Higher-order topological superconductor in ${\mathcal P}, {\mathcal T}$-odd quadrupolar Dirac metal

Presence or absence of certain symmetries in the normal state (NS) also plays important role in determining the symmetry of the Cooper pairs. We here show that two- and three-dimensional Dirac metals, realized by doping parity (${\mathcal P}$) and time-reversal (${\mathcal T}$) odd topologically trivial Dirac insulators, sustain a local or inter-unit cell pairing that assumes the form of a time-reversal odd and mixed parity (due to the absence of ${\mathcal T}$ and ${\mathcal P}$ in the NS, respectively) pairing around the Fermi surface. When the NS additionally breaks discrete four-fold ($C_4$) symmetry (yielding a ${\mathcal P}$, ${\mathcal T}$-odd, quadrupolar Dirac metal), the system gives birth to a higher-order $p+id$ pairing, hosting corner (in $d=2$) or hinge (in $d=3$) modes of codimension $d_c=2$ of Majorana fermions. While the $p$-wave component stems from the Dirac nature of quasiparticles in the NS, appearance of the $d$-wave component is solely attributed to the lack of $C_4$ symmetry, as its restoration produces $p+is$ pairing.