Heisenberg spin1/2 XXX chain
http://bomber0.myid.net/ (토론)님의 2011년 1월 7일 (금) 06:48 판
introduction
- XXX spin chain can be solved by Bethe ansatz
summary
- Hamiltonian of XXX spin chain with anisotropic parameter \(\Delta=1\)
\(\hat H = \sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \sigma_j^z \sigma_{j+1}^z+1)\) - two body scattering term
\(s_{jl}=1-2\Delta e^{ik_l}+ e^{ik_l+ik_j}=1-2e^{ik_l}+ e^{ik_l+ik_j}\) - phase shift
- 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
- raising and lowering 연산자
\(\sigma_{\pm}=\frac{1}{2}(\sigma_{x}\pm i\sigma_{y})\)
\(\sigma_{+}=\frac{1}{2}(\sigma_{x}+ i\sigma_{y})=\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}\)
\(\sigma_{-}=\frac{1}{2}(\sigma_{x}- i\sigma_{y})=\begin{pmatrix} 0&0\\ 1&0 \end{pmatrix}\)
\([\sigma_{z},\sigma_{\pm}]=\pm 2\sigma_{\pm}\)
\(h=\frac{\sigma_{i}\cdot\sigma_{j}+1}{2}\) acts as the permutation operator
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
history
encyclopedia
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
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- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
- 2010년 books and articles
- http://gigapedia.info/1/
- http://gigapedia.info/1/
- http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
expositions
articles
- Evaluation of Integrals Representing Correlations in XXX Heisenberg Spin Chain
- H.E.Boos, V.E.Korepin, 2001
- H.E.Boos, V.E.Korepin, 2001
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://arxiv.org/
- http://www.pdf-search.org/
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- http://dx.doi.org/
question and answers(Math Overflow)
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experts on the field