The Bethe Ansatz

Let us now attempt to construct explicit \(R\) matrices fulfilling the inversion requirement RRe1 and the Yang-Baxter relation YB. We proceed as systematically as possible. We begin by the simplest nontrivial auxiliary space we can consider, \({\cal A} \sim \mathbb{C}^2\). The smallest interesting \(R\) matrices are thus represented as \(4 \times 4\) matrices,

\begin{equation*} R(\lambda,\mu) = \left( \begin{array}{cccc} r_{11}(\lambda,\mu) & r_{12}(\lambda,\mu) & r_{13}(\lambda,\mu) & r_{14}(\lambda,\mu) \\ r_{21}(\lambda,\mu) & r_{22}(\lambda,\mu) & r_{23}(\lambda,\mu) & r_{24}(\lambda,\mu) \\ r_{31}(\lambda,\mu) & r_{32}(\lambda,\mu) & r_{33}(\lambda,\mu) & r_{34}(\lambda,\mu) \\ r_{41}(\lambda,\mu) & r_{42}(\lambda,\mu) & r_{43}(\lambda,\mu) & r_{44}(\lambda,\mu) \end{array} \right) \end{equation*}

For the monodromy matrix, we can write the representation

\begin{equation} T (\lambda) \equiv \left( \begin{array}{cc} A (\lambda) & B(\lambda) \\ C (\lambda) & D(\lambda) \end{array} \right) \tag{TeABCD}\label{TeABCD} \end{equation}

in which \(A, B, C, D\) are operators in the (as yet unspecified) Hilbert space. Using natural tensor conventions

\begin{equation*} M \otimes N = \left(\begin{array}{cccc} M_{11} N_{11} & M_{11} N_{12} & M_{12} N_{11} & M_{12} N_{12} \\ M_{11} N_{21} & M_{11} N_{22} & M_{12} N_{21} & M_{12} N_{22} \\ M_{21} N_{11} & M_{21} N_{12} & M_{22} N_{11} & M_{22} N_{12} \\ M_{21} N_{21} & M_{21} N_{22} & M_{22} N_{21} & M_{22} N_{22} \end{array} \right) \end{equation*}

we have (for the purposes of RTTeTTR)

\begin{equation*} T_1 (\lambda) = \left( \begin{array}{cccc} A(\lambda) & 0 & B(\lambda) & 0 \\ 0 & A(\lambda) & 0 & B(\lambda) \\ C(\lambda) & 0 & D (\lambda) & 0 \\ 0 & C(\lambda) & 0 & D(\lambda) \end{array} \right) \hspace{1cm} T_2 (\mu) = \left( \begin{array}{cccc} A (\mu) & B(\mu) & 0 & 0 \\ C (\mu) & D(\mu) & 0 & 0 \\ 0 & 0 & A (\mu) & B(\mu) \\ 0 & 0 & C (\mu) & D(\mu) \end{array} \right) \end{equation*} \begin{equation*} T_1 (\lambda) T_2 (\mu) = \left( \begin{array}{cccc} A(\lambda) A(\mu) & A(\lambda) B(\mu) & B(\lambda) A(\mu) & B(\lambda) B(\mu) \\ A(\lambda) C(\mu) & A(\lambda) D(\mu) & B(\lambda) C(\mu) & B(\lambda) D(\mu) \\ C(\lambda) A(\mu) & C(\lambda) B(\mu) & D(\lambda) A(\mu) & D(\lambda) B(\mu) \\ C(\lambda) C(\mu) & C(\lambda) D(\mu) & D(\lambda) C(\mu) & D(\lambda) D(\mu) \end{array} \right) \end{equation*} \begin{equation*} T_2 (\mu) T_1 (\lambda) = \left( \begin{array}{cccc} A(\mu) A(\lambda) & B(\mu) A(\lambda) & A(\mu) B(\lambda) & B(\mu) B(\lambda) \\ C(\mu) A(\lambda) & D(\mu) A(\lambda) & C(\mu) B(\lambda) & D(\mu) B(\lambda) \\ A(\mu) C(\lambda) & B(\mu) C(\lambda) & A(\mu) D(\lambda) & B(\mu) D(\lambda) \\ C(\mu) C(\lambda) & D(\mu) C(\lambda) & C(\mu) D(\lambda) & D(\mu) D(\lambda) \end{array} \right) \end{equation*}

Choosing a form for the \(R\) matrix thus sets the form of the commutation relations between \(A, B, C, D\) operators. Once all entries of the \(R\) matrix are set, all these relations are also set. The logic we follow in this section is to first set the form of possible \(R\) matrices. For each form, we give the commutation relations of \(A, B, C, D\) operators.