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

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<h5>introduction</h5>
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==introduction==
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* special case of [[Heisenberg spin chain model]]
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* {{수학노트|url=하이젠베르크_스핀_1/2_XXX_모형}}
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* {{수학노트|url=하이젠베르크_스핀_1/2_XXX_모형의_좌표_베테_가설_풀이(coordinate_Bethe_ansatz)}}
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* topics
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** emptiness formation probability
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** near neighbor correlations
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* XXX spin chain can be solved by [[Bethe ansatz]]
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==related items==
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<h5>review on spin system</h5>
 
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* [[spin system and Pauli exclusion principle|spin system]]<br>
 
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*  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>
 
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<math>h=\frac{\sigma_{i}\cdot\sigma_{j}+1}{2}</math> acts as the permutation operator
 
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<h5>summary</h5>
 
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*  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>
 
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*  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>
 
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*  phase shift term <math>\theta(p,q)</math><br><math>\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)}}</math><br>
 
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*  equation satisfied by wave numbers<br><math>\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))</math><br>
 
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*  fundamental equation<br><math>k_jN=2\pi I(k_j)+\sum_{l=1}^{N}\theta(k_j,k_l)</math><br>
 
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<h5>wavefunction amplitude</h5>
 
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*  amplitudes <math>A(P)</math> satisfies<br><math>A_{P}=\sigma_{P}\prod_{1\leq i< j\n}s_{P_{j}P_{i}}</math>, where <math>\sigma_{P}</math> = sign of the permutation<br>
 
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* <math>A(312)</math> corresponds to the permutation <math>1\to3, 2\to1, 3\to2</math>
 
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*  n=2 case<br><math>A(12)=s_{21}</math><br><math>A(21)=-s_{12}</math><br>
 
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*  n=3 case<br><math>A(123)=s_{21}s_{31}s_{32}</math><br><math>A(312)=s_{13}s_{23}s_{21}</math><br><math>A(231)=s_{32}s_{12}s_{13}</math><br>  <br>
 
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<h5>Bethe ansatz equation</h5>
 
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<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>
 
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<math>\exp(ik_jL)=(-1)^{n-1}\prod_{l=1, l\neq j}^{n}\frac{s_{l,j}}{s_{j,l}}</math>
 
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n=1
 
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<math>\exp(ik_jL)=1</math>
 
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n=2
 
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<math>\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}}</math>
 
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<math>\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}}</math>
 
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n=3
 
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<math>\exp(ik_1L)=\frac{s_{2,1}s_{3,1}}{s_{1,2}s_{1,3}}</math>
 
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<math>\exp(ik_2L)=\frac{s_{1,2}s_{3,2}}{s_{2,1}s_{2,3}}</math>
 
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<math>\exp(ik_3L)=\frac{s_{1,3}s_{2,3}}{s_{3,1}s_{3,2}}</math>
 
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n=0 analysis
 
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n=1 analysis
 
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ansatz a(x)=e^{ikx}
 
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derive an difference equation
 
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eigenvalue
 
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boudary condition implies e^{ikL}=1
 
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<h5>eigenvalues</h5>
 
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<h5>emptiness formation probability</h5>
 
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<h5>near neighbor correlations</h5>
 
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<h5>history</h5>
 
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
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<h5>related items</h5>
 
  
 
* [[six-vertex model and Quantum XXZ Hamiltonian]]
 
* [[six-vertex model and Quantum XXZ Hamiltonian]]
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* [[talk on Bethe ansatz]]
 
 
* [[Bethe ansatz]]
 
* [[Bethe ansatz]]
  
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia</h5>
 
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* http://en.wikipedia.org/wiki/
 
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* http://www.scholarpedia.org/
 
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* http://www.proofwiki.org/wiki/
 
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* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
 
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<h5>books</h5>
 
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* [[2010년 books and articles]]<br>
 
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* http://gigapedia.info/1/
 
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* http://gigapedia.info/1/
 
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* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
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<h5>expositions</h5>
 
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* [http://pos.sissa.it/archive/conferences/038/006/Solvay_006.pdf XXX Spin Chain: from Bethe Solution to Open Problems]
 
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* [http://arxiv.org/abs/cond-mat/9509183 Bethe Ansatz for Heisenberg XXX Model] Authors: Shao-shiung Lin, Shi-shyr Roan
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5>
 
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* Takahashi, Minoru. 2010. Correlation function and simplified TBA equations for XXZ chain. 1101.0035 (December 29). http://arxiv.org/abs/1101.0035.
 
