Exceptional Symmetry of G₂ in Spin-3/2 Fermion Systems

As the smallest exceptional Lie group and the automorphism group of the non-associative algebra of octonions, G₂ is often employed for describing exotic symmetry structures. We prove a G₂ symmetry in a Hubbard-like model with spin-3/2 fermions in a bipartite lattice, which lies in the intersection of two SO(7) algebras connected by the structure constants of octonions and dual to each other. Depending on the representations of the order parameters, the G₂ symmetry can be spontaneously broken into either an SU(3) one associated with Goldstone manifold S⁶, or into an SU(2) × U(1) with a Grassmannian Goldstone manifold. In the quantum disordered states, quantum fluctuations generate the effective SU(3) and SU(2) × U(1) gauge theories for low energy fermions, both of which are of interest in high energy physics. In (1+1) spacetime dimension, this model shows a triality relation among three different phases, which is inherited from the underlying SO(8) structure.