Fragile topology in nodal-line semimetal superconductors

We study the band topology of the superconducting nodal-line semimetal (SC-NLSM) protected by the inversion symmetry with and without the spin-orbital coupling. Without the spin-orbital coupling, both the $s$-wave SC-NLSM and the chiral $p$-wave SC-NLSM are topologically nontrivial and can be described by the nonzero winding number. Based on the Wilson loop method, we verify that they are both the fragile topological superconductors, namely, their nontrivial band topologies can be moved off by coupling to additional topologically trivial bands. The fragile topological phase persists in spinful system with the time-reversal symmetry when a spin-orbital coupling term is added. For the spinful system, both the $p$-wave SC-NLSM and the $s$-wave SC-NLSM are second-order fragile topological superconductors. We propose that the fragile topology in the SC-NLSM system depends strongly on the degeneracy of the Majorana zero modes and the parity of the superconducting gap function. Interestingly, in presence of a vortex line, the spinful $s$-wave SC-NLSM system hosts two pairs of stable Majorana zero modes in the vortex core.

The concept of the fragile topology was proposed very recently [18,[34][35][36][37][38][39][40][41][42][43]. The Wanniner obstruction of a fragile topological material can be removed by coupling an additional topologically trivial band [34]. Generally, the fragile topological state do not host robust gapless edge state, thereby it may break the bulk and edge correspondence. Several materials, such as the twisted graphene [44], the Dirac semimetal with the higher order fermi arc [45], and the photonic crystal [38], are proposed to exhibit fragile topological properties. An NLSM system with two nodal loops is also proposed to exhibit fragile topology [18,34].
The method of the symmetry indicator has been widely used in searching for various topologically nontrivial sys- * Corresponding author: tzhou@scnu.edu.cn tems [46][47][48][49][50][51][52]. Recently, this method has been developed to search for the topological superconductor [53][54][55][56][57][58]. For the topological superconducting system, based on the symmetry indicator method, the band topology is described through the pairing symmetry and the representation of the filling bands in the normal state. Moreover, in Ref. [58], the concept of the fragile topology was extended to the superconducting system. The topology of the fragile topological superconductor can also be diagnosed by the symmetry indicator method. Similar to the fragile topological insulating system, if a topologically nontrivial superconducting system becomes a topologically trivial superconductor when coupling to one or several topologically trivial bands, then this system is a fragile topological superconductor. It was proposed that the system with two copies of Kitaev chains in presence of the inversion symmetry is a fragile topological [58]. Searching for other fragile topological superconducting systems is of interest. Especially, as far as we know, the possible fragile topological superconducting state in the three dimensional system has not been explored yet.
The Majorana Krammer pairs protected by the mirror symmetry and the rotational symmetry have been studied intensively [59][60][61][62]. However, as proposed in Ref. [18], if the energy bands of the material invert twice or more, the edge states may be not stable. For the superconducting NLSM (SC-NLSM) system, the energy bands naturally invert twice due to the particle-hole symmetry. As a result, the edge states and the Majorana Krammer pairs may be not stable. Thus the SC-NLSM system may be a candidate to realize the fragile topology, with the topological properties being rather different from the stable topological system.
In this paper, we study the fragile topological properties of the SC-NLSM system with and without the spin orbital coupling. In the normal state, the NLSM system we considered has the inversion symmetry, the time rever-sal symmetry and the mirror symmetry. In the superconducting state, previously the preferred pairing symmetry for an SC-NLSM system has not been identified yet. The s-wave pairing symmetry was proposed by a recent experiment [22], but the spin polarized surface state and the nontrivial (pseudo)-spin texture on the torus Fermi surface of the NLSM favor the spin triplet pairing instability [29,33]. So both pairing symmetries are considered in our present work. The fragile topology is verified based on the Wilson loop method. In the past the vortex line has been widely used to probe the topological properties [63][64][65][66][67]. Here a vortex line is introduced to both the s-wave and the p-wave SC-NLSM. The possible vortex states are studied numerically through the zero energy local density of states (LDOS). Our results indicate that the numerical results in presence of a vortex line may depend strongly on the specific systems and no general conclusions can be made for the vortex states in a fragile topological superconductor.
