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

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27번째 줄: 27번째 줄:
 
* finding eigenvalues and eigenvectors of transfer matrix is crucial
 
* finding eigenvalues and eigenvectors of transfer matrix is crucial
 
* Bethe ansatz equation is used to find the eigenvectors and eigenvalues of the transfer matrix
 
* Bethe ansatz equation is used to find the eigenvectors and eigenvalues of the transfer matrix
−
* partition function  is calculated in terms of the eigenvalues of the transfer matrix<br>  <br>
+
* partition function  is calculated in terms of the eigenvalues of the transfer matrix
 +
* the below is from [[Yang-Baxter equation (YBE)|Yang-Baxter equation]]
 +
 
 +
*  transfer matrix is builtup from matrices of  Boltzmann weights<br>
 +
*  we need the trasfer matrices coming from different set of Boltzman weights commute <br>
 +
*  partition function = trace of power of transfer matrices<br>
 +
*  so the problem of solving the model is reduced to the computation of this trace<br>
 +
 
 +
 
  
 
 
 
 
61번째 줄: 69번째 줄:
 
<h5>anistropic one-dimensional Heisenberg model</h5>
 
<h5>anistropic one-dimensional Heisenberg model</h5>
  
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* [[Heisenberg spin chain model|Heisenberg model]]
+
* [[Heisenberg spin chain model]]
 
* XXZ model or XXZ spin chain
 
* XXZ model or XXZ spin chain
 
* first solved by Bethe 
 
* first solved by Bethe 

2010년 8월 3일 (화) 10:05 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
  • 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  is calculated in terms of the eigenvalues of the transfer matrix
  • the below is from Yang-Baxter equation
  • transfer matrix is builtup from matrices of  Boltzmann weights
  • we need the trasfer matrices coming from different set of Boltzman weights commute 
  • partition function = trace of power of transfer matrices
  • so the problem of solving the model is reduced to the computation of this trace

 

 

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

 

 

free energy
  • \(F=-kT \ln Z\)

 

 

partition function

 

 

correlation functions

 

 

anistropic one-dimensional Heisenberg model
  • Heisenberg spin chain model
  • XXZ model or XXZ spin chain
  • first solved by Bethe 
  • Yang and Yang
  • ground state eigevector for Hamiltonian  is a common eigenvector

 

 

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

 

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