Physics Letters A

Exact solution of the one-dimensional isotropic Heisenberg chain with arbitrary spins S

H. Babujian

Published Aug 1, 1982• 383 citations

Publisher

DOI: 10.1016/0375-9601(82)90403-0

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Physics Letters A

Exact solution of the one-dimensional isotropic Heisenberg chain with arbitrary spins S

H. Babujian

Published Aug 1, 1982• 383 citations

Publisher

DOI: 10.1016/0375-9601(82)90403-0

Related papers

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Nuclear Physics BFeb 1, 1983415 citations

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A. G. Antipov, Igor V. Komarov

Physica D Nonlinear PhenomenaAug 25, 20069 citations

The picture of low-lying excitations in the isotropic Heisenberg chain of arbitrary spins

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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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Exact Results for the One-Dimensional, Anisotropic Classical Heisenberg Model

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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.