Heisenberg spin1/2 XXX chain

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imported>Pythagoras0님의 2012년 12월 24일 (월) 06:37 판
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introduction

 

review on spin system

\[\sigma_1 = \sigma_x = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \sigma_2 = \sigma_y = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix},\sigma_3 = \sigma_z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}\]

  • raising and lowering operators

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

  • the permutation operator can be written in terms of Pauli matrices

\[h_{ij}=\frac{\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}+1}{2}\]


 

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

\[A_{P}=\sigma_{P}\prod_{1\le i< j\le n}s_{P_{j}P_{i}}\], 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 denote the number of up spins


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=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 \(A(12)/A(21)=-s_{2,1}/s_{1,2}\)

 

 

eigenvalues

 

 

emptiness formation probability

 

 

 

near neighbor correlations

 

 

 

 

history

 

 

related items

 

 

encyclopedia


 

 

books

 

 

 

expositions

 

 

articles

 

 

question and answers(Math Overflow)


links

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