Refined symmetry indicators for topological superconductors in all space groups.

Topological superconductors are exotic phases of matter featuring robust surface states that could be leveraged for topological quantum computation. A useful guiding principle for the search of topological superconductors is to relate the topological invariants with the behavior of the pairing order parameter on the normal-state Fermi surfaces. The existing formulas, however, become inadequate for the prediction of the recently proposed classes of topological crystalline superconductors. In this work, we advance the theory of symmetry indicators for topological (crystalline) superconductors to cover all space groups. Our main result is the exhaustive computation of the indicator groups for superconductors under a variety of symmetry settings. We further illustrate the power of this approach by analyzing fourfold symmetric superconductors with or without inversion symmetry and show that the indicators can diagnose topological superconductors with surface states of different dimensionalities or dictate gaplessness in the bulk excitation spectrum.

INTRODUCTION

Unconventional pairing symmetry in a superconductor indicates a departure from the well-established Bardeen-Cooper-Schrieffer (BCS) paradigm for superconductivity. Such systems, exemplified by the high-temperature superconductors like the cuprate, typically display a wealth of intricate, oftentimes mysterious, phenomena that are of great theoretical, experimental, and technological interest (). The physics of unconventional superconductors has gained a new dimension in the past decade, thanks to the bloom in the understanding of topological quantum materials (–). A hallmark of topological superconductors (TSCs) is the presence of robust surface states that correspond to Majorana fermions—an exotic emergent excitation that can loosely be described as being half of an ordinary electron. These Majorana excitations might be harvested for topological quantum computation, and much effort has been paid to the experimental realization of such exotic phases of matter ().

The intense research effort on topological quantum materials has resulted in an ever increasing arsenal of experimentally verified topological (crystalline) insulators and semimetals, but the discovery of TSCs has proven to be much more challenging. The theoretical landscape, however, has evolved rapidly in recent years. On the one hand, the complex problem of how the diverse set of spatial symmetries in a crystal can both prohibit familiar topological phases and protect new ones has largely been solved, with the theoretical efforts culminating in the production of general classifications for topological crystalline phases in a variety of symmetry settings (–). On the other hand, general theories for how crystalline symmetries can be used to identify topological materials have been developed (, ). In particular, the method of symmetry indicators (SIs) () has enabled comprehensive surveys of topological materials among existing crystal structure databases, and thousands of materials candidates have been uncovered (–).

It is natural to ask if the theory of SIs could be used to facilitate the discovery of TSCs. There are two main difficulties: First, unconventional superconductivity emerges out of strong electronic correlations, and for such systems, theoretical treatments using different approximation schemes rarely converge to the same answers. Such debates could only be settled by meticulous experimental studies, which could take years to be completed. Second, even within the simplifying assumption that a mean-field Bogoliubov–de Gennes (BdG) provides a satisfactory treatment for the system, the original theory of SIs falls short in identifying key examples of TSCs like the one-dimensional (1D) Kitaev chain (–) and its higher-dimensional analogs like the higher-order TSCs in 2D (, ). We remark that alternative formulas relating the signs of the pairing order parameters on different Fermi surfaces and topological invariants also exist in the literature, but this approach requires more detailed knowledge on the system than just the symmetry representations (). Furthermore, the extension of these formulas for other crystalline and higher-order TSCs has only been achieved for specific examples (–).

In this work, we address the second part of the problem by extending the theory of SIs to the study of TSCs described by a mean-field BdG Hamiltonian in any space group. This is achieved by a refinement of the SI for TSCs, which was previously proposed in (, ) and analyzed explicitly for inversion-symmetric systems. Technically, our results do not rely on the weak pairing assumption, which states that the superconducting gap scale is much smaller than the normal-state bandwidth (, , , ). In practice, however, the prediction from this method is most reliable when the assumption is valid. For such weakly paired superconductors, only two pieces of data are required to diagnose a TSC: (i) the normal-state symmetry representations of the filled bands at the high-symmetry momenta and (ii) the pairing symmetry.

Our key result is the exhaustive computation of the refined SI groups for superconductors with or without time-reversal symmetry and spin-orbit coupling, which are tabulated in section S1. In the main text, we will first review the topology of superconductors (“Topology of superconductors” section), followed by the “Refined symmetry indicators for superconductors” section, in which we give an interpretative elaboration for the SI refinement proposed in (, ). As an example of the results, we will provide an in-depth discussion on the refined SIs for class DIII systems with C4 rotation symmetry in the “Interpretation of computed symmetry indicators for superconductors” section, and a summary of the SIs for other key symmetry groups is provided in section S2.

Curiously, we discover that the refined C4 SI is, like the Fu-Kane parity formula () and the corresponding version for odd-parity TSC (, ), linked to the ℤ2 quantum spin Hall (QSH) index in the 10-fold way classification of TSC. This link is established in the “Indicators for Wannierizable topological superconductors” section and is perhaps surprising given the SI refinement captured TSCs with corner modes in systems with inversion symmetry (, ). To our knowledge, this also represents the first instance of diagnosing a QSH phase using a proper rotation symmetry. Instead of a reduction in the wave function–based formula for the topological index to the symmetry representations, as was done in the original Fu-Kane approach (), our argument relies on an introduction of a class of phases that we dub “Wannierizable TSCs” (WTSCs). We will conclude and highlight a few future directions in Discussion.

THEORETICAL FRAMEWORK

Topology of superconductors

In this section, we review the framework of describing TSCs by BdG Hamiltonians as a preparation for formulating SIs in the “Refined symmetry-indicators for superconductors” section. Our discussion elucidates the possibility of marginally topological SCs, which may be called fragile TSCs.

Symmetry of BdG Hamiltonian

Let us consider the Hamiltonian H of the normal phase, which we assume to be a D-dimensional Hermitian matrix. We take a superconducting gap function Δ that satisfies , which is also a square matrix with the same dimension. The parameter ξ can be either +1 or −1 depending on the physical realization. We then form the 2D-dimensional BdG Hamiltonian

This Hamiltonian always has the particle-hole symmetryHere stands for the D-dimensional identity matrix. Throughout this work, all blank entries of a matrix should be understood as 0. The particle-hole symmetry in Eq. 3 satisfies . To see the consequence of the particle-hole symmetry, suppose that is an eigenstate of with an eigenvalue E. Then, the particle-hole symmetry implies that is an eigenstate of with eigenvalue −E. We call the eigenvalue E the quasiparticle spectrum. The BdG Hamiltonian is gapped when the quasiparticle spectrum has a gap around E = 0 for all .

