Constant coupling approximation for Ising spin systems

Kazuhiro Takahashi, Motoyuki Tanaka

The spin-1 Ising model with bilinear and biquadratic interactions is investigated using a generalized constant-coupling approximation with two parameters. Equations to determine these parameters are derived by minimizing the variational free energy. The averages of the dipolar and quadrupolar moments expressed in terms of these parameters are first ascertained to give exact results for a linear chain, then investigated in detail for the quadrupolar-coupled system in comparison with the molecular-field results. The pair-correlation and the heat capacity of this system are also investigated in connection with the phase transition. For the system with the bilinear and biquadratic interactions the effect of the anisotropy constant to the phase transition is discussed.

Hadey K. Mohamad

T. Kaneyoshi

K.G. Chakraborty

The two-state Ising model with nearest-neighbour bilinear exchange constant J (2) , three-spin coupling constant J (3) and four-spin coupling constant J (4) is studied using the effective-field approximation. The temperature variation in spontaneous magnetization is studied for different coordination numbers z and for various values of the parameters alpha =J (3) /J (2) and alpha '=J (4) /J (2) . It is seen that, in the absence of three-spin coupling, there exists a critical value of alpha ' (say, alpha ' c ) above which the magnetization becomes double valued at and beyond the Curie temperature T c , forming a bulge near T c which indicates the occurrence of a first-order transition. The expressions for alpha ' c for z=3, 4, 6 are derived and it is found that alpha ' c increases as z increases. In the absence of four-spin coupling it is found that first-order transitions appear for alpha >or=0. The magnetization curves are also studied for the case when all the interactions are present. The results are discussed with reference to those obtained from the molecular-field approximation.

J. A. Plascak

Abstract The thermodynamical properties of the site‐ and bond‐diluted transverse Ising model with anisotropic coupling constants are studied by using a variational procedure based on Bogolyubov's inequality. Within this approach the thermal behaviour of the system is obtained in the entire range of temperature (0 ≤ T < ∞). The results for the critical temperature, critical transverse field, and critical concentration are, in several cases, comparable to the constant‐coupling approximation.

Michael Nauenberg