Extremality of Gibbs Measures for the $HC$-Blume–Capel Model on the Cayley Tree

In this paper, we consider translation-invariant Gibbs measures (TIGMs) for the $HC$-Blume–Capel model in case of “wands” with chemical potential with parameters $(\theta,\eta)$ on the Cayley tree. It is proved that, for $\eta\le\theta^{3}$, there is a unique TIGM and, for $\eta>\theta^{3}$, there are exactly three TIGMs in the case of “wands” with chemical potential for the model under consideration. In addition, the problem of the (non)extremality of these measures is studied.