Pressure Induced Topological Superconductivity in the Spin-Orbit Mott Insulator GaTa4Se8

Lacunar spinel GaTa$_4$Se$_8$ is a unique example of spin-orbit coupled Mott insulator described by molecular $j_{\text{eff}}\!=\!3/2$ states. It becomes superconducting at T$_c$=5.8K under pressure without doping. In this work, we show, this pressure-induced superconductivity is a realization of a new type topological phase characterized by spin-2 Cooper pairs. Starting from first-principles density functional calculations and random phase approximation, we construct the microscopic model and perform the detailed analysis. Applying pressure is found to trigger the virtual interband tunneling processes assisted by strong Hund coupling, thereby stabilizing a particular $d$-wave quintet channel. Furthermore, we show that its Bogoliubov quasiparticles and their surface states exhibit novel topological nature. To verify our theory, we propose unique experimental signatures that can be measured by Josephson junction transport and scanning tunneling microscope. Our findings open up new directions searching for exotic superconductivity in spin-orbit coupled materials.


INTRODUCTION
The confluence of spin-orbit coupling (SOC) and strong electron correlation provides a new paradigm of solid-state quantum phenomena [1][2][3][4][5][6] . In particular, the new type of superconductivity that are expected to arise in spin-orbit coupled Mott insulators has drawn great attentions. The representative candidate materials are transition metal dicalchogenides TaS 2 1 and Sr 2 IrO 4 5-11 . Despite of the promising examples, the microscopic superconducting mechanism itself as well as its pairing symmetry remain elusive. The key step forward is to have a concrete material platform for which the unambiguous theoretical description can be provided and tested. In addition, reliable prediction of pairing symmetry and the detailed suggestions for its experimental verification are demanded.
Strikingly, applying pressure induces the phase transition from a spin-orbit coupled Mott insulator to a metal and eventually to a superconductor 22,[30][31][32] . The characteristics of this superconductivity are quite intriguing in many regards. First, GaTa 4 Se 8 does not show any long-range magnetic order down to low temperature 22,30 . Second, there is no experimental signature for struc-tural transition as a function of pressure and no drastic phonon mode change. Third, nevertheless, the anomalies in specific heat as well as magnetic susceptibility are repeatedly identified at around 50K which is an order of magnitude higher than superconducting T c 22,33-36 . Most importantly, it is also noted that superconductivity is only observed in the case of M=Nb and Ta; namely, only when the low energy band structure is of j eff =3/2 character 28 . This observation heavily prompts a speculation that j eff =3/2 nature of the electronic band structure pervades the origin of the superconductivity, being different from the conventional BCS type. However, there has been no firm investigation on its character both theoretically and experimentally.
In this paper, we show that the superconductivity in GaTa 4 Se 8 is attributed to the new type of electronic pairing. Due to the intriguing interplay of multi-band j eff = 3/2 character and inter-band correlation, novel dwave quintet superconductivity with spin-2 Cooper pairs is stabilized. Such high angular momentum Cooper pair state has been also referred to as the quintet pairing states [37][38][39][40][41][42][43] . Utilizing both density functional theory and random phase approximation (RPA), we first show that the system well retains the characteristic of j eff =3/2 under high pressure. Our first-principles calculations also show how intra-, inter-orbital electron interactions and Hund coupling change by pressure. Starting from the constructed many-body Hamiltonian, we analytically show that applying pressure activates many-body interband tunnelings and opens attractive quintet pairing channels assisted by strong Hund coupling. Among the possible quintet pairings, it turns out the system favors a particular d-wave superconductivity with t 2g symmetries. This novel superconductivity is characterized by nodal lines of Bogoliubov quasiparticles and by topologically protected Majorana modes at the surface. Thereby, our work theoretically establishes GaTa 4 Se 8 as a strong candidate of topological d-wave superconductor. In order to facilitate its confirmation, we also propose the concrete experimental setups and the signatures to be identified in Josephson junction transport and scanning tunneling microscopy (STM).

