}^a}%
2$$

where $\underline{b}_{I,\alpha ,\sigma }$ is given by (31).

The Hamiltonian $\overline{\overline{H}}$ (C.6), including all the terms proportional to $t$ and $\frac{t^2}U$, takes on the form

$$\overline{\overline{H}}=\overline{\overline{H}}^{(1)}+\overline{\overline{H}}%
^{(2)}$$

where $$\begin{aligned}
\overline{\overline{H}}^{(1)} &=&-t\sum\limits_{I,\sigma }\left[
a_{I,1,\sigma }^{+}a_{I,2,\sigma }+a_{I,2,\sigma }^{+}a_{I,1,\sigma }\right]
\nonumber \\[0.2cm]
&&-t\sum\limits_{I,\sigma }\left[ \widetilde{a}_{I,2,\sigma }^{+}\widetilde{%
a}_{I+1,1,\sigma }+\widetilde{a}_{I+1,1,\sigma }^{+}\widetilde{a}%
_{I,2,\sigma }\right] \\[0.2cm]
&&-t\sum\limits_{I\neq J,\sigma }\sum\limits_{\alpha =1}^2\widetilde{a}%
_{I,\alpha ,\sigma }^{+}\widetilde{a}_{J,\alpha ,\sigma }  \nonumber\end{aligned}$$

and $$\begin{aligned}
\overline{\overline{H}}^{(2)} &=&\frac{4t^2}U\sum\limits_I[\overrightarrow{S%
}_{I,1}\cdot \overrightarrow{S}_{I,2}-\frac 14n_{I,1}^an_{I,2}^a]  \nonumber
\\
&&-\frac{t^2}U\sum\limits_{I,\sigma }[\widetilde{a}_{I,2,\sigma }^{+}%
\widetilde{\widetilde{a}}_{I+1,1,\sigma }+\widetilde{\widetilde{a}}%
_{I,2,\sigma }^{+}\widetilde{a}_{I+1,1,\sigma }  \nonumber \\
&&+\widetilde{a}_{I+1,1,\sigma }^{+}\widetilde{\widetilde{a}}_{I,2,\sigma }+%
\widetilde{\widetilde{a}}_{I+1,1,\sigma }^{+}\widetilde{a}_{I,2,\sigma }]
\nonumber \\
&&-\frac{t^2}U\sum\limits_{I\neq J,\sigma }\sum\limits_{\alpha =1}^2[%
\widetilde{a}_{I,\alpha ,\sigma }^{+}\widetilde{\widetilde{a}}_{J,\alpha
,\sigma }+\widetilde{\widetilde{a}}_{I,\alpha ,\sigma }^{+}\widetilde{a}%
_{J,\alpha ,\sigma }] \\
&&+\frac{2t^2}U\sum\limits_I[\overrightarrow{S}_{I,2}\cdot (\overrightarrow{%
\underline{\underline{s}}}_{I,1;I+1,1}+\overrightarrow{\underline{s}}%
_{I+1,1;I,1})  \nonumber \\
&&+\overrightarrow{S}_{I+1,1}\cdot (\overrightarrow{\underline{\underline{s}}%
}_{I+1,2;I,2}+\overrightarrow{\underline{s}}_{I,2;I+1,2})]  \nonumber \\
&&+\frac{2t^2}U\sum\limits_{I\neq J}[\overrightarrow{S}_{I,1}\cdot (%
\overrightarrow{\underline{\underline{s}}}_{I,2;J,1}+\overrightarrow{%
\underline{s}}_{J,1;I,2})  \nonumber \\
&&+\overrightarrow{S}_{I,2}\cdot (\overrightarrow{\underline{\underline{s}}}%
_{I,1;J,2}+\overrightarrow{\underline{s}}_{J,2;I,1})].  \nonumber\end{aligned}$$

The operators $\underline{\underline{s}}_{I,\mu ;J,\nu }^{z,\pm }$ and $%
\underline{s}_{I,\mu ;J,\nu }^{z,\pm }$ in (42) retain their forms introduced in (C.1) but $\underline{\underline{a}}_{I,\mu ,\sigma }(%
\underline{\underline{a}}_{I,\mu ,\sigma }^{+})$ in (C.1) should actually be replaced by $\widetilde{a}_{I,\mu ,\sigma }(\widetilde{a}_{I,\mu ,\sigma
}^{+})$, defined by (38). The total Hamiltonian $\overline{H}$ in the large $%
U$ limit (24) including all terms proportional to $t$ and $\frac{t^2}U$ can thus be written in the form

$$\overline{H}=P^{(1)}(\overline{\overline{H}}^{(1)}+\overline{\overline{H}}%
^{(2)})+P^{(2)}\overline{\overline{H}}^{(1)}=P^{(1)}\overline{\overline{H}}%
^{(1)}+(P^{(1)}\overline{\overline{H}}^{(2)}+P^{(2)}\overline{\overline{H}}%
^{(1)}).$$

The first part, 