Recent developments in phase transitions and critical phenomena

Per-Anker Lindgård

Colin J. Thompson

This article traces the development and study of phase transitions from late last century to the present day. We begin with a brief historical sketch and a description of the statistical mechanics of phase transitions. Particular attention is given to the modern era which began in 1944 with Onsager's celebrated solution of the two-dimensional Ising model. Points of development since Onsager, which are highlighted in this article, include the study of critical exponents, the scaling hypothesis, realisation of the universality of critical exponents and the recent renormalization group approach to critical phenomena. The basic idea of the renormalization group method, rather than the detailed application of the recipe to particular cases, is stressed and is discussed critically in some detail.

Michael Ellis Fisher

The aim of this contribution is to draw attention to some recent developments in the theory of critical phenomena and phase transitions that may be of interest to the magnetism community in that magnetic systems have provided a fertile testing ground for modern theoretical ideas concerning ordering and phase transitions while, on the other hand, many aspects of observed magnetic phase behavior have required sophisticated theory for their proper elucidation. The new works in question have been published or will appear shortly: accordingly, only an outline is presented here together with references to the main literature sources which, in turn, include fuller references to earlier work not explicitly mentioned. Four linked topics are addressed: (i) finite-size effects at first-order transitions; (ii) asymptotic degeneracy in transfer matrices, which is intimately related to long-range order; (iii) some consequences of the conformal covariance of fluctuations at a critical point in bulk and in finite systems, especially in two spatial dimensions; (iv) the discovery of ‘‘exactly soluble’’ two-dimensional models exhibiting two infinite classes of distinct critical and multicritical points.

Pierre C. Hohenberg, Bertrand I. Halperin

An introductory review of the central ideas in the modern theory of dynamic critical phenomena is followed by a more detailed account of recent developments in the field. The concepts of the conventional theory, mode-coupling, scaling, universality, and the renormalization group are introduced and are illustrated in the context of a simple example---the phase separation of a symmetric binary fluid. The renormalization group is then developed in some detail, and applied to a variety of systems. The main dynamic universality classes are identified and characterized. It is found that the mode-coupling and renormalization group theories successfully explain available experimental data at the critical point of pure fluids, and binary mixtures, and at many magnetic phase transitions, but that a number of discrepancies exist with data at the superfluid transition of $^{4}\mathrm{He}$.