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

수학노트
둘러보기로 가기 검색하러 가기
imported>Pythagoras0
imported>Pythagoras0
12번째 줄: 12번째 줄:
  
 
*  raising and lowering operators
 
*  raising and lowering operators
−
:<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>
+
:<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>
  
−
* permutation operator
+
* the permutation operator can be written in terms of Pauli matrices
−
:<math>h=\frac{\sigma_{i}\cdot\sigma_{j}+1}{2}</math> acts as the permutation operator
+
:<math>h=\frac{\sigma_{i}\cdot\sigma_{j}+1}{2}</math>
  
−
 
 
  
 
 
 
 
23번째 줄: 23번째 줄:
 
==summary==
 
==summary==
  
−
*  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>
+
*  Hamiltonian of XXX spin chain with  anisotropic parameter <math>\Delta=1</math>
−
*  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>
+
:<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>
−
*  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>
+
*  two body scattering term
−
*  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>
+
:<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>
−
*  fundamental equation<br><math>k_jN=2\pi I(k_j)+\sum_{l=1}^{N}\theta(k_j,k_l)</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>
  
 
 
 
 
35번째 줄: 40번째 줄:
 
==wavefunction amplitude==
 
==wavefunction amplitude==
  
−
*  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>
+
*  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>
 
* <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=2 case<br><math>A(12)=s_{21}</math><br><math>A(21)=-s_{12}</math><br>
48번째 줄: 54번째 줄:
 
<math>\exp(ik_jL)=(-1)^{n-1}\prod_{l=1, l\neq j}^{n}\frac{s_{l,j}}{s_{j,l}}</math>
 
<math>\exp(ik_jL)=(-1)^{n-1}\prod_{l=1, l\neq j}^{n}\frac{s_{l,j}}{s_{j,l}}</math>
  
−
n=1
+
* n denote the number of up spins
 +
 
 +
 
 +
===n=1===
  
 
<math>\exp(ik_jL)=1</math>
 
<math>\exp(ik_jL)=1</math>
  
−
n=2
+
 
 +
===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_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>
60번째 줄: 70번째 줄:
 
 
 
 
  
−
n=3
+
===n=3===
  
 
<math>\exp(ik_1L)=\frac{s_{2,1}s_{3,1}}{s_{1,2}s_{1,3}}</math>
 
<math>\exp(ik_1L)=\frac{s_{2,1}s_{3,1}}{s_{1,2}s_{1,3}}</math>
70번째 줄: 80번째 줄:
 
 
 
 
  
−
n denote the number of up spins
+
===n=0 analysis===
−
 
 
−
 
 
−
 
 
−
n=0 analysis
 
  
 
 
 
 
  
−
n=1 analysis
+
===n=1 analysis===
  
 
ansatz <math>a(x)=e^{ikx}</math>
 
ansatz <math>a(x)=e^{ikx}</math>
90번째 줄: 96번째 줄:
 
 
 
 
  
−
n=2 analysis
+
===n=2 analysis===
  
 
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>
106번째 줄: 112번째 줄:
 
 
 
 
  
−
n=3 analysis
+
===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>
 
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>
224번째 줄: 230번째 줄:
 
* http://mathoverflow.net/search?q=
 
* http://mathoverflow.net/search?q=
  
−
 
 
−
 
−
 
 
−
 
−
==blogs==
 
−
 
−
*  구글 블로그 검색<br>
 
−
**  http://blogsearch.google.com/blogsearch?q=<br>
 
−
** http://blogsearch.google.com/blogsearch?q=
 
−
* http://ncatlab.org/nlab/show/HomePage
 
−
 
−
 
 
−
 
−
 
 
−
 
−
==experts on the field==
 
−
 
−
* http://arxiv.org/
 
−
 
−
 
 
−
 
−
 
 
  
 
==links==
 
==links==

2012년 12월 24일 (월) 05:48 판

introduction

 

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

\[h=\frac{\sigma_{i}\cdot\sigma_{j}+1}{2}\]


 

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

 

 

related items

 

 

encyclopedia


 

 

books

 

 

 

expositions

 

 

articles

 

 

question and answers(Math Overflow)


links

원본 주소 "https://wiki.mathnt.net/index.php?title=Heisenberg_spin1/2_XXX_chain&oldid=36300"