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

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Bethe-Peierls approximation for Lagrangian and Hamiltonian lattice models

Elbio R. Dagotto, Adriana Moreo

Published Jan 15, 1984• 15 citations
Publisher

Abstract

The Bethe-Peierls approximation is applied to the classical Ising model in $D$ dimensions and in a variational treatment of the Hamiltonian $Z(2)$ gauge model in 2+1 dimensions. These analytical and simple calculations are in good agreement with exact results.

DOI
10.1103/physrevd.29.300

Related papers

Expanded Bethe–Peierls approximation for the Ising model with S = 1/2 and 1

An Du, H. J. Liu, Yao Yu

Abstract A expanded Bethe–Peierls approximation has been used to study the Ising model on honeycomb lattice, square lattice, kagomé lattice, triangular lattice and simple cubic lattice with spin S = 1/2 and 1. The spatial dimensionality of system is distinguished for different lattices. It is found that the transition temperatures on different lattices are much closer to the real values than those obtained by the effective field theory and the Bethe lattice approximation, and the transition temperatures of systems on kagomé lattice (z = 4) and trianguilar lattice (z = 6) are lower than those on square lattice (z = 4) and simple cubic lattice (z = 6), respectively. The static magnetic susceptibilities are also calculated. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

physica status solidi (b)•Dec 10, 2003•15 citations

Loop expansion around the Bethe–Peierls approximation for lattice models

Giorgio Parisi A, František Slanina

We develop an effective field theory for lattice models, in which the only non-vanishing diagrams exactly reproduce the topology of the lattice. The Bethe–Peierls approximation appears naturally as the saddle-point approximation. The corrections to the saddle-point result can be obtained systematically. We calculate the lowest loop corrections for magnetization and correlation function.

arXiv:cond-mat/0512529•Journal of Statistical Mechanics Theory and Experiment•Feb 8, 2006•43 citations

Bethe-Peierls approximation for the 2D random Ising model

Copenhagen O Dk

The partition function of the 2D Ising model with random nearest-neighbour coupling is expressed in the dual lattice made of square plaquettes. The dual model is solved in the mean field and in different types of Bethe-Peierls approximations, using the replica method. The application of methods of mean-field type to Ising models allows one to obtain very accurate approximations of the thermodynamic quantities. However, in the presence of quenched disorder this approach is difficult to implement. In this paper, we show that rather good results can be obtained after performing a duality transformation of the Ising model with random nearest-neighbour coupling that assumes the values Jij D1 with equal probability. The model is thus defined on a dual lattice where the spin variables are attached to the square plaquettes. The advantage is that the quadratic term of the dual Hamiltonian has constant coefficients instead of random ones. It is therefore possible to use the standard methods of the mean field to estimate the quenched free energy. Our results can be generalized to higher dimensions, although the approximations become rougher, because the number of plaquette spins over the number of interaction links increases with the dimensionality (1). In particular, we obtain an extremely good estimate of the ground- state energy of the random Ising model, by applying the Bethe-Peierls approximation where part of the short-range order is taken into account. In section 1, we introduce the dual lattice made of elementary square plaquettes. On this lattice the partition function can be expressed as a function of the inverse temperature e D 1 ln tanh.1=T/ whereT D 1 is the temperature of the original lattice. In section 2, we apply the mean-field approximation to the dual model, using the replica method. In section 3, we introduce the Bethe-Peierls approximation. This allows us to obtain a very precise estimate of the ground-state energy of the two-dimensional Ising model with random coupling. In section 4, we show that it is possible to improve the Bethe-Peierls approximation by introducing an interaction between different replicas. In section 5, the reader will find some remarks and conclusions.

•Jan 1, 1996•0 citations

How to compute loop corrections to the Bethe approximation

Andrea Montanari A, Tommaso Rizzo

We introduce a method for computing corrections to the Bethe approximation for spin models on arbitrary lattices. Unlike cluster variational methods, the new approach takes into account fluctuations on all length scales. The derivation of the leading correction is explained and applied to two simple examples: the ferromagnetic Ising model on d -dimensional lattices, and the spin glass on random graphs (both in their high-temperature phases). In the first case we rederive the well known Ginzburg criterion and the upper critical dimension. In the second, we compute finite-size corrections to the free energy.

arXiv:cond-mat/0506769•Journal of Statistical Mechanics Theory and Experiment•Oct 21, 2005•81 citations

Modified Mean Field approximation for the Ising Model

Cayetano Di Bartolo, Lorenzo Leal

We study a modified mean-field approximation for the Ising Model in arbitrary dimension. Instead of taking a "central" spin, or a small "drop" of fluctuating spins coupled to the effective field of their nearest neighbors as in the Mean-Field or the Bethe-Peierls-Weiss methods, we take an infinite chain of fluctuating spins coupled to the mean field of the rest of the lattice. This results in a significative improvement of the Mean-Field approximation with a small extra effort.

