The Bethe Ansatz

Here, we put \(M = N/2 = M_1\), \(M_1^< = N/2-1\) and \(M_1^\infty = 1\). The equations for the limiting quantum numbers then fall back onto the \(M \rightarrow M - 1\) ones, so now \(I^{1,\infty} = \frac{N}{4} + 1\) and \(I^{1,\mbox{max}} = \frac{N}{4}\). We thus have to put \(M_1^<\) quantum numbers in \(N/2 + 1\) slots, yielding \(\left( \begin{array}{c} N/2 + 1 \\ N/2 - 1 \end{array} \right) = \frac{N(N+2)}{8}\) states, which are the \(S = 1, S^z = 0\) two-spinon states.