Gaussian model.

In the absence of perturbations, for ${\bm{g}}=0$, the effective theory given by Eq. (4) in the main text has conformal invariance, and hence $$\label{cw}
2\langle  \cos {\sqrt{8 \pi } \Theta(0)} \cos {\sqrt{8 \pi } \Theta (r)} \rangle_G =r^{-4}.$$ When marginally irrelevant perturbations are included on top of the Gaussian model, $g_{\bot}$ does not contribute to the anomalous dimension of the $\cos {\sqrt{8 \pi } \Theta}$ field to first order, since $$\langle  \cos {\sqrt{8 \pi } \Theta(0)} \int \mathrm{d^2} {{\bm{x}}} \mathcal{H'}_2({\bm{x}})  \cos {\sqrt{8 \pi } \Theta (r)} \rangle_G =0.$$ Hence, at the lowest (first) order in ${\bm{g}}\neq 0$, we can include $g_{||}$ into the quadratic part of the action and obtain $$\label{andimension}
G^{x,x}_{B_2}(r) \sim {r^{-2 ( 2-4\pi g_{||} / \sqrt{3})}}.$$ Perturbation $\mathcal{H'}_1$ can be included into the quadratic part of the action independently of the strength of $g_{||}$. It is the strength of $g_{\bot}$ that must be small in order to use the anomalous dimension obtained from perturbative analyses at the lowest order.

From Eq. (\[andimension\]) we read off the anomalous dimension of the $\cos {\sqrt{8 \pi } \Theta}$ field at the lowest order in ${\bm{g}}$, $$\gamma_{B_2}({\bm{g}})= 2-4\pi g_{||}/\sqrt{3}.$$ Using the fixed-point value of $g_{||}(\infty)=\sqrt{3}(1- \eta)/(2\pi )$ in Eq. 