Abstract This chapter presents the phenomenological description of critical phenomena. It derives inequalities for critical exponents which follow from thermodynamics. It explains the scaling theory of critical phenomena of Ising-like systems, it gives scaling for Heisenberg-like systems, and introduces the concept of universality.
The theory of critical phenomena in systems at equilibrium is reviewed at an introductory level with special emphasis on the values of the critical point exponents α, β, γ,..., and their interrelations. The experimental observations are surveyed and the analogies between different physical systems - fluids, magnets, superfluids, binary alloys, etc. - are developed phenomenologically. An exact theoretical basis for the analogies follows from the equivalence between classical and quantal `lattice gases' and the Ising and Heisenberg-Ising magnetic models. General rigorous inequalities for critical exponents at and below T c are derived. The nature and validity of the `classical' (phenomenological and mean field) theories are discussed, their predictions being contrasted with the exact results for plane Ising models, which are summarized concisely. Padé approximant and ratio techniques applied to appropriate series expansions lead to precise critical-point estimates for the three-dimensional Heisenberg and Ising models (tables of data are presented). With this background a critique is presented of recent theoretical ideas: namely, the `droplet' picture of the critical point and the `homogeneity' and `scaling' hypotheses. These lead to a `law of corresponding states' near a critical point and to relations between the various exponents which suggest that perhaps only two or three exponents might be algebraically independent for any system.
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.
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.
Abstract Chapter 16 explores the statistical mechanics of discrete lattice models and their use in describing critical phenomena. The chapter begins with a review of phase diagrams, coexistence curves, crititical points, and the behavior of certain thermodynamic quantities in the neighborhood of a critical point. The notion of universality classes is introduced. To set up the class of lattice models, magnetic systems are described; simplifications are made to these models to yield the so-called Ising model. Solution of this model is demonstrated within the mean-field theory approximation, which is shown to yield a qualitative description of critical behavior. Exact solutions are derived in one and two dimensions, and predictions of critical exponents from them are obtained. Following this, the renormalization group is introduced, and approaches for understanding critical behavior and universality from renormalization group transformations are discussed. The chapter concludes with a discussion of the use of discrete models in areas ranging from molecular crystals to tissue morphogenesis
In this chapter, we investigate the concept of scaling, in particular the scaling hypothesis that has proved so important in the development of the modern theory of critical phenomena. Scaling refers to the way in which properties such as the susceptibility behave in the critical region as they approach infinity (or zero). In this discussion, it is important to be specific about the thermodynamic field variables used to represent the system as well as the direction in which the critical point is approached. A useful schematic in this regard is shown in figure 9.1, which shows phase coexistence diagrams for both an Ising-type magnet and a typical fluid. The fields for the magnet are (H, T ) while for the fluid they are (μ, T ). We note immediately from this figure that the magnet’s coexistence curve is coincident with the H = 0 axis and symmetric about the M = 0 axis. However, these features do not reflect in the corresponding diagrams for the fluid, with important implications for a scaling analysis in this system, which we take up later in this chapter. We illustrate the basic critical scaling laws using the Ising (magnet) system; however, the fact that similar scaling laws also apply to fluids is an important consequence of critical-point universality, a topic we have often alluded to in this text and which is discussed in more detail in chapter 10 dealing with the renormalization-group method.
Abstract Interacting many-particle systems may undergo phase transitions and exhibit critical phenomena in the limit of infinite system size, while the precursors of these phenomena are studied in the theory of finite-size scaling. After surveying the basic notions of phases, phase diagrams, and phase transitions, this chapter focuses on critical behaviour at a second-order phase transition. The Landau-Ginzburg theory and the concept of scaling prepare readers for an elementary introduction to the concepts of the renormalization group, followed by an introduction into the field of quantum phase transitions where quantum fluctuations take over the role of thermal fluctuations.