Suppose that the Hamiltonian of the normal phase has a space group symmetry G. Each element g ∈ G is represented by a unitary matrix U(g) that satisfiesIf the gap function satisfiesthe spatial symmetry is encoded in the BdG Hamiltonian asThe 1D representation χ of G defines the symmetry property of the superconducting gap Δ.

Last, the BdG Hamiltonian has the time-reversal symmetry if there exists such thatThe representation χ must be either ±1 for all g ∈ G. Then, the representation of the time-reversal symmetry in the BdG Hamiltonian is

When ξ = +1, which is usually the case for electrons, the BdG Hamiltonians without time-reversal symmetry fall into class D of the 10-fold Altland-Zirnbauer (AZ) symmetry classification. When the time-reversal symmetry is present and satisfies , the symmetry class becomes BDI, and when it satisfies instead, the symmetry class is DIII. If we discard the particle-hole symmetry from class D, BDI, and DIII, they respectively reduce to class A, AI, and AII. In the presence of spin SU(2) symmetry for spinful electrons, ξ effectively becomes −1 (, ). Then, the system without time-reversal symmetry is class C, and with the time-reversal symmetry is class CI. We can formally consider the case , which is classified as class CII, but it may be difficult to be realized in electronic systems. The general discussions of this work apply to all of these symmetry classes with the particle-hole symmetry regardless of ξ = +1 or −1.

Stacking of BdG Hamiltonians

To carefully define the trivial SCs, let us introduce the formal stacking of two SCs by the direct sum of two BdG Hamiltonians , in which H and Δ in Eq. 1 are respectively replaced withWhen the dimension of is D′, the stacked BdG Hamiltonian is 2(D + D′)–dimensional and has the particle-hole symmetry Ξ.

We furthermore assume that and have the same spatial symmetry G. Their representations can be different, but χ must be common. We define by replacing U(g) in Eq. 7 withThe possible time-reversal symmetry of the stacked SC is defined in the same way.

Trivial superconductors

Let us now define the topologically trivial class of SCs. Our discussion is inspired by the recent proposal in (, ).

Suppose that the BdG Hamiltonian is gapped. We say is strictly trivial when it can be smoothly deformed to eitherwhich describes the vacuum state where all electronic levels are unoccupied, orwhich represents the fully occupied state. They are physically equivalent to the chemical potential μ = ± ∞ limit of . Here, the smooth deformation is defined by an interpolating BdG Hamiltonian withthat maintains both the gap in the quasiparticle spectrum and all the assumed symmetries for all t ∈ [0,1]. Note that we do not modify the representations, such as Ξ, , and , of assumed symmetries during the process. When a smooth deformation to Hvac exists, we writeSimilarlywhen there is an adiabatic path to Hfull. Under a space group symmetry, conditions Eqs. 17 and 18 are generally inequivalent. The SC is strictly trivial when at least one of the two conditions are fulfilled. For example, the BCS superconductor with SU(2) symmetry, described byis strictly trivial. This can be seen by the interpolating Hamiltonian

The above definition of trivial SCs is, however, sometimes too restrictive, especially under a spatial symmetry. One instead has to allow for adding trivial degrees of freedom (DOFs). Using the notation summarized in the “Stacking of BdG Hamiltonians” section, we ask ifSee Fig. 1 for the illustration. The right-hand side of this equation is the same as Eq. 14, but the identity matrix is enlarged to . We leverage the freedom in the choice of the matrix size D′, D′′ and the symmetry representation , of the trivial DOFs. If, however, there does not exist any smooth path in Eq. 21 for whatever choice of trivial DOFs, then we say is stably topological. It might look unnatural to assign the flipped signs of between the left- and right-hand side of Eq. 21, but this choice is, in fact, necessary in the presence of space group symmetry in general. This point will become clear in the “Symmetry obstructions” section. Although Eq. 21 describes a smooth deformation to the vacuum state, one can equally consider a deformation to the fully occupied state, which is mathematically equivalent to Eq. 21 as long as trivial DOFs are freely chosen.

Fig. 1

Illustration of the equivalence relation in Eq. 21.

Electron-like states are colored in red, and hole-like states are colored in blue. States outside of the dashed box represent trivial DOFs included in the deformation process. See the “Refined symmetry indicators for superconductors” section for the definition of vectors in this figure.

These definitions of “strictly trivial SCs” and “stably topological SCs” leave a possibility of fragile topological phases (, ), which becomes trivial if and only if appropriate trivial DOFs are added. We will discuss examples of these cases in the “Examples” section.

Examples

As an example of what we explained so far, let us discuss the odd-parity SC in the Kitaev chain ().

Class D

The BdG Hamiltonian of a single Kitaev chain is given by Eq. withThis model falls into the ℤ2 nontrivial phase in class D and is stably topological.

Let us take two copies of this model by settingFor the doubled BdG Hamiltonian, there exists an adiabatic path for both Eqs. 17 and 18 given by the interpolating Hamiltonianwhich preserves the particle-hole symmetry and the gap in the quasiparticle spectrum. Therefore, the two copies of the Kitaev chain is strictly trivial.

Class D with inversion symmetry

Let us now take into account the inversion symmetry of the Kitaev chain. For the doubled model, the representation of the inversion symmetry is given by Eq. 7 withwhere χ = −1 indicates odd-parity pairing. Under the inversion symmetry, the adiabatic path in the sense of Eq. 17 or Eq. 18 no longer exists even for the doubled Kitaev model. This can be easily seen by looking at the inversion parity of the quasiparticle spectrum below E < 0. On the one hand, in the initial BdG Hamiltonian specified by Eq. 23, the inversion parity of two E < 0 levels is both +1 at k = 0 and −1 at k = π. (The opposite parity at k = 0 and π is a consequence of χ = −1.) On the other hand, in the final trivial Hamiltonian Hvac with D = 2, the inversion parities of E = −1 levels are all −1. If we use Hfull instead as the trivial Hamiltonian, the inversion parities are all +1. This mismatch of inversion parities serves as an obstruction for any inversion-symmetric adiabatic deformation. The path in Eq. 24 breaks the inversion symmetry for t ∈ (0,1).

