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PHYSICAL REVIEW X 12, 021016 (2022)

Spin-Group Symmetry in Magnetic Materials with Negligible Spin-Orbit Coupling

Pengfei Liu , 1,2 Jiayu Li , 1 Jingzhi Han, 2 Xiangang Wan, 3,* and Qihang Liu 1,4,5,†

1 Shenzhen Institute for Quantum Science and Engineering and Department of Physics,

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

Nanjing 210093, China and Collaborative Innovation Center of Advanced Microstructures, Nanjing University,

Nanjing 210093, China 4 Guangdong Provincial Key Laboratory of Computational Science and Material Design,

Southern University of Science and Technology, Shenzhen 518055, China 5 Shenzhen Key Laboratory of Advanced Quantum Functional Materials and Devices,

Southern University of Science and Technology, Shenzhen 518055, China

(Received 21 May 2021; revised 11 January 2022; accepted 18 February 2022; published 21 April 2022)

Symmetry formulated by group theory plays an essential role with respect to the laws of nature, from fundamental particles to condensed-matter systems. Here, by combining symmetry analysis and model calculations, we elucidate that the crystallographic symmetry groups 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 spin group, is required. We derive the classifications of spin point groups describing coplanar and collinear magnetic structures, and the irreducible corepresentations of spin space groups illustrating more energy degeneracies that are disallowed by magnetic groups. One consequence of the spin group is the new antiunitary symmetries that protect SOC-free Z 2 topological phases with unprecedented surface-node structures. Our work not only manifests the physical reality of materials with weak SOC, but also sheds light on the understanding of all solids with and without SOC by a unified group theory.

DOI: 10.1103/PhysRevX.12.021016 Subject Areas: Condensed Matter Physics,

Topological Insulators

I. INTRODUCTION

The study of symmetry has always been the kernel of condensed-matter physics and materials chemistry, as it dictates the way in which wave functions of elementary excitations behave, including geometric phases, selection rules, and degeneracies. The corresponding wave-function properties thus reflect the physical observables such as polarization, response susceptibility, and band dispersions. 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 * † [email protected] [email protected]

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

2160-3308=22=12(2)=021016(19)

SGs (MSGs), and their double groups with spinor representations [1,2]. They apply to the nonmagnetic materials without and with spin-orbit coupling (SG and double SG) as well as the magnetic materials with spin-orbit coupling (double MSG). The recent prosperity of the symmetryprotected topological phase in condensed-matter systems is based on electronic structures, providing a fertile playground for a survey of various quasiparticles including Weyl, Dirac fermions, and others beyond them [3–15]. Moreover, the theories of the 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]. Notably, topological phases without spin-orbit coupling (SOC) are also widely investigated in nonmagnetic materials as a starting point of the Hamiltonian with relativistic SOC [13,32–34]. However, the symmetry description of the remaining quadrant, i.e., the magnetic materials with negligible SOC, which represents a vast number of compounds with light elements, is much less 021016-1 Published by the American Physical Society

FIG. 1. Four-quadrant diagram describing the symmetry of solids. The general single-electron 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 Cn ð θÞ and spin rotation Um ð φÞ, where m and n denote the rotation axes, and the real scalars φ and θ are the rotation angles. For the materials with SOC, i.e., the quadrants III and IV, the spatial and spin rotations are completely locked. For example, a spatial rotation by 2π=3 requires a simultaneous spin rotation by 2π=3 along the same axis. For the materials without SOC, the spin and spatial rotations are 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 can 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 of 2π=3 and a spin rotation of 4π=3 along the same axis. Such operations are written as fUm ð φÞkCn ð θÞjτg where the spatial and spin rotation axes and angles could be different. LIU, LI, HAN, WAN, and LIU PHYS. REV. X 12, 021016 (2022)

explored (see Fig. 1). The most striking characteristic of the materials in this quadrant is the nontrivial spin degrees of freedom yet decoupled with the orbital part. Thus, the corresponding symmetry is not fully described by any of the abovementioned SGs. Specifically, the symmetry operations of spin, e.g., spin rotations and the symmetry operations of lattice, can combine in the way disallowed by SOC, which form a composite symmetry group applied on both position space and spin space but not necessarily simultaneously. Such groups, dubbed spin group including spin point group (SPG) and spin space group (SSG), were first considered in the 1960–1970s to account for the extra symmetries of the Heisenberg Hamiltonian with application on spin waves [35,36]. Later, spin group was defined in a mathematically rigorous way, where all 598 nontrivial spin point groups (in which pure spin operations are excluded) were classified using an analogous method of classifying 90 type-I and type-III crystallographic MPGs [37,38]. Apart from the original works, there are scattered works discussing the applications of spin-group symmetry in the Landau theory of phase transition [39,40], neutron scattering [41], electronic states with spiral magnetic order [42], etc.

As a complete characterization of symmetry in magnetic materials when SOC is negligible, the theory of spin group is expected to be powerful also in the application of electronic ground-state properties such as spin or orbit polarization, Berry curvature, and linear responses such as anomalous or spin Hall conductance, (inverse) spin galvanic effect, (inverse) Faraday effect, etc. Surprisingly, such topics were seldomly explored until the recent growing interest of antiferromagnetic (AFM) spintronics [43,44]. It is found that certain effects in AFM materials could appear without the assistance of SOC, including transverse spin current in collinear [45–47] and noncollinear AFM systems [48–50], AFM-induced spin splitting [51–56], giant piezomagnetism [47], etc. In Refs. [48,49], spin-group symmetry is used for determining the conductivity tensor of Mn 3 X (X ¼ Sn, Ir, Ga, and Ge) without SOC, showing more symmetry restrictions for allowed spin conductivity compared to the case with SOC. In Refs. [55,56], broken of a combined operation of translation and pure spin rotation 021016-2 SPIN-GROUP SYMMETRY IN MAGNETIC MATERIALS WITH … that flip spin direction gives rise to a type of antiferromagnet with spin-split bands. In the other works concerning nontrivial effects of AFM order, spin-group symmetry may not be taken into account but actually can act as an effective tool in explaining the predicted results.

Indeed, it is known that symmetries concerning spin and orbital degrees of freedom are decoupled in magnetic materials with negligible SOC, but such symmetries can hardly be fully comprehended without spin-group theory. Most previous studies relating to spin-group symmetries consider symmetry operations leaving a specific Hamiltonian invariant, posing limitations in applying a uniform theory to magnetic materials with all kinds of spin arrangements, including collinear, coplanar, and noncoplanar spin configurations. In Ref. [57], for example, the Hamiltonians considering collinear magnetic order are block diagonal, which greatly simplifies the symmetry analysis of the system. However, such an approach is not applicable to noncollinear magnetic configurations without SOC. On the other hand, the recent progress of condensedmatter physics, in which the geometric phase, topological matter, and emergent quasiparticles play an essential role, paves a way for the application of spin groups in describing complicated magnetic phases. Hence, there is an urgent need to establish the connection between the powerful spingroup theory and the frontier of modern condensed-matter studies.

Here, we systematically study the description of spin1=2 electrons in magnetic lattices by spin groups and their applications on the topological electronic structures. Starting from general single-electron Hamiltonians, 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 (Sec. II). Exemplified by a spinful hexagonal molecule model, we pedagogically elucidate that compared with magnetic groups where the spin and spatial rotations are completely locked to each other, in spin groups, more discrete or continuous spin rotations under a spatial rotation are permitted. We further derive comprehensive classifications of SPGs that are direct products of the nontrivial and spinonly part, including 252 and 90 SPGs to describe the coplanar and collinear magnetic structures, respectively (Sec. III). Exemplified by a kagome lattice model, we derive the irreducible corepresentations of the little spin group at high-symmetry momenta and illustrate the resulting energy degeneracies that cannot be understood by the conventional magnetic (double) groups (Sec. IV). Within the regime of spin group, we find more symmetry operations that would lead to new topological phases such as a SOC-free Z 2 topological insulator (TI) with unprecedented surface-node structures, further enriching the existing zoo of topological materials (Sec. V). For materials realization, we show that square-net compounds AMnBi 2 (A ¼ Sr, Ca) could realize such Z 2 topological phases with surface-nodal PHYS. REV. X 12, 021016 (2022)

lines as well as bulk Dirac points at generic momenta protected by spin-group symmetries (Sec. VI). Our work appeals to the general interest in utilizing spin group in various fields with an expanded material pool for potential spintronics applications, which contains candidates with both light elements and nontrivial electronic properties.

