Alternating Linear Heisenberg Antiferromagnet: The Exciton Limit

The one-dimensional alternating antiferromagnet with $\mathcal{H}=2J\Sigma n^{}{\overrightarrow{\mathrm{S}}}_{2n}·{\overrightarrow{\mathrm{S}}}_{2n-1}+2\mathrm{}{J}^{\prime }\Sigma n^{}{\overrightarrow{\mathrm{S}}}_{2n}·{\overrightarrow{\mathrm{S}}}_{2n+1}$ is studied for $J^{\prime }\ll J$. For ${\beta }^{-1\mathrm{}}\equiv k_{B}T\ll J$ the susceptibility is expanded in powers of the exciton density as $\chi T\propto A(\beta J^{\prime },\frac{J^{\prime }}{J})e^{-2\mathrm{}\beta J}+B(\beta J^{\prime },\frac{J^{\prime }}{J})e^{-4\mathrm{}\beta J}+\cdots$ and the coefficients $A$ and $B$ are calculated for $\frac{J^{\prime }}{J}\to 0$. The calculation of $B(\beta J^{\prime },0)$ required the evaluation of the two-exciton scattering matrix. The interactions between excitons which affect the susceptibility are found to be repulsive. As a result, the coefficient $B$ is correctly predicted by the usual assumption that excitons obey localized statistics. A general discussion relating statistics to the on-shell forward-scattering $t$ matrix enables one to understand the difference between the statistical properties of spin waves and excitons. For opposite-spin excitons an attractive bound state is found to exist for all values of total momentum. Perturbation theory in $\frac{J^{\prime }}{J}$ is used to calculate the single-exciton dispersion relation at zero temperature as $E\left(k\right)=(2J+\frac{5J^{\prime 3}}{32J^2})-(\frac{J^{\prime }+J^{\prime 2}}{2J}-\frac{5J^{\prime 3}}{32J})cosk-(\frac{J^{\prime 2}}{4J}+\frac{J^{\prime 3}}{8J^2}){cos}^{2}k-\left(\frac{J^{\prime 3}}{8J^2}\right){cos}^{3}k$.