Variational Approaches to the Antiferromagnetic Linear Chain

Abstract

Within the framework of a somewhat general free-energy variational calculation we find that a solution with an energy gap and spontaneous long-range order is favored. However, if on the basis of recent results one disallows the gap, then Bulaevskii's solution, without spontaneous long-range order, obtains. From Bulaevskii's coupled integral equations and the condition for maximum entropy with respect to variation of the external field $h$ at fixed temperature $\theta$, we obtain a pseudophase boundary in the $h$, $\theta$ plane. The analysis indicates that asymptotically along the boundary, $\frac{\mathrm{}(2-h)\mathrm{}}{\theta }\to \mathrm{positive}$ constant as $\theta \to 0$, and $\frac{h}{{({\theta }_0-\theta )}^{\frac{1}{2}}}\to \mathrm{positive}$ constant as $\theta \to {\theta }_0$, where ($h=0\mathrm{}$, $\theta ={\theta }_0$) is the intersection of the boundary with the $\theta$ axis. Qualitatively similar behavior is displayed by the exactly soluble $X-Y$ model for which the pseudophase boundary is also given here. The boundary curves are compared with one obtained for a finite chain by Bonner and Fisher.