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arXiv:hep-th/9601135 · Journal of Physics A Mathematical and General

Quantum KZ equation with |q| = 1 and correlation functions of theXXZmodel in the gapless regime

Michio Jimbo, Tetsuji Miwa

Published Jun 21, 1996• 268 citations
arXivHTMLPDF

Abstract

An integral solution to the quantum Knizhnik - Zamolodchikov (qKZ) equation with |q| =1 is presented. Upon specialization, it leads to a conjectural formula for correlation functions of the XXZ model in the gapless regime. The validity of this conjecture is verified in special cases, including the nearest-neighbour correlator with an arbitrary coupling constant and general correlators in the XXX and XY limits.

DOI
10.1088/0305-4470/29/12/005

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An integral solution to the quantum Knizhnik - Zamolodchikov (qKZ) equation with |q| =1 is presented. Upon specialization, it leads to a conjectural formula for correlation functions of the XXZ model in the gapless regime. The validity of this conjecture is verified in special cases, including the nearest-neighbour correlator with an arbitrary coupling constant and general correlators in the XXX and XY limits.
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).
Abstract. Correlation functions of the XXZ model in the massive and massless regimes are known to satisfy a system of linear equations. The main relations among them are the difference equations obtained from the qKZ equation by specializing the variables (λ1, · · · , λ2n) as (λ1, · · · , λn, λn + 1, · · · , λ1 + 1). We call it the reduced qKZ equation. In this article we construct a special family of solutions to this system. They can be written as linear combinations of products of two transcendental functions ˜ω, ω with coefficients being rational functions. We show that correlation functions of the XXZ model in the massive regime are given by these formulas with an appropriate choice of ˜ω, ω. We also present a conjectural formula in the massless regime. 1.
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.
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 ().