1$ for $-1\le \lambda \le 1$.

These amplitudes appearing in Eqs. (\[ampl\]) have been checked numerically [@HF98]. Amplitudes $A_0^x$ and $A_1^z$ diverge in the isotropic AFM limit, $\lambda \to 1$, since in this limit mapping of the spin-$\frac{1}{2}$ chain to the Gaussian model becomes singular due to the marginally irrelevant (cosine) term with the scaling dimension $2/\eta$ occurring in the low-energy effective theory from the “spin umklapp” processes [@Haldane; @BE; @dN; @Affleck85].

Effective theory description of the XXZ spin-$\frac{1}{2}$ chain, for $\lambda\to 1$, necessarily contains terms beyond the Gaussian model, $$\label{effectiveH}
H_{eff}=\int \mathrm{d}r\mathcal{H}_{eff}=\int \mathrm{d}r[\mathcal{H}_G(r)+ \mathcal{H'}_1(r)+\mathcal{H'}_2(r)],$$ where $  \mathcal{H'}_1=\frac{\pi g^0_{||}}{\sqrt{3}}  [  (\partial_r \Phi)^2-\Pi^2 ] $ and $\mathcal{H'}_2=\frac{\pi g^0_{\bot}}{\sqrt{3}} \cos{\sqrt{8\pi }\Phi}$. The running coupling constants ${{\bm{g}}}=(g_{||},g_{\bot})$, with the bare values ${{\bm{g}}^0}= (g^0_{||}, g^0_{\bot})$, are governed by the Kosterlitz-Thouless [@Kosterlitz74] RG equations, $$\begin{aligned}
\label{KTRG}
\beta_{||}=\dot  g_{||}=-4\pi g^2_{\bot}/\sqrt{3},\,\, \beta_{\bot}=\dot  g_{\bot}=-4\pi g_{\bot}g_{||}/\sqrt{3},\end{aligned}$$ where the dot indicates a derivative with respect to the RG scale $l=\ln{\tilde r}$ and $\pi \tilde r^{-1}$ is the running ultraviolet cutoff. At the $SU(2)$ AFM point $g_{||}= g_{\bot}=g$, and the exact expression of amplitudes of the asymptotic correlation functions of spin operators were derived [@Affleck98; @Lukyanov98] by combining the expressions of $A_0^x$ [@LZ] and $A_1^z$ [@Lukyanov99] for $\lambda\to 1_-$ with RG 