SO(4) multicriticality of two-dimensional Dirac fermions

We study quantum multicritical behavior in a $(2+1)\text{\ensuremath{-}}\mathrm{dimensional}$ Gross-Neveu-Yukawa field theory with eight-component Dirac fermions coupled to two triplets of order parameters that act as masses and transform $(1,0)+(0,1)$ representation under the $\mathrm{S}\mathrm{O}(4)\ensuremath{\simeq} \mathrm{S}\mathrm{O}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SO}(3)$ symmetry group. This is relevant spin-1/2 on honeycomb or $\ensuremath{\pi}$-flux lattices, for example, near transition points between an $s$-wave superconductor charge-density wave, one side, N\'eel order, other. Two such always allow common pair other would complete them maximal set compatible (anticommuting) orders five. first derive unitary transformation Nambu (particle-hole) space which maps any triplets, possibly containing some superconducting orders, onto purely insulating parameters. allows consider universal SO(4) description multicriticality without doubling. then proceed renormalization-group flow coupling constants at one-loop $4\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions, allowing also more general transforming $\mathrm{S}\mathrm{O}({n}_{a})\ifmmode\times\else\texttimes\fi{}\mathrm{S}\mathrm{O}({n}_{b})$. While ${n}_{a}={n}_{b}>2$ bosonic sector decoupled there stable fixed point flow, Yukawa quickly leads its elimination by generic fixed-point collision range fermion flavor numbers ${N}_{f}$. suggests replacement critical runaway physical case ${n}_{a}={n}_{b}=3$. The structure renormalization group ${n}_{a}\ensuremath{\ne}{n}_{b}$ discussed, nonperturbative arguments favor stability when ${n}_{a}=3$ ${n}_{b}=1$ $D=2+1$ are provided.