Statistical Mechanics of the Anisotropic Linear Heisenberg Model

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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