Periodic Gibbs Measures and Their Extremality for the HC-Blume–Capel Model in the Case of a Wand with a Chemical Potential on a Cayley Tree

Periodic Gibbs measures for the HC-Blume–Capel model with a chemical potential with parameters \((\theta,\eta)\) on a Cayley tree in the case of a wand graph are studied. We prove that in this case for \(\theta^3\le\eta\) there exist precisely three periodic Gibbs measures, all of which are translation-invariant, while for \(\theta^3>\eta\) there exist precisely three periodic Gibbs measures, one of which is translation-invari The (non)extremality of these measures is also studied.