Six-vertex model and Quantum XXZ Hamiltonian

수학노트
imported>Pythagoras0님의 2014년 4월 13일 (일) 00:36 판
둘러보기로 가기 검색하러 가기

introduction

  • six-vertex model, also called ice-type model, R model, Rys model
  • XXZ spin chain and the six-vertex transfer matrix have the same eigenvectors
  • Bethe ansatz can be applied to solve the model


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 and the Yang-Baxter equation

  • $T(u)$ 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 $[Q_n,Q_m]=0$


R-matrix and Boltzmann weights

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


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 Bethe ansatz for the eigenvectors of the transfer matrix
  • let \(f(x_1,\cdots,x_n)\) be the coefficient in an eigenvector $v$ of the state with up arrows at the sites \(x_ 1<x_ 2<\cdots<x_n\) so that

\[v(k_1,\cdots,k_n)= \sum_{\substack{\mathbf{x}=(x_ 1,x_ 2,\cdots,x_n) \\ x_ 1<x_ 2<\cdots<x_n}} f(x_1,\cdots,x_n|k_1,\cdots,k_n)\sigma_{-}^{(x_1)}\cdots\sigma_{-}^{(x_n)}|0\rangle \]

  • Bethe ansatz suggests the following form for $f$

\[f(x_ 1,\cdots,x_n)=\sum_{P\in S_n}A (P)\exp(i\sum_{j=1}^{n}x_jk _{P_j})\]

  • Bethe ansatz equation for wave numbers : there are n conditions

\[\exp(ik_jn)=\prod_{\ell \neq j}B(k_j,k_\ell)=\prod_{\ell=1}^{n}B(k_j,k_\ell),\quad \forall j=1,\cdots, n\] where \[B(k,q)=-\frac{1+e^{ik}e^{iq}-e^{ik}}{1+e^{ik}e^{iq}-e^{iq}}\]

  • eigenvalue $\lambda$ of $v$ is given by

\[\lambda=\frac{1+e^{i(k_{1}+\cdots+k_{n})}}{\prod_{j=1}^{n}1-e^{ik_{j}}}=\frac{1+e^{i(k_{1}+\cdots+k_{n})}}{(1-e^{ik_{j}})\cdots(1-e^{ik_{j}})}\]


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

  • \(F=-kT \ln Z=-\beta \ln Z\)

partition function

correlation functions

computational resource


related items


 

encyclopedia


books


expositions

blogs



articles

원본 주소 "https://wiki.mathnt.net/index.php?title=Six-vertex_model_and_Quantum_XXZ_Hamiltonian&oldid=43781"