The Bethe Ansatz

The simplest \(R\)-matrix which can be defined above diagonal ones takes the generic form

\begin{equation} R (\lambda, \mu) = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & b (\lambda, \mu) & c (\lambda, \mu) & 0 \\ 0 & c (\lambda, \mu) & b (\lambda, \mu) & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \tag{R1}\label{R1} \end{equation}

For calculations, it is convenient to represent this matrix as

\begin{equation} R (\lambda,\mu) = \frac{1 + b(\lambda,\mu)}{2} {\bf 1} \otimes {\bf 1} + \frac{1 - b(\lambda,\mu)}{2} \sigma^z \otimes \sigma^z + c(\lambda,\mu) (\sigma^+ \otimes \sigma^- + \sigma^- \otimes \sigma^+). \tag{R1r}\label{R1r} \end{equation}

In order for this to be a proper \(R\) matrix, we must impose the self-consistency requirements RRe1 and YB.

Constraint RRe1 yields the conditions

\begin{equation} b(\lambda, \mu) b(\mu, \lambda) + c(\lambda, \mu) c(\mu, \lambda) = 1, \hspace{1cm} b(\lambda, \mu) c(\mu, \lambda) + c(\lambda, \mu) b(\mu, \lambda) = 0. \tag{bc12}\label{bc12} \end{equation}

while the Yang-Baxter relation YB leads to

\begin{align} (b(\lambda,\mu) - b(\lambda,\nu)) c(\mu,\nu) + c(\lambda,\mu) c(\lambda,\nu) b(\mu,\nu) = 0, \tag{bc3}\label{bc3} \\ c(\lambda,\mu) (b(\lambda,\nu) - b(\mu,\nu)) - b(\lambda,\mu) c(\lambda,\nu) c(\mu,\nu) = 0, \tag{bc4}\label{bc4} \\ (1 - b(\lambda,\mu) b(\mu,\nu)) c(\lambda,\nu) - c(\lambda,\mu) c(\mu,\nu) = 0. \tag{bc5}\label{bc5} \end{align}