Exact solution of a quantum integrable system associated with the $G_2$ exceptional Lie algebra

A quantum integrable spin chain model associated with the $G_2$ exceptional Lie algebra is studied. By using the fusion technique, the closed recursive relations among the fused transfer matrices are obtained. These identities allow us to derive the exact energy spectrum and Bethe ansatz equations of the system based on polynomial analysis. The present method provides a unified treatment to investigate the Bethe ansatz solutions for both periodic and non-diagonal open boundary conditions associated with exceptional Lie algebras.