Fermi-surface diagnosis for topological superconductivity with s-wave-like pairing symmetries

Abstract

Theoretical prediction of topological superconductivity is key to their discovery. Recently, it is proven that in 199 out of 230 space groups, topological superconductivity coexists with an s-wave-like pairing symmetry, raising the hope of finding more candidates for this exotic phase. However, a comprehensive and efficient method for diagnosing topological superconductivity in realistic materials remains elusive. Here, we derive Fermi-surface formulas for gapped and gapless topological phases of time-reversal symmetric superconductors with s-wave-like pairing symmetries in all layer and space groups, applicable to thin-film and bulk materials. Our diagnosis uses only the sign of the pairing and the Fermi velocity at several Fermi points, and yields complete (partial) diagnosis for gapped and gapless superconducting phases in 159 (71) space groups. This provides a fundamental basis for the first-principles prediction of new topological superconductors.

The authors derive formulas for topological invariants in time-reversal symmetric spinful superconductors invariant under all symmetry operations for a given space group. The formulas require only the sign of the intra-band pairing and sign of the Fermi velocity at several points on the Fermi surface.

Introduction

Over the past decades, the discovery of materials exhibiting bulk topological superconductivity has been one of the central issues in condensed matter physics^1–5. Thanks to intensive studies in recent years, it has been shown that a wide variety of topological superconducting phases exist^6–25. Unconventional superconductors, such as p- and d-wave superconductors, serve as important platforms for realizing topological superconductivity^26–41. However, such non-s-wave superconductors are rare in nature. As a consequence, we have not yet verified the existence of intrinsic topological superconductivity in real materials.

Recently, a class of time-reversal symmetric spinful superconductors has been extensively studied^42–56. In these superconductors, the Cooper pairs are invariant under all spatial symmetry operations for a given space group. We refer to a time-reversal symmetric spinful superconductor with this property as “superconductor with s-wave-like pairing symmetry” (more precise definition is provided below). Notably, most ever-discovered superconductors fall into this class. It is natural to ask how many topological phases exist in this class. In fact, this question is answered by classification of gapped topological superconducting phases in all space groups. It turns out that topological superconductivity is compatible with s-wave-like pairing symmetries in 199 out of the 230 space groups⁴⁵. Importantly, all nontrivial gapped phases in this class exhibit topologically protected gapless states on boundaries. More recently, ref. ⁴⁶ has found topological invariants defined for Bogoliubov-de Gennes (BdG) Hamiltonians, which can characterize gapped and gapless superconductors in this class. Despite these advances, it remains challenging to compute topological invariants for realistic materials following these definitions, as full BdG Hamiltonians are often unavailable for realistic superconductors.

The theories of symmetry indicators and topological quantum chemistry have been introduced to efficiently detect nontrivial topology in insulators and semimetals^57–62. Moreover, symmetry-indicator theory has been generalized to superconductors^63–66, enabling the diagnosis of topology in realistic superconductors using only irreducible representations of normal-state wave functions and pairing symmetries under certain assumptions⁶⁷. However, this approach fails for s-wave-like pairings: all topologically nontrivial phases with s-wave-like pairing symmetries cannot be detected by existing symmetry indicators [see Supplementary Note 2A for more details]. Thus, an efficient diagnostic framework for the topology with s-wave-like pairing symmetries remains elusive.

In this work, we present Fermi-surface formulas of topological invariants in all layer and space groups, applicable to both thin-film and bulk materials. Importantly, all our formulas require only signs of the Fermi velocities and intraband pairings on line segments connecting two high-symmetry points—no information about the entire Fermi surfaces is needed (see Fig. 1 for an illustration). In addition to the efficiency, our formulas can detect a wide range of nontrivial topological phases—including both gapped and gapless—with s-wave-like pairing symmetries. Our formulas provide a complete efficient diagnosis of gapped and stable gapless phases in 159 space groups and partial coverage in the remaining 71 space groups.

Fig. 1. Illustration of our Fermi-surface formulas.

On the left-hand side, the red dots represent intersections of the Fermi surface with certain line segments in the Brillouin zone. All we need for the formulas are the signs of Fermi velocities, v_f and intraband pairings, δ_k at the intersections. Our formulas enable us to compute topological invariants only from the information at the intersections. Nonzero values of the invariants indicate the existence of some gapless states on boundaries, as illustrated by the example on the right-side^21,86.

Results

Fermi surface formulas

In this work, we focus on s-wave-like pairing symmetries, formally defined by the symmetry properties of the order parameter Δ_k. Let U_k(g) be a representation of a space group symmetry g. We say that the pairing symmetry is s-wave-like if $U_{\mathbf{k}}\left(g\right){\mathrm{\Delta }}_{\mathbf{k}}{U}_{-\mathbf{k}}^{\top }\left(g\right)={\mathrm{\Delta }}_{g\mathbf{k}}$ for all symmetries in a given space group. It should be noted that the s-wave-like pairing symmetry does not necessarily mean that the system is a conventional s-wave superconductor. For example, consider a single orbital superconducting order Δ_k = (d(k) ⋅ σ)iσ_y under the mirror symmetry ${\mathcal{M}}_z$ along the z-axis, where $\mathbf{d}\left(\mathbf{k}\right)=\left(0,\mathrm{i}\sin k_x,0\right)$ and σ_i=0,x,y,z are Pauli matrices in spin space. While this is p-wave like, the pairing symmetry is s-wave like, as one can see $\mathrm{i}{\mathrm{\sigma }}_z{\mathrm{\Delta }}_{\mathbf{k}}{\left(\mathrm{i}{\mathrm{\sigma }}_z\right)}^{\top }={\mathrm{\Delta }}_{{\mathcal{M}}_z\mathbf{k}}$. Another representative example is s_±-wave pairing.

Topological invariants are often defined on subregions in momentum space. For convenience, we decompose the Brillouin zone into points, line segments, and polygons in a symmetric manner^68,69. See Fig. 2(b) for an illustration of the decomposition. We assign an orientation to each component. For example, line segment a in Fig. 2(b) is oriented from Γ to X. In this work, we consider topological invariants defined on the line segments.

