The Bethe Ansatz

\(N=2, a=0\)

\begin{align*} &z_1^2 - 2\Delta z_1 + 1 = 0, \\ &~z_1 = \Delta \pm \sqrt{\Delta^2 - 1}, \\ &~z_2 = 1/z_1 = \Delta \mp \sqrt{\Delta^2 - 1}, \end{align*}

so only one distinct solution,

\[ z_1 = \Delta + \sqrt{\Delta^2 - 1}, ~~z_2 = 1/z_1 = \Delta - \sqrt{\Delta^2 - 1}. \]

Wavefunction:

\begin{align*} 1+z_1z_2 - 2\Delta z_1 = 2 (1 - \Delta z_1) = 2 (1-\Delta (\Delta + \sqrt{\Delta^2 - 1}))\\ 1+z_1z_2 - 2\Delta z_2 = 2 (1 - \Delta z_2) = 2 (1-\Delta (\Delta - \sqrt{\Delta^2 - 1})) \end{align*} \begin{align*} \Psi_{12} (k_1, k_2) &= A_{12} e^{ik_1 + 2ik_2} + A_{21} e^{2ik_1 + ik_2} \\ &= A_{12} e^{-ik_1} + A_{21} e^{-ik_2} \\ &= (1 +z_1 z_2 -2\Delta z_1)/z_1 - (1 + z_1 z_2 - 2\Delta z_2)/z_2 \\ &= 1/z_1 + z_2 - 1/z_2 - z_1 = 2 (z_2 - z_1) = -4 \sqrt{\Delta^2 - 1} \end{align*}

so the wavefunction is just (as expected) the single state \(|1,2\rangle\) (up to normalization).

\(N=2, a=1\)

The Bethe polynomial identically vanishes.