length scale $\tilde r$ at which the theory is defined. Since the effective theory is derived from the original microscopic lattice model, the initial length scale is given by the lattice constant $r_0$ and is increased in the RG process of gradually eliminating high-energy degrees of freedom.

The following connection between the bare and renormalized correlation functions is provided by the CS Eq. (\[RGEQ\]), $$\label{BR}
G(r,r_0,{\bm{g}}(r_0))= G(r,r_1,{\bm{g}}(r_1))e^{-2\int^{r_1}_{r_0}  \gamma({\bm{g}}(\tilde r))\mathrm{d}\ln{ \tilde r} }.$$ To see this, observe that the left-hand side of Eq. (\[BR\]) does not depend on some arbitrary scale $r_1$. Applying $r_1\frac{\delta}{\delta r_1}$ to both sides of Eq. (\[BR\]) reproduces Eq. (\[RGEQ\]) for $r_1=\tilde r$.

We will choose $r_1$ large enough, so that $G(r,r_1, {\bm{g}}(r_1))$ can be expanded in powers of $g_{\perp}(r_1)\ll 1$. This step is called the RG improvement of the perturbation theory. The zeroth-order term, evaluated by Gaussian fixed point action, with rescaled cutoff $r_1$ is $$\label{GFPA}
G(r, r_1, {\bm{g}}(r_1))|_{g_{\perp}=0}=Const.\left(\frac{r_1}{r r_0}\right)^{4\eta}.$$ Note that increasing cutoff from $r_0$ to $r_1$ is equivalent to decreasing distance (measured in new units) by the same factor, $r\to rr_0/r_1$.

Since at the lowest order $\gamma_{B_2}({\bm{g}})=2-4\pi g_{||}/\sqrt{3}$, we need the solution of the Kosterlitz-Thouless RG equations (5), presented in the main text, only 