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 $\mathrm{SO}\left(N_A\right)$ and $\mathrm{SO}\left(N_B\right)$ 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 $\mathrm{SO}(N_A+N_B)$ 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 $\mathrm{SO}\left(4\right)$ and $\mathrm{SO}\left(5\right)$ 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 $\mathrm{SO}\left(4\right)\simeq \mathrm{SO}\left(3\right)\times \mathrm{SO}\left(3\right)$ 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.

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• Dirac fermions
• Metal-insulator transition
• Quantum criticality
• Renormalization group
• Graphene
• Critical exponents
• O(N) symmetry