Introduction

Realization of topological superconductivity in a controllable platform holds a promise for the potential applications in quantum information science1,2,3,4,5,6,7,8,9,10,11,12. Diverse Majorana platforms have been suggested, ranging from archetypal approaches that utilize topological superconductors13,14,15,16,17,18,19,20 to more recent approaches that benefit the proximity of superconductivity in heterostructures21,22,23. These platforms have yielded convincing experimental evidence of the Majorana states24,25,26,27,28,29,30,31,32. Nonetheless, searching for more scalable and controllable platforms is a pressing need for future developments. Encouragingly, routes to topological superconductivity are diversified, including more exotic classes that harbor gapless excitations such as the nodal plane Bogoliubov Fermi surface (BFS) superconductor33,34.

The pioneering idea of proximity effect-induced superconductivity of topological surface states17,21 has subsequently generalized to simpler devices in wider varieties. For example, a doped topological insulator or a topological semimetal facilitates a single-material setting by driving bulk superconductivity35,36,37 with the advantage of the intrinsic surface and interband superconducting coherence in the momentum space. Conversely, known superconductors have been identified as topologically nontrivial38,39,40. Furthermore, opportunities have been dilated beyond topological surface states, including the Shockley normal states41 with strong Rashba splitting, which drives Majorana zero modes under an external magnetic field42,43,44.

A more exotic realization is a recently discovered surface superconductor that coexists with the bulk Dirac semimetal45,46,47. A hexagonal nonpolar layered material KZnBi is a paradigmatic example whose bulk state hosts a topological Dirac semimetal in three dimensions. The onset of superconductivity is driven on the surface of KZnBi at Tc = 0.85 K45. This peculiar situation provides a unique platform for intrinsic superconductivity of the three-dimensional massless Dirac fermions, dominated by the surface superconducting condensation. However, the character of the bulk superconducting state in this class of materials has remained unexplored.

In this Letter, we present a density functional theory (DFT)-based low-energy model that sheds light on the possible superconducting states of KZnBi. Our theory proposes a general scenario in which the nonsymmorphic band degeneracy of Dirac semimetals leads to stable nodal line superconductivity fostered by multiband interactions. To complement the experimental observations in KZnBi, we explicitly calculate the quasiparticle density of states and specific heat, which are in excellent agreement with the observed results. In particular, the nodal lines form a hexagonal hosohedron, which easily inflates to the BFS superconductors when time-reversal symmetry is broken, as we demonstrate by applying an external magnetic field. Our results suggest the emergence of an unconventional topological superconductor in KZnBi, offering a feasible route to finding smoking gun evidence of the Bogoliubov Fermi surface in the existing topological Dirac semimetal KZnBi.

Results and discussion

HNL superconductor in KZnBi

The present numerical study shows that KZnBi hosts a nodal superconductor with the hexagonal hosohedron formation of the nodal lines shown in Fig. 1b, which we refer to as the hosohedral nodal line (HNL) superconductor. The hosohedron refers to a geometric structure in which the tessellation of lunes on a spherical surface forms, such that each lune shares the same two polar opposite vertices. Fig. 1a delineates examples of the n-gonal hosohedrons (n = 3, 4, and 6), represented by the schläfli symbol {2, n}. A characteristic of the HNL superconductor is the superconducting gap highly suppressed near the polar vertices. More remarkably, the HNL superconductor features the phase boundary of the BFS superconductors induced by time-reversal breaking perturbations, such as a magnetic field. Fig. 1b exemplifies the phase boundary of the BFS at zero magnetic fields. In the presence of the TR-breaking magnetic field, the HNL is immediately inflated to the BFS. We demonstrate this process in the KZnBi system.

Fig. 1: Schematic figure of the hosohedral nodal lines (HNLs) and the Bogoliubov Fermi surface (BFS).
figure 1

a Illustration of Cnz-symmetric n-gonal HNLs with the Schläfli symbol of {2, n} for n = 3, 4, 6. The blue lines and orange spheroidal surfaces delineate the nodal lines and the Fermi surface, respectively. The top view of the nodal lines near the polar vertex of the hosohedron is shown with the nodal lines (blue) and the corresponding inflated BFS (red surface). in the gray plane. b Phase diagram in magnetic field B space. The HNL superconductors occur at the B = 0 phase boundary of the BFS superconductors, represented by a blue dashed line. The red surface illustrates BFS.

