An exact solution of the homogenous trimer Bose-Hubbard model

It is shown that the homogenous three-site (trimer) Bose-Hubbard model with a periodic boundary condition can be solved exactly by using an extended Bethe ansatz based on the solution of the dimer model and the site-permutation symmetry. A solution of the model within the S (3) symmetric or non-symmetric subspace is presented. Coupled differential equations of a series of extended Heine-Stieltjes polynomials, the zeros of which are related to the solution of the model, are derived. Numerical examples of the solution of the model with N <= 4.