II. GENERAL SINGLE-ELECTRON

HAMILTONIANS

To begin with, we apply spatial and spin rotational operations on the general single-electron Hamiltonians to illustrate the requirements of symmetry operations in various cases. For a nonmagnetic system without SOC [H ¼ ½p̂ 2 =ð2mÞ þ Vðr̂Þ], the wave functions are labeled by the single-valued representations of the Vðr̂Þ-determined SG, of which the rotational elements contain the spatial rotations C n ðθÞ solely. When the general SOC term H SOC ¼ ½1=ð2m 2 c 2 Þ½∇Vðr̂Þ × p̂ · σ̂ is added, neither spatial rotation C n ðθÞ nor spin rotation U m ðφÞ alone, but only a locked combination U n ðθÞC n ðθÞ can keep H SOC invariant. The resulting spinor wave functions furnish the double-valued representations of the Vðr̂Þ-determined double SG. To describe the magnetic systems, we apply H mag ¼ Sðr̂Þ · σ̂ under the framework of single-particle mean-field approximation [58–60], where Sðr̂Þ stands for the r-dependent exchange field due to the distribution of local magnetic moments, and σ̂ is the spin operator. Note that H mag resembles the form of the doubleexchange model widely used to describe the magnetic phase transitions of manganites [61–63]. If H þ H soc þ H mag is considered, the locking between spin and spatial rotations still holds, while the full symmetry description requires double MSG with the inclusion of time-reversal operation T.

If we consider the symmetry operations of H þ H mag , i.e., a magnetic system without SOC, it is straightforward to prove that the complete locking of spatial and spin rotations is no longer required. Instead, partially locked rotations, i.e., U m ðφÞC n ðθÞ, with C n ðθÞ keeping Vðr̂Þ invariant, could keep H mag invariant and thus the total Hamiltonian. Note that Sðr̂Þ presents the effects of spatially distributed magnetic moments residing at the atomic sites with a given magnetization direction, coupling with the spin through exchange-correlation interactions. Unlike SOC, such a spin-spatial coupling is nonrelativistic and constrains U m ðφÞ with C n ðθÞ in various ways according to the specific spin arrangement, forming spin groups, as discussed below (see Appendixes A–C). The Hamiltonians and symmetry groups describing different systems are summarized in Fig. 1, with the derivation of the constraints provided in the Supplemental Material, Sec. II [64].

021016-3 PHYS. REV. X 12, 021016 (2022) LIU, LI, HAN, WAN, and LIU

III. SPIN POINT GROUP: PARTIALLY

DECOUPLED SPIN AND SPATIAL ROTATION

Spin group includes the SPG and SSG. We first discuss SPG, whose elements are denoted by fU m ðφÞ; TU m ðφÞk C n ðθÞ; IC n ðθÞg. While spatial inversion I can combine C n ðθÞ forming improper rotations and mirror reflections, time-reversal symmetry T reverses spin and is thus written in the spin space (T 2 ¼ −1 in spin-1=2 electron system) [72]. For simplicity, we first consider proper rotations only, i.e., fU m ðφÞkC n ðθÞg. The partial coupling between spin and spatial operations implies pure spin rotation fU m ðφÞkEg and coupled spin-spatial rotation, the latter of which forms nontrivial SPGs containing elements of the form fU m ðφÞkC n ðθÞg with C n ðθÞ ≠ E (E is the identity rotation) except identity element fEkEg. Reference [38] constructed all nontrivial SPGs in 3D crystals by combining the factor groups of PGs and their exhaustive isomorphic groups as the spin part. Here, different from mathematical construction, we focus on an exemplified structure with spin arrangements to illustrate how the regime of spin group differentiates the 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.

To illustrate that for certain configurations, SPG generally possesses more symmetry operations than the

generally possesses more symmetry operations than the conventional MPG, we consider a spinful hexagonal mole- cular structure with the spatial rotational group D6 with Consequently, there are seven inequivalent types of spin configurations containing Cz ð π=3Þ and Cx ð πÞ spatial rota- tions, including one collinear ferromagnetic (FM), one 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 C6 z and C2 x spatial rotations. The corresponding nontrivial SPG symbols and generators are shown. We follow the notations of SPG in Ref. [38], with the detailed explanation provided in Appendix C. (i) Symmetry hierarchy from SPG to MPG for various spin arrangements. SOC, spin-orbit coupling; MO, magnetic order; Gp 0, PG for spinless system (containing spatial operations only); Gp , PG with complete locking between spin and spatial degrees of freedom; GM P, MPG; GS P, full SPG; GN SP, nontrivial SPG.

cosðθ=2Þ i sinðθ=2Þ isinðθ=2Þ cosðθ=2Þ

021016-4 SPIN-GROUP SYMMETRY IN MAGNETIC MATERIALS WITH … collinear AFM, one noncollinear FM, and four coplanar AFM configurations. Their nontrivial SPG symbols and generators are shown in Fig. 2. 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 the same angle leaves the eigenvalues of the Hamiltonian invariant. The abovementioned properties explicitly elucidate how the spin and spatial rotation “partially” 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 C n ðθÞ and U m ðφÞ that reduce symmetry. Consequently, spin group itself could serve as a unified 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 the Supplemental Material, Sec. III [64]. The nonmagnetic or paramagnetic phase without SOC, where spin rotation is fully independent of the spatial operations, has the highest symmetry, i.e., the direct product of the spatial part G p0 and spin part SOð3Þ × Z T 2 (Z T 2 ¼ fE; Tg, “×” denotes the internal direct product, while “⊗” denotes the (external) direct product of two groups; see the Supplemental Material, Sec. I [64]). With SOC, the symmetry degrades to subgroups G p × Z T 2 by adding the constraint of the complete spin-space coupling (type-II MPG, 32 gray groups). The further addition of magnetic orders leads to conventional magnetic group G MP , including type I (32 colorless MPGs) and type III (58 black-white MPGs).

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 G NSP [38], which could describe noncoplanar spin arrangements. In addition, for coplanar moments, there exists a bosonlike time-reversal group that forms the spin-only PG Z K 2 ¼ fE; TU m ðπÞ ¼ Kg, where K denotes complex conjugation, rendering the full SPG G SP ¼ G NSP × Z K 2 . For collinear moments, the full SPGs is G SP ¼ G NSP × ½SOð2Þ ⋊ Z K 2  with an additional SO(2) rotational symmetry group along the common direction of the spins (see Appendix B). After a comprehensive classification, we obtain 252 and 90 SPGs to describe the symmetries of coplanar and collinear magnetic structures without SOC, respectively (see Appendix C).

IV. SPIN SPACE GROUP AND BAND

DEGENERACY

By considering the translational symmetry of the lattice, one can easily generalize the operations of SPG to SSG, fU m ðφÞkC n ðθÞjτg, where τ denotes the spatial translation within a primitive cell. Similarly, the analogous symmetry hierarchy for SPG in Fig. 2(i) could be generalized to SSG PHYS. REV. X 12, 021016 (2022)

by involving color Bravais lattices, which certainly include 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 an infinite number of possible types.