The structure of our paper is organized as follows. In Sec. II, we discuss the topological properties of the spinless SC-NLSM system. In Sec. III, we discuss the topological properties of the spinful SC-NLSM system in presence of the spin-orbital coupling. At last, we present a brief summary in Sec. IV.

A. Model and formalism
We start with a model in the three dimensional system including the normal state term (H N ) and the superconducting pairing term (H s/p SC ), H(k) NLSM is a two-band model for the spinless (or preserving SU (2) spin rotational symmetry) NLSM system [7], with, where M (k) = m−(t x k 2 x +t y k 2 y +t z k 2 z ) and λ z (k) = 2βk z .
We consider two kinds of pairing symmetries, namely, the s-wave pairing symmetry and the chiral p-wave pairing symmetry. The s-wave superconducting pairing term is expressed as, and the chiral p-wave one is expressed as, σ i and τ i are the pauli matrices on the orbital channel and the particle-hole channel, respectively. ∆ s/p are the pairing amplitudes. µ represents the chemical potential. t i (i = x, y, z) and β are the hopping constants. The vector basis of Eq.
, with the indices a and b representing two orbitals.
In the normal state, the nodal ring at the Fermi energy is protected by the the inversion symmetry (I), the time reversal symmetry (T ), and the mirror symmetry (M xy ). The operators are expressed as: T = σ x K (K is the complex conjugate operator), To study the symmetry in the superconducting state, we first study the parity of the pairing term. From Eqs. (5) and (6), the momentum dependent pairing functions are rewritten as: ∆ s (k) = ∆ s iσ y , ∆ p (k) = ∆ p (k x σ 0 + k y iσ 0 ). The parities of superconducting pairing term are expressed as, The s-wave pairing term does not break any symmetry of the NLSM. Moreover, it induces some additional symmetries. The whole Hamiltonian with the s-wave pairing term is invariant under the particlehole symmetry (P s ), the time reversal symmetry (T s ), the chiral symmetry (S), the inversion symmetry (I s ) and the mirror symmetry (M s xy ). Based on the parity in Eq.(7)-(10), the symmetries of the whole s-wave SC-NLSM Hamiltonian are expressed as: Based on the ten fold Atland-Zirnbauer classification [68,69], it belongs to the class BDI. As for the chiral p-wave pairing, the time reversal symmetry is naturally broken. The system is protected by the particle-hole symmetry (P p = τ x ⊗ σ 0 K), the mirror symmetry (M p xy = τ 0 ⊗ σ x ) and the inversion symmetry (I p = τ z ⊗ σ x ). Therefore, the topology of a chiral p-wave SC-NLSM system is categorized to the class D. In the lattice system and performing the partial Fourier transformation along the z direction, the Hamiltonian is reexpressed as, The superconducting pairing term is then expressed as, and The whole Hamiltonian in the superconducting state [Eqs. (4)(5)(6)] can be rewritten as a 4N z × 4N z matrix . N z is the number of sites along the zdirection. The Green's function can be obtained through the matrix M . The matrix elements for the Green's function are expressed as below where u in (k) and E n (k) are the eigenvectors and the eigenvalues obtained by diagonalizing the matrix M . The spatial dependent spectral function and the LDOS can then be expressed as, and with m = 4(z − 1). N xy is the number of sites of the x− y plane.
We would also like to study the vortex bound states of the superconducting system [63]. The vortex line is induced to the system through considering ∆(r) = ∆ 0 tanh( x 2 + y 2 /ζ)e iθ , with θ = arctan( y x ). ζ is the coherence distance. The parameters in the following presented results are set as β = t x = t y = t z = 1, m = 3, ∆ s = 0.125, ∆ p = 0.35, and ζ = 4.  [70].
For the s-wave pairing symmetry, as is seen in Fig. 1(a), when k y increases from −π to 0, the valance bands and the conduction bands cross twice at the Fermi energy. There are two topologically nontrivial regions indicated as I and II in Fig. 1(a). As presented in Figs. 1(a) and 1(c), the energy band inverts once and twice at these two regions and the corresponding winding numbers are 1 and 2. For the chiral p-wave pairing state, as is shown in Figs. 1(b) and 1(d), the energy bands are fully gapped. They are two-fold degenerate when at the k z = 0 plane. The winding number is changed directly from 0 to 2 when the momentum crosses the normal state nodal ring.
