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

Unknown eBooks

Critical behavior in many body systems

Cyril Domb

Published Jul 1, 1974• 0 citations
OpenAlex

Abstract

Earlier studies of the simple Ising model have been extended and the techniques applied to a variety of multistate and multispin models. Briefly presented is the authors' research results in which they have developed methods for the derivation of series developments for the Potts model, for which one of the most interesting questions is the order of the transition, for Ising models with four-spin interactions (including the eight-vertex model) and the three-spin triangular Ising model. For the three-spin triangular Ising model it has been found possible to conjecture the exact form of the spontaneous magnetisation. They have also continued to study the general cluster expansion for the high temperature specific heat of the Ising model of a ferromagnet. High field expansions have been derived for the Ising model on the hydrogen peroxide lattice. The exploration of the size and shape of polymer chains using insight gained from critical phenomena has continued.

Related papers

Critical Behaviour in Many Body Systems.

Cyril Domb

This work has been concentrated on a detailed study of low temperature (and high field) expansions for the simple Ising model. Methods were considered for their derivation and derived long series; these have been analyzed to provide evidence on critical indices. Also studied were the derivation of series expansions for the Ising model with spin S > 1/2 and the corresponding low temperature behavior. Also reported is a study of the configurational problem presented by the Potts model and the planar Potts model. The former model in particular has attracted particular attention recently in relation to the properties of liquid crystals. There is evidence that its behavior differs significantly from that of the Ising model and may in certain circumstances lead to a first order transition. Work has continued with a general program of tabulation of lattice constants for all standard two and three dimensional lattices. (Modified author abstract)

•Jul 1, 1974•0 citations

Low-Temperature Critical Exponents from High-Temperature Series: The Ising Model

David S. Gaunt, George Allen Baker Jr.

First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.

Physical review. B, Solid state•Feb 1, 1970•113 citations

Advanced Statistical Mechanics

Barry M. McCoy

Abstract This book begins where elementary books and courses leave off and covers the advances made in statistical mechanics in the past fifty years. The book is divided into three parts. The first part is on general theory which includes a summary of the basic principles of statistical mechanics; a presentation of the physical phenomena covered and the models used to discuss them; theorems on the existence and uniqueness of partition functions; theorems on order; and critical phenomena and scaling theory. The second part is on series and numerical methods which includes derivations of the Mayer and Ree–Hoover expansions of the low density virial equation of state; Groeneveld's theorems; the application to hard spheres and discs; a summary of numerical studies of systems at high density; and the use of high temperature series expansions to estimate critical exponents for magnets. The third part covers exactly solvable models which includes a detailed presentation of the Pfaffian methods of computing the Ising partition function, magnetization, correlation functions, and susceptibility; the star-triangle (Yang–Baxter equation); functional equations and the free energy for the eight-vertex model; and the hard hexagon and chiral Potts models. All needed mathematics is developed in detail and many open questions are discussed. The goal is to guide the reader to the current forefront of research.

•Dec 1, 2009•99 citations

Critical point phenomena

B. L. Smith

During the last decade considerable effort has been expended in the study of critical phenomena. These investigations are especially rewarding because they provide important links between several diverse branches of physics. For example, antiferromagnets, pure fluids, binary solutions and alloys all exhibit analogous critical behaviour. It is now firmly established that the classical phenomenological theories that have been proposed to describe these systems fail in the critical region: mainly because they do not account for the fluctuation phenomena which occur close to critical points, and are a poor approximation for systems with shortrange forces. Considerable progress has been made in formulating more realistic models, for example the king and Heisenberg ferromegnets, and some success has been achieved in obtaining solutions for these models. Similar theories may be postulated to describe the critical behaviour of other systems. Recent progress in theoretical understanding and refinement of experimental techniques have helped to stimulate interest in the subject, and today the study of critical phenomena is one of the most promising and rapidly advancing areas of physics.

Contemporary Physics•Jul 1, 1969•16 citations

High-temperature series expansion analyses of mixed-spin Ising model

S. L. Schofield, Roger Bowers

A high-temperature series expansion in the cumulant form of Brout (1959, 1960) is applied to mixed-spin Ising models on simple quadratic, simple cubic and body-centred cubic lattices. Motivation is provided primarily by an investigation of critical phenomena in Ising models with less than the usual translational symmetry and of relevance to the study of ferrimagnetism. Secondary motivation, appropriate to the simple quadratic lattice, is derived from a desire to obtain results that can be compared with those of renormalisation-group calculations performed elsewhere by the authors. Series expansions of the susceptibility and specific heat are obtained to seventh and tenth order, respectively. The derived expansion coefficients are analysed through both ratio and Pade approximant methods and exponent values are found to be in good agreement with those suggested by the universality hypothesis.

