Symmetry-based approach to nodal structures: Unification of compatibility relations and point-node classifications
SSymmetry-based approach to nodal structures: Unification of compatibility relations andpoint-node classifications
Seishiro Ono ∗ and Ken Shiozaki † Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
Determination of the symmetry property of superconducting gaps has been a central issue in studies to under-stand the mechanisms of unconventional superconductivity. Although it is often difficult to completely achievethe aforementioned goal, the existence of superconducting nodes, one of the few important experimental sig-natures of unconventional superconductivity, plays a vital role in exploring the possibility of unconventionalsuperconductivity. The interplay between superconducting nodes and topology has been actively investigated,and intensive research in the past decade has revealed various intriguing nodes out of the scope of the pioneeringwork to classify order parameters based on the point groups. However, a systematic and unified description ofsuperconducting nodes for arbitrary symmetry settings is still elusive. In this paper, we develop a systematicframework to comprehensively classify superconducting nodes pinned to any line in momentum space. Whilemost previous studies have been based on the homotopy theory, our theory is on the basis of the symmetry-basedanalysis of band topology, which enables systematic diagnoses of nodes in all magnetic space groups. Further-more, our framework can readily provide a highly effective scheme to detect nodes in a given material by usingdensity functional theory, which elucidates the symmetry property of superconductivity. We substantiate thepower of our method through the time-reversal broken and noncentrosymmetric superconductor CaPtAs. Ourwork establishes a unified theory for understanding superconducting nodes and facilitates determining super-conducting gaps in materials combined with experimental observations.
I. INTRODUCTION
Superconducting nodes, geometry of gapless regions in theBogolibov quasi-particle spectrum, are key ingredients to de-termine the symmetry property of Cooper pairings in uncon-ventional superconductivity. For example, while the tempera-ture dependence of the specific heat and the magnetic penetra-tion depth of fully gapped superconductors is exponential, thatof nodal superconductors is power-law. The pairing symme-tries of realistic materials are sometimes controversial [1–25],and therefore predictions of superconducting nodes by theo-retical studies are helpful to clarify the possible properties ofunconventional superconductivity.Inspired by a series of the discovery of heavy-fermion su-perconductors such as CeCu Si [26] and UPt [27], super-conducting order parameters were classified by irreduciblerepresentations of point groups [28–32]. Since order parame-ters were described by basis functions of the irreducible repre-sentations in the theory of Ref. [32], the intersection betweenFermi surfaces and regions where the basis functions vanishwas understood as superconducting nodes. Although the the-ory in Ref. [32] well described the nodes of particular super-conductors like cuprate superconductors [33], recent inten-sive researches revealed that novel nodes were missed in themethod of Ref. [32], which mostly originated from multi-band(orbital) effects [34–40] or the presence of nonsymmorphicsymmetries [41–45]. For example, although Ref. [46] statedthat line-nodes could not exist in odd-parity superconductors,several studies showed counterexamples in the presence ofnonsymmorphic symmetries [41, 42, 47–50]. In fact, such ∗ [email protected] † [email protected] line-nodes in UPt were reported [41–45]. Another exampleis surface-nodes called Bogolibov Fermi surfaces. When thetime-reversal symmetry (TRS) is broken, the Bogolibov Fermisurfaces can be realized by a pseudo magnetic field arisingfrom interband Cooper pairs [35].Recently, two approaches to nodes overcoming the insuf-ficiency of the previous studies have been proposed. Oneis based on the group-theoretical analysis of representa-tions of the Cooper pair wave functions induced by Blochstates [42, 45, 48, 50, 51]. From the representations, onecan know which order parameters are incompatible with fullygapped phases [42, 50]. The other is the topological ap-proach [47, 49, 52–64]. Based on effective internal symme-tries which keep a point in momentum space unchanged, onecan apply a ten-fold way classification, i.e., the homotopy the-ory, to classifications of gapless phases. Despite the signifi-cant progress reported in these works, existing theories coveronly simple symmetry settings such as generic points in thepresence of the inversion or in the mirror planes. In otherwords, high-symmetry settings such as the rotation and thescrew axes in the glide planes are out of the scope of them,which commonly happen in realistic materials. Therefore,a unified theory to predict superconducting nodes in a givensymmetry setting has long been awaited.Meanwhile, the symmetry-based diagnosis of band topol-ogy has been developed [65–74], which recently has beenextended to superconductors [75–81]. Symmetry representa-tions of wave functions play a pivotal role in the theory. Inparticular, when the system is fully gapped, nontrivial con-straints, called compatibility relations, are imposed on thesymmetry representations [65, 66, 82–85]. Namely, if somecompatibility relations are violated, the system should be gap-less. In fact, Refs. [86–91] have performed comprehensivesurveys of topological (crystalline) insulators and semimetalsand detected the violations of compatibility relations in ma- a r X i v : . [ c ond - m a t . s up r- c on ] F e b terials listed in the crystal structure database, which ensuregaplessness although the crossing type is not completely de-termined. It is natural to ask if the symmetry-based approachcan be incorporated into classifications and diagnosis of nodesin superconductors.On the basis of recent progress in the symmetry-basedapproach to band topology, we present a systematic frame-work to classify nodes pinned to lines in momentum spacefor all symmetry classes of superconductors. Unlike previ-ous studies, our method can be applied to any symmetry set-tings, for example, in the absence of the inversion symmetryand the presence of several nonsymmorphic symmetries. Infact, we apply the framework to all magnetic space groups(MSGs) taking into account all the possible one-dimensionalirreducible representations of the superconducting gaps, thepresence/absence of spin-rotation SU(2) symmetry, and thespinful/spinless nature of the systems. The classification ta-bles we obtained are tabulated in Supplementary Materials.Furthermore, our method can diagnose shapes of the nodessuch as points, lines, or surfaces.Our framework leads to an efficient algorithm to detect anddiagnose nodes in realistic materials, requiring only informa-tion of irreducible representations at high-symmetry points.Our results therefore will help to determine the pairing sym-metries in superconducting materials. As a demonstration, weapply our algorithm to CaPtAs, in which the broken TRS isobserved [92]. As a result, we show that this material is ex-pected to have small Bogolibov Fermi surfaces.This paper is organized as follows. In Sec. II, we summa-rize our strategy and key ideas through a simple example MSG P /m (cid:48) . In Sec. III, we introduce several ingredients used toformulate our theory. We devote Sec. IV to establish the clas-sification of point-nodes on the lines in the presence of pointgroup symmetries. In Sec. V, we integrate the point-node clas-sifications into the symmetry-based analysis to classify nodalstructures pinned to the lines. In Sec. VI, we discuss how toapply our theory to detection of nodes in realistic supercon-ductors. We comment on nodes at generic momenta and therelationship between such nodes and symmetry-indicators inSec. VII. We conclude the paper with outlooks for the futureworks in Sec. VIII. Several details are included in appendicesto avoid digressing from the main subjects. II. OVERVIEW OF STRATEGY
Before moving on to the detailed discussions, we providean overview of the key ideas and our strategy. The purposeof this work is to establish a systematic framework to clas-sify various nodes pinned to any line in momentum space. Toachieve this, we will utilize symmetry-based analysis of bandtopology, especially compatibility relations which are non-trivial constraints on symmetry and topology to be gapped.Although compatibility relations are powerful tools for un-derstanding nodes, they alone cannot provide complete infor-mation about geometry of nodes. Then, the classification ofpoint-nodes on the lines in momentum space can compensatefor the incompleteness of compatibility relations. We illustrate the ideas through spinful superconductors inMSG P /m (cid:48) , which possess TRS T , the particle-hole sym-metry (PHS) C , the two-fold rotation C y along the y -axis sat-isfying the anticommutation relations {C , C y } = 0 , and theinversion I holding the commutation relation [ C , I ] = 0 . Let H k and ψ m k be the Hamiltonian and its eigenvectors, and H k ψ m k = E m k ψ m k , where the Bogolibov quasi-particlespectrum E m k is labeled by the band index m and the mo-mentum k . Since the combined symmetries I C and I T donot change k , ( I C ) ψ m k and ( I T ) ψ m k are also eigenvectorsof H k with the energies − E m k and E m k , respectively.Let us begin with a two-dimensional plane invariant underthe mirror symmetry M y = IC y . In this plane, the eigenvec-tors ψ m k of H k are also those of M y with mirror eigenvalues ξ m k = ± i . Then ( I C ) ψ m k and ( I T ) ψ m k have the mirroreigenvalues ξ m k and − ξ m k , respectively. This implies thatthe combined symmetry I C does not change the mirror sec-tor but I T changes, which results in class D as the effectiveAltland-Zirnbauer (AZ) symmetry [93] and furnishes the Pfaf-fian invariants p ± i k [94–96] for each mirror sector (see Fig. 1).Since topological invariants do not change when the systemin the same topological phase during the continuous defor-mation, p ± i k should be the same everywhere in the plane ifthe system is fully gapped. We will refer to this kind of con-straints as “compatibility relations.” Suppose that we have agapless point at a point Λ = ( λ, , for (0 < λ < π ) , whichindicates that the Pfaffian invariants at points on the k x -axischange at Λ . In other words, there are two regions on the k x -axis where p ± i k = 1 and p ± i k = 0 . Because of the com-patibility relations in the plane, these regions are extended tothe plane, whose boundary line is the domain wall of changesof Pfaffian invariants. Next, we discuss whether the line isextended to the outside of the plane, i.e., the domain wallbecomes a surface. Generic momenta are invariant only un-der I T and I C , and the effective internal symmetry classes atgeneric points are class DIII. Since there are no topological in-variants at generic points, compatibility relations between anypoint in the plane and generic points do not exist. Therefore,the domain wall in the plane is the line-node.Finally, we consider the rotation symmetric line, where ψ m k are eigenvectors of C y with the rotation eigenvalues η m k = ± i . As is the case for the mirror plane, the effec-tive AZ classes for each rotation sector are class D, and thePfaffian invariants are defined. We again suppose that we finda gapless point at the line. Since, unlike the case for the mir-ror plane, there are no compatibility relations between the lineand its neighborhood, one might think that this gapless pointis a point-node. However, this is untrue. To show that the gap-less point is not a point-node, we then classify point-nodes inthe line. The classification of point-nodes at a line is equiv-alent to that of two-dimensional gapped Hamiltonians in theplane perpendicular to the line. As a result, we find that theredoes not exist any point-node in the line, which will be ex-plicitly shown in Sec. IV C 1. Therefore, we conclude thatthis gapless point is a part of a line-node with a different topo-logical origin from compatibility relations. k z k x k y C y + E E ψ m k ξ m k = − iξ m k = + i I T I C E ψ m k I C I T + ( η m k = + i )( η m k = − i ) Class DClass DIII
FIG. 1. Illustration of the action of symmetries discussed in Sec. II.White circles represent the gapless points. There are two irreduciblerepresentations at the mirror plane ( k z - k x plane) and the rotation axis( k y -axis), and they are invariant under I C but exchanged by I T . Asa result, Z topological invariants (the Pfaffian invariants) are definedat every point in the mirror plane and the rotation axis. On the otherhand, any zero-dimensional topological invariant cannot be definedat generic points (see Table I), and therefore there do not exist anyconstraints to keep the gap opening between the mirror plane (therotation axis) and generic points. III. FORMALISM
In Sec. II, we have shown the procedures to diagnose nodesat symmetric lines in a particular symmetry setting. In thissection, we explain several ingredients to systematically for-mulate the above processes for arbitrary symmetry classes,which will be discussed in Secs. IV and V.
