Higher-order topological superconductors from Weyl semimetals.

We propose that doped Weyl semimetals with four Weyl points are natural candidates to realize higher-order topological superconductors, which exhibit a fully gapped bulk while the surface hosts robust gapless chiral hinge states. We show that in such a doped Weyl semimetal, a featureless finite-range attractive interaction favors a $p+ip$ pairing symmetry. By analyzing its topological properties, we identify such a chiral pairing state as a higher-order topological superconductor, which depending on the existence of a four-fold roto-inversion symmetry $\mathsf{R}_{4z}$, is either intrinsic (meaning that the corresponding hinge states can only be removed by closing the bulk gap, rather than modifying the surface states) or extrinsic. We achieve this understanding via various methods recently developed for higher-order topology, including Wannier representability, Wannier spectrum, and defect classification approaches. For the $\mathsf{R}_{4z}$ symmetric case, we provide a complete classification of the higher-order topological superconductors. We show that such second-order topological superconductors exhibit chiral hinge modes that are robust in the absence of interaction effects but can be eliminated at the cost of introducing surface topological order.