From $G_2$ to $SO(8)$: Emergence and reminiscence of supersymmetry and triality

We construct a (1+1)-dimension continuum model of 4-component fermions incorporating the exceptional Lie group symmetry $G_2$. Four gapped and five gapless phases are identified via the one-loop renormalization group analysis. The gapped phases are controlled by four different stable $SO(8)$ Gross-Neveu fixed points, among which three exhibit an emergent triality, while the rest one possesses the self-triality, i.e., invariant under the triality mapping. The gapless phases include three $SO(7)$ critical ones, a $G_2$ critical one, and a Luttinger liquid. Three $SO(7)$ critical phases correspond to different $SO(7)$ Gross-Neveu fixed points connected by the triality relation similar to the gapped SO(8) case. The $G_2$ critical phase is controlled by an unstable fixed point described by a direct product of the Ising and tricritical Ising conformal field theories with the central charges $c=\frac{1}{2}$ and $c=\frac{7}{10}$, respectively, while the latter one is known to possess spacetime supersymmetry. In the lattice realization with a Hubbard-type interaction, the triality is broken into the duality between two $SO(7)$ symmetries and the supersymmetric $G_2$ critical phase exhibits the degeneracy between bosonic and fermionic states, which are reminiscences of the continuum model.