New correlated-effective-field theory in the Ising model

Differential Operator Technique in the Ising Spin Systems

“…In the differential operator technique, on the other hand, the concept (44) has been used for evaluating the multispin correlation functions [15]. This is sharply in contrast to the above approach.…”

“…In order to treat the Callen identity (9), he noticed the following exact relations valid for and so on, where K = AI for nearest-neighbor interaction J. For instance, the identity (9) for the honeycomb lattice with coordination number z = 3 can be, upon using the exact relation (15), rewritten as where i+δ (b = 1, 2, 3) denote the nearest-neighbors of the site i. However, when Ei in (9) includes a number of Ising spins, it is not so easy to write the corresponding exact relation.…”

“…As is understood from (16) (or (15)), the use of the kinematic relations for the spin operation is a cucial step in the theory based on the identity (9) (or (12)) as an average over a finite polynomial spin operators belonging to the neighboring sites. This can be systematically and easily achieved by the use of a differential operator technique introduced by Honmura and Kaneyoshi [4]:…”

Expanded Bethe–Peierls approximation for the Ising model with S = 1/2 and 1

“…In the other case, the model must be solved by approximation method or numerical method, such as the molecular field approximation (MFA), the Bethe-Peierls approximation (B-PA) [3], the Bethe lattice approximation (BLA) [4], in which the lattice is simplified as an infinite Cayley tree with the coordination number z, the effective field theory (EFT) [5], the coherent anomaly method (CAM) [6], the double-chain approximation (DCA) [7], which is a natural extension of the pair approximation, the cluster variation method (CVM) [8], the series expansion methods (SE) [1,9], the Monte Carlo technique (MC) [10], and so on. The transition temperatures obtained by the former seven approximation methods are higher than the exact value, the B-PA and BLA give the same transition temperatures for S = 1/2, and the DCA with the large cluster gives the transition temperature closing mostly to the exact value.…”

Effective Field Method for Ising Model with Arbitrary Ferromagnetic Interaction

“…The simplest approximation (effective field approximation) suggested by Honmura and Kaneyoshi [3] neglects the correlation between spin operators on different sites. In [4] the authors omitted in the equation for magnetization all correlations higher than the three-spin ones and decoupled the latter. Another approach was proposed in [2,5].…”