To resolve the obstruction, we introduce trivial DOFs with an appropriate inversion property. Specifically, we set D′ = 2, D′′ = 0 andNow, the inversion parities of E < 0 levels on both sides of Eq. 21 agree: two +1’s and two −1’s at k = 0 and one +1 and three −1’s at k = π. There exists an interpolating HamiltonianThis confirms that the two copies of Kitaev chains with inversion symmetry becomes trivial if and only if proper trivial DOFs are added.

Refined symmetry indicators for superconductors

In this section, we discuss the formalism of SIs for SCs. Our goal is to systematically diagnose the topological properties of SCs described by BdG Hamiltonians using their space group representation. We also clarify the difference between the present approach extending the idea of (, ) and the previous approach in (, ).

Symmetry representations of BdG Hamiltonians

Let us consider a BdG Hamiltonian in Eq. 1 with a space group symmetry G represented by in Eq. 7. We assume that the spectrum of is gapped at the momentum for which we study its symmetry properties. Nevertheless, as we explain later, our framework can also be used to diagnose nodal SCs.

Suppose that is an eigenstate of and belongs to an irreducible representation of the little group G ≤ G of (α labels distinct irreducible representations). Then, the particle-hole symmetry implies that belongs to an irreducible representation of G−, which can be seen in Eq. 8. We write this correspondence among irreducible representations as

The SI is formulated in terms of integers that count the number of irreducible representations of G appearing in the E < 0 quasiparticle spectrum. That is, the little group representation formed by all the eigenstates with E < 0 can be decomposed into irreducible representations asWe say is a high-symmetry point if every point in a neighborhood of has a lower symmetry than (). Similarly, is a point belonging to a high-symmetry line (plane) if a neighborhood of contains a line (plane) that has the same symmetry as . The set of high-symmetry momenta K contains every high-symmetry point and a representative point from each of the high-symmetry lines and planes. Integers are not all independent as they obey compatibility relations as explained in (, , ). We compute for all α and ∈ K and form a vector BdG, whose components are given by .

For a later purpose, let us also define and using the E > 0 quasiparticle spectrum in the same way. Because of the particle-hole symmetry, we find

Next, let us examine a trivial BdG Hamiltonian Hvac in Eq. 14 for which the space group G is represented by the same matrix in Eq. 7 as for . Observe that E > 0 levels of Hvac use U(g) as the representation of G for every ∈ K. Similarly, E < 0 levels of Hvac use χ−(g)* as the representation of G. We define the integers and by the irreducible decompositionand construct vectors and vac, respectively, using integers and . By construction, we havesince both sides of this equation denote the total representation counts in .

Last, we consider additional trivial DOFs described bySuppose that the space group G is represented by in Hfull′.

We can perform the irreducible decomposition as in Eqs. 32 and 33 and define ′ and using the coefficients for U(g)′ and χ−(g)′*, respectively.

Symmetry obstructions

Now, we are ready to derive several obstructions for the smooth deformation in Eqs. 17, 18, and 25. A necessary (but not generally sufficient) condition for the existence of adiabatic paths in Eqs. 17 and 18 is, respectively,When both of these conditions are violated, the representation counts in the E < 0 spectrum of the initial and final BdG Hamiltonian in the deformation process do not agree and a smooth symmetric deformation is prohibited. We have seen this already in the doubled Kitaev model with inversion symmetry in the “Examples” section.

Similarly, comparing the representation counts in the E < 0 spectrum of the two ends of the adiabatic path of Eq. 21, we find the condition (see Fig. 1)Therefore, a necessary condition for this adiabatic deformation is the existence of ′, such thatThat is, the mismatch in Eq. 36 of the form can be resolved by including trivial DOFs. This is also what we have done for the doubled Kitaev model in the “Examples” section. Note that is canceled out from Eq. 39. Therefore, the trivial DOF in Eq. 21 with the same sign of on both sides of the equation does not help as far as space group representations are concerned.

Completeness of trivial limits

The above vector ′ corresponds to the atomic limit of an insulator in class A, AI, or AII depending on the assumption on the time-reversal symmetry in . As discussed in detail in (), there are generally a variety of distinct atomic insulators in the presence of spatial symmetries. An atomic insulator can be specified by the position of the localized orbitals and the orbital character. These choices specify a representation U(g)′ of G for each atomic insulator, and we write its representation count as (j labels distinct atomic insulators). The setis like a finite-dimensional vector space, except that the scalars are integers. We take a basis (i = 1,2, …, d) of {AI}.

Viewed as the representation counts in the valence bands of an insulator, it was proven in () that BdG can always be expanded in terms of ’s using fractional (or integer) coefficientsSince the left-hand side is integer valued, only special values of rational numbers are allowed. The relation Eq. 41 was the fundamental basis of the SIs for topological insulators. Here, we extend the argument for TSCs by proving that BdG − vac can always be expanded in the following formReaders not interested in the detail of the proof can skip to the “Quotient group” section.

To demonstrate Eq. 42, note first that vac belongs to {AI} and thus can be expanded asAlso, Eqs. 41 and 43 imply that and , which can be verified using Eq. 31. Then it follows thatThe second term vanishes because and because of Eq. 34. Therefore, c in Eq. 42 is given by (q − p)/2.

Quotient group

Given a BdG Hamiltonian with a set of assumed symmetries, we can separately compute BdG and vac and deduce BdG − vac. Distinct BdG Hamiltonians with the same symmetry setting may have different values of BdG − vac. Let us introduce the group {BS}BdG as the set of all possible BdG − vac realizable using a BdG Hamiltonian in this symmetry class.

The discussion in the “Symmetry obstructions” section clarified that, as far as the symmetry obstruction in Eq. 39 is concerned, the difference in {BS}BdG by the combination is unimportant. Hence, it makes sense to introduce the following subgroup of {BS}BdGWhen BdG − vac ∈ {BS}BdG does not belong to {AI}BdG, the condition Eq. 39 is violated and any smooth deformation in Eq. 21 is prohibited. Such nontrivial values of BdG − vac can be classified by the quotient groupThis is what we call the refined SI group in this work, which extends the idea in () to more general symmetry classes.

