"Six-vertex model and Quantum XXZ Hamiltonian"의 두 판 사이의 차이

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44번째 줄: 44번째 줄:
 
*  one can regard the up(or down) arrows in a row as 'particles'<br>
 
*  one can regard the up(or down) arrows in a row as 'particles'<br>
 
*  because of the ice rule, their number is conserved and one can try a [[Bethe ansatz]] for the eigenvectors of the transfer matrix<br>
 
*  because of the ice rule, their number is conserved and one can try a [[Bethe ansatz]] for the eigenvectors of the transfer matrix<br>
−
* f(x_1,\cdots,x_n) be the amplitude in an eigenvector of the state with up arrows at the sites <math> x_1<x_2<\cdots<x_n</math><br>
+
* <math>f(x_1,\cdots,x_n)</math> be the amplitude in an eigenvector of the state with up arrows at the sites <math> x_1<x_2<\cdots<x_n</math><br>
−
*   <br> obtain the equation for amplitudes <br><math>f(x_1,\cdots,x_n)=\sum_{P}A(P)\exp(i\sum_{j=1}^{n}x_jk_{P_j})</math><br>
+
*  obtain the equation for amplitudes <br><math>f(x_1,\cdots,x_n)=\sum_{P}A(P)\exp(i\sum_{j=1}^{n}x_jk_{P_j})</math><br>
−
*  Bethe ansatz equation for wave numbers<br><math>\exp(ik_jn)=\prod_{j\neq i}B(k_i,k_j)=\prod_{j=1}^{n}B(k_i,k_j)</math><br> where <br><math>B(k,q)=-\frac{1+e^{ik}e^{iq}-e^{ik}}{1+e^{ik}e^{iq}-e^{iq}}</math><br>
+
*  Bethe ansatz equation for wave numbers : there are n conditions<br><math>\exp(ik_jn)=\prod_{j\neq i}B(k_i,k_j)=\prod_{j=1}^{n}B(k_i,k_j)</math><br> where <br><math>B(k,q)=-\frac{1+e^{ik}e^{iq}-e^{ik}}{1+e^{ik}e^{iq}-e^{iq}}</math><br>
 
*  eigenvalue<br><math>\lambda=\frac{1+e^{i(k_{1}+\cdots+k_{n})}}{\prod_{j=1}^{n}1-e^{ik_{j}}}=\frac{1+e^{i(k_{1}+\cdots+k_{n})}}{(1-e^{ik_{j}})\cdots(1-e^{ik_{j}})}</math><br>
 
*  eigenvalue<br><math>\lambda=\frac{1+e^{i(k_{1}+\cdots+k_{n})}}{\prod_{j=1}^{n}1-e^{ik_{j}}}=\frac{1+e^{i(k_{1}+\cdots+k_{n})}}{(1-e^{ik_{j}})\cdots(1-e^{ik_{j}})}</math><br>
  
58번째 줄: 58번째 줄:
  
 
* [[Heisenberg spin chain model]]
 
* [[Heisenberg spin chain model]]
−
*  Hamiltonian of XXZ model or XXZ spin chain with  anisotropic parameter <math>\Delta=1/2</math><br><math>\hat H = -\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \Delta \sigma_j^z \sigma_{j+1}^z)</math><br>
+
 
 +
*   <br> Hamiltonian of XXZ model or XXZ spin chain with  anisotropic parameter <math>\Delta=1/2</math><br><math>\hat H = -\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \Delta \sigma_j^z \sigma_{j+1}^z)=-\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \frac{1}{2} \sigma_j^z \sigma_{j+1}^z)</math><br>  <br>
 +
*  two body scattering term<br><math>s_{jl}=1-2\Delta e^{ik_l}+ e^{ik_l+ik_j}=1-e^{ik_l}+ e^{ik_l+ik_j}</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-e^{ip}+e^{i(p+q)}}{1- e^{iq}+e^{i(p+q)}}</math><br>
 +
*  fundamental equation<br><math>k_jN=2\pi I(k_j)+\sum_{l=1}^{N}\theta(k_j,k_l)</math><br>
 +
*  eigenvalue<br>
 +
 
 
* ground state eigevector for Hamiltonian  is a common eigenvector although the eigenvalues are different
 
* ground state eigevector for Hamiltonian  is a common eigenvector although the eigenvalues are different
 +
* the maximum eigenstate of the transfer matrix and the ground state of the above Hamiltonian are identical becasue both are characterized by the fact that <math>f(x_1,\cdots,x_n)>0</math>
 
* see '''[YY1966-2]'''
 
* see '''[YY1966-2]'''
  

2010년 8월 3일 (화) 13:32 판

introduction
  • ice-type model, R model, Rys model
  • XXZ spin chain and the six-vertex transfer matrix have the same eigenvectors
  • Boltzmann weights
  • monodromy matrix
  • trace of monodromy matrix is the transfer matrix
  • power of transfer matrix becomes the partition function

