# Second-order Dirac superconductors and magnetic field induced Majorana hinge modes

## TL;DR

This paper explores three-dimensional higher-order superconductors with Majorana hinge states and surface states, identifying models with topological invariants and symmetry-protected hinge modes, including effects of Zeeman fields.

## Contribution

It introduces six s-wave pairing-based models of higher-order superconductors, revealing new hinge Majorana states and their symmetry protections.

## Key findings

- Two models exhibit flat-band hinge Majorana states with surface states.
- Hinge states are protected by rotation and mirror symmetries.
- Zeeman term can switch hinge states between different hinges.

## Abstract

We identify three-dimensional higher-order superconductors characterized by the coexistence of one-dimensional Majorana hinge states and gapless surface sates. We show how such superconductors can be obtained starting from the model of a spinful quadrupolar semimetal with two orbitals and adding an s-wave superconducting pairing term. By considering all the possible s-wave pairings satisfying Fermi-Dirac statistics we obtain six different superconducting models. We find that for two of these models a flat-band of hinge Majorana states coexist with surface states, and that these models have a non-vanishing quadrupole-like topological invariant. Two of the other models, in the presence of a Zeeman term, exhibit helical and dispersive hinge states localized only at two of the four hinges. We find that these states are protected by combinations of rotation and mirror symmetries, and that the pair of corners exhibiting hinge states switches upon changing the sign of the Zeeman term. Furthermore, we show that these states can be localized to a single hinge with suitable perturbations. The remaining two models retain gapless bulk and surface states that spectroscopically obscure any possible hinge states.