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* [http://iopscience.iop.org/0305-4470/35/20/305 Quantum correlations and number theory ]
 
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* [http://arxiv.org/abs/hep-th/0105144 Evaluation of Integrals Representing Correlations in XXX Heisenberg Spin Chain]<br>
 
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**  H.E.Boos, V.E.Korepin, 2001<br>
 
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* [http://dx.doi.org/10.1088/0305-4470/34/26/301 Quantum spin chains and Riemann zeta function with odd arguments]<br>
 
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* http://www.ams.org/mathscinet
 
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* http://www.zentralblatt-math.org/zmath/en/
 
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* http://arxiv.org/
 
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* http://www.pdf-search.org/
 
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* http://pythagoras0.springnote.com/
 
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
 
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* http://dx.doi.org/
 
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<h5>question and answers(Math Overflow)</h5>
 
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* http://mathoverflow.net/search?q=
 
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* http://mathoverflow.net/search?q=
 
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<h5>blogs</h5>
 
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*  구글 블로그 검색<br>
 
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**  http://blogsearch.google.com/blogsearch?q=<br>
 
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** http://blogsearch.google.com/blogsearch?q=
 
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* http://ncatlab.org/nlab/show/HomePage
 
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<h5>experts on the field</h5>
 
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* http://arxiv.org/
 
  
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==articles==
  
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* Takahashi, Minoru. 2011. “Correlation Function and Simplified TBA Equations for XXZ Chain.” Symmetry, Integrability and Geometry: Methods and Applications (January 8). doi:10.3842/SIGMA.2011.004. http://www.emis.de/journals/SIGMA/2011/004/
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* Boos, H. E., V. E. Korepin, Y. Nishiyama, and M. Shiroishi. 2002. “Quantum Correlations and Number Theory.” Journal of Physics A: Mathematical and General 35 (20) (May 24): 4443. doi:[http://dx.doi.org/10.1088/0305-4470/35/20/305 10.1088/0305-4470/35/20/305].
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* Boos, H. E., and V. E. Korepin. 2001. “Quantum Spin Chains and Riemann Zeta Function with Odd Arguments.” Journal of Physics A: Mathematical and General 34 (26) (July 6): 5311. doi:[http://dx.doi.org/10.1088/0305-4470/34/26/301 10.1088/0305-4470/34/26/301].
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<h5>links</h5>
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[[분류:integrable systems]]
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[[분류:math and physics]]
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[[분류:migrate]]
  
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
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==메타데이터==
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* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
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===위키데이터===
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
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* ID :  [https://www.wikidata.org/wiki/Q899196 Q899196]
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* http://functions.wolfram.com/
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===Spacy 패턴 목록===
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* [{'LOWER': 'heisenberg'}, {'LEMMA': 'model'}]
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* [{'LOWER': 'quantum'}, {'LOWER': 'heisenberg'}, {'LEMMA': 'model'}]
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* [{'LOWER': 'heisenberg'}, {'LOWER': 'spin'}, {'LEMMA': 'chain'}]

2021년 2월 17일 (수) 02:04 기준 최신판

introduction


related items


articles

  • Takahashi, Minoru. 2011. “Correlation Function and Simplified TBA Equations for XXZ Chain.” Symmetry, Integrability and Geometry: Methods and Applications (January 8). doi:10.3842/SIGMA.2011.004. http://www.emis.de/journals/SIGMA/2011/004/
  • Boos, H. E., V. E. Korepin, Y. Nishiyama, and M. Shiroishi. 2002. “Quantum Correlations and Number Theory.” Journal of Physics A: Mathematical and General 35 (20) (May 24): 4443. doi:10.1088/0305-4470/35/20/305.
  • Boos, H. E., and V. E. Korepin. 2001. “Quantum Spin Chains and Riemann Zeta Function with Odd Arguments.” Journal of Physics A: Mathematical and General 34 (26) (July 6): 5311. doi:10.1088/0305-4470/34/26/301.

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'heisenberg'}, {'LEMMA': 'model'}]
  • [{'LOWER': 'quantum'}, {'LOWER': 'heisenberg'}, {'LEMMA': 'model'}]
  • [{'LOWER': 'heisenberg'}, {'LOWER': 'spin'}, {'LEMMA': 'chain'}]
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