Six-vertex model and Quantum XXZ Hamiltonian

수학노트
http://bomber0.myid.net/ (토론)님의 2010년 8월 3일 (화) 13:16 판
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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
    \(\exp(ik_jn)=\prod_{j\neq i}B(k_i,k_j)\)
    where 
    \(B(k,q)=-\frac{1+e^{ik}e^{iq}-e^{ik}}{1+e^{ik}e^{iq}-e^{iq}}\)
  • eigenvalue
     

 

 

 

anistropic one-dimensional Heisenberg model (XXZ model)
  • Heisenberg spin chain model
  • Hamiltonian of XXZ model or XXZ spin chain with  anisotropic parameter \(\Delta=1/2\)
    \(\hat H = -\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \Delta \sigma_j^z \sigma_{j+1}^z)\)
  • ground state eigevector for Hamiltonian  is a common eigenvector although the eigenvalues are different
  • 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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