We construct and study the relativistic Gross-Neveu-Yukawa field theory for the $\text{SO}\left(2N\right)$ real symmetric second-rank tensor order parameter coupled to $N_f$ flavors of $4N$-component Majorana fermions in $2+1$ dimensions. Such a tensor order parameter unifies all Lorentz-invariant mass-gap orders for $N$ two-component Dirac fermions in two dimensions except for the $\text{SO}\left(2N\right)$-singlet anomalous quantum Hall state. The value $N_f=1$ corresponds to the canonical Gross-Neveu model. Within the leading-order $\epsilon$-expansion around the upper critical dimension of $3+1$, the field theory exhibits a critical fixed point in its renormalization group flow which describes spontaneous symmetry breaking to $\text{SO}\left(N\right)\times \text{SO}\left(N\right)$ for the number of flavors of Majorana fermions higher than a critical value $N_{f,c2}\approx 2N$. For $N_{f,c1}<N_f<N_{f,c2}$, with $N_{f,c1}\approx N$, the critical fixed point resides in the unstable region of the theory where the effective potential is unbounded from below, whereas for $N_f<N_{f,c1}$ there is no real critical fixed point and the flow runs away. In either case, for $N_f<N_{f,c2}$ the transition should become fluctuation-induced first order, and we discuss the dependence of its size on the parameters $N$ and $N_f$ in the theory. One-loop critical exponents for the universality class at $N_{f,c2}<N_f$ are computed and the flow diagram in various regimes is discussed.

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