As we proved in the previous section, BdG − vac of a given BdG Hamiltonian can be expanded as Eq. 42. Conversely, a vector BdG − vac given in the form of right-hand side of Eq. 42 has a realization using some BdG Hamiltonian as far as BdG − vac is integer valued and is consistent with the time-reversal symmetry. This implies that XBdG takes the form ℤ × ℤ × ⋯ (i.e., it contains only torsion factors) and that the actual calculation of XBdG can be done by the Smith decomposition of {AI}BdG without explicitly constructing {BS}BdG ().

Relation to previous approach

In previous works (, ), BdG was viewed as the representation counts in the valence bands of an insulator and was analyzed in the same way as for class A, AI, or AII. In this approach, BdG is directly compared against atomic limits i (discussed in the “Completeness of trivial limits” section) of the same symmetry setting. When BdG cannot be written as a superposition of ’s with integer coefficients (i.e., BdG ∉ {AI}), then it is said to be nontrivial. This is a sufficient condition for violating all of Eqs. 36, 37, and 39. However, this requirement may be too strong in that, even when BdG ∈ {AI}, it would still be possible that BdG − vac ∉ {AI}BdG and BdG − vac belongs to the nontrivial class of XBdG. We will see an example of this in the “Example” subsection of the “Refined symmetry-indicators for superconductors” section.

Weak pairing assumption

When applying these methods in the actual search for candidate materials of TSCs, it would be more useful if the input data are only the representation count in the band structure of the normal phase described by H, not in the quasiparticle spectrum of . Such a reduction is achieved in (), relying on the weak pairing assumption (, , ). This assumption states that in the superconducting phase does not change even if the limit Δ → 0 is taken.

To explain how it works, let ψ be an eigenstate of H with the energy ϵ belonging to the representation of G. Then, the eigenstate of has the energy −ϵ− and the representation of G, defined in Eq. 29. Thus, representations appearing in the negative-energy quasiparticle spectrum of can be decomposed into the occupied bands (occ) of H and unoccupied bands (unocc) of H−:The last term of this expression is precisely defined in Eq. 33 This was pointed out recently by () for the case of inversion symmetry, and we see here that it applies to more general symmetry setting. After all, components of BdG − vac are given byThe last expression is purely the occupied band contribution of the normal-phase band structure, which may be calculated, for example, using the density functional theory ().

We remark that our sense of “weak pairing” is less stringent than that used in (), in that arbitrary inter-Fermi surface pairing is allowed so long as the normal-state energy at the high-symmetry momenta is sufficiently far away from the Fermi surface when compared with the pairing scale.

Example

As an example of SIs for SCs, let us discuss again the Kitaev chain, focusing on its inversion parities. Similar exercise has already been performed in (, ), but here we repeat it in our notation to clarify the difference in the present and previous approaches.

For the Kitaev chain with the inversion symmetry, the BdG Hamiltonian is given by with Eq. 22, and the symmetry representation is Eq. 7 with U(I) = 1 and χ = −1. For this model, we getwhere α = ± of corresponds to the inversion parity. The vacuum limit uses χ(I)* = −1 at both k = 0 and π and, thus, has vac = (0,1,0,1). Therefore, we find

For inversion symmetric 1D models in class A, {AI} is a 3D space spanned byFor these basis vectors, we findSince , {AI}BdG is a 2D space spanned by and . We findThe fractional coefficients imply the nontrivial topology of . That is, the quotient group in Eq. 46 is XBdG = ℤ2, and BdG − vac of the present model belongs to the nontrivial class of XBdG.

In contrast, we see thatMore generally, the quotient group in one dimension is always trivial for class A, AI, or AII (), meaning that all BdG vectors can be expanded by ’s with integer coefficients. This implies that one cannot detect the nontrivial topology of in 1D based on representations alone in the previous approach.

RESULTS

Interpretation of computed symmetry indicators for superconductors

Using the refined scheme explained in the “Refined symmetry indicators for superconductors” section, we perform a comprehensive computation of XBdG for all space groups G and 1D representations χ of superconducting gap functions. The full lists of the results are included in section S1 for both spinful and spinless electrons with or without time-reversal symmetry. The corresponding AZ symmetry classes are listed in Table 1. Most of the nontrivial entries of XBdG can be understood as supergroup of a countable number of key space groups discussed in section S2.

Table 1

Settings used in the calculation for refined SIs and the corresponding AZ symmetry classes.

Spin	Time-reversal symmetry	AZ classes
Spinful	−1	DIII
Spinless	+1	BDI, CI
Spinful	0	D
Spinless	0	D, C

In this section, we discuss the meaning of XBdG using two illuminating examples of G = P4 and P4/m in class DIII. Below, we write the component of BdG − vac asIn addition, we use the standard labeling of irreducible representations in the literature for χ ().

P4 with B representation

The space group P4 contains the fourfold rotation symmetry C4 in addition to the lattice translation symmetries. The B representation refers to the 1D representation of C4 with χ = −1.

In two spatial dimensions, we find XBdG = ℤ2. To see the meaning of this, let us introducewhere represents the number of irreducible representations in Eq. 57 with the fourfold rotation eigenvalue at Γ = (0,0) and M = (π, π). As we discuss in the “Indicators for Wannierizable topological superconductors” section, ν turns out to measure the ℤ2 QSH index. This was unexpected, because known diagnosis of the QSH index in class AII required either the inversion symmetry I () or the rotoinversion symmetry S4 (, ), and any proper rotation was not sufficient. Second-order TSCs with a corner Majorna zero mode are also stable under this symmetry setting but are not diagnosed by representations alone, as we discuss in the “Indicators for Wannierizable topological superconductors” section.

In three spatial dimensions, XBdG = ℤ2 detects the weak topological phase of 2D TSCs stacked along the rotation axis z. The strong ℤ2 phase of class DIII is prohibited because the ℤ2 index of kz = 0 and kz = π planes are forced to be the same by the rotation symmetry C4.

P4/m

The space group P4/m contains both the inversion I about the origin and the fourfold rotation C4 around the z axis. The mirror symmetry about the xy plane and fourfold rotoinversion symmetry are given as their products. There are four real 1D representations: A (χ = +1, χ = +1), A (χ = +1, χ = −1), B (χ4 = −1, χ = −1), and B (χ = −1, χ = +1). For the A representation, XBdG is trivial. We discuss the other three representations one by one. In this section, we denote six high-symmetry points by Γ = (0,0,0), Z = (0,0, π), X = (π,0,0), R = (π,0, π), and A = (π, π, π) ().

