(\[andimension\]) reproduces the $r$ dependence of $G^{x,x}_{B_2}$ in Eq. (7) of the main text, $\sim r^{-4\eta}$.

Next we provide the details of calculating the exact long-distance asymptotics of $G^{x,x}_{B_2}$ at the $SU(2)$ antiferromagnetic point, given in Eq. (14) of the main text.

RG improved perturbation theory approach for long-distance asymptotics of $G^{x,x}_{B_2}$ 
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Here we will generalize the calculation of exact asymptotic correlation functions of single-spin operators at the $SU(2)$ antiferromagnetic point [@Affleck98; @Lukyanov98] to the case of $G^{x,x}_{B_2}$.

Our aim is to compute the two-point correlation function for the effective action with the bare coupling constants ${\bm{g}}^0$ (which carry information of the initial microscopic lattice model) $G(r)=G(r,r_0, {\bm{g}}(r_0))$. However, since the Hamiltonian is not Gaussian, one has to use some approximate methods for computing correlation functions. If one tries to perform a perturbation theory calculation in coupling constants, a standard method of interacting field theory, because of the logarithmic divergences that occur in the infrared limit, one cannot stop perturbative series at some finite order, even if initially ${\bm{g}}(r_0) \ll 1$. For example, in our case, the first order in the coupling constants contribution in the correlation function $G_{B_2}(r)$ comes with $g_{||}$, $$\begin{aligned}
-2\langle  \cos {\sqrt{8 \pi } \Theta(0)} 