Scaling-field representation of Wilson's exact renormalization-group equation

Abstract

A novel momentum-space renormalization-group (RG) technique, termed the scaling-field method, is proposed for the investigation of critical phenomena in three-dimensional systems. The approach provides a method for solving Wilson's exact functional differential equation for RG Hamiltonians H l [ σ] by successive approximation, and allows the determination of both critical exponents and scaling functions. In other papers the method is used to calculate to high precision the critical exponents of the isotropic N-vector model in three dimensions and to investigate the Potts, percolation, cubic, and random Ising models in ranges of dimensions between the upper critical and three dimensions. The purpose of this article is to present the foundations of the method. The scaling-field representation of the Wilson functional RG equation is derived by using the expansion of the RG Hamiltonian H l [ σ] in terms of the hierarchy of Gaussian operators of appropriate symmetries. The expansion coefficients are termed scaling fields. The algorithm for the computation of the scaling-field coupling coefficients is developed for the isotropic N-vector model. Technical details and examples are given in a series of Appendices. The invariance of the Wilson equation under physically redundant scale changes of the spin variable σ is carefully investigated for both the functional renormalization group and the scaling-field representation. The understanding of this invariance yields the operational definition of the spin-rescaling function that enters the Wilson equation as a parameter. Exact recursion relations for thermodynamic and correlation functions are derived by using the functional-integral representation of the Wilson equation. The recursion relations provide the starting point for the computation of scaling functions. Finally, as an example and a guide for the development of numerical techniques, the scaling-field equations for the isotropic N-vector model are solved by ε expansion and in the spherical-model limit.