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 Mn 3 Ge and Mn 3 Sn [73–76], as shown in Fig. 3(a). 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:

X  X H ¼ ta † R;i;α δ α;β a R 0 ;j;β

α;β

þ J hR;i;R 0 ;ji X

a † R;i;α ðS i · σÞ α;β a R;i;β  : ð2Þ R;i

The first term is the nearest-neighbor hopping, and the second term counts the effect of the local magnetic moment S i . Figures 3(b) and 3(c) show the SOC-free band structures of such a six-band model (including spin) without and with magnetic order, 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 noncollinear AFM order, the spin-degenerate bands in Fig. 3(b) split into two sets of Dirac cones without opening a gap. Apparently, such a twofold degeneracy certainly cannot be interpreted with a MSG Cmm’m’ that supports only 1D irreducible corepresentations at the K point. Such degeneracies were also obtained in band structures of noncollinear AFM kagome models [49,77]. While for collinear AFM order, the Hamiltonian is block diagonal for spin-up and spin-down components [57,78], the noncollinear 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.

First of all, we construct all irreducible corepresentations of the little spin group at a high-symmetry momentum via the similar procedure of deriving irreducible corepresentations for magnetic groups [2]. Thus, the compatibility relations, i.e., the correspondence among the representations of a nonmagnetic group, spin group, and magnetic group, can be obtained (see Appendix D for the methods of obtaining the irreducible corepresentations of little cogroups). Then, starting from the representations of the nonmagnetic site groups furnished by atomic orbitals sitting on a certain Wyckoff position, we obtain the nonmagnetic band representations of the little group at high-symmetry 021016-5

FIG. 3. (a) A spinful kagome lattice with noncollinear AFM spin configuration shown in Fig. 2(e). (b),(c) SOC-free band structure of the kagome lattice without (b) and with (c) magnetic order. The band eigenvalues at K include a 4D irreducible representation K4 and a 2D irreducible representation K1 if considering spin. The numbers in the paratheses represent the dimension of representations. The band eigenvalues of (c) at K include two twofold degenerate points belonging to 2D irreducible corepresentation Ks6 and two nondegenerate points belonging to Ks3 and Ks4 , respectively. (d) Level splitting at K with consecutive addition of magnetic order and spin-orbit coupling. MO and SOC denote magnetic order and spin-orbit coupling, respectively. The irreducible (co)representations of bands are shown in the diagram, with the numbers in the paratheses denoting the dimensionality (considering spin). Note: There are two K points with different little co-groups in the framework of magnetic group, and the one with higher symmetry is chosen for demonstration. LIU, LI, HAN, WAN, and LIU PHYS. REV. X 12, 021016 (2022)

momentum [16]. Finally, the correspondence between single-valued nonmagnetic representations and spin-group corepresentations gives rise to the resulting band representations of a spin group (see the Supplemental Material, Sec. V for the details of the TB model as well as the irreducible corepresentations of little co-groups [64]).

The band degeneracy and compatibility relations at the K valley shown in Fig. 3(c) can be successfully explained by spin-group symmetry. As shown in Fig. 3(b), the d z 2 orbitals on the three kagome sites (Wyckoff position 3g) give rise to two kinds of representations K 1 ð2Þ and K 5 ð4Þ including spin (the number in parentheses denotes the degree of degeneracy). By choosing the eigenstates of spatial rotoinversion C 6z I as the basis functions and adding spin degrees of freedom, we operate the spin-group generators fU z ½ð2πÞ=3jjC z ðπ=3ÞIj0g, fU x ðπÞjjC x ðπÞj0g, and fTU z ðπÞjjIj0g on the Hamiltonian and find that K 5 ð4Þ splits into three levels with two 1D irreducible corepresentation K s 3 (1), K s 4 (1), and a 2D irreducible corepresentation K s 6 (2), consistent with Fig. 3(c) calculated by the TB model. The compatibility of the band splitting is summarized in Fig. 3(d).

V. QUASI-KRAMERS DEGENERACY AND Z 2

MAGNETIC TOPOLOGICAL INSULATOR

Spin-group operations, including an enormous number of combinations of pure spatial operations, time reversal, and pure spin rotations, significantly enhance the symmetries of magnetic materials. As a result, there could be various topological phases unexpected before, protected by spin-group symmetries. Analogous to Z 2 TIs protected by T, we next consider antiunitary operations squared into −1 in spin groups, which could allow Z 2 topological classification in 2D subspaces of the 3D Brillouin zone [79]. We list all such symmetry operations in the regime of 021016-6 SPIN-GROUP SYMMETRY IN MAGNETIC MATERIALS WITH … PHYS. REV. X 12, 021016 (2022)

TABLE I. SSG symmetries supporting quasi-Kramers degeneracy. The SSG symmetries, the momenta with protected twofold degeneracy, the surfaces that maintain the corresponding symmetry, and the possible surface states with various nodal structures are listed. Square of some operators: T 2 ¼ −1, U n ðπÞ 2 ¼ −1, m ½001 2 ¼ 1, τ z=2 2 ¼ 1ð−1Þ for k z ¼ 0ðπÞ.

SSG symmetry fTkEjτ z=2 g fTU z ðπÞkm ½001 jτ x=2 g fTkC z ðπÞj0g fTkjm ½001 0g

fTkjm½ 0010 g fTkm½ 001j τx =2g

fTU n ðπÞkm ½001 jτ x=2 g fTU n ðπÞkEjτ z=2 g Momenta with protected

twofold degeneracy Surfaces with the symmetry TRIM within k z ¼ 0 plane ðπ; 0; k z Þ and ðπ; π; k z Þ lines k z ¼ 0 and k z ¼ π planes ð0; 0; k z Þ, ð0; π; k z Þ, ðπ; 0; k z Þ, and ðπ; π; k z Þ lines ð0; 0; k z Þ and ð0; π; k z Þ lines ðπ; 0; k z Þ and ðπ; π; k z Þ lines TRIM within k z ¼ π plane (xy0) (010) (001) (xy0) (010) (010) (xy0)

SSG in Table I and define the resulting degeneracy at certain high-symmetry k points as “quasi-Kramers degeneracy.” Furthermore, we consider only the symmetries that still persist in certain cleaved surfaces, i.e., possibly having symmetry-protected topological surface states [e.g., fTkIj0g and fTU n ðπÞkC z ðπÞjτ z=2 g are excluded]. Note that there are other Z 2 topological classifications protected by crystalline symmetries that are unitary or squared to 1 [80–82], which can also be extended to spin-group symmetries.

It is found that only two symmetry operations in Table I also exist in MSG, corresponding to AFM Z 2 TI (e.g., MnBi 2 Te 4 ) [83–87] and topological semimetals with symmetry-protected double-helicoid surface states predicted in nonmagnetic systems [88,89], 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 analogs in nonmagnetic materials or magnetic materials with large SOC. Among them, fTkC z ðπÞj0g, fTkm ½001 j0g, fTkm ½001 jτ x=2 g contain both pure spatial rotations and the spin-1=2 time-reversal T. The (001) surface bands with fTkC z ðπÞj0g symmetry are all doubly degenerate, and thus do not protect gapless surface states 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 a double Dirac point, which has been predicted only in bulk bands [10,90]. The three spin-group symmetries containing spatial mirror reflection m ½001 support the surface Dirac nodal line, which has not been reported in magnetic systems. Among them, fTkm ½001 j0g could lead to Z 2 classification for a system and the corresponding 3D quantum spin Hall phases. The last symmetry fTU n ðπÞkEjτ z=2 g also supports Z 2 magnetic TI, similar to fTkEjτ z=2 g, except that the surface Dirac point is located at ð0; πÞ or ðπ; πÞ momenta in momentum space. We note that similar emergent Z 2 topological phases protected by order-two nonsymmorphic antiunitary symmetries were discussed mathematically [91], while Possible surface states Dirac point at ð0; 0Þ or ðπ; 0Þ Dirac nodal line at k x ¼ π Possible double Dirac point Dirac nodal line at k x ¼ 0 or k x ¼ π Dirac nodal line at k x ¼ 0 Dirac nodal line at k x ¼ π Dirac point at ð0; πÞ or ðπ; πÞ fTU n ðπÞkEjτ z=2 g in SSG provides a physical scenario to realize such exotic topological phases.

