"Heisenberg spin1/2 XXX chain"의 두 판 사이의 차이

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34번째 줄: 34번째 줄:
 
n=2
 
n=2
  
−
<math>\exp(ik_1L)=-\frac{s_{1,2}}{s_{2,1}}</math>
+
<math>\exp(ik_1L)=-\frac{s_{2,1}}{s_{1,2}}=-\frac{1-2e^{ik_2}+ e^{ik_1+ik_2}}{1-2e^{ik_1}+ e^{ik_1+ik_2}}</math>
  
−
<math>\exp(ik_2L)=-\frac{s_{2,1}}{s_{1,2}}</math>
+
<math>\exp(ik_2L)=-\frac{s_{1,2}}{s_{2,1}}=-\frac{1-2e^{ik_1}+ e^{ik_1+ik_2}}{1-2e^{ik_2}+ e^{ik_1+ik_2}}</math>
 +
 
 +
 
  
 
n=3
 
n=3
  
−
<math>\exp(ik_1L)=-\frac{s_{1,2}}{s_{2,1}}</math>
+
 
 +
 
 +
<math>\exp(ik_1L)=\frac{s_{2,1}s_{3,1}}{s_{1,2}s_{1,3}}</math>
 +
 
 +
<math>\exp(ik_2L)=\frac{s_{1,2}s_{3,2}}{s_{2,1}s_{2,3}}</math>
  
 
 
 
 
 +
 +
<math>\exp(ik_3L)=\frac{s_{1,3}s_{2,3}}{s_{3,1}s_{3,2}}</math>
  
 
 
 
 

2011년 1월 6일 (목) 10:00 판

introduction
  • Hamiltonian of XXX spin chain with  anisotropic parameter \(\Delta=1\)
    \(\hat H = \sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \sigma_j^z \sigma_{j+1}^z)\)
  • two body scattering term
    \(s_{jl}=1-2\Delta e^{ik_l}+ e^{ik_l+ik_j}=1-2e^{ik_l}+ e^{ik_l+ik_j}\)
  • equation satisfied by wave numbers
    \(\exp(ik_jN)=(-1)^{N-1}\prod_{l=1}^{N}\exp(-i\theta(k_j,k_l))\)
    where
    \(\theta(p,q)\) is defined as
    \(\exp(-i\theta(p,q))=\frac{1-2\Delta e^{ip}+e^{i(p+q)}}{1-2\Delta e^{iq}+e^{i(p+q)}}=\frac{1-2e^{ip}+e^{i(p+q)}}{1-2e^{iq}+e^{i(p+q)}}\)
  • fundamental equation
    \(k_jN=2\pi I(k_j)+\sum_{l=1}^{N}\theta(k_j,k_l)\)

 

review on spin system

spin system

  • raising and lowering 연산자
    \(\sigma_{\pm}=\frac{1}{2}(\sigma_{x}\pm i\sigma_{y})\)
    \(\sigma_{+}=\frac{1}{2}(\sigma_{x}+ i\sigma_{y})=\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}\)
    \(\sigma_{-}=\frac{1}{2}(\sigma_{x}- i\sigma_{y})=\begin{pmatrix} 0&0\\ 1&0 \end{pmatrix}\)
    \([\sigma_{z},\sigma_{\pm}]=\pm 2\sigma_{\pm}\)

 

\(\frac{\sigma_{i}\cdot\sigma_{j}+1}{2}\)

 

 

Bethe ansatz equation

\(s_{jl}=1-2\Delta e^{ik_l}+ e^{ik_l+ik_j}=1-2e^{ik_l}+ e^{ik_l+ik_j}\)

\(\exp(ik_jL)=(-1)^{n-1}\prod_{l=1, l\neq j}^{n}\frac{s_{l,j}}{s_{j,l}}\)

n=1

\(\exp(ik_jL)=1\)

n=2

\(\exp(ik_1L)=-\frac{s_{2,1}}{s_{1,2}}=-\frac{1-2e^{ik_2}+ e^{ik_1+ik_2}}{1-2e^{ik_1}+ e^{ik_1+ik_2}}\)

\(\exp(ik_2L)=-\frac{s_{1,2}}{s_{2,1}}=-\frac{1-2e^{ik_1}+ e^{ik_1+ik_2}}{1-2e^{ik_2}+ e^{ik_1+ik_2}}\)

 

n=3

 

\(\exp(ik_1L)=\frac{s_{2,1}s_{3,1}}{s_{1,2}s_{1,3}}\)

\(\exp(ik_2L)=\frac{s_{1,2}s_{3,2}}{s_{2,1}s_{2,3}}\)

 

\(\exp(ik_3L)=\frac{s_{1,3}s_{2,3}}{s_{3,1}s_{3,2}}\)

 

 

 

 

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