number of bonds in the system. The energy ${E}_0^{(p)}$, due to its variational origin, serves as an upper bound of the true ground-state energy per bond ${\cal E}_{0}^{(p)}$.

Since the structure of every local Hamiltonian $G_{k}^{(p)}$ does not depend on $k$ (we investigate the system in the thermodynamic limit), the variational problem in  is equivalent to minimization of the local energy per bond of an arbitrary polygon in the lattice center (in order to avoid boundary effects) $${E}_0^{(p)} = \min\limits_{\Psi_p}
            \frac{2}{p}\frac{\langle \Psi_p | {G_\ell^{(p)}} | \Psi_p \rangle}
                 {\langle \Psi_p            | \Psi_p \rangle} \, ,
\label{Egbond}$$ where $\ell$ is the index of the selected polygon and the normalization factor $2/p$ reflects the fact that the $p$ bonds of each polygon are shared with its neighbors. Moreover, if we utilize the tensor product structure of the state $|\Psi_p\rangle$, we can express the denominator $\langle \Psi_p | \Psi_p \rangle \equiv {\cal D}(W_p(\{\sigma\}))$ and the numerator $\langle \Psi_p | {G_\ell^{(p)}} | \Psi_p \rangle \equiv {\cal N}(W_p(\{\sigma\}))$ as functions of the tensor elements $W_p(\{\sigma\})$ only. Consequently, our variational problem transforms onto a multi-dimensional minimization over $2^p$ tensor elements $W_p(\{\sigma\})$ $${E}_0^{(p)} =  \min\limits_{W_p(\{\sigma\})} 
               \frac{2}{p}\frac{{\cal N}(W_p(\{\sigma\}))}{{\cal D}(W_p(\{\sigma\}))} \, .
\label{e0}$$

Furthermore, symmetries of the local Hamiltonian $G_{\ell}^{(p)}$ 