Exceptional Symmetry of G$_2$ in Spin-$frac{3}{2}$ Fermion Systems

As the smallest exceptional Lie group and the automorphism group of the non-associative algebra of octonions, G$_2$ is often employed for describing exotic symmetry structures. We prove a G$_2$ symmetry in a Hubbard-like model with spin-$frac{3}{2}$ 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 $S^6$ Goldstone manifold, or, into an SU(2)$times$U(1) with a Grassmannian Goldstone manifold $mbox{Gr}_5^+(mathbb{R}^7)$. In the quantum disordered states, quantum fluctuations generate the effective SU(3) and SU(2)$times$U(1) gauge theories for low energy fermions.