Effective Field Approximatioin for Random Ordered Phase

The transition temperature of the random ordered phase for the quenched bond problem is calculated by the use of the effective field approximation, which is shown to be naturally derived from the variation principle of the model free energy proposed by Strieb et al. and by Oguchi. As an elementary cluster in this approximation, we take (a) a pair of nearest neighbor spins and (b) n spins consisting a smallest bond loop, and (c) one central spin and its z nearest neighbours. Assuming that the magnitude m of the effective field is the same for all spins, we can express the free energy as a function of m. From this expression we derive the equation determining the transition temperature. This equation still contains the averages of the function of the direction of the effective field. These averages are calculated in two approximations. Thus, the five different approximations are used for calculating the transition temperatures. Two approximations out of five are equivalent to those used by Ono in the cases of the Cayley and the Cactus trees, respectively. The results in these approximations are qualitaively the same, but one of our new approximation is shown to give the best result.