A. BdG Hamiltonian and Symmetry representations
In this work, we always focus on superconductors whichcan be described by the Bogoliubov–de Gennes (BdG) Hamil-tonian H k = (cid:18) h k ∆ k ∆ † k − h ∗− k (cid:19) , (1)where h k and ∆ k denote the normal-phase Hamiltonian andthe superconducting gap function, respectively [97].Suppose that the normal phase is invariant under a magneticspace group (MSG) M = G + A , where G is a space groupand A is an anti-unitary part of M . A MSG M always has asubgroup T composed of lattice translations. An element g ∈M maps a point r in the real space to g r = p g r + t g , where p g is an element of O (3) , and t g is not always a lattice vector.Because of the existence of PHS C in the BdG Hamiltonian,the full symmetry group G is divided by the following fourparts G = M + MC = G + A + P + J (2) where P = GC and J = AC are sets of particle-hole like andchiral like symmetries.We recall symmetry representations of G in momentumspace. We introduce two maps c, φ : G → Z = {− , } .Here, φ g = +1 ( − means g is unitary (anti-unitary), and c g = +1 ( − represents g commutes (anticommutes) withthe Hamiltonian H k . Accordingly, an element g ∈ M trans-forms a point k in momentum space into g k = φ g p g k . Inaddition, the representation ρ k ( g ) is expressed by ρ k ( g ) = (cid:40) U k ( g ) for φ g = +1 ,U k ( g ) K for φ g = − , (3)and ρ k ( g ) satisfies ρ k ( g ) H k = (cid:40) H g k ρ k ( g ) for c g = +1 , − H g k ρ k ( g ) for c g = − , (4)where U k ( g ) and K are a unitary matrix and the conjugationoperator, respectively. Note that U k ( g ) is a projective repre-sentation, i.e., the following relation holds ρ g (cid:48) k ( g ) ρ k ( g (cid:48) ) = z g,g (cid:48) ρ k ( gg (cid:48) ) , (5)where z g,g (cid:48) ∈ U (1) is a projective factor. For spinless sys-tems, we can always choose z g,g (cid:48) = +1 for g, g (cid:48) ∈ G or A .Let us consider a point k in momentum space. For thispoint, we introduce a subgroup G k of G by { h ∈ G| h k = k + ∃ G } , where G is a reciprocal lattice vector, and this subgroupis called “little group.” Then, we can block-diagonalize H k and U k ( h ) as U k ( h ) = diag [ U α k ( h ) ⊗ m , · · · , U α n k ( h ) ⊗ m n ] , (6) H k = diag [ d α ⊗ H α k , · · · , d αn ⊗ H α n k ] , (7)where U α k ( h ) is an irreducible representations of G k . Here, d α and m α are dimensions of U α k ( h ) and H α k , respectively.One often considers the finite group G k /T , where G k isdefined in the same way as G k . In the literature [98], G k /T is referred to as “little co-group.” Importantly, G k /T is iso-morphic to a magnetic point group with PHS. We can alwaysrelate representations of G k to those of G k /T , and we definethe representation σ k ( g ) of G k /T by σ k ( g ) = (cid:40) U k ( g ) e − i k · t g for φ g = +1 ,U k ( g ) e − i k · t g K for φ g = − , (8)where t g is a fractional translation or zeros. Correspondingly,projective factors also change as σ k ( g ) σ k ( h ) = z k g,h σ k ( gh ) , (9)where z k g,h = z g,h e − i k ( p g t h − φ g t h ) . Using these projectivefactors z k g,h , we can obtain irreducible representations u α k of G k /T , which is simply related to irreducible representations U α k of G k by U α k ( g ) = u α k ( g ) e − i k · t g . (10) B. Cell decomposition
Here, we explain the decomposition of the first Brillouinzone (BZ), which is called cell decomposition [99]. Let T d bethe d -dimensional BZ. We introduce a series of subspace of T d such that X ⊂ X ⊂ · · · ⊂ X d = T d , (11)where X p is referred to as p -skelton. Each X p is invariantunder G , i.e., g k ∈ X p for ∀ g ∈ G if and only if k ∈ X p . Weremark that the choice of X p is not unique.In the following, we will explain a way to get X p that en-able us to perform procedures in Sec.V systematically. First,we divide a unit of BZ into the set of p -dimensional open disks { D pi } i for p = 0 , , · · · , d , which are called p -cells. Next, weextend such division to the entire BZ. To achieve this, we actsymmetry operations in G/T on each p -cell, and then we candefine the entire set of p -cells by C p ≡ (cid:91) i (cid:91) g ∈ G/T D pg ( i ) , (12)where D pg ( i ) = gD pi . Note that, in this construction, some p -cells are equivalent or symmetry-related to others. However,we do not identify such p -cells with others in the proceduresof cell decomposition, and we will take into account theseidentifications in the construction of E -pages in Sec. III D.Each p -cell satisfies the following conditions:(i) The intersection of any two p -cells in C p is an emptyset, i.e., D pi ∩ D pj = ∅ ( i (cid:54) = j ) .(ii) Any point in a p -cell D pi is invariant under symmetriesor transformed to points in different p -cells by symme-tries, namely, g k = k + ∃ G or g k ∈ D pg ( i ) if k ∈ D pi .(iii) The boundary ∂D pi consists of ( p − -cells for p ≥ .(iv) Each p -cell is oriented by a symmetric manner.(v) Any two of the boundary p -cells of the ( p + 1) -cell arenot equivalent and symmetry-related to each other.For our purpose to systematically diagnose nodes pinned tolines in BZ, the condition (v) is crucial. In Appendix A, weprovide units of 3D BZ for each type of lattices. Finally, the p -skelton X p is determined by X = C , X p = C p ∪ X p − ( p ≥ . (13)To clarify notions introduced in this section, let us discussthe point group p mm in the two-dimension. We first de-compose an asymmetric unit into three -cells (orange cir-cles), three -cells (solid red lines), and a -cell (pink plane)in Fig. 2, and then we act symmetry opearations on this asym-metric unit. As a result, we get a cell decomposition for p mm as C = { Γ , X , M, X , M , X , M , X , M } , (14) C = { a, b, c, a , b , c , a , b , c , a , b , c , c , c , c , c } , (15) C = { α, α , α , α , α , α , α , α } . (16) TABLE I. The classification of emergent AZ (EAZ) symme-try classes. Here, p α k and N α k denote the Pfaffian invariantand the number of irreducible representation.The triple W [ α ] =( W α k ( T ) , W α k ( C ) , W α k (Γ)) represents the result of Wigner criteria.EAZ W k [ α ] classification indexA (0 , , Z N α k AIII (0 , ,
1) 0
NoneAI (1 , , Z N α k BDI (1 , , Z p α k D (0 , , Z p α k DIII ( − , ,
1) 0
NoneAII ( − , , Z (cid:101) N α k = N α k / CII ( − , − ,
1) 0
NoneC (0 , − ,
0) 0
NoneCI (1 , − ,
1) 0
None
Here, although various p -cells are equivalent or symmetry-related to other p -cells as explained above, we assign differentlabels to them. For example, X = ( − π, is equivalent toX = ( π, and X = (0 , π ) is symmetry-related to X. C. Emergent Altland-Zirnbauer classes
For each p -cell, we determine the effective internal symme-try classes, which is referred to as EAZ classes. Symmetriesin A , P , and J sometimes keep a sector H α k in Eq. (7) un-changed, and other times transform it to another sector. In thefollowing, we discuss how to know the effect of symmetriesin A , P , and J .In our construction of the cell decomposition, the littlegroups G k at any point k in a p -cell D p are in common,and therefore the common little group is denoted by G D p .In the same way as G D p , we define a subset V D p of V by V D p = { v ∈ V| v k = k + ∃ G for ∀ k ∈ D p } , where V = A , P , J . Then, we identify actions of time-reversal like,particle-hole like, and chiral like symmetries on each H α k bythe Wigner criteria [98, 99] W αD p ( P ) = 1 |P k /T | (cid:88) c ∈P k /T z k c,c χ α k ( c ) ∈ { , ± } , (17) W αD p ( A ) = 1 |A k /T | (cid:88) a ∈T k /T z k a,a χ α k ( a ) ∈ { , ± } , (18) W αD p ( J ) = 1 |G k /T | (cid:88) g ∈G k /T z k γ,γ − gγ z k g,γ [ χ α k ( γ − gγ )] ∗ χ α k ( g ) (19) ∈ { , } , where χ α k ( g ) = tr[ u α k ( g )] for k ∈ D p and γ is a chiral likesymmetry. Note that, in fact, it is enough for our purpose toconsider a point k in D p . When W αD p ( V ) = 0 , additionalsymmetries in V D p transform H α k into another sector H β k . Onthe other hand, when W αD p ( V ) = ± , H α k is invariant underthe additional symmetries. Then, the EAZ symmetry class for C M y Γ X M α a b c a b c X M α a α b c M X a α b c M X α α α α c c c c FIG. 2. Cell decomposition for p mm . We first find a unit of BZ illustrated in the left panel. The red and black arrows signify orientationsof 1-cells and 2-cells, respectively. Then, we rotate each p-cell in the unit by the four-fold rotation symmetry. Finally, mapping them by themirror symmetry, we arrive at the cell decomposition shown in the right panel. H α k is determined by W D p [ α ] ≡ ( W αD p ( A ) , W αD p ( P ) , W αD p ( J )) . (20)Depending on the EAZ symmetry classes, we assign the fol-lowing zero-dimensional topological invariants to each sector H α k (see Table I): p α k ≡ iπ log Pf[ U ( H α k )]Pf[ U ( H α k ) vac ] mod 2 , (21) N α k ≡ n α k − ( n α k ) vac , (22)where H vac k is a vacuum Hamiltonian in the same symmetrysetting. Furthermore, we introduce ( H α k ) vac as H α k in Eq. (6)for H vac k , and U is an effective particle-hole like symmetrysuch that ( U U ∗ ) = +1 . We also define n α k and ( n α k ) vac by thenumber of occupied states in H α k and ( H α k ) vac . In practice, wecan always choose H vac k as H k in the limit of infinite chemicalpotential. D. E -pages As seen in the preceding discussions, the Wigner criteria inEqs. (17)-(19) tell us effective internal symmetry classes foreach irreducible representation. Then, let us define abeliangroups E p, in the following, which can be interpreted as theclassification of { H α k } α at points k inside p -cells.The total set C p of p -cells consists of N p subsets (so-called“star” in the literature [98]) defined by S D pi = { D pg ( i ) = gD pi | g ∈ G } , where N p the number of subsets and D pi is arepresentative p -cell of the subset S D pi . The representativesform a set of independent p -cells F p ≡ { D pi } N p i =1 . (23)Then, we can define abelian groups E p, by E p, ≡ (cid:77) i | Dpi ∈ Fp (cid:0) Z ⊕ α ⊕ Z ⊕ β (cid:1) , (24) where we perform the summation about labelsof irreducible representations α and β only when W D pi [ α ] ∈ { (0 , , , (1 , , } and W D pi [ β ] ∈{ (0 , , , (1 , , , ( − , , } (see Eq. (20)).As discussed in Ref. 99, E p, has several different interpre-tations. In particular, E , and E , represent the sets of zero-dimensional gapped Hamiltonians on 0-cells and stable gap-less nodal structure on 1-cells, respectively. Based on theseinterpretations, we can characterize any system by a list ofband labels n ( p ) = ( p α D p , p α D p , · · · N β D p , · · · , p α (cid:48) D p , · · · , N β (cid:48) D p , · · · ) , (25)where p αD pi and N βD pi are Z -valued and Z -valued band labels,respectively. Correspondingly, the abelian group E p, is for-mulated by E p, = (cid:77) i | Dpi ∈ Fp (cid:77) α Z [ b ( p ) D pi ,α ] ⊕ (cid:77) β Z [ b ( p ) D pi ,β ] , (26)where { b ( p ) D pi ,α } denotes the set of basis ‘vectors’ [100] whichcan expand an arbitrary n ( p ) , and the summation about α and β are the same in Eq. (24).In the following, we explain the construction of the basisvector b ( p ) D pi ,α . Each b ( p ) D pi ,α is generated by an irreducible repre-sentation U αD pi at a p -cell D pi in C p . As explained in Sec. III B,we include the equivalence or symmetry relations among p -cells in the basis. We consider a p -cell D pi , and suppose thatwe have a nontrivial band label p αD pi = 1 or N αD pi = 1 for an ir-reducible representation U αD pi . Then, band labels at equivalentor symmetry-related points are determined by those at D pi asfollows. We first derive the relation between irreducible rep-resentations U α (cid:48) D pg ( i ) and U αD pi U α (cid:48) D pg ( i ) ( h (cid:48) ) = (cid:40) z h (cid:48) ,g z g,g − h (cid:48) g U αD pi ( g − h (cid:48) g ) for φ g = +1 z h (cid:48) ,g z g,g − h (cid:48) g [ U αD pi ( g − h (cid:48) g )] ∗ for φ g = − , (27)where g ∈ G and h (cid:48) ∈ G D pg ( i ) . Since the spectrum of H α (cid:48) D pg ( i ) is the same as that of H αD pi , band labels at D pg ( i ) then straight-forwardly follow p α (cid:48) D pg ( i ) = p αD pi , (28) N α (cid:48) D pg ( i ) = (cid:40) N αD pi for c g = +1 − N αD pi for c g = − . (29)As a result, we can obtain the set of band labels such thatonly p αD pi ( N αD pi ) and associated band labels are ( or − ).Indeed, this is exactly what we call b ( p ) D pi ,α .To make our understanding clearer, let us discuss a sim-ple example: a one-dimensional even-parity superconductorin class D. We first decompose BZ into three 0-cells and two1-cells illustrated in Fig. 3: C ≡ { Γ , X , X (cid:48) } , (30) C ≡ { a, a (cid:48) } , (31)Since X (cid:48) is equivalent to X and a (cid:48) is symmetry-related to a , F = { Γ , X } and F = { a } . We first obtain the classifica-tions of each irreducible representation at -cells and -cells.Fig. 3 illustrates the action of the particle-hole like symme-tries to each sector of Hamiltonians at each cell, and we findthat the EAZ classes for each inversion eigenvalue at Γ and Xare class D and the EAZ class at a is also class D. Therefore, E , = ( Z ) and E , = Z .Next, we consider the different interpretation of E -pagesand find them in the form of Eq. (26). We define the Pfaf-fian invariants p α = ± D ∈C [96] for each inversion eigenvaluefor the 0-cells, and the set of band labels for -cells is ( p +Γ , p − Γ , p + X , p − X , p + X (cid:48) , p − X (cid:48) ) . On the other hand, since the 1-cells are invariant under the combination of PHS C and theinversion symmetry I with ( C I ) = +1 , the set of band labelsfor the 1-cells is ( p a , p a (cid:48) ) . We then construct the basis vec-tors of E , and E , . From Eq. (28), we find p ± X (cid:48) = p ± X and p a (cid:48) = p a . Therefore, we obtain E , = Z [ b (0)Γ , + ] ⊕ Z [ b (0)Γ , − ] ⊕ Z [ b (0) X , + ] ⊕ Z [ b (0) X , − ] , (32) E , = Z [ b (1) a ] , (33)where b (0)Γ , + = (1 , , , , , , (34) b (0)Γ , − = (0 , , , , , , (35) b (0) X , + = (0 , , , , , , (36) b (0) X , − = (0 , , , , , , (37) b (1) a = (1 , . (38) E. Compatibility relations
In this subsection, we discuss constraints on band la-bels, which are called “compatibility relations” developed in Refs. [65, 66, 79, 80, 85]. Compatibility relations will be uti-lized in Sec. V.Let D ( p +1) be a ( p + 1) -cell, and let D p be a bound-ary p -cell of D ( p +1) . Since G D ( p +1) ⊆ G D p in our cell de-composition, an irreducible representation U αD p of G D p canbe decomposed into irreducible representations U βD ( p +1) of G D ( p +1) such that U αD p ( g ) = (cid:76) β c αβD p ,D p +1 U βD ( p +1) ( g ) , where c αβD p ,D p +1 = (cid:80) g ∈G D ( p +1) /T (cid:16) χ βD ( p +1) ( g ) (cid:17) ∗ χ αD p ( g ) is a non-negative integer. Once we have the number of an irreduciblerepresentation U αD p contained in U D p ( g ) , denoted by n αD p , wecan know how many an irreducible representation U βD ( p +1) isincluded in U D ( p +1) ( g ) by n βD ( p +1) = (cid:80) α n αD p c αβD p ,D p +1 .Correspondingly, when the system is fully gapped, zero-dimensional topological invariants in Eqs. (21) and (22) at k (cid:48) ∈ D ( p +1) are related to those at k ∈ D p . The relationsbetween them are classified into the following four cases: p β k (cid:48) = (cid:88) α c αβD p ,D p +1 p α k + (cid:88) γ c γβD p ,D p +1 N γ k , mod 2 (39) p β k (cid:48) = 0 mod 2 , (40) N β k (cid:48) = (cid:88) α c αβD p ,D p +1 N α k , (41) N β k (cid:48) = 0 , (42)which we refer to as compatibility relations.Using compatibility relations, we construct a map from E p, to E p +1 , . Band labels at all boundaries D pi of D p +1 contribute to those at D p +1 . Taking into account the orienta-tions of cells, we have the following relations: p βD p +1 = (cid:88) i δ D pi ,D p +1 (cid:34)(cid:88) α c αβD pi ,D p +1 p α k + (cid:88) γ c γβD pi ,D p +1 N γ k (cid:35) , (43) N βD p +1 = (cid:88) i (cid:88) α δ D pi ,D p +1 c αβD pi ,D p +1 N αD pi , (44)where δ D pi ,D p +1 = 0 when D p +1 is not adjacent to D pi and δ D pi ,D p +1 = 1 ( − if D p +1 is adjacent to D pi and the orienta-tion of D pi agrees (disagrees) with that the orientation inducedby ( p + 1) -cell D p +1 . Note that, while all coefficients arenon-negative in Eqs. (39)-(42), some coefficients in Eqs. (43)-(44) can be negative. By computing the above relations forall ( p + 1) -cells, one can construct a matrix in terms of bandlabels at p -cells. When we rewrite this matrix in the basis of E p, , we obtain a map from E p, to E p +1 , d p, : E p, → E p +1 , , (45)which is called “the first differential [99].” One can seethat d p, always satisfies d p +1 , ◦ d p, = 0 , that is, d p +1 , (cid:16) d p, ( n ( p ) ) (cid:17) = .It is worth noting that if d , ( n (0) ) = holds, the set ofband labels at 0-cells satisfies all compatibility relations. On C I I = +1 I = − C E C E E C I C , I C , I Class D Class D Class D I = +1 I = − C E C E Class D Class D C I E C I Class D I = +1 I = − C E C E Class D Class D C , I a (cid:31) a Γ( k = 0)X (cid:31) ( k = − π ) X( k = π ) FIG. 3. Illustration of the cell decomposition and EAZ classes for 1D even-parity superconductors. The red arrows signify orientations of1-cells. the other hand, when d , ( n (0) ) (cid:54) = , some compatibility re-lations are violated, which implies that gapless points exist onthe 1-cells.Again, we discuss the 1D even-parity superconductors dis-cussed in Sec. III D. From Eq. (39), the compatibility relationsare p a = p +Γ + p − Γ = p + X + p − X , (46) p a (cid:48) = p +Γ + p − Γ = p + X (cid:48) + p − X (cid:48) . (47)According to the orientations in Fig. 3, we find δ Γ ,a = δ Γ ,a (cid:48) = − δ X ,a = − δ X (cid:48) ,a (cid:48) = +1 , and then we have (cid:18) p a p a (cid:48) (cid:19) = (cid:18) − − − − (cid:19) p +Γ p − Γ p + X p − X p + X (cid:48) p − X (cid:48) . (48)Correspondingly, d , is expressed by a matrix M d , M d , = b (0)Γ , + b (0)Γ , − b (0) X , + b (0) X , − b (1) a − − . (49) IV. CLASSIFICATION OF POINT NODES ON 1-CELL
In this section, we discuss the method to classify locallystable point-nodes on -cells. Gapless points can be inter-preted as topological phase transition points by consideringmomentum along the 1-cell as a parameter. Therefore, theclassification of the gapless points on 1-cells of 3D systems isequivalent to that of gapped 2D systems perpendicular to the -cells.Ref. 101 has shown that one can redefine any point groupsymmetries as onsite symmetries with classifications of free-fermionic topological phases unchanged, which we will referto as Cornfeld-Chapman’s method. Refs. 101 and 102 alsohave classified 3D topological insulators and superconductorsin the presence of non-magnetic and magnetic point groupsymmetries by using the method. In the following, we will apply Cornfeld-Chapman’smethod [101] to classifications of locally stable point-nodeson 1-cells, and then we will obtain generators of the point-nodes on -cells as elements of E , . This will be integratedinto compatibility relations discussed in Sec. V. A. Cornfeld-Chapman’s method for 2D systems
Suppose that there exists a gapless point at a -cell (denotedby D ). To classify 2D gapped systems perpendicular to the -cells, let us discuss the following 2D massive Dirac Hamil-tonian near D : H ( k ,k ) = k γ + k γ + δk γ , (50)where k and k are momenta in the directions perpendicularto the -cell, and δk is a displacement from the gapless pointin the direction of D . Gamma matrices γ , γ , and γ anti-commute with each others. To apply the Cornfeld-Chapman’smethod to the 2D massive Dirac Hamiltonian in Eq. (50), weconsider the little co-group in the following discussion, andthen Hamiltonian is symmetric under G D /T , i.e., H ( k ,k ) satisfies σ k ( g ) H k = (cid:40) H r g k σ k ( g ) for c g = +1 , − H r g k σ k ( g ) for c g = − , (51)where r g is an element of O (2) . Generally, r g can be writtenby r g = (cid:32) cos θ g − sin θ g sin θ g cos θ g (cid:33) for det r g = +1 , (cid:32) − cos θ g − sin θ g − sin θ g cos θ g (cid:33) for det r g = − . (52)For simplicity, we thereafter use s g = det r g . In the follow-ing, we will make all elements of G D /T onsite.First, we introduce onsite symmetries and define their rep-resentations by (cid:101) σ ( g ) ≡ γ − sg e θg γ γ σ k ( g ) for ∀ g ∈ G D /T. (53) TABLE II. Classification of EAZ symmetry classes. The subscripts T , C , and Γ signify that irreducible representations are related by theonsite anti-unitary and the chiral symmetries.EAZ (cid:102) W [ (cid:101) α ] classifying space π A, A T , A C , A Γ , A T , C (0 , , C Z AIII, AIII T (0 , , C AI, AI C (1 , , R Z BDI (1 , , R D, D T (0 , , R Z DIII ( − , , R AII, AII C ( − , , R CII ( − , − , R C, C T (0 , − , R Z CI (1 , − , R Z By performing explicit calculations, one can verify (cid:101) σ ( g ) H ( k ,k ) = s g c g H ( k ,k ) (cid:101) σ ( g ) , (54) (cid:101) σ ( g ) (cid:101) σ ( h ) = ( s g c g ) − sh z (cid:48) g,h z k g,h (cid:101) σ ( gh ) , (55)where z (cid:48) g,h is determined by γ − sh e θh γ γ γ − sg e θg γ γ = z (cid:48) g,h γ − sgh e θgh γ γ . (56)Note that, when σ k ( g ) with s g = − commutes (anticom-mutes) with H ( k ,k ) , (cid:101) σ ( g ) anticommutes (commutes) with H ( k ,k ) . In other words, unitary (chiral like) symmetriesfor s g = − become onsite chiral (unitary) symmetries.The same thing happens to anti-unitary symmetries. As aresult, we have another decomposition of symmetry group G = (cid:101) G + (cid:101) A + (cid:101) P + (cid:101) J , where each subset is defined by (cid:101) G = { g ∈ G | s g c g = 1 , φ g = 1 } , (cid:101) A = { g ∈ G | s g c g =1 , φ g = − } , (cid:101) P = { g ∈ G | s g c g = − , φ g = − } , and (cid:101) J = { g ∈ G | s g c g = − , φ g = 1 } .Similar to Eqs. (6) and (7), we can block-diagonalize (cid:101) σ ( g ) and H ( k ,k ) . Our next step is to determine effective inter-nal symmetry classes for each sector of block-diagonalized H ( k ,k ) . To achieve this, we again use Wigner criteria replac-ing z k g,g in Eqs. (17)-(19) with ( s g c g ) − sg z (cid:48) g,g z k g,g , and the re-sult is denoted by (cid:102) W k [ (cid:101) α ] ≡ ( (cid:102) W α k ( (cid:101) A ) , (cid:102) W α k ( (cid:101) P ) , (cid:102) W α k ( (cid:101) J )) .Finally, we classify the mass term γ of Eq. (50) in the pres-ence of the onsite symmetry group. Mass terms for each sectorare classified by π ( C s ) or π ( R s ) [52], where C s and R s areclassifying spaces of EAZ classes (see Table II). B. Character decomposition formulas
As explained in the previous subsection, we can classifygapless points on 1-cells. The next step is to obtain the gener-ators of these gapless points as elements of E , . This can beachieved by the orthogonality of irreducible representations.In this subsection, we will derive formulas to map a generatorof a gapless point to an element of E , . The formulas aresummarized in Table III. Let us suppose that we have a generator of a gapless pointon a 1-cell and onsite symmetries in Eq. (53). Then, we canconstruct symmetries of G D /T by σ k ( g ) = e − θg γ γ γ − sg (cid:101) σ ( g ) . (57)What we have to know is the number of band labels are con-tained in the above representation σ k ( g ) in Eq. (57). Using theorthogonality or irreducible representations, we obtain N αD and p αD by N βD = 1 |G k /T | (cid:88) g ∈G k /T χ β k ( g )tr[ γ e − θg γ γ γ − sg (cid:101) σ ( g )] , (58) p βD = 1 / |G k /T | (cid:88) g ∈G k /T χ β k ( g )tr[ e − θg γ γ γ − sg (cid:101) σ ( g )] mod 2 , (59)where occupied and unoccupied bands contribute to band la-bels with different signs by γ in Eq. (58). After performingthe same procedures for all irreducible representations of G k ,we get the generator as an element of E , .For each of EAZ classes, in fact, we can derive the formulasby fixing the form of generating Hamiltonians and representa-tions. Here, we show the formulas for class A C as an example.The generating Hamiltonian and representations can be repre-sented by H ( k ,k ) = k τ + k τ + δk τ , (60) (cid:101) σ ( C ) = (cid:40) iτ σ K for [ (cid:101) σ ( C )] = − ,τ σ K for [ (cid:101) σ ( C )] = +1 , (61) (cid:101) σ ( g ) = τ (cid:18)(cid:101) u (cid:101) α ( g ) (cid:101) u (cid:101) C (cid:101) α ( g ) (cid:19) for ∀ (cid:101) g ∈ (cid:101) G , (62)where (cid:101) u (cid:101) α and (cid:101) u (cid:101) C (cid:101) α denote an irreducible representation of on-site unitary symmetry group (cid:101) G and its particle-hole-related ir-reducible representation, respectively. In addition, (cid:101) C is thegenerator of (cid:101) P , and σ µ and τ µ ( µ = 0 , , , are Pauli matri-ces representing different degrees of freedom. By substitutingEqs. (60) and (62) into Eqs. (58) and (59), we get N βD = − i | G D | (cid:88) g ∈G D /T δ s g , sin θ g χ βD ( g )] ∗ × (cid:16)(cid:101) χ (cid:101) α ( g ) + (cid:101) χ (cid:101) C (cid:101) α ( g ) (cid:17) , (63) p βD = 1 | G D | (cid:88) g ∈G D /T δ s g , cos θ g χ βD ( g )] ∗ × (cid:16)(cid:101) χ (cid:101) α ( g ) + (cid:101) χ (cid:101) C (cid:101) α ( g ) (cid:17) mod 2 , (64)where (cid:101) χ (cid:101) α ( g ) = tr[ (cid:101) u (cid:101) α ( g )] . C. Example
