From G2 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 G₂. 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 G₂ 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 G₂ 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 G₂ critical phase exhibits the degeneracy between bosonic and fermionic states, which are reminiscences of the continuum model.