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

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41번째 줄: 41번째 줄:
 
$$[T(u), T^(v)]=0$$
 
$$[T(u), T^(v)]=0$$
 
which implies $[Q_n,Q_m]=0$
 
which implies $[Q_n,Q_m]=0$
−
* in order to have $[T(u), T^(v)]=0$, the YBE must be satisfied
+
* in order to have $[T(u), T(v)]=0$, the YBE must be satisfied
  
  
64번째 줄: 64번째 줄:
 
*  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>
 
* <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>
 
* <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>
−
*  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 :
−
*  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>
+
<math>f(x_ 1,\cdots,x_n)=\sum_{P}A (P)\exp(i\sum_{j=1}^{n}x_jk _{P_j})</math>
−
*  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>
+
*  Bethe ansatz equation for wave numbers : there are n conditions
 +
:<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  
 +
:<math>B(k,q)=-\frac{1+e^{ik}e^{iq}-e^{ik}}{1+e^{ik}e^{iq}-e^{iq}}</math><br>
 +
*  eigenvalue
 +
:<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>
  
  
72번째 줄: 76번째 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)=-\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>
+
*  Hamiltonian of XXZ model or XXZ spin chain with  anisotropic parameter <math>\Delta=1/2</math>
−
*  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>
+
:<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>
−
*  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>
+
*  two body scattering term
−
*  fundamental equation<br><math>k_jN=2\pi I(k_j)+\sum_{l=1}^{N}\theta(k_j,k_l)</math><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>
 +
*  equation satisfied by wave numbers
 +
:<math>\exp(ik_jN)=(-1)^{N-1}\prod_{l=1}^{N}\exp(-i\theta(k_j,k_l))</math>
 +
where <math>\theta(p,q)</math> is defined as
 +
:<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
 +
:<math>k_jN=2\pi I(k_j)+\sum_{l=1}^{N}\theta(k_j,k_l)</math><br>
 
*  eigenvalue<br>
 
*  eigenvalue<br>
−
 
 
* ground state eigenvector for Hamiltonian  is a common eigenvector although the eigenvalues are different
 
* ground state eigenvector 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 because both are characterized by the fact that <math>f(x_ 1,\cdots,x_n)>0</math>
 
* the maximum eigenstate of the transfer matrix and the ground state of the above Hamiltonian are identical because both are characterized by the fact that <math>f(x_ 1,\cdots,x_n)>0</math>
90번째 줄: 99번째 줄:
 
==Sutherland's observation==
 
==Sutherland's observation==
  
−
*  the eigenvectors of the transfer matrix depended on a,b,c only via the parameter<br><math>\Delta=\frac{a^2+b^2-c^2}{2ab}=\cos \eta</math><br>
+
*  the eigenvectors of the transfer matrix depended on a,b,c only via the parameter
 +
:<math>\Delta=\frac{a^2+b^2-c^2}{2ab}=\cos \eta</math><br>
 
*  $\delta$ = anistropic parameter in [[Heisenberg spin chain model]]<br>
 
*  $\delta$ = anistropic parameter in [[Heisenberg spin chain model]]<br>
  

2012년 12월 23일 (일) 19:02 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


integrability of the model

  • $T(u)$ transfer matrix
  • $\log T(u)=\sum_{n=0}^{\infty}Q_{n}u^n$
  • here $Q_1$ plays the role of the Hamiltonian
  • necessary and sufficient codntion to have infinitely many conserved quantities

$$[T(u), T^(v)]=0$$ which implies $[Q_n,Q_m]=0$

  • in order to have $[T(u), T(v)]=0$, the YBE must be satisfied


YBE

  • R-matrix $$R(u,\eta)=\rho\left( \begin{array}{cccc} \sin (u+\eta ) & 0 & 0 & 0 \\ 0 & \sin (u) & \sin (\eta ) & 0 \\ 0 & \sin (\eta ) & \sin (u) & 0 \\ 0 & 0 & 0 & \sin (u+\eta ) \end{array} \right)$$



transfer matrix formalism and coordinate Bethe ansatz

  • \(M=N^{2}\) number of molecules
  • 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)

\[\hat H = -\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \Delta \sigma_j^z \sigma_{j+1}^z)=-\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \frac{1}{2} \sigma_j^z \sigma_{j+1}^z)\]

  • 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 eigenvector 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 because 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

\[\Delta=\frac{a^2+b^2-c^2}{2ab}=\cos \eta\]


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



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STATISTICAL MECHANICS-A REVIEW OF SELECTED RIGOROUS RESULTS

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.

 

 

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