and to support this nomenclature, we now discuss an interface. For concreteness we take coordinates $(x,y,z)\in{\mathbb{Z}}^3$ for the vertices of the lattice and identify the interface as the $x=0$ plane. We also denote $P_L$ ($P_R$) the plaquettes in the region $x\le 0$ ($x>0$). In the limit $J_2,J_1\to\infty$, $(AAAA)_{\tilde{p}}=1$ for $\tilde{p} \in P_R$. By conservation of flux, we find that for all boundary plaquettes $p \in \partial P$, $(AAAA)_{p}=1$. Consequently since $J_2$ forces $(-)^{\omega_p}(AAAA)_p=1$, $\omega_p=0$ on the $2D$ boundary. The surface partition function in this limit is thus given by $$\begin{aligned}
  \label{eq:Z_surf}
  Z_{\text{surf},0} &= 
  \sum_{\sigma} \prod_{p\in \partial P}\delta({\omega_p}) =  \sum_{\sigma,\tau} \prod_{p\in \partial P}(\tau)^{\omega_p}\, .\end{aligned}$$

The possible domain wall configurations for $\sigma$’s in $2D$ are depicted in Fig. \[fig:conf\_omega0\] where a second mapping to arrow configurations of the eight-vertex model is also discussed. The constraint $\omega_p=0$ implies a two-in two-out ice rule, supporting the vorticity interpretation and mapping the surface theory to the critical six vertex model with an anisotropy parameter $\Delta = \frac{1}{2}$ [@Baxter].

The latter model model is critical and described by a compact free boson $\phi$. This fact can be established with the Coulomb gas method [@Nienhuis1984], which we now briefly recall. Denoted by $S_{\ell}=\pm 1$ the arrow at 