dimer approach to Hubbard-like models is the bond operator theory as an extension of the slave bosonic and fermionic operators (see Refs \[40\], \[41\] and papers cited therein).

[Appendix A]{}

The exact solution of the dimer eigenvalue problem (5) reads:

$$\begin{array}{ll}
E_0=0; & |E_0\rangle =|0\rangle ,
\end{array}$$

$$\begin{array}{ll}
E_{11}=-t; & |E_{11}\rangle =\frac{1}{\sqrt{2}}(|11\rangle +|13\rangle ), \\
E_{12}=t; & |E_{12}\rangle =\frac{1}{\sqrt{2}}(|11\rangle -|13\rangle ), \\
E_{13}=-t; & |E_{13}\rangle =\frac{1}{\sqrt{2}}(|12\rangle +|14\rangle ), \\
E_{14}=t; & |E_{14}\rangle =\frac{1}{\sqrt{2}}(|12\rangle -|14\rangle ),
\end{array}$$

$$\begin{array}{ll}
E_{21}=0; & |E_{21}\rangle =\frac 1{\sqrt{2}}(|23\rangle +|24\rangle ), \\
E_{22}=U; & |E_{22}\rangle =\frac 1{\sqrt{2}}(|21\rangle -|26\rangle ), \\
E_{23}=C+\frac U2; & |E_{23}\rangle =a_1(|21\rangle +|26\rangle
)-a_2(|23\rangle -|24\rangle ), \\
E_{24}=-C+\frac U2; & |E_{24}\rangle =a_2(|21\rangle +|26\rangle
)+a_1(|23\rangle -|24\rangle ), \\
E_{25}=0; & |E_{25}\rangle =|22\rangle , \\
E_{26}=0; & |E_{26}\rangle =|25\rangle ,
\end{array}
\tag{A.1}$$

$$\begin{array}{ll}
E_{31}=t+U; & |E_{31}\rangle =\frac 1{\sqrt{2}}(|31\rangle +|33\rangle ), \\
E_{32}=-t+U; & |E_{32}\rangle =\frac 1{\sqrt{2}}(|31\rangle -|33\rangle ),
\\
E_{33}=t+U; & |E_{33}\rangle =\frac 1{\sqrt{2}}(|32\rangle +|34\rangle ), \\
E_{34}=-t+U; & |E_{34}\rangle =\frac 1{\sqrt{2}}(|32\rangle -|34\rangle ),
\end{array}$$

$$\begin{array}{ll}
E_4=2U; & |E_4\rangle =|4\rangle
\end{array}$$ where

$$C=\sqrt{{\left( \frac U2\right) }^2+4t^2},  \tag{A.2}$$

$$a_1=\frac 12\sqrt{1+\frac U{2C}},  \tag{A.3}$$

$$a_2=\frac 12\sqrt{1-\frac U{2C}}.  \tag{A.4}$$

[Appendix B]{}

The exact dimer representation of the construction operators is given by the following expressions:

$$\begin{array}[t]{ll}
c_{1,\uparrow }= & \frac 