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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.