The Bethe Ansatz

Allowable quantum numbers

Consider looking for set of real rapidities as solution to h.bel. The maximally allowed quantum number \(I^{1,\infty}_M\) associated to a rapidity \(\lambda_a \rightarrow \infty\) in the presence of \(M - 1\) other finite rapidities \(\lambda_b < \infty\) is given by taking the limit \[ \lim_{\lambda_a \rightarrow \infty} 2 ~\mbox{atan}~ 2\lambda_a - \frac{1}{N} \sum_{b=1}^{M} 2~\mbox{atan} ~(\lambda_a - \lambda_b) = \pi (1 - \frac{M-1}{N}) \equiv 2\pi \frac{I^{1,\infty}_M}{N} \] from which we get

\begin{equation} I^{1,\infty}_M = \frac{N-M + 1}{2}, \hspace{1cm} I^{1,\mbox{max}}_M = I^{1,\infty}_M - 1 = \frac{N-M - 1}{2}. \tag{h.im}\label{h.im} \end{equation}

The Bethe equations will therefore have a solution in terms of a set of distinct real, finite rapidities if and only if all quantum numbers obey the inequality \(|I_j| < I^{\infty}_M\).

For a given \(M\), there are thus

\begin{equation*} \left( \begin{array}{c} N - M \\ M \end{array} \right) \end{equation*}

states containing exclusively real, finite rapidities. Since the \(XXX\) chain has a global \(su(2)\) symmetry, its eigenstates can be classified according to the representation of this algebra. Eigenstates containing real, finite rapidities only are highest-weight states of this representation b-Gaudin. Acting with the global spin lowering operator to obtain the other non-highest-weight states is achieved by adding infinite rapidities to a state, which still provides a {\it bona fide} solution to the Bethe equations for the XXX chain. At each value of \(M\), it is possible to add up to \(N/2 - M\) infinite rapidities without crossing the equator. We will denote the number of one-strings as \(M_1\), the number of finite-rapidity one-strings as \(M_1^<\) and the number of infinite rapidities as \(M_1^\infty\), so \(M_1 = M_1^< + M_1^\infty\). From the states with purely real, finite rapidities at a fixed \(M\), we can thus construct

\begin{equation*} \sum_{M_1^\infty=0}^{N/2-M} \left( \begin{array}{c} N-M+M_1^\infty \\ M - M_1^\infty \end{array} \right) \end{equation*}

states which, for a given \(M_1^\infty\) have global \(su(2)\) quantum numbers \(S = N/2-M+M_1^\infty\), \(S^z = N/2 - M\). Let us provide the simplest examples of these.