Master equation for spin-spin correlation functions of the XXZ chain

TL;DR

This paper introduces a master equation that unifies different integral representations of spin-spin correlation functions in the XXZ chain, enabling direct analysis and expansion in form factors, and extends to dynamical correlations.

Contribution

It presents a novel master equation formulation that links existing integral formulas and form factor expansions for the XXZ chain's correlation functions.

Findings

Unified integral representation for static correlation functions

Re-summation of series into a single integral form

Extension to dynamical correlation functions

Abstract

We derive a new representation for spin-spin correlation functions of the finite XXZ spin-1/2 Heisenberg chain in terms of a single multiple integral, that we call the master equation. Evaluation of this master equation gives rise on the one hand to the previously obtained multiple integral formulas for the spin-spin correlation functions and on the other hand to their expansion in terms of the form factors of the local spin operators. Hence, it provides a direct analytic link between these two representations of the correlation functions and a complete re-summation of the corresponding series. The master equation method also allows one to obtain multiple integral representations for dynamical correlation functions.