 

 

types of six vertex models
  • on a square lattice with periodic boundary conditions
  • on a square lattice with domain wall boundary conditions

 

 

transfer matrix
  • borrowed from transfer matrix in statistical mechanics
  • transfer matrix is builtup from matrices of  Boltzmann weights
  • finding eigenvalues and eigenvectors of transfer matrix is crucial
  • Bethe ansatz equation is used to find the eigenvectors and eigenvalues of the transfer matrix
  • partition function = trace of power of transfer matrices
  • so the partition function  is calculated in terms of the eigenvalues of the transfer matrix
  • then the problem of solving the model is reduced to the computation of this trace

 

 

 

transfer matrix formalism and the role of Bethe ansatz
  • one can regard the up(or down) arrows in a row as 'particles'
  • because of the ice rule, their number is conserved and one can try a Bethe ansatz for the eigenvectors of the transfer matrix
  • \(f(x_1,\cdots,x_n)\) be the amplitude in an eigenvector of the state with up arrows at the sites \( x_1<x_2<\cdots<x_n\)
  • obtain the equation for amplitudes 
    \(f(x_1,\cdots,x_n)=\sum_{P}A(P)\exp(i\sum_{j=1}^{n}x_jk_{P_j})\)
  • Bethe ansatz equation for wave numbers : there are n conditions
    \(\exp(ik_jn)=\prod_{j\neq i}B(k_i,k_j)=\prod_{j=1}^{n}B(k_i,k_j)\)
    where 
    \(B(k,q)=-\frac{1+e^{ik}e^{iq}-e^{ik}}{1+e^{ik}e^{iq}-e^{iq}}\)
  • eigenvalue
    \(\lambda=\frac{1+e^{i(k_{1}+\cdots+k_{n})}}{\prod_{j=1}^{n}1-e^{ik_{j}}}=\frac{1+e^{i(k_{1}+\cdots+k_{n})}}{(1-e^{ik_{j}})\cdots(1-e^{ik_{j}})}\)

 

 

 

anistropic one-dimensional Heisenberg model (XXZ model)
  •  
    Hamiltonian of XXZ model or XXZ spin chain with  anisotropic parameter \(\Delta=1/2\)
    \(\hat H = -\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \Delta \sigma_j^z \sigma_{j+1}^z)=-\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \frac{1}{2} \sigma_j^z \sigma_{j+1}^z)\)
     
  • two body scattering term
    \(s_{jl}=1-2\Delta e^{ik_l}+ e^{ik_l+ik_j}=1-e^{ik_l}+ e^{ik_l+ik_j}\)
  • equation satisfied by wave numbers
    \(\exp(ik_jN)=(-1)^{N-1}\prod_{l=1}^{N}\exp(-i\theta(k_j,k_l))\)
    where
    \(\theta(p,q)\) is defined as
    \(\exp(-i\theta(p,q))=\frac{1-2\Delta e^{ip}+e^{i(p+q)}}{1-2\Delta e^{iq}+e^{i(p+q)}}=\frac{1-e^{ip}+e^{i(p+q)}}{1- e^{iq}+e^{i(p+q)}}\)
  • fundamental equation
    \(k_jN=2\pi I(k_j)+\sum_{l=1}^{N}\theta(k_j,k_l)\)
  • eigenvalue
  • ground state eigevector for Hamiltonian  is a common eigenvector although the eigenvalues are different
  • the maximum eigenstate of the transfer matrix and the ground state of the above Hamiltonian are identical becasue both are characterized by the fact that \(f(x_1,\cdots,x_n)>0\)
  • see [YY1966-2]

 

 

 

 

Sutherland's observation
  • the eigenvectors of the transfer matrix depended on a,b,c only via the parameter
    \(\Delta=\frac{a^2+b^2-c^2}{2ab}\)
  • is the \delta = anistropic parameter in Heisenberg spin chain model ?

 

 

 

entropy of two-dimensional ice
  • entropy is given as
    \(Mk\ln W\) where M is the number of molecules and \(W=(4/3)^{3/2}=1.53960\cdots\)

 

 

 

free energy
  • \(F=-kT \ln Z\)

 

 

partition function

 

 

correlation functions

 

 

related items

 

 

books

 

 

encyclopedia

 

 

blogs

 

 

STATISTICAL MECHANICS-A REVIEW OF

SELECTED RIGOROUS RESULTS1•2

By JOEL L. LEBOWITZ

 

 

Method for calculating finite size corrections in Bethe ansatz systems: Heisenberg chain and six-vertex model
de Vega, H. J.; Woynarovich, F.

 

 

articles

 


links
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