It is instructive to discuss concrete symmetry settings. Herewe consider four examples in the presence of PHS C : MSGs TABLE III. Formulas to obtain generators of gapless points as elements of E p, . Here, χ α and (cid:101) χ (cid:101) α are characters of the little co-group G D /T and the onsite symmetry group (cid:101) G , respectively. The first column represent EAZ classes of irreducible representations ˜ u (cid:101) α . In addition, (cid:101) T (cid:101) α , (cid:101) C (cid:101) α , and (cid:101) Γ (cid:101) α are labels of the time-reversal, the particle-hole, and the chiral symmetry related irreducible representations. Derivations of theseformulas are included in Appendix B.EAZ Formula for the map to Z Formula for the map to Z A − i | G D | (cid:80) g ∈ G D δ s g , sin θ g [ χ βD ( g )] ∗ (cid:101) χ (cid:101) α ( g ) | G D | (cid:80) g ∈ G D δ s g , cos θ g [ χ βD ( g )] ∗ (cid:101) χ (cid:101) α ( g ) A T − i | G D | (cid:80) g ∈ G D δ s g , sin θ g [ χ βD ( g )] ∗ (cid:16)(cid:101) χ (cid:101) α ( g ) − (cid:101) χ (cid:101) T (cid:101) α ( g ) (cid:17) | G D | (cid:80) g ∈ G D δ s g , cos θ g [ χ βD ( g )] ∗ (cid:16)(cid:101) χ (cid:101) α ( g ) + (cid:101) χ (cid:101) T (cid:101) α ( g ) (cid:17) A C − i | G D | (cid:80) g ∈ G D δ s g , sin θ g [ χ βD ( g )] ∗ (cid:16)(cid:101) χ (cid:101) α ( g ) + (cid:101) χ (cid:101) C (cid:101) α ( g ) (cid:17) | G D | (cid:80) g ∈ G D δ s g , cos θ g [ χ βD ( g )] ∗ (cid:16)(cid:101) χ (cid:101) α ( g ) + (cid:101) χ (cid:101) C (cid:101) α ( g ) (cid:17) A Γ − i | G D | (cid:80) g ∈ G D δ s g , sin θ g [ χ βD ( g )] ∗ (cid:16)(cid:101) χ (cid:101) α ( g ) − (cid:101) χ (cid:101) Γ (cid:101) α ( g ) (cid:17) | G D | (cid:80) g ∈ G D δ s g , cos θ g [ χ βD ( g )] ∗ (cid:16)(cid:101) χ (cid:101) α ( g ) + (cid:101) χ (cid:101) Γ (cid:101) α ( g ) (cid:17) A T , C − i | G D | (cid:80) g ∈ G D δ s g , sin θ g [ χ βD ( g )] ∗ | G D | (cid:80) g ∈ G D δ s g , cos θ g [ χ βD ( g )] ∗ × (cid:16)(cid:101) χ (cid:101) α ( g ) − (cid:101) χ (cid:101) T (cid:101) α ( g ) + (cid:101) χ (cid:101) C (cid:101) α ( g ) − (cid:101) χ (cid:101) Γ (cid:101) α ( g ) (cid:17) × (cid:16)(cid:101) χ (cid:101) α ( g ) + (cid:101) χ (cid:101) T (cid:101) α ( g ) + (cid:101) χ (cid:101) C (cid:101) α ( g ) + (cid:101) χ (cid:101) Γ (cid:101) α ( g ) (cid:17) C − i | G D | (cid:80) g ∈ G D δ s g , sin θ g [ χ βD ( g )] ∗ (cid:101) χ (cid:101) α ( g ) | G D | (cid:80) g ∈ G D δ s g , cos θ g [ χ βD ( g )] ∗ (cid:101) χ (cid:101) α ( g ) C T − i | G D | (cid:80) g ∈ G D δ s g , sin θ g [ χ βD ( g )] ∗ (cid:16)(cid:101) χ (cid:101) α ( g ) − (cid:101) χ (cid:101) T (cid:101) α ( g ) (cid:17) | G D | (cid:80) g ∈ G D δ s g , cos θ g [ χ βD ( g )] ∗ (cid:16)(cid:101) χ (cid:101) α ( g ) + (cid:101) χ (cid:101) T (cid:101) α ( g ) (cid:17) D − i | G D | (cid:80) g ∈ G D δ s g , sin θ g [ χ βD ( g )] ∗ (cid:101) χ (cid:101) α ( g ) | G D | (cid:80) g ∈ G D δ s g , cos θ g [ χ βD ( g )] ∗ (cid:101) χ (cid:101) α ( g ) D T i | G D | (cid:80) g ∈ G D δ s g , sin θ g [ χ βD ( g )] ∗ (cid:16)(cid:101) χ (cid:101) α ( g ) − (cid:101) χ (cid:101) T (cid:101) α ( g ) (cid:17) | G D | (cid:80) g ∈ G D δ s g , cos θ g [ χ βD ( g )] ∗ (cid:16)(cid:101) χ (cid:101) α ( g ) + (cid:101) χ (cid:101) T (cid:101) α ( g ) (cid:17) AI − | G D | (cid:80) g ∈ G D δ s g , cos θ g [ χ βD ( g )] ∗ (cid:101) χ (cid:101) α ( g ) AI C − | G D | (cid:80) g ∈ G D δ s g , cos θ g [ χ βD ( g )] ∗ (cid:16)(cid:101) χ (cid:101) α ( g ) + (cid:101) χ (cid:101) C (cid:101) α ( g ) (cid:17) CI − | G D | (cid:80) g ∈ G D δ s g , cos θ g [ χ βD ( g )] ∗ (cid:101) χ (cid:101) α ( g ) P /m (cid:48) with B g pairing, P (cid:48) with B pairing, P with E pairing, and P mc (cid:48) with A pairing. We classify locallystable point-nodes on 1-cells which are part of symmetric linesor planes, and we get the generators as elements of E , . Theresults in this subsection will be used in Sec. V B, where thephysical consequences will be also discussed. P /m (cid:48) with B g pairing We fist discuss spinful MSG P /m (cid:48) , and recall that thisMSG has the two-fold rotation C y along the y -axis, the inver-sion I , and the TRS T . For B g pairing, { σ ( C ) , σ ( C y ) } = 0 and [ σ ( C ) , σ ( I )] = 0 hold. Let us consider a two-fold rota-tion symmetric line as the 1-cell D [see Fig. 4 (a)]. The littleco-group is given by the following subsets: G D /T = { e, C y } , (65) A D /T = { I T , ( IC y ) T } , (66) P D /T = { I C , ( IC y ) C} , (67) J D /T = { Γ ≡ T C , C y Γ } , (68)where e denotes the identity element. To perform the proce-dures in Sec. IV A, we define generators of the onsite symme-try group by (cid:101) σ ( C y ) ≡ γ γ σ ( C y ) , (69) (cid:101) σ ( I T ) ≡ σ ( I T ) , (70) (cid:101) σ ( I C ) ≡ σ ( I C ) , (71)One can verify that s g = +1 for all elements in G D /T , andthen the onsite unitary symmetry group is (cid:101) G = { e, C y } . Since [ (cid:101) σ ( C y )] = − [ σ ( C y )] = +1 , there are two one-dimensionalirreducible representations (cid:101) U (cid:101) α ( C y ) = α ( α = ± . Thegenerators in Eqs. (69)-(71) possess the same commutationand anticommutation relations as σ ( C y ) , σ ( I T ) , and σ ( I C ) ,i.e., { (cid:101) σ ( C y ) , (cid:101) σ ( I C ) } = 0 , (72) [ (cid:101) σ ( C y ) , (cid:101) σ ( I T )] = 0 , (73) [ (cid:101) σ ( I T )] = − . (74)As a result, we find EAZ classes for α = ± are class AII C ,and therefore any point node is not stable on this line. P (cid:48) with B pairing We next consider MSG P (cid:48) , which is generated by thetwo-fold rotation C y along the y -axis and the TRS T . For B pairing, PHS anticommutes with the two-fold rotation, i.e., { σ ( C ) , σ ( C y ) } = 0 . Again, let us consider a two-fold rotationsymmetric line as the 1-cell D in Fig. 4 (a). Unlike the caseof MSG P /m (cid:48) , there exist only the following unitary andchiral parts in the little co-group G D /T = { e, C y } , (75) J D /T = { Γ , C y Γ } . (76)To perform the procedures in Sec. IV A, we define generatorsof onsite symmetries by Eq. (69) and (cid:101) σ (Γ) ≡ σ (Γ) , and wefind { (cid:101) σ (Γ) , (cid:101) σ ( C y ) } = 0 . (77)0 k z k x k y k z k x (a) (b) D Z R U Γ Y X Z C y S z UU Γ R (cid:31) X D FIG. 4. Illustrations of cell decomposition for the half BZ in P /m (cid:48) (a) and the quarter BZ in P mc (cid:48) (b). Here we omit orientationsexcept for the 1-cells denoted by D . In both (a) and (b), adjacent2-cells to the 1-cell D are colored by red. In (b), blue and yellowplanes represent the mirror and glide planes of MSG P mc (cid:48) , re-spectively.TABLE IV. Irreducible representations of the onsite symmetry group (cid:101) G and G D /T for MSG P /m (cid:48) and P (cid:48) .EAZ of P /m (cid:48) ( B g ) EAZ of P (cid:48) ( B ) irrep (cid:101) α e C y ˜ G AII C A Γ AII C A Γ − EAZ of P /m (cid:48) ( B g ) EAZ of P (cid:48) ( B ) irrep β e C y G D /T D A i D A − i Since [ (cid:101) σ ( C y )] = +1 , we have two one-dimensional irre-ducible representations (cid:101) U (cid:101) α ( C y ) = α ( α = ± whose EAZclasses are class A Γ . Therefore, point nodes on the 1-cell areclassified into π ( C ) = Z . The final step is to map the gen-erator of Z to an elements of E , . This can be accomplishedby N βD = − i | G D | (cid:88) g ∈ G D δ s g , sin θ g χ βD ( g )] ∗ × (cid:16)(cid:101) χ (cid:101) α ( g ) − (cid:101) χ (cid:101) Γ (cid:101) α ( g ) (cid:17) , (78)where (cid:101) χ (cid:101) Γ (cid:101) α is the charcter of irreducible representation chiral-symmetry-related to (cid:101) χ (cid:101) α . By substituting irreducible represen-tations in Table IV into Eq. (78), we obtain the generator ofpoint nodes as ( N , N ) = (2 , − , (79)which corresponds to one of the bases of E , multiplied bytwo. P with E pairing Next, we discuss the four-fold rotation symmetric line inspinful MSG P , which is the same 1-cell D in Fig. 4 (a)with the axes exchanged. Since this MSG does not have TRS, TABLE V. Irreducible representations of the onsite symmetry group (cid:101) G and G D /T for P .EAZ irrep (cid:101) α e C z ( C z ) ( C z ) ˜ G A A i − − i A − i − i A − − EAZ irrep β e C z ( C z ) ( C z ) G D /T A e i π i e i π A e i π − i e i π A e − i π i e − i π A e − i π − i e − i π the little co-group G D /T has only a unitary part G D /T = { e, C z , ( C z ) , ( C z ) } . Then, the onsite symmetry group alsohas a unitary part generated by (cid:101) σ ( C z ) ≡ e π γ γ σ ( C z ) , (80)where [ (cid:101) σ ( C z )] = +1 . There are four irreducible representa-tions of (cid:101) G in Table V, and therefore gapless points on the lineare classified into Z . We can map the generators to elementsof E , by the following formula N βD = − i | G D | (cid:88) g ∈ G D δ s g , sin θ g χ βD ( g )] ∗ (cid:101) χ (cid:101) α ( g ) , (81)where β represent to labels of irreducible representations of G D /T in Table V. As a result, we obtain the generators as ( N D , N D , N D , N D ) = ( − , , , for (cid:101) α = 1(1 , − , , for (cid:101) α = 2(0 , , , − for (cid:101) α = 3(0 , , − , for (cid:101) α = 4 , (82)which correspond to not any basis of E , but linear combi-nations of them. P mc (cid:48) with A pairing Last, we discuss nonsymmorphic and noncentrosymmetricMSG
P mc (cid:48) with A pairing. Here we consider the 1-cellon the boundary of BZ denoted by D in Fig. 4 (b). The littleco-group consists of the following four parts: G D /T = { e, M y } , (83) A D /T = { C z T , M x T } , (84) P D /T = { C z C , M x C} , (85) J D /T = { Γ , M y Γ } . (86)1We define generators of the onsite symmetry group by (cid:101) σ ( M y ) ≡ γ σ ( M y ) , (87) (cid:101) σ ( M x T ) ≡ γ γ σ ( M y T ) , (88) (cid:101) σ ( M x C ) ≡ γ γ σ ( M y C ) . (89)Then, the onsite symmetry group is composed of the follow-ing symmetries: (cid:101) G = { e, M y Γ } , (90) (cid:101) A = { M x T , M y M x C} , (91) (cid:101) P = { M x C , M y M x T } , (92) (cid:101) J = { Γ , M y } . (93)One can explicitly verify [ (cid:101) σ ( M y Γ)] = − and [ (cid:101) σ ( M y Γ) , (cid:101) σ ( M x T )] = [ (cid:101) σ ( M y Γ) , (cid:101) σ ( M x C )] = 0 . These re-lations imply that EAZ classes for irreducible representations (cid:101) U (cid:101) α ( M y Γ) = iα ( α = ± are class AIII T , and therefore theclassification is π ( C ) = 0 . V. UNIFICATION OF COMPATIBILITY RELATIONS ANDPOINT-NODE CLASSIFICATIONS
In this section, we integrate classifications of point nodesdiscussed in Sec. IV into compatibility relations in Sec. III E,which results in a unified way to diagnose the shapes of nodes.We first explain the general scheme to classify nodes on 1-cells, and then we apply the scheme to several symmetry set-tings: MSGs P /m (cid:48) with B g pairing, P (cid:48) with B pairing, P with E pairing, and P mc (cid:48) with A pairing. A. Classifications of nodes on 1-cell