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KZnBi in hexagonal ABC crystals

We first clarify the crystalline symmetries of KZnBi. Fig. 2a shows the atomic structure of KZnBi as the representative case of the hexagonal non-polar ABC family. The unit cell comprises two formula units with six atoms (2Zn, 2Bi, 2K atoms). A layered structure is formed with two K layers (A-sublattice), which are intercalated by the two additional Zn-Bi layers with alternating Zn-Bi positions. The crystal structure preserves the symmetries of the non-symmorphic and centrosymmetric space group P63/mmc (# 194), generated by inversion \({{{{{{{\mathcal{P}}}}}}}}\), sixfold screw \({S}_{6z}=\{{C}_{6z}| 00\frac{1}{2}\}\), and two-fold rotation C2x operations. Important symmetries include two mirrors mz and a mirror mx, protecting the Dirac crossings in the normal states bands with time-reversal symmetry \({{{{{{{\mathcal{T}}}}}}}}\). Similarly, the inversion \({{{{{{{\mathcal{P}}}}}}}}\) and glide mirror \({G}_{y}=\{{m}_{y}| 00\frac{1}{2}\}\) play a crucial role in protecting the nodal line superconducting phase.

Fig. 2: Atomic and electronic structures of KZnBi.
figure 2

a Crystal structure of KZnBi. The solid black lines show the primitive unit cell. b Brillouin zone (BZ). High-symmetry momenta are colored red. Green solid circles indicate the position of the Dirac points (DPs). c Density functional theory (DFT) band structure of KZnBi. The inset shows the magnified view of the band crossing in the boxed region. The irreducible representations at the high-symmetry A point are provided, which are obtained for the DFT calculations.

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Dirac semimetal in KZnBi

In good agreement with the previous study in Ref. 45, our density functional theory (DFT) calculations reproduce the Dirac bands. Fig. 2c shows the calculated DFT bands along the high-symmetry momenta shown in Fig. 2b. The magnified view of the band crossing in the inset clearly shows that a linear band crossing, referred to as the Dirac point (DP), appears along the high-symmetry Γ-A line in otherwise all gapped momentum space. Our calculation further reveals that the irreducible representations of the conduction and valance Dirac bands at A correspond to the A6 and A4 ⊕ A5 representations, respectively as indicated in Fig. 2c. The fourfold degeneracy at A is enforced by the mx and \({{{{{{{\mathcal{PT}}}}}}}}\) symmetries (See Supplementary Note 1 for the detailed symmetry analysis).

We construct a k ⋅ p minimal Hamiltonian that faithfully distills the Dirac bands of KZnBi. We propose the minimal eight-band model to capture the nonsymmorphic band degeneracy. The model Hamiltonian \({{{{{{{{\mathcal{H}}}}}}}}}_{A}({{{{{{{\boldsymbol{k}}}}}}}})\) harbors the two four-fold degenerate states at A4 ⊕ A5 and A6, such that it respects the symmetry constraints of the little group of A

$${{{{{{{{\mathcal{H}}}}}}}}}_{A}({{{{{{{\boldsymbol{k}}}}}}}})=g{{{{{{{\mathcal{H}}}}}}}}({D}_{g}^{-1}{{{{{{{\boldsymbol{k}}}}}}}}){g}^{-1}.$$
(1)

