QKZ equation with |q|=1 and correlation functions of the XXZ model in the gapless regime

TL;DR

The paper addresses the problem of computing correlation functions for the gapless XXZ chain by constructing an integral solution to the $q$KZ equation at $|q|=1$ and proposing a lattice specialization that yields correlation functions. The authors develop a detailed integral framework involving auxiliary functions $\kappa$, $\rho$, $\varphi$, and $\psi$, and provide an explicit $n$-fold integral representation for $G_n$, along with a one-time integration reduction to an $(n-1)$-fold form. They define lattice correlators via inhomogeneous parameters, and verify the approach by obtaining exact results for the nearest-neighbor correlator, the XXX limit, and the XY limit where the results reproduce free-fermion determinants. The work provides a concrete, integral-analytic route to qKZ-based correlation functions in the gapless regime, linking to known results in special limits and offering a bridge to sine-Gordon form factors and free-fermion theory.

Abstract

An integral solution to the quantum Knizhnik-Zamolodchikov ($q$KZ) 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 neighbor correlator with an arbitrary coupling constant, and general correlators in the XXX and XY limits.

Figures (6)

• Figure 2: The contours for $I_1$, $I_2$ and $I_3$.
• Figure 3: The contour for $I_1-I_2$.
• Figure 5: The contour $C$ and the semi-circle $C_\varepsilon$.
• Figure 6: The contour for the residue calculus.
• Figure 7: The contour $C$.

Theorems & Definitions (19)

• Proposition 3.1
• Proposition 4.1
• Proposition 4.2
• Proposition 4.3
• Proposition 5.1
• Proposition 5.2
• Proposition 5.3
• Proposition 5.4
• Proposition D.1
• Proposition D.2