Multi-effective-field theory: Applications to the CAM analysis of the two-dimensional Ising model

A multi-effective-field theory is formulated and applied to the two-dimensional Ising model from the viewpoint of the coherent-anomaly method (CAM). Two necessary conditions to construct the CAM canonical series are shown. Two series of approximations are derived on 3×3- and 4×4-clusters and with the use of the CAM we estimate the critical exponent of the susceptibility within an error of 0.37 and 0.45 percent, respectively, for the exact value γ = 1.75 where the exact T_c is assumed. This accuracy of the estimation shows the effectiveness of this theory. It is generally proved that certain kinds of approximations with different combinations of effective fields yield exactly the same approximate critical temperature and mean-field coefficient.