We next take fTkm ½001 j0g and fTU n ðπÞkEjτ z=2 g as examples to illustrate the new Z 2 magnetic TIs and various unexpected surface-node structures. We start from a 3D Dirac semimetal model without SOC [70], which is analogous to a 3D version of graphene. Such a phase can easily transform to a Weyl semimetal under local magnetic moments S z along the z direction. We can thus tune the hopping parameters to realize a Chern insulator phase at the k z ¼ π=2 plane of the Brillion zone. Then, by building an AFM structure through cell doubling [Fig. 4 (a)], we can annihilate the Weyl points with opposite chirality and create a gapped insulator. By constructing an eight-band model (see the Supplemental Material, Sec. VI for details [64]), we realize a Z 2 magnetic TI protected by both fTkm ½001 j0g and fTU n ðπÞkEjτ z=2 g symmetries, with gapless Dirac surface states at the boundaries of all 2D planes perpendicular to the k z direction. Consequently, it manifests surface Dirac nodal lines at the k x ¼ 0 or k x ¼ π line for any surfaces perpendicular to m ½001 [Figs. 4(c) and 4(e)], which is impossible in conventional TIs protected by T or fTkEjτ z=2 g. To examine the impact of each symmetry, we apply an in-plane FM canting to break fTkm ½001 j0g [Fig. 4(b)]; then, the surface-node structure becomes a Dirac point at ð0; πÞ [Figs. 4(d) and 4(f)], which is consistent with the symmetry analysis shown in Table I. If fTU n ðπÞkEjτ z=2 g is broken by the dimerization of the two layers, the Dirac point at ð0; πÞ is finally gapped [Figs. 4(g) and 4(h)]. Therefore, we demonstrate that unlike nonmagnetic materials and magnetic materials with SOC, magnetic materials with negligible SOC possess new Z 2 topological classification with unprecedented surface-node structures protected by SSG symmetries. We note that the previous studies of SOC-free TIs focused on spinless systems protected by pure crystalline symmetry without considering spin rotation or time-reversal symmetry, which differs from the situation discussed here [32].

021016-7

FIG. 4. Z2 magnetic topological insulators protected by SSG symmetries. (a),(b) A magnetic system with (a) A-type AFM structure invariant under fTkm½ 001j 0g and fTUn ð πÞkEjτz =2g SSG symmetry and (b) fTUn ð πÞkEjτz =2g SSG symmetry. (c),(d) The corresponding (100) surface nodal structures in the Brillouin zone, including (c) surface nodal line and (d) surface Dirac cone. (e),(f) The corresponding surface band dispersion. (g) The configuration with broken fTUn ð πÞkEjτz =2g by the dimerization of the two layers. (h) the corresponding (100) surface band dispersion with a gapped Dirac cone. LIU, LI, HAN, WAN, and LIU PHYS. REV. X 12, 021016 (2022)

VI. MATERIALS REALIZATION

OF NODAL-LINE SEMIMETALS

A remarkable consequence of symmetry and topology in the electronic structure of materials is the existence of protected degeneracies, leading to various topological semimetals such as Dirac, Weyl, nodal-line, and nodalsurface 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., SrMnBi 2 and CaMnBi 2 , to illustrate their unrevealed bulk and surface nodes that exist only under the regime of spingroup 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 semirealistic setup. The identification of more suitable material candidates described by spin-group symmetries is left for future work.

AMnBi 2 (A ¼ Sr, Ca) are layered materials with anisotropic Dirac fermions [see Fig. 5(a)], inspiring the study of square-net materials as topological semimetals such as ZrSiS [92–94]. Despite a checkerboard-type AFM configuration, the nodal properties in such compounds are typically treated via nonmagnetic models [71]. Without SOC, the diagnosis for nonmagnetic topological semimetals with inversion symmetry indicates that AMnBi 2 are bulk nodal-line semimetals [34] (see the Supplemental Material, Sec. VII [64]). The AFM magnetic order further turns the nodal lines to discrete Dirac points with fourfold degeneracy. Compared with the case with SOC where the Dirac points occur only 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 a peculiar property could be understood by the k · p low-energy Hamiltonian. By applying fTkIj0g (iσ y τ x K) and SO(2) spin rotation along the spin direction (e −iθσ z ), the symmetry-allowed Hamiltonian takes the form HðkÞ ¼ f 0 ðkÞσ 0 τ 0 þ f 1 ðkÞσ 0 τ x þ f 2 ðkÞσ 0 τ y þ f 3 ðkÞσ z τ z with the basis fjψ A ↑i; jψ B ↑i; jψ A ↓i; jψ B ↓ig, where A and B represent two sublattices connected by fTkIj0g. The last three terms of HðkÞ mutually anticommute, leading to stable Dirac nodes that could appear at generic momenta and cannot be gapped by any perturbation that maintains 021016-8

FIG. 5. Z2 topological phase protected by fTkm½ 110j 0g symmetry in square-net materials. (a) Crystal structure of AMnBi2 (A ¼ Sr, Ca). The A atoms are not plotted for clarification. (b) Locations of 16 Dirac points and the corresponding surface nodal lines of pressured SrMnBi2 (with lattice parameter c reduced by 10%). The equivalent Dirac points that are connected by symmetry are denoted by the same color. There are three types of nonequivalent Dirac points, with eight Dirac points located at generic momenta (denoted by purple) represented by ð0.690; 0.066; 0Þ ðÅ− 1 Þ; four Dirac points located along the Γ–X line (denoted by cyan) represented by (0.280,0.280,0), and four Dirac points located along the M–G line (denoted by blue) represented by (0.293,0.293,0.272). The four surface nodal lines terminate at the surface projections of the Dirac points (denoted by the dashed lines). (c) Topological nodal-line surface states of pressured SrMnBi2 . The k path is from ¯Γ0 ð0.287; 0.287Þ to ¯M0 ð0.973; −0.399Þ. (d) Wilson loop calculation on a plane [colored by green in panel (b)] parallel to m½ 110. The integration along ¯Γ0 − ¯M0 is calculated along kz from kz ¼ 0 to the Brillouin zone boundary kz ¼ π=c. (e) The transition of 2D Z2 value defined at the vertical planes in the Brillouin zone parallel to m½ 110, as a function of the momentum along the [110] direction. SPIN-GROUP SYMMETRY IN MAGNETIC MATERIALS WITH … PHYS. REV. X 12, 021016 (2022)

fTkIj0g, SO(2) spin rotation and translation symmetries. The Dirac points of SrMnBi 2 calculated by density-functional theory (DFT) are shown in Fig. 5(b) (see methods of DFT calculations in Appendix E).

According to the above discussion, fTkm ½110 j0g spingroup symmetry in AMnBi 2 protects Z 2 topological classification with unprecedented surface nodal lines. To verify this, we next apply DFT calculations on SrMnBi 2 under uniaxial pressure [the lattice parameter c (along the z direction) is reduced by 10%]. Figure 5(c) plots the surface states on the (001) surface, showing a gapless Dirac cone at both the Γ̄ 0 and M̄ 0 points. The existence of the two Dirac points at (0.280,0.280,0) and (0.293,0.293,0.272), protects a region in which any vertical planes in the Brillouin zone parallel to m ½110 yield a nontrivial 2D Z 2 ¼ 1 phase, as indicated by the Wilson loop of a representative plane [shown in green in Fig. 5(b)] and the transition of Z 2 as a function of the momentum along the [110] direction [see Figs. 5(d) and 5(e)]. Furthermore, the surface nodes form a line between the surface projections of the two Dirac points. The fourfold rotation symmetry in this system transforms the nodal line into four nodal lines, with two protected by fTkm ½110 j0g and the other two protected by fTkm ½11̄0 j0g.

