Investigation of thermodynamic properties of mixed-spin (2, 1/2) Ising and Blum–Capel models on a Cayley tree

Abstract

This study focuses on addressing the practical difficulty of extracting statistical properties from the Ising model, which incorporates mixed spin-(2,1/2) configurations and is defined on a Cayley tree of third order. We study Gibbs measures for the mixed spin-(2,1/2) Ising model by leveraging the self-replicating characteristics of the semi-finite Cayley tree. Our aim is to establish the model’s partition function, identify the point at which a phase transition occurs in this model, and compute various thermodynamic properties. We demonstrate the presence of multiple limiting Gibbs measures in both the ferromagnetic and antiferromagnetic regions, thereby establishing the existence of phase transitions in both domains by a numerical approach. Additionally, we investigate the average magnetization linked to the fixed points in the dynamics of the (2,1/2)-MSIM system. We examine the behavior of these quantities as temperature approaches both zero and infinity, with a particular focus on the critical temperature T→0 and as T→∞. Then, utilizing the self-similarity property inherent in the semi-infinite Cayley tree, we deduce precise expressions for the free energy, entropy, and magnetization of the mixed spin-(2,1/2) Blume–Capel model. At certain critical temperatures, we numerically observe peaks and kinks in thermodynamic quantities, including free energy, entropy, and magnetization, within the context of the Blume–Capel model. In the case of the Blume–Capel model, it is noted that an increase in the crystal field (D) corresponds to an increase in these thermodynamic quantities.