Electronic Structure and Many-Body Hamiltonian
GaTa 4 Se 8 consists of GaSe 4 and Ta 4 Se 4 clusters arranged in NaCl structure (see Fig. 1(a)) 22 that belong to the space group F43m, which forms non-centrosymmetric structure. Due to the short intra-cluster bondings, its electronic band structure is well understood by molecular orbital states, and the states near Fermi level are dominated by triply degenerate molecular t 2 orbitals denoted by (D xy , D yz , D zx ) 22 . Just as the atomic t 2g orbitals, molecular t 2 can also be represented by effective angular momentum l eff = 1 = −L (1) where L (1) is the angular momentum operator with orbital quantum number l = 1 44 . The spin-orbit interaction, H SOC = −λl · S, gives rise to the molecular quartet j eff = 3/2 and the doublet j eff = 1/2. In particular, molecular quartet j eff = 3/2 in the basis of |j, j z > is being represented as |3/2, ±3/2 >= ∓ 1 √ 2 (|D yz,↑↓ ±i|D zx,↑↓ ) and refer to spin directions 28 . The calculated band dispersions and the projected density of states (PDOS) are shown in Fig. 1(c) as a function of pressure (for more details, see Supplementary Information 1 and Ref. 22 for the crystal structure data under the pressures). Note that, not only at the ambient pressure but at the high pressure up to 14.5 GPa, j eff = 3/2 band characters are well maintained and still dominating the near Fermi energy region. It justifies our low energy model containing j eff = 3/2 states.
In order to take into account electronic correlations, we construct many-body Hamiltonian including intra-orbital (U > 0), inter-orbital interaction (U > 0) and Hund coupling (J H > 0): where d iuσ (d † iuσ ) is the annihilation (creation) operator of electrons with orbital u ∈ (D xy , D yz , D zx ) and spin σ. The third and fourth terms are the Hund exchange and Hund's pair hopping interaction, respectively.
It is remarkable that no matter how large is the intraorbital interaction U , g 1 coupling can be attractive and therefore induce superconducting instability if the Hund coupling is comparable to inter-orbital interaction, J H > 3U . In contrast, the singlet pairing channel cannot be attractive (g 0 < 0) since the Hunds coupling and the Hubbard interactions are both positive. This result is irrespective of the interaction parameters, which single out the possibility of the trivial singlet superconductivity. Importantly, this t 2g symmetry d-wave pairing is robust even when the inter-band mixings between j eff = 3/2 and 1/2 are considered. Fig. 1(c) shows the band separation between j eff = 3/2 and 1/2 gradually decreases as the pressure increases the bandwidth. At high enough pressure, the sizable many-body interband tunneling is expected. In this regime, j eff = 1/2 can make an additional contribution to the effective pairing interaction, g, through the virtual tunneling process. This effect can formally be calculated using many-body Schrieffer-Wolff transformation 46 . Interestingly, we find, the leading order contribution of the interband tunneling is always attractive pairing interactions irrespective of the specific values of (U, U , J H ) (See Supplementary Information 3 B for the estimation of the interaction parameters derived using the RPA calculation.) As a result, the tunneling effect, assisted by strong J H , opens up the attractive superconducting channel characterized by g 1 < 0, and results in quintet-spin Cooper pairs with t 2g d-wave symmetry.

