Ordering in Linear Antiferromagnetic Chains with Anisotropic Coupling

Some reasonable conjectures are made concerning the finite-temperature pair correlations of spins with anisotropic antiferromagnetic coupling. These conjectures provide a general description of the ordering. Using them together with the finite value of the zero-temperature susceptibility, one obtains $S_1<S_3<\dots <0<\dots <S_4<S_2,$ where $S_n=1-{\mathrm{}(-1\mathrm{})\mathrm{}}^n{\omega }_{\infty }+2\mathrm{}\Sigma \stackrel{n}{l=1\mathrm{}}{\omega }_l,$ ${\omega }_l$ is the zero-temperature pair correlation, and ${\omega }_{\infty }$ is the infinite-$l$ limit of $\left|{\omega }_{\mathrm{l}}\right|$. Bonner and Fisher's finite-chain extrapolations for ${\omega }_l$ are in agreement with this result. Using their values of ${\omega }_l$ ($l=1, 2, 3, 4, \infty$) and the inequality, bounds are computed for ${\omega }_5$. The further conjecture that the rate of decrease in the absolute value of the correlation with distance is monotonic leads to a contradiction near the Heisenberg limit. The role of ${\omega }_{\infty }$ in the inequality and its derivation is particularly interesting since the limit $l\to \infty$ followed by $T\to 0$ of the pair correlation of spins separated by $l-1\mathrm{}$ spins is probably zero and not ${\omega }_{\infty }$. When the correlations approximate their zero-temperature value out to a distance $\xi$ such that $\left|{\omega }_{\xi }\right|\approx {\omega }_{\infty }$ and decrease slowly thereafter with increasing separation, then $T\chi$ is approximately zero.