Quantum multicriticality and emergent symmetry in Dirac systems with two order parameters at three-loop order

Two-dimensional materials with interacting Dirac excitations can host quantum multicritical behavior near the phase boundaries of the semimetallic and two-ordered phases. We study such behavior in Gross--Neveu--Yukawa field theories where $N_f$ flavors of Dirac fermions are coupled to two order-parameter fields with $SO(N_A)$ and $SO(N_B)$ symmetry, respectively. To that end, we employ the perturbative renormalization group up to three-loop order in $4-\epsilon$ spacetime dimensions. We distinguish two key scenarios: (i) The two orders are compatible as characterized by anticommuting mass terms, and (ii) the orders are incompatible. For the first case, we explore the stability of a quantum multicritical point with emergent $SO(N_A\!+\!N_B)$ symmetry. We find that the stability is controlled by increasing the number of Dirac fermion flavors. Moreover, we extract the series expansion of the leading critical exponents for the chiral $SO(4)$ and $SO(5)$ models up to third order in $\epsilon$. Notably, we find a tendency towards rapidly growing expansion coefficients at higher orders, rendering an extrapolation to $\epsilon=1$ difficult. For the second scenario, we study a model with $SO(4) \simeq SO(3) \times SO(3)$ symmetry, which was recently suggested to describe criticality of antiferromagnetism and superconductivity in Dirac systems. However, it was also argued that a physically admissible renormalization-group fixed point only exists for $N_f$ above a critical number $N_{c}^>$. We determine the corresponding series expansion at three-loop order as $N_{c}^>\approx 16.83-7.14\epsilon-7.12\epsilon^2$. This suggests that the physical choice of $N_f=2$ may be a borderline case, where true criticality and pseudocriticality, as induced by fixed-point annihilation, are extremely challenging to distinguish.