A representation (χ4 = +1, χ = −1)

Let us start with the A representation. Although XBdG in 3D is large (see Table 2), many factors can be attributed to lower dimensions.

Table 2

List of XBdG for class DIII systems with (), P4, and P4/m symmetry in each spatial dimension.

SG (rep of Δk)	1D	2D	3D
P1¯ (Au)	ℤ2	(ℤ2)2 × ℤ4	(ℤ2)3 × (ℤ4)3 × ℤ8
P4 (B)	ℤ1	ℤ2	ℤ2
P4/m (Au)	(ℤ2)2	ℤ2 × ℤ8	(ℤ2)4 × ℤ4 × ℤ8 × ℤ16
P4/m (Bu)	ℤ1	ℤ2 × ℤ8	ℤ2 × (ℤ4)2 × ℤ8
P4/m (Bg)	ℤ1	(ℤ2)2	(ℤ2)3

In a 1D system along the rotation axis, we find XBdG = (ℤ2)2, which can be characterized bywhere represents the number of irreducible representations with the C4 eigenvalue and the inversion parity β = ±1 at Γ and Z. They measure the number of Majorana edge modes with different rotation eigenvalues.

In mirror-invariant 2D planes orthogonal to the z axis, we find XBdG = ℤ2 × ℤ8. The ℤ2 factor is given bywhere represents the number of irreducible representations with the C2 eigenvalue ±i and the inversion parity β = ±1 at X. The phase with corresponds to 1D Kitaev chains stacked along x and y axes. The ℤ8 factor is related to TSCs with mirror Chern number CM and second-order TSCs. It is given by ().This formula rewrites the one for the mirror Chern number in the form of summation. Thus, z8 equals to C mod 4 (). To see the meaning of z8 = 4 mod 8, we generate an example of phases with by the wire construction as illustrated in Fig. 2A. Keeping both the inversion and the rotation symmetry but breaking the translation symmetry, one can gap out edges, realizing a second-order TSC that features two zero modes with different chirality at each of four corners. We can also generate a phase with by stacking four copies of mirror Chern insulator with C = 1. From these observations, we concludewhere ν2 = 0 and 1 is the index for the second-order TSC.

Fig. 2

Examples of wire construction for P4/m realizing higher-order TSCs with Majorana corner modes (circled by green ellipses).

(A) phase in 2D. Orange marks represent translation-breaking perturbations on the edge, gapping out pairs of zero modes. (B) (z8, , z8, , z16) = (4,4,8) phase in 3D.

Last, we discuss 3D systems. The (ℤ2)4 × ℤ4 × ℤ8 part of XBdG originates from lower dimensions. For example, in Eq. 61 for the k = 0 plane gives a ℤ2 factor, and in Eq. 59 and in Eq. 60 for the two fourfold symmetric lines (the Γ − Z line and the M-A line) produce four ℤ2 factors, but only three of them are independent from other indices. The ℤ4 factor can be accounted for by the inversion parityThe ℤ8 factor is given by the index z8 in Eq. 62 for the k = π plane.

To explain the remaining ℤ16 factor, we introduce a strong ℤ16 index defined bywhere K4 represents the set of four high-symmetry momenta invariant under S4. We find that z16 = 0 mod 16 holds for all elements in {AI}BdG. This z16 invariant, when focused on its “mod 8” part, has the same implication as for class AII systems (, ). Namely, z16 mod 4 agrees with the mirror Chern number, and z16 = 4 mod 8 implies a second-order TSC. To investigate the topology of phases with z16 = 8 mod 16, we stack 2D second-order TSCs with z8 = 4 and C = 0 as illustrated in Fig. 2B. The value of z16 depends on how to stack the 2D layers since z16 is related to the z8 index for the k = 0 and π planes asIf the inversion center is contained in a layer (i.e., there exists a single layer left invariant under the inversion), z16 = 8 and the system exhibits Majorana corner states. If, on the other hand, the inversion center is not contained in any layer, z16 = 0 and the surface can be completely gapped without breaking symmetries or closing the bulk gap. Phases with z16 = 8 mod 16 can also be mirror Chern TSCs just like in the 2D case.

B representation (χ4 = −1, χ = −1)

Next, let us consider the B representation. In one dimensions, we found XBdG is trivial. This is because the fourfold rotation symmetry together with the choice χ = −1 implies that the ℤ index of class DIII is trivial.

In two dimensions, the interpretation of XBdG = ℤ2 × ℤ8 is the same as the A representation, but the formula for z8 index must be replaced by

In three dimensions, the ℤ2 × ℤ4 × ℤ8 part of XBdG is weak indices. The remaining ℤ4 factor can be explained by κ1 in Eq. 66.

The QSH indices of the kz = 0 and kz = π planes must be the same since rotation eigenvalues on these planes must coincide due to the compatibility relations along rotation-symmetric lines. Therefore κ1 (defined modulo 8) is restricted to be even and characterizes the ℤ4 factor. For the B representation, κ4 always vanishes and z16 = κ1. Since Eq. 68 still holds, κ1 = 2 mod 4 implies a nontrivial mirror Chern number. As discussed in section S3, there are no third-order TSCs in this symmetry setting. With these results, we conclude that κ1 = 4 mod 8 also indicates a nonzero mirror Chern number.

B representation (χ4= −1, χ =+1)

Last, let us discuss B representation. In this case, the mirror symmetry commutes with the particle-hole symmetry (), and the mirror Chern number must vanish ().

In two dimensions, we find XBdG = ℤ2 × ℤ2. These class are characterized bywhere represents the number of irreducible representations with the C4 eigenvalue and the inversion parity β = ±1 at Γ = (0,0) and M = (π, π). Phases with or (0,1) turn out to be gapless. An example of the quasiparticle spectrum is shown in Fig. 3A. To see why, recall that mod 2 implies the nontrivial ℤ2 QSH index as discussed in the “P4 with B representation” section. However, when χI = + 1, the QSH index cannot be nontrivial (). The only way out is the gap closing of the quasiparticle spectrum, which invalidates the definition of the QSH index. Phases with (1,1) ∈ ℤ2 × ℤ2 can be gapped, and they are second-order TSCs.

