Physical Review Letters

Exact Solution for a Closed Chain of Classical Spins with Arbitrary, Isotropic Nearest-Neighbor Exchange

Luke Lokia Liu, R. I. Joseph

Published May 31, 1971• 28 citations

Abstract

The problem of a system of classical spins of arbitrary dimensionality interacting through an arbitrary, isotropic nearest-neighbor-exchange Hamiltonian is solved exactly for the closed linear chain. The $2m$-uple spin-correlation function for spins a distance $k$ apart is shown to be the $k\mathrm{th}$ power of the ratio of the $m\mathrm{th}$ eigenvalue to the largest one. It is proposed that experimental data be used to find the general functional form of the exchange interaction.

DOI: 10.1103/physrevlett.26.1378

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Physical Review Letters

Exact Solution for a Closed Chain of Classical Spins with Arbitrary, Isotropic Nearest-Neighbor Exchange

Luke Lokia Liu, R. I. Joseph

Published May 31, 1971• 28 citations

Publisher

Abstract

2m

-uple spin-correlation function for spins a distance

k

apart is shown to be the

k\mathrm{th}

power of the ratio of the

m\mathrm{th}

eigenvalue to the largest one. It is proposed that experimental data be used to find the general functional form of the exchange interaction.

DOI: 10.1103/physrevlett.26.1378

Related papers

Exactly soluble model of interacting classical spins in one dimension with random interactions

A. K. Rajagopal

An exact calculation of the thermodynamic properties of a linear chain of classical spins with nearest-neighbor bilinear and biquadratic isotropic random exchange interactions is presented. Both open and closed chains are discussed. Only bond disorder is being considered here. It is shown that in contrast to the uniform system, "dipolar disorder points" as well as "quadrupolar disorder points" with qualitative and quantitative differences from the uniform case occur in this system. This is because the competing dipolar and quadrupolar interactions are further enriched by the random signs of these interactions. It is interesting to point out that much of the analysis can be made by means of a study of the eigenvalues of the "transfer kernel." The method can also be applied to obtain exact results for a linear chain of spins of arbitrary dimensionality interacting through an arbitrary isotropic nearest-neighbor Hamiltonian. In the case of the open chain, the Edwards-Anderson expedient is found to be exact in all the cases considered.An exact calculation of the thermodynamic properties of a linear chain of classical spins with nearest-neighbor bilinear and biquadratic isotropic random exchange interactions is presented. Both open and closed chains are discussed. Only bond disorder is being considered here. It is shown that in contrast to the uniform system, "dipolar disorder points" as well as "quadrupolar disorder points" with qualitative and quantitative differences from the uniform case occur in this system. This is because the competing dipolar and quadrupolar interactions are further enriched by the random signs of these interactions. It is interesting to point out that much of the analysis can be made by means of a study of the eigenvalues of the "transfer kernel." The method can also be applied to obtain exact results for a linear chain of spins of arbitrary dimensionality interacting through an arbitrary isotropic nearest-neighbor Hamiltonian. In the case of the open chain, the Edwards-Anderson expedient is found to be exact in all the cases considered.

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K. M. Leung

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Stephen B. Haley

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Exact Solution for a Linear Chain of Isotropically Interacting Classical Spins of Arbitrary Dimensionality

