The Bethe Ansatz

Slavnov's scalar product formula ssp can be specialized to obtain the norm of Bethe eigenstates, by taking the limit \(\mu_a \rightarrow \lambda_a\), \(a=1, ..., M\). This yields Gaudin's norm formula (writing \(\lambda_{ab} \equiv \lambda_a - \lambda_b\))

\begin{equation} {\mathbb N}_M = \langle 0 | \prod_{a=1}^M C (\lambda_a) \prod_{b=1}^M B (\lambda_b) | 0 \rangle = \varphi^M(\eta) \prod_{a \neq b} \frac{\varphi(\lambda_{ab} + \eta)}{\varphi(\lambda_{ab})} \det {\cal G} (\{ \lambda \}), \tag{gnf}\label{gnf} \end{equation}

with the Gaudin matrix entries being

\begin{equation} {\cal G}_{ab} = -\frac{\partial}{\partial \lambda_b} \ln \left[ \frac{a(\lambda_a)}{d(\lambda_b)} \prod_{c = 1, \neq a}^M \frac{b(\lambda_a, \lambda_c)}{b(\lambda_c, \lambda_a)}\right]. \tag{gm}\label{gm} \end{equation}

Written out explicitly, these entries are

\begin{equation} {\cal G}_{ab} = \left\{ \begin{array}{ll} N \frac{\sinh \eta}{\sinh^2 \lambda_a - \sinh^2 \frac{\eta}{2}} - \sum_{c \neq a} \frac{\sinh 2\eta}{\sinh^2 \lambda_{ac} - \sinh^2 \eta}, \hspace{5mm} & a=b,\\ \frac{\sinh 2\eta}{\sinh^2 \lambda_{ab} - \sinh^2 \eta} & a \neq b \end{array}\right. \tag{gmx}\label{gmx} \end{equation}