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abstract: 'Exact asymptotic expressions of the uniform parts of the two-point correlation functions of bilinear spin operators in the Heisenberg antiferromagnetic spin-$\frac{1}{2}$ chain are obtained. Apart from the algebraic decay, the logarithmic contribution is identified, and the numerical prefactor is determined. We also confirm numerically the multiplicative logarithmic correction of the staggered part of the bilinear spin operators $\langle\langle  S^{a}_0S^{a}_{1}S^{b}_{r}S^{b}_{r+1}  \rangle\rangle=(-1)^rd/(r \ln^{\frac{3}{2}}r)  +(3\delta_{a,b}-1)  \ln^2r /(12 \pi^4 r^4)$, and estimate the numerical prefactor as $d\simeq 0.067$. The relevance of our results for ground state fidelity susceptibility at the Berezinskii-Kosterlitz-Thouless quantum phase transition points in one-dimensional systems is discussed at the end of our work.'
author:
- 'T. Vekua'
- 'G. Sun'
title: 'Exact asymptotic correlation functions of bilinear spin operators of the Heisenberg antiferromagnetic spin-$\frac{1}{2}$ chain'
---

 Introduction
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The XXZ spin-$\frac{1}{2}$ chain is a paradigmatic one-dimensional quantum many-body system which can be studied using exact methods, while simultaneously describing the magnetic properties of real materials [@KM]. The Hamiltonian of the XXZ chain, written in terms of spin-$\frac{1}{2}$ matrices, reads, $$\label{ham-XXZ0}
\hat{H}_{XXZ}= J\sum_{r} \Big\{ S^{x}_{r}S^{x}_{r+1}
+S^{y}_{r}S^{y}_{r+1} +\lambda S^{z}_{r}S^{z}_{r+1}\Big\},$$ where $J$ is an exchange coupling, that will be assumed to be positive, and $\lambda$ is an anisotropy parameter. For $\lambda=1$, the 