Heisenberg spin1/2 XXX chain

introduction

• XXX spin chain can be solved by Bethe ansatz

 

review on spin system

• spin system
  

• Pauli matrices (해밀턴의 사원수 참조)
  \(\sigma_1 = \sigma_x = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} \)
  \(\sigma_2 = \sigma_y = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} \)
  \(\sigma_3 = \sigma_z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}\)
  

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

 

\(h=\frac{\sigma_{i}\cdot\sigma_{j}+1}{2}\) acts as the permutation operator

 

 

summary

• Hamiltonian of XXX spin chain with  anisotropic parameter \(\Delta=1\)
  \(\hat H = \sum_{j=1}^{L} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \sigma_j^z \sigma_{j+1}^z+1)\)
  
• two body scattering term
  \(s_{j,l}=1-2\Delta e^{ik_l}+ e^{ik_l+ik_j}=1-2e^{ik_l}+ e^{ik_l+ik_j}\)
  
• phase shift term \(\theta(p,q)\)
  \(\exp(-i\theta(k_j,k_l))=\frac{s_{l,j}}{s_{j,l}}=\frac{1-2\Delta e^{ik_j}+e^{i(k_j+k_l)}}{1-2\Delta e^{ik_l}+e^{i(k_j+k_l)}}\)
  
• equation satisfied by wave numbers
  \(\exp(ik_jL)=(-1)^{n-1}\prod_{l=1, l\neq j}^{n}\frac{s_{l,j}}{s_{j,l}}=(-1)^{n-1}\prod_{l=1}^{L}\exp(-i\theta(k_j,k_l))\)
  
• fundamental equation
  \(k_jN=2\pi I(k_j)+\sum_{l=1}^{N}\theta(k_j,k_l)\)
  

 

 

wavefunction amplitude

• amplitudes \(A(P)\) satisfies
  \(A_{P}=\sigma_{P}\prod_{1\leq i< j\n}s_{P_{j}P_{i}}\), where \(\sigma_{P}\) = sign of the permutation
  
• \(A(312)\) corresponds to the permutation \(1\to3, 2\to1, 3\to2\)
• n=2 case
  \(A(12)=s_{21}\)
  \(A(21)=-s_{12}\)
  
• n=3 case
  \(A(123)=s_{21}s_{31}s_{32}\)
  \(A(312)=s_{13}s_{23}s_{21}\)
  \(A(231)=s_{32}s_{12}s_{13}\)
   
  

 

Bethe ansatz equation

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

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

n=1

\(\exp(ik_jL)=1\)

n=2

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

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

 

n=3

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

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

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

 

n denote the number of up spins

 

n=0 analysis

 

n=1 analysis

ansatz \(a(x)=e^{ikx}\)

derive difference equations

compute eigenvalue \(E=L-2+2(\cos k)\)

boundary condition \(a(x+L)=a(x)\) implies \(e^{ikL}=1\)

 

n=2 analysis

ansatz \(a(x,y)=A(12)e^{ik_1x+ik_2y}+A(21)e^{ik_2x+ik_1y}\)

derive difference equations to get two-body scattering term

compute eigenvalue \(E=L-4+2(\cos k_1+\cos k_2)\)

use two-body scattering condition \(a(x,x)+a(x+1,x+1)=2a(x,x+1)\) to get \(A(12)/A(21)=-s_{2,1}/s_{1,2}\)

boundary condition \(a(y,x+L)=a(x,y)\) imples \(A(12)/A(21)=e^{ik_1L}\)

 

 

n=3 analysis

ansatz \(a(x,y,z)=A(123)e^{ik_1x+ik_2y+ik_3z}+A(132)e^{ik_1x+ik_3y+ik_2z}+\cdots = \sum _{P}A(P)e^{iP\cdot x}\)

derive difference equations. we get several of them

e.g.

\(a(x,x,z)+a(x+1,x+1,z)=2a(x,x+1,z)\)

compute the eigenvalue \(E=L-4+2(\cos k_1+\cos k_2+\cos k_3)\)

use two-body scattering condition \(a(x,x)+a(x+1,x+1)=2a(x,x+1)\) to get \(A(12)/A(21)=-s_{2,1}/s_{1,2}\)

 

 

 

 

 

 

 

 

 

 

 

history

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