The Application of the Bethe-Peierls Method to Ferromagnetism

Weiß, Peter

doi:10.1103/physrev.74.1493

Cited by 226 publications

(54 citation statements)

References 9 publications

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“…The first studies of the statistics of dimer coverings of a lattice were carried out more than 50 years ago, [6][7][8] obtaining results analogous to those of the Bethe approximation [9][10][11] for the Ising and Heisenberg models. At that time, the relation of this approximation scheme to the structure of the Cayley tree ͑a recursive "lattice," an example of which, with coordination number q, is shown in Fig.…”

Section: Introductionmentioning

confidence: 99%

Dimer statistics on a Bethe lattice

Harris

Cohen

2006

The Journal of Chemical Physics

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We discuss the exact solutions of various models of the statistics of dimer coverings of a Bethe lattice.We reproduce the well-known exact result for noninteracting hard-core dimers by both a very simple geometrical argument and a general algebraic formulation for lattice statistical problems. The algebraic formulation enables us to discuss loop corrections for finite dimensional lattices. For the Bethe lattice we also obtain the exact solution when either (a) the dimers interact via a short-range interaction or (b) the underlying lattice is anisotropic. We give the exact solution for a special limit of dimers on a Bethe lattice in a quenched random potential in which we consider the maximal covering of dimers on random clusters at site occupation probability p. Surprisingly the partition function for "maximal coverage" on the Bethe lattice is identical to that for the statistics of branched polymers when the activity for a monomer unit is set equal to −p. Finally we give an exact solution for the number of residual vacancies when hard-core dimers are randomly deposited on a one dimensional lattice. We discuss the exact solutions of various models of the statistics of dimer coverings of a Bethe lattice. We reproduce the well-known exact result for noninteracting hard-core dimers by both a very simple geometrical argument and a general algebraic formulation for lattice statistical problems. The algebraic formulation enables us to discuss loop corrections for finite dimensional lattices. For the Bethe lattice we also obtain the exact solution when either ͑a͒ the dimers interact via a short-range interaction or ͑b͒ the underlying lattice is anisotropic. We give the exact solution for a special limit of dimers on a Bethe lattice in a quenched random potential in which we consider the maximal covering of dimers on random clusters at site occupation probability p. Surprisingly the partition function for "maximal coverage" on the Bethe lattice is identical to that for the statistics of branched polymers when the activity for a monomer unit is set equal to −p. Finally we give an exact solution for the number of residual vacancies when hard-core dimers are randomly deposited on a one dimensional lattice.

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Section: Introductionmentioning

confidence: 99%

Dimer statistics on a Bethe lattice

Harris

Cohen

2006

The Journal of Chemical Physics

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“…The BPW approximation, which was first applied to the isotropic Heisenberg model by Weiss in Ref. 15, gives a qualitatively incorrect behavior in the low-temperature region. This behavior becomes more pronounced at lower magnetic field strengths.…”

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Correlated cluster mean-field theory for spin systems

Yamamoto

2009

Phys. Rev. B

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A cluster mean-field method is introduced and the applications to the Ising and Heisenberg models are demonstrated. We divide the lattice sites into clusters whose size and shape are selected so that the equivalence of all sites in a cluster is preserved. Since the strength of interactions of a cluster with its surrounding clusters is strongly dependent on the spin configuration of the central cluster itself, we include this contribution in the effective fields acting on the spins. The effects of "correlations" between clusters can be taken into account beyond the standard mean-field level, and as a result our cluster-based method gives qualitatively (and even quantitatively) correct results for the both Ising and Heisenberg models. Especially, for the Ising model on the honeycomb and square lattices, the calculated results of the critical temperature are very close (overestimated by only less than 5 %) to the exact values.

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“…Dentre as muitas tentativas para se construir uma teoria melhor está a aplicação feita por P.R. Weiss [3] (que nãoé o Pierre Weiss) da teoria de Bethe e Peierls [4,5], cuja idéia para descrever o ferromagnetismoé bem simples: considera-se de forma exata a interação entre um determinado momento magnético e seus primeiros vizinhos, enquanto estes por sua vez são submetidos a um campo molecular análogo aquele proposto por Pierre Weiss.…”

Section: Introductionunclassified

A aproximação de campo médio de Bethe-Peierls

Stein-Barana¹,

Yoshida²,

Líbero³

2004

Rev. Bras. Ens. Fis.

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As aproximações de campo médio de Pierre Weiss e de Bethe-Peierls são implementadas para o modelo de Ising para ferromagnetismo, salientando-se o papel das flutuações espaciais dos momentos magnéticos localizados. Na aproximação de Bethe-Peierls, embora a Hamiltoniana não seja de partículas independentes, nós mostramos uma forma simples de se obter a energia média da rede utilizando a função de correlação spinspin. Nós também comparamos a correlação de spins primeiros vizinhos calculada em ambas as aproximações com a solução exata para o sistema bidimensional de spins 1/2. Essa comparação deixa clara a supremacia da aproximação Bethe-Peierls sobre a de Pierre Weiss. Palavras-chave: campo médio, Bethe-Peierls, Ising.The Pierre Weiss and Bethe-Peierls mean-field approximations are implemented for the ferromagnetic Ising model, with emphasis in the spatial fluctuations of the localized magnetic moments. Although in the BethePeierls approximation we do not have a single-particle Hamiltonian, we present a simple way to obtain the mean energy of the lattice using the spin-spin correlation function. We also compare the next-nearest neighbor spin-spin correlation calculated in both mean-field approximations with the exact result for the two-dimensional lattice of 1/2 spin. This comparison shows clearly the supremacy of the Bethe-Peierls method over the Pierre Weiss one. Keywords: mean-field, Bethe-Peierls, Ising. IntroduçãoO começo do século passado marca o início da aplicação da Mecânica Estatística ao estudo dos materiais ferromagnéticos, ou seja, daqueles materiais que abaixo (acima) de determinada temperatura apresentam (não apresentam) magnetização espontânea, istoé, magnetização mesmo na ausência de campo externo. Exemplos eram conhecidos, como Fe, Ni e Co, e ligas contendo esses materiais. O grande nome nesse estudoé o de Pierre Weiss, que introduziu a idéia de que cada domínio magnético num material está sujeito a um campo proporcionalà magnetização total da amostra [1]. Esse campo, denominado molecular ou de Weiss, deve-seàs interações de troca entre os momentos magnéticos do material, algo que naépoca de Weiss era desconhecido e cuja origem foi elucidada somente com a teoria de Heisenberg [2]. A teoria de Weissé aproximada e constata-se que melhorias são necessárias quando se confronta as suas previsões com os resultados experimentais. Dentre as muitas tentativas para se construir uma teoria melhor está a aplicação feita por P.R. Weiss [3] (que nãoé o Pierre Weiss) da teoria de Bethe e Peierls [4,5], cuja idéia para descrever o ferromagnetismoé bem simples: considera-se de forma exata a interação entre um determinado momento magnético e seus primeiros vizinhos, enquanto estes por sua vez são submetidos a um campo molecular análogo aquele proposto por Pierre Weiss.

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The Application of the Bethe-Peierls Method to Ferromagnetism

Cited by 226 publications

References 9 publications

Dimer statistics on a Bethe lattice

Dimer statistics on a Bethe lattice

Correlated cluster mean-field theory for spin systems

A aproximação de campo médio de Bethe-Peierls

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