Here Dg is a symmetry operator defined in the momentum space, and g is the corresponding symmetry operator acting on the Bloch wave functions. The symmetry operators, g, are explicitly represented by \({{{{{{{\mathcal{P}}}}}}}}=-{\sigma }_{x}\), \({C}_{2x}={{{{{{{\rm{i}}}}}}}}(-\frac{\sqrt{3}}{2}{s}_{x}+\frac{1}{2}{s}_{y}){\tau }_{+}-{{{{{{{\rm{i}}}}}}}}{s}_{z}{\tau }_{-}\), \({S}_{6z}=-{{{{{{{\rm{i}}}}}}}}\frac{\sqrt{3}}{2}{s}_{z}{\tau }_{+}{\sigma }_{z}-\frac{1}{2}{\tau }_{-}{\sigma }_{z}+\frac{1}{2}{\tau }_{z}{\sigma }_{z}\), and \({{{{{{{\mathcal{T}}}}}}}}={{{{{{{\rm{i}}}}}}}}{s}_{y}{{{{{{{\mathcal{K}}}}}}}}\). Here, τ± = (1 ± τz)/2 and τz = ± 1 (σz = ± 1) denote A6 and A4 ⊕ A5 (\({A}_{6}^{\pm }\)) states, respectively. Similarly, sz = ± 1 describes spin, and \({{{{{{{\mathcal{K}}}}}}}}\) is the complex conjugation. The detailed derivation of our model Hamiltonian is in Supplementary Note 1. In result, the explicit form of \({{{{{{{{\mathcal{H}}}}}}}}}_{A}\) is given as follows;

$${{{{{{{{\mathcal{H}}}}}}}}}_{A} = \; {m}_{{{{{{{{\boldsymbol{k}}}}}}}}}^{+}{\tau }_{+}+{m}_{{{{{{{{\boldsymbol{k}}}}}}}}}^{-}{\tau }_{-}+\mu {k}_{1}({k}_{1}^{2}-3{k}_{2}^{2}){\sigma }_{y}\\ + {\lambda }_{1}\left[{k}_{1}({s}_{y}+{s}_{z}){\tau }_{y}+{k}_{2}({\tau }_{x}+{s}_{x}{\tau }_{y})\right]{\sigma }_{y}\\ + {\lambda }_{2}{k}_{z}\left[{l}_{1}{s}_{x}{\tau }_{-}+{l}_{2}{s}_{y}{\tau }_{-}+{l}_{3}{s}_{z}{\tau }_{+}\right]{\sigma }_{z}\\ + {\lambda }_{3}\left[({k}_{1}^{2}-{k}_{2}^{2})({\tau }_{x}-{s}_{x}{\tau }_{y})+2{k}_{1}{k}_{2}({s}_{z}{\tau }_{y}-{s}_{y}{\tau }_{y})\right],$$
(2)

\({k}_{1}\equiv \frac{1}{2}{k}_{x}+\frac{\sqrt{3}}{2}{k}_{y}\), \({k}_{2}\equiv \frac{\sqrt{3}}{2}{k}_{x}-\frac{1}{2}{k}_{y}\), and \({m}_{{{{{{{{\boldsymbol{k}}}}}}}}}^{\pm }\equiv {m}_{0}^{\pm }+{m}_{\parallel }^{\pm }({k}_{1}^{2}+{k}_{2}^{2})+{m}_{z}^{\pm }{k}_{z}^{2}\). The parameters \({m}_{0}^{\pm }\), \({m}_{\parallel }^{\pm }\), \({m}_{z}^{\pm }\), λi, μ, and li (i = 1, 2, 3) are determined to reproduce the DFT bands of KZnBi near the Fermi energy (See Supplementary Note 1 for the full Hamiltonian and the parameters that reproduce the DFT bands of KZnBi).

Superconducting phases

We determine the possible parings of the hole-doped bulk KZnBi superconductor, observed at EF = − 50 meV45. We first classify the paring functions based on the symmetries of D6h, which admits the total twenty-eight distinct superconducting channels. The complete list of twenty-eight pairing functions can be found in Supplementary Note 2. Out of these, three stable superconducting phases are found when the density-density type interactions are considered,

$${{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{I}}}}}}}}}=U\mathop{\sum}\limits_{i}\,{n}_{i}^{2}+2V\mathop{\sum}\limits_{i\ne j}\,{n}_{i}{n}_{j},$$
(3)

where U (V) represents the intra-(inter-)band interaction strength, and ni(j) is a density operator for the electrons in the i(j)-th orbital. (See Table 1 for the specific irreps and the corresponding gap structure).

