The Bethe Ansatz

We simply have \(M_1 = N/2-1\), \(M_1^\infty = 0\). We have \(I^{1,\infty} = \frac{N}{4} + 1\) and \(I^{\mbox{max}} = \frac{N}{4}\). The number of states is thus \(\left( \begin{array}{c} N/2 + 1 \\ N/2 - 1 \end{array} \right) = \frac{N(N+2)}{8}\). These are the \(S = 1\), \(S^z = 1\) two-spinon states.