Symmetry indicators for topological superconductors

The systematic diagnosis of band topology enabled by the method of"symmetry indicators"underlies the recent advances in the search for new materials realizing topological crystalline insulators. Such an efficient method has been missing for superconductors because the band structure in the superconducting phase is not usually available. In this work, we establish symmetry indicators for weak-coupling superconductors that detect nontrivial topology based on the representations of the metallic band structure in the normal phase, assuming a symmetry property of the gap function. We demonstrate the applications of our formulas using examples of tight-binding models and density-functional-theory band structures of realistic materials.


I. INTRODUCTION
In recent years, topological superconductors (SCs) have been actively investigated because Majorana fermions that emerge at vortex cores and on surfaces of topological SCs are promising building blocks of quantum computers [1][2][3] . Intensive experimental efforts have obtained strong indications for topological superconductivity realized in artificial structures by superconducting proximity effect [4][5][6] . Further searches for intrinsic topological SCs in crystalline solids are actively ongoing issues. In addition to the topological superconductivity protected by local symmetries 7-9 the topological crystalline superconductivity [10][11][12][13][14] and higher-order topological superconductivity may be realized in crystalline systems. In previous works a method suitable for a candidate material was used [15][16][17] . A systematic theory to clarify topological properties of vast SCs is awaited.
Recently, there have been fundamental advances in the method of symmetry indicators [18][19][20][21][22] and in a similar formalism 23 , which provide an efficient way to diagnose the topology of band insulators and semimetals based on the representations of valence bands at high-symmetry momenta. This scheme can be understood as a generalization of the Fu-Kane formula 24 that computes the Z 2 -indices in terms of inversion parities to arbitrary (magnetic) space groups [25][26][27] and a wider class of topologies including higher-order ones [28][29][30][31][32][33] . It formed the basis of recent extensive material searches based on the density functional theory (DFT) calculation by several groups that resulted in the discovery of an enormous number of new topological materials 34-36 . Up to this moment, however, symmetry indicators are applicable only to insulators and semimetals in which a fixed number of valence bands exist below the Fermi level at every high-symmetry momentum. If one wants to apply this method to SCs, one must examine the representations in the band structure of the Bogoliubov-de Gennes (BdG) Hamiltonian including a gap function. In fact, this is the approach taken in Ref. 37 that recently extended the symmetry indicators to the 10 Altland-Zirnbauer symmetry classes [7][8][9] . However, this is not ideal because such a band structure is not available in the standard DFT calculation. Furthermore, in this way, the total number of bands that have to be taken into account can be huge unless one uses an effective tight-binding model.
In this work, we further develop the theory of symmetry indicators exclusively designed for weak-coupling SCs. It enables us to determine the topology of SCs based on the representations of a finite number of bands below the Fermi surface in the normal phase, although one still has to assume a symmetry transformation property of the gap function. This is a generalization of the famous criterion 15,38,39 that an odd-parity SC with the inversion symmetry is topological when the number of connected Fermi surfaces is odd. Our refined criterion finds that an odd-parity SC can be topological even when the number of Fermi surfaces is even as we demonstrate in Fig. 2 below using a concrete model. We also apply our formulas to DFT band structures of several realistic materials to confirm the usefulness of symmetry indicators in the theoretical and experimental search of topological SCs.

II. SYMMETRY OF BOGOLIUBOV-DE GENNES HAMILTONIAN
Our discussion in this work is based on the BdG Hamiltonian with the particle-hole symmetry Ξ = τ x K: where the gap function ∆ k satisfies ∆ T k = −∆ −k . The BdG Hamiltonian H BdG k describes the band structure of the superconducting phase [ Fig. 1 (b)], while H k encodes the band structure in the normal phase [ Fig. 1 (a)]. We assume a band gap around E = 0 in the superconducting phase at least at every high-symmetry momentum as illustrated in Fig. 1 (b). To simplify the analysis, we also set the Fermi level E F in the normal phase to be 0.
Let us review the spatial symmetry of H BdG k . Let U k (g) be a unitary matrix representing an element g of a space group G. When H k and ∆ k obey the conditions U k (g)H k U k (g) † = H gk and U k (g)∆ k U k (g) t = χ g ∆ gk , the symmetry representation for the BdG Hamiltonian is given by and satisfies U BdG The U (1) phase χ g must form a linear representation of G that char-arXiv:1811.08712v1 [cond-mat.supr-con] 21 Nov 2018 If TRS is unbroken in the superconducting phase, χ g must be either ±1 for all g ∈ G.

