Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Oxford University Press eBooks

Phase Transitions and the Ising Model

Robert H. Swendsen

Published Dec 10, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.1093/oso/9780198853237.003.0031

Related papers

Phase Transitions and the Ising Model

Robert H. Swendsen

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.

Oxford University Press eBooks•Mar 1, 2012•1 citation

Mean-field theory for quantum spin chains

Mylène Martirosyan, Astrid Monin-Baroille, Leïla Moueddene +2 more

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.

arXiv:2609.04717•Sep 4, 2026•0 citations

Introduction to phase transitions

Antoni Planes, Avadh Saxena

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.

Cambridge University Press eBooks•Apr 30, 2025•0 citations

Phase Transition – Mean Field Theories

Nicolas Sator, Nicolas Pavloff, Lénaïc Couëdel

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.

CRC Press eBooks•Jun 9, 2023•0 citations

How Phase Transitions Occur in Practice

Nigel D. Goldenfeld

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.

CRC Press eBooks•Mar 8, 2018•1 citation

Exact solutions and related topics

Hidetoshi Nishimori, Gerardo Ortíz

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.

•Dec 2, 2010•0 citations
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.
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.
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.
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.
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 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.