Coherent-Anomaly Method in Critical Phenomena. III.

Two kinds of systematic mean-field transfer-matrix methods are formulated in the 2-dimensional Ising spin system, by introducing Weiss-like and Bethe-like approximations. All the critical exponents as well as the true critical point can be estimated in these methods following the CAM procedure. The numerical results of the above system are T_c* = 2.271 (J/k_B), γ=γ’ ≃ 1.749, β≃0.131 and δ ≃ 15.1. The specific heat is confirmed to be continuous and to have a logarithmic divergence at the true critical point, i.e., α=α′=0. Thus, the finite-degree-of-approximation scaling ansatz is shown to be correct and very powerful in practical estimations of the critical exponents as well as the true critical point.