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

arXiv:cond-mat/0012079

Ising Meets Ornstein and Zernike, Debye and Huckel, Widom and Rowlinson, and Others

Ronald Dickman

Published Dec 6, 2000• 0 citations
arXivHTMLPDF

Abstract

The name Ising has come to stand not only for a specific model, but for an entire universality class - arguably the most important such class - in the theory of critical phenomena. I review several examples, both in and out of equilibrium, in which Ising universality appears or is pertinent. The "Ornstein-Zernike" connection concerns a thermodynamically self-consistent closure of the eponymous relation, which lies at the basis of the modern theory of liquids, as applied to the Ising lattice gas. Debye and Huckel founded the statistical mechanics of ionic solutions, which, despite the long-range nature of the interaction, now appear to exhibit Ising-like criticality. The model of Widom and Rowlinson involves only excluded-volume interactions between unlike species, but again belongs to the Ising universality class. Far-from-equilibrium models of voting behavior, catalysis, and hysteresis provide further examples of this ubiquitous universality class.

DOI
10.48550/arxiv.cond-mat/0012079

Related papers

Ising meets Ornstein and Zernike, Debye and Hückel, Widom and Rowlinson, and others

Ronald Dickman

The name Ising has come to stand not only for a specific model, but for an entire universality class - arguably the most important such class - in the theory of critical phenomena. I review several examples, both in and out of equilibrium, in which Ising universality appears or is pertinent. The "Ornstein-Zernike" connection concerns a thermodynamically self-consistent closure of the eponymous relation, which lies at the basis of the modern theory of liquids, as applied to the Ising lattice gas. Debye and Hückel founded the statistical mechanics of ionic solutions, which, despite the long-range nature of the interaction, now appear to exhibit Ising-like criticality. The model of Widom and Rowlinson involves only excluded-volume interactions between unlike species, but again belongs to the Ising universality class. Far-from-equilibrium models of voting behavior, catalysis, and hysteresis provide further examples of this ubiquitous universality class.

Brazilian Journal of Physics•Dec 1, 2000•6 citations

Discrete models and critical phenomena

Mark E. Tuckerman

Abstract Chapter 16 explores the statistical mechanics of discrete lattice models and their use in describing critical phenomena. The chapter begins with a review of phase diagrams, coexistence curves, crititical points, and the behavior of certain thermodynamic quantities in the neighborhood of a critical point. The notion of universality classes is introduced. To set up the class of lattice models, magnetic systems are described; simplifications are made to these models to yield the so-called Ising model. Solution of this model is demonstrated within the mean-field theory approximation, which is shown to yield a qualitative description of critical behavior. Exact solutions are derived in one and two dimensions, and predictions of critical exponents from them are obtained. Following this, the renormalization group is introduced, and approaches for understanding critical behavior and universality from renormalization group transformations are discussed. The chapter concludes with a discussion of the use of discrete models in areas ranging from molecular crystals to tissue morphogenesis

•Aug 1, 2023•0 citations

Critical Exponents and the Renormalization Group

Louis A. Girifalco

Abstract The order-disorder transition and ferromagnetism have their origins in quite different manifestations of atomic interactions and are exhibited in quite different kinds of material. But, as shown in chapter 9, both can be described by the Ising model with considerable success. In fact, there are other phenomena that are different from either magnetism or order-disorder that can also be described by similar models. These include adsorption of atoms or molecules on surfaces, absorption of gases in solids, and the attachment of impurities to dislocation cores. A simple description of such phenomena is provided by the lattice gas model, and this model is easily shown to be equivalent to the Ising model. The fact that the same model can describe such varied systems implies that there are some important similarities among them that do not depend on the specific nature of the systems or the interaction energies involved. This, in fact, turns out to be the case and is most clearly apparent in the neighborhood of critical points. The statistical mechanics of the lattice gas is described in appendix 7, and specific applications are given in chapter 12. Here, the model and its connection to the Ising model are described.