As mentioned in Sec. III D, E , can be interpreted as theset of gapless states on 1-cells, which involve the changes ofzero-dimensional topological invariants. However, it is notnecessary that the gapless points in the 1-cells must be point-nodes in BZ. In general, they might be part of line- or surface-nodes. Here, we provide a systematic way to distinguish thesepossibilities.As discussed in Sec. II, compatibility relations tell us if thechange of zero-dimensional topological invariants at a p -cellmake domain walls of the changes at ( p + 1) -cells. This pro-cess is formulated in terms of d p, . Let us suppose that wehave the set of band labels n (1) = b (1) D ,α , where b (1) D ,α is abasis vector of E , generated by an irreducible representa-tion U αD at a 1-cell D (see Sec. III D). Applying the abovestrategy to the 1-cell, there are two cases: (A) d , ( n (1) ) (cid:54) = and (B) d , ( n (1) ) = .We first consider case (A). Since d , ( n (1) ) is an element of E , , d , ( n (1) ) can be expanded by the basis vectors of E , as d , ( n (1) ) = (cid:88) i (cid:88) α r (2) D i ,α b (2) D i ,α + (cid:88) β m (2) D i ,β b (2) D i ,β , (94)where r (2) D i ,α ∈ Z and m (2) D i ,β ∈ Z . This equation tells usthat the gapless point on the 1-cell is extended to adjacent 2-cells with the nontrivial coefficients in Eq. (94), which resultsin the domain walls. As a result, the gapless point on the1-cell is part of line-nodes or surface-nodes. To distinguishbetween these two possibilities, we further examine whether d , is nontrivial. One might recall the relation d , ◦ d , = 0 and think that d , is useless for this purpose. However, whenwe focus on only one domain wall induced by d , , the samediscussion can be applied to the 2-cells. In other words, wefocus on only one basis vector in Eq. (94), and then we cancheck if d , is nontrivial. If there exist the basis vectors suchthat d , ( b (2) D i ,α ) (cid:54) = 0 in Eq. (94), the gapless point on the1-cell is part of a surface-node. Otherwise, it is part of a line-node.Next, we discuss the case (B) where d , ( n (1) ) = . Sincethe relation indicates the absence of any domain walls dis-cussed above, one might expect that the gapless point on the1-cell is a genuine point-node. Indeed, this is not always true.In Sec. IV, we have classified locally stable point-nodes on1-cells and have shown that some generators of stable point-nodes do not correspond to any basis vector of E , . If n (1) isa member of generators of point-nodes derived by formulas inSec. IV B, the gapless point on the 1-cell is a genuine point-node. Otherwise, the gapless point is a part of line-nodes ex-tended from the 1-cell to 3-cells, generic momenta.Using the above scheme, we classify nodes on all 1-cellsfor any MSG M , taking into account all the possible one-dimensional irreducible representations of the superconduct-ing gaps, the presence/absence of spin-rotation SU(2) symme-try, and the spinful/spinless nature of the systems. The resultsare tabulated in the Supplementary Materials. In Appendix A,we explain the cell decomposition for 3D BZ which we usedin the classifications. B. Examples
In the following, we will apply the above scheme to con-crete symmetry settings. As mentioned in Secs. I and II, ourscheme is applicable to complex symmetry settings, e.g., non-centrosymmetric systems and rotation axes in the glide planes,which are out of the scope of previous studies. After we repro-duce the results of previous works for spinful superconductorsin MSG P /m (cid:48) with B g pairing by our method, we showthat our method can detect nodal structures for those in MSG P (cid:48) , P , and P mc (cid:48) , which are noncentrosymmetric, TRbreaking, or nonsymmorphic MSGs.2 P /m (cid:48) with B g pairing Let us begin with the 1-cell D in Fig. 4 (a), which is the ro-tation axis in BZ for P /m (cid:48) with B g pairing. On the 1-cell,there are two irreducible representations listed in Table IV.Ref. [63] has shown that line-nodes pinned to the rotation axescan exist in this symmetry setting, although the derivation hasnot been shown. Here, we show that the line-nodes pinnedto the rotation axes can be stable by d , and our point-nodeclassifications.Let us suppose that we have n (1) = b (1) D in which p D , p D , p T D , and p T D equal . We first define adjacent 2-cellsto the 1-cell D by Fig. 4 (a). The EAZ classes at the 2-cells are class DIII due to the existence of I T and I C with ( I T ) = − and ( I C ) = +1 , and then compatibility rela-tions among them do not exist. This results in d , ( n (1) ) = 0 ,which indicates the gapless point on the 1-cell is not extendedto the 2-cells.Next, we classify stable point-nodes on the 1-cell. As dis-cussed in Sec. IV C 1, we find there are no stable point-nodeson the 1-cell. Therefore, we conclude that the gapless pointis part of a line-node extended from the 1-cell to 3-cells. Thisline-node is protected by one-dimensional winding number W defined by the chiral symmetry at the 3-cells. This is preciselywhat Ref. [63] has proposed.We then discuss the mirror plane in the k y = 0 . Let us focuson the 1-cell b in Fig. 2 and suppose that we have n (1) = b (1) b which has p ± b = 1 for irreducible representations U ± b ( M y ) = ± i . The 2-cells α and α are adjacent to the 1-cell b and thesame symmetry class. Consequently, compatibility relationsamong them exist, and d , ( b (1) b ) = b (2) α + b (2) α . Here, b (2) α ( b (2) α ) is a basis of E , in which p ± α ( p ± α ) and associatedband labels equal . As discussed in Sec. V A, d , ( n (1) ) (cid:54) =0 indicates that the gapless point on the 1-cell b should beextended to the adjacent 2-cells. Since EAZ classes of all 3-cells are class DIII, there are no compatibility relations, i.e., d , = 0 . Therefore, we conclude that the gapless point on the1-cell b is classified into Z and is part of the line-node in themirror plane. Our result is consistent with the result of grouptheoretical analysis in Refs. [50] P (cid:48) with B pairing Next, we consider the same 1-cell as that in Sec. V B 1, butwithout the inversion symmetry. In this case, the system canhave line-nodes pinned to the rotation axes. Irreducible repre-sentations U β =1 , D and their EAZ classes are listed in Table IV.We again assume that we have n (1) = b (1) D in which N D = − N D = +1 and associated band labels equal or − . Unlike the above case, the 2-cells are invariant only un-der Γ , and then their EAZ classes are class AIII. As with thecase of Sec. V B 1, this implies that there are no compatibilityrelations among them, i.e., d , ( n (1) ) = 0 , and the gaplesspoint on the 1-cell is not extended to the 2-cells. As shown inSec. IV C 2, Since ( N D , N D ) = (1 , − is not the genera- k z k x k y C y FIG. 5. The nodal line of the tight-binding model in Eq. (95) for µ = +1 . tor of point-nodes in Eq. (79), we conclude that b D indicatesthe existence of line-nodes pinned to the rotation axes. This isconsistent with the fact that the winding number W does notchange after breaking the inversion symmetry of the system inSec. V B 1.To verify the existence of such line-nodes, let us considerthe following model H k = (3 − cos k x − cos k y − cos k z − µ ) τ z + (sin k x + 2 sin k z ) τ y , (95) ρ ( C y ) = − iτ z σ y , (96) ρ ( T ) = iσ y , (97) ρ ( C ) = τ x , (98)where σ i = x,y,z and τ j = x,y,z are Pauli matrices which repre-sent different degree of freedom. After computing the regionwhere the spectrum is gapless, we find a line-node in Fig. 5.This is the line-node that we have discussed above.The question is whether n (1) = 2 b (1) D is the point-nodeor not. In the following, we show that the above line-nodecan exist even in the case. To explain this, we start with thecase where there are two the above line-nodes generated by n (1) = 2 b (1) D illustrated in Fig. 6 (a). By rotating one of thelines, the winding numbers can be cancelled. Then, we gettwo pair of point-nodes in Fig. 6 (b). However, in the absenceof other symmetries than MSG P (cid:48) with PHS, there are noreasons why two gapless points on the 1-cell exist at the samepoint. Finally, each pair again forms a line-node illustrated inFig. 6 (c). As a result, n (1) = 2 b (1) D indicates the existence ofline-nodes in Fig. 6 (c), and therefore nodes on the 1-cell areclassified into Z , whose elements are line-nodes of case (B) . P with E pairing Next, we discuss MSG P with E pairing, which is gener-ated by the four-fold rotation symmetry C z . We consider the1-cell D in Fig. 7 (a). In the following, we show that a gap-less point on the 1-cell is part of surface-nodes. Irreducible3 k y k y × × k y W = +1 C y W = − W = +1 W = − (a) (b) (c) gapless points T D D D FIG. 6. Deformation of nodal structures in MSG P (cid:48) . Two line-nodes pinned to the rotation axis are protected by 1D winding num-bers (a). These line-nodes can be deformed to point-nodes withoutclosing gap at 0-cells (b). Since there are no reasons why two point-nodes are at the same position, each of two split gapless points isagain part of a line-node. representations U βD ( β = 1 , , , and their EAZ classesare tabulated in Table V.Suppose that we have n (1) = b (1) D ,β =1 which has N D = − N C D = +1 . Although there exist eight adjacent 2-cellsto D [colored in Fig. 7 (a)], only two of them are inde-pendent due to the presence of C z . Here, we choose blueplanes D and D in Fig. 7 (a) as independent adjacent 2-cells. Since the EAZ classes at D , the adjacent 2-cells, and 3-cells are the same, compatibility relations exist. Accordingly, d , ( n (1) ) = b (2) D − b (2) D , in which N D i =1 , = +1 and associ-ated band labels equal or − . We further find d , ( b D ) (cid:54) = 0 and d , ( b D ) (cid:54) = 0 , which implies that a gapless point on the1-cell is part of surface-nodes. Note that the disucssions andresults for other values of β do not change.As shown in Sec. IV C 3, when n (1) is a linear combinationof { b (1) D ,β } β =1 , point-nodes on the 1-cell can exist. However,the same logic in Sec. V B 2 is valid, and therefore the point-nodes can be inflated, which results in sphere nodes (Bogoli-ubov Fermi surfaces) pinned at the 1-cell like the right panelin Fig. 7 (b). P mc (cid:48) with A pairing Finally, we discuss nonsymmorphic and noncentrosymmet-ric MSG
P mc (cid:48) with A pairing. We focus on the 1-cell D in the boundary of BZ [see Fig. 4 (b)], which is invariantunder the glide symmetry G y . There are two irreducible rep-resentations U ± D ( G y ) = ± of G D , and their EAZ classesare class D. Let us consider that we have n (1) = b (1) D in which p ± D = 1 and associated band labels are nontrivial. As shownin Fig. 4 (b), three adjacent 2-cells to D exist. The EAZclasses of the 2-cells in the k y = 0 plane and the k z = π areclass A and class DIII, respectively. Consequently, there areno compatibility relations, i.e., d , ( n (1) ) = . In addition, asshown in Sec. IV C 4, there are no locally stable point-nodes.As a result, we arrive at the line-node pinned to the 1-cell D ,which is extended from the 1-cell D to 3-cells. Interestingly, k z k x k y (a) (b) D Γ Y X Z × × gapless points D D k z k z C z C z FIG. 7. (a) A half BZ in MSG P . Here, the blue planes D and D are adjacent 2-cells to D and red ones are symmetry-related to D and D . (b) Deformation of nodal structures in MSG P . such line-nodes on the 1-cell do not exist in symmorphic MSG P mm (cid:48) with A pairing, whose point group is the same as P mc (cid:48) . In P mm (cid:48) with A pairing, the line-node pinnedto the 1-cell D is understood by the compatibility relations.This is an example where nonsymmorphic symmetries changethe classifications of nodes. As shown in this example, ourmethod can capture the shape of nodes even in the presence ofnonsymmorphic symmetries and in the absence of the inver-sion symmetry. VI. APPLICATIONS TO MATERIALS
In this section, we provide an efficient algorithm to diag-nose the shape of nodes, which needs only band labels at 0-cells as input data. Since the energy scale of the supercon-ducting gaps in most superconductors is believed to be muchsmaller than that of normal phases [75, 78, 80, 103–105], wecan obtain the input from DFT calculations assuming the pair-ing symmetry. We also show that how to apply the algorithmto realistic materials.
