Spontaneous breaking of the SO(2N) symmetry in the Gross-Neveu model

The canonical Gross-Neveu model for N two-component Dirac fermions in $2+1$ dimensions suffers a continuous phase transition at a critical interaction gc1 ~ 1/N at large N, at which its continuous symmetry SO(2N) is preserved and a discrete (Ising) symmetry becomes spontaneously broken. A recent mean-field calculation, however, points to an additional transition at a different critical gc2~ -N gc1, at which SO(2N) → SO(N) × SO(N). To study the latter phase transition we rewrite the Gross-Neveu interaction g (ψ̄ ψ)² in terms of three different quartic terms for the single ($L=1$) $4N$-component real (Majorana) fermion, and then extend the theory to $L>1$. This allows us to track the evolution of the fixed points of the renormalization group transformation starting from L 1, where one can discern three distinct critical points which correspond to continuous phase transitions into (1) SO(2N)-singlet mass-order-parameter, (2) SO(2N)-symmetric-tensor mass-order-parameters, and (3) SO(2N)-adjoint nematic-order-parameters, down to $L=1$ value that is relevant to the standard Gross-Neveu model. Below the critical value of Lc (N)≈ 0.35 N for N1 only the Gross-Neveu critical point (1) still implies a diverging susceptibility for its corresponding (SO(2N)-singlet) order parameter, whereas the two new critical points that existed at large L ultimately become equivalent to the Gaussian fixed point at $L=1$. We interpret this metamorphosis of the SO(2N)-symmetric-tensor fixed point from critical to spurious as an indication that the transition at gc2 in the original Gross-Neveu model is turned first-order by fluctuations.