in (\[prop\]), an $\eta$-dependent factor such that $$\label{correctidenx}
\sum_r S^{x}_{r}S^{x}_{r+1} \to  \int \mathrm{d} r  \{ \sqrt{B_0}+  \hat O^{x}_2  +   \hat O^{x}_3+ \hat O^{x}_4 \}.$$ The scaling dimension of $ \hat O^{x}_2$ is $2\eta$, while those of $\hat O^{x}_3$ and $\hat O^{x}_4$ are $2/\eta$ and $2$, respectively. In the limit of the $SU(2)$ AFM point $\eta\to 1$ and all of them become marginal.

The constant term in Eq. (\[ccor\]) can be easily fixed due to the translational symmetry, $ B_0=\frac{1}{4}(e_0-\lambda \partial e_0/\partial \lambda)^2$, where $e_0$ is the ground state energy density known exactly (together with its dependence on $\lambda$) from the Bethe ansatz. However we will be interested in the following with the reduced correlation function, $$\label{totalbl}
\langle  \langle S_{0}^xS_{1}^x S_{r}^xS_{r+1}^x   \rangle \rangle\! = \! G^{x,x}_r  \!= (-1)^rG^{x,x}_{s}(r)+G^{x,x}_{u}(r).$$ Namely, the uniform part of the above reduced correlation function is the main quantity of our interest, $$\begin{aligned}
\label{bl}
G^{x,x}_{u}(r)&=& G^{x,x}_{B_2}(r)+G^{x,x}_{B_3}(r)+ G^{x,x}_{B_4}(r)\nonumber\\
 &=& \frac{B_2}{r^{4\eta}}+ \frac{B_3}{r^{4/\eta}}+  \frac{B_4}{r^{4}}.\end{aligned}$$ For the XXZ chain, for $-1<\lambda<1$, the exact expression of $B_2$ amplitude has been obtained [@LT], $$\label{LT}
B_2= \frac{[\Gamma(\eta)]^4}{2^{3+4\eta}\pi^{2+2\eta} (1-\eta)^2}\left[   \frac{ \Gamma(\frac{1}{2-2\eta})}{\Gamma(\frac{\eta}{2-2\eta})} \right]^{4-4\eta}$$ and confirmed numerically away from the $SU(2)$ points [@HF04]. In Appendix A we provide details of calculating $B_2$, confirming expression (\[LT\]). However, when $\eta\to 