Topological crystalline superconductivity and second-order topological superconductivity in nodal-loop materials

Abstract: We study the intrinsic fully-gapped odd-parity superconducting order in doped nodal-loop materials with a torus-shaped Fermi surface. We show that the mirror symmetry, which protects the nodal loop in the normal state, also protects the superconducting state as a topological crystalline superconductor. As a result, the surfaces preserving the mirror symmetry host gapless Majorana cones. Moreover, for a Weyl loop system (two-fold degenerate at the nodal loop), the surfaces that break the mirror symmetry (those parallel to the bulk nodal loop) contribute a Chern (winding) number to the quasi-two-dimensional system in a slab geometry, which leads to a quantized thermal Hall effect and a single Majorana zero mode bound at a vortex line penetrating the system. This Chern number can be viewed as a higher-order topological invariant, which supports hinge modes in a cubic sample when mirror symmetry is broken. For a Dirac loop system (four-fold degenerate at the nodal loop), the fully gapped odd-parity state can be either time-reversal symmetry-breaking or symmetric, similar to the $A$- and $B$- phases of $^3$He. In a slab geometry, the $A$-phase has a Chern number two, while the $B$-phase carries a nontrivial $\mathbb{Z}_2$ invariant. We discuss the experimental relevance of our results to nodal-loop materials such as CaAgAs.