Higher-order topological superconductors in P-, T-odd quadrupolar Dirac materials

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 (P-) and time-reversal (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, T-odd (due to the lack of P and 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 (MnBi2Te4), can harbor HOTSCs, while the absence of an external strain should be conducive for the axionic p + is pairing.