Calculation of the Free Energy of the Ising Model on a Cayley Tree via the Self-Similarity Method

Abstract

In this study, an interactive Ising model having the nearest and prolonged next-nearest neighbors defined on a Cayley tree is considered. Inspired by the results obtained for the one-dimensional Ising model, we will construct the partition function and then calculate the free energy of the Ising model having the prolonged next nearest and nearest neighbor interactions and external field on a two-order Cayley tree using the self-similarity of the semi-infinite Cayley tree. The phase transition problem for the Ising system is investigated under the given conditions.

Calculation of Thermodynamic Quantities of 1D Ising Model with Mixed Spin-(s,(2t − 1)/2) by Means of Transfer Matrix

“…There are many methods for calculating the free energy of lattice models on the Bethe lattices or the d-dimensional integer lattice Z d (d ≥ 1). Some of these are the exact recursion relations [9], the cavity method [18], and vector-valued boundary conditions. One of the most widely used methods to examine thermodynamic quantities corresponding to a 1D Ising model is the transfer matrix technique [21,22].…”

“…When we multiply the transfer matrices P and Q given in Equations ( 17) and (18), we obtain the square matrix:…”

“…In the literature, there are numerous methods for determining the free energy of a given model [13,18]. The free energy of the lattice models on the Bethe lattice is calculated while boundary conditions are taken into consideration [13,18,19].…”

“…In the literature, there are numerous methods for determining the free energy of a given model [13,18]. The free energy of the lattice models on the Bethe lattice is calculated while boundary conditions are taken into consideration [13,18,19]. In-depth research has been performed on the thermodynamic properties of the 1D Ising model with single spin-s using the transfer matrix [20][21][22].…”

The Extremality of Disordered Phases for the Mixed Spin-(1,1/2) Ising Model on Cayley Tree of Arbitrary Order

“…will also analyze the phase transition problem for larger k values. It is well-known that as a CT is very sensitive to the boundary conditions [4], depending on the given boundary conditions, the free energy formulas differ (see [30,38,42,43] for details). In our next studies, we will provide precise formulas for the free energy function related to the (1,1/2)-MSIM for a few boundary conditions.…”