Table 1 Symmetry classification of pairing functions.
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Fig. 3a shows an exemplary phase diagram comprising stable channels for given sets of relative interaction strengths between V and U and the spin-orbit couplings λ1 in a unit of the hopping parameter (\({m}_{\parallel }^{+}\)). It is readily manifested from the phase diagram that U < 0 (V < 0) enhances the stability of ΔA1g (ΔB2g and ΔB2u) phases. Furthermore, when the strong inter-band interaction regime, a strong SOC (\({\lambda }_{1}/{m}_{\parallel }^{+} > 0.364\,[{a}^{-1}]\)) induces the HNL superconducting phase (ΔB2g). The k ⋅ p parameters that reproduce the DFT bands satisfy the strong SOC criterion with \({\lambda }_{1}/{m}_{\parallel }^{+} \sim 0.577\,[{a}^{-1}]\), supporting the HNL phase in KZnBi.

Fig. 3: Hosohedral nodal line (HNL) superconductor in ΔB2g Pairing.
figure 3

a Phase diagram of the bulk superconducting phases of KZnBi. The phase diagram is calculated in terms of the V/U and \({\lambda }_{1}/{m}_{\parallel }^{+}\) parameter space, where V, U, λ1, and \({m}_{\parallel }^{+}\) are the inter-band Coulomb interaction, intra-band Coulomb interaction, spin-orbit coupling (SOC), and hopping parameters, respectively. Here, V/U is unitless, and \({\lambda }_{1}/{m}_{\parallel }^{+}\) is represented in units of the inverse lattice parameter a−1. The ΔA2u,ΔA1g, and ΔB2g orders dominate the blue, yellow, and red regions, respectively. b Bogolon quasiparticle energy spectra along the high-symmetry Γ-A-Γ line for the pairing functions ΔB2g. c Nodal lines of ΔB2g in the Brillouin zone. The winding number is calculated along the path \({{{{{{{\mathcal{C}}}}}}}}\) delineated by the green circle. The light orange surfaces represent the Fermi surfaces of the normal state.

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Fig. 3b shows the calculated energy spectra for ΔB2g, where the nodal lines are captured by the gap closing on the high-symmetry A-Γ line. Since each branch is doubly degenerate, enforced by the \({{{{{{{\mathcal{P}}}}}}}}{{{{{{{\mathcal{T}}}}}}}}\) symmetry, the band crossings give rise to four-fold degeneracy. A close inspection of the zero nodes in the entire BZ leads to the hexagonal HNLs, where the nodal loop resides on the Gy invariant planes, including ky = 0 and their C3z-related planes (Fig. 3c).

Stability of the HNL

The symmetry analysis reveals that the glide mirror Gy plays a crucial role in stabilizing the four-fold degeneracy of the HNL phase. To see this, we define the unitary chiral symmetry \(\tilde{\chi }={{{{{{{\rm{i}}}}}}}}\tilde{{{{{{{{\mathcal{T}}}}}}}}}\tilde{C}\) where \({\tilde{\chi }}^{2}=1\). The following mutual commutation relation holds between the symmetry operators of \(\tilde{{G}_{y}}={G}_{y}\oplus -{G}_{y}\), \(\tilde{\chi }\) and the Bogoliubov-de Gennes (BdG) Hamiltonian \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{BdG}}}}}}}}}\) of ΔB2g phase (See Supplementary Note 2 for the detailed construction of the BdG Hamiltonian),

$$\{\tilde{{{{{{{{\mathcal{P}}}}}}}}}\tilde{{{{{{{{\mathcal{T}}}}}}}}},\tilde{{G}_{y}}\} = \; 0,\{\tilde{\chi },\tilde{{G}_{y}}\} = 0, \\ \left[\tilde{{{{{{{{\mathcal{P}}}}}}}}}\tilde{{{{{{{{\mathcal{T}}}}}}}}},{{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{BdG}}}}}}}}}\right] = \; 0,\left \{\tilde{\chi },{{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{BdG}}}}}}}}}\right\}=0.$$
(4)

