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Synergetics is a common feature in interesting statistical-mechanical problems. One of the most important examples of synergetics is the emergence of a second-order phase transition and critical behaviour. It is rich and still far from fully understood. The reason why it is so difficult is due to the increase to infinity of the number of strongly correlated variables in the vicinity of the critical point. For example, the Ising model, which is a model for magnets, exhibits critical behaviour as the temperature comes closer to its critical value; the closer the temperature is to criticality, the more spin variables cooperate with each other to attain the global magnetization. In this regime, neither standard probability theory for independent random variables nor naive perturbation techniques work. The lace expansion, which is the topic of this article, is currently one of the few approaches to rigorous investigation of critical behaviour for various statistical-mechanical models. The chapter summarizes some of the most intriguing lace-expansion results for self-avoiding walk (SAW), percolation, and the Ising model.

This chapter summarizes the many results of the exactly solved Ising model in two dimensions at H = 0 including: the partition function, free energy, spontaneous magnetization, the Toeplitz determinant representations of the correlation functions, the Painlevé VI differential equation for the diagonal two spin correlations, the form factor expansion of the magnetic susceptibility, the scaling limit, the boundary properties of the half plane lattice, and the random layered lattice. For H ≠ 0 the Lee-Yang circle theorem, the free energy and magnetization for H/kBT = iπ/2 expansions for small H, the results of Zamolodchikov at T = Tc and a discussion of extended analyticity are given.

The Tutte polynomial and its relatives play important roles in matroid theory, computational complexity, and models of statistical physics. They provide the natural way to count and relate a variety of objects defined on graphs. This chapter shows that they permit a representation of the two-point correlation function of a ferromagnetic Potts model on a graph G in terms of the flow polynomials of certain related random graphs. This representation extends to general Potts models the so-called random-current expansion for Ising models, and it amplifies the links between the Potts partition function and the Tutte polynomial.

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.

The chapter discusses a multiscale model for a two-phases material. The model is a stochastic process on the finest scale. The effective behaviour on larger scales is governed by deterministic nonlinear evolution equations. Due to the stochasticity on the finest scale, deviations from these limit evolution laws can happen with small probability. The chapter describes the most likely among those deviations in two situations: (i) the switching from one stable equilibrium of the evolution equation to another one, (ii) enforced, fast motion on a manifold of stationary solutions. This chapter is based on joint work with Giovanni Bellettini, Anna DeMasi and Errico Presutti.

This chapter considers systems of interacting agents placed on networks. The agents — spins, oscillators, interacting individuals, etc. — occupy the nodes of networks and interact with each other through network links. In more complicated situations, these agents, in turn, influence their network substrates, and so the pair, a network and the system of agents, co-evolve. The chapter discusses unusual phenomena in these cooperative systems on complex networks. In particular, the Ising model on networks is considered, various synchronization phenomena, and basic game theory models. Finally, avalanches and other abrupt phenomena in many-particle systems on complex networks are touched upon.

This chapter presents the computation of the partition function of the two-dimensional Ising model in terms of a sum of Pfaffians using a combinatorial approach. This method is extended to derive the determinental expressions for the correlation functions.

Actual computations of fixed points and eigenvalues usually involve approximations, often crude ones, except for a very limited number of simple cases such as the one-dimensional Ising model of the previous chapter. In real- and momentum-space renormalization group theory, there are no general prescriptions to systematically improve the degree of the approximation with a modest amount of effort. There are established methods to systematically improve precision, but they usually need a large amount of numerical calculations. The scope of the present chapter is modest as we limit ourselves to basic examples, including the epsilon expansion about the Gaussian fixed-point of the Landau-Ginzburg-Wilson model. Finally, the last section illustrates the extension of the renormalization group framework to study quantum phase transitions.

The objects that are solutions to an NP-complete problem are difficult to count. Counting can be a subtle and complex problem even when the corresponding existence and optimisation problems are in P. Spanning trees and perfect matchings are simple graph-theoretic objects, and the difference between them has deep mathematical roots. A matrix's determinant is the number of spanning trees while its permanent is the number of perfect matchings. Counting is closely associated with sampling. This chapter explores how to generate random matchings, and hence count them approximately, using a Markov chain that mixes in polynomial time. It considers the special case of planar graphs, such as the square lattice, to demonstrate that the number of perfect matchings is in P. It also discusses the implications of this fact for statistical physics and looks at how to find exact solutions for many physical models in two dimensions, including the Ising model.