Fig. 2. Illustration of the model (10) and (11).

a shows the energy dispersion of the BdG Hamiltonian (11). b shows decomposition of the Brillouin zone, Fermi surfaces (colored solid lines), and the pairing nodes (dashed lines). The solid black lines and arrows denote the line segments specified by the gray endpoints and their orientations. The red and blue colors in the figure indicate Fermi surfaces with mirror eigenvalues  + i and  − i, respectively. The intersections of the Fermi surface and the pairing nodes correspond to the gapless points in (a). The parameters are set to be {t, λ, μ, Δ} = { − 1, 0.3, − 1, 0.3}.

Before presenting our main results on the Fermi surfaces of the topological invariants, we introduce our basic assumptions on target systems. We always assume that superconductors are in the weak coupling regime. More precisely, we assume that the target superconducting system can be continuously deformed into a superconductor that possesses the following properties:

• (i)intraband pairings dominate, i.e., all interband pairings are negligible;
• (ii)the superconducting gap is negligible except on Fermi surfaces.

In addition, we further assume that the following conditions are satisfied:

• (iii)all Fermi surfaces are minimally degenerate as allowed by symmetries;
• (iv)normal state energies at high-symmetry points are not at the Fermi level;
• (v)there is no superconducting node on all line segments, including their endpoints.

The assumptions (iii) and (iv) usually hold for realistic systems. The assumption (v) must be satisfied to define topological invariants on line segments.

Under these five assumptions, intraband pairing potentials and Fermi velocities can determine the values of topological invariants defined on the line segments. To define intraband pairing potentials and Fermi velocities, we consider the m-th Fermi point at k_m and momenta k in its neighborhood. Let ${\mathrm{\Phi }}_{\mathbf{k},m}^{\alpha }$ be a matrix whose columns are eigenvectors of the normal conducting Hamiltonian h_k with the energy ${\epsilon }_{\mathbf{k},m}^{\alpha }$, where α is a label of an irreducible representation (irrep) $u_{\mathbf{k}}^{\alpha }\left(g\right)$. In other words, ${\mathrm{\Phi }}_{\mathbf{k},m}^{\alpha }$ satisfies the relations $h_{\mathbf{k}}{\mathrm{\Phi }}_{\mathbf{k},m}^{\alpha }={\epsilon }_{\mathbf{k},m}^{\alpha }{\mathrm{\Phi }}_{\mathbf{k},m}^{\alpha }$ and $U_{\mathbf{k}}\left(g\right){\mathrm{\Phi }}_{\mathbf{k},m}^{\alpha }={\mathrm{\Phi }}_{\mathbf{k},m}^{\alpha }{u}_{\mathbf{k}}^{\alpha }\left(g\right)$. Then, the intraband pairing potential is defined by a projected superconducting order

$${\tilde{\mathrm{\Delta }}}_{\mathbf{k},m}^{\alpha }:={\left[{\mathrm{\Phi }}_{\mathbf{k},m}^{\alpha }\right]}^{†}{\mathrm{\Delta }}_{\mathbf{k}}{\left[U\left(\mathcal{T}\right)\right]}^{*}{\mathrm{\Phi }}_{\mathbf{k},m}^{\alpha }.$$ 1

Here, $U\left(\mathcal{T}\right)$ is a unitary representation of time-reversal symmetry (TRS) satisfying $U\left(\mathcal{T}\right){\left[U\left(\mathcal{T}\right)\right]}^{*}=-\mathbb{1}$, where $\mathbb{1}$ is the identity matrix. The above intraband pairing satisfies the relations

$${\left[{\tilde{\mathrm{\Delta }}}_{\mathbf{k},m}^{\alpha }\right]}^{†}={\tilde{\mathrm{\Delta }}}_{\mathbf{k},m}^{\alpha };u_{\mathbf{k}}^{\alpha }\left(g\right){\tilde{\mathrm{\Delta }}}_{\mathbf{k},m}^{\alpha }{\left[u_{\mathbf{k}}^{\alpha }\left(g\right)\right]}^{†}={\tilde{\mathrm{\Delta }}}_{\mathbf{k},m}^{\alpha }.$$ 2

Since $u_{\mathbf{k}}^{\alpha }\left(g\right)$ is irreducible, the intraband pairing is proportional to the identity matrix. In particular, when we consider the Fermi point k_m on the line segment l, we represent ${\tilde{\mathrm{\Delta }}}_{{\mathbf{k}}_m,m}^{\alpha }$ by

$${\tilde{\mathrm{\Delta }}}_{{\mathbf{k}}_m,m}^{\alpha }={\delta }_m^{\left(l,\alpha \right)}\mathbb{1}\left({\delta }_m^{\left(l,\alpha \right)}\in \mathbb{R}\right),$$ 3

where ${\delta }_m^{\left(l,\alpha \right)}$ represents the intraband pairing potential at the m-th Fermi point with irrep α on line segment l.

The Fermi velocity is defined by the directional derivative of the energy. Suppose that we consider a line segment l connecting from k_A to k_B, and that the m-th Fermi point at k_m lies on this line segment. The Fermi velocity of the energy ${\epsilon }_{\mathbf{k},m}^{\alpha }$ is given by

$$v_m^{\left(l,\alpha \right)}:={\mathit{\nabla }}_{\mathbf{k}}{\epsilon }_{\mathbf{k},m}^{\alpha }{\mid }_{\mathbf{k}={\mathbf{k}}_m}\cdot {\widehat{\mathbf{e}}}_l,$$ 4

where ${\widehat{\mathbf{e}}}_l=\left({\mathbf{k}}_{\mathrm{B}}-{\mathbf{k}}_{\mathrm{A}}\right)/\parallel {\mathbf{k}}_{\mathrm{B}}-{\mathbf{k}}_{\mathrm{A}}\parallel$.

