Statistical Mechanics of the Anisotropic Linear Heisenberg Model

The anisotropic Hamiltonian, $H=-\frac{1}{2}\Sigma (J_x{{\sigma }_l}^x{{\sigma }_{l+1\mathrm{}}}^x+J_y{{\sigma }_l}^y{{\sigma }_{l+1\mathrm{}}}^y+J_z{{\sigma }_l}^z{{\sigma }_{l+1\mathrm{}}}^z)-m\mathcal{H}\Sigma {{\sigma }_l}^z,$ of the linear spin array in the Heisenberg model of magnetism is examined. The eigenstate and the partition function for the case $J_z=0\mathrm{}$ 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 $-\frac{F}{\mathrm{NkT}}=\left(\frac{1}{\pi }\right)\int 0^{\pi }\ln \left\{2\mathrm{}cosh{[{K_x}^2+{K_y}^2+2\mathrm{}{K}_x{K}_{y}cos2\omega -2C(K_x+K_y)cos\omega +C^2]}^{\frac{1}{2}}\right\}d\omega ,$ 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) $J_x=J_y=J$, $J_z=0\mathrm{}$, (ii) $J_x=J$, $J_y=J_z=0\mathrm{}$, (${\mathrm{iii}}_{\mathrm{f}}$) $J_x=J_y=0\mathrm{}$, $J_z=J>\mathrm{}0\mathrm{}$, (${\mathrm{iii}}_{\mathrm{a}}$) $J_x=J_y=0\mathrm{}$, $J_z=J<\mathrm{}0\mathrm{}$, (${\mathrm{iv}}_{\mathrm{f}}$) $J_x=J_y=J_z=J>\mathrm{}0\mathrm{}$, (${\mathrm{iv}}_{\mathrm{a}}$) $J_x=J_y=J_z=J<\mathrm{}0\mathrm{}$ 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.