Translation-invariant Gibbs measures for the mixed spin-$1/2$ and spin-$1$ Ising model with an external field on a Cayley tree

Phase transitions of the mixed spin-$1/2$ and spin-$1$ Ising model under the presence of an external field on the general order Cayley tree are investigated within the framework of the tree-indexed Markov chains. We find the conditions that ensure the existence of at least three translation-invariant Gibbs measures for the model on the Cayley tree of order $k$. We are able to solve the model exactly on the binary tree $(k=2)$ under the specific external field. The main attention is paid to the systematic study of the structure of the set of the Gibbs measures. We find the extremality and non-extremality regions of the disordered phase of the model on the binary tree.