•Sep 21, 2000•0 citations

Universality in Non-Equilibrium Quantum Systems

William Berdanier

The phenomenon of universality is one of the most striking in many-body physics. Despite having sometimes wildly different microscopic constituents, systems can nonetheless behave in precisely the same way, with only the variable names interchanged. The canonical examples are those of liquid boiling into vapor and quantum spins aligning into a ferromagnet; despite their obvious differences, they nonetheless both obey quantitatively the same scaling laws, and are thus in the same universality class. Remarkable though this is, universality is generally a phenomenon limited to thermodynamic equilibrium, most commonly present at transitions between different equilibrium phases. Once out of equilibrium, the fate of universality is much less clear. Can strongly non-equilibrium systems behave universally, and are their universality classes different from those familiar from equilibrium? How is quantum mechanics important? This dissertation attempts to address these questions, at least in a small way, by showing and analyzing universal phenomena in several classes of non-equilibrium quantum systems.

arXiv:2009.05706•Sep 12, 2020•1 citation

Integrable field theory and critical phenomena: the Ising model in a magnetic field

Gesualdo Delfino A

The two-dimensional Ising model is the simplest model of statistical mechanics exhibiting a second-order phase transition. While in absence of magnetic field it is known to be solvable on the lattice since Onsager's work of the 1940s, exact results for the magnetic case have been missing until the late 1980s, when A Zamolodchikov solved the model in a field at the critical temperature, directly in the scaling limit, within the framework of integrable quantum field theory. In this paper, we review this field theoretical approach to the Ising universality class, with particular attention to the results obtained starting from Zamolodchikov's scattering solution and to their comparison with the numerical estimates on the lattice. The topics discussed include scattering theory, form factors, correlation functions, universal amplitude ratios and perturbations around integrable directions. Although we restrict our discussion to the Ising model, the emphasis is on the general methods of integrable quantum field theory which can be used in the study of all universality classes of critical behaviour in two dimensions.

arXiv:hep-th/0312119•Journal of Physics A Mathematical and General•Mar 23, 2004•104 citations

Critical phenomena in a binary lattice gas. II. Series analysis

J. Anthony Gualtieri, John C. Wheeler

The critical behavior of a two-component lattice-gas model (the 1-2 model) in two and three dimensions is investigated by means of standard and nonstandard methods of series analysis on exact series expansions for the thermodynamic properties of the model. The model exhibits a critical line that passes from the Ising critical point at finite temperature to a critical point at zero temperature. In two dimensions the Ising values of the critical exponents found along the critical line and at zero temperature support the expectation that the model lies in the Ising universality class. In three dimensions Ising values of the critical exponents are found away from zero temperature, but near and at zero temperature the series analysis suggests a crossover to non-Ising values of the critical exponents. This would imply that the model belongs to the Ising universality class at nonzero temperatures and to a different universality class at zero temperature---a quite unexpected result. Various possible explanations for this unusual behavior of the series results are discussed.

