The Bethe Ansatz

We take \(M = N/2\), \(M_1 = M_1^< = M-4 = N/2-4\), \(M_2 = 2\). The limiting quantum numbers are here

\begin{equation*} \pi (1 - \frac{M-5}{N} - \frac{4}{N}) = \frac{2\pi}{N} \frac{N-M+1}{2} = \frac{2\pi}{N} (\frac{N}{4} + \frac{1}{2}) \rightarrow I^{1,\infty} = \frac{N}{4} + \frac{1}{2}. \end{equation*} \begin{equation*} \pi (1 - 2 \frac{M-4}{N} - \frac{3}{N}) = \frac{2\pi}{N} \frac{N - 2M + 5}{2} \rightarrow I^{2,\infty} = \frac{5}{2} \end{equation*}

There are thus 2 slots for 2 two-strings (1 possibility) and \(N/2\) slots for \(N/2 - 4\) one-strings, giving in total \(\frac{N (N-2) (N-4) (N-6)}{384}\) states, which are the \(S = 0\), \(S^z = 0\) four-spinon states.