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

Abstract

It was recently established that the paradigmatic Gross--Neveu model with $N$ copies of two-dimensional Dirac fermions features an $\mathrm{SO}(2N)$ 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 $\mathrm{SO}(2N)$ model predicts, besides the chiral Ising transition at $g_{c1}$, a second critical point $g_{c2}$ where $\mathrm{SO}(2N)$ is broken down to $\mathrm{SO}(N)\times\mathrm{SO}(N)$. 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 $N_f > N_{f,c}^{\mathrm{ST}}(N) \approx 0.56 + 1.48 N +\mathcal{O}(\epsilon)$, 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.