Physical review. B, Condensed matter•Sep 1, 1979•2 citations
The name Ising has come to stand not only for a specific model, but for an entire universality class - arguably the most important such class - in the theory of critical phenomena. I review several examples, both in and out of equilibrium, in which Ising universality appears or is pertinent. The "Ornstein-Zernike" connection concerns a thermodynamically self-consistent closure of the eponymous relation, which lies at the basis of the modern theory of liquids, as applied to the Ising lattice gas. Debye and Huckel founded the statistical mechanics of ionic solutions, which, despite the long-range nature of the interaction, now appear to exhibit Ising-like criticality. The model of Widom and Rowlinson involves only excluded-volume interactions between unlike species, but again belongs to the Ising universality class. Far-from-equilibrium models of voting behavior, catalysis, and hysteresis provide further examples of this ubiquitous universality class.
The name Ising has come to stand not only for a specific model, but for an entire universality class - arguably the most important such class - in the theory of critical phenomena. I review several examples, both in and out of equilibrium, in which Ising universality appears or is pertinent. The "Ornstein-Zernike" connection concerns a thermodynamically self-consistent closure of the eponymous relation, which lies at the basis of the modern theory of liquids, as applied to the Ising lattice gas. Debye and Hückel founded the statistical mechanics of ionic solutions, which, despite the long-range nature of the interaction, now appear to exhibit Ising-like criticality. The model of Widom and Rowlinson involves only excluded-volume interactions between unlike species, but again belongs to the Ising universality class. Far-from-equilibrium models of voting behavior, catalysis, and hysteresis provide further examples of this ubiquitous universality class.
Abstract Chapter 16 explores the statistical mechanics of discrete lattice models and their use in describing critical phenomena. The chapter begins with a review of phase diagrams, coexistence curves, crititical points, and the behavior of certain thermodynamic quantities in the neighborhood of a critical point. The notion of universality classes is introduced. To set up the class of lattice models, magnetic systems are described; simplifications are made to these models to yield the so-called Ising model. Solution of this model is demonstrated within the mean-field theory approximation, which is shown to yield a qualitative description of critical behavior. Exact solutions are derived in one and two dimensions, and predictions of critical exponents from them are obtained. Following this, the renormalization group is introduced, and approaches for understanding critical behavior and universality from renormalization group transformations are discussed. The chapter concludes with a discussion of the use of discrete models in areas ranging from molecular crystals to tissue morphogenesis
Abstract The order-disorder transition and ferromagnetism have their origins in quite different manifestations of atomic interactions and are exhibited in quite different kinds of material. But, as shown in chapter 9, both can be described by the Ising model with considerable success. In fact, there are other phenomena that are different from either magnetism or order-disorder that can also be described by similar models. These include adsorption of atoms or molecules on surfaces, absorption of gases in solids, and the attachment of impurities to dislocation cores. A simple description of such phenomena is provided by the lattice gas model, and this model is easily shown to be equivalent to the Ising model. The fact that the same model can describe such varied systems implies that there are some important similarities among them that do not depend on the specific nature of the systems or the interaction energies involved. This, in fact, turns out to be the case and is most clearly apparent in the neighborhood of critical points. The statistical mechanics of the lattice gas is described in appendix 7, and specific applications are given in chapter 12. Here, the model and its connection to the Ising model are described.
The phenomenon of universality is one of the most striking in many-body physics. Despite having sometimes wildly different microscopic constituents, systems can nonetheless behave in precisely the same way, with only the variable names interchanged. The canonical examples are those of liquid boiling into vapor and quantum spins aligning into a ferromagnet; despite their obvious differences, they nonetheless both obey quantitatively the same scaling laws, and are thus in the same universality class. Remarkable though this is, universality is generally a phenomenon limited to thermodynamic equilibrium, most commonly present at transitions between different equilibrium phases. Once out of equilibrium, the fate of universality is much less clear. Can strongly non-equilibrium systems behave universally, and are their universality classes different from those familiar from equilibrium? How is quantum mechanics important? This dissertation attempts to address these questions, at least in a small way, by showing and analyzing universal phenomena in several classes of non-equilibrium quantum systems.
The two-dimensional Ising model is the simplest model of statistical mechanics exhibiting a second-order phase transition. While in absence of magnetic field it is known to be solvable on the lattice since Onsager's work of the 1940s, exact results for the magnetic case have been missing until the late 1980s, when A Zamolodchikov solved the model in a field at the critical temperature, directly in the scaling limit, within the framework of integrable quantum field theory. In this paper, we review this field theoretical approach to the Ising universality class, with particular attention to the results obtained starting from Zamolodchikov's scattering solution and to their comparison with the numerical estimates on the lattice. The topics discussed include scattering theory, form factors, correlation functions, universal amplitude ratios and perturbations around integrable directions. Although we restrict our discussion to the Ising model, the emphasis is on the general methods of integrable quantum field theory which can be used in the study of all universality classes of critical behaviour in two dimensions.
The critical behavior of a two-component lattice-gas model (the 1-2 model) in two and three dimensions is investigated by means of standard and nonstandard methods of series analysis on exact series expansions for the thermodynamic properties of the model. The model exhibits a critical line that passes from the Ising critical point at finite temperature to a critical point at zero temperature. In two dimensions the Ising values of the critical exponents found along the critical line and at zero temperature support the expectation that the model lies in the Ising universality class. In three dimensions Ising values of the critical exponents are found away from zero temperature, but near and at zero temperature the series analysis suggests a crossover to non-Ising values of the critical exponents. This would imply that the model belongs to the Ising universality class at nonzero temperatures and to a different universality class at zero temperature---a quite unexpected result. Various possible explanations for this unusual behavior of the series results are discussed.