Appendix D. The summary is made in section VI.

Background
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The DDW Hamiltonian $H _{ddw} = \sum _{ {\bf k} \sigma } \chi _{ {\bf k} \sigma } ^\dag B _{ {\bf k} }  \chi _{ {\bf k} \sigma } $ [@Gerami] has been introduced for the AF quasielectrons with $$\begin{aligned}
B _{ {\bf k} } =  \left[
\begin{array}{c} 
\varepsilon _{ {\bf k} } - \mu   \text{ \ \ \ \ \ \ \ }   \Delta _{ {\bf k} } \text{ \ \ \ \ }  \\  \text{ \ \ \ } \Delta _{ {\bf k} } ^{\ast} \text{ \ \ \ \ }  \varepsilon _{ {\bf k} + {\bf Q} } - \mu   
\end{array}
\right]\end{aligned}$$ and $\chi _{ {\bf k} \sigma }  ^{\dag} = ( c _{ {\bf k} \text{ } \sigma } ^{\dag} , - i c _{ {\bf k}+{\bf Q} \text{ } \sigma } ^{\dag}  )$. Here $\mu$ denotes the chemical potential, the wave-vector [**k**]{} belongs to the reduced Brillouin zone (RBZ), ${\bf Q }$ equals the DDW ordering wavevector, $\sigma$ denotes the spin orientation $\uparrow$ or $\downarrow$, $\varepsilon _{ {\bf k} }$ and $ \varepsilon _{ {\bf k} + {\bf Q} }$ are the energy 