For a given zero-energy eigenstate \(\left\vert {\psi }_{E = 0}\right\rangle\) with \(\tilde{{G}_{y}}\left\vert {\psi }_{E = 0}\right\rangle =\left\vert {\psi }_{E = 0}\right\rangle\), the symmetry relations Eq. (4) ensure four mutually independent zero-energy states {\(\left\vert {\psi }_{E = 0}\right\rangle\), \(\tilde{\chi }\left\vert {\psi }_{E = 0}\right\rangle\), \(\tilde{{{{{{{{\mathcal{P}}}}}}}}}\tilde{{{{{{{{\mathcal{T}}}}}}}}}\left\vert {\psi }_{E = 0}\right\rangle\), \(\tilde{{{{{{{{\mathcal{P}}}}}}}}}\tilde{{{{{{{{\mathcal{T}}}}}}}}}\tilde{\chi }\left\vert {\psi }_{E = 0}\right\rangle\)}. Thus, the presence of four-fold degeneracy at E = 0 is compatible with the symmetries. The proof of mutual independence of these zero modes is given in Supplementary Note 2.

The HNL is topologically stable, protected by a non-zero topological \(2{\mathbb{Z}}\) winding number. One-parameter families of inversion, time-reversal, and chiral symmetric superconductors (DIII+\({{{{{{{\mathcal{P}}}}}}}}\) class) can be characterized by a \(2{\mathbb{Z}}\) winding number \({{{{{{{{\mathcal{N}}}}}}}}}_{{{{{{{{\rm{1D}}}}}}}}}\)48,49

$${{{{{{{{\mathcal{N}}}}}}}}}_{{{{{{{{\rm{1D}}}}}}}}}=\frac{1}{2\pi {{{{{{{\rm{i}}}}}}}}}\,\log \left(P\,\exp {\oint }_{C}d{{{{{{{\boldsymbol{k}}}}}}}}\cdot {\nabla }_{{{{{{{{\boldsymbol{k}}}}}}}}}{{{{{{{\rm{Tr}}}}}}}}\left[\ln {Q}_{{{{{{{{\boldsymbol{k}}}}}}}}}\right]\right),$$
(5)

where P represents the path-ordering operator and Qk is a unitary matrix obtained by flattening the positive energy spectrum of the BdG Hamiltonian (See Supplementary Note 2 for the precise definition of the Qk matrices). Any path \({{{{{{{\mathcal{C}}}}}}}}\) that encloses a nodal line an odd number of times, as shown in Fig. 3c, generates the non-trivial winding number \({{{{{{{{\mathcal{N}}}}}}}}}_{{{{{{{{\rm{1D}}}}}}}}}=2\), which topologically protects the nodal lines. While \({{{{{{{{\mathcal{N}}}}}}}}}_{{{{{{{{\rm{1D}}}}}}}}}=2\) explains the stability of HNLs, hosting surface zero modes associated with the winding number may be improbable due to the general breaking of spatial inversion symmetry on the surface.

HNL state in KZnBi

The observed residual specific heat of KZnBi can be explained by the contributions of the finite density of states at the HNL phase45. Our calculations show that the order parameter ∣ΔB2g∣ ~ 10−1 meV order leads to the largest superconducting gap on the spheroidal surface of 0.04 meV. Fig. 4a shows the calculated DOS, which clearly shows that in stark contrast to the fully gapped superconducting state, the HNL phase exhibits a finite number of states. We also explicitly calculate the corresponding superconducting specific heat Cs in Fig. 4b as a function of normalized temperature T/Tc, where Cs is calculated by50

$${C}_{{{{{{{{\rm{s}}}}}}}}}(T,\Delta )=\frac{1}{T}\mathop{\sum}\limits_{n}\mathop{\sum}\limits_{{{{{{{{\boldsymbol{k}}}}}}}}}\left({E}_{n{{{{{{{\boldsymbol{k}}}}}}}}}^{2}+\beta {E}_{n{{{{{{{\boldsymbol{k}}}}}}}}}\frac{\partial \Delta }{\partial \beta }\frac{\partial {E}_{n{{{{{{{\boldsymbol{k}}}}}}}}}}{\partial \Delta }\right)\left[-\frac{\partial f({E}_{n{{{{{{{\boldsymbol{k}}}}}}}}})}{\partial {E}_{n{{{{{{{\boldsymbol{k}}}}}}}}}}\right],$$
(6)

