The Bethe Ansatz

Similarly as for the isotropic case, we obtain \(\rho_{\rm sp} (\omega) = \frac{1}{1 + a_2 (\omega)}\) so

\begin{equation} \rho_{\rm sp} (\omega) = \frac{\sinh (\frac{\omega \pi}{2})}{2 \sinh(\frac{\omega \pi}{2}(1 - \frac{\zeta}{\pi}))}. \tag{p.rsp}\label{p.rsp} \end{equation}

The energy and momentum of 2-spinon states are again given by h.2spe and h.2spp (using the XXZ kernels), where the single spinon energy and momentum here become

\begin{equation*} \varepsilon_{\rm sp} (\lambda^h) = \frac{J \pi \sin \zeta}{2\zeta \cosh (\frac{\pi \lambda^h}{\zeta})}, \hspace{1cm} p_{\rm sp} (\lambda^h) = -2~\mbox{atan} ~e^{-\pi \lambda^h/\zeta} \in [-\pi, 0]. \end{equation*}

Combining these gives the XXZ spinon dispersion relation,

\begin{equation} \varepsilon_{\rm sp} (p) = \frac{J \pi \sin \zeta}{2\zeta} |\sin p|, \hspace{1cm} p \in [-\pi, 0]. \tag{p.esp}\label{p.esp} \end{equation}