Reduced qKZ equation and correlation functions of the XXZ model

TL;DR

This paper introduces a reduced qKZ equation for the XXZ model's correlation functions, constructs solutions as combinations of transcendental functions, and connects these solutions to the model's massive and massless regimes.

Contribution

It constructs explicit solutions to the reduced qKZ equation using transcendental functions, linking them to XXZ model correlation functions in different regimes.

Findings

Correlation functions satisfy the reduced qKZ equation.

Solutions expressed as linear combinations of transcendental functions.

Formulas applicable to the massive regime; conjectural formulas for the massless regime.

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 (\lambda_1,...,\lambda_{2n}) as (\lambda_1,...,\lambda_n,\lambda_{n}+1,...,\lambda_{1}+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 $\tilde{\omega}, \omega$ 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 $\tilde{\omega}, \omega$. We also present a conjectural formula in the massless regime.