where \(\beta =\frac{1}{{k}_{{{{{{{{\rm{B}}}}}}}}}T}\), kB is the Boltzmann constant, and Enk is the n-th quasiparticle energy at momentum k. Near zero temperature, the Cs/T curve vanishes for the ΔA1g pairing, while finite residual specific heat persists for the ΔB2g pairing. The experiment45 shows that the linear Cs/T versus T2 plot yields a non-zero value near T = 0 K, which implies that the constant Fermionic contribution exists to the heat capacity even at the low temperature T < 0.8 K.

Fig. 4: Unique characteristics of the hosohedral nodal line (HNL) superconductor.
figure 4

a Density of the quasiparticle states N in the unit of the density of the normal states N0. b Temperature-divided specific heat of the superconducting state(Cs/T), normalized with the specific heat of the normal state at the critical temperature(Cn/Tc). Red (black) Dots represent the data points for ΔB2g (ΔA1g). The line plot of the same color was generated by interpolating the data points. c Inflation of an HNL to Bogoliubov Fermi surface (BFS) in momentum space. The green point and the light orange surface indicate the normal state Dirac point (DP) and the Fermi surface, respectively. The grey-boxed section is magnified in the right inset, which shows the cross-sectional view of the BFS in the kx = 0 plane. The blue dot represents the intersecting point of the HNL penetrating the kx = 0 plane. The red lines illustrate the inflated BFS. The ω marked with the green arrows indicates the size of inflation along the kz axis.

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B-field induced BFS

We demonstrate that the HNL phase corresponds to the topological boundary of the BFS superconducting phase. Consequently, the magnetic field response of KZnBi offers further definitive experimental evidence for the HNL phase through the inflation of HNLs into BFS. To demonstrate this, we introduce the additional Zeeman term \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{Zeeman}}}}}}}}}=-{{{{{{{\boldsymbol{B}}}}}}}}\cdot {{{{{{{\boldsymbol{\mu }}}}}}}}\) in the original Hamiltonian \({{{{{{{{\mathcal{H}}}}}}}}}_{A}\) (Eq. (2)), where B = (Bx, By, Bz) is the magnetic field, and μ = − gμBα/ℏ is the magnetic moment. Here, the α = (αx, αy, αz) is given by αx = sx(τ0 + τz)/2, αy = − sy(τ0 + τz)/2, and αz = − sz, where μB is the Bohr magneton and g is the Landé g-factor, which can effectively account for the spin and orbital effects51. We set the g = 2 and ℏ = 1 for simplicity. The detailed derivation of the Zeeman Hamiltonian is in Supplementary Note 3. Fig. 4c shows the superconducting nodal surface calculated from the BdG Hamiltonian for Δ = ΔB2g in the presence of the z-directed magnetic field strength of Bz = 0.5 × 10−5 eV/μB. We judiciously chose a minimal magnetic field value to demonstrate that BFS inflation occurs immediately upon applying the magnetic field. (See Supplementary Note 3 for the superconducting energy spectra calculated at larger B-fields). The nodes occur on a surface resembling the HNL shape.

We explicitly demonstrate the inflation of the nodal lint into a BFS by directly calculating the evolution of the nodal structure as a function of an applied magnetic field. Fig. 4c shows the change of the nodal line at kx = 0 plane as we increase the z-directed magnetic field strength B. We focus on the nodal structure near the upper polar vertices. The magnified view shows the intersection position between the nodal line and the kx = 0 plane (solid blue dot), which occurs without a magnetic field. The red-colored lines show the cross-sectional position of the inflated BFS on the kx = 0 plane that we obtain as we increase the magnetic field strength from zero to 0.5 × 10−5eV/μB. The zero-dimensional point inflates to one-dimensional lines, which demonstrates the inflation of the one-dimensional nodal lines to the two-dimensional BFS. The size of inflation along the kz axis (ω) is indicated by green arrows in Fig. 4c. The inflation width along the polar axis at the upper polar vertice ω is given by (See Supplementary Note 3 for the derivation.)

$$\omega =2\frac{g{\mu }_{{{{{{{{\rm{B}}}}}}}}}{B}_{z}}{\hslash {v}_{{{{{{{{\rm{F}}}}}}}}}},$$
(7)

where vF is the Fermi velocity along the out-of-plane z-direction. The layered geometry of KZnBi suppresses vF into 0.61 × 106m s−1, which is ten times weaker than the in-plane direction. Correspondingly, the BFS should be more prominent and observable in KZnBi.

