The Bethe Ansatz

Let us begin by making a choice for the \(b,c\) functions in R1. The first observation is that we wish for rapidity-dependent operators (to make our model nontrivial), and we thus ignore the simple case of constant functions. To simplify things as much as possible, we also choose these two-parameter functions to be functions of a single parameter, and write (by an abuse of notation) \(b(\lambda,\mu) \equiv b(\lambda - \mu)\) and \(c(\lambda,\mu) \equiv c(\lambda - \mu)\).

We now suppose that \(b(\lambda)\) and \(c(\lambda)\) are analytic, and that they are bounded at \(|\lambda| \rightarrow \infty\). Starting with the off-diagonal function \(c\), we can put it to the simplest such non-constant function, namely a single pole

\begin{equation*} c(\lambda) \equiv \frac{c_0}{\lambda + \eta} \end{equation*}

where \(c_0\) and \(\eta\) are finite generic complex constants. The constraint bc12 (left) means that \(b(\lambda)\) must like \(c(\lambda)\) have a single pole at \(\lambda = -\eta\), and that it must have the asymptotic value \(\lim_{\lambda \rightarrow \infty} b(\lambda) = 1\). We thus must use a form

\begin{equation*} b(\lambda) \equiv 1 + \frac{b_0}{\lambda + \eta}. \end{equation*}

Equating the residues at the \(\lambda = -\eta\) pole in Equations bc12 then yield the constraints \(b_0 = -\eta\), \(c_0 = \eta\). Conditions bc345 are then also fulfilled. We thus obtain our first fully-fledged \(R\)-matrix, the "rational" matrix defined as

\begin{equation} R (\lambda) = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & b (\lambda) & c (\lambda) & 0 \\ 0 & c (\lambda) & b (\lambda) & 0 \\ 0 & 0 & 0 & 1 \end{array} \right), \hspace{1cm} b(\lambda) = \frac{\lambda}{\lambda + \eta}, \hspace{1cm} c(\lambda) = \frac{\eta}{\lambda + \eta}. \tag{Rr}\label{eq:bcR1} \end{equation}

Note that we can write this as

\begin{equation} R (\lambda) = b(\lambda) {\bf 1} + c(\lambda) \mathbb{P}, \tag{R1P}\label{R1P} \end{equation}

where \(\mathbb{P}\) is the permutation matrix,

\begin{equation} \mathbb{P} = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right). \tag{Pmat}\label{Pmat} \end{equation}

Note that Rr can be related to an equivalent form commonly used in the literature:

\begin{equation} R_{ru} (\lambda, \mu) = \left( \begin{array}{cccc} f(\mu, \lambda) & 0 & 0 & 0 \\ 0 & g (\mu, \lambda) & 1 & 0 \\ 0 & 1 & g (\mu, \lambda) & 0 \\ 0 & 0 & 0 & 1 \end{array} \right), \hspace{10mm} f(\mu, \lambda) = 1 + \frac{ic}{\mu - \lambda}, \hspace{5mm} g(\mu, \lambda) = \frac{ic}{\mu - \lambda}. \tag{Rru}\label{Rru} \end{equation}

The exact correspondence with Rr is

\begin{equation} R(\lambda - \mu) = f^{-1} (\mu, \lambda) \left. P R_{ru} (\lambda, \mu) \right|_{c = i\eta}. \tag{RRru}\label{RRru} \end{equation}