We turn to study the edge states through considering the open boundary condition along the z direction and the periodic boundary condition along the x − y plane [Eqs. (11)(12)(13)]. The corresponding energy bands for the s-wave pairing and the chiral p-wave pairing with and 2(f). In presence of an additional chemical potential term, the previous four-fold degenerate edge states shown in Figs.2(a) and 2(b) split into two two-fold degenerate edge states corresponding to the electron states and the hole states. Generally, the chiral symmetry would be broken by the chemical potential term in the two band NLSM Hamiltonian, then the zero energy surface state shifts to the finite energy. However, in the s-wave superconducting state, the chiral symmetry is preserved even when the chemical potential is nonzero because of the particle-hole symmetry. As is shown in Fig. 2(c), the gap of the surface state is partially opened [at the region II of Fig. 1(a)], even though without breaking any symmetry. For the chiral p-wave pairing symmetry, as is shown in Fig. 2(d), the surface state is fully gapped in presence of an additional chemical potential term.
Our results indicate that when the winding number equals to 2, the gapless edge states are not stable. In principle, the nonzero winding number may generate Majorana zero modes at the system edges. However, the Majorana zero modes with equal spin at the same site are usually not stable. They will annihilate into the finite energy quasiparticles with the particle-hole symmetry. As a result, an energy gap will be opened at the region II when an additional chemical potential term is added. This also implies that the system may belong to a fragile topological superconductor.
As defined in Ref. [58], the Wannier obstruction of a fragile topological superconductor can be removed through coupling additional topologically trivial bands.
Here we consider four additional particle-hole symmetric atomic orbitals and couple them into the original Hamiltonian of the SC-NLSM to verify its fragile topological properties. The whole Hamiltonian is expressed as, where the base vector is expressed as Ψ T . The inversion and particle-hole operators are rewritten as These two symmetries preserve in the whole coupled Hamiltonian.
Based on the Wilson loop method [1][2][3]70], the corresponding Wilson loop spectra from the above coupled Hamiltonian are calculated and presented in Fig. 3. As is seen, for both pairing symmetries, the Wilson loop spectra have been unwound, indicating that the systems become topologically trivial. Our numerical results verify directly the fragile topology of the present SC-NLSM system.

C. Vortex bound states
Let us study the possible vortex bound states in the spinless SC-NLSM system. The intensity plots of the zero energy LDOS spectra in presence of a vortex line for different chemical potentials and different pairing symmetries are displayed in Fig. 4. We first discuss the possible vortex bound states in the chiral p-wave superconducting state. When the chemical potential is zero [ Fig. 4(a)], the zero energy states exist at the system surface with z = 1 and z = N z . In this case there are no bound states existing. As the chemical potential increases to µ = 0.2, the vortex bound states emerge in the vortex core, as is seen in Fig. 4(b), which is similar to the surface phase transition proposed in the fully gapped second-order topological superconductor with a vortex line [64]. On the other hand, here the energy of the bound state is not exactly at the zero energy, as is seen in Fig. 4(c). Our numerical results indicate that no Majorana modes exist in the vortex core.
Since the spinless SC-NLSM system has mirror symmetry, the Hamiltonian of the chiral p-wave SC-NLSM can be block diagonalized into two 2 × 2 matrices with, at the k z = 0 plane is expressed as The two subsector Hamitonians have a nontrivial Chern number (1/ − 1) and they both have the particle-hole symmetry P with P = σ x K. Based on the ten fold AZ classification, they both belong to the class D and can transform to each other via the spinless time reversal operation. In presence of a vortex line, in principle two pairs of Majorana zero modes should appear. However, similar to the case of the surface states, here the Majorana states are not stable. They will annihilate and four finite energy bound states with the particle-hole symmetry emerge, as presented in Fig. 4(c).