Based on these two quantities, we define a set of integer-valued quantities ${\left\{{\mathcal{N}}_{\left(l,\alpha \right)}\right\}}_{l,\alpha }$ for a given system:

$${\mathcal{N}}_{\left(l,\alpha \right)}:={\sum }_{m=1}^{n_{\left(l,\alpha \right)}}\mathrm{sgn}\left(v_m^{\left(l,\alpha \right)}\right)\frac{1-\mathrm{sgn}\left({\delta }_m^{\left(l,\alpha \right)}\right)}{2}\in \mathbb{Z},$$ 5

where n_(l, α) is the number of minimally degenerate Fermi points with irrep α on line segment l. In fact, this integer-valued quantity encodes the topological nature of superconducting phases under the above assumptions.

Now, we are ready to provide concrete forms of Fermi-surface formulas of topological invariants. For each space group, there exist three types of topological invariants: (i) $\mathbb{Z}$-valued invariants ${\left\{{\mathcal{W}}_i^{\mathrm{g}apless}\right\}}_{i=1}^{N_{\mathrm{g}apless}}$ to detect gapless points in two-dimensional subregions surrounded by line segments; (ii) $\mathbb{Z}$-valued invariants ${\left\{{\mathcal{W}}_i^{\mathrm{g}apped}\right\}}_{i=1}^{N_{\mathrm{g}apped}}$ to capture gapped topological phases; (iii) ${\mathbb{Z}}_{\lambda }$-valued invariants ${\left\{{\mathcal{X}}_i\right\}}_{i=1}^M$ to identify gapped topological phases. It should be emphasized that both the number of defined topological invariants (N_gapless, N_gapped, M), and topological invariants themselves depend on a given space group. Notably, gapped topological phases detected by ${\left\{{\mathcal{W}}_i^{\mathrm{g}apped}\right\}}_{i=1}^{N_{\mathrm{g}apped}}$ and ${\left\{{\mathcal{X}}_i\right\}}_{i=1}^M$ must have protected gapless surface states. Furthermore, ${\left\{{\mathcal{W}}_i^{\mathrm{g}apless}\right\}}_{i=1}^{N_{\mathrm{g}apless}}$ can detect all stable gapless phases in all space groups with s-wave-like pairing symmetries (See Supplementary Note 2 for more details).

Our Fermi-surface formulas of these invariants are given by

$${\mathcal{W}}_i^{\mathrm{g}apless}={\sum }_{l,\alpha }{w}_{i,\left(l,\alpha \right)}^{\mathrm{g}apless}{\mathcal{N}}_{\left(l,\alpha \right)}={\sum }_{l,\alpha }{w}_{i,\left(l,\alpha \right)}^{\mathrm{g}apless}{\sum }_{m=1}^{n_{\left(l,\alpha \right)}}\mathrm{sgn}\left(v_m^{\left(l,\alpha \right)}\right)\frac{1-\mathrm{sgn}\left({\delta }_m^{\left(l,\alpha \right)}\right)}{2},$$ 6

$${\mathcal{W}}_i^{\mathrm{g}apped}={\sum }_{l,\alpha }{w}_{i,\left(l,\alpha \right)}^{\mathrm{g}apped}{\mathcal{N}}_{\left(l,\alpha \right)}={\sum }_{l,\alpha }{w}_{i,\left(l,\alpha \right)}^{\mathrm{g}apped}{\sum }_{m=1}^{n_{\left(l,\alpha \right)}}\mathrm{sgn}\left(v_m^{\left(l,\alpha \right)}\right)\frac{1-\mathrm{sgn}\left({\delta }_m^{\left(l,\alpha \right)}\right)}{2},$$ 7

$${\mathcal{X}}_i={\sum }_{l,\alpha }{x}_{i,\left(l,\alpha \right)}{\mathcal{N}}_{\left(l,\alpha \right)}\mathrm{mod}\lambda ={\sum }_{l,\alpha }{x}_{i,\left(l,\alpha \right)}{\sum }_{m=1}^{n_{\left(l,\alpha \right)}}\mathrm{sgn}\left(v_m^{\left(l,\alpha \right)}\right)\frac{1-\mathrm{sgn}\left({\delta }_m^{\left(l,\alpha \right)}\right)}{2}\mathrm{mod}\lambda ,$$ 8

where $w_{i,\left(l,\alpha \right)}^{\mathrm{g}apless},w_{i,\left(l,\alpha \right)}^{\mathrm{g}apped},x_{i,\left(l,\alpha \right)}\in \mathbb{Z}$ represent contribution weights of irrep α on line segment l for topological invariants. In other words, these integer-valued quantities tell us about two important things: which line segments and irreps are relevant for topological invariants, and how we should combine Fermi-point data ${\left\{{\mathcal{N}}_{\left(l,\alpha \right)}\right\}}_{l,\alpha }$ to detect the topological nature of superconducting phases. For a given space group, we can systematically determine these contribution weights by Atiyah-Hirzebruch spectral sequence. See ref. ⁴⁶ for more details. For ${\mathbb{Z}}_2$-valued invariants, we can drop $\mathrm{sgn}\left(v_m^{\left(l,\alpha \right)}\right)$ since n = − n mod 2 for an integer n. As a result, the formulas can be further simplified to

$${\mathcal{X}}_i={\sum }_{l,\alpha }{x}_{i,\left(l,\alpha \right)}{\sum }_{m=1}^{n_{\left(l,\alpha \right)}}\frac{1-\mathrm{sgn}\left({\delta }_m^{\left(l,\alpha \right)}\right)}{2}\mathrm{mod}2.$$ 9

These formulas immediately give us the following insight. The quantities (6)–(9) are always trivial for superconductors whose intraband pairings have the same sign everywhere. This means that superconductors in the BCS limit, which can be connected to the vacuum without closing gap, cannot have nontrivial values of Eqs. (6)–(9).

