"Monodromy matrix"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
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| (사용자 3명의 중간 판 19개는 보이지 않습니다) | |||
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| + | ==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> | ||
| + | |||
| + | |||
| + | |||
| + | ==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 | ||
| + | |||
| + | |||
| + | [[분류:integrable systems]] | ||
| + | [[분류:math and physics]] | ||
| + | [[분류:migrate]] | ||
2020년 12월 28일 (월) 04:19 기준 최신판
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 \(V^{\otimes N}\)