Abstract The exact solutions of one-dimensional systems discussed in the previous chapter are particularly simple and mathematically elegant. Unfortunately, they are more an exception than a rule: for higher dimensional lattices, the computation of the partition functions of statistical models usually poses a formidable problem from a mathematical point of view. For that reason, it is of the utmost importance to develop some approximate methods that permit us to analyze the most relevant physical aspects and to extract an estimate of critical exponents or other thermodynamic quantities. In this chapter we will discuss several approximate solutions of the Ising model and its generalizations.
Abstract The exact solutions of one-dimensional systems discussed in the previous chapter are particularly simple and mathematically elegant. Unfortunately, they are more an exception than a rule: for higher dimensional lattices, the computation of the partition functions of statistical models usually poses a formidable problem from a mathematical point of view. For that reason, it is of the utmost importance to develop some approximate methods that permit us to analyze the most relevant physical aspects and to extract an estimate of critical exponents or other thermodynamic quantities. In this chapter we will discuss several approximate solutions of the Ising model and its generalizations.
Abstract Only a limited number of models of phase transitions and critical phenomena can be solved exactly. These examples nevertheless play important roles in many aspects including the verification of the accuracy of approximation theories such as the mean-field theory and renormalization group. Mathematical methods to solve such examples are interesting in their own right and constitute an important subfield of mathematical physics. In particular the exact solution of the two-dimensional Ising model occupies an outstanding status as one of the founding studies of the modern theory of phase transitions and critical phenomena. The present chapter shows simple but typical examples of exact solutions of classical spin systems such as the one-dimensional Ising model with various boundary conditions, the n-vector model, the spherical model, the one-dimensional quantum XY model, and the two-dimensional Ising model. An account on the Yang-Lee theory will also be given as a set of basic rigorous results on phase transitions.
Abstract Chapter 3 discusses the approximation schemes used to approach lattice statistical models that are not exactly solvable. In addition to the mean field approximation, it also considers the Bethe–Peierls approach to the Ising model. Moreover, there is a thorough discussion of the Gaussian model and its spherical version, both of which are two important systems with several points of interest. A chapter appendix provides a detailed analysis of the random walk on different lattices: apart from the importance of the subject on its own, it explains how the random walk is responsible for the critical properties of the spherical model.
As we have seen on several occasions, a great many of the physical phenomena of interest concern collective behaviour or strongly correlated particles. While it is sometimes possible to obtain useful insight using approximation methods, such as perturbation theory, reliable controlled approximations do not exist. In such situations, numerical simulations have become an indispensable tool. We shall not attempt in this chapter to discuss computer programming nor shall we discuss many of the powerful methods, both algorithmic and in data analysis, that have been developed since numerical simulations have become an important research tool. Instead, we shall assume that the reader already has enough familiarity with computer programming to apply the notions which we discuss here. Our goal is to explain enough of the basics of the Monte Carlo method (classical and quantum) to allow the reader to apply it readily to interesting equilibrium statistical physics problems of the type discussed in this book. The problems in this chapter have been tailored with this in mind. The programs needed are relatively simple and the physics illustrates many of the phenomena already seen: phase transitions with and without spontaneous symmetry breaking, critical exponents, diverging susceptibilities, scaling, vortices, superfluids etc. Consequently, some of the problems are somewhat long and may be better considered as mini-projects. Markov chains, convergence and detailed balance Even relatively innocent looking models are impossible to solve exactly. The exact solution of the two-dimensional Ising model is known but not in the presence of an external magnetic field. In three dimensions, even in the absence of a magnetic field, the exact solution is not known. In fact, relatively few models can be solved exactly; see Reference [14] for an excellent review of the subject.
From the exact solution of the Ising model by Onsager in 1944 up to that of the hard hexagon model by Baxter in 1980, the statistical mechanics of two-dimensional systems has been enriched by a number of exact results. One speaks (in quick manner) of exact models once a convenient mathematical expression has been obtained for a physical quantity such as the free energy, an order parameter or some correlation, or at the very least once their evaluation is reduced to a problem of classical analysis. Such solutions, often considered as singular curiosities upon their appearance, often have the interest of illustrating the principles and general theorems rigorously established in the framework of definitive theories, and also enabling the control of approximate or perturbative methods applicable to more realistic and complex models. In the theory of phase transitions, the Ising model and the results of Onsager and Yang have eminently played such a reference role. With the various vertex models, the methods of Lieb and Baxter have extended this role and the collection of critical exponents, providing new useful elements of comparison with extrapolation methods, and forcing a refinement of the notion of universality. Intimately linked to two-dimensional classical models (but of less interest for critical phenomena), one-dimensional quantum models such as the linear magnetic chain and Bethe's famous solution have certainly contributed to the understanding of fundamental excitations in many-body systems. One could also mention the physics of one-dimensional conductors.
Abstract Until the early 1970s all calculations of critical exponents were made either by exactly solving a model for its thermodynamic properties and then examining its behaviour in the critical regime (as with the Ising model in one and two dimensions) or by direct numerical simulations (as with the three-dimensional Ising model) or by extrapolation from approximate solutions which become invalid in the vicinity of a phase transition (high-temperature expansions, for example). However, in a now-famous paper published in 1966, Kadanoff presented arguments which, he maintained, would allow one to simplify calculations in the critical regime to the point at which critical exponents could be extracted, without ever working out the partition function for the problem. However, though Kadanoff’s ideas showed great physical intuition about the processes giving rise to critical phenomena, they lacked the mathematical precision which might have given one faith in his results.
Abstract Chapter 2 discusses one-dimensional statistical models, for example, the Ising model and its generalizations (Potts model, systems with O(n) or Zn-symmetry, etc.). It discusses several methods of solution and covers the recursive method, the transfer matrix approach, and series expansion techniques. General properties of these methods, which are valid on higher-dimensional lattices, are also covered. The contents of this chapter are quite simple and pedagogical but extremely useful for understanding the following sections of the book. One of the appendices at the end of the chapter is devoted to a famous problem of topology, i.e. the four-colour problem, and its relation with the two-dimensional Potts model.