Construction of $G_2$ symmetry in a Hubbard-type model

As the smallest exceptional Lie group and the automorphism group of the non-associative algebra octonions, $G_2$ is often employed for describing exotic symmetry structures. We construct $G_2$ symmetry in a self-dual Hubbard-type model with 4-component fermions in a bipartite lattice, which lies in the intersection of two $SO(7)$ algebras connected by the structure constants of octonions. Depending on the representations of the order parameters, the $G_2$ symmetry can be spontaneously broken into either an $SU(3)$ one associated with an $S^6$ sphere Goldstone manifold, or, into $SU(2)\times U(1)$ with a Grassmannian Goldstone manifold. In the quantum disordered states, quantum fluctuations generate the effective $SU(3)$ and $SU(2)\times U(1)$ gauge theories for low energy fermions.