Topological Superconductivity
The intriguing nature of this d-wave quintet Cooper pair can be found in its non-trivial spin texture originating from the unique topological property of Bogliubov-de Gennes (BdG) energy spectrum. Among the possible superconducting order parameter configurations with t 2g symmetry, we find that the energetically most favorable state, ψ T T γψ = (1, 0, 0)-state where γ = (γ 1 , γ 2 , γ 3 ) represents quintet pairing with t 2g symmetry, is characterized by gapless nodal lines as shown in Fig.2 (a); see Supplementary Information 4 for more details of our calculation. Due to the j eff = 3/2 orbital character, the nodal lines in Fig. 2(a) exhibit robust d xy symmetry even in the presence of small inversion symmetry breaking terms. This nodal lines have a topological origin and are protected by the non-trivial winding number.
It is first noted that the particle-hole and the timereversal symmetry allow us to define the following nonhermitian matrix and its singular value decomposition, is the normal Hamiltonian. D k is now a diagonal matrix containing all the positive energy eigenvalues. Second, we can consider the adiabatic band flattening process by smoothly deforming D k to I 4 without any gap closing. This procedure defines the new unitary matrix, q k ≡ U † k V k = n e iλn(k) |n(k) n(k)|, and the corresponding phase λ n (k). These phases are well-defined as long as the system is fully gapped. Therefore, one can assign Z 2 topological winding number along a line that encircles the nodal line as follow: w = i 2π dk · tr(q † k ∇ k q k ) according to DIII class in the Altland-Zirnbauer classifications 47 . Another experiment we suggest is Josephson junction transport. Fig. 3(a)-(c) show the current-phase relation (CPR) for the planar junction of rotating orientations. The CPR can be expressed as a series of sinusoidal harmonics of the phase difference, φ: I J (φ) = n I n sin(nφ) where I n gives the 2πn periodic Josephson current component. Due to the d xy pairing symmetry, Josephson coupling gains π phase under 90 • rotation of the junction orientation, and therefore the sign of I 1 is inverted as shown in Fig. 3(a) and (c). In between the two angles (i.e., when the junction is formed along the [100]direction), the first harmonics vanishes, I 1 = 0; see Fig. 3(b). The next dominant CPR has π periodicity and the resulting Josephson frequency, 4eV /h, is the twice of the conventional Josephson frequency 51 . This frequency doubling can be directly observed from the measurement of the Shapiro step in the I-V characteristics. One can also make use of the pairing symmetry in this material which results in the unconventional magnetic oscillation pattern 52-56 . Fig. 3(d) shows the schematic setup of the Josephson corner junction which is constructed on the corner of the lacunar spinel crystal. Due to the π-phase difference in CPR with different orientations, Josephson currents at each face destructively interfere with each other. However, because of small inversion symmetry breaking in the system, the critical current does not completely cancel but makes the dips in the Fraunhofer diffraction pattern as shown in Fig.3(e). As a consequence, we find that the overall locations of the peaks and the dips in the Fraunhofer pattern should be reversed compared to the case of conventional superconductors. This unusual magnetic oscillation pattern can be regarded as the signature of the quintet pairing in GaTa 4 Se 8 .

DISCUSSION
We discuss the relevance to another lacunar spinel material GaNb 4 Se 8 which shares many similar features with GaTa 4 Se 8 . At ambient pressure, GaNb 4 Se 8 is known to have a Mott gap of 0.19 eV 22 , and the previous calculation shows that its low energy band character is also well-identified by j eff =3/2 states due to the sizable SOC in Nb atoms 28 . The pressure-induced superconductivity is also found with T c = 2.9K at 13GPa 30 . Such similarity with GaTa 4 Se 8 may indicate GaNb 4 Se 8 as another strong candidate of topological superconductors. Nevertheless, in contrast to GaTa 4 Se 8 , GaNb 4 Se 8 has a sizable band overlap between j eff = 3/2 and j eff = 1/2 bands 28 , which indicates the stronger inter-band tunneling effect that goes beyond the analysis of Schrieffer-Wolff transformation method. While stronger virtual tunneling effect on the superconductivity is expected, detailed correlation effect may be different under pressure, depending on molecular states occupied with either Nb or Ta.
In addition, Guiot et al. have performed Te doping in GaTa 4 Se 8 by substitution of Se atoms 24 . The empirical effect of the Te doping is the reduction of the effec-tive bandwidth followed by the increase of the Mott gap. Similarly, we may expect the increase of superconducting critical temperature. This would solidify our prediction that the superconducting pairing mediated by the electron-electron interaction than the phonon coupling.
Furthermore, the pressure control can be another interesting path to control the superconducting phase transition. Near the superconducting critical point, the transition to the time-reversal broken states is expected (See supplementary information 4). The time-reversal broken states is signatured by the Bogoliubov Fermi surfaces, and it can be measured by the anomalous thermal Hall effect similar to that of the p + ip chiral superconductor. In general, the time-reversal broken phase cannot occur in conventional singlet pairing. Therefore, the thermal Hall effect near the superconducting phase transition would be another smoking gun signature of the quintet superconductivity.
In summary, we have suggested a new superconducting pairing mechanism for which spin-orbit entangled multiband nature plays an essential role together with electron correlation. Our theory is developed for and finds its relevance to GaTa 4 Se 8 and other lacunar spinels, where the origin of pressure induced superconductivity has not been understood for a long time. Starting from the realistic band structure and considering the correlation strengths calculated by first-principles DFT calculations, we have developed the detailed microscopic theory. Superconducting gap is found to have d-wave symmetry and its gapless nodal lines emerge with the non-trivial topological character. Furthermore, we have proposed concrete experiments that can confirm our theoretical suggestion. The unusual I-V characteristics and the magnetic oscillation patterns are expected from Josephson transport and can be regarded as the smoking gun signatures for this quintet paring. STM image can also be compared with our results. Our findings will pave a new way to search for exotic superconductivity in lacunar spinel compounds.