Eugene Stanley

The isotropic Hamiltonian ${\mathcal{H}}^{(\ensuremath{\nu})}=\ensuremath{-}J\ensuremath{\Sigma}{j=1}^{}^{N\ensuremath{-}1}\mathrm{S}_{j}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{j+1}$ is considered for an open linear chain of $N$ $\ensuremath{\nu}$-dimensional vector spins ${\mathrm{S}}_{j}$ ;${\mathcal{H}}^{(\ensuremath{\nu})}$ reduces to the $S=\frac{1}{2}$ Ising, planar, and Heisenberg models for $\ensuremath{\nu}=1,2,\mathrm{and} 3$. The thermodynamic properties (including the susceptibility) of ${\mathcal{H}}^{(\ensuremath{\nu})}$ are found for ferromagnetic ( $J>0$ ) and antiferromagnetic ( $J<0$ ) exchange interactions for all temperatures $T$ and all spin dimensionalities $\ensuremath{\nu}$. The manner in which the various properties depend upon $T$ and $\ensuremath{\nu}$ is studied; in particular we find (a) that although the chain of spins does not display long-range order except at $T=0$ for any value of $\ensuremath{\nu}$ most of the properties vary monotonically with $\ensuremath{\nu}$ (in such a way that, e.g., the degree of "short-range order" decreases with increasing $\ensuremath{\nu}$; and (b) that as the spin dimensionality increases without limit, all of the calculated properties approach precisely those predicted by the Berlin-Kac spherical model.The isotropic Hamiltonian ${\mathcal{H}}^{(\ensuremath{\nu})}=\ensuremath{-}J\ensuremath{\Sigma}{j=1}^{}^{N\ensuremath{-}1}\mathrm{S}_{j}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{j+1}$ is considered for an open linear chain of $N$ $\ensuremath{\nu}$-dimensional vector spins ${\mathrm{S}}_{j}$;${\mathcal{H}}^{(\ensuremath{\nu})}$ reduces to the $S=\frac{1}{2}$ Ising, planar, and Heisenberg models for $\ensuremath{\nu}=1,2,\mathrm{and} 3$. The thermodynamic properties (including the susceptibility) of ${\mathcal{H}}^{(\ensuremath{\nu})}$ are found for ferromagnetic ($J>0$) and antiferromagnetic ($J<0$) exchange interactions for all temperatures $T$ and all spin dimensionalities $\ensuremath{\nu}$. The manner in which the various properties depend upon $T$ and $\ensuremath{\nu}$ is studied; in particular we find (a) that although the chain of spins does not display long-range order except at $T=0$ for any value of $\ensuremath{\nu}$ most of the properties vary monotonically with $\ensuremath{\nu}$ (in such a way that, e.g., the degree of "short-range order" decreases with increasing $\ensuremath{\nu}$; and (b) that as the spin dimensionality increases without limit, all of the calculated properties approach precisely those predicted by the Berlin-Kac spherical model.

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Heisenberg Linear Chain Magnets with Unlike Spins –An Exactly Soluble Model–

Hisao Okamoto, Katsuhiko Nagano, Tahashi Karasudani

Journal of the Physical Society of JapanDec 1, 19760 citations

Magnetism in One-Dimensional Systems—The Heisenberg Model for Infinite Spin

Michael Ellis Fisher

American Journal of PhysicsMay 1, 19641,544 citations

2m

-uple spin-correlation function for spins a distance

k

apart is shown to be the

k\mathrm{th}

power of the ratio of the

m\mathrm{th}

eigenvalue to the largest one. It is proposed that experimental data be used to find the general functional form of the exchange interaction.

N

$\ensuremath{\nu}$-dimensional vector spins

{\mathrm{S}}_{j}

;${\mathcal{H}}^{(\ensuremath{\nu})}$ reduces to the

S=\frac{1}{2}

Ising, planar, and Heisenberg models for $\ensuremath{\nu}=1,2,\mathrm{and} 3$. The thermodynamic properties (including the susceptibility) of ${\mathcal{H}}^{(\ensuremath{\nu})}$ are found for ferromagnetic (

J>0

) and antiferromagnetic (

J<0

) exchange interactions for all temperatures

T

and all spin dimensionalities $\ensuremath{\nu}$. The manner in which the various properties depend upon

T

and $\ensuremath{\nu}$ is studied; in particular we find (a) that although the chain of spins does not display long-range order except at

T=0

for any value of $\ensuremath{\nu}$ most of the properties vary monotonically with $\ensuremath{\nu}$ (in such a way that, e.g., the degree of "short-range order" decreases with increasing $\ensuremath{\nu}$; and (b) that as the spin dimensionality increases without limit, all of the calculated properties approach precisely those predicted by the Berlin-Kac spherical model.