New approach to critical phenomena theory

David Paul Landau

G. Toulouse

Masuo Suzuki, Makoto Katori

A new method is proposed to study critical phenomena, which is a unification of Weiss' mean-field theory and Fisher's finite-size scaling theory. A systematic series of clusters is considered and generalized mean-fields are applied to the boundary spins of such clusters. From the size-dependence of the coefficients of mean-field singularities systematically obtained, the static critical exponents α, β and γ, and critical slowing down exponent Δ can be estimated. The critical point T C can be also evaluated from the size-dependence of mean-field critical points obtained in a systematic series of clusters. Some applications to the two- and three-dimensional Ising models are given explicitly to obtain T C =(2.24(0.036)) J / k B and γ=1.7(0.05) in the two-dimensional square lattice in our preliminary calculations.

James H. Luscombe

The primary goal of modern theories of critical phenomena is the evaluation of critical exponents from first principles, implying the need to get beyond mean field theories which predict the same exponents independent of the nature of the system. This chapter focuses on a new approach emerged in the 1960s based on two ideas—scaling and renormalization. It presents the physical picture developed by L. P. Kadanoff that provides a conceptual basis for scaling and which has become part of the standard language of critical phenomena. Renormalization provides a way to calculate the free energy not involving a direct evaluation of the partition function—it's a new paradigm in statistical mechanics. The chapter analyzes a simplified, approximate set of recursion relations valid at high temperature for a model with nearest and next-nearest-neighbor interactions. In 1971 K. G. Wilson published two articles marking the introduction of the renormalization group to the study of critical phenomena.

Ole G. Mouritsen

Wang ZhiGuo, Haiyan Zhang, Xu Bo-wei

The critical phenomena of the one-dimensional quantum Ashkin–Teller chain are discussed by using the numerical calculation in the quantum conformal field theory. The result shows that there exists a new critical region of the model.