# The bilinear–biquadratic model on the complete graph

We study the spin-1 bilinear–biquadratic model on the complete graph of N sites, i.e. when each spin

is interacting with every other spin with the same strength. Because of its complete permutation

invariance, this Hamiltonian can be rewritten as the linear combination of the quadratic Casimir

operators of ##IMG## [http://ej.iop.org/images/1751-8121/51/10/105201/aaaa92bieqn001.gif] {$

\newcommand{\su}{{\rm su}} \su$} (3) and ##IMG##

[http://ej.iop.org/images/1751-8121/51/10/105201/aaaa92bieqn002.gif] {$ \newcommand{\su}{{\rm su}}

\su$} (2). Using group representation theory, we explicitly diagonalize the Hamiltonian and map out

the ground-state phase diagram of the model. Furthermore, the complete energy spectrum, with

degeneracies, is obtained analytically for any number of sites.