Scaling and Renormalization in Statistical Physics

John Cardy

This book provides an introduction to the concepts which underlie the modern understanding of the behaviour of complicated physical systems which exhibit the property of scale invariance or self-similarity. This is most clearly illustrated in materials, such as magnets or fluids, in the vicinity of a second order phase transition. The theoretical framework for understanding these phenomena, known as the renormalization group, first arose in the late 1960s and has evolved into a common language used by workers in such diverse fields as particle physics, cosmology, neural networks and biophysics, as well as the more conventional aspects of condensed matter physics. Beginning with a brief review of phase transitions in simple systems and of mean field theory, the text then goes on to introduce the core ideas of the renormalization group. Following chapters cover phase diagrams, fixed points, cross-over behaviour, finite-size scaling, perturbative renormalization methods, low-dimensional systems, surface critical behaviour, random systems, percolation, polymer statistics, critical dynamics and conformal symmetry. The book closes with an appendix on Gaussian integration, a selected bibliography, and a detailed index. Many problems are included. The emphasis throughout is on providing an elementary and intuitive approach. In particular, the perturbative method introduced leads, among other applications, to a simple derivation of the epsilon expansion in which all the actual calculations (at least to lowest order) reduce to simple counting, avoiding the need for Feynman diagrams.