III. SYMMETRY INDICATORS FOR SUPERCONDUCTORS
The data required for computing the symmetry indicator in the superconducting phase is the collection of (n α k ) BdG that counts the number of occurrence of u α k (g) below E = 0 at each high-symmetry momentum k [red dots in Fig. 1(b)]. Here, u α k (g) (α = 1, 2, . . .) are irreducible representations of the little group G k . However, when the matrix size of H k is N , the total number of bands below E = 0 in the superconducting phase is also N . This is unfavorable, since N can be arbitrary large because of the existence of irrelevant highenergy bands far above the Fermi level in the normal phase.
To avoid this difficulty we reduce the input data to the representations of the occupied bands in the normal phase [blue dots in Fig. 1(a)]. To this end, we have to introduce the so-called "weak-pairing assumption" following previous works 15,38,39,41,42 , which states that (n α k ) BdG in the superconducting phase does not change even if the limit ∆ k → 0 is taken. (This assumption is usually valid 15,38,39,41,42 ; to our knowledge, there are no exceptions.) In this limit, one sees from Eqs. (1) and (2) that eigenstates of H BdG k and their representations can be deduced from those of H ±k . Let ψ n,k be an eigenstate of H k with the energy n,k belonging to the representation u α k (g) of G k . Then, ψ * n,−k is an eigenstate of −H * −k with the energy − n,−k that belongs to the representation of G k . Equation (3) defines an one-to-one map f k among irreducible representations of G ±k , which can be inverted as The above observation implies that there are two contributions to (n α k ) BdG : one from occupied bands of H k and the other from unoccupied bands of H * −k . Let n α k occ.
(n α k unocc. ) be the number of occurrence of u α k (g) in the occupied (unoccupied) bands of H k . Then, we find Relying on the fact that the band insulator that completely fills all bands is topologically trivial, we can drop the last term, as far as the symmetry indicators are concerned. After all, we get This is the main theoretical result of this work, which enables us to compute (n α k ) BdG of the superconducting phase solely by the occupied bands of the normal phase. Note that we are dealing with a metallic band structure and n α k here by itself does not necessarily satisfy the compatibility relations unlike (n α k ) BdG .

IV. USEFUL FORMULAS
Let us translate the general formula in Eq. (5) into more convenient forms in applications.

A. Inversion with TRS
We start with the inversion symmetry I. According to Eq. (3), in this case we have n , since the eigenvalues of I is either ±1 and −k is equivalent with k at time-reversal invariant momenta (TRIMs). Therefore, for even-parity SCs (χ I = +1), (n α k ) BdG n α k − n αχ I k always vanishes and all indicators are trivial. On the other hand, for odd-parity (χ I = −1) SCs with TRS (class DIII), the Z 2 weak indices ν BdG i (i = 1, 2, 3) 24 and the Z 4 strong index κ BdG 1 20,21 can be computed as k∈2D TRIMs α=±1 αn α k is the sum of the inversion parities of occupied bands over the four appropriate TRIMs (divided by four) andκ 1 ≡ 1 4 k∈3D TRIMs α=±1 αn α k is the same but over all eight TRIMs (divided by four). Note thatν i andκ 1 here can be a half-integer, since the band structure in the normal phase is allowed to be metallic. The even/oddness of κ BdG 1 agrees with the 3D winding number W in class DIII modulo 2.
For example, whenκ 1 is a half-integer, κ BdG 1 is odd and W must also be odd. This occurs when the number of connected Fermi surfaces is odd, which is consistent with the criterion in the previous studies 15,38,39 . More interesting scenario is wheñ κ 1 is an odd integer, leading to κ BdG 1 = 2 (mod 4), while all weak indices vanishes. Although this case has been classified to the trivial category according to the criterion in Refs. 15, 38, and 39, it still exhibits a nontrivial, possibly higher-order topology as we demonstrate now through an example.
Let us introduce a toy lattice model of 3 He B-phase, given by H BdG where σ = (σ x , σ y , σ z ) is the Pauli matrix and σ 0 is the identity matrix. Below we set t = µ = ∆ = 1. The model has the inversion symmetry U k (I) = σ 0 and the TRS U T = −iσ y . Only the Γ ≡ (0, 0, 0) point is occupied by the two evenparity bands. We thus getκ 1 = 1 2 , which implies that W is odd. We indeed find W = ±1 depending on ξ = ±1.
To realize the case withκ 1 = 1, let us take two copies of this model: where V = −im · σ (|m| < 1) represents a perturbation respecting both the inversion and TRS and we set m =  Fig. 2 (a,b)]. When we choose ∆ (+,+) k for the gap function, the winding W = 2 implies that the 2D surface is gapless [see Fig. 2 (c,d)]. In fact, this case has co-existing 2D surface modes together with 1D hinge modes, just like the model  Fig. 2 (e,f) 33 . In either case, we observe symmetry-protected gapless states.

