We study superconducting (SC) phases that are naturally proximate to a spin-orbit-coupled SU(8) Dirac semimetal on a honeycomb lattice. This system, which offers enhanced low-energy symmetries, presents an interesting platform for realizing unconventional superconductivity in $j=3/2$ electrons. In particular, we find 72 superconducting charge-$2e$ fermion bilinears, which, under classification of microscopic symmetries, lead to 12 different SCs—four singlets, two doublets, and six triplets—seven of them are gapped and five are symmetry-protected nodal SCs. The strong spin-orbit coupling leads to locking of the spin of the Cooper pairs with real-space direction—as is evident from the structure of the Cooper-pair wave functions—leading to unusual nonunitary superconductors (even singlets), and with finite-momentum pairing (for the triplets). This results, in many cases, in the magnitude of multiple pairing gaps being intricately dependent on the direction of the SC order parameter. The present classification of SCs along with normal phases [Phys. Rev. B 108, 245106 (2023)] provides the complete list of naturally occurring phases in the vicinity of such a SU(8) Dirac semimetal. This study allows for understanding the global phase diagram of such systems, stimulating further experimental work on candidate materials such as metallic halides ($\mathit{MX}{}_3$ with $M=\mathrm{Zr},\mathrm{Hf}$, and $X=\mathrm{Cl},\mathrm{Br}$). Further, it provides the starting point for the exploration of unconventional phase transitions in such systems.

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• Multiband superconductivity
• Pair density wave
• Superconductivity
• Dirac semimetal