Spontaneous symmetry breaking of $\mathrm{S}\mathrm{O}(2N)$ in Gross--Neveu theory from $2+\epsilon$ expansion

It was recently established that the paradigmatic Gross–Neveu model with $N$ copies of two-dimensional Dirac fermions features an $\mathrm{S}\mathrm{O}(2N)$ symmetry if certain interactions are suppressed. This becomes evident when the theory is rewritten in terms of $2N$ copies of two-dimensional Majorana fermions. Mean-field theory for the $\mathrm{S}\mathrm{O}(2N)$ model predicts, besides the chiral Ising transition at $g_{c1}$, a second critical point $g_{c2}$ where $\mathrm{S}\mathrm{O}(2N)$ is broken down to $\mathrm{S}\mathrm{O}(N)\times \mathrm{S}\mathrm{O}(N)$. A subsequent Wilsonian renormalization group analysis directly in $d=3$ supports its existence in a generalized theory, where $N_f$ copies of the $4N$-component Majorana fermions are introduced. This allows to track the evolution of a (i) quantum anomalous Hall Gross–Neveu–Ising, (ii) symmetric-tensor, and (iii) adjoint-nematic fixed point separately. However, it turns out that (ii) and (iii) lose their criticality when approaching $N_f=1$, suggesting that the transition is first order. In this work, we approach the problem from the lower-critical dimension of two. We construct a Fierz-complete renormalizable Lagrangian, compute the leading order $\beta$ functions, fermion anomalous dimension, as well as the order parameter anomalous dimensions, and resolve the three universality classes corresponding to (i)–(iii). Before becoming equal to the Gaussian fixed point at $N_f=1$, (ii) remains critical for all values of $N_f>N_{f,c}^{\mathrm{S}\mathrm{T}}(N)\approx 0.56+1.48N+\mathcal{O}(\epsilon )$, which compares well with the estimate of previous studies. We further find that (iii) becomes equal to (i) when approaching $N_f=1$. An instability is, however, only present in the susceptibility corresponding to the Gross–Neveu–Ising order parameter.