\Phi(r)}+H.c.$, up to sub-leading contributions. The anomalous dimension of $\hat D$ is $\gamma_D=5/2-\sqrt{3}\pi g/2$, giving $
\langle \hat D(0)\hat D(r)  \rangle \sim   {{\ln^{\frac{3}{2}} r}}/{r^5}.
$

Hence, including the leading oscillatory contribution in $G^{a,a}_r-G^{a,b}_r$, we obtain $$\label{FE}
G^{a,a}_r\!-G^{a,b}_r=(1-\delta_{a,b}) \left[ \frac{ \ln^2{r}}{4\pi^4 r^4}+\tilde c (-1)^r  \frac{{\ln^{\frac{3}{2}} r}}{r^5} \right],$$ where $\tilde c$ is a numerical constant estimated from fitting to DMRG data. We present in Fig. 2 comparison of our analytical curves, obtained separately for even and odd $r$ from Eq. (\[FE\]), with our numerical data.

Relevance for fidelity susceptibility
=====================================

The asymptotically exact expression of the uniform part of the correlation function of the bilinear spin operators at $\lambda=1$, Eq. (\[mainresult\]), confirms our previous work [@Vekua1], showing that computing the ground state fidelity susceptibility of the XXZ spin-$\frac{1}{2}$ chain by the effective Gaussian model gives a qualitatively wrong result at the Berezinskii-Kosterlitz-Thouless quantum phase transition point. Moreover, our approach allows us to explicitly follow the steps on how the divergence in fidelity susceptibility at the Berezinskii-Kosterlitz-Thouless transition point arises in the thermodynamic limit due to the singular nature of the mapping of the Heisenberg spin-$\frac{1}{2}$ AFM chain on the Gaussian model and is not a property of either the microscopic or effective 