models. When taking into account marginally irrelevant corrections to the effective Gaussian model and resumming perturbation series with the help of the RG, the spurious divergence of fidelity susceptibility disappears, as explained in Appendix D.

Summary
=======

Using an effective field-theory approach, exact asymptotic expressions of the uniform parts of the biliniar spin correlation functions of the Heisenberg antiferromagnetic spin-$\frac{1}{2}$ chain, $G^{a,a}_u(r)$ and $G^{a,b}_u(r)$, have been computed. We have checked numerically analytical results and also estimated the numerical prefactor in front of the staggered part of the bilinear spin correlation function and identified the logarithmic contribution in accordance with the previous analytical investigations [@Affleck+89].

As a by-product, our studies confirm the finiteness of the ground state fidelity susceptibility at the Berezinskii-Kosterlitz-Thouless quantum phase transition points in one-dimensional systems.

Acknowledgments
===============

This work has been supported by DFG Research Training Group (Graduiertenkolleg) 1729 and Center for Quantum Engeneering and Space-Time research (QUEST). We thank A. Furusaki, T. Hikihara, A. K. Kolezhuk, S. Lukyanov, and M. Oshikawa for helpful discussions.

Appendix
========

Calculating constant $B_2$ for $-1<\lambda<1$
---------------------------------------------

We provide details of calculating the exact expression of constant $B_2$ appearing in Eq. (12) in the main text. For this we introduce the 