SO(4) multicriticality of two-dimensional Dirac fermions

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