On Ising model with four competing interactions on Cayley tree (Q1029258)

The paper studies a spin model on a Cayley tree of order two. Its Hamiltonian \(H\) includes four competing interactions (external field, nearest neighbor, second neighbors and triples of neighbors). For a sequence \(V_1 \subset V_2 \subset \dots\) with \(\bigcup_{n=1}^{\infty}V_n=V\) and Gibbs measures \(\mu^{(n)}\) on \(\{-1,+1\}^{V_n}\) (\(n\in\mathbb{N}\)) satisfying the consistency conditions one obtains a limit Gibbs measure \(\mu\) on \(\{-1,+1\}^V\) as \(n\to \infty\). The criterion (see equation (5) of the paper) to meet these conditions is given in terms of the model parameters appearing in \(H\) and in the description of measures \(\mu^{(n)}\). In a particular case it is proved that there are three translation-invariant Gibbs measures \(\mu_1,\mu_2,\mu_3\) as equation (5) admits different solutions (i.e. there is a phase transition). The complete description of the periodic Gibbs measures for the model is provided as well. Moreover, the continuum of distinct non-periodic extreme Gibbs measures is constructed.