B. Inversion without TRS
Although our focus in this work is mainly on fully gapped SCs, our theory can also be equally applied to nodal SCs as far as the nodes do not locate at high-symmetry points in the Brillouin zone. For example, let us again discuss the sum of inversion parities over eight TRIMs but this time without assuming TRS (class D) 37 : This Z 4 strong index can be nontrivial only for odd-parity SCs. When µ BdG 1 is odd, there must be at least a pair of Weyl nodes in the gap function, which is the superconducting generalization of the phenomena pointed out in Ref. 44. This happens when the sum of inversion parities is odd in the normal phase in 3D. I. Formulas for diagnosing (mirror) Chern numbers for SCs based on the n-fold rotation eigenvalues (n = 2, 3, 4, and 6). R and ∆ are defined in Table II. When Mxy exists, Rσ and ∆σ for each mirror sector σ = ±i are defined in the same way. If the normal phase has TRS, R+i = (R−i) * and ∆+i = ∆−i. An example is provided by a 3D extension of the chiral pwave SC:

Symmetry (mirror) Chern numbers
There is only one band in the normal phase and it occupies only the Γ point with an even parity [ Fig. 3 (a)]. Hence, the sum of inversion parities in the normal phase is +1. There is, indeed, a pair of Weyl points at k z = ± π 2 as illustrated in Fig. 3 (b).

C. Rotation
Next, let us discuss formulas diagnosing the (mirror) Chern numbers based on n-fold rotation eigenvalues following Refs. 41 and 45. We summarize our results in Tables I and  II, which enable us to determine the (mirror) Chern numbers of SCs modulo n using the rotation eigenvalues in the normal phase. There are additional constrains on mirror Chern numbers, such as C +i = C −i when χ Mxy = +1 and C +i = −C −i when TRS is unbroken in the superconducting phase. If representations are not consistent with them, the gap |∆ k | must vanish at some k resulting in a nodal SC.
The simplest example is given by the k z = 0 plane of the chiral p-wave SC in Eqs. (12) and (13). The model has C 4rotation symmetry with U k (C 4 ) = 1 and χ C4 = i. Recalling that there is only one band and it occupies only Γ, we apply the formula for n = 4 as R = 1×1/1 = 1 and ∆ = 2×0−1−0 = −1. Hence, we find e 2πi 4 C = i −1 ×1 2 = −i. This agrees with the actual value of C = −1.
Next, let us discuss the tight-binding model used in Fig. 1. We present the details of the model and the symmetry representations in Appendix. It has a C 4 -rotation symmetry, a mirror symmetry M xy , and TRS. Based on the band structure in Fig. 1, we get R +i = (R −i ) * = e   Table I. Here, ζ k , θ k , ξ k , and η k respectively represent the product of 2, 3, 4, and 6-fold rotation eigenvalues over all occupied bands at k. N k is the total number of occupied bands at k.

