Mean field theory works well when R/a is large or the co-ordination number is large. In both cases, each degree of freedom is coupled to many neighbors. Mean field theory is exact when each spin is coupled to all other spins. In some complex situations, a systematic and relatively easy way to generate mean field theory is by extending the Hamiltonian to be infinite range. Another way of generating mean field theory is to allow the order parameter to have n components and to consider the case n → ∞. The chapter examines the limitations of the phenomenological theory. It provides the analysis which has been for the Landau theory of the Ising universality class. The chapter discusses the Gaussian approximation to the functional integral, and shows how the critical exponents are calculated the lowest order systematic correction to mean field theory.
Mean field theory works well when R/a is large or the co-ordination number is large. In both cases, each degree of freedom is coupled to many neighbors. Mean field theory is exact when each spin is coupled to all other spins. In some complex situations, a systematic and relatively easy way to generate mean field theory is by extending the Hamiltonian to be infinite range. Another way of generating mean field theory is to allow the order parameter to have n components and to consider the case n → ∞. The chapter examines the limitations of the phenomenological theory. It provides the analysis which has been for the Landau theory of the Ising universality class. The chapter discusses the Gaussian approximation to the functional integral, and shows how the critical exponents are calculated the lowest order systematic correction to mean field theory.
Abstract The present chapter explains the mean-field approximation, the Landau theory, the infinite-range model, and the Bethe approximation, and shows that all these (mean-field) theories are essentially equivalent to each other. The Landau theory is a phenomenological approach that uses the concept of symmetry and the order parameter, a measure of the breaking of that symmetry, as fundamental collective degrees of freedom. Also described are the Landau theory of tricritical behaviour, correlation functions, the limit of applicability of the mean-field theory, known as the Ginzburg criterion, and dynamic critical phenomena. Mean-field theories yield the exact critical exponents for dimensions larger than the upper critical dimension, and their solutions provide a reasonable starting point for more advanced methods including the renormalization group.
Mean-field theory, presented in the preceding chapter, correctly describes the qualitative features of most phase transitions and, in some cases, the quantitative features. Since mean-field theory replaces the actual configurations of the local variables (e.g. spins) by their average value, it neglects the effects of fluctuations about this mean. These fluctuations may or may not be important. The more spins that interact with a particular test spin, the more the test spin sees an effective average or mean field. If the test spin interacts with two neighbors, the averaging is minimal and the fluctuations are large and important. The number of spins producing the effective field increases with the range of the interaction and with the dimension. Thus we will find that mean-field theory is a good approximation in high dimensions but fails to provide a quantitatively correct description of second-order critical points in low dimensions. This chapter will be devoted to the study of second-order phase transitions when mean-field theory is not a good approximation. We will begin by considering fluctuations of the order parameter about its spatially uniform mean-field value. We will find that for spatial dimensions below an upper critical dimension d c (typically d c = 4), fluctuations always become important and invalidate mean-field theory for temperatures sufficiently close to T c . This will motivate a generalization of Landau theory that incorporates spatial fluctuations about a mean-field free energy minimum.
In the previous chapter, we developed the perturbative formulation of the Landau–Ginzburg theory and introduced Feynman diagrams as a systematic way to compute corrections to the Gaussian (mean-field) results. Mean-field theory provides a qualitatively correct picture of phase transitions but fails quantitatively: it neglects fluctuations, leading to incorrect values of the critical exponents, especially in low dimensions ( https://www.w3.org/1998/Math/MathML" display="inline"> D < 4 ). To obtain accurate exponents, one must incorporate fluctuations perturbatively and study how couplings evolve with scale.
This book provides an elementary introduction to the notions of continuum limit and universality in statistical systems with a large number of degrees of freedom. The existence of a continuum limit requires the appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature. In this context, the book emphasizes the role of gaussian distributions and their relations with the mean field approximation and Landau′s theory of critical phenomena. The book shows that quasi-gaussian or mean-field approximations cannot describe correctly phase transitions in three space dimensions. The book assigns this difficulty to the coupling of very different physical length scales, even though the systems we will consider have only local, that is, short range, interactions. To analyze the unusual situation, a new concept is required: the renormalization group, whose fixed points allow understanding the universality of physical properties at large distance, beyond mean-field theory. In the continuum limit, critical phenomena can be described by quantum field theories. In this framework, the renormalization group is directly related to the renormalization process; that is, the necessity to cancel the infinities that arise in straightforward formulations of the theory. The book discusses the renormalization group in the context of various relevant field theories. This leads to proofs of universality and to efficient tools for calculating universal quantities in a perturbative framework. Finally, the book constructs a general functional renormalization group, which can be used when perturbative methods are inadequate.
This chapter shows how two apparently different physical systems can be described by mean field theory. In the Van der Waals gas, each particle interacts with the average density due to all the other gas particles. In the Weiss model of ferromagnetism, each spin interacts with the average magnetic field due to the magnetisation of the other spins. The chapter addresses the question of why the critical exponents are the same for both the Van der Waals fluid and the Weiss ferromagnet. It considers some examples of order parameters, with differing numbers of components. Landau theory may be motivated by a systematic calculation from the microscopic Hamiltonian. One remark about cubic terms in Landau theory. The Landau criterion is only a statement made within the context of mean field theory. It is important to realize that the fluctuation effects omitted by mean field theory can change the order of a transition.
In the preceding two chapters, we have discussed various types of order that can occur in nature and how the ordering process can be quantified by the introduction of order parameters. We also developed a formalism for dealing with the thermodynamics of ordered states. In this chapter, we will use mean-field theory to study phase transitions and the properties of various ordered phases. Mean-field theory is an approximation for the thermodynamic properties of a system based on treating the order parameter as spatially constant. It is a useful description if spatial fluctuations are not important. It becomes an exact theory only when the range of interactions becomes infinite. It, nevertheless, makes quantitatively correct predictions about some aspects of phase transitions (e.g. critical exponents) in high spatial dimensions where each particle or spin has many nearest neighbors, and it makes qualitatively correct predictions in physical dimensions. Mean-field theory has the enormous advantage of being mathematically simple, and it is almost invariably the first approach taken to predict phase diagrams and properties of new experimental systems. Before proceeding, let us review some simple facts about phase transitions. At high temperatures, there is no order, and the order parameter 〈φ〉 is zero. At a critical temperature, T c , order sets in so that, for temperatures below T c , 〈φ〉 is nonzero. If 〈φ〉 rises continuously from zero, as shown in Fig. 4.0.1a, the transition is second order .