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-1=2 021016-9 PHYS. REV. X 12, 021016 (2022) LIU, LI, HAN, WAN, and LIU electronic systems. The main purpose of our work is to build the bridge between the powerful but largely overlooked symmetry group and the frontier of quantummaterial studies. The abovementioned symmetry-protected degeneracy and Z 2 topological classification are merely the tip of the iceberg for the application of spin group, leaving fruitful diversity of topological phases and emergent fermions induced by such an enhanced symmetry group to be further explored. For example, a recent work realizes a condensed-matter counterpart of the SUð2Þ flavor symmetry in particle physics, leading to a new type of AFM Weyl semimetal protected by spin-group symmetries [95]. 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.

We next discuss two physical effects of the spin mirror symmetry fTU z ðπÞkEj0g existing in coplanar and collinear spin groups (the comprehensive list of 252 and 90 SPGs, respectively, is provided in the Supplemental Material, Sec. IV [64]). The first one is spin-momentum locking. For coplanar spin groups, if spin splitting exists owing to broken fTkIj0g symmetry, the spin texture, i.e., the expectation of spin operator with respect to the momentum SðkÞ, has to be ½S x ðkÞ; S y ðkÞ; S z ðkÞ ¼ ½S x ð−kÞ; S y ð−kÞ; −S z ð−kÞ. In addition, for collinear spin groups with moments along the x axis, the existence of the SO(2) group forces nondegenerate energy eigenstates to be the eigenstates of σ x ; hence, only inversion-symmetric S x components survive, i.e., ½S x ðkÞ; 0; 0 ¼ ½S x ð−kÞ; 0; 0. The second one is charge and spin transports. Because the expression of Berry curvature is independent of spin, for coplanar magnets, fTU z ðπÞkEj0g forces the integral of Berry curvature throughout the Brilloun zone to be zero, in analogy to the role of T for nonmagnetic systems. As a result, the anomalous Hall effect cannot be induced by coplanar magnetic orders without the assistance of perturbations such as SOC and magnetic tilting. However, fTU z ðπÞkEj0g, in general, does not forbid transverse spin current and the spin Hall effect in both coplanar [48–50] and collinear AFM systems [45–47]. Particularly, in coplanar AFM systems, both T-odd (transverse spin current) and T-even (spin Hall effect) contributions could be nonzero [48,49], while only T-odd contributions exist in collinear AFM systems.

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, crystal field, etc. (typically in the order of eV). Consequently, one can construct the zeroorder Hamiltonian of a magnetic ground state based on spin group and add SOC as high-order perturbation terms. Such an 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. In addition, each element of the response tensors χ ðnÞ in Kubo formalism for observables like spin-orbit torque correlates the symmetry of crystals [96], determining its zero or nonzero value but not the magnitude. Therefore, the conventional MSG cannot tell if a symmetry-permitted element is tiny or large, even if neglecting 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.

Last but not least, since the symmetries of spin and space degrees of freedom are considered separately, spin group could provide a unified group theory for describing materials in all 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 (co)representation of the 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 and magnetic groups and paves a new way for understanding the emergent properties of magnetic quantum materials.

ACKNOWLEDGMENTS

We thank Chen Fang, Zhida Song, and Zhi Wang for helpful discussions. This work is supported by National Key R&D Program of China under Grant No. 2020YFA0308900, the National Natural Science Foundation of China under Grants No. 11874195, No. 11834006, and No. 12188101, Guangdong Provincial Key Laboratory for Computational Science and Material Design under Grant No. 2019B030301001, the Science, Technology, and Innovation Commission of Shenzhen Municipality (Grant No. ZDSYS20190902092905285), and the Center for Computational Science and Engineering of Southern University of Science and Technology. X. W. also acknowledges support from the Tencent Foundation through the XPLORER PRIZE.

Note added.—Recently, we became aware of some other works on spin-group symmetry [97–100]. In Ref. [97], the 021016-10 SPIN-GROUP SYMMETRY IN MAGNETIC MATERIALS WITH … magnon band structures with nodal point, line, and volume protected by spin-group symmetry are illustrated via Kitaev-Heisenberg models. Reference [98] predicts a new type of collinear antiferromagnet with spin splitting via spin-group symmetry analysis. Reference [99] derives symmetry invariants of SPGs with the form P × Z T 2 , indicating new excitations in the framework of spin group. Reference [100] predicts a kind of eightfold-degenerate fermion in two-dimensional antiferromagnet protected by spin-group symmetry.

APPENDIX A: DEFINITION OF SPIN GROUP

AND ITS CLASSIFICATIONS

Thirty-two crystallographic PGs and 230 SGs are groups that aim to describe 3D nonmagnetic crystals. For magnetic crystals, the ordered spin arrangements in periodic lattices are generally described by 122 MPGs and 1651 MSGs. Such magnetic groups introduce the antisymmetric timereversal operation T that flips double-valued properties like spin, thus enlarging the number of crystallographic group types. However, spin, as a vector in 3D Euclidian vector space, could have more than two values in realistic spin arrangements, the symmetry operations of which include spin rotations and spin inversion forming an orthogonal group O(3) that keeps the 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 PHYS. REV. X 12, 021016 (2022)

G0 ⊗ Os ð3Þ ¼ G0 ⊗ ½SOð3Þ × ZT2 : ðA5Þ

For G 0 being PG G p0 , which is written as

G p0 ¼ fC n ðωÞI m jn

¼ sinðθÞcosðφÞx̂ þ sinðθÞsinðφÞŷ þ cosðθÞẑ;θ ∈ ð0;π;φ ∈ ð0;2π;ω ∈ ð0;2π;m ¼ 0;1g;

ðA6Þ

where C n ðωÞ stands for a spatial rotation operation with rotation axis n and rotation angle ω, while I stands for spatial inversion symmetry operation, every subgroup of G p0 ⊗ O s ð3Þ is called SPG G SP , operations of which are denoted as

fg s kgg; ðA7Þ

with g s ∈ O s ð3Þ and g ∈ G p0 , and

fg s kggfg 0 s kg 0 g ¼ fg s g 0 s kgg 0 g; ðA8Þ

fg s kgg −1 ¼ fg s −1 kg −1 g: ðA9Þ

We define g s and g acting on the spin space and coordinate space, respectively,

through T, for which we write the orthogonal group acting on the spin space as Os ð3Þ ¼ SOð3Þ × ZT2 ; ðA1Þ gs ∶ S → Rðgs Þ S; ðA10Þ g∶ r → RðgÞr; ðA11Þ

Z T 2 ¼ fE; Tg; ðA2Þ where Rðg s Þ and RðgÞ are the representation matrix of g s

and g in 3D Euclidian space, respectively. Then, for the r-dependent spin arrangement SðrÞ, we have SOð3Þ ¼ fU n ðωÞjn

¼ sinðθÞ cosðφÞx̂ þ sinðθÞ sinðφÞŷ

þ cosðθÞẑ; θ ∈ ð0; π; φ ∈ ð0; 2π; ω ∈ ð0; 2πg;

ðA3Þ

where U n ðωÞ stands for spin rotation with rotation axis n and rotation angle ω. Such group O s ð3Þ is termed the orthogonal group of spin symmetries.

