Periodic Gibbs Measures and Their Extremes for the HC–Blume–Capel Model in the Case of a ‘‘Wand’’ on the Cayley Tree

In this paper, we study the periodic Gibbs measures for the HC–Blume–Capel model in the case of a ‘‘wand’’ on a Cayley tree of order two. It is proved that in this case, for $$0<\theta<1$$ there exist exactly three periodic Gibbs measures that are translation-invariant; and for $$\theta>1$$ there exist exactly three periodic Gibbs measures, one of which is translation-invariant, two others are periodic (not translation-invariant) with period two. In addition, the problem of the (non) extremes of these measures has been studied.

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