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arXiv:hep-th/0206224 · Journal of Physics A Mathematical and General

Bootstrap equations and correlation functions for the Heisenberg XYZ antiferromagnet

Yas-Hiro Quano

Published Oct 28, 2002• 5 citations

Abstract

We present two kinds of integral solutions to the quantum Knizhnik–Zamolodchikov equations for the 2 n -point correlation functions of the Heisenberg XYZ antiferromagnet. Our first integral solution can be obtained from those for the cyclic SOS model by using the vertex–face correspondence. By the construction, the sum with respect to the local height variables k 0 , k 1 , …, k 2 n of the cyclic SOS model remains other than the n -fold integral in the first solution. In order to perform these summations, we improve this to find the second integral solution of ( r + 1) n -fold integral for r ∊ ℤ >1 , where r is a parameter of the XYZ model. Furthermore, we discuss the relations among our formula, Lashkevich–Pugai's formula and that of Shiraishi.

DOI: 10.1088/0305-4470/35/45/305

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Integral solutions to the quantum Knizhnik-Zamolodchikov equations for the correlation functions of both the bulk and boundary XXZ models in the anti-ferromagnetic regime are presented. The difference equations can be derived from Smirnov-type master equations for correlation functions on the basis of the CTM bootstrap. Our integral solutions with an appropriate choice of the integral kernel reproduce the formulae previously obtained by using the bosonization of the vertex operators of the quantum affine algebra Uq ().

Integral formulae for polynomial solutions of the quantum Knizhnik–Zamolodchikov equations associated with the R -matrix of the six-vertex model are considered. It is proved that when the deformation parameter q is equal to and the number of vertical lines of the lattice is odd, the solution under consideration is an eigenvector of the inhomogeneous transfer matrix of the six-vertex model. In the homogeneous limit, it is a ground-state eigenvector of the antiferromagnetic XXZ spin chain with the anisotropy parameter Δ equal to −1/2 and an odd number of sites. The obtained integral representations for the components of this eigenvector allow us to prove some conjectures on its properties formulated earlier. A new statement relating the ground-state components of XXZ spin chains and Temperley–Lieb loop models is formulated and proved.

This paper is continuation of our previous papers hep-th/0209246 and hep-th/0304077 . We discuss in more detail a new form of solution to the quantum Knizhnik-Zamolodchikov equation [qKZ] on level -4 obtained in the paper hep-th/0304077 for the Heisenberg XXX spin chain. The main advantage of this form is it's explicit reducibility to one-dimensional integrals. We argue that the deep mathematical reason for this is some special cohomologies of deformed Jacobi varieties. We apply this new form of solution to the correlation functions using the Jimbo-Miwa conjecture. A formula (46) for the correlation functions obtained in this way is in a good agreement with the ansatz for the emptiness formation probability from the paper hep-th/0209246. Our previous conjecture on a structure of correlation functions of the XXX model in the homogeneous limit through the Riemann zeta functions at odd arguments is a corollary of the formula (46).

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.