Nodal higher-order topological superconductivity from a $\mathcal{C}_4$-symmetric Dirac semimetal

TL;DR

This paper investigates higher-order topological superconductivity in a $ ext{Cd}_3 ext{As}_2$-like Dirac semimetal with $ ext{C}_4$ symmetry, revealing stable Majorana zero modes and nodal structures protected by topological charges and symmetry.

Contribution

It demonstrates that certain pairing symmetries in Dirac semimetals lead to higher-order topological superconducting states with stable Majorana modes and complex nodal structures.

Findings

Presence of four BdG Dirac point nodes protected by a $ ext{Z}_2$ monopole charge.

Existence of higher-order Majorana zero modes on hinges in rod geometries.

Stable nodal cages formed by nodal lines in certain pairing channels.

Abstract

We analyze the topological properties of the possible superconducting states emerging from a Cd$_3$As$_2$-like, $\mathcal{C}_4$-symmetric Dirac semimetal, with two four-fold degenerate Dirac points separated in the $k_z$ direction. Unlike the simplest Weyl semimetal for which all pairing orders are topologically obstructed and nodal, we show that the topological obstruction for pairing in Dirac semimetals crucially only exists for certain pairing symmetries. In particular, we focus on odd-parity $B_{1u}$ and $B_{2u}$ pairing states, both of which can be induced by Ising ferromagnetic fluctuations. The $B_{1u}$ and $B_{2u}$ pairing states inherit the topological obstruction from the normal state, which dictates that these states necessarily hosts four Bogolibov- de Gennes (BdG) Dirac point nodes protected by a $\mathbb{Z}_2$ monopole charge. By a Wannier state analysis, we show that the topological obstruction in the superconducting states is of higher-order nature. As a result, in a rod geometry with gapped surfaces, arcs of higher-order Majorana zero modes exist in certain $k_z$ regions of the hinges between the BdG Dirac points. Unlike Fermi arcs in Weyl semimetals, the higher-order Majorana arcs are stable against self-annihilation due to an additional $\mathbb{Z}$-valued monopole charge of the BdG Dirac points protected by $\mathcal{C}_4$ symmetry. We find that the same $\mathbb{Z}$-valued charge is also carried by $B_{1g}$ and $B_{2g}$ channels, where the BdG spectrum hosts bulk ``nodal cages", i.e., cages formed by nodal lines, that are stable against symmetry preserving perturbations.