The above numerical results can be understood further through analysing the topological invariant. The topological invariant of the vortex line for the gapped p-wave system is defined as [6][7][8]70], Here v p = −1/+1 describes the topologically nontrivial/trivial vortex line, with/without the Majorana bound states at the end of the vortex line. Based on our calculation [70], here v p equals to +1, being independent on the chemical potential µ. This indicates that the vortex bound states obtained in Fig. 4(b) are indeed not Majorana bound states, while on the other hand, we cannot conclude that here the vortex line is topologically trivial. It was proposed in Ref. [58] that two copies of Kitaev chains protected by the inversion symmetry is a fragile topological superconductor and hosts fragile end states.
Here is qualitatively the same with that of the chiral p-wave compound, namely, the zero energy states appear at the system surfaces and no vortex bound states exist. Interestingly, when the chemical potential increases, as is seen in Fig. 4(d), the result is significantly different from that in the chiral p-wave system [77]. There are no bound states in the vortex core. Instead, the intensity of the zero energy LDOS is nearly zero in the vortex region.
Outside of the vortex region, the zero energy LDOS increases and it reaches the maximum value at the system corners.
The above results for the s-wave compound can be understood through exploring the superconducting gap magnitudes. In the normal state, the flat bands at the system surface will shift to the energy E = µ when an additional chemical potential µ is added to the system. On the other hand, in the s-wave superconducting state, the flat bands still exist due to the particle-hole symmetry. The size of the flat band region will increase monotonously when the gap magnitude increases [70]. As a result, the zero energy LDOS depends strongly on the superconducting order parameter magnitudes when the chemical potential is nonzero. Here in presence of a vortex, the gap magnitude is suppressed in the vortex core and is largest at the system corners, leading to the distribution of the zero energy LDOS presented in Fig. 4(d).
Let us discuss the numerical results of the s-wave SC-NLSM system in more detail. In presence of a vortex line along the z direction, the momentum along the z direction (k z ) is still a good quantum number. Considering the open boundaries along the x and y directions with the system size 41 × 41 and putting a vortex line at the site (x, y) = (20, 20), the band structure as a function of k z is presented in Fig. 5. As is seen, the system is indeed gapless with the energy bands crossing the Fermi energy at k z = 0. Actually, for a s-wave SC-NLSM, the gapless states are robust and protected by the mirror symmetry.
We block diagonalize the Hamiltonian for the s-wave For the whole 4 × 4 Hamiltonian, two nodal rings exist at the mirror plane (k z = 0), consistent with the numerical results of the band structure presented in Fig. 1. When a vortex line along the z direction is induced to the system and considering the open boundaries along the x and y directions, the Hamiltonian Eq.(1) can be rewritten as: In the presence of vortex line, the mirror symmetry and the inversion symmetry preserve [70]. At the mirror plane k z = 0, the Hamiltonian can be expressed as, (k z = 0)) will cross to each other and generate nodal points at k z = 0.

A. Model and formalism
Generally the spin-orbital coupling (or the broken SU (2) rotational symmetry) will directly change the topological classification of the system and lead to different topological properties. In a real material system, the spin-orbital coupling usually cannot be ignored. Here we start to discuss the topological properties of the spinful SC-NLSM system in the presence of the spin orbital coupling. The Hamiltonian is expressed as, (29) h(k) is the Hamiltonian of NLSM in presence of the spinorbital coupling, with where R is the Rashba spin orbital coupling strength. h s/p soc (k) represents the superconducting pairing, with and σ i , s i , τ i are pauli matrices and in the orbital, spin, and particle-hole channels, respectively.
The base vector is expressed as The parameters are set as β = 1, t x,y,z = 1, m = 3, R = 0.4, ∆ soc In the normal state with ∆ soc s/p = 0, the above Hamiltonian is protected by the inversion symmetry (g 1 = s 0 ⊗ σ x ), the time reversal symmetry (g 2 = is y K ⊗ σ x ), and the mirror symmetry (g z = is z ⊗ σ x ).
