"Heisenberg spin1/2 XXX chain"의 두 판 사이의 차이
| 2번째 줄: | 2번째 줄: | ||
* XXX spin chain can be solved by [[Bethe ansatz]] | * XXX spin chain can be solved by [[Bethe ansatz]] | ||
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| 10번째 줄: | 8번째 줄: | ||
* [[spin system and Pauli exclusion principle|spin system]]<br> | * [[spin system and Pauli exclusion principle|spin system]]<br> | ||
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| + | * Pauli matrices ([http://pythagoras0.springnote.com/pages/3063330 해밀턴의 사원수] 참조)<br><math>\sigma_1 = \sigma_x = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} </math><br><math>\sigma_2 = \sigma_y = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} </math><br><math>\sigma_3 = \sigma_z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}</math><br> | ||
| + | |||
* raising and lowering operators<br><math>\sigma_{\pm}=\frac{1}{2}(\sigma_{x}\pm i\sigma_{y})</math><br><math>\sigma_{+}=\frac{1}{2}(\sigma_{x}+ i\sigma_{y})=\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}</math><br><math>\sigma_{-}=\frac{1}{2}(\sigma_{x}- i\sigma_{y})=\begin{pmatrix} 0&0\\ 1&0 \end{pmatrix}</math><br><math>[\sigma_{z},\sigma_{\pm}]=\pm 2\sigma_{\pm}</math><br> | * raising and lowering operators<br><math>\sigma_{\pm}=\frac{1}{2}(\sigma_{x}\pm i\sigma_{y})</math><br><math>\sigma_{+}=\frac{1}{2}(\sigma_{x}+ i\sigma_{y})=\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}</math><br><math>\sigma_{-}=\frac{1}{2}(\sigma_{x}- i\sigma_{y})=\begin{pmatrix} 0&0\\ 1&0 \end{pmatrix}</math><br><math>[\sigma_{z},\sigma_{\pm}]=\pm 2\sigma_{\pm}</math><br> | ||
| 93번째 줄: | 94번째 줄: | ||
ansatz <math>a(x,y)=A(12)e^{ik_1x+ik_2y}+A(21)e^{ik_2x+ik_1y}</math> | ansatz <math>a(x,y)=A(12)e^{ik_1x+ik_2y}+A(21)e^{ik_2x+ik_1y}</math> | ||
| − | derive difference equations | + | derive difference equations to get two-body scattering term |
| − | + | compute eigenvalue <math>E=L-6+2(\cos k_1+\cos k_2)</math> | |
| − | + | use two-body scattering condition <math>a(x,x)+a(x+1,x+1)=2a(x,x+1)</math> to get <math>A(12)/A(21)=-s_{2,1}/s_{1,2}</math> | |
| − | + | boundary condition <math>a(y,x+L)=a(x,y)</math> imples <math>A(12)/A(21)=e^{ik_1L}</math> | |
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| 107번째 줄: | 108번째 줄: | ||
n=3 analysis | 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} | + | ansatz <math>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}</math> |
| − | derive difference equations | + | derive difference equations. we get several of them |
| − | + | e.g. | |
| − | + | <math>a(x,x,z)+a(x+1,x+1,z)=2a(x,x+1,z)</math> | |
compute the eigenvalue <math>E=L-6+2(\cos k_1+\cos k_2+\cos k_3)</math> | compute the eigenvalue <math>E=L-6+2(\cos k_1+\cos k_2+\cos k_3)</math> | ||
| − | + | use two-body scattering condition <math>a(x,x)+a(x+1,x+1)=2a(x,x+1)</math> to get <math>A(12)/A(21)=-s_{2,1}/s_{1,2}</math> | |
2011년 1월 13일 (목) 13:49 판
introduction
- XXX spin chain can be solved by Bethe ansatz
review on 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 \(E=L-6+2(\cos k)\)
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 \(E=L-6+2(\cos k_1+\cos k_2)\)
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 \(E=L-6+2(\cos k_1+\cos k_2+\cos k_3)\)
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
encyclopedia
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- http://www.proofwiki.org/wiki/
- 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
- 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
- H.E.Boos, V.E.Korepin, 2001
- Quantum spin chains and Riemann zeta function with odd arguments
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://arxiv.org/
- http://www.pdf-search.org/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/
question and answers(Math Overflow)
blogs
- 구글 블로그 검색
- http://ncatlab.org/nlab/show/HomePage
experts on the field