Coherent Anomaly Method in Critical Phenomena. I.

A coherent anomaly method (CAM) is formulated for critical phenomena on the basis of the general idea of CAM for cooperative phenomena proposed by Suzuki. The coherent anomalies of response functions are derived phenomenologically using Kubo's linear response theory and Fisher's scaling form of correlation functions. This yields the relation between the coherent anomaly exponents and non-classical critical exponents. The convergence of the CAM critical points is proven rigorously. Some explicit applications of the CAM theory based on systematic cluster-mean-field approximations and on the mean-field transfer-matrix method are presented to show how useful the CAM is in studying critical phenomena. The CAM theory of critical dynamics is also formulated to give the dynamical critical exponent Δ ≃ 1.84 in the two-dimensional kinetic Ising model.