The Bethe Ansatz

A crucial result by N. Slavnov concerns the scalar product of states constructed by the action of the operators \(B(\lambda)\) on the vacuum. Namely, we wish to compute

\begin{equation} S_M (\{\mu \}, \{ \lambda \}) = \langle 0 | \prod_{j=1}^M C(\mu_j) \prod_{k=1}^M B(\lambda_k) | 0 \rangle \tag{sp}\label{sp} \end{equation}

in general, when at least one of the two sets \(\{ \mu \}\), \(\{ \lambda \}\) is solution to the Bethe equations. For definiteness, we consider \(\{ \lambda \}\) as such a set.

Slavnov's theorem 1989.Slavnov.TMP.79 then states that for \(\{ \lambda \}\) solution to the Bethe equations BER1 and for \(\{ \mu \}\) an arbitrary set of parameters, the scalar product \(S_M\) sp can be represented as a determinant,

\begin{equation} S_M (\{ \mu \}, \{ \lambda \}) = S_M (\{ \lambda \}, \{ \mu \}) = \frac{\prod_{j=1}^M \prod_{k=1}^M \varphi (\mu_j - \lambda_k)}{\prod_{j < k} \varphi (\mu_j - \mu_k) \prod_{j > k} \varphi(\lambda_j - \lambda_k)} \det T(\{ \mu \}, \{ \lambda \}), \tag{ssp}\label{ssp} \end{equation}

where

\begin{equation*} T_{ab} = \frac{\partial}{\partial \lambda_a} \tau (\mu_b| \{ \lambda \}). %\hspace{0.3cm} V_{ab} = \frac{1}{\varphi(\mu_b - \lambda_a)}, \hspace{0.3cm} 1 \leq a, b \leq M. \end{equation*}

Here,

\begin{equation*} \varphi (\lambda) \equiv \left\{ \begin{array}{cc} \lambda & \Delta = 1 \\ \sinh \lambda & |\Delta| < 1. \end{array} \right. \label{eq:phidef} \end{equation*}

The proof of this is omitted here, and goes either through the direct use of the commutation relations between the operators to obtain recursion relations for the scalar products, or through Maillet's \(F\)-basis representation 1999.Kitanine.NPB.554.

In particular, specializing to the norm of Bethe eigenstates, a proof of Gaudin's norm formula is obtained through taking the limit \(\mu_a \rightarrow \lambda_a\), \(a=1, ..., M\):

\begin{equation} {\mathbb N}_M = \langle 0 | \prod_{j=1}^M C (\lambda_j) \prod_{k=1}^M B (\lambda_k) | 0 \rangle = \varphi^M(\eta) \prod_{j \neq k} \frac{\varphi(\lambda_j - \lambda_k + \eta)}{\varphi(\lambda_j - \lambda_k)} \det \Phi (\{ \lambda \}), \tag{gnf}\label{gnf} \end{equation}

with the Gaudin matrix entries being

\begin{equation} \Phi_{jk} = -\frac{\partial}{\partial \lambda_k} \ln \left[ \frac{a(\lambda_j)}{d(\lambda_j)} \prod_{l = 1, \neq j}^M \frac{b(\lambda_j, \lambda_l)}{b(\lambda_l, \lambda_j)}\right]. \tag{gm}\label{gm} \end{equation}