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Chapman & Hall/CRC eBooks

Foundations of the Theory of Critical Phenomena

A.N. Vasil'ev

Published Apr 28, 2004• 0 citations
Publisher

Abstract

In this chapter we give a general overview of the modern theory of critical phenomena. In the first section we review the history of the field, and in the sections that follow we discuss the basic concepts and ideas.
DOI
10.1201/9780203483565-5

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This brief overview is designed to introduce some of the advances that have occurred in our understanding of phase transitions and critical phenomena. The presentation is organized around three simple questions: (i) What are the basic phenomena under consideration? (ii) Why do we care? (iii) What do we actually do? To answer the third question, the author shall briefly review scaling, universality, and renormalization, three of the many important themes which have served to provide the framework of much of our current understanding of critical phenomena. The style is that of a colloquium, not that of a mini-review article.
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This review covers several topics involving renormalization group ideas. The solution of the sss-wave Kondo Hamiltonian, describing a single magnetic impurity in a nonmagnetic metal, is explained in detail. See Secs. VII-IX. "Block spin" methods, applied to the two dimensional Ising model, are explained in Sec. VI. The first three sections give a relatively short review of basic renormalization group ideas, mainly in the context of critical phenomena. The relationship of the modern renormalization group to the older problems of divergences in statistical mechanics and field theory and field theoretic renormalization is discussed in Sec. IV. In Sec. V the special case of "marginal variables" is discussed in detail, along with the relationship of the modern renormalization group to its original formulation by Gell-Mann and Low and others.