Abstract We obtain some simple eigenvectors of the transfer matrix of the zero-field eight-vertex model. These are also eigenvectors of the Hamiltonian of the one-dimensional anisotropic Heisenberg chain. We also obtain new equations for the matrix Q ( v ) introduced in earlier papers.

Advanced series in mathematical physicsMar 1, 19901 citation

Bulletin of the American Physical Society

Statistical quantum mechanics and entanglement in anisotropic Heisenberg model.

You-Ling Chiang, ARMEN N. KOCHARIAN, Chee Yang

Published Mar 9, 2007• 0 citations

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The anisotropic Hamiltonian, $H=\ensuremath{-}\frac{1}{2}\ensuremath{\Sigma}({J}_{x}{{\ensuremath{\sigma}}_{l}}^{x}{{\ensuremath{\sigma}}_{l+1}}^{x}+{J}_{y}{{\ensuremath{\sigma}}_{l}}^{y}{{\ensuremath{\sigma}}_{l+1}}^{y}+{J}_{z}{{\ensuremath{\sigma}}_{l}}^{z}{{\ensuremath{\sigma}}_{l+1}}^{z})\ensuremath{-}m\mathcal{H}\ensuremath{\Sigma}{{\ensuremath{\sigma}}_{l}}^{z},$ of the linear spin array in the Heisenberg model of magnetism is examined. The eigenstate and the partition function for the case ${J}_{z}=0$ are obtained exactly for a finite system and for an infinite system with the aid of annihilation and creation operators, and the free energy $F$ of the latter is given by $\ensuremath{-}\frac{F}{\mathrm{NkT}}=(\frac{1}{\ensuremath{\pi}})\ensuremath{\int}{0}^{\ensuremath{\pi}}\mathrm{ln}{2cosh{[{{K}_{x}}^{2}+{{K}_{y}}^{2}+2{K}_{x}{K}_{y}cos2\ensuremath{\omega}\ensuremath{-}2C({K}_{x}+{K}_{y})cos\ensuremath{\omega}+{C}^{2}]}^{\frac{1}{2}}}d\ensuremath{\omega},$ where ${K}_{x}=\frac{{J}_{x}}{2kT}$ , ${K}_{y}=\frac{{J}_{y}}{2kT}$ , $C=\frac{m\mathcal{H}}{\mathrm{kT}}$ . The case ${J}_{x}={J}_{y}={J}_{z}=J$ is discussed with the aid of a high-temperature expansion and of analysis of small systems. Specific heats and susceptibilities in special cases: (i) ${J}_{x}={J}_{y}=J$ , ${J}_{z}=0$ , (ii) ${J}_{x}=J$ , ${J}_{y}={J}_{z}=0$ , ( ${\mathrm{iii}}_{\mathrm{f}}$ ) ${J}_{x}={J}_{y}=0$ , ${J}_{z}=J>0$ , ( ${\mathrm{iii}}_{\mathrm{a}}$ ) ${J}_{x}={J}_{y}=0$ , ${J}_{z}=J<0$ , ( ${\mathrm{iv}}_{\mathrm{f}}$ ) ${J}_{x}={J}_{y}={J}_{z}=J>0$ , ( ${\mathrm{iv}}_{\mathrm{a}}$ ) ${J}_{x}={J}_{y}={J}_{z}=J<0$ are compared and it is shown that (i), ( ${\mathrm{iii}}_{\mathrm{a}}$ ), and ( ${\mathrm{iv}}_{\mathrm{a}}$ ) have the characteristic features of the observed parallel susceptibility of an antiferromagnetic substance, (ii) those of perpendicular susceptibility, and ( ${\mathrm{iii}}_{\mathrm{f}}$ ) and ( ${\mathrm{iv}}_{\mathrm{f}}$ ) those of paramagnetic susceptibility, even though they have no singularities. The distribution of the zeros of the partition function is also discussed.The anisotropic Hamiltonian, $H=\ensuremath{-}\frac{1}{2}\ensuremath{\Sigma}({J}_{x}{{\ensuremath{\sigma}}_{l}}^{x}{{\ensuremath{\sigma}}_{l+1}}^{x}+{J}_{y}{{\ensuremath{\sigma}}_{l}}^{y}{{\ensuremath{\sigma}}_{l+1}}^{y}+{J}_{z}{{\ensuremath{\sigma}}_{l}}^{z}{{\ensuremath{\sigma}}_{l+1}}^{z})\ensuremath{-}m\mathcal{H}\ensuremath{\Sigma}{{\ensuremath{\sigma}}_{l}}^{z},$ of the linear spin array in the Heisenberg model of magnetism is examined. The eigenstate and the partition function for the case ${J}_{z}=0$ are obtained exactly for a finite system and for an infinite system with the aid of annihilation and creation operators, and the free energy $F$ of the latter is given by $\ensuremath{-}\frac{F}{\mathrm{NkT}}=(\frac{1}{\ensuremath{\pi}})\ensuremath{\int}{0}^{\ensuremath{\pi}}\mathrm{ln}{2cosh{[{{K}_{x}}^{2}+{{K}_{y}}^{2}+2{K}_{x}{K}_{y}cos2\ensuremath{\omega}\ensuremath{-}2C({K}_{x}+{K}_{y})cos\ensuremath{\omega}+{C}^{2}]}^{\frac{1}{2}}}d\ensuremath{\omega},$ where ${K}_{x}=\frac{{J}_{x}}{2kT}$, ${K}_{y}=\frac{{J}_{y}}{2kT}$, $C=\frac{m\mathcal{H}}{\mathrm{kT}}$. The case ${J}_{x}={J}_{y}={J}_{z}=J$ is discussed with the aid of a high-temperature expansion and of analysis of small systems. Specific heats and susceptibilities in special cases: (i) ${J}_{x}={J}_{y}=J$, ${J}_{z}=0$, (ii) ${J}_{x}=J$, ${J}_{y}={J}_{z}=0$, (${\mathrm{iii}}_{\mathrm{f}}$) ${J}_{x}={J}_{y}=0$, ${J}_{z}=J>0$, (${\mathrm{iii}}_{\mathrm{a}}$) ${J}_{x}={J}_{y}=0$, ${J}_{z}=J<0$, (${\mathrm{iv}}_{\mathrm{f}}$) ${J}_{x}={J}_{y}={J}_{z}=J>0$, (${\mathrm{iv}}_{\mathrm{a}}$) ${J}_{x}={J}_{y}={J}_{z}=J<0$ are compared and it is shown that (i), (${\mathrm{iii}}_{\mathrm{a}}$), and (${\mathrm{iv}}_{\mathrm{a}}$) have the characteristic features of the observed parallel susceptibility of an antiferromagnetic substance, (ii) those of perpendicular susceptibility, and (${\mathrm{iii}}_{\mathrm{f}}$) and (${\mathrm{iv}}_{\mathrm{f}}$) those of paramagnetic susceptibility, even though they have no singularities. The distribution of the zeros of the partition function is also discussed.

