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CRC Press eBooks

Ising Model and Phase Transitions

Francesco Caravelli, Diego A. R. Dalvit

Published Jun 19, 2026• 0 citations
Publisher

Abstract

We now focus on the two-dimensional ferromagnetic Ising model, which provides one of the most fundamental and celebrated examples of a system exhibiting a finite-temperature phase transition. Although the exact solution was obtained only later by Lars Onsager (1944) , the existence of a phase transition in two dimensions was first rigorously argued by Rudolf Peierls in 1936 [Pei36]. Here, we introduce the essential ideas of the low-temperature expansion and the Peierls argument.

DOI
10.1201/9781003603528-16

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We now focus on the two-dimensional ferromagnetic Ising model, which provides one of the most fundamental and celebrated examples of a system exhibiting a finite-temperature phase transition. Although the exact solution was obtained only later by Lars Onsager (1944) , the existence of a phase transition in two dimensions was first rigorously argued by Rudolf Peierls in 1936 [Pei36]. Here, we introduce the essential ideas of the low-temperature expansion and the Peierls argument.
Though the idea to use numerical techniques, in order to solve complex three-dimensional problems, has become quite old, computational techniques have gained immense importance in past few decades because of the advent of new generation fast and efficient computers and development of algorithms as parallel computing.Many mathematical problems have no exact solutions.Depending on the complexity of the equations, one needs to use approximate methods.But there are problems, which are beyond our limits, and need support of computers.Ernst Ising published his PhD dissertation in the form of a scientific report in 1925.He used a string of magnetic moments; spin up (+1/2) and spin down (À1/2), and applied periodic boundary conditions to prove that magnetic phase transition does not exist in one dimensions.Lars Onsager, latter, exactly solved the phase transition problem in two dimensions in 1944.It is going to be a century-old problem now.A variety of potential applications of Ising model are possible now a days; classified as Ising universality class models.It has now become possible to solve phase transition problems in complex three-dimensional geometries.Though the area of spinotronics still needs more engagements of computational techniques, its limited use have provided good insights at molecular scale in recent past.This chapter gives a brief introduction to Ising model and its applications, highlighting the developments in the field of magnetism relevant to the area of solid state physics.
Abstract In this chapter we will begin our study of the Ising model on the two-dimensional lattice. In two dimensions the model has a phase transition, with critical exponents that have different values from those obtained in the mean field approximation. For this reason, it provides an important example of critical phenomena. As we will see in great detail in this chapter and in the next, among all exactly solved models of statistical mechanics, the two-dimensional Ising model is not only the one that has been most studied but it is also the model that has given a series of deep mathematical and physical results. Many solutions of the model stand out for the ingenious methods used, such as the theory of determinants, combinatorial approaches, Grassmann variables, or elliptic functions. Many results have deeply influenced the understanding of critical phenomena and have strongly stimulated new fields of research. Ideas that have matured within the study of the two-dimensional Ising model, such as the duality between its highand low-temperature phases, have been readily generalized to other systems of statistical mechanics and have also found important and fundamental applications in other important areas such as, for instance, quantum field theory. Equally fundamental is the discovery that in the vicinity of the critical point, the dynamics of the model can be described from the relativistic Dirac equation for Majorana fermions.
Abstract 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.
The anisotropic two-dimensional Ising model in the presence of a magnetic field is studied within two different approaches: the effective-field theory (EFT) with correlation and the Bethe-Peierls (BP) approximation. The model consists of ferromagnetic interaction (Jx)({J}_{x})(Jx​) in the xxx direction and antiferromagnetic interaction (Jy)({J}_{y})(Jy​) in the yyy direction. The phase diagram in the $T\text{\ensuremath{-}}H$ plane is obtained for the particular case Jx=Jy{J}_{x}={J}_{y}Jx​=Jy​. Special focus is given in the low-temperature region of the phase diagram, where a first-order phase transition is observed using the mean-field approximation, which is in disagreement with the linear chain approximation (LCA). Our results indicate a second-order phase transition for all values of H∕J∊[0,2]H∕J∊[0,2]H∕J∊[0,2], with the presence of a reentrant behavior only observed in the BP approximation in accordance with the results of the LCA and exact solution. The null field critical temperature is an increasing function of r=Jy∕Jxr={J}_{y}∕{J}_{x}r=Jy​∕Jx​, and in the $r\ensuremath{\rightarrow}0$ limit we have found, using BP and EFT, the approximate form ${k}_{B}{T}_{N}∕{J}_{x}\ensuremath{\simeq}A∕\mathrm{ln}(1∕r)$ in accordance with the exact result of Onsager.
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 XYXYXY 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.