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

2013년 2월 28일 (목) 14:06 판

introduction

special case of Heisenberg spin chain model
틀:수학노트
틀:수학노트

emptiness formation probability

near neighbor correlations

related items

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
Quantum spin chains and Riemann zeta function with odd arguments

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
+* special case of [[Heisenberg spin chain model]]
 * {{수학노트|url=하이젠베르크_모형(Heisenberg_model)}}
-* special case of [[Heisenberg spin chain model]]
+* {{수학노트|url=좌표_베테_가설_풀이(coordinate_Bethe_ansatz)}}
-* XXX spin chain can be solved by [[Bethe ansatz]]
-==review on spin system==
-* [[spin system and Pauli exclusion principle|spin system]]
-* {{수학노트|url=파울리_행렬}}
-:<math>\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}</math><br>
-*  raising and lowering operators
-:<math>\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}</math>
-:<math>[\sigma_{z},\sigma_{\pm}]=\pm 2\sigma_{\pm}</math>
-* the two-site permutation operator can be written in terms of Pauli matrices
-:<math>P_{ij}=\frac{\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}+1}{2}</math>
-==summary==
-*  Hamiltonian of XXX spin chain with  anisotropic parameter <math>\Delta=1</math>
-:<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)=\sum_{j=1}^{L-1}P_{i,i+1}+P_{L,1}</math>
-*  two body scattering term
-:<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 term <math>\theta(p,q)</math>
-:<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>
-*  equation satisfied by wave numbers
-:<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>
-*  fundamental equation
-:<math>k_jN=2\pi I(k_j)+\sum_{l=1}^{N}\theta(k_j,k_l)</math><br>
-==wavefunction amplitude==
-*  amplitudes <math>A(P)</math> satisfies
-:<math>A_{P}=\sigma_{P}\prod_{1\le i< j\le n}s_{P_{j}P_{i}}</math>, where <math>\sigma_{P}</math> = sign of the permutation<br>
-* <math>A(312)</math> corresponds to the permutation <math>1\to3, 2\to1, 3\to2</math>
-*  n=2 case<br><math>A(12)=s_{21}</math><br><math>A(21)=-s_{12}</math><br>
-*  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>
-==Bethe ansatz equation==
-<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>
-<math>\exp(ik_jL)=(-1)^{n-1}\prod_{l=1, l\neq j}^{n}\frac{s_{l,j}}{s_{j,l}}</math>
-* n denote the number of up spins
-===n=1===
-<math>\exp(ik_jL)=1</math>
-===n=2===
-<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>
-<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>
-===n=3===
-<math>\exp(ik_1L)=\frac{s_{2,1}s_{3,1}}{s_{1,2}s_{1,3}}</math>
-<math>\exp(ik_2L)=\frac{s_{1,2}s_{3,2}}{s_{2,1}s_{2,3}}</math>
-<math>\exp(ik_3L)=\frac{s_{1,3}s_{2,3}}{s_{3,1}s_{3,2}}</math>
-===n=0 analysis===
-===n=1 analysis===
-ansatz <math>a(x)=e^{ikx}</math>
-derive difference equations
-compute eigenvalue <math>E=L-2+2(\cos k)</math>
-boundary condition <math>a(x+L)=a(x)</math> implies <math>e^{ikL}=1</math>
-===n=2 analysis===
-ansatz <math>a(x,y)=A(12)e^{ik_1x+ik_2y}+A(21)e^{ik_2x+ik_1y}</math>
-derive difference equations to get two-body scattering term
-compute eigenvalue <math>E=L-4+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>
-===n=3 analysis===
-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. 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-4+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>
-==eigenvalues==
@@ 144번째 줄: / 14번째 줄: @@
 ==near neighbor correlations==
-==history==
-* http://www.google.com/search?hl=en&tbs=tl:1&q=
@@ 167번째 줄: / 25번째 줄: @@
 * [[Bethe ansatz]]
-==expositions==
-* [http://pos.sissa.it/archive/conferences/038/006/Solvay_006.pdf XXX Spin Chain: from Bethe Solution to Open Problems]<br>  <br>
-*  Nepomechie, Rafael I. 1998. A Spin Chain Primer. hep-th/9810032 (October 5). http://arxiv.org/abs/hep-th/9810032. <br>  <br>
-* 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.
@@ 182번째 줄: / 34번째 줄: @@
 * 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, [http://iopscience.iop.org/0305-4470/35/20/305 Quantum correlations and number theory] , 2002
-* [http://arxiv.org/abs/hep-th/0105144 Evaluation of Integrals Representing Correlations in XXX Heisenberg Spin Chain]<br>
+* [http://arxiv.org/abs/hep-th/0105144 Evaluation of Integrals Representing Correlations in XXX Heisenberg Spin Chain]
 **  H.E.Boos, V.E.Korepin, 2001<br>
-* [http://dx.doi.org/10.1088/0305-4470/34/26/301 Quantum spin chains and Riemann zeta function with odd arguments]<br>
+* [http://dx.doi.org/10.1088/0305-4470/34/26/301 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/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
-* http://dx.doi.org/
-==question and answers(Math Overflow)==
-* http://mathoverflow.net/search?q=
-* http://mathoverflow.net/search?q=
-==links==
-* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
-* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
-* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
-* http://functions.wolfram.com/
 [[분류:integrable systems]]
-[[분류:math and physics]]
 [[분류:math and physics]]

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

2013년 2월 28일 (목) 14:06 판

목차

introduction

emptiness formation probability

near neighbor correlations

related items

articles

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