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 = 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{1}{2} $$\end{document} and c = 710\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{7}{10} $$\end{document}, 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.