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Texts and readings in physical sciences

Critical Phenomena: An Introduction from a Modern Perspective

Somendra M. Bhattacharjee

Published Jan 1, 2001• 4 citations
Publisher

Abstract

Our aim in this set of lectures is to give an introduction to critical phenomena that emphasizes the emergence of and the role played by diverging length-scales. It is now accepted that renormalization group gives the basic understanding of these phenomena and so, instead of following the traditional historical trail, we try to develop the subject in a way that lays stress on the length-scale based approach.

DOI
10.1007/978-93-86279-07-1_3

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Our aim in this set of lectures is to give an introduction to critical phenomena that emphasizes the emergence of and the role played by diverging length-scales. It is now accepted that renormalization group gives the basic understanding of these phenomena and so, instead of following the traditional historical trail, we try to develop the subject in a way that lays stress on the length-scale based approach.
Abstract The approaches described so far in this book have given a broad phenomenological understanding of critical phenomena. However, although a substantial framework of results and connections has been built up, we have, as yet, no explanations for the following: What is needed is a theory, based on the physics of what is happening at the critical point. we argued in Chapter 1 that the special feature of criticality is that the correlation length is infinite and that the critical system is invariant on all length scales. The aim is to write down a (hopefully short, elegant, and comprehensible) mathematical theory which embodies this physics and explains all the observations listed above. A useful theory will also allow the calculation of critical exponents and transition temperatures, if not exactly, then within an accurate and well-controlled approximation scheme.
The basic idea of the renormalization group is introduced and illustrative examples are presented. Emphasis is put on the application to the theory of critical phenomena. This article is prepared for pedagogical purposes. It is written at a level that a second-year graduate student in physical sciences can understand. No previous knowledge of critical phenomena or field theory is needed. We make no attempt to survey the field or cover a wide range of subjects. On the contrary, we limit the scope to the most basic aspects. We choose to elaborate at length to make the basic idea clear and the definitions precise, and to go through the examples very carefully. We feel that once these basic aspect are understood, there will be no difficulty in confronting the rapidly expanding literature on this subject.
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.
Power laws are distinctive features of critical phenomena and can be ascribed to diverging length-scales. One of the most important contributions of the studies of critical phenomena is the shift in point of view to a length-scale based analysis and classifying terms of a Hamiltonian as relevant or irrelevant rather than classifying them as numerically weak or strong. The idea of universality transcends the domain of critical phenomena. Critical points are classified by the number of relevant variables required to describe them. A universality class would be described by the exponents and also the amplitude ratios of the various singulax quantities on both sides of the critical point. In a correlation based approach, we need to compute the critical correlations of all possible combinations of the degrees of freedom. Scaling is the rule for criticality, originating from a diverging length scale, while anomalous dimension is seen for fluctuation dominated cases.
Abstract In the third and last part of this book we concentrate on renormalization group (or RG) in its various forms. As the great success of RG has been in the context of critical phenomena, it seems sensible to consider the general theoretical situation in that subject as it existed just prior to the coming of RG. Moreover, as the application of RG to critical phenomena was to some extent inspired by the concepts of scaling and “blocking the lattice,” it will be helpful to examine these ideas also.