When considering 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:

SðrÞ ¼ ½S x ðrÞ; S y ðrÞ; S z ðrÞ T : ðA4Þ

Then, spin group G s is defined as any subgroup of the external direct product of a group with elements exerting on three-dimensional spatial coordinates (either PG or SG) denoted as G 0 , and the orthogonal groups of spin symmetries, i.e., O s ð3Þ:

g s ∶ SðrÞ → Rðg s ÞSðrÞ; ðA12Þ

g∶ SðrÞ → S(Rðg− 1 Þr): ðA13Þ

Thus, it is natural to define the action of SPG operations on the spin arrangement as

fg s kgg∶ SðrÞ → Rðg s ÞS(Rðg −1 Þr): ðA14Þ

If we are interested in the symmetry groups of realistic molecule or crystal, we must consider another scalar-valued function fðrÞ, which stands for the electric potential function contributed by atomic nuclei. Since spin rotation has no influence on electric potential, the action of SPG operations is defined as

fgs k gg∶fðrÞ → f(Rðg− 1 Þr) ðA15Þ

Note that for g s ¼ U n ðωÞ=g ¼ C n ðωÞ, we denote R(U n ðωÞ) ¼ R n ðωÞ=R(C n ðωÞ) ¼ R n ðωÞ with R n ðωÞ 021016-11 PHYS. REV. X 12, 021016 (2022) LIU, LI, HAN, WAN, and LIU being the rotation matrix in 3D Euclidian space with rotation axis and rotation direction represented by n and rotation angle represented by ω. For g s ¼ T=g ¼ I, we denote RðTÞ ¼ −I=RðIÞ ¼ −I, with I being the identity matrix.

For G being space group G s0 , then every subgroup of G s0 ⊗ O s ð3Þ is called a SSG G SS . We can further write SSG operations as fg s kgjtg;

APPENDIX B: GROUPS CONSISTING

OF PURE SPIN OPERATIONS

A spin-only group consists of elements of the form fg s kEj0g (or fg s kEg), which act on the spin configuration SðrÞ as

fgs k Ej0g∶SðrÞ → Rðgs Þ SðrÞ: ðB1Þ

ðA16Þ

where fgjtg ∈ G with g denoting the PG operation, and t is a three-component real vector denoting the translation operation. We have fg s kgjtgfg 0 s kg 0 jt 0 g ¼ fg s g 0 s kgg 0 jRðgÞt 0 þ tg; ðA17Þ fg s kgjtg −1 ¼ fg s −1 kg −1 j − Rðg −1 Þtg: ðA18Þ The actions of fg s kgjtg on spin arrangement and scalar potential satisfy fg s jjgjtg∶ SðrÞ → Rðg s ÞS(Rðg −1 Þðr − tÞ); ðA19Þ

fgs j jgjtg∶fðrÞ → f(Rðg− 1 Þðr − tÞ): ðA20Þ

MSGs and MPGs include an antisymmetry (or timereversal) operation in addition to spatial symmetry operations while neglecting spin rotation operations. Thus, they are incomplete, in a sense, for describing the full symmetry of a general spin arrangement. When spin-orbit coupling is included, MSGs and MPGs are accurate for describing the symmetry of the Hamiltonian in physics because the spin and lattice degrees of freedom must rotate synchronously. 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.

Reference [101] shows that every spin group G s (either SPG G SP or SSG G SS ) can be written as a direct product of a so-called spin-only group G SO and a nontrivial spin group G NS . Spin-only group stands for the group formed by pure spin operations fg s kEj0g (or fg s kEg), while the nontrivial spin group stands for the group that contains no pure spin operations, i.e., all group elements contain spatial operations except the identity.

In Appendix B, we analyze all possible spin-only groups for different types of spin arrangements. In Appendix C, we comprehensively develop all possible combinations between nontrivial SPGs G NSP and spin-only PGs G SOP to provide a full description of the possible SPGs.

To analyze the spin-only groups for different spin arrangements, we divide all of the pure spin operations into four types, i.e., g s ¼ U n ðωÞ (ω ≠ 0), g s ¼ T, g s ¼ U n ðπÞT, and g s ¼ U n ðωÞT (ω ≠ 0 and ω ≠ π) and analyze the conditions for SðrÞ to be invariants under them separately. (We choose specific axes of spin rotations to simplify our analysis.)

Type 1: g s ¼ U n ðωÞ for ω ≠ 0. In this case, we have

SðrÞ ¼ R(U n ðωÞ)SðrÞ. Then, there are two types of spin arrangements that have U n ðωÞ symmetry: (1) SðrÞ is zero for any r corresponding to nonmagnetic spin arrangements. (2) If SðrÞ does not belong to a nonmagnetic spin arrangement, SðrÞ should be parallel to n corresponding to collinear spin arrangements. Type 2: g s ¼ T. If g s ¼ T, then the spin arrangements

invariant under fTkEj0g should satisfy SðrÞ ¼ −SðrÞ, indicating nonmagnetic spin arrangements. Type 3: g s ¼ U n ðπÞT. If n ¼ z, then the spin arrange-

ments invariant under fU z ðπÞTkEj0g should satisfy 

1 0 0  SðrÞ ¼ 0 1 0 SðrÞ. Then, the spin arrange-

0 0 −1 ments SðrÞ that have no z component have such symmetry. We define all spin arrangements satisfying this condition that are not nonmagnetic or collinear spin arrangements as coplanar spin arrangements. Apparently, each spin arrangement that is nonmagnetic, collinear, or coplanar is invariant under fU n ðπÞTkEj0g for certain n. Consequently, the spin arrangements that are not coplanar, collinear, or nonmagnetic are known as noncoplanar spin arrangements. Type 4: g s ¼ U n ðωÞT (ω ≠ 0 and ω ≠ π). If

n ¼ z, then the spin arrangement that is invariant under fU z ðωÞTkEj0g satisfies SðrÞ ¼  − cosðωÞ sinðωÞ 0  − sinðωÞ − cosðωÞ 0 SðrÞ. It is easy to check

0 0 −1 that such an equation has no solution for SðrÞ unless SðrÞ ¼ 0. In conclusion, there are four types of spin arrangements that have different spin-only groups. Since the spatial part is always identitfied in spin-only groups, we write down only the spin part of G SO for simplicity in the following.

021016-12 SPIN-GROUP SYMMETRY IN MAGNETIC MATERIALS WITH …

1. Nonmagnetic spin arrangements

The spin-only group is invariant under all the pure spin operations:

G SO ¼ O s ð3Þ: ðB2Þ

2. Collinear spin arrangements

Any spin rotation along the direction of spin arrangement (e.g., z direction) could leave this type of spin arrangement invariant, indicating SOð2Þ ≡ fU z ðωÞjω ∈ ð0; 2πg spin rotation group. Furthermore, such a spin arrangement is also invariant under U n ðπÞT, where n could be any direction perpendicular to z. We then have a binary spin-only group defined as Z K 2 ≡ fE; U n ðπÞTg. Then, the full spin-only group is the internal semidirect product of SO(2) and Z K 2 , i.e.,

G SO ¼ SOð2Þ ⋊ Z K 2 : ðB3Þ

Note that the internal semidirect product is because SO(2) is a normal subgroup of G SO while Z K 2 is not.

3. Coplanar spin arrangements

From the discussion above, the spin-only group of such a spin arrangement is

G SO ¼ Z K 2 ≡ fE; U n ðπÞTg; ðB4Þ where n denotes the direction perpendicular to the plane of the spin arrangements.