A. Efficient algorithm for detection of nodal structures
In Sec. V, we have classified nodes on the 1-cells, and wehave shown that the basis of E , can largely determine theshape of nodes. Here we recall that d , is a map from E , to E , . This enable us to know nodal structures on the 1-cells from information at the 0-cells. First, let us assumethat we have the set of band labels at the 0-cells n (0) and d , ( n (0) ) (cid:54) = 0 . By expanding d , ( n (0) ) by the basis of E , ,we find which coefficients are nontrivial. Using the resultsof classifications in Sec. V, we diagnose the shape of nodalstructures, i.e., gapless points on the 1-cells are point-node orpart of line/surface-nodes.To demonstrate the scheme, we consider a simple tight-4 k z k x k y C y k z k x k y C C C y FIG. 8. The nodal lines of the tight-binding model in Eq. (99) for µ = +1 . The blue plane is the mirror symmetric plane. binding model of MSG P /m (cid:48) with B g pairing: H k = (3 − cos k x − cos k y − cos k z − µ ) τ z + (sin k x + 2 sin k z ) sin k y τ y σ y , (99) ρ ( I ) = , (100) ρ ( C y ) = − iτ z σ y , (101) ρ ( T ) = iσ y , (102) ρ ( C ) = τ x , (103)where σ i = x,y,z and τ j = x,y,z are Pauli matrices which rep-resent different degree of freedom. Using this model, weshow that the above algorighm can detect nodal structuresdiscussed in Sec. V B 1. After computing Pfaffian invari-ants in Eq. (21) for all 0-cells, we find p = p = 1 andothers equal zero, where p and p are band labels for ir-reducible representations ( U ( I ) , U ( C y )) = (1 , + i ) and ( U ( I ) , U ( C y )) = (1 , − i ) . This set of band labels cor-respond to a basis of E , denoted by b (0)Γ , , and we get d , ( b (0)Γ , ) = b (1) a + b (1) b + b (1) a + b (1) b + b (1) D , where we usethe same labels of 1-cells in Figs. 2 and 4(a). This indicatesthat gapless points exist on the 1-cells a, b, a , b , and D . Asdiscussed in Sec. V B 1, the gapless point on the 1-cell b is partof line-nodes in the mirror plane. Similar to the case, gaplesspoints on the 1-cells a, a , and b are also extended to theiradjacent 2-cells in the plane. Taking into account symmetryrelations among 2-cells, we find that a line-node in the mirrorplane encircles Γ point. On the other hand, we have shownthat a gapless point in the rotation axis is also part of a line-node pinned to the axis. We verify that our method correctlycaptures the nodes of the tight-binding model shown in Fig. 8. B. Material example
In this subsection, we apply the above algorithm to real-istic superconductors CaPtAs, whose MSG is I md (cid:48) . Arecent experiment [92] has reported the time-reversal break-ing and the signature of point-nodes. Breaking TRS indicates that the order parameter belongs to E or E representationsof the point group C . Then, MSG I md (cid:48) is reduced to I . Here, we assume that the superconducting gap belongsto E representation. Ref. [80] has computed irreducible rep-resentations by QUANTUM-ESPRESSO [106, 107] and qeir-reps [108] and found that p = 1 and N = − N = − ,where the labels of irreducible representations follow Table V.Then, the set of band labels n (0) corresponds to − b (0)Γ , + b (0)Γ , .In the following, we show that this superconducting materialis expected to have small Bogoliubov Fermi surfaces.We check if this material satisfies compatibility relations,i.e., d , ( n (0) ) = . After computing d , ( n (0) ) , we find d , ( n (0) ) = − b (1) D , + b (1) D , , where D denotes the rota-tion symmetric line between Γ = (0 , , and Z = (0 , , π ) .In fact, the symmetry setting in this line is the completelysame as that in Sec. V B 3, and then the nodal structures arealso the same. Since d , ( n (0) ) correspond to a generator ofpoint-nodes listed in Eq. (82), we expect that this material hassmall Bogoliubov Fermi surfaces as discussed in Sec. V B 3(see Fig. 7 (b)).Our result might not contradict the experimental observa-tion. Since the superconducting gaps in most superconduc-tors are considered to be much small, it is natural to thinkthe Bogoliubov Fermi surfaces are also small. Further exper-iments to distinguish between this case and exact point-nodesare awaited. VII. FURTHER EXTENSION TO NODES AT GENERICPOINTS
Thus far, we have focused on nodes pinned to 1-cells. How-ever, in general, nodes can exist at generic points. In this sec-tion, we discuss how to extend our symmetry-based approachto nodes at generic points through the mirror plane in MSG P /m (cid:48) with B u pairing.Here, we decompose the mirror plane into the cell decom-position in Fig. 2 and discuss the 1-cell denoted by b in Fig. 2.After applying the method in Sec. IV to the 1-cell, we findthat the classification of gapless points is Z . The generatingHamiltonian is H ( k ,k ) = k τ y + k τ x σ z + δk τ z , (104) σ ( I C ) = iτ y K (105) σ ( I T ) = iτ z σ y K (106) σ ( M y ) = iτ z σ x , (107)where k is perpendicular to both the mirror plane, the 1-cell, k is perpendicular to the 1-cell but parallel to the mirrorplane, and δk is a displacement from the gapless point in thedirection of the 1-cell. The gapless point is protected by themirror winding number [109]. On the other hand, since theEAZ class at the 1-cell is class AIII, there are no topologi-cal invariants, which implies that gapless points pinned to the1-cell do not exist. In fact, we can add the symmetric pertur-bation terms which shift the gapless point to the k -direction.Therefore, gapless points can locally exist everywhere in themirror plane.5 ++ FIG. 9. Illustration of annihilation process of gapless points. Herewhite solids circles denote gapless points and ± represent the sign ofthe winding numbers. The question is whether these gapless points are globallystable. In the following, we show that there can globallyexist only two gapless points in the plane. To explain this,let us suppose that there are four gapless points in the planeas shown in Fig. 9. Since C y anticommutes with PHS, C y changes the sign of the winding number (see Appendix C).As discussed above, the gapless points can freely move in theplane, and therefore two winding numbers with opposite signscan be canceled. This indicates that only one pair of gaplesspoints can globally exist.Symmetry-indicators in this symmetry class can detect theglobally stable gapless points. The symmetry indicator groupis ( Z ) × Z , whose Z -parts originate from lower dimen-sions. The Z index is defined by z = 14 (cid:88) K ∈ TRIMs (cid:0) N + K − N − K (cid:1) mod 4 , (108)where N ± K is the band label for irreducible representa-tions U ± K ( I ) = ± at the time-reversal invariant momenta(TRIMs). If the system is fully gapped, z = +1 indicates themirror Chern number modulo equals . However, the non-trivial mirror Chern numbers are forbidden in this symmetrysetting [110]. Therefore, we conclude that z = +1 indicatesthe existence of gapless points.Actually, the above annihilation procedure can be under-stood as “second differential” d p, in the theory of Atiyah-Hirzebruch Spectral Sequence [99]. Although establishingfull classifications of nodes at generic points and relationshipbetween symmetry-indicators and the nodes are interesting is-sues, they are out of scope of this paper. VIII. CONCLUSION AND OUTLOOK
In this work, we have established a systematic framework toclassify superconducting nodes pinned to any line in momen-tum space. After decomposing BZ of all MSGs into points(0-cells), lines (1-cells), planes (2-cells), and polyhedrons (3-cells), we have applied our method to the lines and obtainedcomprehensive classifications of nodes pinned to the lines.Moreover, our theory has resulted in a highly efficient way todiagnose the superconducting nodes in superconducting mate- rials. As a demonstration, we have analyzed the nodes in CaP-tAs assuming the time-reversal broken pairing and pointed outthat this material can have small Bogolibov Fermi surfaces.Our work opens up various possibilities for future stud-ies. Although our results cover a wide range of nodes,nodes at generic points are missing as discussed in Sec. VII.The symmetry-based approach can be more refined to detectsuch nodes, and we leave deriving full relationships betweensymmetry-indicators and the nodes as future works. This typeof study will give us more information of nodes pinned tolines as follows. Suppose that a system violates compatibilityrelations, which indicates the existence of nodes pinned to 1-cells as discussed in Sec. VI. Since we can always forget aboutsymmetries that impose the violated compatibility relations onthe system, we can apply symmetry-indicators for lower sym-metry classes to the system as discussed in Ref. [111]. Then,the symmetry-indicators will clarify topological nature behindthe nodes.The integration of our algorithm with DFT calculations en-ables a comprehensive investigation of nodes in the materialslisting in the database. Such studies help to find the possiblepairings of unconventional superconductivity compatible withexperimental observations. We hope that our study will leadto a deep understanding of superconductivity in discoveredsuperconductors.
ACKNOWLEDGMENTS
We thank Hoi Chun Po, Shuntaro Sumita, Takuya Nomoto,and Haruki Watanabe for valuable discussions. In particular,KS thanks Takuya Nomoto for sharing ideas on how the firstdifferential detects the nodal structure in the early stages of theproject. The work of SO is supported by The ANRI Fellow-ship and KAKENHI Grant No. JP20J21692 from the JSPS.The work of KS is supported by PRESTO, JST (Grant No.JPMJPR18L4) and CREST, JST (Grant No. JPMJCR19T2).
Appendix A: Cell decomposition for representative space groups
In this appendix, we present units of BZ for each typesof lattices, which can fill entire BZ by symmetry opera-tions. In fact, it is enough to define the units for SG
P m ¯3 m, Cmmm, P /mmm, F m ¯3 m, and Im ¯3 m . Note thatthe cell decomposition for Amm is the same as that for Cmmm with axes exchanged and the cell decomposition for R ¯3 m is constructed by { k j,x b + k j,y b + k j,z b } j , where ( k j,x , k j,y , k j,z ) is a cell for P m ¯3 m and b i is a primitive re-ciprocal lattice vector. When we discuss a lower symmetrysetting than them, we use the cell decomposition of one whoselattice is the same as the system. Appendix B: Derivation other formulas
In this appendix, we derive formulas to obtain generators ofgapless points as elements of E p, . To achieve this, we find6 (a) (b) P m ¯3 m R = ( π, π, π ) M = ( π, π, Γ = (0 , , X = ( π, , (c) (d)(e) Z = (2 π, , R = ( π, π, π ) M = ( π, π, Γ = (0 , , Im ¯3 m Γ = (0 , , R = ( π, π, π ) S = ( π, π, Y = (2 π, , Z = (0 , , π ) T = (0 , π, π ) Cmmm
F m ¯3 m P /mmm Γ = (0 , , A = (0 , , / H = (1 / , / , / K = (1 / , / , L = (0 , / , / M = (0 , / , Γ = (0 , , M = (2 π, π, W = ( π, π, X = (0 , π, L = ( π, π, π ) Y = (0 , π, T = (2 π, , π ) FIG. 10. Units of BZ for
P m ¯3 m (a), Cmmm (b), P /mmm (c), F m ¯3 m (d), and Im ¯3 m (e). Note that coordinates in P /mmm aredenoted by coefficients of primitive reciprocal lattice vectors. Orien-tations except for blue and red lines in Cmmm and
F m ¯3 m can bearbitrarily chosen. Orientations of these colored lines are choosen bysymmetric manners. the generating Hamiltonian and symmetries like Eqs. (60)-(62). We tabulate Gamma matrices γ , , and symmetryrepresentations (cid:101) σ ( g ) in Table VI. By substituting them intoEqs. (58) and (59), one can obtain formulas in Table III. Appendix C: Symmetry property of the winding number
