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CRC Press eBooks

Phase Transition – Mean Field Theories

Nicolas Sator, Nicolas Pavloff, Lénaïc Couëdel

Published Jun 9, 2023• 0 citations

Abstract

This and the four next chapters are devoted to the study of several aspects of the theory of phase transition. This has been for more than one century, and still remains, a subject of intense experimental and theoretical activity, with important breakthroughs having impacts on different fields in physics but also on other disciplines such as mathematics, economics, sociology, biology In the first section of this chapter, the transition between the ferromagnetic and paramagnetic phases of the Ising system will be studied in the framework of a mean field approximation. The problem will first be addressed by means of the intuitive Weiss molecular field approximation ( Section 9.1.2 ) and then by means of the more detailed Bragg-Williams approximation ( Section 9.1.3 ) which will serve as a base to develop, in Section 9.2 , a general theory of continuous phase transitions: the Landau mean field theory for which the concept of order parameter will first be introduced. Other types of phase transitions will then be studied using the mean field approximation. Phase transitions in liquid crystals will be studied in Section 9.3 . These are systems with a tensor order parameter which exhibit a rich phenomenology. Finally, in Section 9.4 , a transition of a fairly more elaborate nature than a simple first or second order phase transition will be discussed.

DOI: 10.1201/9781003272427-9

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This and the four next chapters are devoted to the study of several aspects of the theory of phase transition. This has been for more than one century, and still remains, a subject of intense experimental and theoretical activity, with important breakthroughs having impacts on different fields in physics but also on other disciplines such as mathematics, economics, sociology, biology In the first section of this chapter, the transition between the ferromagnetic and paramagnetic phases of the Ising system will be studied in the framework of a mean field approximation. The problem will first be addressed by means of the intuitive Weiss molecular field approximation ( Section 9.1.2 ) and then by means of the more detailed Bragg-Williams approximation ( Section 9.1.3 ) which will serve as a base to develop, in Section 9.2 , a general theory of continuous phase transitions: the Landau mean field theory for which the concept of order parameter will first be introduced. Other types of phase transitions will then be studied using the mean field approximation. Phase transitions in liquid crystals will be studied in Section 9.3 . These are systems with a tensor order parameter which exhibit a rich phenomenology. Finally, in Section 9.4 , a transition of a fairly more elaborate nature than a simple first or second order phase transition will be discussed.

In the previous chapters, we discussed mean-field approaches such as the Curie–Weiss and Bragg–Williams theories, which provide a microscopic justification for macroscopic order parameters and spontaneous symmetry breaking. However, these approaches were based on specific microscopic Hamiltonians. Lev Davidovich Landau introduced a far more general and elegant framework: one that does not depend on microscopic details but on symmetry principles and analyticity of the free energy near the transition.

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.

Mean-field theory is a widely used approximation for describing phase transitions, particularly effective above the upper critical dimension. Its origins can be traced back to the van der Waals theory of the liquid-gas transition and Weiss's molecular field theory of the paramagnetic-ferromagnetic transition. However, it was Lev D. Landau who provided a unifying and general framework applicable to a broad class of physical systems. The mean-field approach typically involves neglecting thermal fluctuations, which is a reasonable assumption in many classical contexts. However, its application to quantum phase transitions at zero temperature is less common. The aim of this short pedagogical paper is to explore and clarify the use of the mean-field approach in the less familiar domain of quantum phase transitions. We specifically consider the Ising model and the Blume-Capel model.

This chapter presents a simple approximation scheme — the mean field approximation — which allows a study of the neighbourhood of the transition temperature. It distinguishes between first and second order phase transitions. In the latter case, the correlation length diverges at the critical (transition) temperature. This has important physical consequences: in the mean field approximation, the singular behaviour of thermodynamic quantities at the critical temperature, and more generally in the neighbourhood of the critical temperature and in small magnetic field, is universal, that is, does not depend on the details of the interactions, on the dimension of space and, in the high temperature phase at least, on the symmetry of the models.

Chapter 17 presented one example of a phase transition, the van der Waals gas. This chapter provides another, the Ising model, a widely studied model of phase transitions. We first give the solution for the Ising chain (one-dimensional model), including the introduction of the transfer matrix method. Higher dimensions are treated in the Mean Field Approximation (MFA), which is also extended to Landau theory. The Ising model is deceptively simple. It can be defined in a few words, but it displays astonishingly rich behavior. It originated as a model of ferromagnetism in which the magnetic moments were localized on lattice sites and had only two allowed values.

Abstract Phase transitions and critical phenomena have consistently been among the principal subjects of active studies in statistical physics. The simple act of transforming one state of matter or phase into another, for instance by changing the temperature, has always captivated the curious mind. This book provides an introductory account on the theory of phase transitions and critical phenomena, a subject now recognized to be indispensable for students and researchers from many fields of physics and related disciplines. The first five chapters are very basic and quintessential, and cover standard topics such as mean-field theories, the renormalization group and scaling, universality, and statistical field theory methods. The remaining chapters develop more advanced concepts, including conformal field theory, the Kosterlitz-Thouless transition, the effects of randomness, percolation, exactly solvable models, series expansions, duality transformations, and numerical techniques. Moreover, a comprehensive series of appendices expand and clarify several issues not developed in the main text. The important role played by symmetry and topology in understanding the competition between phases and the resulting emergent collective behaviour, giving rise to rigidity and soft elementary excitations, is stressed throughout the book. Serious attempts have been directed toward a self-contained modular approach so that the reader does not have to refer to other sources for supplementary information. Accordingly, most of the concepts and calculations are described in detail, sometimes with additional/auxiliary descriptions given in appendices and exercises. The latter are presented as the topics develop with solutions found at the end of the book, thus giving the text a self-learning character.