4. Noncoplanar spin arrangements

The spin-only group of this type has only the identity element

G SO ¼ fEg: ðB5Þ APPENDIX C: FULL CRYSTALLOGRAPHIC SPIN

POINT GROUPS FOR COLLINEAR AND

COPLANAR SPIN ARRANGEMENTS

Crystallographic SPGs are SPGs being subgroups of G p0 ⊗ O s ð3Þ where G p0 is one of the 32 crystallographic PGs. Construction of nontrivial SPGs from 32 PGs is similar to obtaining the MPGs. However, the orthogonal group of spin symmetries SOð3Þ × Z T 2 has an infinite number of operations, the introduction of which into crystallographic PGs requires us to find all normal subgroups of the 32 PGs and find all of the groups that are subgroups of SOð3Þ × Z T 2 and isomorphic to the corresponding quotient groups. Reference [37] shows that all 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 PHYS. REV. X 12, 021016 (2022)

all subgroups of SOð3Þ × Z T 2 that are isomorphic to the quotient groups through all possible isomorphic relations. By applying such a procedure, Ref. [38] obtains 598 types of nontrivial SPGs, with two groups defined to belong to the same type if they are conjugate subgroups of the direct product of general linear group in spin space and affine group in physical space, i.e., GLð3Þ ⊗ GILð3Þ, which are sufficient to describe noncoplanar spin arrangements according to the discussion in Appendix B. However, for collinear and coplanar spin arrangements, there are pure spin operations; i.e., we have to consider SPGs that are products of nontrivial SPGs and spin-only PGs, i.e., G SP ¼ G NSP G SOP . Such products could always be written as semidirect products G NSP ⋉ G SOP because G SOP is always a normal subgroup of G NSP G SOP . This could also be written as a direct product G NSP × G SOP by proper selection of G NSP such that G NSP is also a normal subgroup of G NSP G SOP . Thus, to classify the point groups of the form G NSP × G SOP , we have to find all types of nontrivial SPGs (G NSP ) that could perform an internal direct product with spin-only PGs G SOP . Since 598 types of nontrivial SPGs are complete, and any SPG can be written as the direct product of nontrivial SPGs and spin-only PGs, such a classification should lead to a complete set of SPGs to describe coplanar and collinear spin arrangements. [We neglect nonmagnetic spin arrangements because they obviously have symmetry groups of the form G 0 ⊗ O s ð3Þ with G 0 being one of the PGs or SGs]. We conduct such a classification in the following two-step procedure:

Step 1: We find all types of nontrivial SPGs that allow

both G NSP and G SOP to be invariant under each other, i.e., satisfy g −1 G NSP g ¼ G NSP for all g ∈ G SOP and h −1 G SOP h ¼ G SOP for all h ∈ G NSP , and then determine the corresponding G NSP G SOP 0 s (one nontrivial SPG might correspond to several full SPGs). Since both G NSP and G SOP are subgroups of G p0 ⊗ O s ð3Þ, and G NSP ∩ G SOP ¼ ffEkEgg, the condition g −1 G NSP g ¼ G NSP for all g ∈ G SOP and h −1 G SOP h ¼ G SOP for all h ∈ G NSP , implies that G NSP G SOP is a group and G NSP G SOP ¼ G NSP × G SOP . Then, we get all types of full SPGs represented by G NSP × G SOP . Step 2: We consider full SPGs obtained from step 1

represented as groups G 1 NSP × G SOP ; G 2 NSP × G SOP ; …; G n NSP × G SOP , with G 1 NSP ; G 2 NSP ; …, and G n NSP belonging to different types of nontrivial SPGs, but with G 1 NSP × G SOP , G 2 NSP × G SOP , …, and G n NSP × G SOP actually belonging to the same types of full SPGs. Then, we choose one of the G i NSP × G SOP to represent this full SPG. In other words, we eliminate multiple counting of the equivalent types of full SPGs. As discussed in Appendix B, the full SPGs for coplanar spin arrangements can be written as G NSP × Z K 2 , while the full SPGs for collinear spin arrangements can be described 021016-13 PHYS. REV. X 12, 021016 (2022) LIU, LI, HAN, WAN, and LIU by G NSP × ½Z K 2 ⋉ SOð2Þ. Next, we separately classify full SPGs for coplanar spin arrangements and for collinear spin arrangements.

Then, we get all of the types of SPGs that can be written as G NSP × Z K 2 with some types possibly being identical.

b. Step 2

1. Classification of full SPGs

for coplanar spin arrangements For coplanar spin arrangements, we have G SOP ¼ Z K 2 . Note that we do not consider the relative directions of the spin rotation axis and space rotation axis in the following discussion because the variation of the relative direction of the spin rotation axis and space rotation axis will not give rise to the different types of spin group.

a. Step 1 Step 1 outlined above implies that if we write K Z 2 ≡ ffEkEg; fTU n ðπÞkEgg, the rotation axis n should be either parallel or perpendicular to every spin rotation axis of the spin rotation part of G NSP , i.e., G s NSP . Otherwise, the condition that G SOP ¼ Z K 2 should be invariant under G NSP will not be satisfied. Thus, this step excludes the nontrivial SPGs whose spin part G s NSP is polyhedral group, including T, T h , T d , O, and O h , because we cannot find a direction that is either parallel or perpendicular to all the rotation axis. Thus, the options left for the spin part G s NSP are 27 axial groups: C n ðn ¼ 1; 2; 3; 4; 6Þ, D n ðn ¼ 2; 3; 4; 6Þ, S n ðn ¼ 2; 4; 6ÞðS 2 ¼ C i ; S 6 ¼ C 3i Þ, C nh ðn ¼ 1; 2; 3; 4; 6ÞðC 1h ¼ C s ¼ C 1v Þ, D nh ðn ¼ 2; 3; 4; 6Þ, C nv ðn ¼ 2; 3; 4; 6Þ, and D nd ðn ¼ 2; 3Þ. For G s NSP being different groups, the direction n is constrained differently in order for the condition that g −1 Z K 2 g ¼ Z K 2 for all g ∈ G NSP to be satisfied. We classify the ways n is constrained into the following five cases.

Case 1: For G s NSP being C 1 or S 2 ðC i Þ, n is not con-

strained. Case 2: For G s NSP being C 1h ¼ ffEkEg; fTU z ðπÞkEgg,

C 2 ¼ ffEkEg; fU z ðπÞkEgg, or C 2h ¼ ffEkEg; fU z ðπÞkEg; fTU z ðπÞkEg; fTkEgg, there is only one twofold spin rotation U z ðπÞ in G s NSP , thus, n which could be either parallel or perpendicular to z. (When n is perpendicular to z, then groups corresponding to different n belong to the same types up to conjugate transformations.) Case 3: For G s NSP being the groups that have rotations

of order larger than two, n should be parallel to the principal axis of G s NSP . Case 4: For G s NSP being D 2 or D 2h , n should be parallel

to one of the twofold spin rotation axes. Case 5: For G s NSP being C 2v ; n should either be

perpendicular to one mirror or parallel to the twofold rotation axis. It is easy to see that for the five cases, the condition that g −1 G NSP g ¼ G NSP for all g ∈ Z K 2 is also satisfied.

Some types obtained from step 1 are identical because the operations that implicitly contain T in G NSP could always be changed to the product of fTU n ðπÞkEg with those operations, for the full group being G NSP × Z K 2 . That is to say, if we have a spin point group G 1 NSP × Z K 2 with G 1 NSP being a nontrivial SPG that could be written as the form G 1 NSP ¼ H þ fTkEgðG 1 NSP − HÞ with H being the subgroup of G 1 NSP of order two that does not contain T, then, we have

G 1 NSP × Z K 2 ¼ ½H þ fTkEgðG 1 NSP − HÞ × ffEkEg; fTU n ðπÞkEgg ¼ ½H þ fU n ðπÞkEgðG 1 NSP − HÞ × ffEkEg; fTU n ðπÞkEgg

n N SP n ¼ G2N SP × ZK2 ; ðC1Þ

with G 2 NSP containing no T. This implies that G NSP in all SPGs of the form G NSP × Z K 2 can be chosen such that the G s NSP corresponding to G NSP is formed by pure spin rotations. Furthermore, it is obvious that two SPGs G A NSP × Z K 2 and G B NSP × Z K 2 with G A NSP and G B NSP being different types and containing no T should be different types of SPGs. Thus, we can use all nontrivial SPGs G NSP that have spin part G s NSP being nine axial PGs of pure spin rotation C n ðn ¼ 1; 2; 3; 4; 6Þ and D n ðn ¼ 2; 3; 4; 6Þ to construct all types of SPGs of the form G NSP × Z K 2 to avoid multiple counting of the same type of group. Then, there are four cases left:

