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Cambridge University Press eBooks

Phase diagrams and fixed points

John Cardy

Published Apr 26, 1996• 1 citation
Publisher

Abstract

In the previous chapter, the ferromagnetic Ising model provided a simple example of a phase diagram with an associated fixed point structure of the renormalization group flows. There were stable fixed points corresponding to low and high temperature phases, and a critical fixed point controlling the behaviour of critical Ising Systems. However, more realistic systems often have more complicated phase diagrams, and therefore a richer fixed point structure. In this chapter we study some of these examples, and show how, even with a rather qualitative description of renormalization group flows, it is possible to understand phase diagrams from the renormalization group viewpoint. More importantly, when more than one non-trivial fixed point is present, the question arises as to which is the dominant one in a particular region of the phase space. The renormalization group answers this question through the theory of cross-over behaviour . The existence of such phenomena, whereby different fixed points may influence the properties of the same system on different length scales, is totally absent in mean field treatments. Ising model with vacancies As a first example, consider a generalisation of the Ising model in which the spin variables s ( r ) may take the value 0 as well as ±1. This may be viewed as the classical version of a quantum spin-1 magnet, or as a lattice gas of magnetic particles, with | s ( r )| playing the role of the occupation number.

DOI
10.1017/cbo9781316036440.005

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In the previous chapter, the ferromagnetic Ising model provided a simple example of a phase diagram with an associated fixed point structure of the renormalization group flows. There were stable fixed points corresponding to low and high temperature phases, and a critical fixed point controlling the behaviour of critical Ising Systems. However, more realistic systems often have more complicated phase diagrams, and therefore a richer fixed point structure. In this chapter we study some of these examples, and show how, even with a rather qualitative description of renormalization group flows, it is possible to understand phase diagrams from the renormalization group viewpoint. More importantly, when more than one non-trivial fixed point is present, the question arises as to which is the dominant one in a particular region of the phase space. The renormalization group answers this question through the theory of cross-over behaviour . The existence of such phenomena, whereby different fixed points may influence the properties of the same system on different length scales, is totally absent in mean field treatments. Ising model with vacancies As a first example, consider a generalisation of the Ising model in which the spin variables s ( r ) may take the value 0 as well as ±1. This may be viewed as the classical version of a quantum spin-1 magnet, or as a lattice gas of magnetic particles, with | s ( r )| playing the role of the occupation number.
The global phase diagram for a three-component lattice gas or spin-one Ising model with general single-site and nearest-neighbor "ferromagnetic" interactions is worked out for two-dimensional lattices using a Migdal-Kadanoff recursion relation. It differs in important qualitative respects from the corresponding mean-field phase diagram. The set of fixed points and flows provides the characteristic phase diagrams of the three-state Potts multicritical point and the ordinary (n=1n=1n=1) tricritical point in a complete set of symmetry-breaking fields. The latter is associated, in this renormalization-group scheme, with seven distinct critical fixed points, a number which is surprisingly large.
This chapter describes the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, the general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Phase transitions occur in macroscopic systems when one or more external parameters like temperature, pressure, electrical field, magnetic field, etc are varied. The chapter demonstrates the utility of the alternative formulations of thermodynamics and statistical mechanics in the context of phase transitions and shows how their use clarifies and unifies the theory of phase transitions. It deals with the basic thermodynamic description of a single component fluid. To understand different phases and associated transitions, one studies how the couplings flow in the parameter space under repeated Renormalization Group (RG) transformations. The flows around the fixed point are easily studied by linearizing the RG equations around the fixed points and determining the eigenvalues and eigenvectors of the linearized RG transformations.
Abstract In Chapter 8 we gave an introduction to the formalism of the renormalization group. A renormalization transformation changes the scale of a system by a factor b. Systems at criticality flow to fixed points under the transformation and their critical exponents and scaling behaviour are determined by linearizing the recursion equations about the fixed point. The aim of this chapter is two-fold: firstly to illustrate these ideas by giving several examples of renormalization group transformations and secondly to demonstrate the ways in which they can be used to calculate critical properties. Our examples are divided into three classes. In Section 9.1, we shall consider a one-dimensional model where the recursion relations can be written down exactly. The drawback of this as a canonical example is that any critical behaviour is at zero temperature. The decimation transformation presented in one dimension is an example of a real-space transformation and the second task, in Section 9.2, is to show how these may be used approximately to treat models in higher dimensions. Thirdly, in Section 9.5, we shall mention the E-expansion where the renormalization takes place in momentum space. This enables an understanding of the crossover to mean-field behaviour. However, the formalism is somewhat involved and the details are beyond the scope of this book.
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
Introduction The real space renormalisation group has a wide application in the study of classical critical phenomena. For quantum systems, several versions of the real space renormalisation group have been proposed (Jullien, 1981; Pfeuty et al ., 1982; dos Santos, 1982). The real space RG is the most direct way of implementing the scaling ideas presented in Chapter 1. The operation of elimination of degrees of freedom as the system is reduced to blocks of smaller sizes under a change of length scale is directly and intuitively represented as mathematical equations. In particular, the block method has been used to study different problems of strongly correlated systems (Pfeuty et al ., 1982) and spin chains (White, 1992, 1993) with different degrees of sophistication. We show here in detail a simple form of this technique applied to the case of the one-dimensional Ising model in a transverse field. The purpose is to illustrate the ideas introduced earlier in Chapter 1 with a concrete example where a quantum critical point is associated with an unstable fixed point. Also, we show how the stable attractors or fixed points can characterise the different phases and how the critical exponents can be obtained. Although the actual values for the quantum critical point and critical exponents that we obtain using a small block are not exact, a procedure is outlined which can be used to improve these results. In this sense, the block method presented here represents a controlled approximation, which can be systematically improved to yield in the limit of very large blocks essentially the correct results. When compared to the k -space version of the RG (to be presented in the next two chapters), the real space version turns out to be in general more difficult to control, in the sense of producing a systematic improvement of the results. On the other hand, this method yields without much effort the attractive fixed points of the different phases, allowing for their characterisation. This turns out to be useful since the nature of the phase transition, whether first or second order, can be determined by exponents calculated in the attractor of this phase (Nienhuis and Naunberg, 1975).
The chapter is an introduction to basic equilibrium aspects of phase transitions. It starts by reviewing thermodynamics and the thermodynamic description of phase transitions. Next, lattice models, such as the paradigmatic Ising model, are introduced as simple physical models that permit a mechano-statistical study of phase transitions from a more microscopic point of view. It is shown that the Ising model can quite faithfully describe many different systems after suitable interpretation of the lattice variables. Special emphasis is placed on the mean-field concept and the mean-field approximations. The deformable Ising model is then studied as an example that illustrates the interplay of different degrees of freedom. Subsequently, the Landau theory of phase transitions is introduced for continuous and first-order transitions, as well as critical and tricritical behaviour are analysed. Finally, scaling theories and the notion of universality within the framework of the renormalization group are briefly discussed.