Topological characterization of BFS

To further demonstrate the presence and stability of the BFS, we calculate the topological invariant associated with the nodal surface. The TR-broken superconductor with particle-hole symmetry belongs to the D symmetry class. In the three-dimensional momentum space, the nodal surfaces of the D class are protected by the \({{\mathbb{Z}}}_{2}\) invariant defined by a Pfaffian of the anti-symmetrized BdG Hamiltonian33. Topological stability is attained when the Pfaffian at the nodal surface is zero, while the Pfaffian in momentum space separated by the nodal surface has a different sign. Accordingly, our calculations result in the negative (positive) sign of the Pfaffian in the interior (exterior) of the nodal surface. (See Supplementary Note 3 for the detailed calculation). Therefore, our explicit calculation demonstrates the magnetic field induces the topological BFS in the HNL superconducting. Thus, the observation of the BFS may be allowed in KZnBi under the magnetic field.

More ABC compounds

Our first-principles calculations predict more materials should realize the HNL superconductor from the family of ABC ternary compounds. Encouragingly, we have tested known non-polar hexagonal ABC materials obtained from the inorganic crystal structure database52 and found eight and four compounds from the I-XII-V and I-II-V groups that harbor the k ⋅ p effective model. The complete list of the candidate compounds is given in Supplementary Note 4. For example, the AZnBi (A = Na and Li) and KMgBi are found to be a Dirac semimetal from I-XII-V and I-II-V groups, respectively. Moreover, NaZnBi(1−x)Sbx should provide a tunable platform for engineering the band gap and topological properties as in the case of NaZnSb(1−x)Bix in a tetragonal phase53.

Conclusion

In summary, we have studied the HNL superconducting phase in hexagonal ABC compounds based on the mean-field calculation using the k ⋅ p Hamiltonian. We find that the BFS can be achieved from the HNLs, by applying a time-reversal symmetry-breaking magnetic field. Furthermore, we attribute the microscopic mechanism of the superconductivity observed in Dirac semimetal KZnBi to the HNL superconductor. In this regard, the recent discovery of topological Dirac semimetal in KZnBi should provide an exciting opportunity for further study45. Our theoretical prediction applies to the family of hexagonal ABC compounds, providing a catalog of candidate materials to realize the BFS.

Methods

Density functional theory calculations

The first-principles calculations based on the density functional theory were performed using the VASP package54. The modified Becke-Johnson (mBJ) meta-generalized gradient approximation(GGA)55 was used to describe the exchange-correlation energy, with the corresponding mBJ parameter CMBJ = 1.2865. The non-colinear spin-orbit coupling was included, and the plane-wave energy cutoff of 278 eV was given throughout the calculations. The lattice constants of KZnBi were set to be a = 4.68 Å and c = 10.65 Å, adapting the results of the previous work45. The self-consistent charge density was obtained on a uniformly sampled 12 × 12 × 4 momentum grid, and the total energy successfully converged below 10−7eV criterion.

Mean-field calculations

To calculate the superconducting phase diagram, we first fixed the hopping parameter \({m}_{\parallel }^{+}\) and varied the SOC parameter λ1. For a given λ1, we determined the intra-band interaction strength U that yields ∣ΔA1g∣ = 5 meV. This strength ranged from − 4.93 eV to − 5.32 eV. With this fixed value of U as a reference, we varied the magnitude of the inter-band interaction V and solved the gap equation to identify the energetically favored gap function. To solve the multi-band gap equation and determine the magnitude of the superconducting order parameter, we used the methods described by Kim et al.56.