physica status solidi (b)

Cluster Identities for the Ising Model as a Basis for Simple Effective Field Approximations Discerning between Various Lattices with the Same Coordination Number

Piotr Tomczak, Enaldo F. Sarmento, Aline Franco Siqueira, A. Ryszard Ferchmin

Published Aug 1, 1987• 26 citations

Abstract

Abstract Identities are proposed for clusters of N interacting Ising spins. For a single spin they reduce to Callen identity, for a spin pair to that of Bobák and Jaščur. It is shown that effective field approximations for spin correlations lead to different Curie temperatures for lattices of the same coordination number but different dimension.

DOI: 10.1002/pssb.2221420224

Related papers

Contribution to the new type of effective-field theory of the Ising model

R. Honmura, T. Kaneyoshi

Journal of Physics C Solid State PhysicsOct 14, 1979622 citations

General solution of spin-1 Ising model in the effective field theory approximation: critical temperatures and spontaneous magnetization

E. Jurčišinová, Marian Jurčisin

Phase TransitionsNov 7, 20213 citations

Effective-field model with correlations for Ising systems

G. Bruce Taggart, I. P. Fittipaldi

Physical review. B, Condensed matterJun 1, 198244 citations

Averaging over interaction fields and spin correlations

S. V. Semkin, Victor Smagin

Территория новых возможностей. Вестник Владивостокского государственного университета экономики и сервисаJan 1, 20200 citations

New correlated-effective-field theory in the Ising model

T. Kaneyoshi, I. P. Fittipaldi, R. Honmura +1 more

Physical review. B, Condensed matterJul 1, 1981145 citations

Diluted Ising spin‐1/2 lattice with an arbitrary coordination number

Bach Thanh Cong, Y. El Amraoui

Abstract A useful representation for the Callen identity in the case of spin‐1/2 is introduced by a simple technique. The phase diagrams and percolation problems of the diluted Ising lattice with arbitrary coordination number z are also discussed.

physica status solidi (b)Sep 1, 19962 citations

physica status solidi (b)

Cluster Identities for the Ising Model as a Basis for Simple Effective Field Approximations Discerning between Various Lattices with the Same Coordination Number

Piotr Tomczak, Enaldo F. Sarmento, Aline Franco Siqueira, A. Ryszard Ferchmin

Published Aug 1, 1987• 26 citations

Publisher

Abstract

DOI: 10.1002/pssb.2221420224

Related papers

Contribution to the new type of effective-field theory of the Ising model

R. Honmura, T. Kaneyoshi

A new type of effective-field theory of the Ising model is presented. The differential-operator method is introduced into the exact spin correlation function identity obtained by Callen. The Curie temperatures are evaluated by using two different types of effective Hamiltonians. It is also shown how the Zernike and the Bethe-Peierls equations can be reproduced within one framework depending on the choice of effective fields in an effective Hamiltonian. The spin correlation function and the specific heat are presented.A new type of effective-field theory of the Ising model is presented. The differential-operator method is introduced into the exact spin correlation function identity obtained by Callen. The Curie temperatures are evaluated by using two different types of effective Hamiltonians. It is also shown how the Zernike and the Bethe-Peierls equations can be reproduced within one framework depending on the choice of effective fields in an effective Hamiltonian. The spin correlation function and the specific heat are presented.

Journal of Physics C Solid State PhysicsOct 14, 1979622 citations

General solution of spin-1 Ising model in the effective field theory approximation: critical temperatures and spontaneous magnetization

E. Jurčišinová, Marian Jurčisin

The system of two polynomial equations is derived for direct simultaneous determination of the critical temperatures of the spin-1 Ising model on lattices with arbitrary values of the coordination number in the single-site cluster effective field theory approximation. Analysis of the critical temperatures allows to determine simple linear function 2z/3−1/2 that approximates the critical temperatures of the model with very high precision for all values of the coordination number z. It is demonstrated that the system of two equations for the critical temperatures of the model can be reduced to the corresponding single equation in the cases with z=3,4, and 5. In addition, the system of entirely algebraic polynomial equations for the determination of the spontaneous magnetization of the model simultaneously for all values of the coordination number is also derived and discussed.The system of two polynomial equations is derived for direct simultaneous determination of the critical temperatures of the spin-1 Ising model on lattices with arbitrary values of the coordination number in the single-site cluster effective field theory approximation. Analysis of the critical temperatures allows to determine simple linear function 2z/3−1/2 that approximates the critical temperatures of the model with very high precision for all values of the coordination number z. It is demonstrated that the system of two equations for the critical temperatures of the model can be reduced to the corresponding single equation in the cases with z=3,4, and 5. In addition, the system of entirely algebraic polynomial equations for the determination of the spontaneous magnetization of the model simultaneously for all values of the coordination number is also derived and discussed.

