Phase transitions on the dark side of the Gross-Neveu model: Spontaneous O ( 4 N ) symmetry breaking at repulsive coupling

Gross-Neveu model in 2+1 dimensions exhibits a continuous transition from gapless Dirac semimetal to the gapped quantum anomalous Hall (QAH) insulator at a finite (attractive) coupling, at which the inversion and time-reversal symmetry become spontaneously broken, and the flavor O($M$) symmetry remains preserved. A unification of leading order parameters of 2+1 dimensional $N$ four-component Dirac fermions collects all Lorentz-singlet mass-like fermion bilinears, except the one condensing in the QAH state, into an irreducible representation of the O($M=4N$), and predicts another phase transition in the Gross-Neveu model to occur at a strong (repulsive) coupling. Here, a fermionic auxiliary-field quantum Monte Carlo algorithm is employed in order to study a lattice realization of the Gross-Neveu field theory in the repulsive regime, where the sign problem is absent. We indeed find the O($4N$) symmetry breaking transition out of Dirac semimetal to occur and to be weakly first-order for $N=2$, relevant to graphene. The size of the discontinuity and the magnitude of the critical coupling, however, both grow with $N$. Adding a finite chemical potential is found to break the symmetry and cause superconductivity. These results are in broad agreement with the predictions of the unified field theory. Our lattice model also displays an interesting exact O($2N$) symmetry, a subgroup of the low-energy O($4N$), and has the ordered ground state with the order parameter that belongs to its $N(2N-1)$-dimensional representation. Other order parameters are also examined, and a certain hierarchy among those that belong to different representations of the exact $O(2N)$ is observed.