Journal of Physics A Mathematical and General•Aug 1, 1981•25 citations

Zero-temperature renormalization-group method for quantum systems. III. Ising model in a transverse field in two dimensions

K. A. Penson, Rémi Jullien, Pierre M. Pfeuty

A newly developed zero-temperature real-space renormalization-group method is applied to the quantum Ising model with a transverse field on different two-dimensional lattices: hexagonal, triangular, and quadratic. The order parameter, spin-spin correlation functions, the critical fields, and the critical exponents describing the low-field-large-field transition are calculated. The influence of size and shape of the spin blocks used in the iterative process is discussed. The results for triangular and hexagonal lattices are new; for the quadratic lattice the calculations show an encouraging agreement with previous series-expansion results. The expected equivalence to the critical behavior of the three-dimensional Ising model is discussed: A reasonable agreement is obtained for the hexagonal lattice.

Physical review. B, Condensed matter•May 1, 1979•61 citations
Earlier studies of the simple Ising model have been extended and the techniques applied to a variety of multistate and multispin models. Briefly presented is the authors' research results in which they have developed methods for the derivation of series developments for the Potts model, for which one of the most interesting questions is the order of the transition, for Ising models with four-spin interactions (including the eight-vertex model) and the three-spin triangular Ising model. For the three-spin triangular Ising model it has been found possible to conjecture the exact form of the spontaneous magnetisation. They have also continued to study the general cluster expansion for the high temperature specific heat of the Ising model of a ferromagnet. High field expansions have been derived for the Ising model on the hydrogen peroxide lattice. The exploration of the size and shape of polymer chains using insight gained from critical phenomena has continued.
This work has been concentrated on a detailed study of low temperature (and high field) expansions for the simple Ising model. Methods were considered for their derivation and derived long series; these have been analyzed to provide evidence on critical indices. Also studied were the derivation of series expansions for the Ising model with spin S > 1/2 and the corresponding low temperature behavior. Also reported is a study of the configurational problem presented by the Potts model and the planar Potts model. The former model in particular has attracted particular attention recently in relation to the properties of liquid crystals. There is evidence that its behavior differs significantly from that of the Ising model and may in certain circumstances lead to a first order transition. Work has continued with a general program of tabulation of lattice constants for all standard two and three dimensional lattices. (Modified author abstract)
First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
Abstract This book begins where elementary books and courses leave off and covers the advances made in statistical mechanics in the past fifty years. The book is divided into three parts. The first part is on general theory which includes a summary of the basic principles of statistical mechanics; a presentation of the physical phenomena covered and the models used to discuss them; theorems on the existence and uniqueness of partition functions; theorems on order; and critical phenomena and scaling theory. The second part is on series and numerical methods which includes derivations of the Mayer and Ree–Hoover expansions of the low density virial equation of state; Groeneveld's theorems; the application to hard spheres and discs; a summary of numerical studies of systems at high density; and the use of high temperature series expansions to estimate critical exponents for magnets. The third part covers exactly solvable models which includes a detailed presentation of the Pfaffian methods of computing the Ising partition function, magnetization, correlation functions, and susceptibility; the star-triangle (Yang–Baxter equation); functional equations and the free energy for the eight-vertex model; and the hard hexagon and chiral Potts models. All needed mathematics is developed in detail and many open questions are discussed. The goal is to guide the reader to the current forefront of research.
During the last decade considerable effort has been expended in the study of critical phenomena. These investigations are especially rewarding because they provide important links between several diverse branches of physics. For example, antiferromagnets, pure fluids, binary solutions and alloys all exhibit analogous critical behaviour. It is now firmly established that the classical phenomenological theories that have been proposed to describe these systems fail in the critical region: mainly because they do not account for the fluctuation phenomena which occur close to critical points, and are a poor approximation for systems with shortrange forces. Considerable progress has been made in formulating more realistic models, for example the king and Heisenberg ferromegnets, and some success has been achieved in obtaining solutions for these models. Similar theories may be postulated to describe the critical behaviour of other systems. Recent progress in theoretical understanding and refinement of experimental techniques have helped to stimulate interest in the subject, and today the study of critical phenomena is one of the most promising and rapidly advancing areas of physics.
A high-temperature series expansion in the cumulant form of Brout (1959, 1960) is applied to mixed-spin Ising models on simple quadratic, simple cubic and body-centred cubic lattices. Motivation is provided primarily by an investigation of critical phenomena in Ising models with less than the usual translational symmetry and of relevance to the study of ferrimagnetism. Secondary motivation, appropriate to the simple quadratic lattice, is derived from a desire to obtain results that can be compared with those of renormalisation-group calculations performed elsewhere by the authors. Series expansions of the susceptibility and specific heat are obtained to seventh and tenth order, respectively. The derived expansion coefficients are analysed through both ratio and Pade approximant methods and exponent values are found to be in good agreement with those suggested by the universality hypothesis.
A newly developed zero-temperature real-space renormalization-group method is applied to the quantum Ising model with a transverse field on different two-dimensional lattices: hexagonal, triangular, and quadratic. The order parameter, spin-spin correlation functions, the critical fields, and the critical exponents describing the low-field-large-field transition are calculated. The influence of size and shape of the spin blocks used in the iterative process is discussed. The results for triangular and hexagonal lattices are new; for the quadratic lattice the calculations show an encouraging agreement with previous series-expansion results. The expected equivalence to the critical behavior of the three-dimensional Ising model is discussed: A reasonable agreement is obtained for the hexagonal lattice.