The Bethe Ansatz

Here, we put \(M_1^< = N/2-2\) and \(M_1^\infty = 2\). The equations for the limiting quantum numbers of the one-string fall back onto the \(M \rightarrow M-2\) ones, so now \(I^{1,\infty} = \frac{N}{4} + \frac{3}{2}\) and \(I^{\mbox{max}} = \frac{N}{4} + \frac{1}{2}\). We thus have to put \(N/2-2\) quantum numbers in \(N/2 + 2\) slots, yielding \(\left( \begin{array}{c} N/2 + 2 \\ N/2 - 2 \end{array} \right) = \frac{(N+4) (N+2) N (N-2)}{384}\) states, which are the four-spinon states in this sector.

Note that for a given \(S\), the number of states at any \(S^z\) is the same due to the invariance of counting under the simultaneous shift \(M \rightarrow M-1\), \(M_1^\infty \rightarrow M_1^\infty + 1\).