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 $V^{\otimes N}$

related items

• RTT=TTR relation in spin chains
• A Spin Chain Primer
• Transfer matrix in statistical mechanics

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxNWJ5NWNIXzRieGc/edit