V. DFT CALCULATIONS
Let us apply our method to more realistic DFT band structures. Our ab initio calculations are performed using WIEN2K 46 and all material information is taken from "Materials Project" 47 . We include the band structure and n α k used in the calculation in Appendix.
Our first example is β-PdBi 2 . Let us assume that (i) the inversion and TRS remain unbroken and that (ii) a full gap 48 with odd inversion parity (χ I = −1) opens 49 in the superconducting phase. Then, by summing up the inversion parities in the normal phase, we get κ BdG 1 = 3 (mod 4). This implies a nontrivial 3D winding number W of class DIII and is consistent with the observation of topological surface states 50,51 .
Similarly, we find κ BdG 1 = 3 (mod 4) for a doped material Ir 1−x Pt x Te 2 (x = 0.05) 52-55 under the same assumptions. In the DFT calculation we set x = 0. Although the undoped compound (IrTe 2 ) is reported as a topological SC 56 , there exists a structural phase transition before the superconducting transition [56][57][58] . Thus the indicator for x = 0 may not be valid in the undoped case.
Finally, let us discuss Sr 2 RuO 4 . There are many proposals of the specific form of the gap function for this material 17,59-61 .
Here we consider the two possibilities studied in Ref. 17. The space group of the two superconducting phases is I4/m. In order to avoid complications using the body-centered lattice, let us use the Brillouin zone of the primitive lattice. Assuming χ Mxy = +1 and χ C4 = −i frist, we get C +i = C −i = +2 (mod 4) on both k z = 0 and k z = π planes. Using χ Mxy = −1 and χ C4 = +1 instead, we get C +i = −C −i = +2 (mod 4) on the k z = 0 plane and C +i = C −i = 0 (mod 4) on the k z = π plane. These results are consistent with the understanding as stacked layers of RuO 2 planes with mirror Chern numbers discussed before.

VI. CONCLUSION
In this work, we extended the theory of symmetry indicators for weak-coupling SCs and derived several useful formulas in the search for new topological SCs. Our general results, such as Eqs. (6) and (11) and Tables I and II, enable us to determine the topology of SCs based on the information of representations, n α k , of occupied bands in the normal phase and the symmetry property χ g of the assumed gap function.
In addition to the comprehensive material investigations through DFT calculation [34][35][36] , the field of materials informatics has been developing rapidly [62][63][64] due to the progress of machine learning and used to identify new SCs 65,66 . Our sym-metry indicators for SCs established in this work can be easily combined with these techniques and should lead to the discovery of many more topological SCs.

Appendix B: Sr2RuO4
Here provide the three-orbit tight-binding model on a 2D square lattice 40 used in Fig. 1 in the main text.
The band structure is computed with the choice of parameters taken from Ref. 17: The four-fold rotation symmetry C 4 , the mirror symmetry M xy , and inversion symmetry I are represented by which satisfy In addition, the tight-binding model has two more mirror symmetries broken in the superconducting phase: In Table III, we summarize the rotation eigenvalues of the occupied bands in the normal phase.
Appendix C: Transformation properties under spin rotation Here we explain the transformation property of the perturbation term V = −im · σ considered in the main text. Let us consider a rotation by an angle θ about an axis n. The SO(3) matrix representation of the rotation is given by p = e −iθL·n , where L = (L 1 , L 2 , L 3 ) t is the matrix representation of the angular momentum and (L i ) jk = −i ijk ( is the Levi-Civita tensor). The corresponding spin rotation is given by p sp = e −iθS·n , where S = 1 2 σ and σ is the Pauli matrix. Let us define V m ≡ m · σ. It satisfies p sp V m p −1 sp = V pm , meaning that m transforms as a (pseudo-)vector. Similarly, ∆ d ≡ (d · σ)iσ y satisfies p sp ∆ d p t sp = ∆ pd . Thus d in ∆ d also transforms as a (pseudo-)vector (but remember p −1 sp is replaced by p t sp ) .