# Higher-order topological odd-parity superconductors

## TL;DR

This paper identifies key conditions involving Fermi surfaces and Berry phases that determine the topological nature of odd-parity superconductors, providing a principle to design higher-order topological superconductors and insulators.

## Contribution

It reveals the conditions for realizing higher-order topological odd-parity superconductors and offers a simple principle for designing such materials.

## Key findings

- Contractibility of Fermi surfaces without crossing time-reversal invariant momenta is crucial.
- Presence of nontrivial Berry phase on Fermi surfaces is essential.
- Removable Dirac pairing nodes are necessary in normal metals.

## Abstract

The topological property of a gapped odd-parity superconductor is jointly determined by its pairing nodes and Fermi surfaces in normal state. We reveal that the contractibility of Fermi surfaces without crossing any time-reversal invariant momentum and the presence of nontrivial Berry phase on Fermi surfaces are two key conditions for the realization of higher-order topological odd-parity superconductors (HOTOPSCs). When the normal state is a normal metal, we reveal the necessity of removable Dirac pairing nodes and provide a general and simple principle to realize HOTOPSCs. Our findings can not only be applied to analyze the topological property of odd-parity superconductors, but also be used as a guiding principle to design new platforms of higher-order topological superconductors, as well as higher-order topological insulators owing to their direct analogy in Hamiltonian description.