Fig. 3

Quasiparticle spectrum (top), symmetry representations of E < 0 states (middle), and the surface band structure (bottom) of with Eqs. 72 and 73. In (A), each bands are doubly degenerate due to the inversion and the time-reversal symmetry. For (B) and (C), inversion-breaking perturbations in Eq. 76 are included. Symmetry representations colored in red is for , and those in blue is for Hvac. In the surface band structure, states localized near edges are colored in red.

Last, we discuss 3D systems. For B representation, κ1 should always be 0, while κ4 can be nonzero. It is restricted to be even, explaining the strong ℤ2 factor in XBdG. As explained in the Supplementary Materials, κ4 = 2 mod 4 indicates second-order TSCs.

To demonstrate the prediction of SIs, let us discuss a simple 2D model with P4/m symmetry. The BdG Hamiltonian is given by Eq. 1 withHere, σ’s are the Pauli matrices. The inversion symmetry and the fourfold rotation symmetry are represented by

Thus, the model belongs to the B representation of P4/m discussed in the “B representation (χ = −1, χ = +1)” section. We compute the indices in Eqs. 70 and 71 and get , which suggests nodal points in the quasiparticle spectrum (see Fig. 3A) as discussed in the “B representation (χ = −1, χ = +1)” section.

If inversion-breaking perturbations are added, the space group symmetry is reduced to P4. Here, we consider the following termThe fourfold rotation symmetry remains intact and χ = −1. Thus, is still well defined. For small perturbations, the bulk spectrum remains gapless and flat surface bands [Andreev bound states ()] appear in the surface spectrum (Fig. 3B). As the perturbation strength is increased, the system gets gapped without closing gap at high-symmetry points and becomes a helical TSC (Fig. 3C).

Application to CuxBi2Se3

One of the most studied bulk TSCs is CuBi2Se3, whose space group is . The corresponding point group D3 contains the inversion I, the threefold rotation about z axis C3, and the mirror symmetry about the yz plane M. There are several proposals for possible odd-parity pairings. One-dimensional representations A1 (χ = − 1, χ = −1, χ = +1) and A2 (χ = −1, χ = + 1, χ = + 1) of produce the superconducting gap Δ2 and Δ3, respectively. When the 2D irreducible representation E is used, the point group D3 is reduced to C2 (). Several recent studies have reported a nematic order that spontaneously breaks the threefold rotation C3, supporting the E pairing (). Then, the 2D representation of D3 splits into two 1D representations: A (χ = −1, χ = −1) and B (χ = −1, χ = +1) of C2, corresponding to the superconducting gap Δ4 and Δ4, respectively (, ).

Let us discuss the implication of SIs for each of these odd-parity pairings. First of all, κ1 (the sum of the inversion parities divided by four; see section S2A) is odd for all of these cases. This can be seen by focusing on the small Fermi surface around Γ that originates from the Cu doping to the topological insulator Bi2Se3. On the one hand, this value of κ1 indicates that the 3D winding number νw is odd for gapped SCs. On the other hand, as proven in section S5, there is a relation νw = − χw among νw and χ. Therefore, we conclude that Δ2 and Δ4 can be a gapped TSC with a nontrivial winding, and Δ3 and Δ4 must contain SC nodes. According to (), the Δ4 pairing contains Dirac nodes protected by the mirror symmetry.

Indicators for WTSCs

So far, we have mostly focused on the formalism and physical meaning of the refined SIs for superconductors. From the discussions on the Kitaev chain in (, ) and the “Refined symmetry indicators for superconductors” section, one might expect the main power of the SI refinement is to capture TSCs with zero-dimensional surface states. This is untrue: In the “P4 with B representation” section, we have already asserted that the C4-refined SI, ν, actually detects either a gapless phase or the helical TSC in class DIII in two dimensions.

We will substantiate our claim in this section. A general approach for physically interpreting the SIs is to first construct a general set of topological phases protected by the symmetries and then evaluate their SIs to establish the relations between the two (, ). We will follow a similar scheme: First, we introduce the notion of WTSC, which, like the Kitaev chain, reduces to an atomic state once the particle-hole symmetries are broken; next, we discuss how particle-hole symmetry restricts the possible associated atomic states that could correspond to a WTSC; and last, we specialize our discussion to 2D superconductors with χ = −1 and show that the nontrivial refined SI does not indicate a WTSC. Our claim follows when the arguments above are combined with the established classification of class AII topological (crystalline) insulators (, , –).

Before moving on, we remark that, insofar as our claim on the physical meaning of ν is concerned, there is probably a simpler approach in which one relates the nontrivial SI ν = 1 to the Fermi-surface invariant in () under the weak-pairing assumption. Our approach, however, is more general in that the weak-pairing assumption is not required and that the analysis of the SIs corresponding to WTSCs also helps one understand the physical meaning of the SIs, as can be seen in the P4/m examples.

Wannierizable TSCs

Let us begin by introducing the notion of WTSCs. Consider a gapped Hamiltonian. To investigate the possible topological nature of the system, we ask if it is possible to remove all quantum entanglement in the many-body ground state while respecting all symmetries. In the context of noninteracting insulators, this question can be rephrased in the notion of Wannier functions, and we say a phase is topological if there is an obstruction for constructing symmetric, exponentially localized Wannier functions out of the Bloch states below the energy gap (, , ).

For superconductors, the question of ground-state quantum entanglement is more subtle even within a mean-field BdG treatment. As a partial diagnosis, we could still apply the same Wannier criterion to the Bloch states below the gap at E = 0, and we say a BdG Hamiltonian is “Wannierizable” when no Wannier obstruction exists. [There is a technical question of whether the addition of trivial states below the gap is allowed, which differentiates “stable” topological phases from “fragile” ones (). Since our starting point is a BdG Hamiltonian, the corresponding physical system does not have charge conservation symmetry, and it is more natural to focus on stable topological phases. We will take this perspective and always assume appropriate trivial DOF could be supplied to resolve any possible fragile obstructions in a model.] A non-Wannierizable BdG Hamiltonian is necessarily topological, and phases like the 2D helical TSC in class DIII can be diagnosed that way. However, as the mentioned Wannier criterion uses only Bloch states with energy E < 0, it inherently ignores the presence of particle-hole symmetry when we consider obstructions to forming localized, symmetric Wannier functions, i.e., in the Wannierization, we only demand the subgroup of symmetries that commute with the single-particle Hamiltonian. Because of this limitation, the Wannier criterion does not detect TSCs whose BdG Hamiltonians become trivial when the particle-hole symmetry is ignored, like the Kitaev chain. When a Wannierizable BdG Hamiltonian is topological (in the sense defined in the “Topology of superconductors” section), we call it a WTSC.

