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

Abstract

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.