Ground states and phase transition of the $\lambda$ model on the Cayley tree

We consider the $\lambda$ model, a generalization of the Potts model, with spin values $\{1,2,3\}$ on the order-two Cayley tree. We describe the model ground states and prove that translation-invariant Gibb measures exist, which means that a phase transition exists. We establish that two-periodic Gibbs measures exist.