"Heisenberg spin1/2 XXX chain"의 두 판 사이의 차이

2011년 1월 7일 (금) 05:54 판

introduction

XXX spin chain can be solved by Bethe ansatz

summary

Hamiltonian of XXX spin chain with anisotropic parameter \(\Delta=1\)
\(\hat H = \sum_{j=1}^{L} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \sigma_j^z \sigma_{j+1}^z+1)\)
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 \(\theta(p,q)\)
\(\exp(-i\theta(p,q))=\frac{s_{q,p}}{s_{p,q}}=\frac{1-2\Delta e^{ip}+e^{i(p+q)}}{1-2\Delta e^{iq}+e^{i(p+q)}}=\frac{1-2e^{ip}+e^{i(p+q)}}{1-2e^{iq}+e^{i(p+q)}}\)
equation satisfied by wave numbers
\(\exp(ik_jL)=(-1)^{n-1}\prod_{l=1}^{L}\exp(-i\theta(k_j,k_l))\)
\(\exp(ik_jL)=(-1)^{n-1}\prod_{l=1, l\neq j}^{n}\frac{s_{l,j}}{s_{j,l}}\)
fundamental equation
\(k_jN=2\pi I(k_j)+\sum_{l=1}^{N}\theta(k_j,k_l)\)

review on spin system

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

http://www.google.com/search?hl=en&tbs=tl:1&q=

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

articles

question and answers(Math Overflow)

blogs

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- http://blogsearch.google.com/blogsearch?q=
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http://ncatlab.org/nlab/show/HomePage

experts on the field

http://arxiv.org/

@@ 9번째 줄: / 9번째 줄: @@
 <h5>summary</h5>
-*  Hamiltonian of XXX spin chain with  anisotropic parameter <math>\Delta=1</math><br><math>\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)</math><br>
+*  Hamiltonian of XXX spin chain with  anisotropic parameter <math>\Delta=1</math><br><math>\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)</math><br>
-*  two body scattering term<br><math>s_{jl}=1-2\Delta e^{ik_l}+ e^{ik_l+ik_j}=1-2e^{ik_l}+ e^{ik_l+ik_j}</math><br>
+*  two body scattering term<br><math>s_{j,l}=1-2\Delta e^{ik_l}+ e^{ik_l+ik_j}=1-2e^{ik_l}+ e^{ik_l+ik_j}</math><br>
-*  phase shift<br>  <br>
+*  phase shift term <math>\theta(p,q)</math><br><math>\exp(-i\theta(p,q))=\frac{s_{q,p}}{s_{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)}}</math><br>
-*  equation satisfied by wave numbers<br><math>\exp(ik_jN)=(-1)^{N-1}\prod_{l=1}^{N}\exp(-i\theta(k_j,k_l))</math><br> where<br><math>\theta(p,q)</math> is defined as<br><math>\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)}}</math><br>
+*   <br> equation satisfied by wave numbers<br><math>\exp(ik_jL)=(-1)^{n-1}\prod_{l=1}^{L}\exp(-i\theta(k_j,k_l))</math><br><math>\exp(ik_jL)=(-1)^{n-1}\prod_{l=1, l\neq j}^{n}\frac{s_{l,j}}{s_{j,l}}</math><br>
 *  fundamental equation<br><math>k_jN=2\pi I(k_j)+\sum_{l=1}^{N}\theta(k_j,k_l)</math><br>