\Psi_p\rangle = \lim\limits_{N\to\infty}
      \sum\limits_{\sigma_1^{~}\sigma_2^{~}\cdots\sigma_{N}^{~}}^{~}
      \prod\limits_{{\langle k \rangle}_p}^{~} W_p(\{\sigma_k\}) 
      |\sigma_1^{~}\sigma_2^{~}\cdots\sigma_N^{~}\rangle \, ,
\label{Psi}$$ where $N$ stands for the total number of the lattice spins. The basis $\sigma_j$ for $j=1, \dots, N$ denotes a binary state, for which we use the arrow notation ${\downarrow}$ or ${\uparrow}$ in the following. The summation runs over the $2^N$ base spin states $|\sigma_1^{~}\sigma_2^{~}\cdots\sigma_N^{~}\rangle$, and $W_p(\{ \sigma_k \})$ are the elements of the $p$-rank tensor $W_p$ depending on $p$ spins $\sigma_{k_1}, \dots, \sigma_{k_p}$ on the $k^{th}$ lattice polygon. The symbol $\{\sigma_k\}$ stands for one of the $2^p$ base configurations of a multi-spin variable representing the group of spins $\sigma_{k_1}, \dots, \sigma_{k_p}$. All the tensors $W_p$ are considered to be identical, therefore, the set of $2^p$ tensor elements $W_p(\{\sigma\})$, where the subscript $k$ has been omitted due to the uniformity of the tensors $W_p$, uniquely describes the state $ |\Psi_p\rangle$, i.e. $|\Psi_p\rangle = |\Psi_p[W_p(\{\sigma\})]\rangle$.

We regard $|\Psi_p^{*}\rangle$ as the best approximation of the ground-state within the class of TPS $|\Psi_p\rangle$, if the minimum of the energy normalized per bond, $${ E}_0^{(p)} \equiv \min\limits_{\Psi_p}\lim\limits_{N_b\to\infty}\frac{1}{N_b}
            \frac{\langle \Psi_p | {\cal H} | \Psi_p \rangle}
                 {\langle \Psi_p            | \Psi_p \rangle}                
                  \, ,
\label{Eg}$$ is obtained for $|\Psi_p^{*}\rangle$. Here, $N_b$ denotes the total 