We make three remarks on our formulas. First, our Fermi-surface formulas are well-defined for a given decomposition, including the orientations of line segments, of the Brillouin zone in a space group. In Supplementary Information, we provide our formulas together with all line segments we used for all layer groups and space groups. If we adopt a different decomposition, we may have physically equivalent but different forms of the formulas. This is because changing the decomposition can change the contribution weights in general. Next, topological invariants for gapped phases, ${\left\{{\mathcal{W}}_i^{\mathrm{gapped}}\right\}}_{i=1}^{N_{\mathrm{gapped}}}$ and ${\left\{{\mathcal{X}}_i\right\}}_{i=1}^M$, are meaningful only when invariants for gapless points, ${\left\{{\mathcal{W}}_i^{\mathrm{gapless}}\right\}}_{i=1}^{N_{\mathrm{gapless}}}$, are all trivial. Lastly, topological invariants defined on higher-dimensional subspaces would be required for gapped topological phases that cannot be detected by our formulas. For example, detecting the nonzero three-dimensional winding number $w_{3\mathrm{D}}\in \mathbb{Z}$ requires information on the entire three-dimensional Brillouin zone⁷⁰. Nonetheless, our Fermi surface formula can still detect $w_{3\mathrm{D}}\mathrm{mod}n$ (where n depends on the space group), as seen below. This is similar to the situation for symmetry indicators^57,58,71.

Examples

Here, we demonstrate how it works through three theoretical models constructed by the real-space construction method and one realistic model from real material.

First, we consider a quasi-two-dimensional gapless superconductor in the layer group p11m. This layer group is generated by the mirror symmetry ${\mathcal{M}}_z=\left\{M_z\mid \left(000\right)\right\}$ and translation symmetries along x- and y-direction. The decomposition of the Brillouin zone is shown in Fig. 2(b). The normal state Hamiltonian for the superconductor is

$$h_{\mathbf{k}}=t\left(\cos k_x+\cos k_y\right)s_0+\lambda \left(\sin k_x+\sin k_y\right)s_3,$$ 10

and the corresponding BdG Hamiltonian takes the form

$$H_{\mathbf{k}}^{\mathrm{BdG}}=\left[h_{\mathbf{k}}-\mu \right]{\kappa }_3+{\mathrm{\Delta }}_{\mathrm{sc}}\left(\sin k_x+\sin k_y\right)s_1{\kappa }_1,$$ 11

where the Pauli matrices s_j and κ_j (j = 0, 1, 2, 3) stand for spins and the Nambu spinor. In addition to TRS and mirror symmetry ${\mathcal{M}}_z$, the BdG Hamiltonian inherently possesses particle-hole symmetry $\mathcal{C}$. For the above BdG Hamiltonian, their representations are given by

$$U^{\mathrm{BdG}}\left(\mathcal{T}\right)=\mathrm{i}{s}_2,U^{\mathrm{BdG}}\left(\mathcal{C}\right)={\kappa }_1,U^{\mathrm{BdG}}\left({\mathcal{M}}_z\right)=\mathrm{i}{\kappa }_3{s}_3.$$ 12

The pairing term ${\mathrm{\Delta }}_{\mathbf{k}}={\mathrm{\Delta }}_{\mathrm{sc}}\left(\sin k_x+\sin k_y\right)s_1$ remains invariant under mirror symmetry ${\mathcal{M}}_z$: $U_{\mathbf{k}}\left({\mathcal{M}}_z\right){\mathrm{\Delta }}_{\mathbf{k}}{U}_{\mathbf{k}}^{\top }{=\mathrm{\Delta }}_{{\mathcal{M}}_z\mathbf{k}}$, where $U_{\mathbf{k}}\left(M_z\right)$ is the representation of the mirror symmetry in the normal phase. Therefore, the pairing exhibits the s-wave-like pairing symmetry. As shown in Fig. 2(a), the energy spectrum of $H_{\mathbf{k}}^{\mathrm{BdG}}$(11) has two superconducting gapless points.

Our formula for $\mathbb{Z}$-valued invariants can detect these gapless points. Using Eq. (6), we find the formula

$${\mathcal{W}}_1^{\mathrm{g}apless}={\sum }_{\alpha =\pm }\alpha \left[{\sum }_{m=1}^{n_{\left(c,\alpha \right)}}\mathrm{sgn}\left(v_m^{\left(c,\alpha \right)}\right)\frac{1-\mathrm{sgn}\left({\delta }_m^{\left(c,\alpha \right)}\right)}{2}\right)\left(-{\sum }_{m=1}^{n_{\left(a,\alpha \right)}}\mathrm{sgn}\left(v_m^{\left(a,\alpha \right)}\right)\frac{1-\mathrm{sgn}\left({\delta }_m^{\left(a,\alpha \right)}\right)}{2}\right],$$ 13

where α = ± means the mirror eigenvalue  ± i for eigenstates of the normal state Hamiltonian (10). The signs of Fermi velocities v^(a, ±) on the line segment a are positive. According to the sign of δ_k shown in Fig. 2(b), we have ${\mathcal{W}}^{\mathrm{gapless}}=-1$. In fact, the invariant is equivalent to the mirror winding number defined on a loop ( − π, 0) → (π, 0) → (π, π) → ( − π, π) → ( − π, 0). The nontrivial mirror winding number indicates the existence of gapless points inside the loop⁷².

Next, we consider a gapped topological superconductor (TSC) in layer group p2₁/b11 to show that our formulas can be used even when the formula in ref. ⁷⁰ cannot. Generators of this layer group are glide symmetry ${\mathcal{G}}_x=\left\{M_x\mid \left(\frac{1}{2}\frac{1}{2}0\right)\right\}$, inversion symmetry $\mathcal{I}=\left\{I\mid \left(000\right)\right\}$, and translation symmetry along x-direction. In the presence of inversion symmetry and TRS, Fermi surfaces in spinful electronic systems must be at least twofold degenerate. Furthermore, topological invariants for class DIII in the tenfold classification^73–75 are also trivial for even-parity pairings. Therefore, this symmetry setting is out of the scope of ref. ⁷⁰.

