The Bethe Ansatz

We take \(M = N/2\), \(M_1 = M_1^< = N/2-3\), \(M_3 = 1\). The limiting quantum numbers are

\begin{equation*} I^{1,\infty} = \frac{N}{4} + 1, \hspace{5mm} I^{3,\infty} = 3 \end{equation*}

meaning that there is one available slot for the three-string, and \(N/2 +1\) slots for the \(N/2-3\) finite rapidity one-strings. This gives \((N + 2) N (N-2) (N-4)/384\) states, which are the four-spinon states in this \(S=0\), \(S^z=0\) sector.