Monodromy matrix
introduction
- monodromy matrix
- <math>
T(\lambda)= \left( \begin{array}{cc} A(\lambda ) & B(\lambda ) \\ C(\lambda ) & D(\lambda ) \end{array} \right) </math>
- describes the transport of the spin around the circular chain
- YBE implies the following RTT=TTR relation in spin chains
- <math>
RTT=TTR </math>
- transfer matrix
- <math>
t=\operatorname{tr} T=A+D </math>
definition
- <math>\lambda</math> : spectral parameter
- <math>R(\lambda)</math> : R-matrix
- define the Lax matrix
- <math>
\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} </math> where <math>\alpha_{n}</math>, <math>\beta_{n}</math>, <math>\gamma_{n}</math>, <math>\delta_{n}</math> are operators on
- <math>
\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} </math>
- monodromy matrix
- <math>
\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} </math> where <math>A(\lambda ) ,B(\lambda ) , C(\lambda ) , D(\lambda )</math> are operators acting on <math>V^{\otimes N}</math>