Abstract
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 (1). 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 (2–4). 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 (5).
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 (6–13). On the other hand, general theories for how crystalline symmetries can be used to identify topological materials have been developed (14, 15). In particular, the method of symmetry indicators (SIs) (14) has enabled comprehensive surveys of topological materials among existing crystal structure databases, and thousands of materials candidates have been uncovered (16–18).
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 (19–21) and its higher-dimensional analogs like the higher-order TSCs in 2D (22, 23). 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 (24). Furthermore, the extension of these formulas for other crystalline and higher-order TSCs has only been achieved for specific examples (25–27).
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 (20, 21) 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 (19, 24, 28, 29). 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 (20, 21). 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 (30) and the corresponding version for odd-parity TSC (28, 29), 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 (20, 21). 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 (30), 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 Hk of the normal phase, which we assume to be a D-dimensional Hermitian matrix. We take a superconducting gap function Δk that satisfies
This Hamiltonian always has the particle-hole symmetry
Suppose that the Hamiltonian of the normal phase has a space group symmetry G. Each element g ∈ G is represented by a unitary matrix Uk(g) that satisfies
Last, the BdG Hamiltonian has the time-reversal symmetry if there exists
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
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
We furthermore assume that
Trivial superconductors. Let us now define the topologically trivial class of SCs. Our discussion is inspired by the recent proposal in (20, 21).
Suppose that the BdG 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 if
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 (33, 34), 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 (35).
Class D
The BdG Hamiltonian of a single Kitaev chain is given by Eq. 1 with
Let us take two copies of this model by setting
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 with
To resolve the obstruction, we introduce trivial DOFs with an appropriate inversion property. Specifically, we set D′ = 2, D′′ = 0 and
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 (21, 22) and the previous approach in (19, 36).
Symmetry representations of BdG Hamiltonians. Let us consider a BdG Hamiltonian
Suppose that
The SI is formulated in terms of integers
For a later purpose, let us also define
Next, let us examine a trivial BdG Hamiltonian Hvac in Eq. 14 for which the space group G is represented by the same matrix
Last, we consider additional trivial DOFs described by
We can perform the irreducible decomposition as in Eqs. 32 and 33 and define a′ and
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,
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)
Completeness of trivial limits. The above vector a′ corresponds to the atomic limit of an insulator in class A, AI, or AII depending on the assumption on the time-reversal symmetry in
Viewed as the representation counts in the valence bands of an insulator, it was proven in (14) that bBdG can always be expanded in terms of ai’s using fractional (or integer) coefficients
To demonstrate Eq. 42, note first that bvac belongs to {AI} and thus can be expanded as
Quotient group. Given a BdG Hamiltonian
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
As we proved in the previous section, bBdG − bvac of a given BdG Hamiltonian
Relation to previous approach. In previous works (19, 36), bBdG 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, bBdG is directly compared against atomic limits ai (discussed in the “Completeness of trivial limits” section) of the same symmetry setting. When bBdG cannot be written as a superposition of ai’s with integer coefficients (i.e., bBdG ∉ {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 bBdG ∈ {AI}, it would still be possible that bBdG − bvac ∉ {AI}BdG and bBdG − bvac 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 Hk, not in the quasiparticle spectrum of
To explain how it works, let ψk be an eigenstate of Hk with the energy ϵk belonging to the representation
We remark that our sense of “weak pairing” is less stringent than that used in (24), 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 (20, 21), 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
For inversion symmetric 1D models in class A, {AI} is a 3D space spanned by
In contrast, we see that
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 χg 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.
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 bBdG − bvac as
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 χC4 = −1.
In two spatial dimensions, we find XBdG = ℤ2. To see the meaning of this, let us introduce
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: Ag (χC4 = +1, χI = +1), Au (χC4 = +1, χI = −1), Bu (χC4 = −1, χI = −1), and Bg (χC4 = −1, χI = +1). For the Ag 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) (39).
Au representation (χC4 = +1, χI = −1)
Let us start with the Au representation. Although XBdG in 3D is large (see Table 2), many factors can be attributed to lower dimensions.
