Phase transition and Gibbs measures of Vannimenus model on semi-infinite Cayley tree of order three

Ising model with competing nearest neighbors (NN) and prolonged next nearest neighbors (NNN) interactions on a Cayley tree has long been studied, but there are still many problems untouched. This paper tackles new Gibbs measures of Ising-Vannimenus model with competing NN and prolonged NNN interactions on a Cayley tree (or Bethe lattice) of order three. By using a new approach, we describe the translation-invariant Gibbs measures (TIGMs) for the model. We show that some of the measures are extreme Gibbs distributions. In this paper, we try to determine when phase transition does occur.