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CRC Press eBooks

How Phase Transitions Occur in Principle

Nigel D. Goldenfeld

Published Mar 8, 2018• 5 citations
Publisher

Abstract

This chapter explains how phase transitions can occur in principle. It examines how phase transition is possible for a phase transition to occur in a simple model of a magnet-the Ising model. The chapter explores some of the wider and more complex issues associated with phase transitions, using other examples. It focuses on ergodicity breaking, using disordered systems as an example. The goal of statistical mechanics is to compute the partition function Z. In order for uniform bulk behaviour to exist, the forces in the system must satisfy certain properties, and the thermodynamic limit must be taken carefully. The casual reader of any textbook or research paper on phase transitions and statistical mechanics cannot help being struck by the frequency of the term “model”. The phase transition literature is replete with models: the Ising model, the Heisenberg model, the Potts model, the Baxter model, the F model, and even such unlikely sounding names as the non-linear sigma model!.

DOI
10.1201/9780429493492-2

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This chapter explains how phase transitions can occur in principle. It examines how phase transition is possible for a phase transition to occur in a simple model of a magnet-the Ising model. The chapter explores some of the wider and more complex issues associated with phase transitions, using other examples. It focuses on ergodicity breaking, using disordered systems as an example. The goal of statistical mechanics is to compute the partition function Z. In order for uniform bulk behaviour to exist, the forces in the system must satisfy certain properties, and the thermodynamic limit must be taken carefully. The casual reader of any textbook or research paper on phase transitions and statistical mechanics cannot help being struck by the frequency of the term “model”. The phase transition literature is replete with models: the Ising model, the Heisenberg model, the Potts model, the Baxter model, the F model, and even such unlikely sounding names as the non-linear sigma model!.
This chapter explores the basic notions of phase transitions, and the manner in which non-analytic thermodynamic behaviour may occur in principle. It presents some concrete calculations of phase transition behaviour in magnetic systems. The chapter is concerned with the transfer matrix method, first introduced by Hendrik Kramers, which reduces the problem of calculating the partition function to the problem of finding the eigenvalues of a certain matrix. It describes the low temperature expansion and Weiss’ mean field theory for magnets. Mean field theory is the simplest treatment of an interacting statistical mechanical system, apart from the approximation of ignoring the interactions all together. The chapter provides a simple, non-systematic derivation of mean field theory for the Ising model. This mean field theory can be easily improved by applying it to clusters of spins rather than individual spins.
Abstract Methods of statistical mechanics have been enormously successful in clarifying the macroscopic properties of many-body systems. Typical examples are found in magnetic systems, which have been a test bed for a variety of techniques. This chapter introduces the Ising model of magnetic systems and explains its mean-field treatment, a very useful technique of analysis of many-body systems by statistical mechanics. Mean-field theory explained here forms the basis of the methods used repeatedly throughout this book. The arguments in the present chapter represent a general mean-field theory of phase transitions in the Ising model with uniform ferromagnetic interactions. Special features of spin glasses and related disordered systems are taken into account in subsequent chapters.
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.
Among the many applications of statistical mechanics, some of the most intriguing and challenging theoretical problems arise in connection with phase transitions. These are abrupt changes of state such as occur, for example, when a liquid is transformed into a vapour, a ferromagnet loses its magnetization upon heating to its Curie temperature, or at the onset at sufficiently low temperatures of superfluidity or superconductivity. It is within the theory of phase transitions, too, that the mathematical relationships between statistical mechanics and relativistic field theories are most powerful. Indeed, the idea of spontaneous symmetry breaking, which lies at the heart of the theory of phase transitions, is the crucial ingredient that turns the gauge theories of chapter 8 into a real working model of the fundamental forces of nature, to be discussed in the next chapter.
The ferromagnetic transition in the Ising model is the paradigmatic example of ergodicity breaking accompanied by symmetry breaking. It is routinely assumed that the thermodynamic limit is taken with free or periodic boundary conditions. More exotic symmetry-preserving boundary conditions, like cylindrical antiperiodic, are less frequently used for special tasks, such as the study of phase coexistence or the roughening of an interface. Here we show, instead, that when the thermodynamic limit is taken with these boundary conditions, a novel type of transition takes place below T_{c} (the usual Ising transition temperature) without breaking either ergodicity or symmetry. Then the low-temperature phase is characterized by a regime (condensation) of strong magnetization's fluctuations which replaces the usual ferromagnetic ordering. This is due to critical correlations perduring for all T below T_{c}. The argument is developed exactly in the d=1 case and numerically in the d=2 case.
This chapter reviews a few properties, from the point of view of phase transitions, of simple ferromagnetic lattice models. In systems with finite range interactions a transfer matrix can be defined. It first examines its properties in a finite volume. In the infinite volume limit, low and high temperature considerations provide convincing evidence of the existence of phase transitions in Ising-like systems. The notion of order parameter is related to cluster properties in the low temperature broken phase. It is shown in a simple example that phase transitions indeed correspond to breaking of ergodicity. The chapter extends the analysis to ferromagnetic systems with continuous symmetries. The appendix contains a brief discussion of quenched disorder.