# The Bethe Ansatz

### Constructing $$R$$ matricesa.R

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

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

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.