ‘šπ‘š 1 οΏ½π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 6/ 1οΏ½6/1οΏ½ 1 1οΏ½ 1οΏ½6/1οΏ½ 1 1οΏ½ βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6/ 1οΏ½6/1 1οΏ½ 1 1οΏ½6/1 1οΏ½ 1 βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6/ 16/1 1οΏ½ 1οΏ½ 16/1 1οΏ½ 1οΏ½ βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 6/ 16/1οΏ½ 1 1 16/1οΏ½ 1 1 βˆžπ‘šπ‘š1 π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 2/ 3 1 21οΏ½/ 1οΏ½ 31οΏ½ 1 21οΏ½/ 1οΏ½ 31οΏ½ ∞1π‘šπ‘š π‘šπ‘šπ‘šπ‘šπ‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š π‘šπ‘š 43 1 4 1οΏ½3 1οΏ½ 1 4 1οΏ½3 1οΏ½ βˆžπ‘šπ‘š1 π‘šπ‘šοΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ 432 1οΏ½ 41 31οΏ½ 2 1οΏ½ 41 31οΏ½ 2 βˆžπ‘šπ‘š1 οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š οΏ½ π‘šπ‘š 1οΏ½ 1 1οΏ½ 1οΏ½ 1 1οΏ½ βˆžπ‘šπ‘š 4/ 32/ 4/ 3 2/ 4/ 3 2/ 1 4/ 32/ 1οΏ½4/1 131οΏ½2/1οΏ½ 1οΏ½4/1 131οΏ½2/1οΏ½ βˆžπ‘šπ‘š1 π‘šπ‘šοΏ½ π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š 4/ 32/ 1οΏ½4/1οΏ½ 1οΏ½31οΏ½2/1 1οΏ½4/1οΏ½ 1οΏ½31οΏ½2/1 βˆžπ‘šπ‘š1 π‘šπ‘šοΏ½ π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š 1 1οΏ½ 1οΏ½ 1 1οΏ½ 1 1οΏ½ 1οΏ½ 1 1οΏ½ βˆžπ‘šπ‘š π‘šπ‘šοΏ½ π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š π‘šπ‘š οΏ½ π‘šπ‘š

47