In this appendix, we show that C y which anticommuteswith PHS changes the sign of the winding number. The wind- ing number is defined by W [ C ] ≡ (cid:73) C tr[ U (Γ) (cid:0) H − k ∂ k H k (cid:1) ] ds, (C1)where we consider Γ − X − M − Y − Γ in Fig. 11 as C . Wefirst compute the integrand tr[ U (Γ) (cid:0) H − k ∂ k i H k (cid:1) ]= tr[ U ( C y ) U (Γ) (cid:0) H − k ∂ k i H k (cid:1) U − ( C y )]= − tr[ U (Γ) U ( C y ) (cid:0) H − k ∂ k i H k (cid:1) U − ( C y )]= − tr[ U (Γ) (cid:18) H − − k ∂∂k i H − k (cid:19) ] . (C2)Using the identity, we derive the following relation (cid:90) X Γ tr[ U (Γ) (cid:16) H − k x , ∂ k x H ( k x , (cid:17) ] dk x = − (cid:90) X (cid:48) Γ tr[ U (Γ) (cid:16) H − k x , ∂ k x H ( k x , (cid:17) ] dk x (C3)For other integral intervals, one finds the same transforma-tion. As a result, we obtain the relation W [ C ] = − W [ C y C ] .Actually, the relation can be generalized to other point groupsymmetries as W [ C ] = χ g det p g W [ gC ] ( χ g = ± , where U ( g ) U ( C ) = χ g U ( C )[ U ( g )] ∗ . + Γ M X Y X Y M (cid:31)(cid:31) (cid:31) FIG. 11. Illustration of the interval of integral in the winding number.Note that the path colored by blue is symmetry-related to the redpath. Here, white solids circles denote gapless points and ± representthe sign of the winding numbers.[1] K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z. Q. Mao,Y. Mori, and Y. Maeno, “Spin-triplet superconductivity inSr RuO identified by O Knight shift,” Nature , 658–660 (1998).[2] G. M. Luke, Y. Fudamoto, K. M. Kojima, M. I. Larkin, J. Mer-rin, B. Nachumi, Y. J. Uemura, Y. Maeno, Z. Q. Mao, Y. Mori,H. Nakamura, and M. Sigrist, “Time-reversal symmetry-breaking superconductivity in Sr RuO ,” Nature , 558–561 (1998). [3] Shingo Yonezawa, Tomohiro Kajikawa, and YoshiteruMaeno, “First-Order Superconducting Transition of Sr RuO ,” Phys. Rev. Lett. , 077003 (2013).[4] Shunichiro Kittaka, Akira Kasahara, Toshiro Sakakibara,Daisuke Shibata, Shingo Yonezawa, Yoshiteru Maeno,Kenichi Tenya, and Kazushige Machida, “Sharp magneti-zation jump at the first-order superconducting transition in Sr RuO ,” Phys. Rev. B , 220502(R) (2014).[5] E. Hassinger, P. Bourgeois-Hope, H. Taniguchi, S. Ren´e deCotret, G. Grissonnanche, M. S. Anwar, Y. Maeno, N. Doiron- TABLE VI. Gamma matrices in Eq. (50) and onsite unitary symmetries. Here, (cid:101) u (cid:101) α is an irreducible representation of the onsite unitarysymmetry group (cid:101) G . In addition, (cid:101) T (cid:101) α , (cid:101) C (cid:101) α , and (cid:101) Γ (cid:101) α are labels of the time-reversal, the particle-hole, and the chiral symmetry related irreduciblerepresentations. EAZ γ γ γ (cid:101) σ ( g ) A σ z σ x σ y ⊗ (cid:101) u (cid:101) α ( g ) A T τ z τ x τ y σ z τ diag ( (cid:101) u (cid:101) α ( g ) , (cid:101) u (cid:101) T (cid:101) α ( g )) A C τ z τ x τ y τ diag ( (cid:101) u (cid:101) α ( g ) , (cid:101) u (cid:101) C (cid:101) α ( g )) A Γ τ z τ x τ y σ z τ diag ( (cid:101) u (cid:101) α ( g ) , (cid:101) u (cid:101) Γ (cid:101) α ( g )) A T , C τ z τ x τ y ( ⊗ σ z ) τ diag ( (cid:101) u (cid:101) α ( g ) , (cid:101) u (cid:101) T (cid:101) α ( g ) , (cid:101) u (cid:101) C (cid:101) α ( g ) , (cid:101) u (cid:102) Γ α ( g )) C σ z σ x σ y ⊗ (cid:101) u (cid:101) α ( g ) C T τ z τ y τ x σ z τ diag ( (cid:101) u (cid:101) α ( g ) , (cid:101) u (cid:101) T (cid:101) α ( g )) D τ z τ x σ y τ y σ y τ diag ( (cid:101) u (cid:101) α ( g ) , (cid:101) u (cid:101) α ( g )) D T s z s y τ y s x τ y σ z s τ diag ( (cid:101) u (cid:101) α ( g ) , (cid:101) u (cid:101) T (cid:101) α ( g )) AI σ z σ x τ y σ y τ diag ( (cid:101) u (cid:101) α ( g ) , (cid:101) u (cid:101) α ( g )) AI C s z s x s y τ z s τ diag ( (cid:101) u (cid:101) α ( g ) , (cid:101) u (cid:101) C (cid:101) α ( g )) CI τ z τ x τ y σ z τ diag ( (cid:101) u (cid:101) α ( g ) , (cid:101) u (cid:101) α ( g )) Leyraud, and Louis Taillefer, “Vertical Line Nodes in the Su-perconducting Gap Structure of Sr RuO ,” Phys. Rev. X ,011032 (2017).[6] Yuuki Yasui, Kaveh Lahabi, Muhammad Shahbaz Anwar,Yuji Nakamura, Shingo Yonezawa, Takahito Terashima, JanAarts, and Yoshiteru Maeno, “Little-Parks oscillations withhalf-quantum fluxoid features in Sr RuO microrings,” Phys.Rev. B , 180507(R) (2017).[7] Shunichiro Kittaka, Shota Nakamura, Toshiro Sakakibara,Naoki Kikugawa, Taichi Terashima, Shinya Uji, Dmitry A.Sokolov, Andrew P. Mackenzie, Koki Irie, Yasumasa Tsut-sumi, Katsuhiro Suzuki, and Kazushige Machida, “Search-ing for Gap Zeros in Sr RuO via Field-Angle-DependentSpecific-Heat Measurement,” Journal of the Physical Societyof Japan , 093703 (2018).[8] A. Pustogow, Yongkang Luo, A. Chronister, Y. S. Su, D. A.Sokolov, F. Jerzembeck, A. P. Mackenzie, C. W. Hicks,N. Kikugawa, S. Raghu, E. D. Bauer, and S. E. Brown, “Con-straints on the superconducting order parameter in Sr RuO from oxygen-17 nuclear magnetic resonance,” Nature ,72–75 (2019).[9] Satoshi Kashiwaya, Kohta Saitoh, Hiromi Kashiwaya, MasaoKoyanagi, Masatoshi Sato, Keiji Yada, Yukio Tanaka, andYoshiteru Maeno, “Time-reversal invariant superconductivityof Sr RuO revealed by Josephson effects,” Phys. Rev. B ,094530 (2019).[10] Kenji Ishida, Masahiro Manago, Katsuki Kinjo, and YoshiteruMaeno, “Reduction of the O Knight Shift in the Super-conducting State and the Heat-up Effect by NMR Pulseson Sr RuO ,” Journal of the Physical Society of Japan ,034712 (2020).[11] Steven Allan Kivelson, Andrew Chang Yuan, Brad Ramshaw,and Ronny Thomale, “A proposal for reconciling diverseexperiments on the superconducting state in Sr RuO ,” npjQuantum Materials , 43 (2020).[12] Aaron Chronister, Andrej Pustogow, Naoki Kikugawa,Dmitry A. Sokolov, Fabian Jerzembeck, Clifford W. Hicks,Andrew P. Mackenzie, Eric D. Bauer, and Stuart E. Brown,“Evidence for even parity unconventional superconductivity inSr RuO ,” (2020), arXiv:2007.13730 [cond-mat.supr-con]. [13] Sheng Ran, Chris Eckberg, Qing-Ping Ding, Yuji Furukawa,Tristin Metz, Shanta R. Saha, I-Lin Liu, Mark Zic, HyunsooKim, Johnpierre Paglione, and Nicholas P. Butch, “Nearly fer-romagnetic spin-triplet superconductivity,” Science , 684–687 (2019).[14] Jun Ishizuka, Shuntaro Sumita, Akito Daido, and YouichiYanase, “Insulator-Metal Transition and Topological Super-conductivity in UTe from a First-Principles Calculation,”Phys. Rev. Lett. , 217001 (2019).[15] Yuanji Xu, Yutao Sheng, and Yi-feng Yang, “Quasi-Two-Dimensional Fermi Surfaces and Unitary Spin-Triplet Pairingin the Heavy Fermion Superconductor UTe ,” Phys. Rev. Lett. , 217002 (2019).[16] Tristin Metz, Seokjin Bae, Sheng Ran, I-Lin Liu, Yun Suk Eo,Wesley T. Fuhrman, Daniel F. Agterberg, Steven M. Anlage,Nicholas P. Butch, and Johnpierre Paglione, “Point-node gapstructure of the spin-triplet superconductor UTe ,” Phys. Rev.B , 220504(R) (2019).[17] Lin Jiao, Sean Howard, Sheng Ran, Zhenyu Wang, Jorge Oli-vares Rodriguez, Manfred Sigrist, Ziqiang Wang, Nicholas P.Butch, and Vidya Madhavan, “Chiral superconductivity inheavy-fermion metal UTe2,” Nature , 523–527 (2020).[18] Shunichiro Kittaka, Yusei Shimizu, Toshiro Sakakibara,Ai Nakamura, Dexin Li, Yoshiya Homma, Fuminori Honda,Dai Aoki, and Kazushige Machida, “Orientation of pointnodes and nonunitary triplet pairing tuned by the easy-axismagnetization in UTe ,” Phys. Rev. Research , 032014(R)(2020).[19] Ian M. Hayes, Di S. Wei, Tristin Metz, Jian Zhang, Yun SukEo, Sheng Ran, Shanta R. Saha, John Collini, Nicholas P.Butch, Daniel F. Agterberg, Aharon Kapitulnik, and John-pierre Paglione, “Weyl Superconductivity in UTe2,” (2020),arXiv:2002.02539 [cond-mat.str-el].[20] Jun Ishizuka and Youichi Yanase, “A Periodic AndersonModel for Magnetism and Superconductivity in UTe2,”(2020), arXiv:2008.01945 [cond-mat.supr-con].[21] Jun Goryo, Mark H. Fischer, and Manfred Sigrist, “Possiblepairing symmetries in srptas with a local lack of inversion cen-ter,” Phys. Rev. B , 100507(R) (2012).[22] P. K. Biswas, H. Luetkens, T. Neupert, T. St¨urzer, C. Baines,G. Pascua, A. P. Schnyder, M. H. Fischer, J. Goryo, M. R. Lees, H. Maeter, F. Br¨uckner, H.-H. Klauss, M. Nicklas, P. J.Baker, A. D. Hillier, M. Sigrist, A. Amato, and D. Johrendt,“Evidence for superconductivity with broken time-reversalsymmetry in locally noncentrosymmetric srptas,” Phys. Rev.B , 180503(R) (2013).[23] Mark H. Fischer, Titus Neupert, Christian Platt, Andreas P.Schnyder, Werner Hanke, Jun Goryo, Ronny Thomale, andManfred Sigrist, “Chiral d -wave superconductivity in srptas,”Phys. Rev. B , 020509(R) (2014).[24] K. Matano, K. Arima, S. Maeda, Y. Nishikubo, K. Kudo,M. Nohara, and Guo-qing Zheng, “Spin-singlet superconduc-tivity with a full gap in locally noncentrosymmetric srptas,”Phys. Rev. B , 140504(R) (2014).[25] Mark H. Fischer and Jun Goryo, “Symmetry and gap classi-fication of non-symmorphic srptas,” Journal of the PhysicalSociety of Japan , 054705 (2015).[26] F. Steglich, J. Aarts, C. D. Bredl, W. Lieke, D. Meschede,W. Franz, and H. Sch¨afer, “Superconductivity in the Presenceof Strong Pauli Paramagnetism: Ce Cu Si ,” Phys. Rev. Lett. , 1892–1896 (1979).[27] G. R. Stewart, Z. Fisk, J. O. Willis, and J. L. Smith, “Possibil-ity of Coexistence of Bulk Superconductivity and Spin Fluc-tuations in U Pt ,” Phys. Rev. Lett. , 679–682 (1984).[28] G. E. Volovik and L. P. Gor’kov, “Superconducting classes inheavy-fermion systems,” Zh. Eksp. Teor. Fiz. , 1412–1428(1985).[29] P. W. Anderson, “Structure of ”triplet” superconducting en-ergy gaps,” Phys. Rev. B , 4000–4002 (1984).[30] Masa-aki Ozaki, Kazushige Machida, and Tetsuo Ohmi, “Onp-Wave Pairing Superconductivity under Cubic Symmetry,”Progress of Theoretical Physics , 221–235 (1985).[31] Masa-aki Ozaki, Kazushige Machida, and Tetsuo Ohmi,“On p-Wave Pairing Superconductivity under Hexagonal andTetragonal Symmetries,” Progress of Theoretical Physics ,442–444 (1986).[32] Manfred Sigrist and Kazuo Ueda, “Phenomenological theoryof unconventional superconductivity,” Rev. Mod. Phys. ,239–311 (1991).[33] C. C. Tsuei and J. R. Kirtley, “Pairing symmetry in cupratesuperconductors,” Rev. Mod. Phys. , 969–1016 (2000).[34] P. M. R. Brydon, Limin Wang, M. Weinert, and D. F. Agter-berg, “Pairing of j = 3 / fermions in half-heusler supercon-ductors,” Phys. Rev. Lett. , 177001 (2016).[35] D. F. Agterberg, P. M. R. Brydon, and C. Timm, “Bogoli-ubov Fermi Surfaces in Superconductors with Broken Time-Reversal Symmetry,” Phys. Rev. Lett. , 127001 (2017).[36] C. Timm, A. P. Schnyder, D. F. Agterberg, and P. M. R.Brydon, “Inflated nodes and surface states in superconductinghalf-heusler compounds,” Phys. Rev. B , 094526 (2017).[37] Lucile Savary, Jonathan Ruhman, J¨orn W. F. Venderbos,Liang Fu, and Patrick A. Lee, “Superconductivity in three-dimensional spin-orbit coupled semimetals,” Phys. Rev. B ,214514 (2017).[38] Hyunsoo Kim, Kefeng Wang, Yasuyuki Nakajima, Rong-wei Hu, Steven Ziemak, Paul Syers, Limin Wang, HalynaHodovanets, Jonathan D. Denlinger, Philip M. R. Brydon,Daniel F. Agterberg, Makariy A. Tanatar, Ruslan Prozorov,and Johnpierre Paglione, “Beyond triplet: Unconventional su-perconductivity in a spin-3/2 topological semimetal,” ScienceAdvances , eaao4513 (2018).[39] Igor Boettcher and Igor F. Herbut, “Unconventional supercon-ductivity in luttinger semimetals: Theory of complex tensororder and the emergence of the uniaxial nematic state,” Phys.Rev. Lett. , 057002 (2018). [40] J¨orn W. F. Venderbos, Lucile Savary, Jonathan Ruhman,Patrick A. Lee, and Liang Fu, “Pairing states of spin- fermions: Symmetry-enforced topological gap functions,”Phys. Rev. X , 011029 (2018).[41] M. R. Norman, “Odd parity and line nodes in heavy-fermionsuperconductors,” Phys. Rev. B , 15093–15094 (1995).[42] T. Micklitz and M. R. Norman, “Odd parity and linenodes in nonsymmorphic superconductors,” Phys. Rev. B ,100506(R) (2009).[43] Takuya Nomoto and Hiroaki Ikeda, “Exotic Multigap Struc-ture in UPt Unveiled by a First-Principles Analysis,” Phys.Rev. Lett. , 217002 (2016).[44] Youichi Yanase, “Nonsymmorphic Weyl superconductivity in