First-principles calculation
Electronic structures calculations were performed with OPENMX software package based on linear combination of pseudo-atomic-orbital basis 57 and within local density approximation (LDA) 58,59 . The SOC was treated within the fully relativistic j-dependent pseudopotential scheme 60 . We used the 12 × 12 × 12 k-grids for momentum-space integration and the experimental crystal structures at different pressures 22 . For the estimation of tight-binding hopping and interaction parameters, we used maximally localized Wannier function (MLWF) method 61,62 and constrained RPA (cRPA) technique [63][64][65] as implmented in ECALJ code 66 .

I. DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author on reasonable request.

II. CODE AVAILABILITY
The computer code used for this study is available upon reasonable request.  The electronic structure calculations by using OPENMX were carried out with 400 Ry energy cutoff. In order to take into account of pressure effect, we used the experimental lattice parameters measured at 0 (ambient), 5, 10, and 14.5 GPa 22 . The tight-binding hopping parameters and the interaction parameters were estimated in between t 2 molecular orbitals. As shown in Fig. S1(a), MLWF-based tight-binding bands well reproduce the DFT-LDA results. Fig. S2(a) visualizes the calculated MLWFs denoted by D xy , D yz , and D zx each of which is composed of four atomic d xy , d yz , and d zx orbitals, respectively. Four major hopping parameters are presented in Table. I where we present the values obtained from ECALJ code 66 . This set of hopping parameters were double checked with OPENMX, and the deviations are found to be less than 1 meV.  In order to consider electron interactions, we performed cRPA calculations which can properly take into account of screening effects in solids [63][64][65][67][68][69][70][71] and give rise to the reliable estimation of effective 'on-site' interaction strengths being much smaller than the 'bare' interactions, V. Within RPA, the fully screened interaction U can be calculated from where = 1 − VP and P is the polarization 63 . In order to cooperate with correlated electron models (e.g., Hubbard model), the correlated orbitals or subspaces need to be defined properly. The effective Coulomb interaction U in such a model can be represented by 63 where P d and P r refers to the polarization within the correlated orbitals and the other ('rest') space, respectively; see Fig. S1. The relationship between the fully screened interaction, U, and the partially-screened ('constrained') U can be found by 63 Our calculation results of these values are presented in Table. II. It is noted that the strengths of 'on-site' Coulomb and Hund's interaction are smaller than the typical values for 5d transition metal ions. It is is reasonably well understood from the nature of molecular orbitals which are distributed over the four atomic Ta sites. According to a recent study, these interaction parameters get reduced by a factor of ∼1/4 72 . The effect beyond this simple estimation such as the screenings of other molecular orbitals have been taken into account by our cRPA calculation.  Table. I.

TIGHT-BINDING MODEL DESCRIPTION
In this section, we construct the tight binding model description for the completeness. We mainly repeat the description of Ref. 28 here. The starting point of our tight-binding model is the molecular t 2 orbitals (D xy , D yz , D zx ) basis. In the absence of the SOC, the nearest-neighbor hopping matrix from i site to j site can be generally written as,T Here, we separate the inversion even and odd hopping components as S and A respectively. According to the Wannier function analysis, there exist four distinct hopping channels t 1 , t 2 , t 3 , and t . The inversion even hopping terms, t 1 , t 2 , and t 3 , correspond to t dd1 (σ-type), t pd (π-type), and t dd2 (δ-type) hopping integrals (See Figure S2). In addition, the inversion odd term t is allowed due to the lack of inversion symmetry. In terms of the hopping matrix defined in Eq. (S5), the matrix elements are explicitly given as, (n 1 , n 2 , n 3 ) = (±1, 0, 0) S 11 = t 1 , S 22 = S 33 = t 2 , S 23 = −t 3 , A 13 = −A 12 = ∓t (S6) (n 1 , n 2 , n 3 ) = (0, ±1, 0) S 11 = S 33 = t 2 , S 22 = t 1 , S 13 = −t 3 , A 12 = −A 23 = ∓t (n 1 , n 2 , n 3 ) = (0, 0, ±1) S 11 = S 22 = t 2 , S 33 = t 1 , S 12 = −t 3 , where (n 1 , n 2 , n 3 ) characterizes the direction of the hopping, r ij = n 1 a 1 + n 2 a 2 + n 1 a 2 . a 1,2,3 are the unit vectors of the FCC lattice. We now include the SOC effect in the Hamiltonian as, where λ SO is the strength of the SOC. and L and S are the orbital and the spin angular momentum operators, respectively. We can rewrite the Hamiltonian in Eq. (S5) in terms of j eff basis: where the arrows indicate the electron spin. In j eff basis, the hopping matrix and the on-site SOC term transforms as,T where I n are n-dimensional identity matrix. T 1/2(3/2) describes the intraband hopping terms of j eff = 1/2 (3/2) bands, and Θ represents the interband tunnelings. The explicit forms of the hopping matrices follow Eq. (S5). If the energy splitting between the j eff = 1/2 and 3/2 bands, 3 2 λ SO , is large compared to the inter-orbital hopping terms Θ, j eff = 1/2 and 3/2 subsectors effectively decouples.

