The Bethe Ansatz

The simplest form of \(R\) matrix we can begin with is a diagonal one,

\begin{equation*} ^{0}R (\lambda, \mu) = \left( \begin{array}{cccc} r_{11} (\lambda,\mu) & 0 & 0 & 0 \\ 0 & r_{22} (\lambda, \mu) & 0 & 0 \\ 0 & 0 & r_{33} (\lambda, \mu) & 0 \\ 0 & 0 & 0 & r_{44} (\lambda,\mu) \end{array} \right) \end{equation*}

which trivially satisfies the Yang-Baxter relation for any functions \(r_{ii}\). Invariance under rescaling by an arbitrary function allows us to set \(r_{11} (\lambda,\mu) = 1\). The inversion relation RRe1 requires \(r_{22} (\lambda,\mu) r_{33} (\mu,\lambda) = 1\) and \(r_{44} (\lambda,\mu) r_{44} (\mu,\lambda) = 1\), so we write

\begin{equation*} r_{22} (\lambda,\mu) = e^{s (\lambda,\mu)}, \hspace{5mm} r_{33} (\lambda,\mu) = e^{-s(\lambda,\mu)}, \hspace{5mm} r_{44} (\lambda,\mu) = (-1)^\Theta e^{t (\lambda,\mu)}, \nonumber \\ s(\lambda,\mu) ~\mbox{arbitrary}, \hspace{1cm} t(\lambda,\mu) = -t(\mu,\lambda), \hspace{1cm} \Theta = 0,1. \end{equation*}