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

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imported>Pythagoras0
imported>Pythagoras0
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==YBE==
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==R-matrix and Yang-Baxter equation==
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*R-matrix $$R(u,\eta)=\rho\left(
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* [[Exact S-matrices in ATFT]]
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* [[Yang-Baxter equation (YBE)]]
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*R-matrix  
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$$
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R(u,\eta)=\rho\left(
 
\begin{array}{cccc}
 
\begin{array}{cccc}
 
  \sin (u+\eta ) & 0 & 0 & 0 \\
 
  \sin (u+\eta ) & 0 & 0 & 0 \\
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  0 & 0 & 0 & \sin (u+\eta )
 
  0 & 0 & 0 & \sin (u+\eta )
 
\end{array}
 
\end{array}
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\right)$$
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\right)
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$$
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==transfer matrix formalism and coordinate Bethe ansatz==
 
==transfer matrix formalism and coordinate Bethe ansatz==

2013년 1월 5일 (토) 08:19 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


R-matrix and Yang-Baxter equation

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

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



books

 

 

encyclopedia


 

 

blogs

 

 

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.

 

 

expositions

 

 

articles

 

links

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