LAUSR

The presence or absence of certain symmetries in the normal state (NS) also determines the symmetry of the Cooper pairs. Here, we show that parity ($\mathcal{P}$-) and time-reversal ($\mathcal{T}$-) odd Dirac insulators (trivial or topological) or metals, sustain a local or intraunit cell pairing that supports corner (in $d=2$) or hinge (in $d=3$) modes of Majorana fermions and stands as a higher-order topological superconductor (HOTSC), when the NS additionally breaks discrete fourfold (${C}_{4}$) symmetry. Although these outcomes do not rely on the existence of a Fermi surface, around it (when the system is doped) the HOTSC takes the form of a mixed parity, $\mathcal{T}$-odd (due to the lack of $\mathcal{P}$ and $\mathcal{T}$ in the NS, respectively) $p+id$ pairing, where the $p(d$)-wave component stems from the Dirac nature of quasiparticles (lack of ${C}_{4}$ symmetry) in the NS. Thus, when strained, magnetically ordered Dirac materials, such as doped magnetic topological insulators (${\mathrm{MnBi}}_{2}{\mathrm{Te}}_{4}$), can harbor HOTSCs, while the absence of an external strain should be conducive for the axionic $p+is$ pairing.