Translation-Invariant Extreme Gibbs Measures for the Blume–Capel Model Withwand on a Cayley Tree

Abstract

We study translation-invariant Gibbs measures for the Blume–Capel model with wand on a Cayley tree of order k. We find the exact critical value 𝜃cr = 1 such that, for 𝜃 ≥ 𝜃cr , there exists a unique translation-invariant Gibbs measure and, for 0 < 𝜃 < 𝜃cr , there are exactly three translation-invariant Gibbs measures in the presence of a wand in the analyzed model. In addition, we study the problem of (no) extremes for these measures.