On the Semi-Classical Limit of Scalar Products of the XXZ Spin Chain

TL;DR

This paper analyzes the semi-classical limit of scalar products in the XXZ spin chain with anisotropy greater than one, deriving a compact integral representation involving quantum dilogarithm functions.

Contribution

It extends methods from the XXX spin chain to the XXZ model, providing a new integral formula for scalar products in the semi-classical regime.

Findings

Derived a contour integral expression for scalar products

Connected quantum dilogarithm to classical dilogarithm in the isotropic limit

Generalized semi-classical analysis to anisotropic XXZ chain

Abstract

We study the scalar products between Bethe states in the XXZ spin chain with anisotropy $|\Delta|>1$ in the semi-classical limit where the length of the spin chain and the number of magnons tend to infinity with their ratio kept finite and fixed. Our method is a natural yet non-trivial generalization of similar methods developed for the XXX spin chain. The final result can be written in a compact form as a contour integral in terms of Faddeev's quantum dilogarithm function, which in the isotropic limit reduces to the classical dilogarithm function.