Gibbs measures for the HC Blume–Capel model with countably many states on a Cayley tree

We study the Blume–Capel model with a countable set $$\mathbb Z$$ of spin values and a force $$J\in \mathbb R$$ of interaction between the nearest neighbors on a Cayley tree of order $$k\geq 2$$ . The following results are obtained. Let $$\theta=e^{-J/T}$$ , $$T>0$$ , be the temperature. For $$\theta\geq 1$$ , there exist no translation invariant Gibbs measures or $$2$$ -periodic Gibbs measures. For $$0<\theta< 1$$ , we prove the uniqueness of a translation-invariant Gibbs measure. Let $$\Theta=\sum_i\theta^{(k+1)i^2}$$ and $$\Theta_\mathrm{cr}(k)=k^k/(k-1)^{k+1}$$ . If $$0<\Theta\leq\Theta_\mathrm{cr}$$ , then there exists exactly one $$2$$ -periodic Gibbs measure that is translation invariant. For $$\Theta>\Theta_\mathrm{cr}$$ , there exist exactly three $$2$$ -periodic Gibbs measures, one of which is a translation-invariant Gibbs measure.

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