}\left( bP_{24,31}+aP_{24,32}\right) \\
& +\frac 1{\sqrt{2}}\left( P_{25,33}+P_{25,34}\right) -\frac 1{\sqrt{2}%
}P_{33,4}+\frac 1{\sqrt{2}}P_{34,4}
\end{array}
\tag{B.4}$$

where

$$P_{\alpha ,\beta }=|E_{\alpha} \rangle \langle E_{\beta} |  \tag{B.5}$$

and

$$\begin{array}{l}
a=a_1+a_2 \\
b=a_1-a_2.
\end{array}
\tag{B.6}$$

To obtain the annihilation operators of the $I-$th dimer, the index $I$ should be added $(c_{1(2),\sigma }\longrightarrow c_{I,1(2),\sigma }$, $%
P_{\alpha ,\beta }\rightarrow P_{\alpha ,\beta }^{(I)})$ in (B.1)-(B.4).

[Appendix C]{}

To present the Hamiltonian $\overline{\overline{H}}$ in a compact form we first introduce the following operators $\left( \mu ,\nu =1,2\right) $ :

$$\begin{array}{ll}
s_{I,\mu ;J,\nu }^{+}= & a_{I,\mu ,\uparrow }^{+}a_{J,\nu ,\downarrow }, \\
&  \\
s_{I,\mu ;J,\nu }^{-}= & a_{J,\nu ,\downarrow }^{+}a_{I,\mu ,\uparrow }, \\
&  \\
n_{I,\mu ;J,\nu ;\sigma }= & a_{I,\mu ,\sigma }^{+}a_{J,\nu ,\sigma }, \\
&  \\
n_{I,\mu ;J,\nu }= & \sum\limits_{\sigma} n_{I,\mu ;J,\nu ;\sigma }, \\
&  \\
s_{I,\mu ;J,\nu }^z= & \frac 12\left( n_{I,\mu ;J,\nu ;\uparrow }-n_{I,\mu
;J,\nu ;\downarrow }\right) ; \\
&
\end{array}$$

$$\begin{array}{ll}
\underline{s}_{I,\mu ;J,\nu }^{+}= & \underline{\underline{a}}_{I,\mu
,\uparrow }^{+}a_{J,\nu ,\downarrow }, \\
&  \\
\underline{s}_{I,\mu ;J,\nu }^{-}= & a_{J,\nu ,\downarrow }^{+}\underline{%
\underline{a}}_{I,\mu ,\uparrow }, \\
&  \\
\underline{n}_{I,\mu ;J,\nu ;\sigma }= & \underline{\underline{a}}_{I,\mu
,\sigma }^{+}a_{J,\nu ,\sigma }, \\
&  \\
\underline{n}_{I,\mu ;J,\nu }= & \sum\limits_{\sigma} \underline{n}_{I,\mu
;J,\nu ;\sigma }, \\
&  \\
\underline{s}_{I,\mu ;J,\nu 