Case 1: For G s NSP ¼ C 1 , n could be random directions

which result in the same types of SPGs. Case 2: For G s NSP ¼ C 2 ¼ ffEkEg; fU z ðπÞkEgg, n

could be either parallel or perpendicular to the z axis. Case 3: For G s NSP ¼ D 2 ¼ ffEkEg; fU x ðπÞkEg;

fU y ðπÞkEg; fU z ðπÞkEgg, n could be parallel to one of the three twofold rotation axes. These three twofold rotation axes are equivalent for G s NSP , but they are not necessarily equivalent for the whole nontrivial SPG G NSP . Thus, the groups which separately have n parallel to the x, y, and z axes could belong to the same types of SPGs or different types of SPGs. Case 4: For G s NSP being one of the six groups left

C n ðn ¼ 3; 4; 6Þ and D n ðn ¼ 3; 4; 6Þ, n has to be parallel to the principal axis z. Thus, we can classify the types of full SPGs of the form G NSP × Z K 2 into 11 categories shown in Table II. Finally, we get 252 types of crystallographic SPGs of the form G NSP × Z K 2 . These SPGs are listed in Tables S1–S11 of Supplemental Material [64].

021016-14 SPIN-GROUP SYMMETRY IN MAGNETIC MATERIALS WITH … PHYS. REV. X 12, 021016 (2022)

TABLE II. Classifications of the SPGs of the form G NSP × Z K 2 , with Z K 2 ≡ ffEkEg; fTU n ðπÞkEgg and the corresponding G s NSP being one of the nine groups C n ðn ¼ 1; 2; 3; 4; 6Þ, D n ðn ¼ 2; 3; 4; 6Þ. The principal axes of G s NSP , if they exist, are assumed to be along the z direction.

Category I Category II Category III Category IV Category V Category VI Category VII Category VIII Category IX Category X Category XI G s NSP Direction of n Number of types C 1 C 2 C 2 C 3 C 4 C 6 D 2 D 2 D 3 D 4 D 6 Random Parallel to the z axis Perpendicular to the x axis Parallel to the z axis Parallel to the z axis Parallel to the z axis Parallel to x, y, or z axes (when the three directions lead to identical types) Parallel to the x, y, or z axes (when the three directions lead to different types) Parallel to the z axis Parallel to the z axis Parallel to the z axis 32 58 58 6 4 7 2 62 10 5 8

Total number of types: 252

2. Classification of full SPGs for collinear

spin arrangements APPENDIX D: IRREDUCIBLE COREPRESENTATIONS OF SPIN GROUPS For collinear spin arrangements, we then consider the form G NSP × ½Z K 2 ⋉ SOð2Þ.

For a Mn 3 Sn kagome lattice, we provide the specific band corepresentations of the little co-groups at different momenta (K and M). The procedure is as follows: (1) Find the maximal unitary subgroups (MUSGs) of particular little co-groups. (2) Get the full list of irreducible representations of MUSGs from single-valued representations of their corresponding spatial part. (3) Construct the irreducible corepresentations of little co-groups from those representations obtained in (2) based on the method derived by Wigner [102].

Tables S14 and S16 in the Supplemental Material [64] list the representation matrices of all irreducible corepresentations of K and M. The correspondences among the irreducible corepresentations of the spin group, the single-valued representations of the corresponding nonmagnetic group, and double-valued corepresentations of the corresponding magnetic group are separately listed in Tables S15 and S17 in the Supplemental Material [64]. Since the character tables are defined only for unitary groups, we list the character tables of MUSGs, all symmetry operations, and conjugate classes in Tables S18 and S21 in the Supplemental Material [64]. Since each nontrivial spin point group is isomorphic to a nonmagnetic or magnetic point group by neglecting the spin rotations, the conjugate classes of MUSGs can be directly obtained by resorting to the conjugate classes of the corresponding nonmagnetic point group.

a. Step 1 We first write SO(2) as SOð2Þ ¼ ffU m ðωÞTjjEj0gjω ∈ ð0; 2πg. Because of the presence of SOð2Þ ¼ ffU m ðωÞTjjEj0gjω ∈ ð0; 2πg, all rotations contained in G s NSP should have rotation axis parallel to m, so that the condition that h −1 G NSP h ¼ G NSP for all h ∈ Z K 2 ⋉ SOð2Þ is satisfied. Thus, G s NSP should have no more than one rotation axis. Then, the options left for G s NSP are 13 PGs: C n ðn ¼ 1; 2; 3; 4; 6Þ, C nh ðn ¼ 1;2;3;4;6Þ, S n ðn ¼ 2; 4; 6Þ. For G s NSP being these PGs, the condition that g −1 ½Z K 2 ⋉ SOð2Þg ¼ Z K 2 ⋉ SOð2Þ for all g ∈ G NSP is also satisfied.

b. Step 2 Since m is parallel to the axis of rotation of G s NSP , and SO(2) contains spin rotations along m with arbitrary rotation angle if there is a spin group of the form G NSP × ½Z K 2 ⋉ SOð2Þ, then G NSP could be chosen such that there is no spin rotation at all in G NSP , similar to the choice of G NSP in G NSP × Z K 2 such that G NSP contains no T, as discussed in Appendix C1. Therefore, the options left for G s NSP are C 1 and S 2 ð¼ C i ¼ Z T 2 Þ so that there is no multiple counting of the same types of full SPGs represented by G NSP × ½Z K 2 ⋉ SOð2Þ. The remaining G NSP ‘s to be considered are actually 90 types, including 32 type-I MPGs and 58 type-II MPGs. Overall, there are 90 crystallographic SPGs of the form G NSP × ½Z K 2 ⋉ SOð2Þ shown in Tables S12–S13 of the Supplemental Material [64].

APPENDIX E: METHODS OF FIRST-PRINCIPLES

CALCULATIONS

First-principles calculations of SrMnBi 2 and CaMnBi 2 are conducted within the framework of density-functional theory [103,104] using the projector-augmented-wave 021016-15 PHYS. REV. X 12, 021016 (2022) LIU, LI, HAN, WAN, and LIU (PAW) method [105], implemented in the Vienna ab initio simulation package [106]. Pseudopotentials under the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof formalism [107,108] are used. The ground state is obtained from a self-consistent calculation with an energy cutoff of 500 eV and 5 × 5 × 5 (7 × 5 × 5) Monkhorst-Pack grid for the symmetrized primitive cells of SrMnBi 2 (CaMnBi 2 ). Because of the local magnetic moments contributed from 3d electrons in Mn atoms, the GGA þ U approach within the Dudarev scheme is applied with U ¼ 3 eV on 3d orbitals of Mn. To obtain the topological properties of SrMnBi 2 and CaMnBi 2 , e.g., Dirac points, surface states, Wilson loop, etc., we construct tight-binding Hamiltonians based on maximally localized Wannier functions [109,110] of Sr 4d, Mn 3d, Bi 6p for SrMnBi 2 and Ca 3d, Mn 3d, Bi 6p in CaMnBi 2 . The iterative Green’s function implemented in WannierTools package is used for surface states calculations [111].

In our calculations, the lattice parameters and atomic positions of SrMnBi 2 and CaMnBi 2 are chosen from the experimental values [71] except the z-direction lattice parameter. To simulate the uniaxial pressure effect, we set lattice parameter c of SrMnBi 2 to be 21.13 Å (rather than 23.13 Å) to realize the Z 2 topological phase in the subspaces of the Brillouin zone.

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