

After using the Poisson identity for each integer and performing the integrations over $\phi_i$ and $A$, the resulting expression becomes Z=\_[{J\_[1,ij]{}},{J\_[2,ij]{}}, {J\_[z,i]{}}]{} [e]{}\^[-H\_J]{}, \[Z2\] H\_J= \_[ij ; a,b]{} (K\^[-1]{})\_[ab]{} J\_[a,ij]{} J\_[b,ij]{} + \_i J\_i\^2, \[H\_J\] where $a,b=1,2$ labels layers and $(K^{-1})_{ab}$ is the matrix inverse to $K_{ab}$ introduced in Eqs.(\[2N0\]),(\[Ktg\]); The summation runs over the integer bond currents $ J_{1,ij}, \, J_{2,ij}$ (oriented from site $i$ to site $j$ so that $ J_{a,ij}=-  J_{a,ji},\, a=1,2$) within each corresponding layer 1,2 as well as over the integer currents $J_{i}$ oriented along the bond connecting the site $i$ in the layer 1 to the site $i$ in the layer 2. All the configurations are restricted by the Kirchhoff’s current conservation rule – the total of all J-currents incoming to any site must be equal to the total of all outcoming currents from the same site.

It is useful to note that Eqs.(\[Z2\],\[H\_J\]) can also be obtained directly from Eq.(\[2N0\]) by reinstating the compact nature of the variables: $ \nabla_{ij} \phi_1 \to \nabla_{ij} \phi_1 +2\pi m_{1, ij}$ and $ \nabla_{ij} \phi_2 \to \nabla_{ij} \phi_2 +2\pi m_{2, ij}$ so that the matrix $K_{ab} $ is viewed as being independent from the parameters in the action (\[2N\]).

The system (\[Z2\],\[H\_J\]) features statistics of closed loops. If $u=0$, there are two sorts of loops – one in each layer. Thus, each configuration is characterized by definite values of the windings $W_{a, \alpha}$ in the $a$th layer along the $\alpha=\hat{x},\hat{y}$ directions of the planes. It is straightforward to show that statistics of these windings determine the renormalized values $\tilde{K}_{ab}$ of the matrix $K_{ab}$ along the line of the approach [@Ceperley]. More specifically \_[ab]{} = \_[=,]{} W\_[a,]{} W\_[b,]{}. \[KR\] This expressions are valid for periodic boundary conditions (PBC). It is important to note that $\tilde{K}_{ab}$ represents an exact linear response (at