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Exact Form-Factor Results for the Longitudinal Structure Factor of the Massless XXZ Model in Zero Field

We consider the XXZ quantum spin chain in its massless, disordered regime at zero field. We derive an exact expression for the two-spinon form-factor of $S^z=1/2σ^z$ by taking a limit of the massive XYZ form-factors found by Lashkevich and by Lukyanov and Terras. This result is used to find the two-spinon contribution to the spectral decomposition of the longitudinal structure factor $S^{zz}(k,w)$. We find that this contribution provides an accurate approximation to the full structure factor over a wide range of the anisotropy parameter. The asymptotic behaviour of $S^{zz}(k,w)$ is computed as the upper and lower $w$ thresholds of the two-spinon $(w,k)$ band are approached, and an analysis of the region of validity of this threshold behaviour is performed. Our results reproduce and refine existing threshold behaviour predictions and extend these results to an accurate description throughout the two-spinon continuum.