Here, we describe our model in which there are four sublattice degrees of freedom labeled by (τ_z, σ_z) = ( ± 1, ± 1). Their coordinates in a unit cell are specified by $\left(x,y\right)=\frac{1}{4}\left[\left(2-{\tau }_z\right)\left(1,0\right)+\left(1-{\sigma }_z\right)\left(0,1\right)\right]$ [see Fig. 3(a)]. According to ref. ⁴⁵, the classification of gapped topological phases is ${\mathbb{Z}}_2$, whose generator is constructed by placing one-dimensional TSCs along x = 1/4 and x = 3/4 in each unit cell. To realize this, we consider a model whose normal state Hamiltonian is given by

$$h_{\mathbf{k}}=t\left({\mathrm{\Gamma }}_{010}+\cos k_y{\mathrm{\Gamma }}_{010}+\sin k_y{\mathrm{\Gamma }}_{020}\right),$$ 14

and the BdG Hamiltonian is

$$H_{\mathbf{k}}^{\mathrm{BdG}}=\left[h_{\mathbf{k}}-\mu \right]{\kappa }_3+2{\mathrm{\Delta }}_{\mathrm{sc}}\sin k_y{\mathrm{\Gamma }}_{301}{\kappa }_1,$$ 15

where Γ_ijk:= τ_i ⊗ σ_j ⊗ s_k(i, j, k = 0, 1, 2, 3). As mentioned above, the minimal degeneracy of states is twofold at each momentum. However, the Fermi surfaces of h_k in Eq. (14) are fourfold degenerate, which violates assumption (iii), as indicated by the gray dashed lines in Fig. 3(b). This violation can be easily resolved by introducing symmetry-allowed perturbations that are small enough not to change any topology, as shown by the solid gray lines in Fig. 3(b).

Fig. 3. Illustration of the model in layer group p2₁/b11.

a Real-space description of the model. The red shaded region indicates the unit cell. The red and blue solid circles denote the four sublattices labeled by (τ_z, σ_z) = ( ± 1, ± 1) in the unit cell. The solid (dashed) arrows represent the hopping between the sublattice degrees in the normal (pairing) part of the BdG Hamiltonian (15). b Fermi surfaces for the Hamiltonian (14). The gray solid (dashed) lines show the normal-state Fermi surfaces with (without) symmetry-allowed perturbations. The red and blue colors mean states with glide eigenvalue $+i{e}^{-i{k}_y/2}$ and $-i{e}^{-i{k}_y/2}$, respectively. The parameters are set to be {t, μ} = { − 1, 1}.

The ${\mathbb{Z}}_2$ topology can be diagnosed by our formula for ${\mathbb{Z}}_2$-valued invariant in Eq. (9). The decomposition of the Brillouin zone is shown in Fig. 3(b). Then, the formula is given by

$${\mathcal{X}}_1={\sum }_{\alpha =\pm }{\sum }_{m=1}^{n_{\left(c,\alpha \right)}}\frac{1-\mathrm{sgn}\left({\delta }_m^{\left(c,\alpha \right)}\right)}{2}+{\sum }_{\beta =\pm }{\sum }_{m=1}^{n_{\left(d,\beta \right)}}\frac{1-\mathrm{sgn}\left({\delta }_m^{\left(d,\beta \right)}\right)}{2},$$ 16

where α = ± represents the glide eigenvalue $\pm {\mathrm{ie}}^{-i{k}_y/2}$ on the line segment c, and β = ± denotes the screw symmetry ${\mathcal{S}}_x={\mathcal{G}}_x\mathcal{I}$ eigenvalue $\pm {\mathrm{ie}}^{-i{k}_x/2}$ on the line segment d. According to the sign of δ_k shown in Fig. 3(b), we have $\mathcal{X}=1$ mod 2.

Last, we discuss a strong TSC in space group P4₁ to show that our formula can inform the three-dimensional winding number w_3D modulo four. This space group is generated by fourfold screw symmetry ${\mathcal{S}}_{4z}=\left\{C_{4z}\mid \left(00\frac{1}{4}\right)\right\}$ and translation symmetries along x- and y-directions.

Our model is described as follows. There are four sublattice degrees of freedom in a unit cell, whose coordinates in a unit cell are (x, y, z) = (0, 0, 0), (0, 0, 1/4), (0, 0, 1/2), and (0, 0, 3/4), labeled as 1, 2, 3, and 4, respectively. The normal state Hamiltonian is constructed as

$${\widehat{\mathcal{H}}}_0={\sum }_{s,\mathbf{R}}{\sum }_{i=1}^4-\frac{t}{2}\left({\widehat{c}}_{si\mathbf{R}+\mathbf{a}}^{†}{\widehat{c}}_{si\mathbf{R}}+{\widehat{c}}_{si\mathbf{R}+\mathbf{b}}^{†}{\widehat{c}}_{si\mathbf{R}}\right)+h.c.+{\sum }_{s,\mathbf{R}}-\frac{t}{2}\left({\widehat{c}}_{s1\mathbf{R}}^{†}{\widehat{c}}_{s2\mathbf{R}}+{\widehat{c}}_{s2\mathbf{R}}^{†}{\widehat{c}}_{s3\mathbf{R}}+{\widehat{c}}_{s4\mathbf{R}}^{†}{\widehat{c}}_{s3\mathbf{R}}\right)+h.c.+{\sum }_{s,\mathbf{R}}-\frac{t}{2}{\widehat{c}}_{s4\mathbf{R}+\mathbf{c}}^{†}{\widehat{c}}_{s1\mathbf{R}}+h.c.,{\sum }_{s,\mathbf{R},i}{\widehat{c}}_{si\mathbf{R}+\mathbf{c}}^{†}{\widehat{c}}_{si\mathbf{R}},$$ 17

where the ${\widehat{c}}_{si\mathbf{R}}\left({\widehat{c}}_{si\mathbf{R}}^{†}\right)$ is the annihilation (creation) operator of an electron at the i-th sublattice with spin s in the unit cell at R. Also, a, b, and c are the primitive lattice vectors along x-, y-, and z-direction, respectively. The pairing term is given by