During the last 100 years, magnetism has made a giant step forward. However, many questions remained unanswered or were not even asked at that time. What is the atomic origin of the magnetisation, and how does it involve quantum mechanics and relativistic physics? What determines the hard or soft character of a steel magnet? How can magnetic properties be tuned by systematically varying crystal structure, chemical composition, and nanostructure? Which ways are there to exploit magnetism in computer science and in other areas of advanced technology? Myriads of questions like these have arisen every decade and turned magnetism into a field of intense research. The modeling of magnetic phenomena and materials is a crucial aspect of this research. This book looks at different models of magnetism, including those developed for magnetic field and magnetisation, circular current, paramagnetic spins, Ising model and exchange, and the viscoelastic model of magnetisation dynamics.

The Ising model provides an extremely useful example for the investigation of phase transitions. This chapter provides both an introduction to the properties of the Ising model and an overview of the complex phenomena exhibited at phase transitions in general. The one-dimensional model is solved exactly using transfer matrices. Models in higher dimensions are treated in the mean field approximation.

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.

In order to acquire a simple physical picture of the dynamic mechanism of a phase transition it is necessary to use the simplest of many-body approximations. It is instructive, in particular, to study the mean-field response of the model system to a time-dependent applied field. In this way, one can obtain considerable insight into the nature of the collective excitations and into the relationship between the static aspects of a phase transition and the occurrence of temperature-dependent (that is, soft) modes and of critical fluctuations. This chapter discusses the static aspects of mean-field theory and the nature of the static singularities which accompany second-order phase transitions. Mean-field dynamics are then described in terms of deviations from the equilibrium mean-field state. Correlated effective-field theory, the quasi-harmonic limit and self-consistent phonons, the deep double-well limit and the Ising model, and the pseudo-spin formalism and tunnel mode are also considered.

Chapter 14 discusses how the identification of a class of universality is one of the central questions needing an answer for those in the field of statistical physics. This chapter discusses in detail the class of universality of several models, and provides examples that include the Ising model, the tricritical Ising model and its structure constants, the Yang–Lee model and the 3-state Potts model. This chapter also covers the study of the statistical models of geometric type (as, for instance, those that describe the self-avoiding walks) and their formulation in terms of conformal minimal models, including conformal models with O(n) symmetry.

Chapter 1 introduces various essential ideas on second-order phase transitions and the theoretical challenges that accompany them. Furthermore, it focuses on important issues, for example, correlation length, correlation functions, scaling laws and behaviour, energy minimization, entropy maximization, dimensionality of space and order parameters and critical exponents, etc. It introduces and also devotes a short discussion to the Ising model and its most significant developments during the years of its study, as well as a short background about Ising himself. The chapter also contains two appendices that summarize all relevant results of ensembles of classical statistical mechanics and quantum statistical mechanics.

The Ising model in a magnetic field is one of the most beautiful examples of an integrable model. This chapter presents its exact S-matrix and the exact spectrum of its excitations, which consist of eight particles of different masses. Similarly, it discusses the exact scattering theory behind the thermal deformation of the tricritical Ising model and the unusual features of the exact S-matrix of the non-unitary Yang–Lee model. Other examples are provided by O(n) invariant models, including the important Sine–Gordon model. It also discusses multiple poles, magnetic deformation, the E₈ Toda theory, bootstrap fusion rules, non-relativistic limits and quantum group symmetry of the Sine–Gordon model.

These lecture notes deal with Ising models with interactions containing products of more than two spins. After a number of examples have been given, some general statements are presented. Of particular interest is a gauge-invariant Ising model in four dimensions. This has important properties in common with models for quantum chromodynamics as developed by Ken Wilson. One phase yields an area law for the Wilson loop, yielding an interaction that increases proportionally to the distance and thus corresponds to quark confinement. The other phase yields a perimeter law allowing for a quark–gluon plasma.

This chapter deals with the simulation of short range ordered crystals. Correlations are introduced as a convenient way to describe short-range order (SRO). The most common way to create structures showing SRO is through Monte Carlo (MC) simulations. The chapter explains the usual interaction potentials and algorithms to minimize the corresponding energy. Interactions for chemical short range order based on an Ising model as well as harmonic and Lennard-Jones potentials for distortions are discussed. Two detailed examples are given. In the first example, a structure showing chemical SRO is created. The second example introduces local distortions, demonstrating the effect of the different potentials on the local structure and the diffraction pattern.

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.

Chapter 4 begins by discussing the Peierls argument, which allows us to prove the existence of a phase transition in the two-dimensional Ising model. The remaining sections of the chapter deal with duality transformations (duality in square, hexagonal and triangular lattices) that link the low- and high-temperature phases of several statistical models. Particularly important is the proof of the so-called star-triangle identity. This identity will be crucial in the later discussion of the transfer matrix of the Ising model. Finally, it covers the aspect of duality in two dimensions. An appendix provides information about the Poisson sum formula.