"Monodromy matrix"의 두 판 사이의 차이
imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) |
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| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
* monodromy matrix | * monodromy matrix | ||
| − | + | :<math> | |
T(\lambda)= | T(\lambda)= | ||
\left( | \left( | ||
| 9번째 줄: | 9번째 줄: | ||
\end{array} | \end{array} | ||
\right) | \right) | ||
| − | + | </math> | |
* describes the transport of the spin around the circular chain | * describes the transport of the spin around the circular chain | ||
* YBE implies the following [[RTT=TTR relation in spin chains]] | * YBE implies the following [[RTT=TTR relation in spin chains]] | ||
| − | + | :<math> | |
RTT=TTR | RTT=TTR | ||
| − | + | </math> | |
* transfer matrix | * transfer matrix | ||
| − | + | :<math> | |
t=\operatorname{tr} T=A+D | t=\operatorname{tr} T=A+D | ||
| − | + | </math> | |
==definition== | ==definition== | ||
| − | * | + | * <math>\lambda</math> : spectral parameter |
| − | * | + | * <math>R(\lambda)</math> : [[R-matrix]] |
* define the Lax matrix | * define the Lax matrix | ||
| − | + | :<math> | |
\begin{eqnarray} | \begin{eqnarray} | ||
L_{0 n}(\lambda) &=& R_{0 n}(\lambda - {i\over 2}) \\ | L_{0 n}(\lambda) &=& R_{0 n}(\lambda - {i\over 2}) \\ | ||
| 34번째 줄: | 34번째 줄: | ||
\,, \qquad n = 1 \,, 2 \,, \ldots \,, N \,, | \,, \qquad n = 1 \,, 2 \,, \ldots \,, N \,, | ||
\end{eqnarray} | \end{eqnarray} | ||
| − | + | </math> | |
where | where | ||
| − | + | <math>\alpha_{n}</math>, <math>\beta_{n}</math>, <math>\gamma_{n}</math>, <math>\delta_{n}</math> are | |
operators on | operators on | ||
| − | + | :<math> | |
\begin{eqnarray} | \begin{eqnarray} | ||
\stackrel{\stackrel{1}{\downarrow}}{V} \otimes \cdots \otimes | \stackrel{\stackrel{1}{\downarrow}}{V} \otimes \cdots \otimes | ||
| 44번째 줄: | 44번째 줄: | ||
\stackrel{\stackrel{N}{\downarrow}}{V} | \stackrel{\stackrel{N}{\downarrow}}{V} | ||
\end{eqnarray} | \end{eqnarray} | ||
| − | + | </math> | |
* monodromy matrix | * monodromy matrix | ||
| − | + | :<math> | |
\begin{eqnarray} | \begin{eqnarray} | ||
T_{0}(\lambda) &=& L_{0 N}(\lambda) \cdots L_{0 1}(\lambda) \\ | T_{0}(\lambda) &=& L_{0 N}(\lambda) \cdots L_{0 1}(\lambda) \\ | ||
| 68번째 줄: | 68번째 줄: | ||
\label{monodromy} | \label{monodromy} | ||
\end{eqnarray} | \end{eqnarray} | ||
| − | + | </math> | |
where | where | ||
| − | + | <math>A(\lambda ) ,B(\lambda ) , C(\lambda ) , D(\lambda )</math> are operators acting on <math>V^{\otimes N}</math> | |
2020년 11월 13일 (금) 23:08 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 \(V^{\otimes N}\)
computational resource