Properties of linear integral equations related to the six-vertex model with disorder parameter

TL;DR

This paper explores linear integral equations related to the six-vertex model with a disorder parameter, providing new representations and proofs for key functions like the dressed charge and magnetization density.

Contribution

It introduces a new expression for the generalized dressed charge using Baxter's Q-functions and offers a novel proof for the asymptotics of the magnetization density.

Findings

Generalized dressed charge expressed via Q-functions

New proof for asymptotics of magnetization density

Linear integral equations depend on the disorder parameter

Abstract

One of the key steps in recent work on the correlation functions of the XXZ chain was to regularize the underlying six-vertex model by a disorder parameter $\alpha$. For the regularized model it was shown that all static correlation functions are polynomials in only two functions. It was further shown that these two functions can be written as contour integrals involving the solutions of a certain type of linear and non-linear integral equations. The linear integral equations depend parametrically on $\alpha$ and generalize linear integral equations known from the study of the bulk thermodynamic properties of the model. In this note we consider the generalized dressed charge and a generalized magnetization density. We express the generalized dressed charge as a linear combination of two quotients of $Q$-functions, the solutions of Baxter's $t$-$Q$-equation. With this result we give a new proof of a lemma on the asymptotics of the generalized magnetization density as a function of the spectral parameter.