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Eight-Vertex Model in Lattice Statistics and One-Dimensional Anisotropic Heisenberg Chain: I. Some Fundamental Eigenvectors

R. J. Baxter

Advanced series in mathematical physicsMar 1, 19901 citation

{J}_{z}=0

are obtained exactly for a finite system and for an infinite system with the aid of annihilation and creation operators, and the free energy

F

of the latter is given by $\ensuremath{-}\frac{F}{\mathrm{NkT}}=(\frac{1}{\ensuremath{\pi}})\ensuremath{\int}{0}^{\ensuremath{\pi}}\mathrm{ln}{2cosh{[{{K}_{x}}^{2}+{{K}_{y}}^{2}+2{K}_{x}{K}_{y}cos2\ensuremath{\omega}\ensuremath{-}2C({K}_{x}+{K}_{y})cos\ensuremath{\omega}+{C}^{2}]}^{\frac{1}{2}}}d\ensuremath{\omega},$ where

{K}_{x}=\frac{{J}_{x}}{2kT}

{K}_{y}=\frac{{J}_{y}}{2kT}

C=\frac{m\mathcal{H}}{\mathrm{kT}}

. The case

{J}_{x}={J}_{y}={J}_{z}=J

is discussed with the aid of a high-temperature expansion and of analysis of small systems. Specific heats and susceptibilities in special cases: (i)

{J}_{x}={J}_{y}=J

{J}_{z}=0

, (ii)

{J}_{x}=J

{J}_{y}={J}_{z}=0

, (

{\mathrm{iii}}_{\mathrm{f}}

)

{J}_{x}={J}_{y}=0

{J}_{z}=J>0

, (

{\mathrm{iii}}_{\mathrm{a}}

)

{J}_{x}={J}_{y}=0

{J}_{z}=J<0

, (

{\mathrm{iv}}_{\mathrm{f}}

)

{J}_{x}={J}_{y}={J}_{z}=J>0

, (

{\mathrm{iv}}_{\mathrm{a}}

)

{J}_{x}={J}_{y}={J}_{z}=J<0

are compared and it is shown that (i), (

{\mathrm{iii}}_{\mathrm{a}}

), and (

{\mathrm{iv}}_{\mathrm{a}}

) have the characteristic features of the observed parallel susceptibility of an antiferromagnetic substance, (ii) those of perpendicular susceptibility, and (

{\mathrm{iii}}_{\mathrm{f}}

) and (

{\mathrm{iv}}_{\mathrm{f}}

) those of paramagnetic susceptibility, even though they have no singularities. The distribution of the zeros of the partition function is also discussed.