We study the Gross-Neveu-Yukawa field theory for the $\mathrm{SO}(N$) symmetric traceless rank-2 tensor order parameter coupled to Majorana fermions using the $\epsilon$-expansion around upper critical dimensions of $3+1$ to two loops. Previously we established in the one-loop calculation that the theory does not exhibit a critical fixed point for $N\ge 4$, but that nevertheless the stable fixed point inevitably emerges at a large number $N_f$ of fermion flavors. For $N_f<N_{f,c1}\approx N/2$, no critical fixed point exists; for $N_{f,c1}<N_f<N_{f,c2}$, a real critical fixed point emerges from the complex plane but fails to satisfy the additional stability conditions necessary for a continuous phase transition; and finally only for $N_f>N_{f,c2}\approx N$ does the fixed point satisfy the stability conditions as well. In the present work we compute the $O\left(\epsilon \right)$ (two-loop) corrections to the critical flavor numbers $N_{f,c1}$ and $N_{f,c2}$. Most importantly, we observe a sharp decrease in $N_{f,c2}$ from its one-loop value, which brings it closer to the point $N_f=1$ relevant to the standard Gross-Neveu model. Some three-loop results are also presented and discussed.

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• Critical phenomena
• Effective field theory
• Lower-dimensional field theories
• Majorana fermions
• Quantum criticality
• Quantum field theory
• Quantum phase transitions
• Two-dimensional electron system
• Gross-Neveu model
• Landau-Ginzburg theory
• Large-N expansion in field theory
• Renormalization group