"Heisenberg spin1/2 XXX chain"의 두 판 사이의 차이
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==introduction== | ==introduction== | ||
| + | * {{수학노트|url=하이젠베르크_모형(Heisenberg_model)}} | ||
* special case of [[Heisenberg spin chain model]] | * special case of [[Heisenberg spin chain model]] | ||
* XXX spin chain can be solved by [[Bethe ansatz]] | * XXX spin chain can be solved by [[Bethe ansatz]] | ||
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| − | + | ==computational resource== | |
| − | + | * https://docs.google.com/file/d/0B8XXo8Tve1cxY2lmcy1tNFQ2dlU/edit | |
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==expositions== | ==expositions== | ||
2012년 12월 24일 (월) 05:51 판
introduction
- 틀:수학노트
- special case of Heisenberg spin chain model
- XXX spin chain can be solved by Bethe ansatz
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
\[P_{ij}=\frac{\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}+1}{2}\]
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
computational resource
expositions
- XXX Spin Chain: from Bethe Solution to Open Problems
- Nepomechie, Rafael I. 1998. A Spin Chain Primer. hep-th/9810032 (October 5). http://arxiv.org/abs/hep-th/9810032.
- Lin, Shao-shiung, and Shi-shyr Roan. 1995. Bethe Ansatz for Heisenberg XXX Model. cond-mat/9509183 (October 2). http://arxiv.org/abs/cond-mat/9509183.
articles
- Takahashi, Minoru. 2010. Correlation function and simplified TBA equations for XXZ chain. 1101.0035 (December 29). http://arxiv.org/abs/1101.0035.
- H.E.Boos, V.E.Korepin, Quantum correlations and number theory , 2002
- Evaluation of Integrals Representing Correlations in XXX Heisenberg Spin Chain
- H.E.Boos, V.E.Korepin, 2001
- H.E.Boos, V.E.Korepin, 2001
- Quantum spin chains and Riemann zeta function with odd arguments
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question and answers(Math Overflow)