Critical Temperature of a Quasi‐One‐Dimensional Ising Model. The Bethe‐Peierls Approximation

Abstract

The critical temperature of a three-dimensional spatially anisotropic Ising model is computed by the Bethe-Peierls method applied to the five-chain cluster.

On the critical temperature of a quasi-one-dimensional Ising model

“…For comparison we have constructed table 2. From this table it is seen that our calculation perceptibly improves the estimates given by EBPA [9]. So, in the standard case J' = J the error in the determination of a phase transition temperature decreases from 6.6% (EBPA) to 1.5% (phenomenological renormalizationgroup; pair (3.4)).…”

“…Thus, owing to the small number of terms in the high-temperature series, the results for kTcfJgiven in [4] are restricted to J'/J 3 0.01; because of the rather small stretch of the parallelepipeds (n'/n ,c 6) the estimates obtained in [5] for the temperature of a phase transition have a considerable error when J'/J < 0.1 and the renormalization group values [6] have been found only forJ'/J 2 0.1. On the other hand, there are calculations of the critical temperature based on the mean-field theory: the linear king chain in a mean field (here the intrachain interactions are taken exactly and the interchain interactions approximately) [7], the double king chain in the analogous field (part of interchain interactions is described rigorously) [SI and finally the extended Bethe-Peierls approximation (EBPA) when a cluster of the central chain and the four chains nearest to it is placed in a molecular field [9]. These approaches, unlike the previous ones, lead to satisfactory results only for high anisotropy of interaction.…”

Extended chain approximation in quasi‐one‐dimensional ising model

“…For the critical temperature, these approximations are inferior or nearly equal to the Bethe-Peierls approximation (BPA) ( Table 1) and the accuracy of the DCA depends on the dirnension- ality, that is, the number of spins treated by the molecular field. On the other hand, there are the extended Bethe-Peierls approximation (EBPA) in the simple cubic lattice [9], the triple chain approximation (TCA) in the square lattice [lo], and the extended variational method (EVM) [ll] as approximations superior to the BPA. The EBPA treats five chains (Fig.…”