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

The exact two-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 in the algebraic analysis based on (infinite-dimensional) quantum group symmetries of this model and the related vertex models. The two-spinon excitations form a two-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 two-spinon density of states has square-root divergences at the lower and upper continuum boundaries. For the two-spinon transition rates; the two regimes 0 less than or equal to Q<Q(kappa) (near the zone center) and Q(kappa)<Q less than or equal to pi/2 (near the zone boundary) must be distinguished, where Q(kappa)-->0 in the Heisenberg limit and Q(kappa)-->pi/2 in the Ising limit. In the regime Q(kappa)<Q less than or equal to pi/2, the two-spinon transition rates relevant for S-xx(Q,omega) are finite at the lower boundary of each sheet,decrease monotonically with increasing omega, and approach zero linearly at the upper boundary. The resulting two-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)less than or equal to pi/2, in contrast, the two-spinon transition rates have a smooth maximum inside the continuum and vanish linearly at either boundary. In the associated two-spinon line shapes of S-xx(Q,omega), the linear cusps at the continuum boundaries are replaced by square-root cusps. Existing perturbation studies have been unable to capture the physics of the regime Q(kappa)<Q less than or equal to pi/2. However, their line-shape predictions for the regime 0 less than or equal to Q<Q(kappa) are in good agreement with the exact results if the anisotropy is very strong. For weak anisotropies, the exact line shapes are more asymmetric.