Bilinear-Biquadratic model on the complete graph

We study the spin-$1$ bilinear-biquadratic model on the complete graph of $N$ sites. First, we show that due to the fully connected structure, the bilinear-biquadratic Hamiltonian can be mapped to the sum of the $\SU(3)$ and $\SU(2)$ Casimir operators. With the help of representation theory, we explicitly diagonalize the Hamiltonian and find the phase diagram of the model. We discuss the ground state degeneracy and the low lying states of the spectrum.