In a 1D system along the rotation axis, we find XBdG = (ℤ2)2, which can be characterized by
In mirror-invariant 2D planes orthogonal to the z axis, we find XBdG = ℤ2 × ℤ8. The ℤ2 factor is given by
(A)
Last, we discuss 3D systems. The (ℤ2)4 × ℤ4 × ℤ8 part of XBdG originates from lower dimensions. For example,
To explain the remaining ℤ16 factor, we introduce a strong ℤ16 index defined by
Bu representation (χC4 = −1, χI = −1)
Next, let us consider the Bu representation. In one dimensions, we found XBdG is trivial. This is because the fourfold rotation symmetry together with the choice χC4 = −1 implies that the ℤ2 index of class DIII is trivial.
In two dimensions, the interpretation of XBdG = ℤ2 × ℤ8 is the same as the Au 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 Bu 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.
Bg representation (χC4= −1, χI =+1)
Last, let us discuss Bg representation. In this case, the mirror symmetry commutes with the particle-hole symmetry (
In two dimensions, we find XBdG = ℤ2 × ℤ2. These class are characterized by
Quasiparticle spectrum (top), symmetry representations of E < 0 states (middle), and the surface band structure (bottom) of
Last, we discuss 3D systems. For Bg 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.
Example. To demonstrate the prediction of SIs, let us discuss a simple 2D model with P4/m symmetry. The BdG Hamiltonian
Thus, the model belongs to the Bg representation of P4/m discussed in the “Bg representation (χC4 = −1, χI = +1)” section. We compute the indices in Eqs. 70 and 71 and get
If inversion-breaking perturbations are added, the space group symmetry is reduced to P4. Here, we consider the following term
Application to CuxBi2Se3. One of the most studied bulk TSCs is CuxBi2Se3, whose space group is
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 = − χMxνw among νw and χMx. Therefore, we conclude that Δ2 and Δ4y can be a gapped TSC with a nontrivial winding, and Δ3 and Δ4x must contain SC nodes. According to (45), the Δ4x 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 (20, 21) 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, νC4, 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 (11, 38). 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 χC4 = −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 (6, 7, 9–13).
Before moving on, we remark that, insofar as our claim on the physical meaning of νC4 is concerned, there is probably a simpler approach in which one relates the nontrivial SI νC4 = 1 to the Fermi-surface invariant in (24) 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 (14, 15, 33).
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 (33). 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 Ek < 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 (7, 9, 10, 46), 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 ∈ CAI 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 have
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
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
Observe that SI[XWTSC] is a subgroup of XBdG. If HBdG is Wannierizable, its representation vector bBdG − bvac 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 νC4. We can now apply the formalism to show that a 2D BdG Hamiltonian in class DIII with νC4 = 1 cannot be Wannierizable, and hence, it must be either gapless or has a nontrivial ℤ2 QSH index (11–13). Following the general plan described above, we will first compute the group 𝒞AI classifying the associated atomic insulators, construct the map Ξχ corresponding to χC4 = −1, and, lastly, show that a phase with νC4 = 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: Wa = {(0,0)}, Wb = {(1/2,1/2)}, Wc = {(1/2,0), (0,1/2)}, and Wd = {(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
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 Wd 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 ψd 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 (15, 46). We can perform a similar analysis by collapsing the sites in Wd to the other two Wyckoff positions, and altogether, we find the equivalence relations
We are now ready to construct the map Ξχ. With the choice of χC4 = −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 follows
We can now compute XWTSC. On the one hand, we can parameterize elements in
Last, we evaluate SI[XWTSC]. The corresponding representation vectors of the atomic states satisfy the relations
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
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 (20, 21) 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 (20, 21). 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 (30), 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 νC4. We argued that νC4 = 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 (47), 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 χC4 = −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 (48). 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 (48) 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 (49) assuming C4 symmetry. Given the vast majority of TI candidates discovered from materials database searches (16–18) 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 (21), 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 (21).
Last, we note that in our analysis for the physical meaning of νC4 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.
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/18/eaaz8367/DC1
This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.
REFERENCES AND NOTES
- Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).