This chapter shows that mean field theory in its simplest form can be improved by inclusion of Gaussian fluctuations. The transition temperature is shifted, and the heat capacity does indeed exhibit a divergence rather than the discontinuity predicted for a uniform system. The expansion, in which the Gaussian and mean field theories are the lowest terms, has an expansion parameter which diverges as the critical point is approached. The problem of anomalous dimensions in critical phenomena was discovered and solved apparently without knowledge or recognition of these other phenomena exhibiting case asymptotics. The chapter briefly discusses the notion of renormalisation in field theory. It is an interesting historical point that Landau had first-hand knowledge of anomalous dimensions in both critical phenomena and the problem of the converging shock wave, but apparently made no reference to a connection between them.
This chapter shows that mean field theory in its simplest form can be improved by inclusion of Gaussian fluctuations. The transition temperature is shifted, and the heat capacity does indeed exhibit a divergence rather than the discontinuity predicted for a uniform system. The expansion, in which the Gaussian and mean field theories are the lowest terms, has an expansion parameter which diverges as the critical point is approached. The problem of anomalous dimensions in critical phenomena was discovered and solved apparently without knowledge or recognition of these other phenomena exhibiting case asymptotics. The chapter briefly discusses the notion of renormalisation in field theory. It is an interesting historical point that Landau had first-hand knowledge of anomalous dimensions in both critical phenomena and the problem of the converging shock wave, but apparently made no reference to a connection between them.
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.
Introduction Our discussion of critical phenomena surrounding a second-order phase transition has thus far focused only on qualitative features. We have now examined three systems, fluids, magnets and random percolation, each of which displays an abruptly sharp transition from a less ordered to a more ordered phase as the respective transition point is encountered. Each shows a similar pattern of developing structure just in advance of the critical point. Fluctuations in the respective order parameter display a self-similar structure that is limited only by a single relevant length scale, the correlation length ξ, which diverges on approach to the transition point. For fluids and magnets, this structure arises from inherent fluctuations that are amplified by either a diverging compressibility or susceptibility, respectively. In this chapter, we explore more quantitative, theoretical approaches taken to understand the features of second-order phase transitions. The simplest of these are the mean field theories , in which the pairwise interaction (needed to produce a phase transition) is introduced in the form of an average field. In this approach, the effects of the growing fluctuations of the order parameter near the critical point are ignored. Although the mean field approach does meet many of the requirements and does predict divergences of certain quantities near the transition, the critical exponents predicted by the theory do not match those seen experimentally. Obtaining correct exponents requires a more advanced approach involving renormalization techniques that exploit the self-similar structure of the fluctuations near the critical point and allow all of the various critical exponents to be inter-related, such that knowledge of any two yields all the others.
It has been seen in the previous chapters that Landau mean field approach relies on an incomplete partition function, the use of which is only justified if the fluctuations are Gaussian. This approach is manifestly invalid in the vicinity of a second order phase transition, since the second derivative of the Landau free energy with respect to the order parameter is equal to zero at the critical temperature https://www.w3.org/1998/Math/MathML" display="inline"> T C 1 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003272427/5ca4d37a-9786-43a3-8b38-8fa9eb3d1a19/content/math_26.tif "/> and the magnitude of the fluctuations near the critical point become prohibitive for the use of the theory. This is a real paradox, since Landau theory is designed to be valid exactly in the vicinity of second order phase transition points (see Section 9.2.2 ). The present chapter focuses first on the discussion of this problem and the precise evaluation of the domain of validity of mean field theories. In a second part, mean field predictions are compared with experimental data and the notion of scale invariance in the vicinity of the critical point is discussed.
Abstract This chapter is devoted to a brief review of general properties of phase transitions in macroscopic physics and, in particular in lattice models. Some of these lattice models actually appear as lattice regularizations of Euclidean (imaginary time) quantum physics theory (QFT). Most of the transitions considered in this work have the following character: spins on the lattice, or macroscopic particles in the continuum, interact through short-range forces, assumed, for simplicity, to decay exponentially. For simple systems, it is possible to find a local observable, called order parameter, whose expectation values depend on the phase in the several phase region, for example, the spin in ferromagnetic systems. In the disordered phase, the connected two-point function decreases exponentially at large distance, at a rate characterized by the correlation length (the inverse of the smallest physical mass in particle physics). In continuous transitions, the correlation length diverges at the critical temperature. Within the mean-field approximation (consistent with Landau's theory of critical phenomena), it can be shown that the singular behaviour of thermodynamic quantities at the critical temperature is universal. These properties can also be reproduced by calculating correlation functions with a perturbed Gaussian measure. It is then shown that the leading corrections to the mean-field approximation, in Ising-like systems, diverge at the critical temperature for dimensions smaller than or equal to 4.
In the previous chapter, we developed the Landau theory of phase transitions, emphasizing how macroscopic order parameters and symmetry considerations determine the general form of the free energy near criticality. There, spatial variations of the order parameter were neglected, corresponding to the mean-field approximation . However, as we have seen, close to the critical temperature T c the correlation length ξ diverges, and fluctuations occur over arbitrarily large scales. In this regime, the assumption of spatial uniformity breaks down, and a local field description becomes necessary.
Abstract Mean-field theory is usually taken as a first step toward understanding critical phenomena, providing an overview that reveals qualitative behaviour of physical quantities. However, it is necessary to proceed beyond the mean-field theory to better understand the situation, both qualitatively and quantitatively, when fluctuations play vital roles leading to exponents that cannot be explained by dimensional analysis, thus introducing anomalous dimensions. The present chapter explains the basic concepts of the renormalization group and scaling theory, which allow us to analyze critical phenomena with fluctuations systematically taken into account. The essential step in a renormalization group calculation consists of establishing recursion relations between the parameters defining the Hamiltonian of the system. These recursion or renormalization group equations define a flow with well-defined fixed points. Details other than the values of the relevant operators have no influence on the critical exponents and this represents universality.