Exact two-spin dynamic structure factor of the one-dimensional s=1/2 Heisenberg-Ising antiferromagnet

The exact 2-spinon part of the dynamic spin structure factor $S_{xx}(Q,\omega)$ for the one-dimensional $s$=1/2 $XXZ$ model at $T$=0 in the antiferromagnetically ordered phase is calculated using recent advances by Jimbo and Miwa in the algebraic analysis based on (infinite-dimensional) quantum group symmetries of this model and the related vertex models. The 2-spinon excitations form a 2-parameter continuum consisting of two partly overlapping sheets in $(Q,\omega)$-space. The spectral threshold has a smooth maximum at the Brillouin zone boundary $(Q=\pi/2)$ and a smooth minimum with a gap at the zone center $(Q=0)$. The 2-spinon density of states has square-root divergences at the lower and upper continuum boundaries. For the 2-spinon transition rates, the two regimes $0 \leq Q < Q_\kappa$ (near the zone center) and $Q_\kappa < Q \leq \pi/2$ (near the zone boundary) must be distinguished, where $Q_\kappa \to 0$ in the Heisenberg limit and $Q_\kappa \to \pi/2$ in the Ising limit. The resulting 2-spinon part of $S_{xx}(Q,\omega)$ is then square-root divergent at the spectral threshold and vanishes in a square-root cusp at the upper boundary. In the regime $0 < Q_\kappa \leq \pi/2$, by contrast, the 2-spinon transition rates have a smooth maximum inside the continuum and vanish linearly at either boundary. Existing perturbation studies have been unable to capture the physics of the regime $Q_\kappa < Q \leq \pi/2$. However, their line shape predictions for the regime $0 \leq Q < Q_\kappa$ are in good agreement with the new exact results if the anisotropy is very strong.