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.

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