To study the symmetry of the system in the superconducting state, we rewrite the s-wave and the p-wave gap functions as, ∆ soc s (k) = 2∆ soc s is y ⊗ σ z and ∆ soc p (k) = 2∆ soc p [sin(k x )s 0 ⊗ σ 0 + i sin(k y )s z ⊗ σ 0 ]. The parities of the superconducting order parameters are expressed as, Based on the above results, The s-wave/p-wave SC-NLSM both belongs to the class DIII. For both systems, the superconducting parameter is odd with the inversion symmetry operator. Note that in presence of the spinorbital coupling, the energy bands are both fully gapped.

B. Fragile topology of the spinful SC-NLSM system
Let us study the topological properties of the spinful SC-NLSM system based on the Wilson loop method [1][2][3]70]. The numerical results of Wilson loop spectra for the s-wave and p-wave pairing symmetries with the winding direction along the k z direction are presented in Figs. 6(a) and 6(b), respectively. As is seen, for both pairing symmetries, the Wilson loop spectra are winding, indicating that the spinful SC-NLSM system is indeed topologically nontrivial.
We now consider additional atomic bands coupling to the present spinful SC-NLSM system and make sure that the symmetry of the whole system does not change. The whole Hamiltonian is expressed as, The numerical results of the Wilson loop spectra from the above coupled Hamiltonian for the s-wave pairing symmetry and the p-wave pairing symmetry are presented in Figs. 6(c) and 6(d). As is seen, the Wilson loop spectra are unwound when coupling additional atomic bands, indicating that the spinful SC-NLSM compounds should indeed be fragile superconductors. Actually, previously the fragile topology is usually proposed in the spinless system. It is of rather interest to realize it in a spinful system. We have also checked numerically that when the winding direction is along the k x direction or the k y direction, the results are qualitatively the same with those along the k z direction.
Considering the open boundaries, we present the zero energy LDOS with the s-wave and p-wave pairing symmetries in Fig. 7. As is seen, the zero energy states emerge at the system hinges for both two pairing symmetries we considered, indicating that these two systems may be second-order topological superconductors.
For the p-wave system, the edge states emerge at the boundaries of the z = 1 and z = N z planes. It seems that these two planes can be seen as the two-dimensional Chern insulator, indicating that the topology of the pwave SC-NLSM system may be described by the mirror Chern number. The mirror symmetry operator g z has two eigenvalues i and −i. The spinful p-wave SC-NLSM Hamiltonian can be block diagonalized as, The Chern numbers of the Hamiltonian in the subsector of system (H p,gz −i and H p,gz i ) are 2 and −2, respectively. Then the mirror Chern number should be 2, consistent with the nontrivial topology of this system. Based on the bulk and edge correspondence, those nonzero Chern number (2 and −2) in the two subsectors should in principle generate two spin-up edge states and two spin-down ones. Because of the particle-hole symmetry, each subsector will generate a pair of spin polarized Majorana zero modes. Two spin polarized Majorana zero modes at the same site are not stable, leading to the fragile topology of this system.
For the s-wave pairing symmetry, the second-order topology can be investigated through the Wannier spectra [4,70]. Considering the open boundary condition along the x-direction and periodic boundary condition along the y −z plane, the Wannier spectra with the winding direction along the k y -direction is plotted in Fig. 8. Our numerical results indicate that a quantized Wannier spectra with v x y,kz=0 = 0.5 exist, corresponding to the hinge polarization. The Wannier spectra are highly degenerate at this value. Based on the argument provided in Ref. [4], the degeneracy determines the number of majorana zero modes per hinge. Therefore, here each hinge may host mutiple channels of Majorana zero modes, leading to the fragile topology of this system. Usually, the fragile topological insulator is expected to be realized in the spinless system or the topological crystalline insulator. Here we have verified that in a superconducting spinful system with the time reversal symmetry, the fragile topology can also be realized. Our results indicate that the fragile topology depends strongly on the parity of the gap function under the inversion operation.
Here both pairing functions are odd under the inversion operation, as is seen in Eqs. (33) and (36). We have also checked numerically the topological properties with other pairing symmetries. For a chiral p-wave pairing symmetry, the pairing function is also odd under the inversion operation. The system is also a fragile topological superconducting system. For the s-wave pairing symmetry with ∆ s (k) = ∆ s is y ⊗ (σ x /σ 0 ), the gap function is even under the inversion operation. We have checked and confirmed that in this case the system is topologically trivial. Besides the inversion symmetry, the fragile topology may also exist in other symmetry configurations. The fragile topology of the SC-NLSM in a spinful system and the band topology with other possible symmetry configurations are of interest and worth further studies.