$$\widehat{\mathrm{\Delta }}={\sum }_{s,\mathbf{R}}-\frac{i}{2}{\mathrm{\Delta }}_{\mathrm{sc}}\left({\widehat{c}}_{s1\mathbf{R}}^{†}{\widehat{c}}_{\overline{s}2\mathbf{R}}^{†}+{\widehat{c}}_{s4\mathbf{R}}^{†}{\widehat{c}}_{\overline{s}3\mathbf{R}}^{†}+{\widehat{c}}_{s2\mathbf{R}}^{†}{\widehat{c}}_{\overline{s}3\mathbf{R}}^{†}\right)+h.c.+{\sum }_{s,\mathbf{R}}\frac{i}{2}{\mathrm{\Delta }}_{\mathrm{sc}}\left({\widehat{c}}_{s1\mathbf{R}}^{†}{\widehat{c}}_{\overline{s}4\mathbf{R}+\mathbf{c}}^{†}-{\sum }_{i=1}^{4}s{\widehat{c}}_{si\mathbf{R}}^{†}{\widehat{c}}_{si\mathbf{R}+\mathbf{a}}^{†}\right)+h.c.+{\sum }_{s,\mathbf{R}}{\sum }_{i=1}^4\frac{1}{2}{\mathrm{\Delta }}_{\mathrm{sc}}{\widehat{c}}_{si\mathbf{R}}^{†}{\widehat{c}}_{si\mathbf{R}+\mathbf{b}}^{†}-h.c.$$ 18

with s and $\overline{s}$ being the opposite spin. It is easy to check the system described by ${\widehat{\mathcal{H}}}_{\mathrm{B}dG}={\widehat{\mathcal{H}}}_0+\widehat{\mathrm{\Delta }}$ has the nonzero three-dimensional winding number w_3D = 1.

Here, we compute the ${\mathbb{Z}}_8$-valued invariant by our formula (8):

$${\mathcal{X}}_3={\sum }_{l=e,g}{\sum }_{\alpha =1,3,5,7}\left(4-\alpha \right){\sum }_{m=1}^{n_{\left(l,\alpha \right)}}\mathrm{sgn}\left(v_m^{\left(l,\alpha \right)}\right)\frac{1-\mathrm{sgn}\left({\delta }_m^{\left(l,\alpha \right)}\right)}{2}+{\sum }_{l=c,d}{\sum }_{m=1}^{n_l}\mathrm{sgn}\left(v_m^l\right)\left(1-\mathrm{sgn}\left({\delta }_m^l\right)\right)+{\sum }_{\beta =\pm }\beta {\sum }_{m=1}^{n_{\left(f,\beta \right)}}\mathrm{sgn}\left(v_m^{\left(f,\beta \right)}\right)\left(1-\mathrm{sgn}\left({\delta }_m^{\left(f,\beta \right)}\right)\right),$$ 19

where for line segment l = e and g, $v_m^{\left(l,\alpha \right)}$ and ${\delta }_m^{\left(l,\alpha \right)}\left(\alpha =1,3,5,7\right)$ are defined for screw eigenvalue $e^{-i\left(\alpha \pi +k_z\right)/4}$; for line segment f, $v_m^{\left(f,\beta \right)}$ and ${\delta }_m^{\left(f,\beta \right)}\left(\beta =\pm \right)$ are defined for twofold rotation eigenvalues βi. After adding symmetry allowed perturbations to lift the accidental degeneracy in ${\widehat{\mathcal{H}}}_0$, the energy spectrum is as shown in Fig. 4. According to the sign of δ_k shown in Fig. 4, we find ${\mathcal{X}}_3=3$. In Methods, we show that ${\mathcal{X}}_3$ mod 4 serves as a ${\mathbb{Z}}_4$-valued indicator of w_3D.

Fig. 4. The band structure of the model (17) with symmetry-allowed perturbations in space group P4₁.

Solid lines (dashed lines) represent bands crossing (not crossing) the Fermi level. The arrows represent the positive direction of the line segments c, d, e, f, and g. The red and blue colors mean states with screw eigenvalue $e^{-i\left(3\pi +k_z\right)/4}$ and $e^{-i\left(\pi +k_z\right)/4}$, respectively. The  + and  − represent the sign of δ_k at the corresponding Fermi point. The parameters are set to be {t, μ} = {1, 1.8}.

Application to realistic materials

We discuss CaFeAs₂, whose space group is P2₁, to demonstrate how to apply our formula to realistic materials based on first-principles calculations.

In Fig. 5, we show the band structure and Fermi surfaces obtained by DFT calculations. From the raw data of Fig. 5(a), we notice that there are eight Fermi surfaces around the Γ point and four Fermi surfaces around the M point. This implies that although the Fermi surfaces in Fig. 5(b) and (c) seem to be degenerate, there is no Fermi surface degeneracy. This is consistent with the symmetry analysis.

Fig. 5. First-principle calculation results for CaFeAs₂.

a The band structure of CaFeAs₂. The bands near the Fermi energy E_F are marked in red. M, X, Γ, Y, Z, R, A, L represent (π/2, π/2, 0), (π/2, 0, 0), (0, 0, 0), (0, π/2, 0), (0, 0, π/2), (0, π/2, π/2), (π/2, π/2, π/2), (π/2, 0, π/2) in the Brillouin zone, respectively. b, c The Fermi surface of CaFeAs₂ with different viewpoints. The Fermi surfaces are visualized by XCrySDen⁸⁷.

One can see that the k_z-dependence of Fermi surfaces is sufficiently weak, and topological properties inherit the quasi-two-dimensional nature of its electronic structure. Therefore, we focus on k_z = 0 plane with the exchange of k_x- and k_y-axes to illustrate the workflow of our diagnostic scheme, which results in Layer group p2₁11 (No. 9).

