Heisenberg spin1/2 XXX chain

introduction

XXX spin chain can be solved by Bethe ansatz

review on spin system

spin system
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

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

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

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

eigenvalues

emptiness formation probability

near neighbor correlations

history

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related items

encyclopedia

http://en.wikipedia.org/wiki/
http://www.scholarpedia.org/
http://www.proofwiki.org/wiki/
Princeton companion to mathematics(Companion_to_Mathematics.pdf)

books

expositions

XXX Spin Chain: from Bethe Solution to Open Problems
Bethe Ansatz for Heisenberg XXX Model Authors: Shao-shiung Lin, Shi-shyr Roan

articles

Takahashi, Minoru. 2010. Correlation function and simplified TBA equations for XXZ chain. 1101.0035 (December 29). http://arxiv.org/abs/1101.0035.
Quantum correlations and number theory
Evaluation of Integrals Representing Correlations in XXX Heisenberg Spin Chain
- H.E.Boos, V.E.Korepin, 2001
Quantum spin chains and Riemann zeta function with odd arguments
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