Nodal topological superconductivity in nodal-line semimetals

We analyze possible nodal superconducting phases that emerge from a doped nodal-line semimetal. We show that nodal-line superconducting phases are favored by interactions mediated by short-range ferromagnetic fluctuations or Hund's coupling. It is found that the leading pairing channels are momentum-independent, orbital-singlet and spin-triplet. In the pairing state, we show that the Bogoliubov-de Gennes (BdG) Hamiltonian hosts a pair of topologically protected nodal rings on the equators of the torus Fermi surface (FS). Using a topological classification for gapless systems with inversion symmetry, we find that these nodal rings are topologically nontrivial and protected by integer-valued monopole charges $\nu = \pm 2$. In the scenario of pairing driven by ferromagnetic fluctuations, we analyze the fate of superconductivity in the magnetically ordered phase. Based on Ginzburg-Landau free energy analysis, we find the energetically favored superconducting state is characterized by the coexistence of two pairing orders whose $\bf d$-vectors are perpendicular to the magnetization axis $\bf M$ with their phases unfixed. In this case, each nodal loop in the pairing state splits into two, carrying a $\pm 1$ monopole charge. For bulk-boundary correspondence, these nodal rings enclose flat-band Majorana zero modes on top and bottom surface Brillouin Zones with distinct $\mathbb{Z}$-valued topological invariants.