Constraints on the associated atomic insulators

By definition, given any Wannierizable BdG Hamiltonian in class DIII, we can define an associated atomic insulator in class AII. On the basis of the recently developed paradigms for the classifications of topological crystalline insulators (, , , ), we can consider the associated atomic insulator ψ as an element of a finitely generated Abelian group CAI. More concretely, let HBdG be Wannierizable, and let ψBdG ∈ C be the associated atomic insulator. Similar to the formalism for the refined SI, we also consider the limit when the chemical potential approaches −∞. The vacuum is Wannierizable, and so we can also define ψvac ∈ CAI. In the following, we will again be focusing on the difference δψ ≡ ψBdG − ψvac ∈ CAI.

Although we have ignored the particle-hole symmetry Ξ in discussing the Wannierizability of a BdG Hamiltonian, it casts important constraints on the possible states δψ ∈ CAI. Physically, ψvac can be identified with the states forming the hole bands (E < 0) of an empty lattice, and it is determined by the sites, orbitals, as well as the choice on the superconducting pairing symmetry denoted by χ. We can also consider the states forming the electron bands (E > 0) of the empty lattice, which are related to ψ by the particle-hole symmetry. More generally, we can define a linear map Ξχ : CAI → CAI which relates an atomic insulator with its particle-hole conjugate. Noticing that ψvac + Ξχ[ψvac] describes the full Hilbert space in our BdG description, we must haveWe remark that Eq. 77 parallels Eq. 34 in defining the refined SI. Equation 77 can be rearranged into a condition that δψ has to satisfywhere we denote the identity map by and the trivial element of 𝒞AI by 0. Note that .

An obvious class of solutions to Eq. 78 is to take δψ = ψ − Ξχ[ψ] for any ψ ∊ 𝒞AI. Such solutions arise when we take ψBdG = Ξχ[ψvac], the fully filled state of the system, in the definition of δψ. Mathematically, we can view them as elements in the image of the map , and it is natural for us to quotient out these trivial solutionsIf δψ belongs to a nontrivial class in XWTSC, the gap must close when we change the chemical potential to either one of the limits μ → ±∞, and so the BdG Hamiltonian cannot be trivial. Physically, we interpret XWTSC as an indicator for WTSC. Note that, generally, XWTSC is only an indicator, not a classification, of WTSCs. This is because ψ1 = ψ2 is only a necessary, but not generally sufficient, c∈ondition for the existence of a symmetric, adiabatic deformation between two Wannierizable BdG Hamiltonians.

We can now relate XWTSC to the refined SI by evaluating the momentum-space symmetry representations of δψ. If δψ belongs to the trivial class of XWTSC, we can write δψ = ψ − Ξχ[ψ] for some ψ ∈ CAI. Correspondingly, its representation vector takes the form for some ∊ {AI}, and so its SI will also be trivial. This implies if two atomic mismatches δψ1 and δψ2 belong to the same class in the quotient group XWTSC, they will have the same refined SI. That is, the evaluation of the refined SI gives a well-defined map SI : XWTSC → XBdG. Note that the symmetry representations may not detect all topological distinctions between atomic states, and so SI[XWTSC] generally contains less information than XWTSC.

Observe that SI[XWTSC] is a subgroup of XBdG. If HBdG is Wannierizable, its representation vector BdG − vac must have an SI in the subgroup SI[XWTSC]. Conversely, any SI that does not belong to this subgroup is inconsistent with any WTSC.

Interpretation of ν

We can now apply the formalism to show that a 2D BdG Hamiltonian in class DIII with ν = 1 cannot be Wannierizable, and hence, it must be either gapless or has a nontrivial ℤ2 QSH index (–). Following the general plan described above, we will first compute the group 𝒞AI classifying the associated atomic insulators, construct the map Ξχ corresponding to χ = −1, and, lastly, show that a phase with ν = 1 cannot be Wannierizable as SI[XWTSC] = ℤ1, the trivial group.

To classify the associated atomic insulators, we first consider the set of possible lattices and orbitals. In 2D with C4 rotation symmetry, there are four Wyckoff positions: W = {(0,0)}, W = {(1/2,1/2)}, Wc = {(1/2,0), (0,1/2)}, and W = {(x, y), ( −y, x), ( −x, −y), (y, −x)} being the general position. A site in Wa or Wb is symmetric under C4 rotation, and for spinful fermions with time-reversal symmetry, we can label the orbitals by α = ±1 or ±3 characterizing the C4 eigenvalue , where the ± states form a Kramers pair. When the site filling is two, we fill one of the two types of orbitals, and we denote the corresponding atomic insulators by and . Generally, the site-filling may be larger than two, and we denote a state with 2n fermions filling orbitals with α = ±1 and 2m with α = ±3, both in Wa, by the expression . We can perform the same analysis for Wc and Wd. A site in Wc only has C2 rotation symmetry, and the two possible rotation eigenvalues form a Kramers pair. Since there is only one orbital type, we will denote the corresponding atomic insulator by ψ. Similarly, we will let ψ denote the atomic insulator living on the general position.

While we have listed a total of six possible atomic insulators with the minimal filling of two fermions per site, these states are not completely independent. To see why, consider setting the free parameters in the general position W to x = y = 0, which corresponds to moving all four sites in the unit cell to the point-group origin. As the deformation of sites can be done in a smooth manner, the atomic insulator ψ must be equivalent to an appropriate stack of atomic insulators defined on Wa. Such equivalence can be deduced by studying the point-group symmetry representation furnished by the collapsing sites (, ). We can perform a similar analysis by collapsing the sites in W to the other two Wyckoff positions, and altogether, we find the equivalence relationsAs such, any atomic insulator ψ in our setting can be formally expanded aswhere n ϵ ℤ and ξ ϵ ℤ2. That is, the atomic insulators are classified by the group CAI = ℤ3 × (ℤ2)2. In this language, we represent any (class of) atomic insulator by the collection of integers (n, n, n, ξ, ξb). For instance,

We are now ready to construct the map Ξχ. With the choice of χ = −1, the C4 rotation eigenvalues of local orbitals related by Ξ differ by −1. As such, the particle-hole acts on the atomic insulators as followsand recall that , the identity. We can equally represent the action of Ξχ by a matrix

We can now compute XWTSC. On the one hand, we can parameterize elements in bywhere each of m, ξ corresponds to a generator, i.e.,This shows that . On the other hand, an element takes the formand so we can writewhich is abstractly the group ℤ2. Comparing Eq. 87 against Eq. 86, we find the quotient groupand we may take (0,0,0,1,0) and (0,0,0,0,1) as representatives of the generating elements.

