On some Gibbs measures of a coupled Ising-Ising model
Keywords:
Gibbs measures, coupled Ising-Ising model, Cayley tree, translation invariance, Hamiltonian, phase transition, statistical mechanics.Abstract
In this paper, we study the translation-invariant Gibbs measures of a
coupled Ising-Ising model defined on a Cayley tree of order k ≥ 2. The model is derived as
a special case of the coupled Ising-Potts model when the number of Potts states q = 2. We
consider a Hamiltonian with spin configurations represented by (s,σ) ∈ {−1,1} × {1,2} and
obtain a system of nonlinear equations characterizing the corresponding Gibbs measures.
For the particular case k = 2, the system is analyzed in detail. We prove that if the
interaction parameter satisfies the condition θ + θ−1 > 2 + 2, then the system admits at least
three distinct translation-invariant Gibbs measures. This result highlights the occurrence of
a phase transition in the model and contributes to the understanding of multi-component
spin systems on hierarchical structures.
References
Rozikov U.A, Gibbs measures in biology and physics: The Potts model., World
Scientific, Singapore, 2023.
Haydarov F.H, Omirov B.A, Rozikov U.A, Coupled Ising-Potts Model: Rich set of
critical
temperatures
arXiv:2502.12014v11. 2025.
