Six-vertex model and Quantum XXZ Hamiltonian

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
  
  • this is related to the Alternating sign matrix theorem

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
  

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
  \(\exp(ik_jn)=\prod_{j \n eq i}B(k_i,k_j)=\prod_{j=1}^{n}B(k_i,k_j)\)
  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)

• Heisenberg spin chain model
• Hamiltonian of XXZ model or XXZ spin chain with anisotropic parameter \(\Delta=1/2\)
  \(\hat H = -\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \Delta \sigma_j^z \sigma_{j+1}^z)=-\sum_{j=1}^{N} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \frac{1}{2} \sigma_j^z \sigma_{j+1}^z)\)
  
• 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]