Last, we evaluate SI[XWTSC]. The corresponding representation vectors of the atomic states satisfy the relationsFrom this, we conclude SI[XWTSC] = ℤ1, and so ν = 1 implies the BdG Hamiltonian cannot be Wannierizable.

While the discussion above focuses on a 2D system with C4 rotation symmetry, one can perform the same analysis for any other symmetry setting. In particular, we tabulate the results for space group and P4/m under different SC representations in Table 3. For and P4/m, we found SI[XWTSC] = XWTSC, and nontrivial entries correspond to WTSCs like stacked Kitaev chains and higher-order TSCs. For and P4/m with the A representation, XWTSC coincides with the maximal (ℤ2) subgroup of XBdG. (For P4/m, m = 2, 2, 7 in one, two, and three dimensions.) For P4/m with B and B representations, XWTSC is only a subgroup of the maximal (ℤ2) subgroups of XBdG, and we explain the correspondence in the Supplementary Materials.

Table 3

List of XWTSC for class DIII systems with , P4, and P4/m symmetry in each spatial dimension.

SG (rep of Δk)	1D	2D	3D
P1¯ (Au)	ℤ2	(ℤ2)3	(ℤ2)7
P4 (B)	ℤ1	(ℤ2)2	(ℤ2)2
P4/m (Au)	(ℤ2)2	(ℤ2)2	(ℤ2)7
P4/m (Bu)	ℤ1	(ℤ2)2	(ℤ2)3
P4/m (Bg)	ℤ1	ℤ2	ℤ2

DISCUSSION

We advanced the theory of SIs for TSCs and computed the indicator groups explicitly for all space groups and pairing symmetries. We showed that the refinement proposed in (, ) enables the detection of a variety of phases, including both “first-order” (i.e., conventional) and higher-order TSCs. This is perhaps surprising, as the refinement only captures phases with zero-dimensional Majorana modes in the case of inversion symmetry studied in (, ). Furthermore, we found that the same indicator could correspond to a possibly gapped or a necessarily gapless phase depending on the additional spatial symmetries that are present. Such observations should be contrasted with the familiar case of the Fu-Kane parity criterion for topological insulators (), which is valid independent of the other spatial symmetries in the system. This suggests that caution must be used in diagnosing a TSC using only part of the spatial symmetries, and it is desirable to perform a more comprehensive analysis taking into account the entire space group preserved by the superconductor, as is done in the present work.

As a concrete example, our analysis for systems with C4 rotation symmetry revealed a new ℤ2-valued index, which we denote by ν. We argued that ν = 1 implies the system is a helical TSC when the system is gapped or indicates a gapless phase when inversion symmetry is present and the superconducting pairing has even parity. Within the weak pairing assumption, this nontrivial index can be realized in systems with d-wave pairing and an odd number of filled Kramers pairs in the normal state (section S4). When inversion symmetry is broken such that mixed-parity pairing becomes possible, one could gap out the nodes of the superconducting gap by increasing the p-wave component, and the end result will be a helical TSC. A similar picture was proposed in (), although the role of the SI was not recognized there. Such mechanism may be possible for the (proximitized) superconductivity on the surfaces of 3D materials, where the surface termination breaks inversion symmetry and can give rise to Rashba spin-orbit coupling. If the system has C4 rotational symmetry and a SC pairing with χ = −1 (e.g, D wave) is realized in the bulk, the induced surface superconductor on a C4-preserving surface will be topological when the number of filled surface-Kramers pair at the momenta Γ and M is odd in the normal state. The surface SC, if viewed as a stand-alone system, will be either a nodal or helical TSC.

Alternatively, one could also replace the innate surface state in the proposal above by an independent 2D system in which superconductivity is induced by proximity coupling to a d-wave superconductor.

More generally, it is interesting to ask how our theory could be applied to surface superconductivity, especially for the anomalous surface states arising from a topological bulk (). Conceptually, one can also compute the refined SI of a nonsuperconducting insulator by assuming an arbitrarily weak pairing amplitude with a chosen pairing symmetry. If the insulator is atomic to begin with (i.e., its ground state is smoothly deformable to a product state of localized electrons), the refined SI is trivial by definition. However, if the insulator is topological, its refined SI may be nontrivial. As the pairing can be arbitrarily weak in the bulk, this nontrivial refined SI is a statement on the nature of the TSC realized at the surface. As a concrete example, consider an inversion-symmetric strong TI. If we assume an odd-parity pairing is added to the system, one sees that the refined SI will be nontrivial. This setup is formally realized for an S-TI-S junction with a π phase shift, and the helical Majorana mode that appears () is consistent with the refined SI discussed above. This correspondence between a strong TI and a (higher-order) TSC is quite general and has been noted earlier in () assuming C4 symmetry. Given the vast majority of TI candidates discovered from materials database searches (–) are in fact (semi-)metallic, they may have superconducting instability and could realize a TSC based on the analysis above.

On a more technical note, we remark that our theory does not incorporate the Pfaffian invariant discussed in (), although this invariant can be readily related to the number of filled states in the normal-state band structure within the weak-pairing assumption. While it will be interesting to incorporate it into our formalism, the Pfaffian invariant is different from the usual representation counts as it is ℤ2 valued. This will bring about some technical differences in the computation of the SI group, although a systematic computation is still possible ().

Last, we note that in our analysis for the physical meaning of ν we introduced the notion of WTSCs, examples of which include the 1D Kitaev chain and 2D higher-order TSCs, as well as weak phases constructed by stacks of them. As a more nontrivial example, we note that the set of WTSCs also includes “first-order” examples like the even entries for the ℤ-valued classification of class DIII superconductors in 3D. While we have developed a formalism for the partial diagnosis of such TSCs, our analysis does not result in a full classification for WTSCs. It will be interesting to explore how the full classification can be obtained, as well as the unique physical properties, if any, that are tied to the notion of WTSCs.