# 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.

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