Monodromy matrix
introduction
- monodromy matrix
\[ T(\lambda)= \left( \begin{array}{cc} A(\lambda ) & B(\lambda ) \\ C(\lambda ) & D(\lambda ) \end{array} \right) \]
- describes the transport of the spin around the circular chain
- YBE implies the following RTT=TTR relation in spin chains
\[ RTT=TTR \]
- transfer matrix
\[ t=\operatorname{tr} T=A+D \]
definition
- \(\lambda\) : spectral parameter
- \(R(\lambda)\) : R-matrix
- define the Lax matrix
\[ \begin{eqnarray} L_{0 n}(\lambda) &=& R_{0 n}(\lambda - {i\over 2}) \\ &=& \left( \begin{array}{cc} \alpha_{n} & \beta_{n} \\ \gamma_{n} & \delta_{n} \end{array} \right) \,, \qquad n = 1 \,, 2 \,, \ldots \,, N \,, \end{eqnarray} \] where \(\alpha_{n}\), \(\beta_{n}\), \(\gamma_{n}\), \(\delta_{n}\) are operators on \[ \begin{eqnarray} \stackrel{\stackrel{1}{\downarrow}}{V} \otimes \cdots \otimes \stackrel{\stackrel{n}{\downarrow}}{V} \otimes \cdots \otimes \stackrel{\stackrel{N}{\downarrow}}{V} \end{eqnarray} \]
- monodromy matrix
\[ \begin{eqnarray} T_{0}(\lambda) &=& L_{0 N}(\lambda) \cdots L_{0 1}(\lambda) \\ &=& \left(\begin{array}{cc} \alpha_{N} & \beta_{N} \\ \gamma_{N} & \delta_{N} \end{array} \right) \cdots \left(\begin{array}{cc} \alpha_{1} & \beta_{1} \\ \gamma_{1} & \delta_{1} \end{array} \right) \\ &=& \left( \begin{array}{cc} A(\lambda ) & B(\lambda ) \\ C(\lambda ) & D(\lambda ) \end{array} \right) \label{monodromy} \end{eqnarray} \] where \(A(\lambda ) ,B(\lambda ) , C(\lambda ) , D(\lambda )\) are operators acting on \(V^{\otimes N}\)