PROJECTION TO j eff = 3/2 BASIS
In this section, we project the interacting Hamiltonian to j eff = 3/2 basis. We start our analysis by writing down the interacting Hamiltonian as, where i is the site index, u, v ∈ (xy, yz, zx) denote the orbital indices (D xy , D yz , D zx ) and σ, σ ∈ (↑, ↓) are spin indices.
Here, n iuσ = d † iuσ d iuσ is the number operator and d iuσ (d † iuσ ) are the annihilation (creation) operator of electrons at site i and orbital u with spin σ. U and U represent the intra-orbital and inter-orbital interaction strengths respectively. The third and fourth terms are the Hund exchange interaction and Hund pair hopping interaction respectively parametrized by J. With (U, U , J) > 0, these interaction terms are repulsive in nature. A. j eff = 3/2 intra-band contribution Due to spin orbit coupling, in the absence of interactions, the degenerate t 2 orbital states split into J = 1/2 doublet with energy λ and J = 3/2 quartet with energy −λ/2. Large λ leads to the large energy gap between these states with negligible mixing. Hence, we restrict to J = 3/2 manifold and project the interacting Hamiltonian H I , given in Eq. (S9), to the J = 3/2 basis states. Any operator O expressed in terms of spins and orbitals are projected to the J = 3/2 subspace through the projection operator P 3/2 and the projected operator is denoted asÕ ≡ P 3/2 OP 3/2 . Thus, the projection of the number operators n αβ , spin operators S αβ = (S x αβ , S y αβ , S z αβ ) to the J = 3/2 subspace can be written asñ where α, β, γ ∈ (x, y, z) with α = β = γ and the orbital index u in Eq. (S9) can be represented as u = αβ. J = (J x , J y , J z ) and I is the 4×4 identity matrix. is the annihilation operator of electrons in angular momentum state |J = 3/2, m j and m j = (3/2, 1/2, −1/2, −3/2) with J z being diagonal in this basis. The projected interacting Hamiltonian is written as, which is a 4 × 4 matrix. Hence, we can express the projected interacting Hamiltonian in terms of the Dirac gamma matrices. In our work, the gamma matrices are explicitly given as following: where σ = (σ x , σ y , σ z ) are the Pauli matrices. Using the gamma matrices, we can define the time-reversal operator as, T = γ 1 γ 3 K where K is the complex conjugate operator. The projected interacting Hamiltonian can, thus, be written as

Fierz transformation
Using Fierz identity we can decompose the particle-hole channel interactions into the pairing channel interactions. We will show that the repulsive particle hole channel interactions can be written in the form of attractive pairing channel terms through Fierz transformation. 73,74 The required Fierz identity is, Eq. (S14) are non-zero only for antisymmetric Γ matrices, i.e. Γ ∈ γ 13 , iγ 13 γ 1 , γ 13 γ 2 , iγ 13 γ 3 , γ 13 γ 4 , γ 13 γ 5 . Using these pieces of information, we construct a Table IV giving Table IV, we can write the projected interaction Hamiltonian terms in I γ1 γ2 γ3 γ4 γ5 γ12 γ13 γ14 γ15 γ23 γ24 γ25 γ34 γ35 γ45 γ13 (S13) in pairing channel form as, The total projected pairing interaction can be written as (S16) The first term in Eq. (S16) give rise to the singlet pairing while the remaining terms here correspond to the quintet pairing channel. We notice that the quintet pairing channel can be attractive in the strong Hund's coupling limit, whereas the single pairing channel is always repulsive.
B. Effect of Interband coupling between j eff = 3/2 and j eff = 1/2 bands We now derive the effective many-body Hamiltonian induced by the interband coupling between j eff = 3/2 and j eff = 1/2 bands. To systematically calculate the effective Hamiltonian of j eff = 3/2 band, we employ the many-body Schrieffer-Wolff transformation 46 and exact diagonalization technique.

