"Monodromy matrix"의 두 판 사이의 차이
imported>Pythagoras0 |
imported>Pythagoras0 |
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| 2번째 줄: | 2번째 줄: | ||
* monodromy matrix | * monodromy matrix | ||
$$ | $$ | ||
| − | T= | + | T(\lambda)= |
\left( | \left( | ||
\begin{array}{cc} | \begin{array}{cc} | ||
| − | A & B \\ | + | A(\lambda ) & B(\lambda ) \\ |
| − | C & D | + | C(\lambda ) & D(\lambda ) |
\end{array} | \end{array} | ||
\right) | \right) | ||
| 15번째 줄: | 15번째 줄: | ||
RTT=TTR | RTT=TTR | ||
$$ | $$ | ||
| − | |||
* transfer matrix | * transfer matrix | ||
$$ | $$ | ||
| 21번째 줄: | 20번째 줄: | ||
$$ | $$ | ||
| + | |||
| + | ==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 $V^{\otimes N}$ | ||
| 33번째 줄: | 83번째 줄: | ||
==related items== | ==related items== | ||
* [[RTT=TTR relation in spin chains]] | * [[RTT=TTR relation in spin chains]] | ||
| − | + | * [[A Spin Chain Primer]] | |
2013년 8월 19일 (월) 05:50 판
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 $V^{\otimes N}$
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