Chapter 8. Critical Phenomena and Continuous Phase Transition

Michel Le Bellac, Fabrice Mortessagne, G. G. Batrouni

In Chapter 3, we examined two examples of phase transitions, the paramagnetic–ferromagnetic transition (Section 3.1.4) and the liquid–vapour transition (Section 3.5.4). In the latter case, thermodynamic functions such as entropy or specific volume are discontinuous at the transition. When, for example, one varies the temperature T at constant pressure, the transition takes place at a temperature T c where the Gibbs potentials of the two phases are equal, but the two phases coexist at T c and are either stable or metastable in the vicinity of T c . Each of the phases carries with it its own entropy, specific volume, etc., which are in general different at T = T c , hence the discontinuities. Such a phase transition is called a first order phase transition . The picture is quite different in the paramagnetic–ferromagnetic phase transition: the thermodynamic quantities are continuous at the transition, the transition is not linked to the crossing of two thermodynamic potentials and one never observes metastability. Such a transition is called a second order, or continuous, phase transition . The transition temperature is called the critical temperature and is denoted by T c . The new and remarkable feature which we shall encounter in continuous phase transitions is the existence of cooperative phenomena. To be specific, think of a spin system, for example an Ising or a Heisenberg model, where the interactions between spins are limited to nearest neighbours. More generally we shall consider short range interactions , which decrease rapidly as a function of the distance between spins, and we exclude from our study all long range interactions.

Jean Zinn‐Justin

This chapter reviews a few properties, from the point of view of phase transitions, of simple ferromagnetic lattice models. In systems with finite range interactions a transfer matrix can be defined. It first examines its properties in a finite volume. In the infinite volume limit, low and high temperature considerations provide convincing evidence of the existence of phase transitions in Ising-like systems. The notion of order parameter is related to cluster properties in the low temperature broken phase. It is shown in a simple example that phase transitions indeed correspond to breaking of ergodicity. The chapter extends the analysis to ferromagnetic systems with continuous symmetries. The appendix contains a brief discussion of quenched disorder.

Hidetoshi Nishimori, Gerardo Ortíz

Abstract As an introduction to the physics of phase transitions and critical phenomena, this chapter explains a number of basic and fundamental ideas such as phases, phase transitions, phase diagrams, universality, and critical phenomena. Especially important is the concept of order parameter, a quantity that measures the degree of asymmetry in the broken symmetry phase. Intuitive accounts are given to the concepts of coarse-graining, and scale and renormalization group transformations, which are powerful, systematic tools to analyze critical behaviour of macroscopic systems. Also explained are several spin and lattice gas model systems, on the basis of which phase transitions and critical phenomena will be studied.

Mário J. de Oliveira