C. Vortex bound states
Let us turn to study the vortex bound state for the SC-NLSM system with the spin-orbital coupling. The intensity plots of the zero energy LDOS in presence of a vortex line for the s-wave SC-NLSM system and the p-wave SC-NLSM system are presented in Figs. 9(a) and 9(b), respectively. The corresponding low energy eigenvalues of the Hamiltonian are presented in Figs. 9(c) and 9(d). As is shown in Fig. 9(a), the bound states emerge in the vortex core for the s-wave SC-NLSM system. The energy of the bound states is exactly at the zero energy, as is seen in Fig. 9(c). Note that, we have checked numerically that the four zero energy eigenvalues are rather stable and does not depend on the chemical potential. On the other hand, for the p-wave SC-NLSM system, no obvious bound states are obtained. We have checked numerically that the results are qualitatively the same with 0 ≤ µ ≤ 2.
In the normal state, the Hamiltonian h(k) has the C 4 rotational symmetry. In the superconducting state, the s/p pairing term breaks the C 4 rotational symmetry and only the C 2 rotational symmetry exists with C 2 = exp(iπs z /2)⊗σ 0 . The pairing terms are odd under the C 2 rotational operation, with C 2 ∆ soc s (k)/∆ soc p (k)(C 2 ) −1 = −∆ s (−k x , −k y , k z )/∆ p (−k x , −k y , k z ). As is discussed in Ref. [59], the symmetry protected Majorana Krammer pairs in a topological superconductor should satisfy the condition that its superconducting gap is odd under certain symmetry operation. Here the four zero energy vortex bound states in the s-wave SC-NLSM system can be stabilized by the C 2 rotational symmetry, which is similar to the double Majorana vortex bound states studied in Refs. [60,79]. Note that, the C 2 operator has two eigenvalues, as a result, two pairs of Majorana vortex bound states can survive. For the p-wave SC-NLSM system, the two subsectors H p,gz i and H p,g=z −i in Eqs. (39)(40)(41) have a Chern number 2 and −2, respectively. In presence of a vortex, in principle, each subsector should generate two pairs of majorana zero modes. Here the C 2 rotational symmetry cannot stabilize four pairs of Majorana zero modes. Thus it is understandable that for the pwave SC-NLSM system, the Majorana zero modes cannot survive. In a fragile topological superconducting system with multiple topologically nontrivial bands, the vortex line topology is a rather complicated issue and still needs further studies.

IV. SUMMARY
In summary, we have studied the topological properties of the SC-NLSM system considering the s-wave pairing symmetry and the p-wave pairing symmetry. For the spinless system, we have verified that the system should be a fragile topological superconductor for both two pairing symmetries we considered. For the spinful system in presence of the spin-orbital coupling and with the time reversal symmetry, the system is a second-order fragile topological superconductor, with the fragile topology is protected by the inversion symmetry. Our results indicate that the fragile topology in the SC-NLSM system comes from the multiple deneracy of Majorana zero modes. When the symmetry in a system cannot stablize these zero modes, the system may become topologically trival one or a fragile topological one. This criteria may be applied to search for other fragile topological superconductors.
In presence of a vortex line, the possible vortex states in the SC-NLSM systems are discussed. For the spinless p-wave system, the vortex bound states exist at the vortex core while no zero energy states exist. For the spinless s-wave system, the energy bands are gapless even in presence of a vortex line and no vortex bound states are obtained. Here the zero modes may exist at the system surface, depending on the parameters, while for all of the parameters we considered, no zero modes are obtained in the vortex core. These results are rather different from those obtained in usual stable topological superconductors. For the spinful system in presence of the spin-orbital coupling, the robust zero energy modes exist in the vortex core for the s-wave system, while no vortex bound states exist for the p-wave system. Therefore, in a fragile topological superconductor, whether the bound states and Majorana zero modes exist in the vortex core may depend strongly on the real system. Our results may help to understand the fragile topology in the superconducting system further.