Emergent SU(8) Dirac semimetal and proximate phases of spin-orbit coupled fermions on a honeycomb lattice

Emergent Dirac fermions provide the starting point for understanding the plethora of novel condensed matter phases. The nature of the associated phases and phase transitions crucially depends on both the emergent symmetries as well as the implementation of the microscopic ones on the low-energy Dirac fermions. Here, we show that j = 3/2 electrons in spin-orbit coupled materials on honeycomb lattice can give rise to SU(8) symmetric Dirac semimetals with symmetry implementation very different from that of graphene. This nontrivial embedding of the microscopic symmetries in the low energy is reflected in the nature of phases proximate to the Dirac semimetal. Such phases can arise from finite short-range electron-electron interactions. In particular, we identify 24 such phases-divided into three classes-and their low-energy properties obtained by condensing particle-number conserving fermion bilinears that break very different microscopic symmetries and/or are topologically protected by symmetries. The latter includes interesting generalizations of quantum spin-Hall phases. Remarkably some of the resultant phases still support a subset of gapless fermions-protected by a subgroup of SU(8)-resulting in interesting density wave semimetals. Near the phase transitions to such density wave semimetals, the surviving gapless fermions strongly interact with the bosonic order parameter field and give rise to novel quantum critical points. Our study is applicable to a wide class of d1 and d3 transition metals with strong spin-orbit coupling and predicts that such materials can harbor a very rich interplay of symmetries and competing interactions in the intermediate correlation regime.