arXiv:0906.1576•Jun 8, 2009•0 citations

How to Compute Loop Corrections to Bethe Approximation

Andrea Montanari A, Tommaso Rizzo

We introduce a method for computing corrections to Bethe approximation for spin models on arbitrary lattices. Unlike cluster variational methods, the new approach takes into account fluctuations on all length scales. The derivation of the leading correction is explained and applied to two simple examples: the ferromagnetic Ising model on d-dimensional lattices, and the spin glass on random graphs (both in their high-temperature phases). In the first case we rederive the well-known Ginzburg criterion and the upper critical dimension. In the second, we compute finite-size corrections to the free energy. UMR 8549, Unité Mixte de Recherche du Centre National de la Recherche Scientifique et de l ’ Ecole

•Jan 1, 2005•56 citations
The Bethe-Peierls approximation is applied to the classical Ising model in DDD dimensions and in a variational treatment of the Hamiltonian Z(2)Z(2)Z(2) gauge model in 2+1 dimensions. These analytical and simple calculations are in good agreement with exact results.
Abstract A expanded Bethe–Peierls approximation has been used to study the Ising model on honeycomb lattice, square lattice, kagomé lattice, triangular lattice and simple cubic lattice with spin S = 1/2 and 1. The spatial dimensionality of system is distinguished for different lattices. It is found that the transition temperatures on different lattices are much closer to the real values than those obtained by the effective field theory and the Bethe lattice approximation, and the transition temperatures of systems on kagomé lattice (z = 4) and trianguilar lattice (z = 6) are lower than those on square lattice (z = 4) and simple cubic lattice (z = 6), respectively. The static magnetic susceptibilities are also calculated. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
The partition function of the 2D Ising model with random nearest-neighbour coupling is expressed in the dual lattice made of square plaquettes. The dual model is solved in the mean field and in different types of Bethe-Peierls approximations, using the replica method. The application of methods of mean-field type to Ising models allows one to obtain very accurate approximations of the thermodynamic quantities. However, in the presence of quenched disorder this approach is difficult to implement. In this paper, we show that rather good results can be obtained after performing a duality transformation of the Ising model with random nearest-neighbour coupling that assumes the values Jij D1 with equal probability. The model is thus defined on a dual lattice where the spin variables are attached to the square plaquettes. The advantage is that the quadratic term of the dual Hamiltonian has constant coefficients instead of random ones. It is therefore possible to use the standard methods of the mean field to estimate the quenched free energy. Our results can be generalized to higher dimensions, although the approximations become rougher, because the number of plaquette spins over the number of interaction links increases with the dimensionality (1). In particular, we obtain an extremely good estimate of the ground- state energy of the random Ising model, by applying the Bethe-Peierls approximation where part of the short-range order is taken into account. In section 1, we introduce the dual lattice made of elementary square plaquettes. On this lattice the partition function can be expressed as a function of the inverse temperature e D 1 ln tanh.1=T/ whereT D 1 is the temperature of the original lattice. In section 2, we apply the mean-field approximation to the dual model, using the replica method. In section 3, we introduce the Bethe-Peierls approximation. This allows us to obtain a very precise estimate of the ground-state energy of the two-dimensional Ising model with random coupling. In section 4, we show that it is possible to improve the Bethe-Peierls approximation by introducing an interaction between different replicas. In section 5, the reader will find some remarks and conclusions.
We introduce a method for computing corrections to the Bethe approximation for spin models on arbitrary lattices. Unlike cluster variational methods, the new approach takes into account fluctuations on all length scales. The derivation of the leading correction is explained and applied to two simple examples: the ferromagnetic Ising model on d -dimensional lattices, and the spin glass on random graphs (both in their high-temperature phases). In the first case we rederive the well known Ginzburg criterion and the upper critical dimension. In the second, we compute finite-size corrections to the free energy.
We study a modified mean-field approximation for the Ising Model in arbitrary dimension. Instead of taking a "central" spin, or a small "drop" of fluctuating spins coupled to the effective field of their nearest neighbors as in the Mean-Field or the Bethe-Peierls-Weiss methods, we take an infinite chain of fluctuating spins coupled to the mean field of the rest of the lattice. This results in a significative improvement of the Mean-Field approximation with a small extra effort.
We introduce a method for computing corrections to Bethe approximation for spin models on arbitrary lattices. Unlike cluster variational methods, the new approach takes into account fluctuations on all length scales. The derivation of the leading correction is explained and applied to two simple examples: the ferromagnetic Ising model on d-dimensional lattices, and the spin glass on random graphs (both in their high-temperature phases). In the first case we rederive the well-known Ginzburg criterion and the upper critical dimension. In the second, we compute finite-size corrections to the free energy. UMR 8549, Unité Mixte de Recherche du Centre National de la Recherche Scientifique et de l ’ Ecole