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

TL;DR

This paper explores how mirror symmetry in doped nodal-loop materials leads to topological crystalline superconductivity, hosting Majorana modes, with implications for higher-order topology and experimental detection.

Contribution

It reveals the topological nature of superconductivity in nodal-loop materials protected by mirror symmetry, including the emergence of Majorana cones, Chern numbers, and hinge modes.

Findings

Mirror symmetry protects topological superconducting states.

Surface Majorana cones are present in certain phases.

Quantized thermal Hall effect and hinge modes are predicted.

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.