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CRC Press eBooks

Landau Theory

Nigel D. Goldenfeld

Published Mar 8, 2018• 1 citation
Publisher

Abstract

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.

DOI
10.1201/9780429493492-5

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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.
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.
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.
In this chapter, the author shows the theory of phase transition using some systems of interacting spins. Note that various systems of different nature can be mapped into spin systems and solved using the spin language. However, the mean-field theory gives some artifacts at low dimensions and cannot determine with precision the critical exponents which characterize the nature of a phase transition. A transition from one phase to another may take place when an external parameter varies. Such a parameter can be the temperature or an external applied magnetic field. Renormalization group analysis shows that the universality class depends only on a few very general parameters such as the space dimension, the symmetry of the order parameter and the nature of the interaction. The Landau–Ginzburg theory is an extension of the mean-field theory which includes a great part of fluctuations so far neglected near the transition.
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.
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.
To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify d c = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.