UPt based on E u representation,” Phys. Rev. B , 174502(2016).[45] T. Micklitz and M. R. Norman, “Nodal lines and nodal loopsin nonsymmorphic odd-parity superconductors,” Phys. Rev. B , 024508 (2017).[46] E. I. Blount, “Symmetry properties of triplet superconduc-tors,” Phys. Rev. B , 2935–2944 (1985).[47] Shingo Kobayashi, Ken Shiozaki, Yukio Tanaka, andMasatoshi Sato, “Topological blount’s theorem of odd-paritysuperconductors,” Phys. Rev. B , 024516 (2014).[48] T. Micklitz and M. R. Norman, “Symmetry-enforced linenodes in unconventional superconductors,” Phys. Rev. Lett. , 207001 (2017).[49] Shingo Kobayashi, Shuntaro Sumita, Youichi Yanase, andMasatoshi Sato, “Symmetry-protected line nodes and majo-rana flat bands in nodal crystalline superconductors,” Phys.Rev. B , 180504(R) (2018).[50] Shuntaro Sumita and Youichi Yanase, “Unconventional super-conducting gap structure protected by space group symmetry,”Phys. Rev. B , 134512 (2018).[51] Takuya Nomoto and Hiroaki Ikeda, “Symmetry-ProtectedLine Nodes in Non-symmorphic Magnetic Space Groups: Ap-plications to UCoGe and UPd Al ,” Journal of the PhysicalSociety of Japan , 023703 (2017).[52] Jeffrey C. Y. Teo and C. L. Kane, “Topological defects andgapless modes in insulators and superconductors,” Phys. Rev.B , 115120 (2010).[53] Keiji Yada, Masatoshi Sato, Yukio Tanaka, and TakehitoYokoyama, “Surface density of states and topological edgestates in noncentrosymmetric superconductors,” Phys. Rev. B , 064505 (2011).[54] Masatoshi Sato, Yukio Tanaka, Keiji Yada, and TakehitoYokoyama, “Topology of Andreev bound states with flat dis-persion,” Phys. Rev. B , 224511 (2011).[55] Yukio Tanaka, Masatoshi Sato, and Naoto Nagaosa, “Symme-try and topology in superconductors –odd-frequency pairingand edge states–,” Journal of the Physical Society of Japan ,011013 (2012).[56] Shunji Matsuura, Po-Yao Chang, Andreas P Schnyder, andShinsei Ryu, “Protected boundary states in gapless topologicalphases,” New Journal of Physics , 065001 (2013).[57] Y. X. Zhao and Z. D. Wang, “Topological classification andstability of fermi surfaces,” Phys. Rev. Lett. , 240404(2013).[58] Ching-Kai Chiu and Andreas P. Schnyder, “Classificationof reflection-symmetry-protected topological semimetals andnodal superconductors,” Phys. Rev. B , 205136 (2014).[59] Bo Lu, Keiji Yada, Masatoshi Sato, and Yukio Tanaka,“Crossed Surface Flat Bands of Weyl Semimetal Supercon-ductors,” Phys. Rev. Lett. , 096804 (2015). [60] Shingo Kobayashi, Yukio Tanaka, and Masatoshi Sato, “Frag-ile surface zero-energy flat bands in three-dimensional chiralsuperconductors,” Phys. Rev. B , 214514 (2015).[61] Shingo Kobayashi, Youichi Yanase, and Masatoshi Sato,“Topologically stable gapless phases in nonsymmorphic su-perconductors,” Phys. Rev. B , 134512 (2016).[62] Tom´aˇs Bzduˇsek and Manfred Sigrist, “Robust doubly chargednodal lines and nodal surfaces in centrosymmetric systems,”Phys. Rev. B , 155105 (2017).[63] Shuntaro Sumita, Takuya Nomoto, Ken Shiozaki, and YouichiYanase, “Classification of topological crystalline supercon-ducting nodes on high-symmetry lines: Point nodes, linenodes, and bogoliubov fermi surfaces,” Phys. Rev. B ,134513 (2019).[64] Sunje Kim and Bohm-Jung Yang, “Linking structures of dou-bly charged nodal surfaces in centrosymmetric superconduc-tors,” (2020), arXiv:2012.08908 [cond-mat.mes-hall].[65] Hoi Chun Po, Ashvin Vishwanath, and Haruki Watanabe,“Symmetry-based indicators of band topology in the 230space groups,” Nat. Commun. , 50 (2017).[66] Barry Bradlyn, L. Elcoro, Jennifer Cano, M. G. Vergniory,Zhijun Wang, C. Felser, M. I. Aroyo, and B. AndreiBernevig, “Topological quantum chemistry,” Nature , 298–305 (2017).[67] Haruki Watanabe, Hoi Chun Po, and Ashvin Vishwanath,“Structure and topology of band structures in the 1651 mag-netic space groups,” Sci. Adv. , eaat8685 (2018).[68] Zhida Song, Tiantian Zhang, and Chen Fang, “Diagnosis forNonmagnetic Topological Semimetals in the Absence of Spin-Orbital Coupling,” Phys. Rev. X , 031069 (2018).[69] Eslam Khalaf, Hoi Chun Po, Ashvin Vishwanath, and HarukiWatanabe, “Symmetry Indicators and Anomalous SurfaceStates of Topological Crystalline Insulators,” Phys. Rev. X ,031070 (2018).[70] Zhida Song, Tiantian Zhang, Zhong Fang, and Chen Fang,“Quantitative mappings between symmetry and topology insolids,” Nat. Commun. , 3530 (2018).[71] Seishiro Ono and Haruki Watanabe, “Unified understanding ofsymmetry indicators for all internal symmetry classes,” Phys.Rev. B , 115150 (2018).[72] Hoi Chun Po, “Symmetry indicators of band topology,” Jour-nal of Physics: Condensed Matter , 263001 (2020).[73] Jennifer Cano and Barry Bradlyn, “Band Representations andTopological Quantum Chemistry,” Annual Review of Con-densed Matter Physics , null (2021).[74] Luis Elcoro, Benjamin J. Wieder, Zhida Song, Yuanfeng Xu,Barry Bradlyn, and B. Andrei Bernevig, “Magnetic topolog-ical quantum chemistry,” (2020), arXiv:2010.00598 [cond-mat.mes-hall].[75] Seishiro Ono, Youichi Yanase, and Haruki Watanabe, “Sym-metry indicators for topological superconductors,” Phys. Rev.Res. , 013012 (2019).[76] Anastasiia Skurativska, Titus Neupert, and Mark H. Fischer,“Atomic limit and inversion-symmetry indicators for topolog-ical superconductors,” Phys. Rev. Research , 013064 (2020).[77] Ken Shiozaki, “Variants of the symmetry-based indicator,”(2019), arXiv:1907.13632 [cond-mat.mes-hall].[78] Seishiro Ono, Hoi Chun Po, and Haruki Watanabe, “Re-fined symmetry indicators for topological superconductors inall space groups,” Science Advances , eaaz8367 (2020).[79] Max Geier, Piet W. Brouwer, and Luka Trifunovic,“Symmetry-based indicators for topological Bogoliubov–deGennes Hamiltonians,” Phys. Rev. B , 245128 (2020). [80] Seishiro Ono, Hoi Chun Po, and Ken Shiozaki, “ Z -enrichedsymmetry indicators for topological superconductors in the1651 magnetic space groups,” (2020), arXiv:2008.05499[cond-mat.supr-con].[81] Sheng-Jie Huang and Yi-Ting Hsu, “Faithful derivation ofsymmetry indicators: A case study for topological supercon-ductors with time-reversal and inversion symmetries,” (2020),arXiv:2010.05947 [cond-mat.supr-con].[82] L. Michel and J. Zak, “Connectivity of energy bands in crys-tals,” Phys. Rev. B , 5998–6001 (1999).[83] Michel, L. and Zak, J., “Elementary energy bands in crys-talline solids,” Europhys. Lett. , 519–525 (2000).[84] L. Michel and J. Zak, “Elementary energy bands in crystals areconnected,” Physics Reports , 377 – 395 (2001), symmetry,invariants, topology.[85] Jorrit Kruthoff, Jan de Boer, Jasper van Wezel, Charles L.Kane, and Robert-Jan Slager, “Topological Classification ofCrystalline Insulators through Band Structure Combinatorics,”Phys. Rev. X , 041069 (2017).[86] Feng Tang, Hoi Chun Po, Ashvin Vishwanath, and XiangangWan, “Efficient topological materials discovery using symme-try indicators,” Nat. Phys. , 470–476 (2019).[87] Feng Tang, Hoi Chun Po, Ashvin Vishwanath, and XiangangWan, “Topological materials discovery by large-order symme-try indicators,” Sci. Adv. , eaau8725 (2019).[88] Tiantian Zhang, Yi Jiang, Zhida Song, He Huang, YuqingHe, Zhong Fang, Hongming Weng, and Chen Fang, “Cata-logue of topological electronic materials,” Nature , 475–479 (2019).[89] M. G. Vergniory, L. Elcoro, Claudia Felser, Nicolas Regnault,B. Andrei Bernevig, and Zhijun Wang, “A complete catalogueof high-quality topological materials,” Nature , 480–485(2019).[90] Feng Tang, Hoi Chun Po, Ashvin Vishwanath, and XiangangWan, “Comprehensive search for topological materials usingsymmetry indicators,” Nature , 486–489 (2019).[91] Yuanfeng Xu, Luis Elcoro, Zhi-Da Song, Benjamin J. Wieder,M. G. Vergniory, Nicolas Regnault, Yulin Chen, ClaudiaFelser, and B. Andrei Bernevig, “High-throughput calcula-tions of magnetic topological materials,” Nature , 702–707(2020).[92] T. Shang, M. Smidman, A. Wang, L.-J. Chang, C. Baines,M. K. Lee, Z. Y. Nie, G. M. Pang, W. Xie, W. B. Jiang,M. Shi, M. Medarde, T. Shiroka, and H. Q. Yuan, “Simul-taneous Nodal Superconductivity and Time-Reversal Symme-try Breaking in the Noncentrosymmetric Superconductor CaP-tAs,” Phys. Rev. Lett. , 207001 (2020).[93] Alexander Altland and Martin R. Zirnbauer, “Nonstandardsymmetry classes in mesoscopic normal-superconducting hy-brid structures,” Phys. Rev. B , 1142–1161 (1997).[94] Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki, and An-dreas W. W. Ludwig, “Classification of topological insulatorsand superconductors in three spatial dimensions,” Phys. Rev.B , 195125 (2008).[95] Alexei Kitaev, “Periodic table for topological insulators andsuperconductors,” AIP Conference Proceedings , 22–30(2009).[96] Shinsei Ryu, Andreas P Schnyder, Akira Furusaki, and An-dreas W W Ludwig, “Topological insulators and superconduc-tors: tenfold way and dimensional hierarchy,” New Journal ofPhysics , 065010 (2010).[97] (), Our method can be applied to the normal phases withoutany modification. To make the presentation coherent, we focuson the superconducting phases in this paper. [98] Christopher John Bradley and Arthur P Cracknell, The Math-ematical Theory of Symmetry in Solids (Oxford UniversityPress, 1972).[99] Ken Shiozaki, Masatoshi Sato, and Kiyonori Gomi, “Atiyah-Hirzebruch Spectral Sequence in Band Topology: GeneralFormalism and Topological Invariants for 230 Space Groups,”(2018), arXiv:1802.06694 [cond-mat.str-el].[100] (), Mathematically, the set of Z - and Z -valued quantities isnot true vector, but we nevertheless refer to it as vector.[101] Eyal Cornfeld and Adam Chapman, “Classification of crys-talline topological insulators and superconductors with pointgroup symmetries,” Phys. Rev. B , 075105 (2019).[102] Ken Shiozaki, “The classification of surface states of topolog-ical insulators and superconductors with magnetic point groupsymmetry,” (2019), arXiv:1907.09354 [cond-mat.mes-hall].[103] Xiao-Liang Qi, Taylor L. Hughes, and Shou-Cheng Zhang,“Topological invariants for the Fermi surface of a time-reversal-invariant superconductor,” Phys. Rev. B , 134508(2010).[104] Masatoshi Sato, “Topological odd-parity superconductors,”Phys. Rev. B , 220504(R) (2010).[105] Liang Fu and Erez Berg, “Odd-Parity Topological Supercon-ductors: Theory and Application to Cu x Bi Se ,” Phys. Rev.Lett. , 097001 (2010).[106] Paolo Giannozzi, Stefano Baroni, Nicola Bonini, MatteoCalandra, Roberto Car, Carlo Cavazzoni, Davide Ceresoli,Guido L Chiarotti, Matteo Cococcioni, Ismaila Dabo, An-drea Dal Corso, Stefano de Gironcoli, Stefano Fabris, GuidoFratesi, Ralph Gebauer, Uwe Gerstmann, Christos Gougous-sis, Anton Kokalj, Michele Lazzeri, Layla Martin-Samos,Nicola Marzari, Francesco Mauri, Riccardo Mazzarello, Ste-fano Paolini, Alfredo Pasquarello, Lorenzo Paulatto, Carlo Sbraccia, Sandro Scandolo, Gabriele Sclauzero, Ari P Seit-sonen, Alexander Smogunov, Paolo Umari, and Renata MWentzcovitch, “QUANTUM ESPRESSO: a modular andopen-source software project for quantum simulations of ma-terials,” Journal of Physics: Condensed Matter , 395502(2009).[107] P Giannozzi, O Andreussi, T Brumme, O Bunau, M Buon-giorno Nardelli, M Calandra, R Car, C Cavazzoni, D Ceresoli,M Cococcioni, N Colonna, I Carnimeo, A Dal Corso,S de Gironcoli, P Delugas, R A DiStasio, A Ferretti,A Floris, G Fratesi, G Fugallo, R Gebauer, U Gerstmann,F Giustino, T Gorni, J Jia, M Kawamura, H-Y Ko, A Kokalj,E K¨uc¸ ¨ukbenli, M Lazzeri, M Marsili, N Marzari, F Mauri,N L Nguyen, H-V Nguyen, A Otero de-la Roza, L Paulatto,S Ponc´e, D Rocca, R Sabatini, B Santra, M Schlipf, A PSeitsonen, A Smogunov, I Timrov, T Thonhauser, P Umari,N Vast, X Wu, and S Baroni, “Advanced capabilities formaterials modelling with Quantum ESPRESSO,” Journal ofPhysics: Condensed Matter , 465901 (2017).[108] Akishi Matsugatani, Seishiro Ono, Yusuke Nomura, andHaruki Watanabe, arXiv:2006.00194.[109] Shengyuan A. Yang, Hui Pan, and Fan Zhang, “Dirac andWeyl Superconductors in Three Dimensions,” Phys. Rev. Lett. , 046401 (2014).[110] Fan Zhang, C. L. Kane, and E. J. Mele, “Topological mirrorsuperconductivity,” Phys. Rev. Lett. , 056403 (2013).[111] Tiantian Zhang, Ling Lu, Shuichi Murakami, Zhong Fang,Hongming Weng, and Chen Fang, “Diagnosis schemefor topological degeneracies crossing high-symmetry lines,”Phys. Rev. Research2