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arXiv:nlin/0001038 · International Journal of Modern Physics A

DIFFERENCE EQUATIONS FOR THE HIGHER RANK XXZ MODEL WITH A BOUNDARY

TAKEO KOJIMA, YAS-HIRO QUANO

Published Sep 20, 2000• 5 citations
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Abstract

The higher rank analog of the XXZ model with a boundary is considered on the basis of the vertex operator approach. We derive difference equations of the quantum Knizhnik–Zamolodchikov type for 2N-point correlations of the model. We present infinite product formulae of two point functions with free boundary condition by solving those difference equations with N=1.

DOI
10.1142/s0217751x00001816

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Abstract The open XXZ spin chain with the anisotropy parameter Δ = − 1 2 and diagonal boundary magnetic fields that depend on a parameter x is studied. For real x > 0, the exact finite-size ground-state eigenvalue of the spin-chain Hamiltonian is explicitly computed. In a suitable normalisation, the ground-state components are characterised as polynomials in x with integer coefficients. Linear sum rules and special components of this eigenvector are explicitly computed in terms of determinant formulas. These results follow from the construction of a contour-integral solution to the boundary quantum Knizhnik–Zamolodchikov equations associated with the R -matrix and diagonal K -matrices of the six-vertex model. A relation between this solution and a weighted enumeration of totally-symmetric alternating sign matrices is conjectured.
A direct proof is given for the fact that the integral formula for the XXX limit of the trace of the type I q-vertex operators satisfies the deformed Knizhnik-Zamolodchikov (dKZ) equation with level −4. We have also carried out one-time integration by taking the residue at infinity. As a corollary of these we can construct a family of the integral formulas for solutions to the dKZ equation. Another corollary is the integral formula for the correlation function of the inhomogeneous XXX model, whose number of integrals is less than that of the previously obtained correlator for the XXZ model.