Su(1,2) Algebraic Structure of the XYZ Antiferromagnetic Model in Linear Spin-Wave Frame

The $XYZ$ antiferromagnetic model in linear spin-wave frame is shown explicitly to have an $su(1,2)$ algebraic structure: the Hamiltonian can be written as a linear function of the $su(1,2)$ algebra generators. Based on it, the energy eigenvalues are obtained by making use of the similar transformations, and the algebraic diagonalization method is investigated. Some numerical solutions are given, and the results indicate that only one group solution could be accepted in physics.