According to our Fermi surface formula, there exist a $\mathbb{Z}$-valued invariant for gapless points, a ${\mathbb{Z}}_2$-valued invariant, and a ${\mathbb{Z}}_4$-valued invariant, whose expressions are given by

$${\mathcal{W}}_1^{\mathrm{g}apless}={\sum }_{m=1}^{n_{a_1}}\mathrm{sgn}\left(v_m^{a_1}\right)\frac{1-\mathrm{sgn}\left({\delta }_m^{a_1}\right)}{2}+{\sum }_{m=1}^{n_{a_2}}\mathrm{sgn}\left(v_m^{a_2}\right)\frac{1-\mathrm{sgn}\left({\delta }_m^{a_2}\right)}{2}-{\sum }_{m=1}^{n_{b_1}}\mathrm{sgn}\left(v_m^{b_1}\right)\frac{1-\mathrm{sgn}\left({\delta }_m^{b_1}\right)}{2}+2{\sum }_{m=1}^{n_{c_1}}\mathrm{sgn}\left(v_m^{c_1}\right)\frac{1-\mathrm{sgn}\left({\delta }_m^{c_1}\right)}{2}-{\sum }_{m=1}^{n_{d_1}}sgn\left(v_m^{d_1}\right)\frac{1-\mathrm{sgn}\left({\delta }_m^{d_1}\right)}{2}-{\sum }_{m=1}^{n_{d_2}}\mathrm{sgn}\left(v_m^{d_2}\right)\frac{1-\mathrm{sgn}\left({\delta }_m^{d_2}\right)}{2};$$ 20

$${\mathcal{X}}_1={\sum }_{m=1}^{n_{b_1}}\frac{1-\mathrm{sgn}\left({\delta }_m^{b_1}\right)}{2}+{\sum }_{m=1}^{n_{d_1}}\frac{1-\mathrm{sgn}\left({\delta }_m^{d_1}\right)}{2}+{\sum }_{m=1}^{n_{d_2}}\frac{1-\mathrm{sgn}\left({\delta }_m^{d_2}\right)}{2}\mathrm{mod}2;$$ 21

$${\mathcal{X}}_2=2{\sum }_{m=1}^{n_{a_2}}\mathrm{sgn}\left(v_m^{a_2}\right)\frac{1-\mathrm{sgn}\left({\delta }_m^{a_2}\right)}{2}-{\sum }_{m=1}^{n_{b_1}}\mathrm{sgn}\left(v_m^{b_1}\right)\frac{1-\mathrm{sgn}\left({\delta }_m^{b_1}\right)}{2}-2{\sum }_{m=1}^{n_{d_1}}\mathrm{sgn}\left(v_m^{d_1}\right)\frac{1-\mathrm{sgn}\left({\delta }_m^{d_1}\right)}{2}\mathrm{mod}4.$$ 22

Here, a₁ and d₁ (a₂ and d₂) are symbols of the irreducible representations whose character of screw symmetry along x-axis is $-\mathrm{i}{e}^{-\mathrm{i}{k}_x/2}\left(+\mathrm{i}{e}^{-\mathrm{i}{k}_x/2}\right)$ (see Section L9 in Supplementary Information for more detailed information).

As a demonstration, here we discuss some scenarios out of all possibilities, in which sign changes happen between the Fermi surfaces near the M point while fixing the pairing signs of the Fermi surfaces near the Γ point to be positive. Although a global gauge transformation can give rise to the global minus sign of the superconducting order parameter, this does not affect the topology. As a result, there are six inequivalent pairing sign choices for possible fully gapped superconducting states (whose ${\mathcal{W}}_1^{\mathrm{gapless}}$ must be trivial), as shown in Fig. 6(b–g). According to the formulas of the invariants, we find that four of these configurations correspond to topologically nontrivial states, $\left({\mathcal{W}}_1^{\mathrm{gapless}},{\mathcal{X}}_1,{\mathcal{X}}_2\right)=\left(0,0,2\right)$. This phase is equivalent to x-direction stacking copies of one-dimensional class DIII topological superconductors along the y-direction, as shown in Fig. 6(h) (see Supplementary Note 3A for more details). The model in Fig. 6(h) with a small perturbation is a second-order topological superconductor with Majorana corner modes, as shown in Fig. 6(i).

Fig. 6. Demonstration of our diagnosis of topological phases based on first-principles calculation results.

a Fermi surface of quasi-2D CaFeAs₂ based on the DFT-calculated data. a, b, c, d are the one-cell, whose directions are labeled by arrows. c_i and d_i represent the irreducible representations on the one-cell c and d. Γ, X, Y, M represent (0, 0), (π, 0), (0, π) and (π, π). b–g The six inequivalent pairing sign configurations satisfying ${\mathcal{W}}_1^{\mathrm{gapless}}=0$.  + and  − represent the pairing sign ${\delta }_i^l$ of i-th Fermi points on the line segment l. $\mathcal{W}$, ${\mathcal{X}}_1$ and ${\mathcal{X}}_2$ represent the value of the topological invariants defined in Eqs. (20), (21) and (22). h The real-space construction of the topological nontrivial state with $\left(\mathcal{W},{\mathcal{X}}_1,{\mathcal{X}}_2\right)=\left(0,0,2\right)$. i The corner modes of the topological nontrivial state with $\left(\mathcal{W},{\mathcal{X}}_1,{\mathcal{X}}_2\right)=\left(0,0,2\right)$.

Finally, let us return to the discussion of the three-dimensional nature. Due to its quasi-two-dimensional electronic nature, three-dimensional topological properties can be understood by stacking the aforementioned two-dimensional topological states along the z-direction. As a result, we conclude that this three-dimensional topological phase exhibits hinge modes along the z-direction.

Discussion

We introduce Fermi-surface formulas of topological invariants for time-reversal symmetric superconductors with s-wave-like pairing symmetries in all layer and space groups. Most of the superconductors ever discovered belong to this class, and within it, many potential candidates for topological superconductors remain to be verified, such as iron-based superconductors⁷⁶, nickelate-based superconductors^77,78, highly overdoped CuO₂ monolayer⁷⁹ and heavy fermion superconductors⁸⁰. This new method requires minimal information on the material, namely the sign of pairing and Fermi velocity at several Fermi points, and yet provides complete topological properties for 159 space groups. While our diagnosis for gapped phases remains partial for 71 space groups, a large part of the nontrivial phases can still be detected.

