Scattering matrix of elementary excitations in the antiperiodic XXZ spin chain with \eta=i\pi/3

We study the thermodynamic limit of the antiperiodic XXZ spin chain with the anisotropic parameter $\eta =\frac{\pi i}{3}$. We parameterize eigenvalues of the transfer matrix by their zero points instead of Bethe roots. We obtain patterns of the distribution of zero points. Based on them, we calculate the ground state energy and the elementary excitations in the thermodynamic limit. We also obtain the two-body scattering matrix of elementary excitations. Two types of elementary excitations and three types of scattering processes are discussed in detailed.