A general self-consistency approach allows for a thorough treatment of the corrections to the mean-field approximation (MFA). The natural extension of standard MFA with the help of a cumulant expansion leads to a point of view on the effective field theories. The proposed approach can be used for a systematic treatment of fluctuation effects of various length scales and, perhaps, for the development of a coarse-graining procedure. We outline and justify our method by some preliminary calculations. Results are given for the critical temperature and the Landau parameters of the phi(4) theory--the field counterpart of the Ising model. An important unresolved problem of the modern theory of phase transitions--the problem for the calculation of the true critical temperature--is considered within the framework of the present approach. A comprehensive description of the ground-state properties of many-body systems is also demonstrated.
A general self-consistency approach allows for a thorough treatment of the corrections to the mean-field approximation (MFA). The natural extension of standard MFA with the help of a cumulant expansion leads to a point of view on the effective field theories. The proposed approach can be used for a systematic treatment of fluctuation effects of various length scales and, perhaps, for the development of a coarse-graining procedure. We outline and justify our method by some preliminary calculations. Results are given for the critical temperature and the Landau parameters of the phi(4) theory--the field counterpart of the Ising model. An important unresolved problem of the modern theory of phase transitions--the problem for the calculation of the true critical temperature--is considered within the framework of the present approach. A comprehensive description of the ground-state properties of many-body systems is also demonstrated.
A general self-consistency approach allows a thorough treatment of the corrections to the standard mean-field approximation (MFA). The natural extension of standard MFA with the help of a cumulant expansion leads to a new point of view on the effective field theories. The proposed approach can be used for a systematic treatment of fluctuation effects of various length scales and, perhaps, for the development of a new coarse graining procedure. We outline and justify our method by some preliminary calculations. Concrete results are given for the critical temperature and the Landau parameters of the ϕd4-theory - the field counterpart of the Ising model. An important unresolved problem of the modern theory of phase transitions - the problem for the calculation of the true critical temperature, is considered within the framework of the present approach. A comprehensive description of the ground state properties of many-body systems is also demonstrated.
A general self-consistency approach allows a thorough treatment of the corrections to the standard mean-field approximation (MFA). The natural extension of standard MFA with the help of a cumulant expansion leads to a new point of view on the MF theories. The proposed approach can be used for a systematic treatment of fluctuation effects of various length scales and, perhaps, for the development of a new coarse graining procedure. We outline and justify our method by some preliminary calculations. Concrete results are given for the critical temperature and the Landau parameters of the theory -- the field counterpart of the Ising model. An important unresolved problem of the modern theory of phase transitions -- the problem for the calculation of the true critical temperature, is considered within the framework of the present approach.
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.
The sophistication of modern tools used in the study of statistical mechanics and field theory is often an obstacle to the easy understanding of new important current results reported in journals. The main purpose of this book is to introduce the reader to the methods of the fluctuation (field) theory of phase transitions and critical phenomena so as to provide a good source for research. The introductory contents are concerned with ideas of description, thermodynamic stability theory related to phase transitions, major experimental facts, basic models and their relationships. Special attention is paid to the mean field approximation and to the Landau expansion for simple and complex models of critical and multicritical phenomena. An instructive representation of the modern perturbation theory and the method of the renormalization group is developed for field models of phase transitions. The essential influence of the fluctuations on the critical behaviour is established together with the theory of correlation functions, Gaussian approximation, the Ginzburg criterion, [symbol]- and 1/n- expansions as practical realizations of the renormalization group ideas. Applications of the theory to concrete aspects of condensed matter physics are considered: quantum effects, Bose condensation, crystal anisotropy, superconductors and liquid crystals, effects of disorder of type randomly distributed quenched impurities and random fields. This volume can be used as an advanced University course book for students with a basic knowledge of statistical physics and quantum mechanics. It could be considered as a complementary text to a standard University course on statistical physics
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 .
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.