Phase TransitionsNov 7, 20213 citations

Effective-field model with correlations for Ising systems

G. Bruce Taggart, I. P. Fittipaldi

Using the techniques introduced by Honmura and Kaneyoshi, we show that two sets of identities for the two-state Ising model with $z$ nearest neighbors can be used as a basis for various approximation schemes which can systematically include correlation effects. In one dimension the results in zero field are exact for the magnetization, nearest-neighbor correlation function, and critical temperature. By neglecting correlations of more than three sites we introduce a simple approximation in two and three dimensions for the critical temperature. This simple approximation yields results which improve on those of the Bethe approximation. We discuss how the approximation may be improved and extended to other Ising problems.Using the techniques introduced by Honmura and Kaneyoshi, we show that two sets of identities for the two-state Ising model with $z$ nearest neighbors can be used as a basis for various approximation schemes which can systematically include correlation effects. In one dimension the results in zero field are exact for the magnetization, nearest-neighbor correlation function, and critical temperature. By neglecting correlations of more than three sites we introduce a simple approximation in two and three dimensions for the critical temperature. This simple approximation yields results which improve on those of the Bethe approximation. We discuss how the approximation may be improved and extended to other Ising problems.

Physical review. B, Condensed matterJun 1, 198244 citations

Averaging over interaction fields and spin correlations

S. V. Semkin, Victor Smagin

Averaging over interaction fields and spin correlations in the Ising modelIn the theory of phase transitions in magnetic systems, there are a small number of exact results.One of them is the well-known Onsager solution for the Ising model on a square lattice.The problem of phase transitions in magnets is a special case of the problem of many bodies.There are many approaches and methods in this area, including the method of averaging over exchange fields.The application of the distribution function over interaction fields to the study of the properties of a system of many interacting particles has been used for a long time.In previous works of the authors, the method of averaging over exchange fields was applied to the analysis of the magnetic properties of pure and diluted magnets.In this paper, we formulate and prove the relations on which the method of averaging over exchange fields can be based on clusters of several spins.It turns out that based only on the possibility of representing thermodynamic means as means of the distribution function of the fields, some nontrivial exact relations can be obtained.The paper considers the Ising model on simple lattices with coordination numbers of three and four.Exact expressions are found that connect the average products of three spins with spontaneous magnetization in the system.Inequalities are obtained that limit the limiting value of the Curie temperature and the maximum value of spontaneous magnetization at a given temperature.Averaging over interaction fields and spin correlations in the Ising modelIn the theory of phase transitions in magnetic systems, there are a small number of exact results.One of them is the well-known Onsager solution for the Ising model on a square lattice.The problem of phase transitions in magnets is a special case of the problem of many bodies.There are many approaches and methods in this area, including the method of averaging over exchange fields.The application of the distribution function over interaction fields to the study of the properties of a system of many interacting particles has been used for a long time.In previous works of the authors, the method of averaging over exchange fields was applied to the analysis of the magnetic properties of pure and diluted magnets.In this paper, we formulate and prove the relations on which the method of averaging over exchange fields can be based on clusters of several spins.It turns out that based only on the possibility of representing thermodynamic means as means of the distribution function of the fields, some nontrivial exact relations can be obtained.The paper considers the Ising model on simple lattices with coordination numbers of three and four.Exact expressions are found that connect the average products of three spins with spontaneous magnetization in the system.Inequalities are obtained that limit the limiting value of the Curie temperature and the maximum value of spontaneous magnetization at a given temperature.

New correlated-effective-field theory in the Ising model

T. Kaneyoshi, I. P. Fittipaldi, R. Honmura +1 more

A new method incorporating the effects of many-body static spin correlations into an effective-field theory is discussed. The method is based on the introduction of a differential operator and the concept of the correlated effective field into two exact Callen identities of the Ising model. The resulting statistical theory is shown to have an accuracy equivalent to that of the Bethe-Peierls method. It is shown that the correlated-effective-field parameter at the transition temperature has a universal function of $\frac{1}{(z\ensuremath{-}1)}$, where $z$ is the number of nearest neighbors.A new method incorporating the effects of many-body static spin correlations into an effective-field theory is discussed. The method is based on the introduction of a differential operator and the concept of the correlated effective field into two exact Callen identities of the Ising model. The resulting statistical theory is shown to have an accuracy equivalent to that of the Bethe-Peierls method. It is shown that the correlated-effective-field parameter at the transition temperature has a universal function of $\frac{1}{(z\ensuremath{-}1)}$, where $z$ is the number of nearest neighbors.

Physical review. B, Condensed matterJul 1, 1981145 citations

Diluted Ising spin‐1/2 lattice with an arbitrary coordination number

Bach Thanh Cong, Y. El Amraoui

physica status solidi (b)Sep 1, 19962 citations

Using the techniques introduced by Honmura and Kaneyoshi, we show that two sets of identities for the two-state Ising model with

z

nearest neighbors can be used as a basis for various approximation schemes which can systematically include correlation effects. In one dimension the results in zero field are exact for the magnetization, nearest-neighbor correlation function, and critical temperature. By neglecting correlations of more than three sites we introduce a simple approximation in two and three dimensions for the critical temperature. This simple approximation yields results which improve on those of the Bethe approximation. We discuss how the approximation may be improved and extended to other Ising problems.

z

is the number of nearest neighbors.