Schrieffer-Wolff transformation
The Schrieffer-Wolff transformation decomposes the many-body interaction, H I , into diagonal and off-diagonal component, D(H I ) and O(H I ) respectively, where each of them acts with the projection operator, P , to a target subsystem and its complementary space, Q = 1 − P . The diagonal and off-diagonal part can be written as, D(H I ) = P H I P + QH I Q (S17) O(H I ) = P H I Q + QH I P In our case, we aim to derive the effective many-body interacting Hamiltonian of j eff = 3/2 bands. Therefore, P projects to the target subspace, characterized by the quarter-filled electrons in j eff = 3/2 bands and zero electrons in j eff = 1/2 bands. The complementary space corresponds to the states with N electrons filled in j eff = 1/2 bands and N electrons removed from quarter filled j eff = 3/2 bands.
The effective Hamiltonian can be perturbatively expanded as 46 , where we define a superoperator as L(X) = i,j i|O(X)|j Ei−Ej |i j|. The first-order term is just equal to the intra-band contributions considered in the previous section. The second-order describes the virtual many-body hopping processes as shown in Fig.S3(a). Other higher-order terms describe the more complicated virtual processes.
More specifically, we rewrite the interacting Hamiltonian in j eff basis, which is written as, indicates the j eff basis. The second order effective Hamiltonian can be written as, where the coupling constants can be computed from the eigenstates and the energy eigenvalues of the non-interacting Hamiltonian as, where W (k 1 , k 2 , q)(k 1 , k 2 , q) is the interaction coefficient in the energy eigenstate basis. θ is the heavi-side step function. i is the energy eigenvalue of i-th non-interacting band. By numerically plugging in the information of the non-interacting bands, we find that the second-order interband tunneling always contributes as the attractive singlet and quintet pairing interactions. This result becomes analytically apparent if we consider the flat band limit. In this limit, H eff,2 can be explicitly calculated as, where ∆E is the band splitting between j eff = 3/2 and 1/2 bands. As a result, we find that the leading order corrections are all negative, which contributes as an attractive pairing channel. This result is irrespective of the specific values of (U, U , J H ), since they are the complete square form. The exact interband tunneling contribution can be numerically computed up to all orders using the exactdiagonalization technique in the flat band limit. Fig. S3(b) shows the effective pairing strength derived from the exact diagonalization technique as a function of the interband splitting between j eff = 3/2 and j eff = 1/2. We find that the decrease of the interband spacing increases the effect of the interband tunneling, finally contributing as negative correction of g 1 Especially, when the fully screened interaction is taken into account, we find that the pairing interaction strength can turn to a negative value. Eventually, the attractive superconducting pairing channels open. In results, we conclude that the reduction of the band splitting in addition to the strong Hund's coupling may induce the quintet pairing superconductivity in j eff = 3/2 bands. Nevertheless, we also note that the time-reversal broken states can be realizable near the phase transition where the order parameter is suppressed. In such a case, the gap structure of the time-reversal broken states are generally characterized by the Bogoliubov Fermi surfaces. This time-reversal broken states have been similarly found in the previous studies of Luttinger semimetal [37][38][39][40][41][42][43] . However, we also note the important difference with the Luttinger semimetal that GaTa 4 Se 8 is a quarter filled material where the Fermi level lies at the center of the j eff = 3/2 valence band.
As explained in the main text, (1, 0, 0) pairing state is characterized by d xy -wave nodal line gap structure. Fig. S4 (b) shows the density of states(DOS) profile of (1, 0, 0) pairing state with T c = 5.8K as a function fo the temperature. Due to the presence of the nodal lines, the DOS profile shows a linear nodal behavior rather than the full gap. This difference with the BCS superconductivity can be directly observed from the tunneling spectroscopy.