We emphasize the distinction between our formulas and the existing formulas introduced in refs. ^70,81. They introduce formulas for class DIII without any spatial symmetries and for the one-dimensional winding number, respectively. In their derivation, they assume that there is no Fermi surface degeneracy. However, this assumption is often invalid in the presence of space group symmetries, where such Fermi surface degeneracy occurs. In addition, space group symmetries often prevent certain topological invariants from being nontrivial. For example, topological invariants in the scope of ref. ⁷⁰ are always trivial for even-parity pairings with inversion symmetry. On the other hand, our formulas can detect nontrivial topology even in such cases, as shown in Sec. “ Examples.” Thus, our formulas have broader applicability than refs. ^70,81.

Furthermore, the diagnostic scheme proposed in this work identifies a broader range of topological phases based on much less information on the superconductor than all previous works. In Sec. “Application to realistic materials,” we actually demonstrate the effectiveness of our formulas by applying them to a density functional theory (DFT) calculation for a realistic material. Given the success of large-scale investigations of topological insulators using symmetry indicators and topological quantum chemistry^60–62, we anticipate that our formulas will play a crucial role in the discovery of realistic topological superconductors.

Methods

Relation between ${\mathcal{X}}_3$ and w_3D

Here, we argue that an odd value of ${\mathcal{X}}_3$ indicates oddness of w_3D. To show this, we compute ${\mathcal{X}}_3$ for all representatives of topological phases. In addition to the strong TSC, we have ${\left({\mathbb{Z}}_2\right)}^2$ phases constructed from one-dimensional TSCs and ${\left({\mathbb{Z}}_2\right)}^2$ phases constructed from two-dimensional TSCs⁴⁵, whose illustrations are shown in Fig. 7. In the Supplementary Note 3, we calculate all the gapless and gapped topological invariants for all representative models for these phases, which are summarized in Table 1. We find that all models with w_3D = 0 have ${\mathcal{X}}_3=0\mathrm{mod}4$. Also, we find ${\mathcal{X}}_3=-w_{3\mathrm{D}}\mathrm{mod}4$ for the generator of strong TSCs. Therefore, we conclude that ${\mathcal{X}}_3\mathrm{mod}4$ faithfully reflects $w_{3\mathrm{D}}\mathrm{mod}4$. This argument can be generalized to Pn₁(n = 2, 3, 4, 6). In Supplementary Note 4, we show that our invariants provide a ${\mathbb{Z}}_2$-valued indicator of w_3D in P2₁ and a ${\mathbb{Z}}_6$-valued indicator for P3₁ and P6₁.

Fig. 7. Illustration of real-space constructions of representative topological superconducting phases in P4₁.

a Unit cell and its cell complex for space group P4₁ (No. 76). The dashed lines denote the unit cell. The symmetry-inequivalent 2-cells, 1-cells and 0-cells are represented as colored faces, bold line segments, and red dots, respectively. The other cells can be obtained from them by acting with symmetry operations in SG P4₁. b, c Two generators of ${\mathbb{Z}}_2^2$-topological phases constructed from 1D TSCs. The black bold lines represent the nontrivial 1D TSCs in DIII class, and the blue dots represent the corresponding Majorana zero modes at the endpoints of 1D TSCs. d, e Two generators of ${\mathbb{Z}}_2^2$ constructed from 2D TSCs. The blue faces represent the nontrivial 2D TSCs in DIII class. The regions enclosed by solid lines represent the decorations on the symmetry-inequivalent 2-cells. f The illustration for the strong TSC in DIII class.

Table 1.

The quantitative mappings between all representative models and the topological invariants $\left({\mathcal{W}}_1^{\mathrm{gapless}},{\mathcal{W}}_2^{\mathrm{gapless}},{\mathcal{X}}_1,{\mathcal{X}}_2,{\mathcal{X}}_3\right)$ computed by our Fermi surface formula

Classification of ref. 45	w3D	$\left({\mathcal{W}}_1^{gapless},{\mathcal{W}}_2^{gapless}\right)$	$\left({\mathcal{X}}_1,{\mathcal{X}}_2,{\mathcal{X}}_3\right)$
1D TSCs: $\left(1,0\right)\in {\mathbb{Z}}_2\times {\mathbb{Z}}_2$	0	(0, 0)	(1, 2, 4)
1D TSCs: $\left(0,1\right)\in {\mathbb{Z}}_2\times {\mathbb{Z}}_2$	0	(0, 0)	(1, 0, 4)
2D TSCs: $\left(1,0\right)\in {\mathbb{Z}}_2\times {\mathbb{Z}}_2$	0	(0, 0)	(0, 3, 4)
2D TSCs: $\left(0,1\right)\in {\mathbb{Z}}_2\times {\mathbb{Z}}_2$	0	(0, 0)	(0, 1, 0)
Strong TSC: $1\in \mathbb{Z}$	1	(0, 0)	(1, 1, 3)

Here, ${\mathcal{W}}_1^{\mathrm{gapless}}$ and ${\mathcal{W}}_2^{\mathrm{gapless}}$ are $\mathbb{Z}$-valued invariants; ${\mathcal{X}}_1,{\mathcal{X}}_2$, and ${\mathcal{X}}_3$ are ${\mathbb{Z}}_2$-, ${\mathbb{Z}}_4$- and ${\mathbb{Z}}_8$-valued ones. The formulas are listed in Supplementary Information.

Detailed information on DFT calculations

In our DFT calculations, we employ QUANTUM ESPRESSO^82,83 and qeirreps⁸⁴ to obtain irreducible representations defined on line segments. The k-mesh is chosen as 24 × 24 × 24 in our self-consistent field calculations. The lattice structure is obtained from Cambridge Crystallographic Data Center (CCDC 961741)⁸⁵ for La_xCa_1−xFeAs₂⁷⁶ by taking x = 0.

Author contributions

Z.Z. and S.O. performed the theoretical and numerical calculations. K.S., C.F., and S.O. conceived the project and provided the theoretical understanding. All authors contributed to the design of the research, discussed the results, and wrote the manuscript.

Peer review

Peer review information

Nature Communications thanks Harley Scammell, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

All data needed to evaluate the conclusions in the study are present in the paper and/or the Supplementary Information.

Code availability

The computer code used for numerical calculation and theoretical analysis is not publicly available because of the high cost of preparation and maintenance. It is available upon request from the corresponding authors.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-72811-z.